{"id":100166,"date":"2026-02-15T20:21:59","date_gmt":"2026-02-15T20:21:59","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100166"},"modified":"2026-02-15T20:21:59","modified_gmt":"2026-02-15T20:21:59","slug":"natural-frequency","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/bn\/calculators\/natural-frequency\/","title":{"rendered":"\u09aa\u09cd\u09b0\u09be\u0995\u09c3\u09a4\u09bf\u0995 \u09ab\u09cd\u09b0\u09bf\u0995\u09cb\u09af\u09bc\u09c7\u09a8\u09cd\u09b8\u09bf \u0995\u09cd\u09af\u09be\u09b2\u0995\u09c1\u09b2\u09c7\u099f\u09b0 (\u09ad\u09b0-\u09ac\u09b8\u09a8\u09cd\u09a4 \u09b8\u09bf\u09b8\u09cd\u099f\u09c7\u09ae)"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Natural Frequency Calculator (Mass-Spring System)\",\"alternateName\":[\"Mass-Spring Frequency Calculator\",\"Vibration Frequency Calculator\",\"Resonance Frequency Calculator\"],\"description\":\"Free online engineering calculator for the natural frequency of a mass-spring system. Computes frequency in Hz, angular frequency, period, equivalent RPM, and static deflection.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/natural-frequency-mass-spring\/\",\"applicationCategory\":\"Engineering Calculator\",\"operatingSystem\":\"Any (Web Browser)\",\"offers\":{\"@type\":\"Offer\",\"price\":\"0\",\"priceCurrency\":\"EUR\"},\"creator\":{\"@type\":\"Organization\",\"name\":\"Vibromera\",\"url\":\"https:\/\/vibromera.eu\/\"},\"datePublished\":\"2024-01-01\",\"dateModified\":\"2025-02-14\",\"inLanguage\":\"en\",\"isAccessibleForFree\":true,\"featureList\":[\"Natural frequency in Hz\",\"Angular frequency in rad\/s\",\"Period and equivalent RPM\",\"Static deflection\",\"Metric and Imperial units\",\"Auto-calculation on input change\",\"URL parameter sharing\",\"Calculation history\"],\"keywords\":\"natural frequency, mass-spring, spring stiffness, resonance, vibration period, angular frequency\"}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[\n{\"@type\":\"Question\",\"name\":\"What is natural frequency?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Natural frequency is the frequency at which a system oscillates freely after being displaced from its equilibrium position and released. For a mass-spring system, it depends only on the mass and the spring stiffness. Every mechanical structure has one or more natural frequencies, and they are fundamental properties used in vibration analysis, structural design, and machine diagnostics.\"}},\n{\"@type\":\"Question\",\"name\":\"Why is natural frequency important for machines?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Knowing the natural frequency of a machine or its components is critical because if the operating speed (excitation frequency) coincides with the natural frequency, resonance occurs. Resonance causes dramatically amplified vibrations that can lead to excessive wear, fatigue failure, bearing damage, and even catastrophic structural collapse. Engineers design machines so that operating speeds are well separated from natural frequencies.\"}},\n{\"@type\":\"Question\",\"name\":\"How to measure spring stiffness?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Spring stiffness (rate) can be measured by applying a known force to the spring and measuring the resulting deflection. Stiffness k = Force \/ Deflection. For example, if a 100 N force compresses a spring by 2 mm, the stiffness is 100\/2 = 50 N\/mm. You can also use a compression testing machine for accurate results, or consult the manufacturer's datasheet for the spring rate specification.\"}},\n{\"@type\":\"Question\",\"name\":\"What if the operating speed equals the natural frequency?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"When the operating speed (excitation frequency) matches the natural frequency of the system, resonance occurs. 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font-size:12px; }\n.vc-footer-links a { margin:0 8px; }\n\n@media print {\n    .vc-section-body { max-height:none!important; }\n    .vc-faq-a { max-height:none!important; }\n    .vc-results { max-height:none!important; padding:24px!important; }\n    .vc-copy-btn, .vc-section-chevron, .vc-faq-chevron, .vc-presets, .vc-promo { display:none!important; }\n}\n<\/style>\n\n<div class=\"vc-calculator\" id=\"vc-natural-frequency\">\n\n<!-- Header -->\n<header class=\"vc-header\">\n    <p class=\"vc-header-eyebrow\">Free Engineering Tool<\/p>\n    <h1 class=\"vc-header-title\">Natural Frequency Calculator<\/h1>\n    <p class=\"vc-header-subtitle\">Calculate the natural frequency of a mass-spring system. Enter mass and spring stiffness to get frequency in Hz, angular frequency, period, equivalent RPM, and static deflection.<\/p>\n    <div class=\"vc-badges\">\n        <span class=\"vc-badge\">Mass-Spring<\/span>\n        <span class=\"vc-badge\">1-DOF<\/span>\n        <span class=\"vc-badge\">Undamped<\/span>\n    <\/div>\n<\/header>\n\n<!-- Calculator Card -->\n<div class=\"vc-card\">\n    <div class=\"vc-unit-bar\">\n        <button type=\"button\" class=\"vc-unit-btn vc-active\" data-system=\"metric\">Metric (kg, N\/mm)<\/button>\n        <button type=\"button\" class=\"vc-unit-btn\" data-system=\"imperial\">Imperial (lb, lbf\/in)<\/button>\n    <\/div>\n\n    <form class=\"vc-form\" id=\"vc-form\" autocomplete=\"off\">\n        <div class=\"vc-form-grid\">\n            <div class=\"vc-field\">\n                <label class=\"vc-label\" for=\"vc-mass\">Mass (m) <span class=\"vc-label-hint\" id=\"vc-mass-unit\">(kg)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-mass\" inputmode=\"decimal\" step=\"any\" min=\"0.001\" placeholder=\"e.g. 150\" value=\"150\">\n            <\/div>\n            <div class=\"vc-field\">\n                <label class=\"vc-label\" for=\"vc-stiffness\">Spring Stiffness (k) <span class=\"vc-label-hint\" id=\"vc-stiffness-unit\">(N\/mm)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-stiffness\" inputmode=\"decimal\" step=\"any\" min=\"0.001\" placeholder=\"e.g. 300\" value=\"300\">\n            <\/div>\n        <\/div>\n\n        <!-- Presets -->\n        <div class=\"vc-presets\">\n            <div class=\"vc-presets-label\">Quick presets<\/div>\n            <button type=\"button\" class=\"vc-preset-btn\" data-m=\"20\" data-k=\"50\">Small rotor 20 kg, k=50 N\/mm<\/button>\n            <button type=\"button\" class=\"vc-preset-btn\" data-m=\"200\" data-k=\"500\">Pump 200 kg, k=500 N\/mm<\/button>\n            <button type=\"button\" class=\"vc-preset-btn\" data-m=\"1000\" data-k=\"2000\">Compressor 1000 kg, k=2000 N\/mm<\/button>\n        <\/div>\n    <\/form>\n\n    <!-- Results -->\n    <div class=\"vc-results\" id=\"vc-results\">\n        <div class=\"vc-results-head\">\n            <h2 class=\"vc-results-title\">Results<\/h2>\n            <button type=\"button\" class=\"vc-copy-btn\" id=\"vc-copy-btn\">\n                <svg width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><rect x=\"9\" y=\"9\" width=\"13\" height=\"13\" rx=\"2\"\/><path d=\"M5 15H4a2 2 0 01-2-2V4a2 2 0 012-2h9a2 2 0 012 2v1\"\/><\/svg>\n                Copy\n            <\/button>\n        <\/div>\n        <div class=\"vc-result-grid\">\n            <div class=\"vc-rcard vc-rcard-primary\">\n                <div class=\"vc-rcard-label\">Natural Frequency<\/div>\n                <div class=\"vc-rcard-value\" id=\"vc-r-freq\">\u2014<\/div>\n            <\/div>\n            <div class=\"vc-rcard\">\n                <div class=\"vc-rcard-label\">Angular Frequency (\u03c9\u2099)<\/div>\n                <div class=\"vc-rcard-value\" id=\"vc-r-omega\">\u2014<\/div>\n            <\/div>\n            <div class=\"vc-rcard\">\n                <div class=\"vc-rcard-label\">Period (T)<\/div>\n                <div class=\"vc-rcard-value\" id=\"vc-r-period\">\u2014<\/div>\n            <\/div>\n            <div class=\"vc-rcard\">\n                <div class=\"vc-rcard-label\">Equivalent RPM<\/div>\n                <div class=\"vc-rcard-value\" id=\"vc-r-rpm\">\u2014<\/div>\n            <\/div>\n            <div class=\"vc-rcard\">\n                <div class=\"vc-rcard-label\">Static Deflection (\u03b4_static)<\/div>\n                <div class=\"vc-rcard-value\" id=\"vc-r-deflection\">\u2014<\/div>\n            <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n\n<!-- Theory & Reference -->\n<div class=\"vc-section vc-open\" id=\"vc-sec-theory\">\n    <button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"true\">\n        <span class=\"vc-section-toggle-text\">\n            <span class=\"vc-section-icon\">\ud83d\udcd8<\/span>\n            <span class=\"vc-section-title\">Theory &amp; Formulas<\/span>\n        <\/span>\n        <svg class=\"vc-section-chevron\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><path d=\"M6 9l6 6 6-6\"\/><\/svg>\n    <\/button>\n    <div class=\"vc-section-body\">\n        <div class=\"vc-section-inner vc-theory\">\n            <h3>Natural Frequency of a Mass-Spring System<\/h3>\n            <p>The natural frequency of an undamped single-degree-of-freedom mass-spring system:<\/p>\n            <div class=\"vc-formula-box\" id=\"vc-katex-1\"><\/div>\n            <ul>\n                <li><strong>k<\/strong> \u2014 spring stiffness (N\/m). If given in N\/mm, multiply by 1000.<\/li>\n                <li><strong>m<\/strong> \u2014 mass (kg)<\/li>\n            <\/ul>\n\n            <h3>Angular Frequency<\/h3>\n            <div class=\"vc-formula-box\" id=\"vc-katex-2\"><\/div>\n\n            <h3>Period and Static Deflection<\/h3>\n            <div class=\"vc-formula-box\" id=\"vc-katex-3\"><\/div>\n            <div class=\"vc-formula-box\" id=\"vc-katex-4\"><\/div>\n            <p>Where <strong>g<\/strong> = 9.81 m\/s\u00b2 and k is in N\/mm for the deflection result in mm.<\/p>\n\n            <h3>Practical Example<\/h3>\n            <div class=\"vc-example-box\">\n                <div class=\"vc-example-title\">Example \u2014 Machine on Spring Mounts<\/div>\n                <p><strong>Given:<\/strong> Machine mass = 150 kg, Total spring stiffness k = 300 N\/mm<\/p>\n                <p>k in N\/m = 300 \u00d7 1000 = 300,000 N\/m<\/p>\n                <p>f_n = (1\/2\u03c0) \u00d7 \u221a(300,000\/150) = (1\/6.2832) \u00d7 44.72 = <strong>7.12 Hz<\/strong><\/p>\n                <p>\u03c9_n = 2\u03c0 \u00d7 7.12 = <strong>44.7 rad\/s<\/strong><\/p>\n                <p>T = 1\/7.12 = <strong>0.140 s<\/strong><\/p>\n                <p>Equivalent RPM = 7.12 \u00d7 60 = <strong>427 RPM<\/strong><\/p>\n                <p>\u03b4_static = 150 \u00d7 9.81 \/ 300,000 \u00d7 1000 = <strong>4.9 mm<\/strong><\/p>\n            <\/div>\n\n            <div class=\"vc-warning-box\">\n                <p style=\"margin:0;font-size:14px;color:var(--vc-ink-secondary);\"><strong>\u26a0\ufe0f Note:<\/strong> Ensure the operating speed is at least 20\u201330% away from the natural frequency to avoid resonance. If operating near resonance, add damping or modify stiffness\/mass.<\/p>\n            <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n\n<!-- Related Calculators -->\n<div class=\"vc-section\" id=\"vc-sec-related\">\n    <button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\">\n        <span class=\"vc-section-toggle-text\">\n            <span class=\"vc-section-icon\">\ud83d\udd17<\/span>\n            <span class=\"vc-section-title\">Related Calculators<\/span>\n        <\/span>\n        <svg class=\"vc-section-chevron\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><path d=\"M6 9l6 6 6-6\"\/><\/svg>\n    <\/button>\n    <div class=\"vc-section-body\">\n        <div class=\"vc-section-inner\">\n            <div class=\"vc-related\">\n                <a class=\"vc-related-link\" href=\"\/calculators\/critical-speed\/\">Spring Critical Speed<\/a>\n                <a class=\"vc-related-link\" href=\"\/calculators\/spring-selection\/\">Spring Selection by Frequency<\/a>\n                <a class=\"vc-related-link\" href=\"\/calculators\/equivalent-spring-stiffness\/\">Equivalent Spring Stiffness<\/a>\n                <a class=\"vc-related-link\" href=\"\/calculators\/residual-unbalance-iso1940\/\">Residual Unbalance (ISO 1940)<\/a>\n            <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n\n<!-- FAQ -->\n<div class=\"vc-section\" id=\"vc-sec-faq\">\n    <button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\">\n        <span class=\"vc-section-toggle-text\">\n            <span class=\"vc-section-icon\">\u2753<\/span>\n            <span class=\"vc-section-title\">Frequently Asked Questions<\/span>\n        <\/span>\n        <svg class=\"vc-section-chevron\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><path d=\"M6 9l6 6 6-6\"\/><\/svg>\n    <\/button>\n    <div class=\"vc-section-body\">\n        <div class=\"vc-section-inner\">\n            <div class=\"vc-faq-list\" id=\"vc-faq-list\"><\/div>\n        <\/div>\n    <\/div>\n<\/div>\n\n<!-- Promo -->\n<div class=\"vc-promo\">\n    <div class=\"vc-promo-text\">\n        <strong>Vibromera \u2014 Vibration Diagnostics &amp; Balancing Equipment<\/strong><br>\n        Professional portable balancers, vibration analyzers, and condition monitoring systems used in 50+ countries. Identify resonance problems and balance rotors on-site.\n    <\/div>\n    <a class=\"vc-promo-link\" href=\"https:\/\/vibromera.eu\/\" target=\"_blank\" rel=\"noopener\">Learn More<\/a>\n<\/div>\n\n<!-- Footer -->\n<footer class=\"vc-footer\">\n    <p>\u00a9 2024\u20132025 <a href=\"https:\/\/vibromera.eu\/\">Vibromera<\/a> \u2014 Engineering Calculators<\/p>\n    <p style=\"font-size:12px;margin-top:4px;\">Last updated: February 2025<\/p>\n    <div class=\"vc-footer-links\">\n        <a href=\"\/calculators\/engineering-calculators\/\">All 252 Calculators<\/a>\n        <a href=\"https:\/\/vibromera.eu\/contacts\/\">Contact<\/a>\n    <\/div>\n<\/footer>\n\n<\/div>\n\n<script src=\"https:\/\/cdn.jsdelivr.net\/npm\/katex@0.16.11\/dist\/katex.min.js\"><\/script>\n<script>\n(function() {\n    'use strict';\n\n    var SLUG = 'natural-frequency-mass-spring';\n    var unitSystem = 'metric';\n    var KG2LB = 2.20462;\n    var NMM2LBFIN = 5.71015;\n    var MM2IN = 25.4;\n    var G = 9.81;\n\n    function $(id) { return document.getElementById(id); }\n    function fmt(v, d) {\n        if (typeof d === 'undefined') d = 4;\n        if (isNaN(v)) return '\u2014';\n        if (v === 0) return '0';\n        if (Math.abs(v) < 0.001) return v.toExponential(2);\n        if (Math.abs(v) < 1) return v.toFixed(d);\n        if (Math.abs(v) < 10) return v.toFixed(3);\n        if (Math.abs(v) < 100) return v.toFixed(2);\n        if (Math.abs(v) < 10000) return v.toFixed(1);\n        return Math.round(v).toLocaleString('en-US');\n    }\n\n    \/* \u2500\u2500 CALCULATION \u2500\u2500 *\/\n    function calculate() {\n        var massVal = parseFloat($('vc-mass').value);\n        var kVal = parseFloat($('vc-stiffness').value);\n\n        if (!massVal || massVal <= 0 || !kVal || kVal <= 0) {\n            $('vc-results').classList.remove('vc-visible');\n            return;\n        }\n\n        \/* Convert to metric (kg, N\/mm) *\/\n        var massKg = unitSystem === 'imperial' ? massVal \/ KG2LB : massVal;\n        var kNmm = unitSystem === 'imperial' ? kVal \/ NMM2LBFIN : kVal;\n\n        \/* Core calculations \u2014 k in N\/m for frequency *\/\n        var kNm = kNmm * 1000;\n        var omega = Math.sqrt(kNm \/ massKg);\n        var fn = omega \/ (2 * Math.PI);\n        var T = 1 \/ fn;\n        var rpm = fn * 60;\n        var deflMm = massKg * G \/ kNm * 1000; \/* mm *\/\n\n        \/* Display deflection in user units *\/\n        var deflDisplay = unitSystem === 'imperial' ? (deflMm \/ MM2IN) : deflMm;\n        var deflUnit = unitSystem === 'imperial' ? 'in' : 'mm';\n\n        $('vc-r-freq').innerHTML = fmt(fn) + ' <span class=\"vc-rcard-unit\">Hz<\/span>';\n        $('vc-r-omega').innerHTML = fmt(omega) + ' <span class=\"vc-rcard-unit\">rad\/s<\/span>';\n        $('vc-r-period').innerHTML = fmt(T) + ' <span class=\"vc-rcard-unit\">s<\/span>';\n        $('vc-r-rpm').innerHTML = fmt(rpm) + ' <span class=\"vc-rcard-unit\">RPM<\/span>';\n        $('vc-r-deflection').innerHTML = fmt(deflDisplay) + ' <span class=\"vc-rcard-unit\">' + deflUnit + '<\/span>';\n\n        $('vc-results').classList.add('vc-visible');\n\n        \/* URL params *\/\n        var u = new URL(window.location);\n        u.searchParams.set('mass', $('vc-mass').value);\n        u.searchParams.set('stiffness', $('vc-stiffness').value);\n        history.replaceState(null, '', u.toString());\n\n        \/* History *\/\n        saveHistory({\n            mass: $('vc-mass').value,\n            stiffness: $('vc-stiffness').value,\n            _summary: 'm=' + $('vc-mass').value + (unitSystem === 'imperial' ? ' lb' : ' kg') + ', k=' + $('vc-stiffness').value + (unitSystem === 'imperial' ? ' lbf\/in' : ' N\/mm') + ' \u2192 ' + fmt(fn) + ' Hz'\n        });\n    }\n\n    \/* \u2500\u2500 UNIT TOGGLE \u2500\u2500 *\/\n    var unitBtns = document.querySelectorAll('.vc-unit-btn');\n    unitBtns.forEach(function(btn) {\n        btn.addEventListener('click', function() {\n            unitBtns.forEach(function(b) { b.classList.remove('vc-active'); });\n            this.classList.add('vc-active');\n            unitSystem = this.getAttribute('data-system');\n            $('vc-mass-unit').textContent = unitSystem === 'imperial' ? '(lb)' : '(kg)';\n            $('vc-stiffness-unit').textContent = unitSystem === 'imperial' ? 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' lbf\/in' : ' N\/mm') + '\\n' +\n            'Natural Frequency: ' + $('vc-r-freq').textContent + '\\n' +\n            'Angular Frequency: ' + $('vc-r-omega').textContent + '\\n' +\n            'Period: ' + $('vc-r-period').textContent + '\\n' +\n            'Equivalent RPM: ' + $('vc-r-rpm').textContent + '\\n' +\n            'Static Deflection: ' + $('vc-r-deflection').textContent + '\\n' +\n            '\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\\n' +\n            'vibromera.eu\/calculators\/natural-frequency-mass-spring\/';\n\n        if (navigator.clipboard) {\n            navigator.clipboard.writeText(text).then(function() {\n                var btn = $('vc-copy-btn');\n                btn.classList.add('vc-copied');\n                btn.innerHTML = '<svg width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><polyline points=\"20 6 9 17 4 12\"\/><\/svg> Copied!';\n                setTimeout(function() {\n                    btn.classList.remove('vc-copied');\n                    btn.innerHTML = '<svg width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><rect x=\"9\" y=\"9\" width=\"13\" height=\"13\" rx=\"2\"\/><path d=\"M5 15H4a2 2 0 01-2-2V4a2 2 0 012-2h9a2 2 0 012 2v1\"\/><\/svg> Copy';\n                }, 2000);\n            });\n        }\n    });\n\n    \/* \u2500\u2500 COLLAPSIBLE SECTIONS \u2500\u2500 *\/\n    document.querySelectorAll('.vc-section-toggle').forEach(function(btn) {\n        btn.addEventListener('click', function() {\n            var section = this.closest('.vc-section');\n            var isOpen = section.classList.contains('vc-open');\n            section.classList.toggle('vc-open');\n            this.setAttribute('aria-expanded', !isOpen);\n        });\n    });\n\n    \/* \u2500\u2500 FAQ \u2500\u2500 *\/\n    var faqData = [\n        { q: 'What is natural frequency?', a: '<p>Natural frequency is the frequency at which a system oscillates freely after being displaced from its equilibrium position and released. For a mass-spring system, it depends only on the mass and the spring stiffness.<\/p><p>Every mechanical structure has one or more natural frequencies, and they are fundamental properties used in vibration analysis, structural design, and machine diagnostics.<\/p>' },\n        { q: 'Why is natural frequency important for machines?', a: '<p>Knowing the natural frequency of a machine or its components is critical because if the operating speed (excitation frequency) coincides with the natural frequency, resonance occurs.<\/p><p>Resonance causes dramatically amplified vibrations that can lead to excessive wear, fatigue failure, bearing damage, and even catastrophic structural collapse. Engineers design machines so that operating speeds are well separated from natural frequencies.<\/p>' },\n        { q: 'How to measure spring stiffness?', a: '<p>Spring stiffness (rate) can be measured by applying a known force to the spring and measuring the resulting deflection. Stiffness k = Force \/ Deflection.<\/p><p>For example, if a 100 N force compresses a spring by 2 mm, the stiffness is 100\/2 = 50 N\/mm. You can also use a compression testing machine for accurate results, or consult the manufacturer\\'s datasheet for the spring rate specification.<\/p>' },\n        { q: 'What if the operating speed equals the natural frequency?', a: '<p>When the operating speed (excitation frequency) matches the natural frequency of the system, resonance occurs. At resonance, vibration amplitudes increase dramatically \u2014 theoretically to infinity in an undamped system, and to very high levels in real (lightly damped) systems.<\/p><p>This can cause rapid fatigue failure, excessive noise, bearing damage, and structural collapse. This is why vibration analysis always checks for proximity to resonance.<\/p>' },\n        { q: 'How to avoid resonance in mechanical systems?', a: '<p>There are several strategies to avoid resonance:<\/p><ul><li>Change the operating speed so it is at least 20\u201330% away from any natural frequency<\/li><li>Modify the stiffness by adding braces, stiffeners, or changing spring rates<\/li><li>Change the mass of the vibrating component<\/li><li>Add damping using vibration dampers, rubber isolators, or tuned mass dampers<\/li><li>Use variable-speed drives to avoid dwelling at critical speeds during startup or shutdown<\/li><\/ul>' }\n    ];\n\n    var faqList = $('vc-faq-list');\n    faqData.forEach(function(faq) {\n        faqList.innerHTML += '<div class=\"vc-faq-item\"><button type=\"button\" class=\"vc-faq-q\"><span>' + faq.q + '<\/span><svg class=\"vc-faq-chevron\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><path d=\"M6 9l6 6 6-6\"\/><\/svg><\/button><div class=\"vc-faq-a\"><div class=\"vc-faq-a-inner\">' + faq.a + '<\/div><\/div><\/div>';\n    });\n    faqList.addEventListener('click', function(e) {\n        var btn = e.target.closest('.vc-faq-q');\n        if (!btn) return;\n        btn.closest('.vc-faq-item').classList.toggle('vc-open');\n    });\n\n    \/* \u2500\u2500 HISTORY \u2500\u2500 *\/\n    function saveHistory(entry) {\n        var key = 'vc_history_' + SLUG;\n        var arr = [];\n        try { arr = JSON.parse(localStorage.getItem(key)) || []; 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