Natural Frequency Calculator \u2014 Resonance, Critical Speed & Mass-Spring Systems \u2014 Vibromera<\/title>\n<meta name=\"description\" content=\"Interactive natural frequency calculator for mass-spring, beam, and shaft systems. Understand resonance, critical speeds, dynamic magnification factor. Formulas, identification methods, and industrial case studies.\">\n<meta name=\"keywords\" content=\"natural frequency, natural frequency calculator, mass spring system, resonance, critical speed, eigenfrequency, vibration, fn formula, bump test, dynamic magnification factor, Balanset, vibration analysis\">\n<meta name=\"author\" content=\"Vibromera\">\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large\">\n\n\n\n\n\n\n\n\n\n\n\n<meta property=\"og:type\" content=\"article\">\n<meta property=\"og:title\" content=\"Natural Frequency Calculator \u2014 Resonance, Critical Speed & Mass-Spring Systems\">\n<meta property=\"og:description\" content=\"Interactive calculator for 5 system types. Resonance curve visualization, critical speed analysis, industrial case studies. ISO-compliant vibration analysis.\">\n<meta property=\"og:url\" content=\"https:\/\/vibromera.eu\/glossary\/natural-frequency\/\">\n<meta property=\"og:site_name\" content=\"Vibromera \u2014 Vibration Analysis & Balancing Equipment\">\n<meta property=\"og:locale\" content=\"en_US\">\n<meta property=\"og:image\" content=\"https:\/\/vibromera.eu\/wp-content\/uploads\/natural-frequency-og.jpg\">\n<meta property=\"article:publisher\" content=\"https:\/\/vibromera.eu\/\">\n<meta property=\"article:section\" content=\"Glossary\">\n<meta property=\"article:tag\" content=\"Natural Frequency\">\n<meta property=\"article:tag\" content=\"Resonance\">\n<meta property=\"article:tag\" content=\"Critical Speed\">\n\n\n<meta name=\"twitter:card\" content=\"summary_large_image\">\n<meta name=\"twitter:title\" content=\"Natural Frequency Calculator \u2014 Mass-Spring, Beam, Shaft Systems\">\n<meta name=\"twitter:description\" content=\"Calculate natural frequency, critical speed, and resonance risk for 5 system types. Formulas, DMF chart, bump test procedures.\">\n\n\n<meta name=\"geo.region\" content=\"EU\">\n<meta name=\"geo.placename\" content=\"Porto, Portugal\">\n<meta name=\"geo.position\" content=\"41.1579;-8.6291\">\n<meta name=\"ICBM\" content=\"41.1579, -8.6291\">\n\n\n<script type=\"application\/ld+json\">\n{\n \"@context\": \"https:\/\/schema.org\",\n \"@graph\": [\n {\n \"@type\": \"TechArticle\",\n \"@id\": \"https:\/\/vibromera.eu\/glossary\/natural-frequency\/#article\",\n \"headline\": \"Natural Frequency: Definition, Calculator, Resonance, and Critical Speeds in Rotating Machinery\",\n \"description\": \"Comprehensive guide to natural frequency in mechanical systems. Interactive calculator for mass-spring, beam, cantilever, and shaft systems. Resonance curves, critical speed analysis, bump test procedures, and industrial case studies.\",\n \"author\": {\"@type\": \"Organization\", \"name\": \"Vibromera\", \"url\": \"https:\/\/vibromera.eu\/\"},\n \"publisher\": {\"@type\": \"Organization\", \"name\": \"Vibromera\", \"url\": \"https:\/\/vibromera.eu\/\", \"logo\": {\"@type\": \"ImageObject\", \"url\": \"https:\/\/vibromera.eu\/wp-content\/uploads\/vibromera-logo.png\"}},\n \"mainEntityOfPage\": \"https:\/\/vibromera.eu\/glossary\/natural-frequency\/\",\n \"datePublished\": \"2024-03-10\",\n \"dateModified\": \"2026-02-07\",\n \"inLanguage\": \"en\",\n \"about\": [\n {\"@type\": \"Thing\", \"name\": \"Natural Frequency\"},\n {\"@type\": \"Thing\", \"name\": \"Resonance\"},\n {\"@type\": \"Thing\", \"name\": \"Critical Speed\"},\n {\"@type\": \"Thing\", \"name\": \"Vibration Analysis\"}\n ]\n },\n {\n \"@type\": \"FAQPage\",\n \"@id\": \"https:\/\/vibromera.eu\/glossary\/natural-frequency\/#faq\",\n \"mainEntity\": [\n {\n \"@type\": \"Question\",\n \"name\": \"What is natural frequency?\",\n \"acceptedAnswer\": {\n \"@type\": \"Answer\",\n \"text\": \"Natural frequency is the frequency at which a mechanical system oscillates freely after being displaced from equilibrium, without any external periodic forcing. Every structure has one or more natural frequencies determined by its mass and stiffness. The fundamental formula is fn = (1\/2\u03c0) \u00d7 \u221a(k\/m) where k is stiffness (N\/m) and m is mass (kg). Natural frequency is measured in Hertz (Hz) or expressed as critical speed in RPM for rotating machinery.\"\n }\n },\n {\n \"@type\": \"Question\",\n \"name\": \"How do you calculate natural frequency of a mass-spring system?\",\n \"acceptedAnswer\": {\n \"@type\": \"Answer\",\n \"text\": \"For a simple mass-spring system: fn = (1\/2\u03c0) \u00d7 \u221a(k\/m), where fn is natural frequency in Hz, k is spring stiffness in N\/m, and m is mass in kg. For example, a 10 kg mass on a spring with stiffness 40,000 N\/m has fn = (1\/2\u03c0) \u00d7 \u221a(40000\/10) = 10.07 Hz. The critical speed equivalent is fn \u00d7 60 = 604 RPM.\"\n }\n },\n {\n \"@type\": \"Question\",\n \"name\": \"What happens at resonance?\",\n \"acceptedAnswer\": {\n \"@type\": \"Answer\",\n \"text\": \"Resonance occurs when an external forcing frequency equals or closely matches a system's natural frequency. At resonance, vibration amplitude is amplified dramatically \u2014 by a factor of 10-50\u00d7 for lightly damped steel structures (Q-factor = 10-50). This can cause catastrophic structural failure, bearing damage, and fatigue cracking. The amplification is described by the Dynamic Magnification Factor: DMF = 1\/\u221a[(1-r\u00b2)\u00b2 + (2\u03b6r)\u00b2] where r is the frequency ratio and \u03b6 is the damping ratio.\"\n }\n },\n {\n \"@type\": \"Question\",\n \"name\": \"What is the difference between natural frequency and critical speed?\",\n \"acceptedAnswer\": {\n \"@type\": \"Answer\",\n \"text\": \"Critical speed is the rotational speed (in RPM) at which a rotating shaft's operating frequency coincides with a natural frequency of the rotor-bearing system, causing resonance. Critical speed (RPM) = natural frequency (Hz) \u00d7 60. API standards require a minimum 15-20% separation margin between operating speed and the nearest critical speed to avoid resonance amplification.\"\n }\n },\n {\n \"@type\": \"Question\",\n \"name\": \"How do you find natural frequency in the field?\",\n \"acceptedAnswer\": {\n \"@type\": \"Answer\",\n \"text\": \"The most common field method is the impact test (bump test): strike the structure with an instrumented hammer while measuring the vibration response with an accelerometer. The natural frequency appears as a peak in the Frequency Response Function (FRF). Other methods include run-up\/coast-down tests (Bode plot), waterfall\/cascade analysis, and Operational Modal Analysis (OMA). The Balanset-1A portable analyzer from Vibromera can perform FFT spectrum analysis to identify resonant frequencies.\"\n }\n },\n {\n \"@type\": \"Question\",\n \"name\": \"How can you change the natural frequency of a structure?\",\n \"acceptedAnswer\": {\n \"@type\": \"Answer\",\n \"text\": \"Natural frequency is determined by fn = (1\/2\u03c0)\u221a(k\/m). To increase fn: add stiffness (bracing, thicker supports, stiffer mounts) or reduce mass. To decrease fn: add mass or reduce stiffness (softer mounts, isolation). Common industrial solutions include adding stiffening ribs to foundations, changing from rigid to flexible mounts, adding mass to shift resonance below operating speed, or redesigning support structures.\"\n }\n },\n {\n \"@type\": \"Question\",\n \"name\": \"What is the static deflection method for calculating natural frequency?\",\n \"acceptedAnswer\": {\n \"@type\": \"Answer\",\n \"text\": \"The static deflection shortcut calculates natural frequency from the deflection under gravity: fn \u2248 15.76 \/ \u221a\u03b4, where fn is in Hz and \u03b4 is static deflection in millimeters. This works because both fn and \u03b4 depend on the same stiffness-to-mass ratio. For example, a beam that deflects 1 mm under its own weight has fn \u2248 15.76 \/ \u221a1 = 15.76 Hz. 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}\n.resonance-indicator.danger .ri-status { color: #f87171; }\n.resonance-indicator .ri-detail {\n font-size: 13px;\n color: rgba(255,255,255,0.5);\n}\n\n\/* ===== CONCEPT CARDS ===== *\/\n.concepts-section {\n padding: 40px 0 48px;\n background: var(--beige-light);\n}\n.section-header {\n margin-bottom: 28px;\n}\n.section-header h2 {\n font-family: 'DM Serif Display', serif;\n font-size: 30px;\n color: var(--navy);\n margin-bottom: 6px;\n}\n.section-header p {\n font-size: 16px;\n color: var(--text-secondary);\n}\n\n.concept-cards {\n display: grid;\n grid-template-columns: repeat(3, 1fr);\n gap: 20px;\n margin-bottom: 36px;\n}\n.concept-card {\n background: var(--white);\n border: 1px solid var(--border-light);\n border-radius: var(--radius);\n padding: 24px;\n box-shadow: var(--shadow-sm);\n transition: all 0.3s;\n position: relative;\n overflow: hidden;\n}\n.concept-card:hover {\n box-shadow: var(--shadow-lg);\n transform: translateY(-3px);\n}\n.concept-card::before {\n content: 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*\/\n.procedure-steps {\n counter-reset: step;\n margin: 24px 0;\n}\n.proc-step {\n display: flex;\n gap: 20px;\n padding: 20px 0;\n border-bottom: 1px solid var(--border-light);\n}\n.proc-step:last-child { border-bottom: none; }\n.proc-step::before {\n counter-increment: step;\n content: counter(step);\n flex-shrink: 0;\n width: 40px;\n height: 40px;\n background: var(--blue);\n color: var(--white);\n border-radius: 50%;\n display: flex;\n align-items: center;\n justify-content: center;\n font-weight: 700;\n font-size: 16px;\n}\n.proc-step .step-content h4 {\n margin: 0 0 4px;\n font-size: 16px;\n color: var(--navy);\n}\n.proc-step .step-content p {\n margin: 0;\n font-size: 14px;\n color: var(--text-secondary);\n}\n\n\/* ===== SHOP CTA ===== *\/\n.shop-cta {\n background: linear-gradient(135deg, var(--navy) 0%, var(--navy-light) 100%);\n padding: 48px 0;\n color: var(--white);\n text-align: center;\n position: relative;\n overflow: hidden;\n}\n.shop-cta::before {\n content: '';\n position: absolute; inset: 0;\n background: radial-gradient(circle at 50% 50%, rgba(37,99,235,0.15) 0%, transparent 60%);\n pointer-events: none;\n}\n.shop-cta h2 {\n font-family: 'DM Serif Display', serif;\n font-size: 32px;\n margin-bottom: 12px;\n position: relative;\n}\n.shop-cta p {\n font-size: 17px;\n color: rgba(255,255,255,0.7);\n margin-bottom: 28px;\n max-width: 640px;\n margin-left: auto;\n margin-right: auto;\n position: relative;\n}\n.shop-cta .cta-btn {\n display: inline-flex;\n align-items: center;\n gap: 8px;\n padding: 14px 36px;\n background: var(--blue);\n color: var(--white);\n text-decoration: none;\n border-radius: var(--radius-sm);\n font-weight: 600;\n font-size: 16px;\n transition: all 0.2s;\n position: relative;\n}\n.shop-cta .cta-btn:hover {\n background: var(--blue-light);\n transform: translateY(-2px);\n box-shadow: 0 8px 24px rgba(37,99,235,0.3);\n}\n\n\/* Footer *\/\n.page-footer {\n background: var(--beige);\n border-top: 1px solid var(--beige-dark);\n padding: 32px 0;\n text-align: center;\n}\n.page-footer a { color: var(--blue); text-decoration: none; font-weight: 500; }\n.page-footer p { font-size: 14px; color: var(--text-muted); }\n\n@media print {\n .quick-nav, .shop-cta { display: none; }\n body { min-width: auto; }\n}\n<\/style>\n<\/head>\n<body>\n\n\n<header class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"breadcrumb\">\n <a href=\"https:\/\/vibromera.eu\/\">Home<\/a> \u2192 <a href=\"https:\/\/vibromera.eu\/glossary\/\">Glossary<\/a> \u2192 Natural Frequency\n <\/div>\n <h1>Understanding <span>Natural Frequency<\/span><\/h1>\n <div style=\"display:inline-flex;align-items:center;gap:6px;padding:4px 12px;background:rgba(37,99,235,0.2);border:1px solid rgba(37,99,235,0.3);border-radius:20px;font-size:12px;font-weight:600;letter-spacing:0.5px;color:rgba(255,255,255,0.85);margin-bottom:12px;text-transform:uppercase;\">\n <span style=\"font-size:14px;\">\ud83d\udccc<\/span> Canonical Reference Article \u2014 vibromera.eu\n <\/div>\n <p class=\"subtitle\">The inherent vibration frequency of every physical structure \u2014 and why its relationship with resonance is one of the most critical concepts in vibration analysis and rotating machinery engineering.<\/p>\n <\/div>\n<\/header>\n\n\n<nav class=\"quick-nav\">\n <div class=\"quick-nav-inner\">\n <a href=\"#calculator\">\u2699 Calculator<\/a>\n <a href=\"#key-concepts\">\ud83d\udcca Key Concepts<\/a>\n <a href=\"#definition\">\ud83d\udcd0 Definition<\/a>\n <a href=\"#mass-stiffness\">\ud83d\udd22 Mass & Stiffness<\/a>\n <a href=\"#resonance\">\u26a1 Resonance<\/a>\n <a href=\"#critical-speeds\">\ud83d\udd04 Critical Speeds<\/a>\n <a href=\"#identification\">\ud83d\udd0d Identification Methods<\/a>\n <a href=\"#practical\">\ud83d\udee0 Practical Examples<\/a>\n <a href=\"#faq\">\u2753 FAQ<\/a>\n <a href=\"#avoidance\">\ud83d\udee1 Avoidance Strategies<\/a>\n <\/div>\n<\/nav>\n\n\n<section class=\"summary-dashboard\" id=\"calculator\">\n <div class=\"container\">\n <div class=\"dashboard-grid\">\n\n \n <div class=\"calc-panel\">\n <h2 class=\"panel-title\">Natural Frequency Calculator<\/h2>\n <p class=\"panel-subtitle\">Calculate f<sub>n<\/sub> for simple systems + check resonance risk against operating speed<\/p>\n\n <div class=\"calc-form\">\n <div class=\"form-group full-width\">\n <label>System Type<\/label>\n <select id=\"systemType\" onchange=\"toggleInputs()\">\n <option value=\"spring-mass\">Spring\u2013Mass System (k + m)<\/option>\n <option value=\"beam-center\">Simply Supported Beam (center load)<\/option>\n <option value=\"beam-cantilever\">Cantilever Beam (end load)<\/option>\n <option value=\"shaft-disc\">Shaft with Central Disc (Rayleigh)<\/option>\n <option value=\"static-deflection\">From Static Deflection (\u03b4)<\/option>\n <\/select>\n <\/div>\n\n <div class=\"form-group\" id=\"fg-stiffness\">\n <label>Stiffness, k <span class=\"unit\">(N\/m)<\/span><\/label>\n <input type=\"number\" id=\"stiffness\" value=\"50000\" min=\"0.1\" step=\"100\">\n <\/div>\n <div class=\"form-group\" id=\"fg-mass\">\n <label>Mass, m <span class=\"unit\">(kg)<\/span><\/label>\n <input type=\"number\" id=\"mass\" value=\"5\" min=\"0.001\" step=\"0.1\">\n <\/div>\n <div class=\"form-group\" id=\"fg-length\" style=\"display:none\">\n <label>Beam Length, L <span class=\"unit\">(m)<\/span><\/label>\n <input type=\"number\" id=\"beamLength\" value=\"1\" min=\"0.01\" step=\"0.01\">\n <\/div>\n <div class=\"form-group\" id=\"fg-ei\" style=\"display:none\">\n <label>EI <span class=\"unit\">(N\u00b7m\u00b2)<\/span><\/label>\n <input type=\"number\" id=\"beamEI\" value=\"1000\" min=\"0.1\" step=\"10\">\n <\/div>\n <div class=\"form-group\" id=\"fg-deflection\" style=\"display:none\">\n <label>Static Deflection, \u03b4 <span class=\"unit\">(mm)<\/span><\/label>\n <input type=\"number\" id=\"staticDefl\" value=\"1\" min=\"0.001\" step=\"0.1\">\n <\/div>\n <div class=\"form-group\" id=\"fg-shaft-dia\" style=\"display:none\">\n <label>Shaft Diameter <span class=\"unit\">(mm)<\/span><\/label>\n <input type=\"number\" id=\"shaftDia\" value=\"50\" min=\"1\" step=\"1\">\n <\/div>\n <div class=\"form-group\" id=\"fg-shaft-len\" style=\"display:none\">\n <label>Bearing Span <span class=\"unit\">(mm)<\/span><\/label>\n <input type=\"number\" id=\"shaftLen\" value=\"800\" min=\"10\" step=\"10\">\n <\/div>\n <div class=\"form-group\" id=\"fg-disc-mass\" style=\"display:none\">\n <label>Disc Mass <span class=\"unit\">(kg)<\/span><\/label>\n <input type=\"number\" id=\"discMass\" value=\"20\" min=\"0.01\" step=\"0.1\">\n <\/div>\n <div class=\"form-group\">\n <label>Operating Speed <span class=\"unit\">(RPM, optional)<\/span><\/label>\n <input type=\"number\" id=\"operatingRPM\" value=\"3000\" min=\"0\" step=\"10\">\n <\/div>\n\n <button class=\"calc-btn\" onclick=\"calculate()\">Calculate Natural Frequency \u2192<\/button>\n <\/div>\n <\/div>\n\n \n <div class=\"results-panel\">\n <h2 class=\"panel-title\">Results<\/h2>\n <p class=\"panel-subtitle\">Natural frequency and resonance risk assessment<\/p>\n <div id=\"resultsArea\">\n <div class=\"results-empty\">\n <span class=\"icon\">\u3030\ufe0f<\/span>\n Enter system parameters and click Calculate<br>to see the natural frequency\n <\/div>\n <\/div>\n <\/div>\n\n <\/div>\n <\/div>\n<\/section>\n\n\n<section class=\"concepts-section\" id=\"key-concepts\">\n <div class=\"container\">\n <div class=\"section-header\">\n <h2>Key Concepts \u2014 At a Glance<\/h2>\n <p>The three fundamental properties that govern every vibrating system<\/p>\n <\/div>\n\n <div class=\"concept-cards\">\n <div class=\"concept-card mass\">\n <div class=\"cc-icon\">\u2696\ufe0f<\/div>\n <div class=\"cc-title\">Mass (Inertia)<\/div>\n <div class=\"cc-body\">\n Mass resists changes in motion. <strong>Increasing mass lowers the natural frequency<\/strong> \u2014 the system becomes slower and heavier to move. Think of a heavy pendulum swinging slowly versus a light one swinging quickly.\n <\/div>\n <div class=\"cc-formula\">f<sub>n<\/sub> \u221d 1 \/ \u221am \u2192 Double mass = f<sub>n<\/sub> \u00d7 0.71<\/div>\n <\/div>\n <div class=\"concept-card stiff\">\n <div class=\"cc-icon\">\ud83d\udd29<\/div>\n <div class=\"cc-title\">Stiffness (Elasticity)<\/div>\n <div class=\"cc-body\">\n Stiffness is the restoring force that pulls the system back to equilibrium. <strong>Increasing stiffness raises the natural frequency<\/strong> \u2014 the system snaps back faster. Like tightening a guitar string raises its pitch.\n <\/div>\n <div class=\"cc-formula\">f<sub>n<\/sub> \u221d \u221ak \u2192 Double stiffness = f<sub>n<\/sub> \u00d7 1.41<\/div>\n <\/div>\n <div class=\"concept-card damp\">\n <div class=\"cc-icon\">\ud83d\udee2\ufe0f<\/div>\n <div class=\"cc-title\">Damping (Energy Dissipation)<\/div>\n <div class=\"cc-body\">\n Damping removes energy from the system, causing vibration to decay over time. <strong>Damping doesn't significantly change f<sub>n<\/sub><\/strong>, but it controls the amplitude at resonance \u2014 the only thing preventing infinite amplitude.\n <\/div>\n <div class=\"cc-formula\">f<sub>d<\/sub> = f<sub>n<\/sub> \u00d7 \u221a(1 \u2212 \u03b6\u00b2) \u2248 f<sub>n<\/sub> for \u03b6 < 0.2<\/div>\n <\/div>\n <\/div>\n\n \n <div class=\"resonance-visual\">\n <div class=\"rv-title\">Interactive Resonance Response Curve<\/div>\n <div class=\"rv-subtitle\">Amplitude magnification vs. frequency ratio (\u03c9\/\u03c9<sub>n<\/sub>) for different damping ratios (\u03b6)<\/div>\n <div class=\"rv-canvas-wrap\">\n <canvas id=\"resonanceCanvas\"><\/canvas>\n <\/div>\n <\/div>\n\n \n <div class=\"table-wrap\">\n <div class=\"table-title\">\ud83d\udcca Typical Natural Frequency Ranges for Common Industrial Structures<\/div>\n <table>\n <thead>\n <tr>\n <th>Structure \/ Component<\/th>\n <th>Typical f<sub>n<\/sub> Range<\/th>\n <th>Typical Operating RPM<\/th>\n <th>Resonance Risk<\/th>\n <th>Notes<\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td>Large concrete foundation<\/td>\n <td class=\"mono\">15\u201340 Hz<\/td>\n <td class=\"mono\">900\u20132400<\/td>\n <td><span class=\"tag safe\">Low<\/span><\/td>\n <td>Very stiff; typically well above operating speed<\/td>\n <\/tr>\n <tr>\n <td>Steel baseplate \/ skid<\/td>\n <td class=\"mono\">20\u201380 Hz<\/td>\n <td class=\"mono\">1200\u20134800<\/td>\n <td><span class=\"tag caution\">Medium<\/span><\/td>\n <td>Can coincide with 2-pole or 4-pole motor speed<\/td>\n <\/tr>\n <tr>\n <td>Piping system (span)<\/td>\n <td class=\"mono\">5\u201350 Hz<\/td>\n <td class=\"mono\">300\u20133000<\/td>\n <td><span class=\"tag danger\">High<\/span><\/td>\n <td>Long unsupported spans are very susceptible<\/td>\n <\/tr>\n <tr>\n <td>Pump pedestal<\/td>\n <td class=\"mono\">25\u201360 Hz<\/td>\n <td class=\"mono\">1500\u20133600<\/td>\n <td><span class=\"tag caution\">Medium<\/span><\/td>\n <td>Vertical pumps are especially problematic<\/td>\n <\/tr>\n <tr>\n <td>Fan housing \/ shroud<\/td>\n <td class=\"mono\">15\u2013120 Hz<\/td>\n <td class=\"mono\">900\u20137200<\/td>\n <td><span class=\"tag caution\">Medium<\/span><\/td>\n <td>Sheet metal panels can have many modes<\/td>\n <\/tr>\n <tr>\n <td>Electric motor frame<\/td>\n <td class=\"mono\">40\u2013200 Hz<\/td>\n <td class=\"mono\">2400\u201312000<\/td>\n <td><span class=\"tag safe\">Low<\/span><\/td>\n <td>Typically designed above 1\u00d7 operating speed<\/td>\n <\/tr>\n <tr>\n <td>Shaft (1st critical)<\/td>\n <td class=\"mono\">20\u2013500 Hz<\/td>\n <td class=\"mono\">1200\u201330000<\/td>\n <td><span class=\"tag danger\">High<\/span><\/td>\n <td>Must be known; crossing critical = severe vibration<\/td>\n <\/tr>\n <tr>\n <td>Bearing housing<\/td>\n <td class=\"mono\">100\u20131000 Hz<\/td>\n <td class=\"mono\">\u2014<\/td>\n <td><span class=\"tag safe\">Low<\/span><\/td>\n <td>Excited by bearing fault impacts, not by 1\u00d7 speed<\/td>\n <\/tr>\n <tr>\n <td>Gearbox casing<\/td>\n <td class=\"mono\">200\u20132000 Hz<\/td>\n <td class=\"mono\">\u2014<\/td>\n <td><span class=\"tag safe\">Low<\/span><\/td>\n <td>Excited by gear meshing frequencies<\/td>\n <\/tr>\n <tr>\n <td>Spring isolators (installed)<\/td>\n <td class=\"mono\">2\u20138 Hz<\/td>\n <td class=\"mono\">120\u2013480<\/td>\n <td><span class=\"tag caution\">Medium<\/span><\/td>\n <td>Must be well below operating speed to isolate<\/td>\n <\/tr>\n <tr>\n <td>Rubber mounts<\/td>\n <td class=\"mono\">5\u201325 Hz<\/td>\n <td class=\"mono\">300\u20131500<\/td>\n <td><span class=\"tag caution\">Medium<\/span><\/td>\n <td>Stiffness varies with temperature and age<\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n <\/div>\n\n \n <div class=\"table-wrap\">\n <div class=\"table-title\">\u26a1 Resonance Risk Assessment \u2014 Operating Speed vs. Natural Frequency Ratio<\/div>\n <table>\n <thead>\n <tr>\n <th>Frequency Ratio (f<sub>op<\/sub> \/ f<sub>n<\/sub>)<\/th>\n <th>Zone<\/th>\n <th>Amplification Factor<\/th>\n <th>Practical Meaning<\/th>\n <th>Recommendation<\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td class=\"mono\">0 \u2013 0.7<\/td>\n <td><span class=\"tag safe\">Safe Below<\/span><\/td>\n <td class=\"mono\">1.0 \u2013 2.0\u00d7<\/td>\n <td>Vibration force transmitted almost 1:1; structure moves in-phase with forcing<\/td>\n <td>Acceptable; normal operating zone for rigidly mounted equipment<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">0.7 \u2013 0.85<\/td>\n <td><span class=\"tag caution\">Approach Zone<\/span><\/td>\n <td class=\"mono\">2 \u2013 5\u00d7<\/td>\n <td>Amplitude begins to amplify significantly; early resonance effects<\/td>\n <td>Avoid steady-state operation; acceptable for brief run-up\/coast-down transit<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">0.85 \u2013 1.15<\/td>\n <td><span class=\"tag danger\">Resonance Band<\/span><\/td>\n <td class=\"mono\">5 \u2013 50\u00d7<\/td>\n <td>Severe amplification; amplitude limited only by damping; structural damage possible<\/td>\n <td>Never operate here; transit through quickly if unavoidable<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">1.15 \u2013 1.4<\/td>\n <td><span class=\"tag caution\">Exit Zone<\/span><\/td>\n <td class=\"mono\">2 \u2013 5\u00d7<\/td>\n <td>Amplitude decreasing but still elevated; phase shifting rapidly<\/td>\n <td>Avoid steady-state; brief transit acceptable<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">1.4 \u2013 2.5<\/td>\n <td><span class=\"tag safe\">Safe Above<\/span><\/td>\n <td class=\"mono\">0.3 \u2013 1.0\u00d7<\/td>\n <td>Vibration is attenuated; structure's inertia resists movement; phase inversion<\/td>\n <td>Good isolation zone for flexibly mounted equipment<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">> 2.5<\/td>\n <td><span class=\"tag safe\">Isolation Zone<\/span><\/td>\n <td class=\"mono\">< 0.3\u00d7<\/td>\n <td>Excellent vibration isolation; very little force transmitted<\/td>\n <td>Ideal for spring\/rubber-mounted machines<\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n <\/div>\n\n \n <div class=\"table-wrap\">\n <div class=\"table-title\">\ud83d\udd0d Comparison of Natural Frequency Identification Methods<\/div>\n <table>\n <thead>\n <tr>\n <th>Method<\/th>\n <th>Equipment Needed<\/th>\n <th>Machine State<\/th>\n <th>Accuracy<\/th>\n <th>Best For<\/th>\n <th>Limitations<\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td>Impact Test (Bump Test)<\/td>\n <td>Modal hammer + accelerometer + FFT analyzer<\/td>\n <td>Stopped<\/td>\n <td><span class=\"tag safe\">High<\/span><\/td>\n <td>Structures, baseplates, piping, bearing housings<\/td>\n <td>Machine must be stopped; can miss speed-dependent effects<\/td>\n <\/tr>\n <tr>\n <td>Run-Up \/ Coast-Down<\/td>\n <td>Vibration sensor + tachometer + order tracking<\/td>\n <td>Running (variable speed)<\/td>\n <td><span class=\"tag safe\">High<\/span><\/td>\n <td>Shaft critical speeds, foundation resonances<\/td>\n <td>Requires variable speed; 1\u00d7 unbalance force excites primarily shaft criticals<\/td>\n <\/tr>\n <tr>\n <td>Operating Deflection Shape (ODS)<\/td>\n <td>Multi-channel analyzer + many sensors<\/td>\n <td>Running (normal)<\/td>\n <td><span class=\"tag caution\">Medium<\/span><\/td>\n <td>Visualizing how structure moves at specific frequency<\/td>\n <td>Shows deflection shape, not true mode shape (multiple modes contribute)<\/td>\n <\/tr>\n <tr>\n <td>Experimental Modal Analysis (EMA)<\/td>\n <td>Modal hammer or shaker + roving sensors + modal software<\/td>\n <td>Stopped<\/td>\n <td><span class=\"tag safe\">Very High<\/span><\/td>\n <td>Complete modal model (frequencies, shapes, damping)<\/td>\n <td>Time-consuming; requires expertise; complex data processing<\/td>\n <\/tr>\n <tr>\n <td>Finite Element Analysis (FEA)<\/td>\n <td>Computer + FEA software + model<\/td>\n <td>N\/A (simulation)<\/td>\n <td><span class=\"tag caution\">Depends on model<\/span><\/td>\n <td>Design phase; what-if analysis; complex geometries<\/td>\n <td>Accuracy depends on model quality; boundary conditions critical<\/td>\n <\/tr>\n <tr>\n <td>Waterfall \/ Cascade Plot<\/td>\n <td>Vibration analyzer with order tracking<\/td>\n <td>Running (variable speed)<\/td>\n <td><span class=\"tag safe\">High<\/span><\/td>\n <td>Identifying multiple resonances during speed changes<\/td>\n <td>Requires speed change; only finds resonances excited by operating forces<\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n <\/div>\n\n <\/div>\n<\/section>\n\n\n<main class=\"main-content\" id=\"definition\">\n <div class=\"container\">\n <div class=\"content-layout\">\n <article class=\"article-content\">\n\n <h2>Definition: What is Natural Frequency?<\/h2>\n\n <div class=\"info-box\" style=\"border-left-color: var(--navy); background: var(--beige-light);\">\n <div class=\"box-title\" style=\"font-size: 16px;\">Quick Answer<\/div>\n <p style=\"font-size: 15px;\"><strong>Natural frequency<\/strong> is the frequency at which a mechanical system oscillates freely after being displaced from equilibrium. It is determined by the system's <strong>mass<\/strong> and <strong>stiffness<\/strong>: <strong>f<sub>n<\/sub> = (1\/2\u03c0) \u00d7 \u221a(k\/m)<\/strong>, where k is stiffness (N\/m) and m is mass (kg). When an external force frequency matches a natural frequency, <strong><a href=\"https:\/\/vibromera.eu\/glossary\/resonance\/\">resonance<\/a><\/strong> occurs \u2014 vibration amplitude can increase 10\u201350\u00d7 and cause catastrophic failure. In rotating machinery, the <strong>critical speed<\/strong> (RPM) = f<sub>n<\/sub> \u00d7 60. A quick field estimate uses static deflection: <strong>f<sub>n<\/sub> \u2248 15.76 \/ \u221a\u03b4<sub>mm<\/sub><\/strong>.<\/p>\n <\/div>\n\n <p>A <strong>natural frequency<\/strong> is the specific frequency at which a physical object or system will oscillate when disturbed from its equilibrium position and then allowed to vibrate freely, without any ongoing external driving force. It is an inherent, fundamental property of the object, determined entirely by its physical characteristics \u2014 principally its <strong>mass<\/strong> (inertia) and its <strong>stiffness<\/strong> (elasticity). Every physical object, from a guitar string to a bridge span to a machine's support pedestal, possesses one or more natural frequencies.<\/p>\n <p>Natural frequencies are sometimes called <em>eigenfrequencies<\/em> (from the German word \"eigen\" meaning \"own\" or \"characteristic\"), and the corresponding vibration patterns are called <em>mode shapes<\/em> or <em>eigenmodes<\/em>. A complex structure like a machine base may have hundreds of natural frequencies, each associated with a unique deformation pattern \u2014 bending, twisting, breathing, rocking, and so on.<\/p>\n\n <div class=\"info-box\">\n <div class=\"box-title\">Why Natural Frequency Matters in Vibration Analysis<\/div>\n <p>In rotating machinery, vibration problems are frequently caused not by excessive excitation forces (such as unbalance), but by the unlucky coincidence of an excitation frequency matching a structural natural frequency. A perfectly acceptable amount of unbalance can produce destructive vibration if the machine operates at or near a structural resonance. Identifying natural frequencies is therefore one of the most important diagnostic steps when investigating unexplained high vibration.<\/p>\n <\/div>\n\n <h2 id=\"mass-stiffness\">The Relationship Between Mass, Stiffness, and Natural Frequency<\/h2>\n <p>The fundamental relationship between mass, stiffness, and natural frequency is one of the most important concepts in vibration engineering. It is both intuitive and mathematically precise.<\/p>\n\n <h3>Intuitive Understanding<\/h3>\n <ul>\n <li><strong>Stiffness (k):<\/strong> A stiffer object has a <em>higher<\/em> natural frequency. Think of a guitar string: tightening the string (increasing tension\/stiffness) raises the pitch (frequency). A thick steel beam vibrates at a much higher frequency than a thin aluminum strip of the same length.<\/li>\n <li><strong>Mass (m):<\/strong> A more massive object has a <em>lower<\/em> natural frequency. Think of a ruler extending off the edge of a desk: a longer, heavier ruler oscillates more slowly (lower frequency) than a shorter, lighter one. Adding weight to a structure always lowers its natural frequencies.<\/li>\n <\/ul>\n\n <h3>The Fundamental Formula<\/h3>\n <p>For a simple single-degree-of-freedom (SDOF) system \u2014 a mass connected to a spring \u2014 the undamped natural frequency is:<\/p>\n\n <div class=\"formula-box\">\n <div class=\"formula-label\">Undamped Natural Frequency<\/div>\n <div class=\"formula-main\">f<sub>n<\/sub> = (1 \/ 2\u03c0) \u00d7 \u221a(k \/ m)<\/div>\n <div class=\"formula-note\">f<sub>n<\/sub> in Hz, k in N\/m, m in kg. Also: \u03c9<sub>n<\/sub> = \u221a(k\/m) in rad\/s<\/div>\n <\/div>\n\n <p>This formula has profound practical implications:<\/p>\n <ul>\n <li>To <strong>increase<\/strong> f<sub>n<\/sub> by 2\u00d7, you must increase stiffness by 4\u00d7 (because of the square root) \u2014 or reduce mass by 4\u00d7<\/li>\n <li>To <strong>decrease<\/strong> f<sub>n<\/sub> by 2\u00d7, you must reduce stiffness by 4\u00d7 \u2014 or increase mass by 4\u00d7<\/li>\n <li>Changes to stiffness and mass have <em>diminishing returns<\/em>: each doubling of f<sub>n<\/sub> requires a 4\u00d7 change in the parameter<\/li>\n <\/ul>\n\n <h3>The Static Deflection Shortcut<\/h3>\n <p>One of the most useful practical formulas in vibration engineering relates natural frequency directly to static deflection under gravity:<\/p>\n\n <div class=\"formula-box\">\n <div class=\"formula-label\">Natural Frequency from Static Deflection<\/div>\n <div class=\"formula-main\">f<sub>n<\/sub> = (1 \/ 2\u03c0) \u00d7 \u221a(g \/ \u03b4) \u2248 15.76 \/ \u221a\u03b4<\/div>\n <div class=\"formula-note\">f<sub>n<\/sub> in Hz, \u03b4 in mm, g = 9810 mm\/s\u00b2. Very handy for quick estimates!<\/div>\n <\/div>\n\n <p>This is remarkably useful because static deflection is often easy to measure or estimate: simply measure how much a structure deflects under the weight of the machine. A machine that sags 1 mm on its supports has a vertical natural frequency of about 15.8 Hz (948 RPM). A machine that sags 0.25 mm has f<sub>n<\/sub> \u2248 31.5 Hz (1890 RPM).<\/p>\n\n <div class=\"info-box success\">\n <div class=\"box-title\">Quick Field Estimate<\/div>\n <p>Need a quick natural frequency estimate without instruments? Place a dial indicator under the machine's bearing housing and observe the static deflection when the machine weight is applied (e.g., during installation). The formula f<sub>n<\/sub> \u2248 15.76\/\u221a\u03b4<sub>mm<\/sub> gives a remarkably good first approximation of the fundamental vertical natural frequency.<\/p>\n <\/div>\n\n <h3>Multiple Degrees of Freedom<\/h3>\n <p>Real structures are not simple SDOF systems \u2014 they have many masses connected through distributed stiffness, resulting in many natural frequencies. A simple rigid body on elastic supports has six natural frequencies corresponding to six degrees of freedom: three translational (vertical, lateral, axial) and three rotational (roll, pitch, yaw). A flexible structure has infinitely many modes, though only the lowest several are usually of practical concern.<\/p>\n <p>The key principle is: <strong>the number of natural frequencies equals the number of degrees of freedom in the model<\/strong>. A simple beam modeled with 10 lumped masses has 10 natural frequencies; a finite element model with 10,000 nodes has 30,000 (3 DOF per node) natural frequencies, though only a few dozen may be in the frequency range of interest.<\/p>\n\n <h3>The Effect of Damping<\/h3>\n <p>Real systems always have some damping \u2014 friction, material hysteresis, radiation into surrounding structure, fluid drag, etc. Damping has two effects:<\/p>\n <ul>\n <li><strong>Slightly lowers the actual resonant frequency:<\/strong> The damped natural frequency is f<sub>d<\/sub> = f<sub>n<\/sub> \u00d7 \u221a(1 \u2212 \u03b6\u00b2), where \u03b6 is the damping ratio. For typical mechanical structures (\u03b6 = 0.01\u20130.05), this effect is negligible \u2014 less than 0.1% reduction.<\/li>\n <li><strong>Limits the amplitude at resonance:<\/strong> Without damping, resonance amplitude would theoretically be infinite. The amplification factor Q (quality factor) at resonance is approximately Q = 1\/(2\u03b6). For a lightly damped structure with \u03b6 = 0.02, Q = 25 \u2014 meaning the vibration amplitude at resonance is 25\u00d7 what it would be away from resonance. This is why even small amounts of unbalance can produce enormous vibration at critical speeds.<\/li>\n <\/ul>\n\n <h2 id=\"resonance\">Natural Frequency and Resonance: The Critical Connection<\/h2>\n <p>The concept of natural frequency is critically important in engineering specifically because of its direct connection to the phenomenon of <strong><a href=\"https:\/\/vibromera.eu\/glossary\/resonance\/\">resonance<\/a><\/strong>.<\/p>\n\n <h3>What is Resonance?<\/h3>\n <p>Resonance occurs when a periodic external force is applied to a system at a frequency that is equal to or very close to one of its natural frequencies. When this happens, the system absorbs energy from the external force with maximum efficiency, causing the vibration amplitude to grow dramatically. Each cycle of the forcing function adds energy to the system in exact synchronization with the system's natural oscillation, building amplitude cycle after cycle until either damping limits further growth or the structure fails.<\/p>\n\n <h3>The Amplification Factor<\/h3>\n <p>The magnification of vibration at resonance depends critically on the system's damping. The dynamic magnification factor (DMF) describes how much larger the dynamic response is compared to the static deflection that the same force would produce:<\/p>\n\n <div class=\"formula-box\">\n <div class=\"formula-label\">Dynamic Magnification Factor<\/div>\n <div class=\"formula-main\">DMF = 1 \/ \u221a[(1 \u2212 r\u00b2)\u00b2 + (2\u03b6r)\u00b2]<\/div>\n <div class=\"formula-note\">r = f<sub>forcing<\/sub>\/f<sub>n<\/sub> (frequency ratio), \u03b6 = damping ratio. At r = 1: DMF \u2248 1\/(2\u03b6)<\/div>\n <\/div>\n\n <table class=\"article-table\">\n <thead>\n <tr>\n <th>Damping Ratio (\u03b6)<\/th>\n <th>Typical System<\/th>\n <th>Q Factor (\u2248 1\/2\u03b6)<\/th>\n <th>Amplification at Resonance<\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td class=\"mono\">0.005<\/td>\n <td>Welded steel structure, undamped<\/td>\n <td class=\"mono\">100<\/td>\n <td>100\u00d7 static deflection<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">0.01<\/td>\n <td>Steel frame, bolted connections<\/td>\n <td class=\"mono\">50<\/td>\n <td>50\u00d7 static deflection<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">0.02<\/td>\n <td>Typical machinery structure<\/td>\n <td class=\"mono\">25<\/td>\n <td>25\u00d7 static deflection<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">0.05<\/td>\n <td>Concrete foundation, bolted joints<\/td>\n <td class=\"mono\">10<\/td>\n <td>10\u00d7 static deflection<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">0.10<\/td>\n <td>Rubber-mounted, well-damped<\/td>\n <td class=\"mono\">5<\/td>\n <td>5\u00d7 static deflection<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">0.20<\/td>\n <td>Highly damped (viscous damper)<\/td>\n <td class=\"mono\">2.5<\/td>\n <td>2.5\u00d7 static deflection<\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n\n <h3>Why Resonance Is Dangerous<\/h3>\n <p>Resonance is particularly treacherous because the vibration amplitude can be 10\u2013100\u00d7 larger than expected based on the forcing magnitude alone. A rotor with 50 \u00b5m of unbalance eccentricity that produces 1 mm\/s vibration at non-resonant speed could produce 25\u201350 mm\/s at resonance \u2014 enough to destroy bearings, fatigue bolts, crack welds, and cause cascading equipment failure.<\/p>\n\n <div class=\"info-box danger\">\n <div class=\"box-title\">Historical Example \u2014 Tacoma Narrows Bridge (1940)<\/div>\n <p>The collapse of the Tacoma Narrows Bridge remains one of the most dramatic demonstrations of resonance in engineering history. Wind forces at a frequency near the bridge's torsional natural frequency caused the bridge deck to oscillate with increasing amplitude until structural failure occurred. The event led to fundamental changes in bridge engineering and is studied in every structural dynamics course worldwide. Modern engineers routinely perform modal analysis to ensure structures are designed away from foreseeable excitation frequencies.<\/p>\n <\/div>\n\n <h2 id=\"critical-speeds\">Critical Speeds of Rotating Machinery<\/h2>\n <p>In rotating machinery, the most important manifestation of natural frequency is the <strong><a href=\"https:\/\/vibromera.eu\/glossary\/critical-speed\/\">critical speed<\/a><\/strong> \u2014 the rotational speed at which the shaft's rotation frequency (1\u00d7 RPM) coincides with a natural frequency of the rotor-bearing-support system. When a machine operates at a critical speed, the 1\u00d7 unbalance force excites the natural frequency, producing severe resonant vibration.<\/p>\n\n <h3>Types of Critical Speeds<\/h3>\n <ul>\n <li><strong>Rigid body criticals:<\/strong> Occur when the shaft speed matches a natural frequency of the rotor on its bearing supports, with the shaft itself remaining essentially straight. These are typically the first and second criticals (bounce and rock modes) and occur at lower speeds. Rigid body criticals can be modified by changing bearing stiffness or support structure mass.<\/li>\n <li><strong>Flexible rotor criticals (bending criticals):<\/strong> Occur when the shaft speed matches a natural frequency associated with shaft bending deformation. The first bending critical typically involves the shaft bowing into a half-sine shape. These are more dangerous because they involve large deflections at the shaft midspan and cannot be controlled by bearing changes alone \u2014 the shaft geometry itself must be modified.<\/li>\n <\/ul>\n\n <h3>Separation Margin<\/h3>\n <p>Industry standards (e.g., API 610, API 617) require a minimum <strong>separation margin<\/strong> between operating speed and critical speeds:<\/p>\n <ul>\n <li><strong>API typical requirement:<\/strong> Operating speed must be at least 15\u201320% away from any lateral critical speed (undamped)<\/li>\n <li><strong>General good practice:<\/strong> A 20% margin is considered minimum; 30% is preferred for critical equipment<\/li>\n <li><strong>VFD-driven equipment:<\/strong> Variable frequency drives change operating speed, potentially sweeping through criticals. The entire operating range must be checked, and criticals within the range must be identified and excluded or rapid transit programmed.<\/li>\n <\/ul>\n\n <div class=\"info-box\">\n <div class=\"box-title\">Practical Implication for Field Balancing<\/div>\n <p>When field balancing a machine that operates near (but safely above) a critical speed, the phase relationship between unbalance and vibration response will differ from what is expected for a \"below-resonance\" machine. The vibration signal may be 90\u2013180\u00b0 ahead of the heavy spot rather than in-phase. Good <a href=\"https:\/\/vibromera.eu\/product\/balanset-1\/\">balancing equipment<\/a> handles this automatically through trial-weight response measurement, but the analyst should be aware that near-critical operation complicates simple vector analysis.<\/p>\n <\/div>\n\n <h2 id=\"identification\">How Are Natural Frequencies Identified?<\/h2>\n <p>Identifying the natural frequencies of a machine or structure is a fundamental diagnostic skill. Several methods are available, ranging from simple to sophisticated:<\/p>\n\n <h3>1. Impact Testing (Bump Test)<\/h3>\n <p>The most common and practical experimental method for identifying structural natural frequencies. The procedure involves striking the machine or structure (while it is <em>not<\/em> running) with an instrumented impact hammer and measuring the resulting vibration with an accelerometer. The hammer blow inputs energy across a broad frequency range simultaneously, and the structure naturally \"rings\" at its natural frequencies, producing clear peaks in the resulting FFT spectrum.<\/p>\n\n <h4>Practical Procedure<\/h4>\n <div class=\"procedure-steps\">\n <div class=\"proc-step\">\n <div class=\"step-content\">\n <h4>Prepare the Equipment<\/h4>\n <p>Mount an accelerometer on the structure at the point of interest (typically the bearing housing or support structure). Connect to an FFT analyzer or data collector configured for impact testing (time-domain trigger, appropriate frequency range, typically 0\u20131000 Hz for structural resonances).<\/p>\n <\/div>\n <\/div>\n <div class=\"proc-step\">\n <div class=\"step-content\">\n <h4>Select Hammer Tip<\/h4>\n <p>Impact hammer tips of different hardness excite different frequency ranges. Soft rubber tips excite 0\u2013200 Hz; medium plastic tips excite 0\u2013500 Hz; hard steel tips excite 0\u20135000 Hz. Choose the tip that covers the frequency range of interest for the specific test.<\/p>\n <\/div>\n <\/div>\n <div class=\"proc-step\">\n <div class=\"step-content\">\n <h4>Strike and Record<\/h4>\n <p>Strike the structure firmly with a single, clean blow. Avoid double-hits (bouncing). The analyzer should capture the time waveform showing the impact and the resulting free vibration decay. The FFT of this response reveals the natural frequencies as peaks.<\/p>\n <\/div>\n <\/div>\n <div class=\"proc-step\">\n <div class=\"step-content\">\n <h4>Average Multiple Hits<\/h4>\n <p>Take 3\u20135 averages to improve signal-to-noise ratio and confirm consistency. If the Frequency Response Function (FRF) varies significantly between hits, check for double-hits, poor accelerometer mounting, or changing boundary conditions.<\/p>\n <\/div>\n <\/div>\n <div class=\"proc-step\">\n <div class=\"step-content\">\n <h4>Identify Natural Frequencies<\/h4>\n <p>Natural frequencies appear as peaks in the FRF magnitude plot. Confirm using the phase plot (natural frequencies show 180\u00b0 phase shift) and the coherence function (should be near 1.0 at natural frequencies). Record the frequencies and compare with operating speed and harmonics.<\/p>\n <\/div>\n <\/div>\n <\/div>\n\n <div class=\"info-box success\">\n <div class=\"box-title\">Bump Test Tips from the Field<\/div>\n <p>Always perform the bump test with the machine <em>assembled<\/em> but <em>not running<\/em>. The natural frequencies can change significantly when the rotor is removed (mass changes) or when the machine is running (gyroscopic effects, bearing stiffness changes with speed, thermal effects). Test in multiple directions (vertical, horizontal, axial) to find all relevant modes. Repeat after any structural modification to verify the change achieved the desired effect.<\/p>\n <\/div>\n\n <h3>2. Run-Up \/ Coast-Down Test<\/h3>\n <p>For running machines, a run-up or coast-down test is the most practical way to identify natural frequencies that are excited by rotating forces. As the machine's speed changes, the 1\u00d7 unbalance force (and any other speed-dependent forces) sweeps through a range of frequencies. When a forcing frequency crosses a natural frequency, the vibration amplitude shows a distinct peak \u2014 identifying that natural frequency as a <a href=\"https:\/\/vibromera.eu\/glossary\/critical-speed\/\">critical speed<\/a>.<\/p>\n <p>The test requires simultaneous vibration measurement and tachometer signal (keyphasor) to correlate vibration amplitude and phase with shaft speed. The data is typically displayed as a Bode plot (amplitude and phase vs. RPM) or a polar plot (amplitude \u00d7 phase vector vs. RPM). Both clearly show critical speeds as amplitude peaks accompanied by ~180\u00b0 phase shifts.<\/p>\n\n <h3>3. Waterfall \/ Cascade Plot Analysis<\/h3>\n <p>A waterfall (or cascade) plot is a 3D representation of multiple FFT spectra taken at different machine speeds during a run-up or coast-down. It displays frequency (horizontal), amplitude (vertical), and speed (depth axis). In this format:<\/p>\n <ul>\n <li><strong>Speed-dependent lines<\/strong> (orders) appear as diagonal lines: 1\u00d7, 2\u00d7, 3\u00d7 etc., moving to the right as speed increases<\/li>\n <li><strong>Natural frequencies<\/strong> appear as vertical peaks (fixed frequency regardless of speed) \u2014 they don't move as speed changes<\/li>\n <li><strong>Resonances<\/strong> are visible where a speed-dependent order line crosses a natural frequency, producing a localized amplitude spike<\/li>\n <\/ul>\n <p>This is one of the most powerful diagnostic tools for distinguishing speed-dependent vibration (from unbalance, misalignment, etc.) from structural resonance problems.<\/p>\n\n <h3>4. Finite Element Analysis (FEA)<\/h3>\n <p>During the design phase, engineers use computer models to predict the natural frequencies of components, machines, and support structures before they are built. FEA discretizes the structure into thousands of small elements, applies the correct material properties (density, elastic modulus, Poisson's ratio), models the boundary conditions (bolt connections, bearing supports, foundation), and solves the eigenvalue problem to extract natural frequencies and mode shapes.<\/p>\n <p>FEA is invaluable for:<\/p>\n <ul>\n <li>Designing structures to avoid resonance issues before fabrication<\/li>\n <li>Performing \"what-if\" analysis: what happens if we add a stiffener? Change the bearing span? Use a different material?<\/li>\n <li>Predicting modal behavior of complex geometries that are difficult to test experimentally<\/li>\n <li>Validating experimental results by correlating measured and predicted natural frequencies<\/li>\n <\/ul>\n\n <h3>5. Operational Modal Analysis (OMA)<\/h3>\n <p>A relatively modern technique that extracts natural frequencies and mode shapes from a running machine using only the response data \u2014 no controlled excitation (hammer or shaker) required. OMA uses advanced algorithms (e.g., stochastic subspace identification) that treat the machine's operating forces as \"white noise\" excitation. This is particularly valuable for large or critical equipment that cannot be shut down for bump testing or where operational boundary conditions differ significantly from stopped conditions.<\/p>\n\n <h2 id=\"practical\">Practical Examples in Industrial Machinery<\/h2>\n\n <div class=\"example-block\">\n <div class=\"example-title\">Case 1: Vertical Pump Excessive Vibration<\/div>\n <p><strong>Problem:<\/strong> A vertical turbine pump running at 1780 RPM (29.7 Hz) shows 12 mm\/s vibration at 1\u00d7 RPM on the motor top. Balancing attempts reduce vibration temporarily but it returns within weeks.<\/p>\n <p><strong>Investigation:<\/strong> A bump test on the motor\/pump assembly reveals a natural frequency at 28.5 Hz \u2014 only 4% below operating speed. The system is operating in the resonance band.<\/p>\n <p><strong>Solution:<\/strong> A steel support brace is added to the motor stool, increasing stiffness. Post-modification bump test shows the natural frequency has moved to 42 Hz (42% above operating speed). Vibration drops to 2.5 mm\/s without any balancing correction \u2014 confirming the root cause was resonance, not unbalance.<\/p>\n <\/div>\n\n <div class=\"example-block\">\n <div class=\"example-title\">Case 2: Fan Foundation Resonance<\/div>\n <p><strong>Problem:<\/strong> A large induced-draft fan on a steel-frame foundation runs at 990 RPM (16.5 Hz). The foundation shows 8 mm\/s vibration at 1\u00d7 RPM, while the fan itself shows only 2 mm\/s at the bearing housing.<\/p>\n <p><strong>Investigation:<\/strong> The fact that the foundation vibrates more than the source (fan) is a classic resonance indicator. A bump test reveals the foundation's lateral natural frequency is 17.2 Hz \u2014 within 4% of operating speed.<\/p>\n <p><strong>Solution:<\/strong> Two options considered: (1) add mass to the foundation (lower f<sub>n<\/sub>), or (2) add stiffness (raise f<sub>n<\/sub>). Cross-bracing is added to the foundation frame, raising f<sub>n<\/sub> to 24 Hz. Foundation vibration drops to 1.8 mm\/s.<\/p>\n <\/div>\n\n <div class=\"example-block\">\n <div class=\"example-title\">Case 3: Piping Resonance at Pump BPF<\/div>\n <p><strong>Problem:<\/strong> Piping connected to a 5-vane centrifugal pump running at 1480 RPM shows severe vibration at 123 Hz (= 5 \u00d7 24.7 Hz, the blade pass frequency). Pipe clamps loosen and fatigue cracks appear at welded supports.<\/p>\n <p><strong>Investigation:<\/strong> A bump test on the affected pipe span reveals a natural frequency at 120 Hz \u2014 almost exactly at the pump's blade pass frequency (5\u00d7 RPM = 123 Hz).<\/p>\n <p><strong>Solution:<\/strong> An additional pipe support is installed at the midspan, raising the span's natural frequency to 185 Hz. Alternatively, for some installations, adding a tuned vibration absorber (dynamic absorber) at the pipe's antinode can be effective. After the support addition, piping vibration drops by 85%.<\/p>\n <\/div>\n\n <h2 id=\"avoidance\">Strategies for Avoiding Resonance Problems<\/h2>\n <p>The best time to address resonance is during design, but it can also be corrected in the field. There are three fundamental strategies:<\/p>\n\n <h3>1. Detune \u2014 Change the Natural Frequency<\/h3>\n <p>Move the natural frequency away from the excitation frequency. Require a minimum separation margin (typically 20\u201330%). Options include:<\/p>\n <ul>\n <li><strong>Increase stiffness:<\/strong> Add bracing, stiffeners, gussets, thicker plates, or concrete fill. This raises f<sub>n<\/sub>. Most common fix for structures that resonate below operating speed.<\/li>\n <li><strong>Add mass:<\/strong> Attach additional mass (steel plates, concrete). This lowers f<sub>n<\/sub>. Used when the natural frequency is just above the excitation frequency and it's easier to move it lower.<\/li>\n <li><strong>Modify bearing stiffness:<\/strong> For shaft criticals, changing bearing clearance, preload, or type can shift the critical speed. Stiffer bearings raise criticals; softer bearings lower them.<\/li>\n <li><strong>Change shaft geometry:<\/strong> For bending criticals, increasing shaft diameter raises critical speed (stiffness increases faster than mass). Shortening bearing span also raises criticals.<\/li>\n <\/ul>\n\n <h3>2. Damp \u2014 Reduce Amplitude at Resonance<\/h3>\n <p>If the natural frequency cannot be moved away from the excitation, add damping to limit the resonant amplitude. Options include:<\/p>\n <ul>\n <li><strong>Constrained layer damping:<\/strong> Viscoelastic material sandwiched between structural plates \u2014 highly effective for panel and housing resonances<\/li>\n <li><strong>Viscous dampers:<\/strong> Squeeze-film or viscous dashpot dampers, commonly used in bearing supports for turbomachinery<\/li>\n <li><strong>Tuned vibration absorbers:<\/strong> A mass-spring system tuned to the problem frequency, attached to the vibrating structure. The absorber vibrates in anti-phase, canceling the structure's motion at the target frequency<\/li>\n <li><strong>Bolted joints:<\/strong> Increasing the number of bolted joints (vs. welded) introduces friction damping through micro-slip at joint interfaces<\/li>\n <\/ul>\n\n <h3>3. Reduce the Exciting Force<\/h3>\n <p>If neither detuning nor damping is practical, reduce the forcing magnitude:<\/p>\n <ul>\n <li><strong>Better balancing:<\/strong> Reduce 1\u00d7 excitation by balancing to a tighter <a href=\"https:\/\/vibromera.eu\/glossary\/balance-quality-grade\/\">G-grade<\/a> \u2014 even if not at resonance, this reduces the force available to excite any resonance<\/li>\n <li><strong>Precision alignment:<\/strong> Reduce 2\u00d7 excitation from misalignment<\/li>\n <li><strong>Speed change:<\/strong> If the machine is VFD-driven, exclude the resonant speed from the operating range or program rapid transit through the resonance band<\/li>\n <li><strong>Isolation:<\/strong> Install vibration isolators to prevent the excitation from reaching the resonant structure<\/li>\n <\/ul>\n\n <div class=\"info-box success\">\n <div class=\"box-title\">The 20% Rule of Thumb<\/div>\n <p>In practice, aim for at least 20% separation between any natural frequency and any significant excitation frequency. For critical applications (power generation, offshore, aerospace), 30% or more is preferred. This applies not just to 1\u00d7 RPM but also to 2\u00d7 (misalignment), blade\/vane pass frequencies, gear mesh frequencies, and any other periodic excitation. A comprehensive resonance avoidance analysis compares <em>all<\/em> excitation frequencies against <em>all<\/em> natural frequencies in the system.<\/p>\n <\/div>\n\n <p>Understanding natural frequency \u2014 and its dangerous relationship with resonance \u2014 is fundamental to the practice of vibration analysis and machinery reliability engineering. Every vibration analyst should be competent in identifying natural frequencies through testing, interpreting their relationship to operating conditions, and recommending appropriate corrective actions when resonance is found to be contributing to a vibration problem.<\/p>\n\n <hr style=\"margin: 48px 0 24px; border: none; border-top: 1px solid var(--border-light);\">\n <p><a href=\"https:\/\/vibromera.eu\/glossary\/\">\u2190 Back to Glossary Index<\/a><\/p>\n\n <\/article>\n\n \n <aside class=\"toc-sidebar\">\n <div class=\"toc-box\">\n <h3>On This Page<\/h3>\n <a href=\"#calculator\">Natural Frequency Calculator<\/a>\n <a href=\"#key-concepts\">Key Concepts<\/a>\n <a href=\"#definition\">Definition<\/a>\n <a href=\"#mass-stiffness\">Mass, Stiffness & f<sub>n<\/sub><\/a>\n <a class=\"sub\" href=\"#mass-stiffness\">Fundamental Formula<\/a>\n <a class=\"sub\" href=\"#mass-stiffness\">Static Deflection Shortcut<\/a>\n <a class=\"sub\" href=\"#mass-stiffness\">Multiple DOF<\/a>\n <a class=\"sub\" href=\"#mass-stiffness\">Effect of Damping<\/a>\n <a href=\"#resonance\">Resonance<\/a>\n <a class=\"sub\" href=\"#resonance\">Amplification Factor<\/a>\n <a class=\"sub\" href=\"#resonance\">Why It's Dangerous<\/a>\n <a href=\"#critical-speeds\">Critical Speeds<\/a>\n <a class=\"sub\" href=\"#critical-speeds\">Separation Margin<\/a>\n <a href=\"#identification\">Identification Methods<\/a>\n <a class=\"sub\" href=\"#identification\">Impact \/ Bump Test<\/a>\n <a class=\"sub\" href=\"#identification\">Run-Up \/ Coast-Down<\/a>\n <a class=\"sub\" href=\"#identification\">Waterfall Plot<\/a>\n <a class=\"sub\" href=\"#identification\">FEA<\/a>\n <a class=\"sub\" href=\"#identification\">OMA<\/a>\n <a href=\"#practical\">Practical Examples<\/a>\n <a href=\"#avoidance\">Avoidance Strategies<\/a>\n <a href=\"#faq\">FAQ (7 Questions)<\/a>\n <\/div>\n\n <div class=\"toc-box\" style=\"margin-top: 24px; background: var(--navy); border-color: var(--navy);\">\n <h3 style=\"color: var(--white);\">Vibromera Equipment<\/h3>\n <p style=\"color: rgba(255,255,255,0.6); font-size: 13px; margin-bottom: 12px;\">Portable vibration analysis and balancing devices with built-in spectrum analysis for resonance identification.<\/p>\n <a href=\"https:\/\/vibromera.eu\/product\/balanset-1\/\" style=\"display: block; padding: 8px 14px; background: var(--blue); color: white; border-radius: 6px; text-align: center; font-weight: 600; font-size: 14px; text-decoration: none; margin-bottom: 8px; border-left: none;\">Balanset-1A \u2192<\/a>\n <a href=\"https:\/\/vibromera.eu\/product\/balanset-4\/\" style=\"display: block; padding: 8px 14px; background: rgba(255,255,255,0.1); color: white; border-radius: 6px; text-align: center; font-weight: 600; font-size: 14px; text-decoration: none; border: 1px solid rgba(255,255,255,0.2); border-left: none;\">Balanset-4 \u2192<\/a>\n <\/div>\n <\/aside>\n <\/div>\n <\/div>\n<\/main>\n\n\n<section class=\"grades-section\" id=\"faq\" style=\"background: var(--white); border-top: 1px solid var(--border-light);\">\n <div class=\"container\" style=\"max-width:900px;\">\n <div class=\"section-header\">\n <h2>Frequently Asked Questions \u2014 Natural Frequency<\/h2>\n <p>Common questions about natural frequency, resonance, and critical speeds<\/p>\n <\/div>\n <div style=\"display:flex;flex-direction:column;gap:16px;\">\n\n <details style=\"background:var(--beige-light);border:1px solid var(--beige-dark);border-radius:var(--radius);overflow:hidden;\">\n <summary style=\"padding:18px 24px;font-weight:700;font-size:16px;cursor:pointer;color:var(--navy);list-style:none;display:flex;align-items:center;gap:10px;\">\n <span style=\"color:var(--blue);font-size:18px;\">\u25b8<\/span> What is natural frequency in simple terms?\n <\/summary>\n <div style=\"padding:0 24px 18px 52px;font-size:15px;line-height:1.7;color:var(--text);\">\n Natural frequency is the rate at which an object \"wants\" to vibrate when disturbed and released. Think of a tuning fork: strike it and it always vibrates at the same pitch \u2014 that's its natural frequency. Every mechanical structure has natural frequencies determined by its mass (how heavy it is) and stiffness (how rigid it is). The formula is: <strong>f<sub>n<\/sub> = (1\/2\u03c0) \u00d7 \u221a(k\/m)<\/strong>. Heavier objects vibrate slower; stiffer objects vibrate faster.\n <\/div>\n <\/details>\n\n <details style=\"background:var(--beige-light);border:1px solid var(--beige-dark);border-radius:var(--radius);overflow:hidden;\">\n <summary style=\"padding:18px 24px;font-weight:700;font-size:16px;cursor:pointer;color:var(--navy);list-style:none;display:flex;align-items:center;gap:10px;\">\n <span style=\"color:var(--blue);font-size:18px;\">\u25b8<\/span> How do you calculate natural frequency of a mass-spring system?\n <\/summary>\n <div style=\"padding:0 24px 18px 52px;font-size:15px;line-height:1.7;color:var(--text);\">\n Use the fundamental SDOF formula: <strong>f<sub>n<\/sub> = (1\/2\u03c0) \u00d7 \u221a(k\/m)<\/strong>, where k is spring stiffness in N\/m and m is mass in kg. Example: a 10 kg mass on a 40,000 N\/m spring \u2192 f<sub>n<\/sub> = (1\/2\u03c0) \u00d7 \u221a(40000\/10) = (1\/6.283) \u00d7 63.25 = <strong>10.07 Hz<\/strong>. The corresponding critical speed is 10.07 \u00d7 60 = <strong>604 RPM<\/strong>.\n <\/div>\n <\/details>\n\n <details style=\"background:var(--beige-light);border:1px solid var(--beige-dark);border-radius:var(--radius);overflow:hidden;\">\n <summary style=\"padding:18px 24px;font-weight:700;font-size:16px;cursor:pointer;color:var(--navy);list-style:none;display:flex;align-items:center;gap:10px;\">\n <span style=\"color:var(--blue);font-size:18px;\">\u25b8<\/span> What happens at resonance? Why is it dangerous?\n <\/summary>\n <div style=\"padding:0 24px 18px 52px;font-size:15px;line-height:1.7;color:var(--text);\">\n Resonance occurs when a forcing frequency matches a natural frequency. The system absorbs energy maximally, amplifying vibration by 10\u201350\u00d7 for typical steel structures. This can cause bearing failure, fatigue cracking, foundation damage, and catastrophic structural collapse. The most famous example is the Tacoma Narrows Bridge (1940), which was destroyed by wind-induced resonance. In industrial machinery, resonance is the #1 cause of unexplained high vibration that doesn't respond to <a href=\"https:\/\/vibromera.eu\/glossary\/rotor-balancing\/\">balancing<\/a>.\n <\/div>\n <\/details>\n\n <details style=\"background:var(--beige-light);border:1px solid var(--beige-dark);border-radius:var(--radius);overflow:hidden;\">\n <summary style=\"padding:18px 24px;font-weight:700;font-size:16px;cursor:pointer;color:var(--navy);list-style:none;display:flex;align-items:center;gap:10px;\">\n <span style=\"color:var(--blue);font-size:18px;\">\u25b8<\/span> What is critical speed and how does it relate to natural frequency?\n <\/summary>\n <div style=\"padding:0 24px 18px 52px;font-size:15px;line-height:1.7;color:var(--text);\">\n Critical speed is the rotational speed (RPM) at which a shaft's operating frequency matches a natural frequency, causing resonance. <strong>Critical speed = f<sub>n<\/sub> \u00d7 60<\/strong>. For safe operation, maintain at least 15\u201320% separation margin between operating speed and any critical speed (per API standards). Machines that operate above the first critical speed are called \"flexible rotors\" and require special balancing techniques.\n <\/div>\n <\/details>\n\n <details style=\"background:var(--beige-light);border:1px solid var(--beige-dark);border-radius:var(--radius);overflow:hidden;\">\n <summary style=\"padding:18px 24px;font-weight:700;font-size:16px;cursor:pointer;color:var(--navy);list-style:none;display:flex;align-items:center;gap:10px;\">\n <span style=\"color:var(--blue);font-size:18px;\">\u25b8<\/span> How do you measure natural frequency in the field?\n <\/summary>\n <div style=\"padding:0 24px 18px 52px;font-size:15px;line-height:1.7;color:var(--text);\">\n The most common method is the <strong>impact test (bump test)<\/strong>: mount an accelerometer on the structure, strike it with a hammer, and analyze the resulting FFT spectrum. Natural frequencies appear as peaks. Other methods: run-up\/coast-down tests (Bode plots), waterfall analysis during speed changes, and Operational Modal Analysis. The <a href=\"https:\/\/vibromera.eu\/product\/balanset-1\/\">Balanset-1A<\/a> portable analyzer can perform FFT spectrum analysis to identify natural frequencies directly in the field.\n <\/div>\n <\/details>\n\n <details style=\"background:var(--beige-light);border:1px solid var(--beige-dark);border-radius:var(--radius);overflow:hidden;\">\n <summary style=\"padding:18px 24px;font-weight:700;font-size:16px;cursor:pointer;color:var(--navy);list-style:none;display:flex;align-items:center;gap:10px;\">\n <span style=\"color:var(--blue);font-size:18px;\">\u25b8<\/span> How can you change a structure's natural frequency to avoid resonance?\n <\/summary>\n <div style=\"padding:0 24px 18px 52px;font-size:15px;line-height:1.7;color:var(--text);\">\n Since f<sub>n<\/sub> = (1\/2\u03c0)\u221a(k\/m): <strong>to increase f<sub>n<\/sub><\/strong> \u2014 add stiffness (welded bracing, concrete grouting, thicker baseplate) or reduce mass. <strong>To decrease f<sub>n<\/sub><\/strong> \u2014 add mass (concrete filling, steel plates) or reduce stiffness (softer isolators). The \"20% rule\" requires operating speed to be at least 20% away from any natural frequency. Adding damping (rubber pads, viscous dampers) won't change f<sub>n<\/sub> but reduces the amplification at resonance.\n <\/div>\n <\/details>\n\n <details style=\"background:var(--beige-light);border:1px solid var(--beige-dark);border-radius:var(--radius);overflow:hidden;\">\n <summary style=\"padding:18px 24px;font-weight:700;font-size:16px;cursor:pointer;color:var(--navy);list-style:none;display:flex;align-items:center;gap:10px;\">\n <span style=\"color:var(--blue);font-size:18px;\">\u25b8<\/span> What is the static deflection shortcut for natural frequency?\n <\/summary>\n <div style=\"padding:0 24px 18px 52px;font-size:15px;line-height:1.7;color:var(--text);\">\n The static deflection formula provides a quick estimate: <strong>f<sub>n<\/sub> \u2248 15.76 \/ \u221a\u03b4<\/strong>, where \u03b4 is static deflection in millimeters under the system's own weight. This works because both f<sub>n<\/sub> and \u03b4 depend on the same k\/m ratio. A system that deflects 1 mm under gravity has f<sub>n<\/sub> \u2248 15.8 Hz; 4 mm deflection gives f<sub>n<\/sub> \u2248 7.9 Hz. This is extremely useful for quick field estimates without knowing exact stiffness values.\n <\/div>\n <\/details>\n\n <\/div>\n <\/div>\n<\/section>\n\n\n<section style=\"padding:32px 0;background:var(--beige-light);border-top:1px solid var(--border-light);\">\n <div class=\"container\" style=\"max-width:900px;\">\n <h3 style=\"font-family:'DM Serif Display',serif;font-size:22px;color:var(--navy);margin-bottom:20px;\">Related Glossary Articles<\/h3>\n <div style=\"display:grid;grid-template-columns:repeat(3,1fr);gap:16px;\">\n <a href=\"https:\/\/vibromera.eu\/glossary\/resonance\/\" style=\"display:block;padding:18px;background:var(--white);border:1px solid var(--border-light);border-radius:var(--radius-sm);text-decoration:none;transition:all 0.2s;box-shadow:var(--shadow-sm);\">\n <div style=\"font-weight:700;color:var(--navy);font-size:15px;margin-bottom:4px;\">Resonance<\/div>\n <div style=\"font-size:13px;color:var(--text-secondary);\">What happens when forcing frequency meets natural frequency \u2014 amplification, damage, and avoidance<\/div>\n <\/a>\n <a href=\"https:\/\/vibromera.eu\/glossary\/critical-speed\/\" style=\"display:block;padding:18px;background:var(--white);border:1px solid var(--border-light);border-radius:var(--radius-sm);text-decoration:none;transition:all 0.2s;box-shadow:var(--shadow-sm);\">\n <div style=\"font-weight:700;color:var(--navy);font-size:15px;margin-bottom:4px;\">Critical Speed<\/div>\n <div style=\"font-size:13px;color:var(--text-secondary);\">Rotor critical speeds, separation margins, and rigid vs. flexible rotor classification<\/div>\n <\/a>\n <a href=\"https:\/\/vibromera.eu\/glossary\/balance-quality-grade\/\" style=\"display:block;padding:18px;background:var(--white);border:1px solid var(--border-light);border-radius:var(--radius-sm);text-decoration:none;transition:all 0.2s;box-shadow:var(--shadow-sm);\">\n <div style=\"font-weight:700;color:var(--navy);font-size:15px;margin-bottom:4px;\">Balance Quality Grade (G-Grade)<\/div>\n <div style=\"font-size:13px;color:var(--text-secondary);\">ISO 21940-11 balancing tolerances \u2014 why balanced rotors still vibrate at resonance<\/div>\n <\/a>\n <a href=\"https:\/\/vibromera.eu\/glossary\/harmonics-vibration\/\" style=\"display:block;padding:18px;background:var(--white);border:1px solid var(--border-light);border-radius:var(--radius-sm);text-decoration:none;transition:all 0.2s;box-shadow:var(--shadow-sm);\">\n <div style=\"font-weight:700;color:var(--navy);font-size:15px;margin-bottom:4px;\">Harmonics in Vibration<\/div>\n <div style=\"font-size:13px;color:var(--text-secondary);\">1\u00d7, 2\u00d7, 3\u00d7 patterns \u2014 harmonics can excite higher-order natural frequencies<\/div>\n <\/a>\n <a href=\"https:\/\/vibromera.eu\/glossary\/unbalance\/\" style=\"display:block;padding:18px;background:var(--white);border:1px solid var(--border-light);border-radius:var(--radius-sm);text-decoration:none;transition:all 0.2s;box-shadow:var(--shadow-sm);\">\n <div style=\"font-weight:700;color:var(--navy);font-size:15px;margin-bottom:4px;\">Unbalance<\/div>\n <div style=\"font-size:13px;color:var(--text-secondary);\">The most common excitation source \u2014 and why its effect depends on proximity to resonance<\/div>\n <\/a>\n <a href=\"https:\/\/vibromera.eu\/glossary\/fft\/\" style=\"display:block;padding:18px;background:var(--white);border:1px solid var(--border-light);border-radius:var(--radius-sm);text-decoration:none;transition:all 0.2s;box-shadow:var(--shadow-sm);\">\n <div style=\"font-weight:700;color:var(--navy);font-size:15px;margin-bottom:4px;\">FFT Spectrum Analysis<\/div>\n <div style=\"font-size:13px;color:var(--text-secondary);\">How FFT is used in bump tests and run-up analysis to identify natural frequencies<\/div>\n <\/a>\n <\/div>\n <\/div>\n<\/section>\n\n\n<section class=\"shop-cta\">\n <div class=\"container\">\n <h2>Professional Vibration Analysis Equipment<\/h2>\n <p>Identify resonance problems and balance rotors in the field with Vibromera's portable devices \u2014 spectrum analysis, phase measurement, and ISO-compliant balancing in one instrument.<\/p>\n <a href=\"https:\/\/vibromera.eu\/shop\/\" class=\"cta-btn\">Browse Equipment \u2192<\/a>\n <\/div>\n<\/section>\n\n\n<footer class=\"page-footer\">\n <div class=\"container\">\n <p><a href=\"https:\/\/vibromera.eu\/glossary\/\">\u2190 Back to Glossary<\/a> | <a href=\"https:\/\/vibromera.eu\/\">vibromera.eu<\/a><\/p>\n <\/div>\n<\/footer>\n\n<script>\n\/\/ ===== CALCULATOR =====\nfunction toggleInputs() {\n const type = document.getElementById('systemType').value;\n const show = (id, vis) => document.getElementById(id).style.display = vis ? '' : 'none';\n\n show('fg-stiffness', type === 'spring-mass');\n show('fg-mass', type === 'spring-mass' || type === 'beam-center' || type === 'beam-cantilever');\n show('fg-length', type === 'beam-center' || type === 'beam-cantilever');\n show('fg-ei', type === 'beam-center' || type === 'beam-cantilever');\n show('fg-deflection', type === 'static-deflection');\n show('fg-shaft-dia', type === 'shaft-disc');\n show('fg-shaft-len', type === 'shaft-disc');\n show('fg-disc-mass', type === 'shaft-disc');\n}\n\nfunction calculate() {\n const type = document.getElementById('systemType').value;\n const rpm = parseFloat(document.getElementById('operatingRPM').value) || 0;\n let fn = 0;\n let description = '';\n\n if (type === 'spring-mass') {\n const k = parseFloat(document.getElementById('stiffness').value);\n const m = parseFloat(document.getElementById('mass').value);\n if (!k || !m) return alert('Enter stiffness and mass.');\n fn = (1 \/ (2 * Math.PI)) * Math.sqrt(k \/ m);\n description = `Spring\u2013Mass: k=${k} N\/m, m=${m} kg`;\n } else if (type === 'beam-center') {\n const m = parseFloat(document.getElementById('mass').value);\n const L = parseFloat(document.getElementById('beamLength').value);\n const EI = parseFloat(document.getElementById('beamEI').value);\n if (!m || !L || !EI) return alert('Enter mass, length, and EI.');\n \/\/ Simply supported beam, center point mass: k_eff = 48*EI\/L^3\n const k_eff = 48 * EI \/ Math.pow(L, 3);\n fn = (1 \/ (2 * Math.PI)) * Math.sqrt(k_eff \/ m);\n description = `Simply supported beam: EI=${EI} N\u00b7m\u00b2, L=${L} m, m=${m} kg`;\n } else if (type === 'beam-cantilever') {\n const m = parseFloat(document.getElementById('mass').value);\n const L = parseFloat(document.getElementById('beamLength').value);\n const EI = parseFloat(document.getElementById('beamEI').value);\n if (!m || !L || !EI) return alert('Enter mass, length, and EI.');\n \/\/ Cantilever beam, end mass: k_eff = 3*EI\/L^3\n const k_eff = 3 * EI \/ Math.pow(L, 3);\n fn = (1 \/ (2 * Math.PI)) * Math.sqrt(k_eff \/ m);\n description = `Cantilever beam: EI=${EI} N\u00b7m\u00b2, L=${L} m, m=${m} kg`;\n } else if (type === 'shaft-disc') {\n const d = parseFloat(document.getElementById('shaftDia').value) \/ 1000; \/\/ mm to m\n const L = parseFloat(document.getElementById('shaftLen').value) \/ 1000;\n const m = parseFloat(document.getElementById('discMass').value);\n if (!d || !L || !m) return alert('Enter shaft diameter, span, and disc mass.');\n const E = 210e9; \/\/ Steel\n const I = Math.PI * Math.pow(d, 4) \/ 64;\n const k_eff = 48 * E * I \/ Math.pow(L, 3);\n fn = (1 \/ (2 * Math.PI)) * Math.sqrt(k_eff \/ m);\n description = `Steel shaft \u2300${(d*1000).toFixed(0)}mm, span ${(L*1000).toFixed(0)}mm, disc ${m} kg`;\n } else if (type === 'static-deflection') {\n const delta = parseFloat(document.getElementById('staticDefl').value);\n if (!delta) return alert('Enter static deflection.');\n fn = 15.76 \/ Math.sqrt(delta);\n description = `Static deflection \u03b4 = ${delta} mm`;\n }\n\n const fn_rpm = fn * 60;\n const omega_n = fn * 2 * Math.PI;\n const period = 1 \/ fn;\n\n let ratio = 0;\n let riskStatus = '';\n let riskClass = '';\n let riskDetail = '';\n\n if (rpm > 0) {\n const fop = rpm \/ 60;\n ratio = fop \/ fn;\n if (ratio < 0.7 || ratio > 1.4) {\n riskStatus = '\u2713 Safe \u2014 Outside Resonance Band';\n riskClass = 'safe';\n riskDetail = `Frequency ratio = ${ratio.toFixed(3)} (operating ${ratio < 1 ? 'below' : 'above'} f_n). Margin: ${(Math.abs(1 - ratio) * 100).toFixed(1)}%`;\n } else if (ratio >= 0.85 && ratio <= 1.15) {\n riskStatus = '\u2717 Danger \u2014 In Resonance Band!';\n riskClass = 'danger';\n riskDetail = `Frequency ratio = ${ratio.toFixed(3)}. Operating within \u00b115% of f_n \u2014 expect severe amplification!`;\n } else {\n riskStatus = '\u26a0 Caution \u2014 Near Resonance';\n riskClass = 'caution';\n riskDetail = `Frequency ratio = ${ratio.toFixed(3)}. Within approach\/exit zone \u2014 elevated vibration expected.`;\n }\n }\n\n const area = document.getElementById('resultsArea');\n area.innerHTML = `\n <div class=\"result-primary\">\n <div class=\"res-label\">Natural Frequency (f<sub>n<\/sub>)<\/div>\n <div class=\"res-value\">${fn >= 100 ? fn.toFixed(1) : fn >= 1 ? fn.toFixed(2) : fn.toFixed(3)}<\/div>\n <div class=\"res-unit\">Hz<\/div>\n <\/div>\n <div class=\"results-secondary\">\n <div class=\"res-card\">\n <div class=\"rc-label\">Critical Speed<\/div>\n <div class=\"rc-value\">${fn_rpm >= 100 ? fn_rpm.toFixed(0) : fn_rpm.toFixed(1)}<\/div>\n <div class=\"rc-unit\">RPM<\/div>\n <\/div>\n <div class=\"res-card\">\n <div class=\"rc-label\">Angular Frequency (\u03c9<sub>n<\/sub>)<\/div>\n <div class=\"rc-value\">${omega_n >= 100 ? omega_n.toFixed(1) : omega_n.toFixed(2)}<\/div>\n <div class=\"rc-unit\">rad\/s<\/div>\n <\/div>\n <div class=\"res-card\">\n <div class=\"rc-label\">Period<\/div>\n <div class=\"rc-value\">${period >= 0.1 ? period.toFixed(3) : period.toFixed(4)}<\/div>\n <div class=\"rc-unit\">seconds<\/div>\n <\/div>\n <div class=\"res-card\">\n <div class=\"rc-label\">System<\/div>\n <div class=\"rc-value\" style=\"font-size:13px; font-family: 'Source Sans 3', sans-serif;\">${description}<\/div>\n <\/div>\n <\/div>\n ${rpm > 0 ? `\n <div class=\"resonance-indicator ${riskClass}\">\n <div class=\"ri-status\">${riskStatus}<\/div>\n <div class=\"ri-detail\">${riskDetail}<\/div>\n <\/div>` : ''}\n `;\n}\n\n\/\/ ===== RESONANCE CURVE CANVAS =====\nfunction drawResonanceCurve() {\n const canvas = document.getElementById('resonanceCanvas');\n const ctx = canvas.getContext('2d');\n const dpr = window.devicePixelRatio || 1;\n const rect = canvas.parentElement.getBoundingClientRect();\n\n canvas.width = rect.width * dpr;\n canvas.height = rect.height * dpr;\n canvas.style.width = rect.width + 'px';\n canvas.style.height = rect.height + 'px';\n ctx.scale(dpr, dpr);\n\n const W = rect.width;\n const H = rect.height;\n const pad = { top: 30, right: 40, bottom: 50, left: 70 };\n const plotW = W - pad.left - pad.right;\n const plotH = H - pad.top - pad.bottom;\n\n \/\/ Clear\n ctx.fillStyle = '#faf7f2';\n ctx.fillRect(0, 0, W, H);\n\n \/\/ Grid\n ctx.strokeStyle = '#e2e8f0';\n ctx.lineWidth = 0.5;\n for (let i = 0; i <= 10; i++) {\n const x = pad.left + (i \/ 10) * plotW;\n ctx.beginPath(); ctx.moveTo(x, pad.top); ctx.lineTo(x, pad.top + plotH); ctx.stroke();\n }\n for (let i = 0; i <= 8; i++) {\n const y = pad.top + (i \/ 8) * plotH;\n ctx.beginPath(); ctx.moveTo(pad.left, y); ctx.lineTo(pad.left + plotW, y); ctx.stroke();\n }\n\n \/\/ Resonance zone highlight\n const x07 = pad.left + (0.7 \/ 3.0) * plotW;\n const x14 = pad.left + (1.4 \/ 3.0) * plotW;\n ctx.fillStyle = 'rgba(220,38,38,0.06)';\n ctx.fillRect(x07, pad.top, x14 - x07, plotH);\n\n \/\/ r=1 line\n const x1 = pad.left + (1.0 \/ 3.0) * plotW;\n ctx.strokeStyle = 'rgba(220,38,38,0.3)';\n ctx.lineWidth = 1;\n ctx.setLineDash([4, 4]);\n ctx.beginPath(); ctx.moveTo(x1, pad.top); ctx.lineTo(x1, pad.top + plotH); ctx.stroke();\n ctx.setLineDash([]);\n\n \/\/ DMF function\n function dmf(r, zeta) {\n const denom = Math.sqrt(Math.pow(1 - r * r, 2) + Math.pow(2 * zeta * r, 2));\n return 1 \/ denom;\n }\n\n const maxDMF = 30;\n const dampings = [\n { zeta: 0.02, color: '#dc2626', label: '\u03b6 = 0.02 (Q=25)' },\n { zeta: 0.05, color: '#d97706', label: '\u03b6 = 0.05 (Q=10)' },\n { zeta: 0.10, color: '#2563eb', label: '\u03b6 = 0.10 (Q=5)' },\n { zeta: 0.20, color: '#059669', label: '\u03b6 = 0.20 (Q=2.5)' },\n { zeta: 0.50, color: '#94a3b8', label: '\u03b6 = 0.50 (Q=1)' },\n ];\n\n const rMax = 3.0;\n const steps = 400;\n\n dampings.forEach(d => {\n ctx.strokeStyle = d.color;\n ctx.lineWidth = 2;\n ctx.beginPath();\n let started = false;\n for (let i = 0; i <= steps; i++) {\n const r = (i \/ steps) * rMax;\n if (r === 0) continue;\n let amp = dmf(r, d.zeta);\n if (amp > maxDMF) amp = maxDMF;\n const x = pad.left + (r \/ rMax) * plotW;\n const y = pad.top + plotH - (amp \/ maxDMF) * plotH;\n if (!started) { ctx.moveTo(x, y); started = true; }\n else ctx.lineTo(x, y);\n }\n ctx.stroke();\n });\n\n \/\/ Axes\n ctx.strokeStyle = var_navy;\n ctx.lineWidth = 1.5;\n ctx.beginPath();\n ctx.moveTo(pad.left, pad.top);\n ctx.lineTo(pad.left, pad.top + plotH);\n ctx.lineTo(pad.left + plotW, pad.top + plotH);\n ctx.stroke();\n\n \/\/ X-axis labels\n ctx.fillStyle = '#475569';\n ctx.font = '12px \"JetBrains Mono\", monospace';\n ctx.textAlign = 'center';\n for (let r = 0; r <= 3; r += 0.5) {\n const x = pad.left + (r \/ rMax) * plotW;\n ctx.fillText(r.toFixed(1), x, pad.top + plotH + 18);\n }\n ctx.font = '13px \"Source Sans 3\", sans-serif';\n ctx.fillText('Frequency Ratio (\u03c9 \/ \u03c9\u2099)', pad.left + plotW \/ 2, pad.top + plotH + 42);\n\n \/\/ Y-axis labels\n ctx.textAlign = 'right';\n ctx.font = '12px \"JetBrains Mono\", monospace';\n for (let a = 0; a <= maxDMF; a += 5) {\n const y = pad.top + plotH - (a \/ maxDMF) * plotH;\n ctx.fillText(a.toString(), pad.left - 8, y + 4);\n }\n ctx.save();\n ctx.translate(16, pad.top + plotH \/ 2);\n ctx.rotate(-Math.PI \/ 2);\n ctx.textAlign = 'center';\n ctx.font = '13px \"Source Sans 3\", sans-serif';\n ctx.fillText('Amplification (DMF)', 0, 0);\n ctx.restore();\n\n \/\/ Legend\n const legendX = pad.left + plotW - 170;\n let legendY = pad.top + 10;\n ctx.fillStyle = 'rgba(255,255,255,0.9)';\n ctx.fillRect(legendX - 8, legendY - 4, 178, dampings.length * 22 + 8);\n ctx.strokeStyle = '#e2e8f0';\n ctx.lineWidth = 1;\n ctx.strokeRect(legendX - 8, legendY - 4, 178, dampings.length * 22 + 8);\n\n dampings.forEach(d => {\n ctx.strokeStyle = d.color;\n ctx.lineWidth = 2.5;\n ctx.beginPath();\n ctx.moveTo(legendX, legendY + 8);\n ctx.lineTo(legendX + 24, legendY + 8);\n ctx.stroke();\n ctx.fillStyle = '#1e293b';\n ctx.font = '12px \"Source Sans 3\", sans-serif';\n ctx.textAlign = 'left';\n ctx.fillText(d.label, legendX + 30, legendY + 12);\n legendY += 22;\n });\n\n \/\/ \"Resonance Zone\" label\n ctx.fillStyle = 'rgba(220,38,38,0.6)';\n ctx.font = '11px \"Source Sans 3\", sans-serif';\n ctx.textAlign = 'center';\n ctx.fillText('RESONANCE', (x07 + x14) \/ 2, pad.top + 15);\n ctx.fillText('ZONE', (x07 + x14) \/ 2, pad.top + 28);\n}\n\nconst var_navy = '#0a2540';\n\n\/\/ Init\ntoggleInputs();\ncalculate();\nsetTimeout(drawResonanceCurve, 100);\nwindow.addEventListener('resize', drawResonanceCurve);\n<\/script>\n\n<\/body>\n<\/html><\/div><\/div><\/div><\/div><\/div>","protected":false},"excerpt":{"rendered":"<p>L\u00e6r om egenfrekvens, frekvensen et system naturlig vibrerer med. Forst\u00e5 hvordan det forholder seg til masse og stivhet, og hvorfor det er n\u00f8kkelfaktoren for \u00e5 for\u00e5rsake resonans.<\/p>","protected":false},"featured_media":0,"template":"","meta":{"ai_generated_summary":"","footnotes":""},"categories":[114,109],"tags":[],"class_list":["post-172","glossary","type-glossary","status-publish","hentry","category-analysis","category-glossary"],"_links":{"self":[{"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/glossary\/172","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/glossary"}],"about":[{"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/types\/glossary"}],"version-history":[{"count":6,"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/glossary\/172\/revisions"}],"predecessor-version":[{"id":101730,"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/glossary\/172\/revisions\/101730"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/media?parent=172"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/categories?post=172"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/tags?post=172"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}

\n\n\n\n\nNatural Frequency Calculator \u2014 Resonance, Critical Speed & Mass-Spring Systems \u2014 Vibromera<\/title>\n<meta name=\"description\" content=\"Interactive natural frequency calculator for mass-spring, beam, and shaft systems. Understand resonance, critical speeds, dynamic magnification factor. Formulas, identification methods, and industrial case studies.\">\n<meta name=\"keywords\" content=\"natural frequency, natural frequency calculator, mass spring system, resonance, critical speed, eigenfrequency, vibration, fn formula, bump test, dynamic magnification factor, Balanset, vibration analysis\">\n<meta name=\"author\" content=\"Vibromera\">\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large\">\n\n\n\n\n\n\n\n\n\n\n\n<meta property=\"og:type\" content=\"article\">\n<meta property=\"og:title\" content=\"Natural Frequency Calculator \u2014 Resonance, Critical Speed & Mass-Spring Systems\">\n<meta property=\"og:description\" content=\"Interactive calculator for 5 system types. Resonance curve visualization, critical speed analysis, industrial case studies. ISO-compliant vibration analysis.\">\n<meta property=\"og:url\" content=\"https:\/\/vibromera.eu\/glossary\/natural-frequency\/\">\n<meta property=\"og:site_name\" content=\"Vibromera \u2014 Vibration Analysis & Balancing Equipment\">\n<meta property=\"og:locale\" content=\"en_US\">\n<meta property=\"og:image\" content=\"https:\/\/vibromera.eu\/wp-content\/uploads\/natural-frequency-og.jpg\">\n<meta property=\"article:publisher\" content=\"https:\/\/vibromera.eu\/\">\n<meta property=\"article:section\" content=\"Glossary\">\n<meta property=\"article:tag\" content=\"Natural Frequency\">\n<meta property=\"article:tag\" content=\"Resonance\">\n<meta property=\"article:tag\" content=\"Critical Speed\">\n\n\n<meta name=\"twitter:card\" content=\"summary_large_image\">\n<meta name=\"twitter:title\" content=\"Natural Frequency Calculator \u2014 Mass-Spring, Beam, Shaft Systems\">\n<meta name=\"twitter:description\" content=\"Calculate natural frequency, critical speed, and resonance risk for 5 system types. Formulas, DMF chart, bump test procedures.\">\n\n\n<meta name=\"geo.region\" content=\"EU\">\n<meta name=\"geo.placename\" content=\"Porto, Portugal\">\n<meta name=\"geo.position\" content=\"41.1579;-8.6291\">\n<meta name=\"ICBM\" content=\"41.1579, -8.6291\">\n\n\n<script type=\"application\/ld+json\">\n{\n \"@context\": \"https:\/\/schema.org\",\n \"@graph\": [\n {\n \"@type\": \"TechArticle\",\n \"@id\": \"https:\/\/vibromera.eu\/glossary\/natural-frequency\/#article\",\n \"headline\": \"Natural Frequency: Definition, Calculator, Resonance, and Critical Speeds in Rotating Machinery\",\n \"description\": \"Comprehensive guide to natural frequency in mechanical systems. Interactive calculator for mass-spring, beam, cantilever, and shaft systems. Resonance curves, critical speed analysis, bump test procedures, and industrial case studies.\",\n \"author\": {\"@type\": \"Organization\", \"name\": \"Vibromera\", \"url\": \"https:\/\/vibromera.eu\/\"},\n \"publisher\": {\"@type\": \"Organization\", \"name\": \"Vibromera\", \"url\": \"https:\/\/vibromera.eu\/\", \"logo\": {\"@type\": \"ImageObject\", \"url\": \"https:\/\/vibromera.eu\/wp-content\/uploads\/vibromera-logo.png\"}},\n \"mainEntityOfPage\": \"https:\/\/vibromera.eu\/glossary\/natural-frequency\/\",\n \"datePublished\": \"2024-03-10\",\n \"dateModified\": \"2026-02-07\",\n \"inLanguage\": \"en\",\n \"about\": [\n {\"@type\": \"Thing\", \"name\": \"Natural Frequency\"},\n {\"@type\": \"Thing\", \"name\": \"Resonance\"},\n {\"@type\": \"Thing\", \"name\": \"Critical Speed\"},\n {\"@type\": \"Thing\", \"name\": \"Vibration Analysis\"}\n ]\n },\n {\n \"@type\": \"FAQPage\",\n \"@id\": \"https:\/\/vibromera.eu\/glossary\/natural-frequency\/#faq\",\n \"mainEntity\": [\n {\n \"@type\": \"Question\",\n \"name\": \"What is natural frequency?\",\n \"acceptedAnswer\": {\n \"@type\": \"Answer\",\n \"text\": \"Natural frequency is the frequency at which a mechanical system oscillates freely after being displaced from equilibrium, without any external periodic forcing. Every structure has one or more natural frequencies determined by its mass and stiffness. The fundamental formula is fn = (1\/2\u03c0) \u00d7 \u221a(k\/m) where k is stiffness (N\/m) and m is mass (kg). Natural frequency is measured in Hertz (Hz) or expressed as critical speed in RPM for rotating machinery.\"\n }\n },\n {\n \"@type\": \"Question\",\n \"name\": \"How do you calculate natural frequency of a mass-spring system?\",\n \"acceptedAnswer\": {\n \"@type\": \"Answer\",\n \"text\": \"For a simple mass-spring system: fn = (1\/2\u03c0) \u00d7 \u221a(k\/m), where fn is natural frequency in Hz, k is spring stiffness in N\/m, and m is mass in kg. For example, a 10 kg mass on a spring with stiffness 40,000 N\/m has fn = (1\/2\u03c0) \u00d7 \u221a(40000\/10) = 10.07 Hz. The critical speed equivalent is fn \u00d7 60 = 604 RPM.\"\n }\n },\n {\n \"@type\": \"Question\",\n \"name\": \"What happens at resonance?\",\n \"acceptedAnswer\": {\n \"@type\": \"Answer\",\n \"text\": \"Resonance occurs when an external forcing frequency equals or closely matches a system's natural frequency. At resonance, vibration amplitude is amplified dramatically \u2014 by a factor of 10-50\u00d7 for lightly damped steel structures (Q-factor = 10-50). This can cause catastrophic structural failure, bearing damage, and fatigue cracking. The amplification is described by the Dynamic Magnification Factor: DMF = 1\/\u221a[(1-r\u00b2)\u00b2 + (2\u03b6r)\u00b2] where r is the frequency ratio and \u03b6 is the damping ratio.\"\n }\n },\n {\n \"@type\": \"Question\",\n \"name\": \"What is the difference between natural frequency and critical speed?\",\n \"acceptedAnswer\": {\n \"@type\": \"Answer\",\n \"text\": \"Critical speed is the rotational speed (in RPM) at which a rotating shaft's operating frequency coincides with a natural frequency of the rotor-bearing system, causing resonance. Critical speed (RPM) = natural frequency (Hz) \u00d7 60. API standards require a minimum 15-20% separation margin between operating speed and the nearest critical speed to avoid resonance amplification.\"\n }\n },\n {\n \"@type\": \"Question\",\n \"name\": \"How do you find natural frequency in the field?\",\n \"acceptedAnswer\": {\n \"@type\": \"Answer\",\n \"text\": \"The most common field method is the impact test (bump test): strike the structure with an instrumented hammer while measuring the vibration response with an accelerometer. The natural frequency appears as a peak in the Frequency Response Function (FRF). Other methods include run-up\/coast-down tests (Bode plot), waterfall\/cascade analysis, and Operational Modal Analysis (OMA). The Balanset-1A portable analyzer from Vibromera can perform FFT spectrum analysis to identify resonant frequencies.\"\n }\n },\n {\n \"@type\": \"Question\",\n \"name\": \"How can you change the natural frequency of a structure?\",\n \"acceptedAnswer\": {\n \"@type\": \"Answer\",\n \"text\": \"Natural frequency is determined by fn = (1\/2\u03c0)\u221a(k\/m). To increase fn: add stiffness (bracing, thicker supports, stiffer mounts) or reduce mass. To decrease fn: add mass or reduce stiffness (softer mounts, isolation). Common industrial solutions include adding stiffening ribs to foundations, changing from rigid to flexible mounts, adding mass to shift resonance below operating speed, or redesigning support structures.\"\n }\n },\n {\n \"@type\": \"Question\",\n \"name\": \"What is the static deflection method for calculating natural frequency?\",\n \"acceptedAnswer\": {\n \"@type\": \"Answer\",\n \"text\": \"The static deflection shortcut calculates natural frequency from the deflection under gravity: fn \u2248 15.76 \/ \u221a\u03b4, where fn is in Hz and \u03b4 is static deflection in millimeters. This works because both fn and \u03b4 depend on the same stiffness-to-mass ratio. For example, a beam that deflects 1 mm under its own weight has fn \u2248 15.76 \/ \u221a1 = 15.76 Hz. This method is widely used for quick field estimates.\"\n }\n }\n ]\n },\n {\n \"@type\": \"BreadcrumbList\",\n \"itemListElement\": [\n {\"@type\": \"ListItem\", \"position\": 1, \"name\": \"Home\", \"item\": \"https:\/\/vibromera.eu\/\"},\n {\"@type\": \"ListItem\", \"position\": 2, \"name\": \"Glossary\", \"item\": \"https:\/\/vibromera.eu\/glossary\/\"},\n {\"@type\": \"ListItem\", \"position\": 3, \"name\": \"Natural Frequency\"}\n ]\n }\n ]\n}\n<\/script>\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=DM+Serif+Display&family=Source+Sans+3:wght@300;400;500;600;700&family=JetBrains+Mono:wght@400;500;600&display=swap\" rel=\"stylesheet\">\n<style>\n:root {\n --navy: #0a2540;\n --navy-light: #1a3a5c;\n --blue: #2563eb;\n --blue-light: #3b82f6;\n --blue-pale: #dbeafe;\n --blue-ghost: #eff6ff;\n --beige: #f5f0e8;\n --beige-dark: #e8dfd3;\n --beige-light: #faf7f2;\n --white: #ffffff;\n --text: #1e293b;\n --text-secondary: #475569;\n --text-muted: #94a3b8;\n --border: 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uppercase;\n letter-spacing: 1px;\n color: var(--blue-light);\n margin-bottom: 8px;\n}\n.result-primary .res-value {\n font-family: 'JetBrains Mono', monospace;\n font-size: 42px;\n font-weight: 700;\n line-height: 1;\n margin-bottom: 4px;\n}\n.result-primary .res-unit {\n font-size: 16px;\n color: rgba(255,255,255,0.5);\n}\n\n.results-secondary {\n display: grid;\n grid-template-columns: 1fr 1fr;\n gap: 12px;\n position: relative;\n z-index: 1;\n}\n.res-card {\n background: rgba(255,255,255,0.06);\n border: 1px solid rgba(255,255,255,0.08);\n border-radius: var(--radius-sm);\n padding: 16px;\n}\n.res-card .rc-label {\n font-size: 11px;\n font-weight: 600;\n text-transform: uppercase;\n letter-spacing: 0.5px;\n color: rgba(255,255,255,0.45);\n margin-bottom: 4px;\n}\n.res-card .rc-value {\n font-family: 'JetBrains Mono', monospace;\n font-size: 20px;\n font-weight: 600;\n}\n.res-card .rc-unit {\n font-size: 12px;\n color: rgba(255,255,255,0.4);\n}\n.results-empty {\n text-align: center;\n padding: 60px 20px;\n color: rgba(255,255,255,0.35);\n font-size: 15px;\n position: relative;\n z-index: 1;\n}\n.results-empty .icon { font-size: 36px; margin-bottom: 12px; display: block; }\n\n\/* Resonance indicator *\/\n.resonance-indicator {\n margin-top: 16px;\n padding: 16px;\n border-radius: var(--radius-sm);\n text-align: center;\n position: relative;\n z-index: 1;\n}\n.resonance-indicator.safe {\n background: rgba(5,150,105,0.15);\n border: 1px solid rgba(5,150,105,0.3);\n}\n.resonance-indicator.caution {\n background: rgba(217,119,6,0.15);\n border: 1px solid rgba(217,119,6,0.3);\n}\n.resonance-indicator.danger {\n background: rgba(220,38,38,0.15);\n border: 1px solid rgba(220,38,38,0.3);\n}\n.resonance-indicator .ri-status {\n font-weight: 700;\n font-size: 14px;\n text-transform: uppercase;\n letter-spacing: 0.5px;\n margin-bottom: 4px;\n}\n.resonance-indicator.safe .ri-status { color: #34d399; }\n.resonance-indicator.caution .ri-status { color: #fbbf24; }\n.resonance-indicator.danger .ri-status { color: #f87171; }\n.resonance-indicator .ri-detail {\n font-size: 13px;\n color: rgba(255,255,255,0.5);\n}\n\n\/* ===== CONCEPT CARDS ===== *\/\n.concepts-section {\n padding: 40px 0 48px;\n background: var(--beige-light);\n}\n.section-header {\n margin-bottom: 28px;\n}\n.section-header h2 {\n font-family: 'DM Serif Display', serif;\n font-size: 30px;\n color: var(--navy);\n margin-bottom: 6px;\n}\n.section-header p {\n font-size: 16px;\n color: var(--text-secondary);\n}\n\n.concept-cards {\n display: grid;\n grid-template-columns: repeat(3, 1fr);\n gap: 20px;\n margin-bottom: 36px;\n}\n.concept-card {\n background: var(--white);\n border: 1px solid var(--border-light);\n border-radius: var(--radius);\n padding: 24px;\n box-shadow: var(--shadow-sm);\n transition: all 0.3s;\n position: relative;\n overflow: hidden;\n}\n.concept-card:hover {\n box-shadow: var(--shadow-lg);\n transform: translateY(-3px);\n}\n.concept-card::before {\n content: '';\n position: absolute;\n top: 0; left: 0; right: 0;\n height: 3px;\n}\n.concept-card.mass::before { background: var(--blue); }\n.concept-card.stiff::before { background: var(--success); }\n.concept-card.damp::before { background: var(--warning); }\n\n.cc-icon {\n font-size: 28px;\n margin-bottom: 12px;\n}\n.cc-title {\n font-family: 'DM Serif Display', serif;\n font-size: 20px;\n color: var(--navy);\n margin-bottom: 8px;\n}\n.cc-body {\n font-size: 14px;\n color: var(--text-secondary);\n line-height: 1.6;\n}\n.cc-body strong { color: var(--text); }\n.cc-formula {\n margin-top: 12px;\n font-family: 'JetBrains Mono', monospace;\n font-size: 13px;\n background: var(--beige-light);\n padding: 8px 12px;\n border-radius: 6px;\n color: var(--navy);\n}\n\n\/* ===== TABLES ===== *\/\n.table-wrap {\n background: var(--white);\n border-radius: var(--radius);\n border: 1px solid var(--border-light);\n overflow: hidden;\n box-shadow: var(--shadow-sm);\n margin-bottom: 32px;\n}\n.table-wrap .table-title {\n padding: 16px 24px;\n font-weight: 600;\n font-size: 15px;\n background: var(--beige);\n border-bottom: 1px solid var(--border-light);\n color: var(--navy);\n}\ntable {\n width: 100%;\n border-collapse: collapse;\n}\ntable th {\n background: var(--navy);\n color: var(--white);\n padding: 12px 16px;\n font-size: 13px;\n font-weight: 600;\n text-transform: uppercase;\n letter-spacing: 0.5px;\n text-align: left;\n}\ntable td {\n padding: 11px 16px;\n font-size: 14px;\n border-bottom: 1px solid var(--border-light);\n color: var(--text);\n}\ntable tr:last-child td { border-bottom: none; }\ntable tr:hover td { background: var(--blue-ghost); }\ntable td:first-child { font-weight: 600; }\n.mono {\n font-family: 'JetBrains Mono', monospace;\n font-size: 13px;\n}\n.tag {\n display: inline-block;\n padding: 2px 8px;\n border-radius: 4px;\n font-size: 11px;\n font-weight: 600;\n text-transform: uppercase;\n letter-spacing: 0.3px;\n}\n.tag.safe { background: var(--success-light); color: #065f46; }\n.tag.caution { background: var(--warning-light); color: #92400e; }\n.tag.danger { background: var(--danger-light); color: #991b1b; }\n.tag.info { background: var(--blue-pale); color: var(--navy); }\n\n\/* ===== VISUAL: Resonance Curve ===== *\/\n.resonance-visual {\n background: var(--white);\n border: 1px solid var(--border-light);\n border-radius: var(--radius);\n padding: 32px;\n margin-bottom: 32px;\n box-shadow: var(--shadow-sm);\n}\n.resonance-visual .rv-title {\n font-family: 'DM Serif Display', serif;\n font-size: 20px;\n color: var(--navy);\n margin-bottom: 8px;\n}\n.resonance-visual .rv-subtitle {\n font-size: 14px;\n color: var(--text-muted);\n margin-bottom: 20px;\n}\n.rv-canvas-wrap {\n position: relative;\n width: 100%;\n height: 320px;\n}\n.rv-canvas-wrap canvas {\n width: 100%;\n height: 100%;\n}\n\n\/* ===== MAIN CONTENT ===== *\/\n.main-content {\n padding: 48px 0 64px;\n background: var(--white);\n}\n.content-layout {\n display: grid;\n grid-template-columns: 1fr 300px;\n gap: 48px;\n}\n\n\/* TOC sidebar *\/\n.toc-sidebar {\n position: sticky;\n top: 80px;\n align-self: start;\n}\n.toc-box {\n background: var(--beige-light);\n border: 1px solid var(--beige-dark);\n border-radius: var(--radius);\n padding: 24px;\n}\n.toc-box h3 {\n font-size: 14px;\n font-weight: 700;\n text-transform: uppercase;\n letter-spacing: 0.5px;\n color: var(--navy);\n margin-bottom: 16px;\n}\n.toc-box a {\n display: block;\n padding: 6px 0;\n font-size: 14px;\n color: var(--text-secondary);\n text-decoration: none;\n border-left: 2px solid transparent;\n padding-left: 12px;\n transition: all 0.2s;\n}\n.toc-box a:hover {\n color: var(--blue);\n border-left-color: var(--blue);\n}\n.toc-box a.sub {\n padding-left: 28px;\n font-size: 13px;\n color: var(--text-muted);\n}\n\n\/* Article *\/\n.article-content h2 {\n font-family: 'DM Serif Display', serif;\n font-size: 28px;\n color: var(--navy);\n margin: 48px 0 16px;\n padding-top: 20px;\n}\n.article-content h2:first-child { margin-top: 0; }\n.article-content h3 {\n font-size: 20px;\n font-weight: 700;\n color: var(--navy-light);\n margin: 32px 0 12px;\n}\n.article-content h4 {\n font-size: 16px;\n font-weight: 700;\n color: var(--text);\n margin: 24px 0 8px;\n}\n.article-content p {\n margin-bottom: 16px;\n color: var(--text);\n line-height: 1.8;\n}\n.article-content ul, .article-content ol {\n margin: 0 0 16px 24px;\n line-height: 1.8;\n}\n.article-content li { margin-bottom: 6px; }\n.article-content li strong { color: var(--navy); }\n.article-content a {\n color: var(--blue);\n text-decoration: none;\n border-bottom: 1px solid transparent;\n transition: border-color 0.2s;\n}\n.article-content a:hover { border-bottom-color: var(--blue); }\n\n\/* Info boxes *\/\n.info-box {\n background: var(--blue-ghost);\n border-left: 4px solid var(--blue);\n border-radius: 0 var(--radius-sm) var(--radius-sm) 0;\n padding: 20px 24px;\n margin: 24px 0;\n}\n.info-box.warning {\n background: #fffbeb;\n border-left-color: var(--warning);\n}\n.info-box.success {\n background: #ecfdf5;\n border-left-color: var(--success);\n}\n.info-box.danger {\n background: #fef2f2;\n border-left-color: var(--danger);\n}\n.info-box .box-title {\n font-weight: 700;\n font-size: 15px;\n margin-bottom: 6px;\n color: var(--navy);\n}\n.info-box p { margin-bottom: 8px; font-size: 14px; }\n.info-box p:last-child { margin-bottom: 0; }\n\n\/* Example block *\/\n.example-block {\n background: var(--beige-light);\n border: 1px solid var(--beige-dark);\n border-radius: var(--radius);\n padding: 28px;\n margin: 24px 0;\n}\n.example-block .example-title {\n font-family: 'DM Serif Display', serif;\n font-size: 18px;\n color: var(--navy);\n margin-bottom: 16px;\n}\n\n\/* Formula box *\/\n.formula-box {\n background: var(--navy);\n color: var(--white);\n border-radius: var(--radius);\n padding: 28px 32px;\n margin: 24px 0;\n text-align: center;\n position: relative;\n overflow: hidden;\n}\n.formula-box::before {\n content: '';\n position: absolute; inset: 0;\n background: radial-gradient(circle at 80% 50%, rgba(37,99,235,0.15) 0%, transparent 60%);\n pointer-events: none;\n}\n.formula-box .formula-label {\n font-size: 12px;\n font-weight: 700;\n text-transform: uppercase;\n letter-spacing: 1px;\n color: var(--blue-light);\n margin-bottom: 12px;\n position: relative;\n}\n.formula-box .formula-main {\n font-family: 'JetBrains Mono', monospace;\n font-size: 26px;\n font-weight: 600;\n position: relative;\n line-height: 1.4;\n}\n.formula-box .formula-note {\n font-size: 13px;\n color: rgba(255,255,255,0.5);\n margin-top: 12px;\n position: relative;\n}\n\n\/* Article table *\/\n.article-table {\n width: 100%;\n margin: 20px 0;\n border: 1px solid var(--border-light);\n border-radius: var(--radius-sm);\n overflow: hidden;\n}\n.article-table th { padding: 10px 14px; font-size: 12px; }\n.article-table td { padding: 10px 14px; font-size: 14px; }\n\n\/* Procedure steps *\/\n.procedure-steps {\n counter-reset: step;\n margin: 24px 0;\n}\n.proc-step {\n display: flex;\n gap: 20px;\n padding: 20px 0;\n border-bottom: 1px solid var(--border-light);\n}\n.proc-step:last-child { border-bottom: none; }\n.proc-step::before {\n counter-increment: step;\n content: counter(step);\n flex-shrink: 0;\n width: 40px;\n height: 40px;\n background: var(--blue);\n color: var(--white);\n border-radius: 50%;\n display: flex;\n align-items: center;\n justify-content: center;\n font-weight: 700;\n font-size: 16px;\n}\n.proc-step .step-content h4 {\n margin: 0 0 4px;\n font-size: 16px;\n color: var(--navy);\n}\n.proc-step .step-content p {\n margin: 0;\n font-size: 14px;\n color: var(--text-secondary);\n}\n\n\/* ===== SHOP CTA ===== *\/\n.shop-cta {\n background: linear-gradient(135deg, var(--navy) 0%, var(--navy-light) 100%);\n padding: 48px 0;\n color: var(--white);\n text-align: center;\n position: relative;\n overflow: hidden;\n}\n.shop-cta::before {\n content: '';\n position: absolute; inset: 0;\n background: radial-gradient(circle at 50% 50%, rgba(37,99,235,0.15) 0%, transparent 60%);\n pointer-events: none;\n}\n.shop-cta h2 {\n font-family: 'DM Serif Display', serif;\n font-size: 32px;\n margin-bottom: 12px;\n position: relative;\n}\n.shop-cta p {\n font-size: 17px;\n color: rgba(255,255,255,0.7);\n margin-bottom: 28px;\n max-width: 640px;\n margin-left: auto;\n margin-right: auto;\n position: relative;\n}\n.shop-cta .cta-btn {\n display: inline-flex;\n align-items: center;\n gap: 8px;\n padding: 14px 36px;\n background: var(--blue);\n color: var(--white);\n text-decoration: none;\n border-radius: var(--radius-sm);\n font-weight: 600;\n font-size: 16px;\n transition: all 0.2s;\n position: relative;\n}\n.shop-cta .cta-btn:hover {\n background: var(--blue-light);\n transform: translateY(-2px);\n box-shadow: 0 8px 24px rgba(37,99,235,0.3);\n}\n\n\/* Footer *\/\n.page-footer {\n background: var(--beige);\n border-top: 1px solid var(--beige-dark);\n padding: 32px 0;\n text-align: center;\n}\n.page-footer a { color: var(--blue); text-decoration: none; font-weight: 500; }\n.page-footer p { font-size: 14px; color: var(--text-muted); }\n\n@media print {\n .quick-nav, .shop-cta { display: none; }\n body { min-width: auto; }\n}\n<\/style>\n<\/head>\n<body>\n\n\n<header class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"breadcrumb\">\n <a href=\"https:\/\/vibromera.eu\/\">Home<\/a> \u2192 <a href=\"https:\/\/vibromera.eu\/glossary\/\">Glossary<\/a> \u2192 Natural Frequency\n <\/div>\n <h1>Understanding <span>Natural Frequency<\/span><\/h1>\n <div style=\"display:inline-flex;align-items:center;gap:6px;padding:4px 12px;background:rgba(37,99,235,0.2);border:1px solid rgba(37,99,235,0.3);border-radius:20px;font-size:12px;font-weight:600;letter-spacing:0.5px;color:rgba(255,255,255,0.85);margin-bottom:12px;text-transform:uppercase;\">\n <span style=\"font-size:14px;\">\ud83d\udccc<\/span> Canonical Reference Article \u2014 vibromera.eu\n <\/div>\n <p class=\"subtitle\">The inherent vibration frequency of every physical structure \u2014 and why its relationship with resonance is one of the most critical concepts in vibration analysis and rotating machinery engineering.<\/p>\n <\/div>\n<\/header>\n\n\n<nav class=\"quick-nav\">\n <div class=\"quick-nav-inner\">\n <a href=\"#calculator\">\u2699 Calculator<\/a>\n <a href=\"#key-concepts\">\ud83d\udcca Key Concepts<\/a>\n <a href=\"#definition\">\ud83d\udcd0 Definition<\/a>\n <a href=\"#mass-stiffness\">\ud83d\udd22 Mass & Stiffness<\/a>\n <a href=\"#resonance\">\u26a1 Resonance<\/a>\n <a href=\"#critical-speeds\">\ud83d\udd04 Critical Speeds<\/a>\n <a href=\"#identification\">\ud83d\udd0d Identification Methods<\/a>\n <a href=\"#practical\">\ud83d\udee0 Practical Examples<\/a>\n <a href=\"#faq\">\u2753 FAQ<\/a>\n <a href=\"#avoidance\">\ud83d\udee1 Avoidance Strategies<\/a>\n <\/div>\n<\/nav>\n\n\n<section class=\"summary-dashboard\" id=\"calculator\">\n <div class=\"container\">\n <div class=\"dashboard-grid\">\n\n \n <div class=\"calc-panel\">\n <h2 class=\"panel-title\">Natural Frequency Calculator<\/h2>\n <p class=\"panel-subtitle\">Calculate f<sub>n<\/sub> for simple systems + check resonance risk against operating speed<\/p>\n\n <div class=\"calc-form\">\n <div class=\"form-group full-width\">\n <label>System Type<\/label>\n <select id=\"systemType\" onchange=\"toggleInputs()\">\n <option value=\"spring-mass\">Spring\u2013Mass System (k + m)<\/option>\n <option value=\"beam-center\">Simply Supported Beam (center load)<\/option>\n <option value=\"beam-cantilever\">Cantilever Beam (end load)<\/option>\n <option value=\"shaft-disc\">Shaft with Central Disc (Rayleigh)<\/option>\n <option value=\"static-deflection\">From Static Deflection (\u03b4)<\/option>\n <\/select>\n <\/div>\n\n <div class=\"form-group\" id=\"fg-stiffness\">\n <label>Stiffness, k <span class=\"unit\">(N\/m)<\/span><\/label>\n <input type=\"number\" id=\"stiffness\" value=\"50000\" min=\"0.1\" step=\"100\">\n <\/div>\n <div class=\"form-group\" id=\"fg-mass\">\n <label>Mass, m <span class=\"unit\">(kg)<\/span><\/label>\n <input type=\"number\" id=\"mass\" value=\"5\" min=\"0.001\" step=\"0.1\">\n <\/div>\n <div class=\"form-group\" id=\"fg-length\" style=\"display:none\">\n <label>Beam Length, L <span class=\"unit\">(m)<\/span><\/label>\n <input type=\"number\" id=\"beamLength\" value=\"1\" min=\"0.01\" step=\"0.01\">\n <\/div>\n <div class=\"form-group\" id=\"fg-ei\" style=\"display:none\">\n <label>EI <span class=\"unit\">(N\u00b7m\u00b2)<\/span><\/label>\n <input type=\"number\" id=\"beamEI\" value=\"1000\" min=\"0.1\" step=\"10\">\n <\/div>\n <div class=\"form-group\" id=\"fg-deflection\" style=\"display:none\">\n <label>Static Deflection, \u03b4 <span class=\"unit\">(mm)<\/span><\/label>\n <input type=\"number\" id=\"staticDefl\" value=\"1\" min=\"0.001\" step=\"0.1\">\n <\/div>\n <div class=\"form-group\" id=\"fg-shaft-dia\" style=\"display:none\">\n <label>Shaft Diameter <span class=\"unit\">(mm)<\/span><\/label>\n <input type=\"number\" id=\"shaftDia\" value=\"50\" min=\"1\" step=\"1\">\n <\/div>\n <div class=\"form-group\" id=\"fg-shaft-len\" style=\"display:none\">\n <label>Bearing Span <span class=\"unit\">(mm)<\/span><\/label>\n <input type=\"number\" id=\"shaftLen\" value=\"800\" min=\"10\" step=\"10\">\n <\/div>\n <div class=\"form-group\" id=\"fg-disc-mass\" style=\"display:none\">\n <label>Disc Mass <span class=\"unit\">(kg)<\/span><\/label>\n <input type=\"number\" id=\"discMass\" value=\"20\" min=\"0.01\" step=\"0.1\">\n <\/div>\n <div class=\"form-group\">\n <label>Operating Speed <span class=\"unit\">(RPM, optional)<\/span><\/label>\n <input type=\"number\" id=\"operatingRPM\" value=\"3000\" min=\"0\" step=\"10\">\n <\/div>\n\n <button class=\"calc-btn\" onclick=\"calculate()\">Calculate Natural Frequency \u2192<\/button>\n <\/div>\n <\/div>\n\n \n <div class=\"results-panel\">\n <h2 class=\"panel-title\">Results<\/h2>\n <p class=\"panel-subtitle\">Natural frequency and resonance risk assessment<\/p>\n <div id=\"resultsArea\">\n <div class=\"results-empty\">\n <span class=\"icon\">\u3030\ufe0f<\/span>\n Enter system parameters and click Calculate<br>to see the natural frequency\n <\/div>\n <\/div>\n <\/div>\n\n <\/div>\n <\/div>\n<\/section>\n\n\n<section class=\"concepts-section\" id=\"key-concepts\">\n <div class=\"container\">\n <div class=\"section-header\">\n <h2>Key Concepts \u2014 At a Glance<\/h2>\n <p>The three fundamental properties that govern every vibrating system<\/p>\n <\/div>\n\n <div class=\"concept-cards\">\n <div class=\"concept-card mass\">\n <div class=\"cc-icon\">\u2696\ufe0f<\/div>\n <div class=\"cc-title\">Mass (Inertia)<\/div>\n <div class=\"cc-body\">\n Mass resists changes in motion. <strong>Increasing mass lowers the natural frequency<\/strong> \u2014 the system becomes slower and heavier to move. Think of a heavy pendulum swinging slowly versus a light one swinging quickly.\n <\/div>\n <div class=\"cc-formula\">f<sub>n<\/sub> \u221d 1 \/ \u221am \u2192 Double mass = f<sub>n<\/sub> \u00d7 0.71<\/div>\n <\/div>\n <div class=\"concept-card stiff\">\n <div class=\"cc-icon\">\ud83d\udd29<\/div>\n <div class=\"cc-title\">Stiffness (Elasticity)<\/div>\n <div class=\"cc-body\">\n Stiffness is the restoring force that pulls the system back to equilibrium. <strong>Increasing stiffness raises the natural frequency<\/strong> \u2014 the system snaps back faster. Like tightening a guitar string raises its pitch.\n <\/div>\n <div class=\"cc-formula\">f<sub>n<\/sub> \u221d \u221ak \u2192 Double stiffness = f<sub>n<\/sub> \u00d7 1.41<\/div>\n <\/div>\n <div class=\"concept-card damp\">\n <div class=\"cc-icon\">\ud83d\udee2\ufe0f<\/div>\n <div class=\"cc-title\">Damping (Energy Dissipation)<\/div>\n <div class=\"cc-body\">\n Damping removes energy from the system, causing vibration to decay over time. <strong>Damping doesn't significantly change f<sub>n<\/sub><\/strong>, but it controls the amplitude at resonance \u2014 the only thing preventing infinite amplitude.\n <\/div>\n <div class=\"cc-formula\">f<sub>d<\/sub> = f<sub>n<\/sub> \u00d7 \u221a(1 \u2212 \u03b6\u00b2) \u2248 f<sub>n<\/sub> for \u03b6 < 0.2<\/div>\n <\/div>\n <\/div>\n\n \n <div class=\"resonance-visual\">\n <div class=\"rv-title\">Interactive Resonance Response Curve<\/div>\n <div class=\"rv-subtitle\">Amplitude magnification vs. frequency ratio (\u03c9\/\u03c9<sub>n<\/sub>) for different damping ratios (\u03b6)<\/div>\n <div class=\"rv-canvas-wrap\">\n <canvas id=\"resonanceCanvas\"><\/canvas>\n <\/div>\n <\/div>\n\n \n <div class=\"table-wrap\">\n <div class=\"table-title\">\ud83d\udcca Typical Natural Frequency Ranges for Common Industrial Structures<\/div>\n <table>\n <thead>\n <tr>\n <th>Structure \/ Component<\/th>\n <th>Typical f<sub>n<\/sub> Range<\/th>\n <th>Typical Operating RPM<\/th>\n <th>Resonance Risk<\/th>\n <th>Notes<\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td>Large concrete foundation<\/td>\n <td class=\"mono\">15\u201340 Hz<\/td>\n <td class=\"mono\">900\u20132400<\/td>\n <td><span class=\"tag safe\">Low<\/span><\/td>\n <td>Very stiff; typically well above operating speed<\/td>\n <\/tr>\n <tr>\n <td>Steel baseplate \/ skid<\/td>\n <td class=\"mono\">20\u201380 Hz<\/td>\n <td class=\"mono\">1200\u20134800<\/td>\n <td><span class=\"tag caution\">Medium<\/span><\/td>\n <td>Can coincide with 2-pole or 4-pole motor speed<\/td>\n <\/tr>\n <tr>\n <td>Piping system (span)<\/td>\n <td class=\"mono\">5\u201350 Hz<\/td>\n <td class=\"mono\">300\u20133000<\/td>\n <td><span class=\"tag danger\">High<\/span><\/td>\n <td>Long unsupported spans are very susceptible<\/td>\n <\/tr>\n <tr>\n <td>Pump pedestal<\/td>\n <td class=\"mono\">25\u201360 Hz<\/td>\n <td class=\"mono\">1500\u20133600<\/td>\n <td><span class=\"tag caution\">Medium<\/span><\/td>\n <td>Vertical pumps are especially problematic<\/td>\n <\/tr>\n <tr>\n <td>Fan housing \/ shroud<\/td>\n <td class=\"mono\">15\u2013120 Hz<\/td>\n <td class=\"mono\">900\u20137200<\/td>\n <td><span class=\"tag caution\">Medium<\/span><\/td>\n <td>Sheet metal panels can have many modes<\/td>\n <\/tr>\n <tr>\n <td>Electric motor frame<\/td>\n <td class=\"mono\">40\u2013200 Hz<\/td>\n <td class=\"mono\">2400\u201312000<\/td>\n <td><span class=\"tag safe\">Low<\/span><\/td>\n <td>Typically designed above 1\u00d7 operating speed<\/td>\n <\/tr>\n <tr>\n <td>Shaft (1st critical)<\/td>\n <td class=\"mono\">20\u2013500 Hz<\/td>\n <td class=\"mono\">1200\u201330000<\/td>\n <td><span class=\"tag danger\">High<\/span><\/td>\n <td>Must be known; crossing critical = severe vibration<\/td>\n <\/tr>\n <tr>\n <td>Bearing housing<\/td>\n <td class=\"mono\">100\u20131000 Hz<\/td>\n <td class=\"mono\">\u2014<\/td>\n <td><span class=\"tag safe\">Low<\/span><\/td>\n <td>Excited by bearing fault impacts, not by 1\u00d7 speed<\/td>\n <\/tr>\n <tr>\n <td>Gearbox casing<\/td>\n <td class=\"mono\">200\u20132000 Hz<\/td>\n <td class=\"mono\">\u2014<\/td>\n <td><span class=\"tag safe\">Low<\/span><\/td>\n <td>Excited by gear meshing frequencies<\/td>\n <\/tr>\n <tr>\n <td>Spring isolators (installed)<\/td>\n <td class=\"mono\">2\u20138 Hz<\/td>\n <td class=\"mono\">120\u2013480<\/td>\n <td><span class=\"tag caution\">Medium<\/span><\/td>\n <td>Must be well below operating speed to isolate<\/td>\n <\/tr>\n <tr>\n <td>Rubber mounts<\/td>\n <td class=\"mono\">5\u201325 Hz<\/td>\n <td class=\"mono\">300\u20131500<\/td>\n <td><span class=\"tag caution\">Medium<\/span><\/td>\n <td>Stiffness varies with temperature and age<\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n <\/div>\n\n \n <div class=\"table-wrap\">\n <div class=\"table-title\">\u26a1 Resonance Risk Assessment \u2014 Operating Speed vs. Natural Frequency Ratio<\/div>\n <table>\n <thead>\n <tr>\n <th>Frequency Ratio (f<sub>op<\/sub> \/ f<sub>n<\/sub>)<\/th>\n <th>Zone<\/th>\n <th>Amplification Factor<\/th>\n <th>Practical Meaning<\/th>\n <th>Recommendation<\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td class=\"mono\">0 \u2013 0.7<\/td>\n <td><span class=\"tag safe\">Safe Below<\/span><\/td>\n <td class=\"mono\">1.0 \u2013 2.0\u00d7<\/td>\n <td>Vibration force transmitted almost 1:1; structure moves in-phase with forcing<\/td>\n <td>Acceptable; normal operating zone for rigidly mounted equipment<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">0.7 \u2013 0.85<\/td>\n <td><span class=\"tag caution\">Approach Zone<\/span><\/td>\n <td class=\"mono\">2 \u2013 5\u00d7<\/td>\n <td>Amplitude begins to amplify significantly; early resonance effects<\/td>\n <td>Avoid steady-state operation; acceptable for brief run-up\/coast-down transit<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">0.85 \u2013 1.15<\/td>\n <td><span class=\"tag danger\">Resonance Band<\/span><\/td>\n <td class=\"mono\">5 \u2013 50\u00d7<\/td>\n <td>Severe amplification; amplitude limited only by damping; structural damage possible<\/td>\n <td>Never operate here; transit through quickly if unavoidable<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">1.15 \u2013 1.4<\/td>\n <td><span class=\"tag caution\">Exit Zone<\/span><\/td>\n <td class=\"mono\">2 \u2013 5\u00d7<\/td>\n <td>Amplitude decreasing but still elevated; phase shifting rapidly<\/td>\n <td>Avoid steady-state; brief transit acceptable<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">1.4 \u2013 2.5<\/td>\n <td><span class=\"tag safe\">Safe Above<\/span><\/td>\n <td class=\"mono\">0.3 \u2013 1.0\u00d7<\/td>\n <td>Vibration is attenuated; structure's inertia resists movement; phase inversion<\/td>\n <td>Good isolation zone for flexibly mounted equipment<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">> 2.5<\/td>\n <td><span class=\"tag safe\">Isolation Zone<\/span><\/td>\n <td class=\"mono\">< 0.3\u00d7<\/td>\n <td>Excellent vibration isolation; very little force transmitted<\/td>\n <td>Ideal for spring\/rubber-mounted machines<\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n <\/div>\n\n \n <div class=\"table-wrap\">\n <div class=\"table-title\">\ud83d\udd0d Comparison of Natural Frequency Identification Methods<\/div>\n <table>\n <thead>\n <tr>\n <th>Method<\/th>\n <th>Equipment Needed<\/th>\n <th>Machine State<\/th>\n <th>Accuracy<\/th>\n <th>Best For<\/th>\n <th>Limitations<\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td>Impact Test (Bump Test)<\/td>\n <td>Modal hammer + accelerometer + FFT analyzer<\/td>\n <td>Stopped<\/td>\n <td><span class=\"tag safe\">High<\/span><\/td>\n <td>Structures, baseplates, piping, bearing housings<\/td>\n <td>Machine must be stopped; can miss speed-dependent effects<\/td>\n <\/tr>\n <tr>\n <td>Run-Up \/ Coast-Down<\/td>\n <td>Vibration sensor + tachometer + order tracking<\/td>\n <td>Running (variable speed)<\/td>\n <td><span class=\"tag safe\">High<\/span><\/td>\n <td>Shaft critical speeds, foundation resonances<\/td>\n <td>Requires variable speed; 1\u00d7 unbalance force excites primarily shaft criticals<\/td>\n <\/tr>\n <tr>\n <td>Operating Deflection Shape (ODS)<\/td>\n <td>Multi-channel analyzer + many sensors<\/td>\n <td>Running (normal)<\/td>\n <td><span class=\"tag caution\">Medium<\/span><\/td>\n <td>Visualizing how structure moves at specific frequency<\/td>\n <td>Shows deflection shape, not true mode shape (multiple modes contribute)<\/td>\n <\/tr>\n <tr>\n <td>Experimental Modal Analysis (EMA)<\/td>\n <td>Modal hammer or shaker + roving sensors + modal software<\/td>\n <td>Stopped<\/td>\n <td><span class=\"tag safe\">Very High<\/span><\/td>\n <td>Complete modal model (frequencies, shapes, damping)<\/td>\n <td>Time-consuming; requires expertise; complex data processing<\/td>\n <\/tr>\n <tr>\n <td>Finite Element Analysis (FEA)<\/td>\n <td>Computer + FEA software + model<\/td>\n <td>N\/A (simulation)<\/td>\n <td><span class=\"tag caution\">Depends on model<\/span><\/td>\n <td>Design phase; what-if analysis; complex geometries<\/td>\n <td>Accuracy depends on model quality; boundary conditions critical<\/td>\n <\/tr>\n <tr>\n <td>Waterfall \/ Cascade Plot<\/td>\n <td>Vibration analyzer with order tracking<\/td>\n <td>Running (variable speed)<\/td>\n <td><span class=\"tag safe\">High<\/span><\/td>\n <td>Identifying multiple resonances during speed changes<\/td>\n <td>Requires speed change; only finds resonances excited by operating forces<\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n <\/div>\n\n <\/div>\n<\/section>\n\n\n<main class=\"main-content\" id=\"definition\">\n <div class=\"container\">\n <div class=\"content-layout\">\n <article class=\"article-content\">\n\n <h2>Definition: What is Natural Frequency?<\/h2>\n\n <div class=\"info-box\" style=\"border-left-color: var(--navy); background: var(--beige-light);\">\n <div class=\"box-title\" style=\"font-size: 16px;\">Quick Answer<\/div>\n <p style=\"font-size: 15px;\"><strong>Natural frequency<\/strong> is the frequency at which a mechanical system oscillates freely after being displaced from equilibrium. It is determined by the system's <strong>mass<\/strong> and <strong>stiffness<\/strong>: <strong>f<sub>n<\/sub> = (1\/2\u03c0) \u00d7 \u221a(k\/m)<\/strong>, where k is stiffness (N\/m) and m is mass (kg). When an external force frequency matches a natural frequency, <strong><a href=\"https:\/\/vibromera.eu\/glossary\/resonance\/\">resonance<\/a><\/strong> occurs \u2014 vibration amplitude can increase 10\u201350\u00d7 and cause catastrophic failure. In rotating machinery, the <strong>critical speed<\/strong> (RPM) = f<sub>n<\/sub> \u00d7 60. A quick field estimate uses static deflection: <strong>f<sub>n<\/sub> \u2248 15.76 \/ \u221a\u03b4<sub>mm<\/sub><\/strong>.<\/p>\n <\/div>\n\n <p>A <strong>natural frequency<\/strong> is the specific frequency at which a physical object or system will oscillate when disturbed from its equilibrium position and then allowed to vibrate freely, without any ongoing external driving force. It is an inherent, fundamental property of the object, determined entirely by its physical characteristics \u2014 principally its <strong>mass<\/strong> (inertia) and its <strong>stiffness<\/strong> (elasticity). Every physical object, from a guitar string to a bridge span to a machine's support pedestal, possesses one or more natural frequencies.<\/p>\n <p>Natural frequencies are sometimes called <em>eigenfrequencies<\/em> (from the German word \"eigen\" meaning \"own\" or \"characteristic\"), and the corresponding vibration patterns are called <em>mode shapes<\/em> or <em>eigenmodes<\/em>. A complex structure like a machine base may have hundreds of natural frequencies, each associated with a unique deformation pattern \u2014 bending, twisting, breathing, rocking, and so on.<\/p>\n\n <div class=\"info-box\">\n <div class=\"box-title\">Why Natural Frequency Matters in Vibration Analysis<\/div>\n <p>In rotating machinery, vibration problems are frequently caused not by excessive excitation forces (such as unbalance), but by the unlucky coincidence of an excitation frequency matching a structural natural frequency. A perfectly acceptable amount of unbalance can produce destructive vibration if the machine operates at or near a structural resonance. Identifying natural frequencies is therefore one of the most important diagnostic steps when investigating unexplained high vibration.<\/p>\n <\/div>\n\n <h2 id=\"mass-stiffness\">The Relationship Between Mass, Stiffness, and Natural Frequency<\/h2>\n <p>The fundamental relationship between mass, stiffness, and natural frequency is one of the most important concepts in vibration engineering. It is both intuitive and mathematically precise.<\/p>\n\n <h3>Intuitive Understanding<\/h3>\n <ul>\n <li><strong>Stiffness (k):<\/strong> A stiffer object has a <em>higher<\/em> natural frequency. Think of a guitar string: tightening the string (increasing tension\/stiffness) raises the pitch (frequency). A thick steel beam vibrates at a much higher frequency than a thin aluminum strip of the same length.<\/li>\n <li><strong>Mass (m):<\/strong> A more massive object has a <em>lower<\/em> natural frequency. Think of a ruler extending off the edge of a desk: a longer, heavier ruler oscillates more slowly (lower frequency) than a shorter, lighter one. Adding weight to a structure always lowers its natural frequencies.<\/li>\n <\/ul>\n\n <h3>The Fundamental Formula<\/h3>\n <p>For a simple single-degree-of-freedom (SDOF) system \u2014 a mass connected to a spring \u2014 the undamped natural frequency is:<\/p>\n\n <div class=\"formula-box\">\n <div class=\"formula-label\">Undamped Natural Frequency<\/div>\n <div class=\"formula-main\">f<sub>n<\/sub> = (1 \/ 2\u03c0) \u00d7 \u221a(k \/ m)<\/div>\n <div class=\"formula-note\">f<sub>n<\/sub> in Hz, k in N\/m, m in kg. Also: \u03c9<sub>n<\/sub> = \u221a(k\/m) in rad\/s<\/div>\n <\/div>\n\n <p>This formula has profound practical implications:<\/p>\n <ul>\n <li>To <strong>increase<\/strong> f<sub>n<\/sub> by 2\u00d7, you must increase stiffness by 4\u00d7 (because of the square root) \u2014 or reduce mass by 4\u00d7<\/li>\n <li>To <strong>decrease<\/strong> f<sub>n<\/sub> by 2\u00d7, you must reduce stiffness by 4\u00d7 \u2014 or increase mass by 4\u00d7<\/li>\n <li>Changes to stiffness and mass have <em>diminishing returns<\/em>: each doubling of f<sub>n<\/sub> requires a 4\u00d7 change in the parameter<\/li>\n <\/ul>\n\n <h3>The Static Deflection Shortcut<\/h3>\n <p>One of the most useful practical formulas in vibration engineering relates natural frequency directly to static deflection under gravity:<\/p>\n\n <div class=\"formula-box\">\n <div class=\"formula-label\">Natural Frequency from Static Deflection<\/div>\n <div class=\"formula-main\">f<sub>n<\/sub> = (1 \/ 2\u03c0) \u00d7 \u221a(g \/ \u03b4) \u2248 15.76 \/ \u221a\u03b4<\/div>\n <div class=\"formula-note\">f<sub>n<\/sub> in Hz, \u03b4 in mm, g = 9810 mm\/s\u00b2. Very handy for quick estimates!<\/div>\n <\/div>\n\n <p>This is remarkably useful because static deflection is often easy to measure or estimate: simply measure how much a structure deflects under the weight of the machine. A machine that sags 1 mm on its supports has a vertical natural frequency of about 15.8 Hz (948 RPM). A machine that sags 0.25 mm has f<sub>n<\/sub> \u2248 31.5 Hz (1890 RPM).<\/p>\n\n <div class=\"info-box success\">\n <div class=\"box-title\">Quick Field Estimate<\/div>\n <p>Need a quick natural frequency estimate without instruments? Place a dial indicator under the machine's bearing housing and observe the static deflection when the machine weight is applied (e.g., during installation). The formula f<sub>n<\/sub> \u2248 15.76\/\u221a\u03b4<sub>mm<\/sub> gives a remarkably good first approximation of the fundamental vertical natural frequency.<\/p>\n <\/div>\n\n <h3>Multiple Degrees of Freedom<\/h3>\n <p>Real structures are not simple SDOF systems \u2014 they have many masses connected through distributed stiffness, resulting in many natural frequencies. A simple rigid body on elastic supports has six natural frequencies corresponding to six degrees of freedom: three translational (vertical, lateral, axial) and three rotational (roll, pitch, yaw). A flexible structure has infinitely many modes, though only the lowest several are usually of practical concern.<\/p>\n <p>The key principle is: <strong>the number of natural frequencies equals the number of degrees of freedom in the model<\/strong>. A simple beam modeled with 10 lumped masses has 10 natural frequencies; a finite element model with 10,000 nodes has 30,000 (3 DOF per node) natural frequencies, though only a few dozen may be in the frequency range of interest.<\/p>\n\n <h3>The Effect of Damping<\/h3>\n <p>Real systems always have some damping \u2014 friction, material hysteresis, radiation into surrounding structure, fluid drag, etc. Damping has two effects:<\/p>\n <ul>\n <li><strong>Slightly lowers the actual resonant frequency:<\/strong> The damped natural frequency is f<sub>d<\/sub> = f<sub>n<\/sub> \u00d7 \u221a(1 \u2212 \u03b6\u00b2), where \u03b6 is the damping ratio. For typical mechanical structures (\u03b6 = 0.01\u20130.05), this effect is negligible \u2014 less than 0.1% reduction.<\/li>\n <li><strong>Limits the amplitude at resonance:<\/strong> Without damping, resonance amplitude would theoretically be infinite. The amplification factor Q (quality factor) at resonance is approximately Q = 1\/(2\u03b6). For a lightly damped structure with \u03b6 = 0.02, Q = 25 \u2014 meaning the vibration amplitude at resonance is 25\u00d7 what it would be away from resonance. This is why even small amounts of unbalance can produce enormous vibration at critical speeds.<\/li>\n <\/ul>\n\n <h2 id=\"resonance\">Natural Frequency and Resonance: The Critical Connection<\/h2>\n <p>The concept of natural frequency is critically important in engineering specifically because of its direct connection to the phenomenon of <strong><a href=\"https:\/\/vibromera.eu\/glossary\/resonance\/\">resonance<\/a><\/strong>.<\/p>\n\n <h3>What is Resonance?<\/h3>\n <p>Resonance occurs when a periodic external force is applied to a system at a frequency that is equal to or very close to one of its natural frequencies. When this happens, the system absorbs energy from the external force with maximum efficiency, causing the vibration amplitude to grow dramatically. Each cycle of the forcing function adds energy to the system in exact synchronization with the system's natural oscillation, building amplitude cycle after cycle until either damping limits further growth or the structure fails.<\/p>\n\n <h3>The Amplification Factor<\/h3>\n <p>The magnification of vibration at resonance depends critically on the system's damping. The dynamic magnification factor (DMF) describes how much larger the dynamic response is compared to the static deflection that the same force would produce:<\/p>\n\n <div class=\"formula-box\">\n <div class=\"formula-label\">Dynamic Magnification Factor<\/div>\n <div class=\"formula-main\">DMF = 1 \/ \u221a[(1 \u2212 r\u00b2)\u00b2 + (2\u03b6r)\u00b2]<\/div>\n <div class=\"formula-note\">r = f<sub>forcing<\/sub>\/f<sub>n<\/sub> (frequency ratio), \u03b6 = damping ratio. At r = 1: DMF \u2248 1\/(2\u03b6)<\/div>\n <\/div>\n\n <table class=\"article-table\">\n <thead>\n <tr>\n <th>Damping Ratio (\u03b6)<\/th>\n <th>Typical System<\/th>\n <th>Q Factor (\u2248 1\/2\u03b6)<\/th>\n <th>Amplification at Resonance<\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td class=\"mono\">0.005<\/td>\n <td>Welded steel structure, undamped<\/td>\n <td class=\"mono\">100<\/td>\n <td>100\u00d7 static deflection<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">0.01<\/td>\n <td>Steel frame, bolted connections<\/td>\n <td class=\"mono\">50<\/td>\n <td>50\u00d7 static deflection<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">0.02<\/td>\n <td>Typical machinery structure<\/td>\n <td class=\"mono\">25<\/td>\n <td>25\u00d7 static deflection<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">0.05<\/td>\n <td>Concrete foundation, bolted joints<\/td>\n <td class=\"mono\">10<\/td>\n <td>10\u00d7 static deflection<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">0.10<\/td>\n <td>Rubber-mounted, well-damped<\/td>\n <td class=\"mono\">5<\/td>\n <td>5\u00d7 static deflection<\/td>\n <\/tr>\n <tr>\n <td class=\"mono\">0.20<\/td>\n <td>Highly damped (viscous damper)<\/td>\n <td class=\"mono\">2.5<\/td>\n <td>2.5\u00d7 static deflection<\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n\n <h3>Why Resonance Is Dangerous<\/h3>\n <p>Resonance is particularly treacherous because the vibration amplitude can be 10\u2013100\u00d7 larger than expected based on the forcing magnitude alone. A rotor with 50 \u00b5m of unbalance eccentricity that produces 1 mm\/s vibration at non-resonant speed could produce 25\u201350 mm\/s at resonance \u2014 enough to destroy bearings, fatigue bolts, crack welds, and cause cascading equipment failure.<\/p>\n\n <div class=\"info-box danger\">\n <div class=\"box-title\">Historical Example \u2014 Tacoma Narrows Bridge (1940)<\/div>\n <p>The collapse of the Tacoma Narrows Bridge remains one of the most dramatic demonstrations of resonance in engineering history. Wind forces at a frequency near the bridge's torsional natural frequency caused the bridge deck to oscillate with increasing amplitude until structural failure occurred. The event led to fundamental changes in bridge engineering and is studied in every structural dynamics course worldwide. Modern engineers routinely perform modal analysis to ensure structures are designed away from foreseeable excitation frequencies.<\/p>\n <\/div>\n\n <h2 id=\"critical-speeds\">Critical Speeds of Rotating Machinery<\/h2>\n <p>In rotating machinery, the most important manifestation of natural frequency is the <strong><a href=\"https:\/\/vibromera.eu\/glossary\/critical-speed\/\">critical speed<\/a><\/strong> \u2014 the rotational speed at which the shaft's rotation frequency (1\u00d7 RPM) coincides with a natural frequency of the rotor-bearing-support system. When a machine operates at a critical speed, the 1\u00d7 unbalance force excites the natural frequency, producing severe resonant vibration.<\/p>\n\n <h3>Types of Critical Speeds<\/h3>\n <ul>\n <li><strong>Rigid body criticals:<\/strong> Occur when the shaft speed matches a natural frequency of the rotor on its bearing supports, with the shaft itself remaining essentially straight. These are typically the first and second criticals (bounce and rock modes) and occur at lower speeds. Rigid body criticals can be modified by changing bearing stiffness or support structure mass.<\/li>\n <li><strong>Flexible rotor criticals (bending criticals):<\/strong> Occur when the shaft speed matches a natural frequency associated with shaft bending deformation. The first bending critical typically involves the shaft bowing into a half-sine shape. These are more dangerous because they involve large deflections at the shaft midspan and cannot be controlled by bearing changes alone \u2014 the shaft geometry itself must be modified.<\/li>\n <\/ul>\n\n <h3>Separation Margin<\/h3>\n <p>Industry standards (e.g., API 610, API 617) require a minimum <strong>separation margin<\/strong> between operating speed and critical speeds:<\/p>\n <ul>\n <li><strong>API typical requirement:<\/strong> Operating speed must be at least 15\u201320% away from any lateral critical speed (undamped)<\/li>\n <li><strong>General good practice:<\/strong> A 20% margin is considered minimum; 30% is preferred for critical equipment<\/li>\n <li><strong>VFD-driven equipment:<\/strong> Variable frequency drives change operating speed, potentially sweeping through criticals. The entire operating range must be checked, and criticals within the range must be identified and excluded or rapid transit programmed.<\/li>\n <\/ul>\n\n <div class=\"info-box\">\n <div class=\"box-title\">Practical Implication for Field Balancing<\/div>\n <p>When field balancing a machine that operates near (but safely above) a critical speed, the phase relationship between unbalance and vibration response will differ from what is expected for a \"below-resonance\" machine. The vibration signal may be 90\u2013180\u00b0 ahead of the heavy spot rather than in-phase. Good <a href=\"https:\/\/vibromera.eu\/product\/balanset-1\/\">balancing equipment<\/a> handles this automatically through trial-weight response measurement, but the analyst should be aware that near-critical operation complicates simple vector analysis.<\/p>\n <\/div>\n\n <h2 id=\"identification\">How Are Natural Frequencies Identified?<\/h2>\n <p>Identifying the natural frequencies of a machine or structure is a fundamental diagnostic skill. Several methods are available, ranging from simple to sophisticated:<\/p>\n\n <h3>1. Impact Testing (Bump Test)<\/h3>\n <p>The most common and practical experimental method for identifying structural natural frequencies. The procedure involves striking the machine or structure (while it is <em>not<\/em> running) with an instrumented impact hammer and measuring the resulting vibration with an accelerometer. The hammer blow inputs energy across a broad frequency range simultaneously, and the structure naturally \"rings\" at its natural frequencies, producing clear peaks in the resulting FFT spectrum.<\/p>\n\n <h4>Practical Procedure<\/h4>\n <div class=\"procedure-steps\">\n <div class=\"proc-step\">\n <div class=\"step-content\">\n <h4>Prepare the Equipment<\/h4>\n <p>Mount an accelerometer on the structure at the point of interest (typically the bearing housing or support structure). Connect to an FFT analyzer or data collector configured for impact testing (time-domain trigger, appropriate frequency range, typically 0\u20131000 Hz for structural resonances).<\/p>\n <\/div>\n <\/div>\n <div class=\"proc-step\">\n <div class=\"step-content\">\n <h4>Select Hammer Tip<\/h4>\n <p>Impact hammer tips of different hardness excite different frequency ranges. Soft rubber tips excite 0\u2013200 Hz; medium plastic tips excite 0\u2013500 Hz; hard steel tips excite 0\u20135000 Hz. Choose the tip that covers the frequency range of interest for the specific test.<\/p>\n <\/div>\n <\/div>\n <div class=\"proc-step\">\n <div class=\"step-content\">\n <h4>Strike and Record<\/h4>\n <p>Strike the structure firmly with a single, clean blow. Avoid double-hits (bouncing). The analyzer should capture the time waveform showing the impact and the resulting free vibration decay. The FFT of this response reveals the natural frequencies as peaks.<\/p>\n <\/div>\n <\/div>\n <div class=\"proc-step\">\n <div class=\"step-content\">\n <h4>Average Multiple Hits<\/h4>\n <p>Take 3\u20135 averages to improve signal-to-noise ratio and confirm consistency. If the Frequency Response Function (FRF) varies significantly between hits, check for double-hits, poor accelerometer mounting, or changing boundary conditions.<\/p>\n <\/div>\n <\/div>\n <div class=\"proc-step\">\n <div class=\"step-content\">\n <h4>Identify Natural Frequencies<\/h4>\n <p>Natural frequencies appear as peaks in the FRF magnitude plot. Confirm using the phase plot (natural frequencies show 180\u00b0 phase shift) and the coherence function (should be near 1.0 at natural frequencies). Record the frequencies and compare with operating speed and harmonics.<\/p>\n <\/div>\n <\/div>\n <\/div>\n\n <div class=\"info-box success\">\n <div class=\"box-title\">Bump Test Tips from the Field<\/div>\n <p>Always perform the bump test with the machine <em>assembled<\/em> but <em>not running<\/em>. The natural frequencies can change significantly when the rotor is removed (mass changes) or when the machine is running (gyroscopic effects, bearing stiffness changes with speed, thermal effects). Test in multiple directions (vertical, horizontal, axial) to find all relevant modes. Repeat after any structural modification to verify the change achieved the desired effect.<\/p>\n <\/div>\n\n <h3>2. Run-Up \/ Coast-Down Test<\/h3>\n <p>For running machines, a run-up or coast-down test is the most practical way to identify natural frequencies that are excited by rotating forces. As the machine's speed changes, the 1\u00d7 unbalance force (and any other speed-dependent forces) sweeps through a range of frequencies. When a forcing frequency crosses a natural frequency, the vibration amplitude shows a distinct peak \u2014 identifying that natural frequency as a <a href=\"https:\/\/vibromera.eu\/glossary\/critical-speed\/\">critical speed<\/a>.<\/p>\n <p>The test requires simultaneous vibration measurement and tachometer signal (keyphasor) to correlate vibration amplitude and phase with shaft speed. The data is typically displayed as a Bode plot (amplitude and phase vs. RPM) or a polar plot (amplitude \u00d7 phase vector vs. RPM). Both clearly show critical speeds as amplitude peaks accompanied by ~180\u00b0 phase shifts.<\/p>\n\n <h3>3. Waterfall \/ Cascade Plot Analysis<\/h3>\n <p>A waterfall (or cascade) plot is a 3D representation of multiple FFT spectra taken at different machine speeds during a run-up or coast-down. It displays frequency (horizontal), amplitude (vertical), and speed (depth axis). In this format:<\/p>\n <ul>\n <li><strong>Speed-dependent lines<\/strong> (orders) appear as diagonal lines: 1\u00d7, 2\u00d7, 3\u00d7 etc., moving to the right as speed increases<\/li>\n <li><strong>Natural frequencies<\/strong> appear as vertical peaks (fixed frequency regardless of speed) \u2014 they don't move as speed changes<\/li>\n <li><strong>Resonances<\/strong> are visible where a speed-dependent order line crosses a natural frequency, producing a localized amplitude spike<\/li>\n <\/ul>\n <p>This is one of the most powerful diagnostic tools for distinguishing speed-dependent vibration (from unbalance, misalignment, etc.) from structural resonance problems.<\/p>\n\n <h3>4. Finite Element Analysis (FEA)<\/h3>\n <p>During the design phase, engineers use computer models to predict the natural frequencies of components, machines, and support structures before they are built. FEA discretizes the structure into thousands of small elements, applies the correct material properties (density, elastic modulus, Poisson's ratio), models the boundary conditions (bolt connections, bearing supports, foundation), and solves the eigenvalue problem to extract natural frequencies and mode shapes.<\/p>\n <p>FEA is invaluable for:<\/p>\n <ul>\n <li>Designing structures to avoid resonance issues before fabrication<\/li>\n <li>Performing \"what-if\" analysis: what happens if we add a stiffener? Change the bearing span? Use a different material?<\/li>\n <li>Predicting modal behavior of complex geometries that are difficult to test experimentally<\/li>\n <li>Validating experimental results by correlating measured and predicted natural frequencies<\/li>\n <\/ul>\n\n <h3>5. Operational Modal Analysis (OMA)<\/h3>\n <p>A relatively modern technique that extracts natural frequencies and mode shapes from a running machine using only the response data \u2014 no controlled excitation (hammer or shaker) required. OMA uses advanced algorithms (e.g., stochastic subspace identification) that treat the machine's operating forces as \"white noise\" excitation. This is particularly valuable for large or critical equipment that cannot be shut down for bump testing or where operational boundary conditions differ significantly from stopped conditions.<\/p>\n\n <h2 id=\"practical\">Practical Examples in Industrial Machinery<\/h2>\n\n <div class=\"example-block\">\n <div class=\"example-title\">Case 1: Vertical Pump Excessive Vibration<\/div>\n <p><strong>Problem:<\/strong> A vertical turbine pump running at 1780 RPM (29.7 Hz) shows 12 mm\/s vibration at 1\u00d7 RPM on the motor top. Balancing attempts reduce vibration temporarily but it returns within weeks.<\/p>\n <p><strong>Investigation:<\/strong> A bump test on the motor\/pump assembly reveals a natural frequency at 28.5 Hz \u2014 only 4% below operating speed. The system is operating in the resonance band.<\/p>\n <p><strong>Solution:<\/strong> A steel support brace is added to the motor stool, increasing stiffness. Post-modification bump test shows the natural frequency has moved to 42 Hz (42% above operating speed). Vibration drops to 2.5 mm\/s without any balancing correction \u2014 confirming the root cause was resonance, not unbalance.<\/p>\n <\/div>\n\n <div class=\"example-block\">\n <div class=\"example-title\">Case 2: Fan Foundation Resonance<\/div>\n <p><strong>Problem:<\/strong> A large induced-draft fan on a steel-frame foundation runs at 990 RPM (16.5 Hz). The foundation shows 8 mm\/s vibration at 1\u00d7 RPM, while the fan itself shows only 2 mm\/s at the bearing housing.<\/p>\n <p><strong>Investigation:<\/strong> The fact that the foundation vibrates more than the source (fan) is a classic resonance indicator. A bump test reveals the foundation's lateral natural frequency is 17.2 Hz \u2014 within 4% of operating speed.<\/p>\n <p><strong>Solution:<\/strong> Two options considered: (1) add mass to the foundation (lower f<sub>n<\/sub>), or (2) add stiffness (raise f<sub>n<\/sub>). Cross-bracing is added to the foundation frame, raising f<sub>n<\/sub> to 24 Hz. Foundation vibration drops to 1.8 mm\/s.<\/p>\n <\/div>\n\n <div class=\"example-block\">\n <div class=\"example-title\">Case 3: Piping Resonance at Pump BPF<\/div>\n <p><strong>Problem:<\/strong> Piping connected to a 5-vane centrifugal pump running at 1480 RPM shows severe vibration at 123 Hz (= 5 \u00d7 24.7 Hz, the blade pass frequency). Pipe clamps loosen and fatigue cracks appear at welded supports.<\/p>\n <p><strong>Investigation:<\/strong> A bump test on the affected pipe span reveals a natural frequency at 120 Hz \u2014 almost exactly at the pump's blade pass frequency (5\u00d7 RPM = 123 Hz).<\/p>\n <p><strong>Solution:<\/strong> An additional pipe support is installed at the midspan, raising the span's natural frequency to 185 Hz. Alternatively, for some installations, adding a tuned vibration absorber (dynamic absorber) at the pipe's antinode can be effective. After the support addition, piping vibration drops by 85%.<\/p>\n <\/div>\n\n <h2 id=\"avoidance\">Strategies for Avoiding Resonance Problems<\/h2>\n <p>The best time to address resonance is during design, but it can also be corrected in the field. There are three fundamental strategies:<\/p>\n\n <h3>1. Detune \u2014 Change the Natural Frequency<\/h3>\n <p>Move the natural frequency away from the excitation frequency. Require a minimum separation margin (typically 20\u201330%). Options include:<\/p>\n <ul>\n <li><strong>Increase stiffness:<\/strong> Add bracing, stiffeners, gussets, thicker plates, or concrete fill. This raises f<sub>n<\/sub>. Most common fix for structures that resonate below operating speed.<\/li>\n <li><strong>Add mass:<\/strong> Attach additional mass (steel plates, concrete). This lowers f<sub>n<\/sub>. Used when the natural frequency is just above the excitation frequency and it's easier to move it lower.<\/li>\n <li><strong>Modify bearing stiffness:<\/strong> For shaft criticals, changing bearing clearance, preload, or type can shift the critical speed. Stiffer bearings raise criticals; softer bearings lower them.<\/li>\n <li><strong>Change shaft geometry:<\/strong> For bending criticals, increasing shaft diameter raises critical speed (stiffness increases faster than mass). Shortening bearing span also raises criticals.<\/li>\n <\/ul>\n\n <h3>2. Damp \u2014 Reduce Amplitude at Resonance<\/h3>\n <p>If the natural frequency cannot be moved away from the excitation, add damping to limit the resonant amplitude. Options include:<\/p>\n <ul>\n <li><strong>Constrained layer damping:<\/strong> Viscoelastic material sandwiched between structural plates \u2014 highly effective for panel and housing resonances<\/li>\n <li><strong>Viscous dampers:<\/strong> Squeeze-film or viscous dashpot dampers, commonly used in bearing supports for turbomachinery<\/li>\n <li><strong>Tuned vibration absorbers:<\/strong> A mass-spring system tuned to the problem frequency, attached to the vibrating structure. The absorber vibrates in anti-phase, canceling the structure's motion at the target frequency<\/li>\n <li><strong>Bolted joints:<\/strong> Increasing the number of bolted joints (vs. welded) introduces friction damping through micro-slip at joint interfaces<\/li>\n <\/ul>\n\n <h3>3. Reduce the Exciting Force<\/h3>\n <p>If neither detuning nor damping is practical, reduce the forcing magnitude:<\/p>\n <ul>\n <li><strong>Better balancing:<\/strong> Reduce 1\u00d7 excitation by balancing to a tighter <a href=\"https:\/\/vibromera.eu\/glossary\/balance-quality-grade\/\">G-grade<\/a> \u2014 even if not at resonance, this reduces the force available to excite any resonance<\/li>\n <li><strong>Precision alignment:<\/strong> Reduce 2\u00d7 excitation from misalignment<\/li>\n <li><strong>Speed change:<\/strong> If the machine is VFD-driven, exclude the resonant speed from the operating range or program rapid transit through the resonance band<\/li>\n <li><strong>Isolation:<\/strong> Install vibration isolators to prevent the excitation from reaching the resonant structure<\/li>\n <\/ul>\n\n <div class=\"info-box success\">\n <div class=\"box-title\">The 20% Rule of Thumb<\/div>\n <p>In practice, aim for at least 20% separation between any natural frequency and any significant excitation frequency. For critical applications (power generation, offshore, aerospace), 30% or more is preferred. This applies not just to 1\u00d7 RPM but also to 2\u00d7 (misalignment), blade\/vane pass frequencies, gear mesh frequencies, and any other periodic excitation. A comprehensive resonance avoidance analysis compares <em>all<\/em> excitation frequencies against <em>all<\/em> natural frequencies in the system.<\/p>\n <\/div>\n\n <p>Understanding natural frequency \u2014 and its dangerous relationship with resonance \u2014 is fundamental to the practice of vibration analysis and machinery reliability engineering. Every vibration analyst should be competent in identifying natural frequencies through testing, interpreting their relationship to operating conditions, and recommending appropriate corrective actions when resonance is found to be contributing to a vibration problem.<\/p>\n\n <hr style=\"margin: 48px 0 24px; border: none; border-top: 1px solid var(--border-light);\">\n <p><a href=\"https:\/\/vibromera.eu\/glossary\/\">\u2190 Back to Glossary Index<\/a><\/p>\n\n <\/article>\n\n \n <aside class=\"toc-sidebar\">\n <div class=\"toc-box\">\n <h3>On This Page<\/h3>\n <a href=\"#calculator\">Natural Frequency Calculator<\/a>\n <a href=\"#key-concepts\">Key Concepts<\/a>\n <a href=\"#definition\">Definition<\/a>\n <a href=\"#mass-stiffness\">Mass, Stiffness & f<sub>n<\/sub><\/a>\n <a class=\"sub\" href=\"#mass-stiffness\">Fundamental Formula<\/a>\n <a class=\"sub\" href=\"#mass-stiffness\">Static Deflection Shortcut<\/a>\n <a class=\"sub\" href=\"#mass-stiffness\">Multiple DOF<\/a>\n <a class=\"sub\" href=\"#mass-stiffness\">Effect of Damping<\/a>\n <a href=\"#resonance\">Resonance<\/a>\n <a class=\"sub\" href=\"#resonance\">Amplification Factor<\/a>\n <a class=\"sub\" href=\"#resonance\">Why It's Dangerous<\/a>\n <a href=\"#critical-speeds\">Critical Speeds<\/a>\n <a class=\"sub\" href=\"#critical-speeds\">Separation Margin<\/a>\n <a href=\"#identification\">Identification Methods<\/a>\n <a class=\"sub\" href=\"#identification\">Impact \/ Bump Test<\/a>\n <a class=\"sub\" href=\"#identification\">Run-Up \/ Coast-Down<\/a>\n <a class=\"sub\" href=\"#identification\">Waterfall Plot<\/a>\n <a class=\"sub\" href=\"#identification\">FEA<\/a>\n <a class=\"sub\" href=\"#identification\">OMA<\/a>\n <a href=\"#practical\">Practical Examples<\/a>\n <a href=\"#avoidance\">Avoidance Strategies<\/a>\n <a href=\"#faq\">FAQ (7 Questions)<\/a>\n <\/div>\n\n <div class=\"toc-box\" style=\"margin-top: 24px; background: var(--navy); border-color: var(--navy);\">\n <h3 style=\"color: var(--white);\">Vibromera Equipment<\/h3>\n <p style=\"color: rgba(255,255,255,0.6); font-size: 13px; margin-bottom: 12px;\">Portable vibration analysis and balancing devices with built-in spectrum analysis for resonance identification.<\/p>\n <a href=\"https:\/\/vibromera.eu\/product\/balanset-1\/\" style=\"display: block; padding: 8px 14px; background: var(--blue); color: white; border-radius: 6px; text-align: center; font-weight: 600; font-size: 14px; text-decoration: none; margin-bottom: 8px; border-left: none;\">Balanset-1A \u2192<\/a>\n <a href=\"https:\/\/vibromera.eu\/product\/balanset-4\/\" style=\"display: block; padding: 8px 14px; background: rgba(255,255,255,0.1); color: white; border-radius: 6px; text-align: center; font-weight: 600; font-size: 14px; text-decoration: none; border: 1px solid rgba(255,255,255,0.2); border-left: none;\">Balanset-4 \u2192<\/a>\n <\/div>\n <\/aside>\n <\/div>\n <\/div>\n<\/main>\n\n\n<section class=\"grades-section\" id=\"faq\" style=\"background: var(--white); border-top: 1px solid var(--border-light);\">\n <div class=\"container\" style=\"max-width:900px;\">\n <div class=\"section-header\">\n <h2>Frequently Asked Questions \u2014 Natural Frequency<\/h2>\n <p>Common questions about natural frequency, resonance, and critical speeds<\/p>\n <\/div>\n <div style=\"display:flex;flex-direction:column;gap:16px;\">\n\n <details style=\"background:var(--beige-light);border:1px solid var(--beige-dark);border-radius:var(--radius);overflow:hidden;\">\n <summary style=\"padding:18px 24px;font-weight:700;font-size:16px;cursor:pointer;color:var(--navy);list-style:none;display:flex;align-items:center;gap:10px;\">\n <span style=\"color:var(--blue);font-size:18px;\">\u25b8<\/span> What is natural frequency in simple terms?\n <\/summary>\n <div style=\"padding:0 24px 18px 52px;font-size:15px;line-height:1.7;color:var(--text);\">\n Natural frequency is the rate at which an object \"wants\" to vibrate when disturbed and released. Think of a tuning fork: strike it and it always vibrates at the same pitch \u2014 that's its natural frequency. Every mechanical structure has natural frequencies determined by its mass (how heavy it is) and stiffness (how rigid it is). The formula is: <strong>f<sub>n<\/sub> = (1\/2\u03c0) \u00d7 \u221a(k\/m)<\/strong>. Heavier objects vibrate slower; stiffer objects vibrate faster.\n <\/div>\n <\/details>\n\n <details style=\"background:var(--beige-light);border:1px solid var(--beige-dark);border-radius:var(--radius);overflow:hidden;\">\n <summary style=\"padding:18px 24px;font-weight:700;font-size:16px;cursor:pointer;color:var(--navy);list-style:none;display:flex;align-items:center;gap:10px;\">\n <span style=\"color:var(--blue);font-size:18px;\">\u25b8<\/span> How do you calculate natural frequency of a mass-spring system?\n <\/summary>\n <div style=\"padding:0 24px 18px 52px;font-size:15px;line-height:1.7;color:var(--text);\">\n Use the fundamental SDOF formula: <strong>f<sub>n<\/sub> = (1\/2\u03c0) \u00d7 \u221a(k\/m)<\/strong>, where k is spring stiffness in N\/m and m is mass in kg. Example: a 10 kg mass on a 40,000 N\/m spring \u2192 f<sub>n<\/sub> = (1\/2\u03c0) \u00d7 \u221a(40000\/10) = (1\/6.283) \u00d7 63.25 = <strong>10.07 Hz<\/strong>. The corresponding critical speed is 10.07 \u00d7 60 = <strong>604 RPM<\/strong>.\n <\/div>\n <\/details>\n\n <details style=\"background:var(--beige-light);border:1px solid var(--beige-dark);border-radius:var(--radius);overflow:hidden;\">\n <summary style=\"padding:18px 24px;font-weight:700;font-size:16px;cursor:pointer;color:var(--navy);list-style:none;display:flex;align-items:center;gap:10px;\">\n <span style=\"color:var(--blue);font-size:18px;\">\u25b8<\/span> What happens at resonance? Why is it dangerous?\n <\/summary>\n <div style=\"padding:0 24px 18px 52px;font-size:15px;line-height:1.7;color:var(--text);\">\n Resonance occurs when a forcing frequency matches a natural frequency. The system absorbs energy maximally, amplifying vibration by 10\u201350\u00d7 for typical steel structures. This can cause bearing failure, fatigue cracking, foundation damage, and catastrophic structural collapse. The most famous example is the Tacoma Narrows Bridge (1940), which was destroyed by wind-induced resonance. In industrial machinery, resonance is the #1 cause of unexplained high vibration that doesn't respond to <a href=\"https:\/\/vibromera.eu\/glossary\/rotor-balancing\/\">balancing<\/a>.\n <\/div>\n <\/details>\n\n <details style=\"background:var(--beige-light);border:1px solid var(--beige-dark);border-radius:var(--radius);overflow:hidden;\">\n <summary style=\"padding:18px 24px;font-weight:700;font-size:16px;cursor:pointer;color:var(--navy);list-style:none;display:flex;align-items:center;gap:10px;\">\n <span style=\"color:var(--blue);font-size:18px;\">\u25b8<\/span> What is critical speed and how does it relate to natural frequency?\n <\/summary>\n <div style=\"padding:0 24px 18px 52px;font-size:15px;line-height:1.7;color:var(--text);\">\n Critical speed is the rotational speed (RPM) at which a shaft's operating frequency matches a natural frequency, causing resonance. <strong>Critical speed = f<sub>n<\/sub> \u00d7 60<\/strong>. For safe operation, maintain at least 15\u201320% separation margin between operating speed and any critical speed (per API standards). Machines that operate above the first critical speed are called \"flexible rotors\" and require special balancing techniques.\n <\/div>\n <\/details>\n\n <details style=\"background:var(--beige-light);border:1px solid var(--beige-dark);border-radius:var(--radius);overflow:hidden;\">\n <summary style=\"padding:18px 24px;font-weight:700;font-size:16px;cursor:pointer;color:var(--navy);list-style:none;display:flex;align-items:center;gap:10px;\">\n <span style=\"color:var(--blue);font-size:18px;\">\u25b8<\/span> How do you measure natural frequency in the field?\n <\/summary>\n <div style=\"padding:0 24px 18px 52px;font-size:15px;line-height:1.7;color:var(--text);\">\n The most common method is the <strong>impact test (bump test)<\/strong>: mount an accelerometer on the structure, strike it with a hammer, and analyze the resulting FFT spectrum. Natural frequencies appear as peaks. Other methods: run-up\/coast-down tests (Bode plots), waterfall analysis during speed changes, and Operational Modal Analysis. The <a href=\"https:\/\/vibromera.eu\/product\/balanset-1\/\">Balanset-1A<\/a> portable analyzer can perform FFT spectrum analysis to identify natural frequencies directly in the field.\n <\/div>\n <\/details>\n\n <details style=\"background:var(--beige-light);border:1px solid var(--beige-dark);border-radius:var(--radius);overflow:hidden;\">\n <summary style=\"padding:18px 24px;font-weight:700;font-size:16px;cursor:pointer;color:var(--navy);list-style:none;display:flex;align-items:center;gap:10px;\">\n <span style=\"color:var(--blue);font-size:18px;\">\u25b8<\/span> How can you change a structure's natural frequency to avoid resonance?\n <\/summary>\n <div style=\"padding:0 24px 18px 52px;font-size:15px;line-height:1.7;color:var(--text);\">\n Since f<sub>n<\/sub> = (1\/2\u03c0)\u221a(k\/m): <strong>to increase f<sub>n<\/sub><\/strong> \u2014 add stiffness (welded bracing, concrete grouting, thicker baseplate) or reduce mass. <strong>To decrease f<sub>n<\/sub><\/strong> \u2014 add mass (concrete filling, steel plates) or reduce stiffness (softer isolators). The \"20% rule\" requires operating speed to be at least 20% away from any natural frequency. Adding damping (rubber pads, viscous dampers) won't change f<sub>n<\/sub> but reduces the amplification at resonance.\n <\/div>\n <\/details>\n\n <details style=\"background:var(--beige-light);border:1px solid var(--beige-dark);border-radius:var(--radius);overflow:hidden;\">\n <summary style=\"padding:18px 24px;font-weight:700;font-size:16px;cursor:pointer;color:var(--navy);list-style:none;display:flex;align-items:center;gap:10px;\">\n <span style=\"color:var(--blue);font-size:18px;\">\u25b8<\/span> What is the static deflection shortcut for natural frequency?\n <\/summary>\n <div style=\"padding:0 24px 18px 52px;font-size:15px;line-height:1.7;color:var(--text);\">\n The static deflection formula provides a quick estimate: <strong>f<sub>n<\/sub> \u2248 15.76 \/ \u221a\u03b4<\/strong>, where \u03b4 is static deflection in millimeters under the system's own weight. This works because both f<sub>n<\/sub> and \u03b4 depend on the same k\/m ratio. A system that deflects 1 mm under gravity has f<sub>n<\/sub> \u2248 15.8 Hz; 4 mm deflection gives f<sub>n<\/sub> \u2248 7.9 Hz. This is extremely useful for quick field estimates without knowing exact stiffness values.\n <\/div>\n <\/details>\n\n <\/div>\n <\/div>\n<\/section>\n\n\n<section style=\"padding:32px 0;background:var(--beige-light);border-top:1px solid var(--border-light);\">\n <div class=\"container\" style=\"max-width:900px;\">\n <h3 style=\"font-family:'DM Serif Display',serif;font-size:22px;color:var(--navy);margin-bottom:20px;\">Related Glossary Articles<\/h3>\n <div style=\"display:grid;grid-template-columns:repeat(3,1fr);gap:16px;\">\n <a href=\"https:\/\/vibromera.eu\/glossary\/resonance\/\" style=\"display:block;padding:18px;background:var(--white);border:1px solid var(--border-light);border-radius:var(--radius-sm);text-decoration:none;transition:all 0.2s;box-shadow:var(--shadow-sm);\">\n <div style=\"font-weight:700;color:var(--navy);font-size:15px;margin-bottom:4px;\">Resonance<\/div>\n <div style=\"font-size:13px;color:var(--text-secondary);\">What happens when forcing frequency meets natural frequency \u2014 amplification, damage, and avoidance<\/div>\n <\/a>\n <a href=\"https:\/\/vibromera.eu\/glossary\/critical-speed\/\" style=\"display:block;padding:18px;background:var(--white);border:1px solid var(--border-light);border-radius:var(--radius-sm);text-decoration:none;transition:all 0.2s;box-shadow:var(--shadow-sm);\">\n <div style=\"font-weight:700;color:var(--navy);font-size:15px;margin-bottom:4px;\">Critical Speed<\/div>\n <div style=\"font-size:13px;color:var(--text-secondary);\">Rotor critical speeds, separation margins, and rigid vs. flexible rotor classification<\/div>\n <\/a>\n <a href=\"https:\/\/vibromera.eu\/glossary\/balance-quality-grade\/\" style=\"display:block;padding:18px;background:var(--white);border:1px solid var(--border-light);border-radius:var(--radius-sm);text-decoration:none;transition:all 0.2s;box-shadow:var(--shadow-sm);\">\n <div style=\"font-weight:700;color:var(--navy);font-size:15px;margin-bottom:4px;\">Balance Quality Grade (G-Grade)<\/div>\n <div style=\"font-size:13px;color:var(--text-secondary);\">ISO 21940-11 balancing tolerances \u2014 why balanced rotors still vibrate at resonance<\/div>\n <\/a>\n <a href=\"https:\/\/vibromera.eu\/glossary\/harmonics-vibration\/\" style=\"display:block;padding:18px;background:var(--white);border:1px solid var(--border-light);border-radius:var(--radius-sm);text-decoration:none;transition:all 0.2s;box-shadow:var(--shadow-sm);\">\n <div style=\"font-weight:700;color:var(--navy);font-size:15px;margin-bottom:4px;\">Harmonics in Vibration<\/div>\n <div style=\"font-size:13px;color:var(--text-secondary);\">1\u00d7, 2\u00d7, 3\u00d7 patterns \u2014 harmonics can excite higher-order natural frequencies<\/div>\n <\/a>\n <a href=\"https:\/\/vibromera.eu\/glossary\/unbalance\/\" style=\"display:block;padding:18px;background:var(--white);border:1px solid var(--border-light);border-radius:var(--radius-sm);text-decoration:none;transition:all 0.2s;box-shadow:var(--shadow-sm);\">\n <div style=\"font-weight:700;color:var(--navy);font-size:15px;margin-bottom:4px;\">Unbalance<\/div>\n <div style=\"font-size:13px;color:var(--text-secondary);\">The most common excitation source \u2014 and why its effect depends on proximity to resonance<\/div>\n <\/a>\n <a href=\"https:\/\/vibromera.eu\/glossary\/fft\/\" style=\"display:block;padding:18px;background:var(--white);border:1px solid var(--border-light);border-radius:var(--radius-sm);text-decoration:none;transition:all 0.2s;box-shadow:var(--shadow-sm);\">\n <div style=\"font-weight:700;color:var(--navy);font-size:15px;margin-bottom:4px;\">FFT Spectrum Analysis<\/div>\n <div style=\"font-size:13px;color:var(--text-secondary);\">How FFT is used in bump tests and run-up analysis to identify natural frequencies<\/div>\n <\/a>\n <\/div>\n <\/div>\n<\/section>\n\n\n<section class=\"shop-cta\">\n <div class=\"container\">\n <h2>Professional Vibration Analysis Equipment<\/h2>\n <p>Identify resonance problems and balance rotors in the field with Vibromera's portable devices \u2014 spectrum analysis, phase measurement, and ISO-compliant balancing in one instrument.<\/p>\n <a href=\"https:\/\/vibromera.eu\/shop\/\" class=\"cta-btn\">Browse Equipment \u2192<\/a>\n <\/div>\n<\/section>\n\n\n<footer class=\"page-footer\">\n <div class=\"container\">\n <p><a href=\"https:\/\/vibromera.eu\/glossary\/\">\u2190 Back to Glossary<\/a> | <a href=\"https:\/\/vibromera.eu\/\">vibromera.eu<\/a><\/p>\n <\/div>\n<\/footer>\n\n<script>\n\/\/ ===== CALCULATOR =====\nfunction toggleInputs() {\n const type = document.getElementById('systemType').value;\n const show = (id, vis) => document.getElementById(id).style.display = vis ? '' : 'none';\n\n show('fg-stiffness', type === 'spring-mass');\n show('fg-mass', type === 'spring-mass' || type === 'beam-center' || type === 'beam-cantilever');\n show('fg-length', type === 'beam-center' || type === 'beam-cantilever');\n show('fg-ei', type === 'beam-center' || type === 'beam-cantilever');\n show('fg-deflection', type === 'static-deflection');\n show('fg-shaft-dia', type === 'shaft-disc');\n show('fg-shaft-len', type === 'shaft-disc');\n show('fg-disc-mass', type === 'shaft-disc');\n}\n\nfunction calculate() {\n const type = document.getElementById('systemType').value;\n const rpm = parseFloat(document.getElementById('operatingRPM').value) || 0;\n let fn = 0;\n let description = '';\n\n if (type === 'spring-mass') {\n const k = parseFloat(document.getElementById('stiffness').value);\n const m = parseFloat(document.getElementById('mass').value);\n if (!k || !m) return alert('Enter stiffness and mass.');\n fn = (1 \/ (2 * Math.PI)) * Math.sqrt(k \/ m);\n description = `Spring\u2013Mass: k=${k} N\/m, m=${m} kg`;\n } else if (type === 'beam-center') {\n const m = parseFloat(document.getElementById('mass').value);\n const L = parseFloat(document.getElementById('beamLength').value);\n const EI = parseFloat(document.getElementById('beamEI').value);\n if (!m || !L || !EI) return alert('Enter mass, length, and EI.');\n \/\/ Simply supported beam, center point mass: k_eff = 48*EI\/L^3\n const k_eff = 48 * EI \/ Math.pow(L, 3);\n fn = (1 \/ (2 * Math.PI)) * Math.sqrt(k_eff \/ m);\n description = `Simply supported beam: EI=${EI} N\u00b7m\u00b2, L=${L} m, m=${m} kg`;\n } else if (type === 'beam-cantilever') {\n const m = parseFloat(document.getElementById('mass').value);\n const L = parseFloat(document.getElementById('beamLength').value);\n const EI = parseFloat(document.getElementById('beamEI').value);\n if (!m || !L || !EI) return alert('Enter mass, length, and EI.');\n \/\/ Cantilever beam, end mass: k_eff = 3*EI\/L^3\n const k_eff = 3 * EI \/ Math.pow(L, 3);\n fn = (1 \/ (2 * Math.PI)) * Math.sqrt(k_eff \/ m);\n description = `Cantilever beam: EI=${EI} N\u00b7m\u00b2, L=${L} m, m=${m} kg`;\n } else if (type === 'shaft-disc') {\n const d = parseFloat(document.getElementById('shaftDia').value) \/ 1000; \/\/ mm to m\n const L = parseFloat(document.getElementById('shaftLen').value) \/ 1000;\n const m = parseFloat(document.getElementById('discMass').value);\n if (!d || !L || !m) return alert('Enter shaft diameter, span, and disc mass.');\n const E = 210e9; \/\/ Steel\n const I = Math.PI * Math.pow(d, 4) \/ 64;\n const k_eff = 48 * E * I \/ Math.pow(L, 3);\n fn = (1 \/ (2 * Math.PI)) * Math.sqrt(k_eff \/ m);\n description = `Steel shaft \u2300${(d*1000).toFixed(0)}mm, span ${(L*1000).toFixed(0)}mm, disc ${m} kg`;\n } else if (type === 'static-deflection') {\n const delta = parseFloat(document.getElementById('staticDefl').value);\n if (!delta) return alert('Enter static deflection.');\n fn = 15.76 \/ Math.sqrt(delta);\n description = `Static deflection \u03b4 = ${delta} mm`;\n }\n\n const fn_rpm = fn * 60;\n const omega_n = fn * 2 * Math.PI;\n const period = 1 \/ fn;\n\n let ratio = 0;\n let riskStatus = '';\n let riskClass = '';\n let riskDetail = '';\n\n if (rpm > 0) {\n const fop = rpm \/ 60;\n ratio = fop \/ fn;\n if (ratio < 0.7 || ratio > 1.4) {\n riskStatus = '\u2713 Safe \u2014 Outside Resonance Band';\n riskClass = 'safe';\n riskDetail = `Frequency ratio = ${ratio.toFixed(3)} (operating ${ratio < 1 ? 'below' : 'above'} f_n). Margin: ${(Math.abs(1 - ratio) * 100).toFixed(1)}%`;\n } else if (ratio >= 0.85 && ratio <= 1.15) {\n riskStatus = '\u2717 Danger \u2014 In Resonance Band!';\n riskClass = 'danger';\n riskDetail = `Frequency ratio = ${ratio.toFixed(3)}. Operating within \u00b115% of f_n \u2014 expect severe amplification!`;\n } else {\n riskStatus = '\u26a0 Caution \u2014 Near Resonance';\n riskClass = 'caution';\n riskDetail = `Frequency ratio = ${ratio.toFixed(3)}. Within approach\/exit zone \u2014 elevated vibration expected.`;\n }\n }\n\n const area = document.getElementById('resultsArea');\n area.innerHTML = `\n <div class=\"result-primary\">\n <div class=\"res-label\">Natural Frequency (f<sub>n<\/sub>)<\/div>\n <div class=\"res-value\">${fn >= 100 ? fn.toFixed(1) : fn >= 1 ? fn.toFixed(2) : fn.toFixed(3)}<\/div>\n <div class=\"res-unit\">Hz<\/div>\n <\/div>\n <div class=\"results-secondary\">\n <div class=\"res-card\">\n <div class=\"rc-label\">Critical Speed<\/div>\n <div class=\"rc-value\">${fn_rpm >= 100 ? fn_rpm.toFixed(0) : fn_rpm.toFixed(1)}<\/div>\n <div class=\"rc-unit\">RPM<\/div>\n <\/div>\n <div class=\"res-card\">\n <div class=\"rc-label\">Angular Frequency (\u03c9<sub>n<\/sub>)<\/div>\n <div class=\"rc-value\">${omega_n >= 100 ? omega_n.toFixed(1) : omega_n.toFixed(2)}<\/div>\n <div class=\"rc-unit\">rad\/s<\/div>\n <\/div>\n <div class=\"res-card\">\n <div class=\"rc-label\">Period<\/div>\n <div class=\"rc-value\">${period >= 0.1 ? period.toFixed(3) : period.toFixed(4)}<\/div>\n <div class=\"rc-unit\">seconds<\/div>\n <\/div>\n <div class=\"res-card\">\n <div class=\"rc-label\">System<\/div>\n <div class=\"rc-value\" style=\"font-size:13px; font-family: 'Source Sans 3', sans-serif;\">${description}<\/div>\n <\/div>\n <\/div>\n ${rpm > 0 ? `\n <div class=\"resonance-indicator ${riskClass}\">\n <div class=\"ri-status\">${riskStatus}<\/div>\n <div class=\"ri-detail\">${riskDetail}<\/div>\n <\/div>` : ''}\n `;\n}\n\n\/\/ ===== RESONANCE CURVE CANVAS =====\nfunction drawResonanceCurve() {\n const canvas = document.getElementById('resonanceCanvas');\n const ctx = canvas.getContext('2d');\n const dpr = window.devicePixelRatio || 1;\n const rect = canvas.parentElement.getBoundingClientRect();\n\n canvas.width = rect.width * dpr;\n canvas.height = rect.height * dpr;\n canvas.style.width = rect.width + 'px';\n canvas.style.height = rect.height + 'px';\n ctx.scale(dpr, dpr);\n\n const W = rect.width;\n const H = rect.height;\n const pad = { top: 30, right: 40, bottom: 50, left: 70 };\n const plotW = W - pad.left - pad.right;\n const plotH = H - pad.top - pad.bottom;\n\n \/\/ Clear\n ctx.fillStyle = '#faf7f2';\n ctx.fillRect(0, 0, W, H);\n\n \/\/ Grid\n ctx.strokeStyle = '#e2e8f0';\n ctx.lineWidth = 0.5;\n for (let i = 0; i <= 10; i++) {\n const x = pad.left + (i \/ 10) * plotW;\n ctx.beginPath(); ctx.moveTo(x, pad.top); ctx.lineTo(x, pad.top + plotH); ctx.stroke();\n }\n for (let i = 0; i <= 8; i++) {\n const y = pad.top + (i \/ 8) * plotH;\n ctx.beginPath(); ctx.moveTo(pad.left, y); ctx.lineTo(pad.left + plotW, y); ctx.stroke();\n }\n\n \/\/ Resonance zone highlight\n const x07 = pad.left + (0.7 \/ 3.0) * plotW;\n const x14 = pad.left + (1.4 \/ 3.0) * plotW;\n ctx.fillStyle = 'rgba(220,38,38,0.06)';\n ctx.fillRect(x07, pad.top, x14 - x07, plotH);\n\n \/\/ r=1 line\n const x1 = pad.left + (1.0 \/ 3.0) * plotW;\n ctx.strokeStyle = 'rgba(220,38,38,0.3)';\n ctx.lineWidth = 1;\n ctx.setLineDash([4, 4]);\n ctx.beginPath(); ctx.moveTo(x1, pad.top); ctx.lineTo(x1, pad.top + plotH); ctx.stroke();\n ctx.setLineDash([]);\n\n \/\/ DMF function\n function dmf(r, zeta) {\n const denom = Math.sqrt(Math.pow(1 - r * r, 2) + Math.pow(2 * zeta * r, 2));\n return 1 \/ denom;\n }\n\n const maxDMF = 30;\n const dampings = [\n { zeta: 0.02, color: '#dc2626', label: '\u03b6 = 0.02 (Q=25)' },\n { zeta: 0.05, color: '#d97706', label: '\u03b6 = 0.05 (Q=10)' },\n { zeta: 0.10, color: '#2563eb', label: '\u03b6 = 0.10 (Q=5)' },\n { zeta: 0.20, color: '#059669', label: '\u03b6 = 0.20 (Q=2.5)' },\n { zeta: 0.50, color: '#94a3b8', label: '\u03b6 = 0.50 (Q=1)' },\n ];\n\n const rMax = 3.0;\n const steps = 400;\n\n dampings.forEach(d => {\n ctx.strokeStyle = d.color;\n ctx.lineWidth = 2;\n ctx.beginPath();\n let started = false;\n for (let i = 0; i <= steps; i++) {\n const r = (i \/ steps) * rMax;\n if (r === 0) continue;\n let amp = dmf(r, d.zeta);\n if (amp > maxDMF) amp = maxDMF;\n const x = pad.left + (r \/ rMax) * plotW;\n const y = pad.top + plotH - (amp \/ maxDMF) * plotH;\n if (!started) { ctx.moveTo(x, y); started = true; }\n else ctx.lineTo(x, y);\n }\n ctx.stroke();\n });\n\n \/\/ Axes\n ctx.strokeStyle = var_navy;\n ctx.lineWidth = 1.5;\n ctx.beginPath();\n ctx.moveTo(pad.left, pad.top);\n ctx.lineTo(pad.left, pad.top + plotH);\n ctx.lineTo(pad.left + plotW, pad.top + plotH);\n ctx.stroke();\n\n \/\/ X-axis labels\n ctx.fillStyle = '#475569';\n ctx.font = '12px \"JetBrains Mono\", monospace';\n ctx.textAlign = 'center';\n for (let r = 0; r <= 3; r += 0.5) {\n const x = pad.left + (r \/ rMax) * plotW;\n ctx.fillText(r.toFixed(1), x, pad.top + plotH + 18);\n }\n ctx.font = '13px \"Source Sans 3\", sans-serif';\n ctx.fillText('Frequency Ratio (\u03c9 \/ \u03c9\u2099)', pad.left + plotW \/ 2, pad.top + plotH + 42);\n\n \/\/ Y-axis labels\n ctx.textAlign = 'right';\n ctx.font = '12px \"JetBrains Mono\", monospace';\n for (let a = 0; a <= maxDMF; a += 5) {\n const y = pad.top + plotH - (a \/ maxDMF) * plotH;\n ctx.fillText(a.toString(), pad.left - 8, y + 4);\n }\n ctx.save();\n ctx.translate(16, pad.top + plotH \/ 2);\n ctx.rotate(-Math.PI \/ 2);\n ctx.textAlign = 'center';\n ctx.font = '13px \"Source Sans 3\", sans-serif';\n ctx.fillText('Amplification (DMF)', 0, 0);\n ctx.restore();\n\n \/\/ Legend\n const legendX = pad.left + plotW - 170;\n let legendY = pad.top + 10;\n ctx.fillStyle = 'rgba(255,255,255,0.9)';\n ctx.fillRect(legendX - 8, legendY - 4, 178, dampings.length * 22 + 8);\n ctx.strokeStyle = '#e2e8f0';\n ctx.lineWidth = 1;\n ctx.strokeRect(legendX - 8, legendY - 4, 178, dampings.length * 22 + 8);\n\n dampings.forEach(d => {\n ctx.strokeStyle = d.color;\n ctx.lineWidth = 2.5;\n ctx.beginPath();\n ctx.moveTo(legendX, legendY + 8);\n ctx.lineTo(legendX + 24, legendY + 8);\n ctx.stroke();\n ctx.fillStyle = '#1e293b';\n ctx.font = '12px \"Source Sans 3\", sans-serif';\n ctx.textAlign = 'left';\n ctx.fillText(d.label, legendX + 30, legendY + 12);\n legendY += 22;\n });\n\n \/\/ \"Resonance Zone\" label\n ctx.fillStyle = 'rgba(220,38,38,0.6)';\n ctx.font = '11px \"Source Sans 3\", sans-serif';\n ctx.textAlign = 'center';\n ctx.fillText('RESONANCE', (x07 + x14) \/ 2, pad.top + 15);\n ctx.fillText('ZONE', (x07 + x14) \/ 2, pad.top + 28);\n}\n\nconst var_navy = '#0a2540';\n\n\/\/ Init\ntoggleInputs();\ncalculate();\nsetTimeout(drawResonanceCurve, 100);\nwindow.addEventListener('resize', drawResonanceCurve);\n<\/script>\n\n<\/body>\n<\/html><\/div><\/div><\/div><\/div><\/div>","protected":false},"excerpt":{"rendered":"<p>L\u00e6r om egenfrekvens, frekvensen et system naturlig vibrerer med. Forst\u00e5 hvordan det forholder seg til masse og stivhet, og hvorfor det er n\u00f8kkelfaktoren for \u00e5 for\u00e5rsake resonans.<\/p>","protected":false},"featured_media":0,"template":"","meta":{"ai_generated_summary":"","footnotes":""},"categories":[114,109],"tags":[],"class_list":["post-172","glossary","type-glossary","status-publish","hentry","category-analysis","category-glossary"],"_links":{"self":[{"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/glossary\/172","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/glossary"}],"about":[{"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/types\/glossary"}],"version-history":[{"count":6,"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/glossary\/172\/revisions"}],"predecessor-version":[{"id":101730,"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/glossary\/172\/revisions\/101730"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/media?parent=172"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/categories?post=172"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/nb\/wp-json\/wp\/v2\/tags?post=172"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}