{"id":100100,"date":"2026-02-15T20:16:38","date_gmt":"2026-02-15T20:16:38","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100100"},"modified":"2026-02-15T20:16:38","modified_gmt":"2026-02-15T20:16:38","slug":"flexible-rotor-balancing","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/bn\/calculators\/flexible-rotor-balancing\/","title":{"rendered":"\u09a8\u09ae\u09a8\u09c0\u09af\u09bc \u09b0\u099f\u09be\u09b0 \u09ad\u09be\u09b0\u09b8\u09be\u09ae\u09cd\u09af \u09b8\u09b9\u09a8\u09b6\u09c0\u09b2\u09a4\u09be \u2014 ISO 21940-14 | \u09ac\u09bf\u09a8\u09be\u09ae\u09c2\u09b2\u09cd\u09af\u09c7 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09b8\u09b0\u099e\u09cd\u099c\u09be\u09ae"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Flexible Rotor Balancing Calculator\",\"description\":\"Calculate per-mode unbalance tolerance for flexible rotors per ISO 21940-14.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/flexible-rotor-balancing\/\",\"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}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[\n{\"@type\":\"Question\",\"name\":\"What is a flexible rotor?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"A flexible rotor operates above one or more of its critical speeds, so it deforms during rotation. The unbalance distribution changes with speed as different mode shapes are excited. Examples include turbine shafts, large generators, and high-speed compressor rotors.\"}},\n{\"@type\":\"Question\",\"name\":\"What is ISO 21940-14?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"ISO 21940-14 provides procedures for evaluating balance quality of flexible rotors. It addresses the additional complexity of modal unbalance and specifies tolerance criteria for each significant mode shape.\"}},\n{\"@type\":\"Question\",\"name\":\"How does flexible rotor balancing differ from rigid rotor balancing?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Rigid rotors need only 2-plane balancing (low-speed). Flexible rotors require N+2 planes for N modes, with speed-dependent correction. Each mode has its own unbalance tolerance based on the modal mass and the mode's critical speed.\"}},\n{\"@type\":\"Question\",\"name\":\"What is modal unbalance?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Modal unbalance is the component of unbalance that excites a specific bending mode of the rotor. Each mode acts like an independent SDOF system with its own effective mass. Total balance requires addressing each significant mode.\"}},\n{\"@type\":\"Question\",\"name\":\"How many correction planes do I need?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"For N flexible modes, you need N+2 correction planes. For example: 1 mode \u2192 3 planes, 2 modes \u2192 4 planes, 3 modes \u2192 5 planes. Plane locations should be chosen to effectively influence each mode shape.\"}}\n]}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"BreadcrumbList\",\"itemListElement\":[\n{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/vibromera.eu\/\"},\n{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Calculators\",\"item\":\"https:\/\/vibromera.eu\/calculators\/\"},\n{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Flexible Rotor Balancing\",\"item\":\"https:\/\/vibromera.eu\/calculators\/flexible-rotor-balancing\/\"}\n]}\n<\/script>\n\n<link rel=\"preconnect\" href=\"https:\/\/fonts.googleapis.com\">\n\n<link rel=\"preconnect\" href=\"https:\/\/fonts.gstatic.com\" crossorigin>\n\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=DM+Sans:ital,opsz,wght@0,9..40,300;0,9..40,400;0,9..40,500;0,9..40,600;0,9..40,700;1,9..40,400&#038;family=JetBrains+Mono:wght@400;500;600&#038;family=Fraunces:opsz,wght@9..144,700;9..144,800&#038;display=swap\" rel=\"stylesheet\">\n\n<link rel=\"stylesheet\" href=\"https:\/\/cdn.jsdelivr.net\/npm\/katex@0.16.11\/dist\/katex.min.css\">\n\n<style>\n:root{--vc-bg:#f4f1ec;--vc-surface:#ffffff;--vc-surface-alt:#f8f6f2;--vc-ink:#1a1a1a;--vc-ink-secondary:#5a5650;--vc-ink-muted:#8a857e;--vc-accent:#c85a2a;--vc-accent-hover:#b04d22;--vc-accent-light:#fdf0ea;--vc-blue:#2a5c8c;--vc-blue-light:#eaf1f8;--vc-green:#2a7a4b;--vc-green-light:#eaf8ef;--vc-yellow:#a67c00;--vc-yellow-light:#fef9e8;--vc-red:#b91c1c;--vc-red-light:#fef2f2;--vc-border:#d9d4cc;--vc-border-light:#e8e4dd;--vc-shadow:0 1px 3px rgba(26,26,26,0.06),0 4px 12px rgba(26,26,26,0.04);--vc-shadow-lg:0 4px 12px rgba(26,26,26,0.08),0 16px 40px rgba(26,26,26,0.06);--vc-radius:8px;--vc-radius-lg:12px;--vc-font:'DM Sans',-apple-system,BlinkMacSystemFont,'Segoe UI',sans-serif;--vc-mono:'JetBrains Mono','Consolas','Monaco',monospace;--vc-display:'Fraunces',Georgia,serif}\n.vc-calculator{font-family:var(--vc-font);font-size:15px;line-height:1.65;color:var(--vc-ink);max-width:960px;margin:0 auto;padding:20px 16px 40px;-webkit-font-smoothing:antialiased}.vc-calculator *,.vc-calculator *::before,.vc-calculator *::after{box-sizing:border-box}\n.vc-header{text-align:center;padding:48px 24px 40px;position:relative}.vc-header::after{content:'';position:absolute;bottom:0;left:50%;transform:translateX(-50%);width:80px;height:3px;background:var(--vc-accent);border-radius:2px}.vc-header-eyebrow{font-family:var(--vc-mono);font-size:11px;font-weight:500;letter-spacing:0.15em;text-transform:uppercase;color:var(--vc-accent);margin:0 0 12px}.vc-header-title{font-family:var(--vc-display);font-size:clamp(24px,4vw,36px);font-weight:800;line-height:1.15;color:var(--vc-ink);margin:0 0 16px;letter-spacing:-0.02em}.vc-header-subtitle{font-size:15px;color:var(--vc-ink-secondary);margin:0 auto 20px;max-width:600px}.vc-badges{display:inline-flex;gap:8px;flex-wrap:wrap;justify-content:center}.vc-badge{font-family:var(--vc-mono);font-size:11px;font-weight:500;padding:4px 10px;border-radius:4px;background:var(--vc-surface);border:1px solid var(--vc-border);color:var(--vc-ink-secondary)}\n.vc-card{background:var(--vc-surface);border:1px solid var(--vc-border);border-radius:var(--vc-radius-lg);box-shadow:var(--vc-shadow);overflow:hidden;margin-top:32px}\n.vc-form{padding:24px}.vc-form-grid{display:grid;grid-template-columns:1fr 1fr;gap:18px}@media(max-width:600px){.vc-form-grid{grid-template-columns:1fr}}.vc-field{display:flex;flex-direction:column}.vc-label{font-size:12px;font-weight:600;letter-spacing:0.04em;text-transform:uppercase;color:var(--vc-ink-secondary);margin-bottom:6px}.vc-label-hint{font-weight:400;text-transform:none;letter-spacing:0;color:var(--vc-ink-muted);font-size:11px}.vc-select,.vc-input{width:100%;padding:10px 12px;border:1.5px solid var(--vc-border);border-radius:6px;font-family:var(--vc-font);font-size:14px;color:var(--vc-ink);background:var(--vc-surface);transition:border-color 0.15s,box-shadow 0.15s}.vc-select{padding-right:36px;appearance:none;-webkit-appearance:none;cursor:pointer;background-image:url(\"data:image\/svg+xml,%3Csvg xmlns='http:\/\/www.w3.org\/2000\/svg' width='16' height='16' fill='none' stroke='%235a5650' stroke-width='2'%3E%3Cpath d='M4 6l4 4 4-4'\/%3E%3C\/svg%3E\");background-repeat:no-repeat;background-position:right 10px center}.vc-select:focus,.vc-input:focus{outline:none;border-color:var(--vc-accent);box-shadow:0 0 0 3px rgba(200,90,42,0.12)}.vc-select:hover,.vc-input:hover{border-color:var(--vc-ink-muted)}\n.vc-presets{display:flex;flex-wrap:wrap;gap:8px;margin-top:18px;padding-top:18px;border-top:1px solid var(--vc-border-light)}.vc-presets-label{font-family:var(--vc-mono);font-size:11px;font-weight:500;letter-spacing:0.1em;text-transform:uppercase;color:var(--vc-ink-muted);width:100%;margin-bottom:2px}.vc-preset-btn{padding:5px 12px;font-family:var(--vc-font);font-size:12px;font-weight:500;border:1px solid var(--vc-border);border-radius:6px;background:var(--vc-surface-alt);color:var(--vc-ink-secondary);cursor:pointer;transition:all 0.15s}.vc-preset-btn:hover{border-color:var(--vc-accent);color:var(--vc-accent);background:var(--vc-accent-light)}\n.vc-results{border-top:1px solid var(--vc-border-light);background:var(--vc-surface-alt);padding:0;max-height:0;overflow:hidden;transition:max-height 0.5s cubic-bezier(0.4,0,0.2,1),padding 0.3s}.vc-results.vc-visible{max-height:4000px;padding:24px}.vc-results-head{display:flex;align-items:center;justify-content:space-between;margin-bottom:18px}.vc-results-title{font-family:var(--vc-display);font-size:20px;font-weight:700;color:var(--vc-ink);margin:0}.vc-copy-btn{display:inline-flex;align-items:center;gap:5px;padding:6px 12px;border:1px solid var(--vc-border);border-radius:6px;background:var(--vc-surface);font-family:var(--vc-font);font-size:12px;font-weight:500;color:var(--vc-ink-secondary);cursor:pointer;transition:all 0.15s}.vc-copy-btn:hover{border-color:var(--vc-accent);color:var(--vc-accent)}.vc-copy-btn.vc-copied{background:var(--vc-green-light);border-color:var(--vc-green);color:var(--vc-green)}\n.vc-result-grid{display:grid;grid-template-columns:1fr 1fr;gap:12px;margin-bottom:20px}@media(max-width:500px){.vc-result-grid{grid-template-columns:1fr}}.vc-rcard{background:var(--vc-surface);border:1px solid var(--vc-border-light);border-radius:var(--vc-radius);padding:16px;transition:transform 0.2s}.vc-rcard:hover{transform:translateY(-1px)}.vc-rcard-primary{border-color:var(--vc-accent);border-width:2px;background:linear-gradient(135deg,#fdf6f2 0%,var(--vc-surface) 100%);grid-column:1\/-1}.vc-rcard-label{font-family:var(--vc-mono);font-size:10px;font-weight:500;letter-spacing:0.1em;text-transform:uppercase;color:var(--vc-ink-muted);margin-bottom:4px}.vc-rcard-value{font-family:var(--vc-mono);font-size:22px;font-weight:600;color:var(--vc-ink);line-height:1.2}.vc-rcard-primary .vc-rcard-value{font-size:30px;color:var(--vc-accent)}.vc-rcard-unit{font-size:14px;font-weight:400;color:var(--vc-ink-secondary)}\n.vc-section{margin-top:24px;border:1px solid var(--vc-border);border-radius:var(--vc-radius-lg);background:var(--vc-surface);box-shadow:var(--vc-shadow);overflow:hidden}.vc-section-toggle{width:100%;display:flex;align-items:center;justify-content:space-between;padding:18px 24px;border:none;background:transparent;cursor:pointer;font-family:var(--vc-font);text-align:left;transition:background 0.15s}.vc-section-toggle:hover{background:var(--vc-surface-alt)}.vc-section-toggle-text{display:flex;align-items:center;gap:10px}.vc-section-icon{font-size:20px;line-height:1}.vc-section-title{font-family:var(--vc-display);font-size:18px;font-weight:700;color:var(--vc-ink);margin:0}.vc-section-chevron{width:20px;height:20px;color:var(--vc-ink-muted);transition:transform 0.3s;flex-shrink:0}.vc-section.vc-open .vc-section-chevron{transform:rotate(180deg)}.vc-section-body{max-height:0;overflow:hidden;transition:max-height 0.4s cubic-bezier(0.4,0,0.2,1)}.vc-section.vc-open .vc-section-body{max-height:20000px}.vc-section-inner{padding:0 24px 24px;border-top:1px solid var(--vc-border-light)}\n.vc-theory h3{font-family:var(--vc-display);font-size:17px;font-weight:700;color:var(--vc-ink);margin:28px 0 10px}.vc-theory h3:first-child{margin-top:20px}.vc-theory p{font-size:14px;color:var(--vc-ink-secondary);margin:0 0 10px}.vc-theory ul,.vc-theory ol{font-size:14px;color:var(--vc-ink-secondary);margin:0 0 12px 18px;padding:0}.vc-theory li{margin-bottom:5px}.vc-formula-box{background:var(--vc-surface-alt);border:2px solid var(--vc-border);border-radius:6px;padding:14px 20px;font-family:var(--vc-mono);font-size:16px;font-weight:500;text-align:center;margin:14px 0;color:var(--vc-ink);overflow-x:auto}.vc-info-box{background:var(--vc-blue-light);border-left:3px solid var(--vc-blue);padding:14px 16px;border-radius:0 6px 6px 0;margin:14px 0}.vc-info-box p{margin:0;font-size:14px;color:var(--vc-ink-secondary)}.vc-warning-box{background:var(--vc-yellow-light);border-left:3px solid var(--vc-yellow);padding:14px 16px;border-radius:0 6px 6px 0;margin:14px 0}.vc-example-box{background:var(--vc-surface-alt);border:1px solid var(--vc-border-light);border-radius:6px;padding:16px;margin:14px 0}.vc-example-title{font-family:var(--vc-mono);font-size:13px;font-weight:600;color:var(--vc-accent);margin-bottom:8px}\n.vc-table{width:100%;border-collapse:collapse;margin:14px 0;font-size:13px}.vc-table th,.vc-table td{padding:10px 12px;border:1px solid var(--vc-border-light);text-align:left}.vc-table th{background:var(--vc-surface-alt);font-size:11px;font-weight:600;text-transform:uppercase;letter-spacing:0.05em;color:var(--vc-ink-secondary)}.vc-table tr:nth-child(even){background:var(--vc-surface-alt)}@media(max-width:600px){.vc-table{font-size:12px}.vc-table th,.vc-table td{padding:8px}}\n.vc-faq-list{display:flex;flex-direction:column;gap:8px;margin-top:20px}.vc-faq-item{border:1px solid var(--vc-border-light);border-radius:6px;overflow:hidden}.vc-faq-q{width:100%;display:flex;align-items:center;justify-content:space-between;gap:12px;padding:14px 16px;border:none;background:var(--vc-surface-alt);cursor:pointer;font-family:var(--vc-font);font-size:15px;font-weight:600;color:var(--vc-ink);text-align:left;transition:background 0.15s}.vc-faq-q:hover{background:#ede9e3}.vc-faq-chevron{width:16px;height:16px;color:var(--vc-ink-muted);flex-shrink:0;transition:transform 0.25s}.vc-faq-item.vc-open .vc-faq-chevron{transform:rotate(180deg)}.vc-faq-a{max-height:0;overflow:hidden;transition:max-height 0.35s}.vc-faq-item.vc-open .vc-faq-a{max-height:1200px}.vc-faq-a-inner{padding:14px 16px;border-top:1px solid var(--vc-border-light);font-size:14px;color:var(--vc-ink-secondary)}.vc-faq-a-inner p{margin:0 0 8px}.vc-faq-a-inner p:last-child{margin-bottom:0}.vc-faq-a-inner ul{margin:4px 0 8px 16px;padding:0}.vc-faq-a-inner li{margin-bottom:3px}\n.vc-related{display:flex;flex-wrap:wrap;gap:10px;margin-top:16px}.vc-related-link{display:inline-block;padding:8px 16px;font-family:var(--vc-font);font-size:13px;font-weight:600;background:var(--vc-surface-alt);color:var(--vc-ink-secondary);border:1px solid var(--vc-border);border-radius:var(--vc-radius);text-decoration:none;transition:all 0.15s}.vc-related-link:hover{border-color:var(--vc-accent);color:var(--vc-accent);background:var(--vc-accent-light)}\n.vc-promo{margin-top:24px;padding:20px;background:linear-gradient(135deg,var(--vc-accent-light),var(--vc-surface));border:1px solid var(--vc-border);border-radius:var(--vc-radius-lg);display:flex;align-items:center;gap:16px}.vc-promo-text{flex:1;font-size:14px;color:var(--vc-ink-secondary)}.vc-promo-text strong{color:var(--vc-ink)}.vc-promo-link{padding:8px 20px;font-size:13px;font-weight:700;color:#fff;background:var(--vc-accent);border-radius:var(--vc-radius);text-decoration:none;white-space:nowrap;transition:background 0.15s}.vc-promo-link:hover{background:var(--vc-accent-hover)}\n.vc-footer{text-align:center;padding:32px 16px;font-size:13px;color:var(--vc-ink-muted)}.vc-footer a{color:var(--vc-accent);text-decoration:none}.vc-footer a:hover{text-decoration:underline}.vc-footer-links{margin-top:8px;font-size:12px}.vc-footer-links a{margin:0 8px}\n@media print{.vc-section-body{max-height:none!important}.vc-faq-a{max-height:none!important}.vc-results{max-height:none!important;padding:24px!important}.vc-copy-btn,.vc-section-chevron,.vc-faq-chevron,.vc-presets,.vc-promo{display:none!important}}\n<\/style>\n\n<div class=\"vc-calculator\" id=\"vc-flexible-rotor\">\n<header class=\"vc-header\">\n    <p class=\"vc-header-eyebrow\">Free Engineering Tool \u2014 #029<\/p>\n    <h1 class=\"vc-header-title\">Flexible Rotor Balancing Tolerance<\/h1>\n    <p class=\"vc-header-subtitle\">Calculate per-mode unbalance tolerance for flexible rotors per ISO 21940-14. Enter rotor mass, maximum speed, balance grade, and number of flexible modes.<\/p>\n    <div class=\"vc-badges\"><span class=\"vc-badge\">ISO 21940-14<\/span><span class=\"vc-badge\">Modal Balancing<\/span><span class=\"vc-badge\">Per-Mode Tolerance<\/span><\/div>\n<\/header>\n\n<div class=\"vc-card\">\n    <form class=\"vc-form\" id=\"vc-form\" autocomplete=\"off\">\n        <div class=\"vc-form-grid\">\n            <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-mass\">Rotor Mass <span class=\"vc-label-hint\">(kg)<\/span><\/label><input class=\"vc-input\" type=\"number\" id=\"vc-mass\" inputmode=\"decimal\" step=\"any\" min=\"0.01\" placeholder=\"e.g. 500\" value=\"500\"><\/div>\n            <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-speed\">Max Operating Speed <span class=\"vc-label-hint\">(RPM)<\/span><\/label><input class=\"vc-input\" type=\"number\" id=\"vc-speed\" inputmode=\"decimal\" step=\"any\" min=\"1\" placeholder=\"e.g. 8000\" value=\"8000\"><\/div>\n            <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-grade\">Balance Grade G<\/label><select class=\"vc-select\" id=\"vc-grade\"><option value=\"6.3\">G6.3 \u2014 General machinery<\/option><option value=\"2.5\" selected>G2.5 \u2014 Turbomachinery<\/option><option value=\"1\">G1.0 \u2014 Precision<\/option><option value=\"0.4\">G0.4 \u2014 Ultra-precision<\/option><\/select><\/div>\n            <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-modes\">Number of Flexible Modes<\/label><select class=\"vc-select\" id=\"vc-modes\"><option value=\"1\">1 mode (3 planes)<\/option><option value=\"2\" selected>2 modes (4 planes)<\/option><option value=\"3\">3 modes (5 planes)<\/option><option value=\"4\">4 modes (6 planes)<\/option><\/select><\/div>\n        <\/div>\n        <div class=\"vc-presets\">\n            <div class=\"vc-presets-label\">Quick presets<\/div>\n            <button type=\"button\" class=\"vc-preset-btn\" data-m=\"500\" data-s=\"8000\" data-g=\"2.5\" data-md=\"2\">Gas turbine 500 kg \/ 8 kRPM<\/button>\n            <button type=\"button\" class=\"vc-preset-btn\" data-m=\"2000\" data-s=\"3600\" data-g=\"2.5\" data-md=\"1\">Generator 2000 kg \/ 3600 RPM<\/button>\n            <button type=\"button\" class=\"vc-preset-btn\" data-m=\"150\" data-s=\"15000\" data-g=\"1\" data-md=\"3\">Compressor 150 kg \/ 15 kRPM<\/button>\n        <\/div>\n    <\/form>\n\n    <div class=\"vc-results\" id=\"vc-results\">\n        <div class=\"vc-results-head\"><h2 class=\"vc-results-title\">Results<\/h2><button type=\"button\" class=\"vc-copy-btn\" id=\"vc-copy-btn\"><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<\/button><\/div>\n        <div class=\"vc-result-grid\">\n            <div class=\"vc-rcard vc-rcard-primary\"><div class=\"vc-rcard-label\">Total Permissible Unbalance (ISO 21940-11)<\/div><div class=\"vc-rcard-value\" id=\"vc-r-total\">\u2014<\/div><\/div>\n            <div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Per-Mode Tolerance<\/div><div class=\"vc-rcard-value\" id=\"vc-r-mode\">\u2014<\/div><\/div>\n            <div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Specific Unbalance (Eccentricity)<\/div><div class=\"vc-rcard-value\" id=\"vc-r-ecc\">\u2014<\/div><\/div>\n            <div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Correction Planes Needed<\/div><div class=\"vc-rcard-value\" id=\"vc-r-planes\">\u2014<\/div><\/div>\n            <div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Angular Velocity \u03c9<\/div><div class=\"vc-rcard-value\" id=\"vc-r-omega\">\u2014<\/div><\/div>\n        <\/div>\n    <\/div>\n<\/div>\n\n<div class=\"vc-section vc-open\" id=\"vc-sec-theory\">\n    <button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"true\"><span class=\"vc-section-toggle-text\"><span class=\"vc-section-icon\">\ud83d\udcd8<\/span><span class=\"vc-section-title\">Theory &amp; Formulas<\/span><\/span><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><\/button>\n    <div class=\"vc-section-body\">\n        <div class=\"vc-section-inner vc-theory\">\n            <h3>Total Permissible Unbalance<\/h3>\n            <p>Same as ISO 21940-11 (rigid rotor baseline):<\/p>\n            <div class=\"vc-formula-box\" id=\"vc-katex-1\"><\/div>\n            <h3>Per-Mode Tolerance Distribution<\/h3>\n            <p>For flexible rotors, the total tolerance is distributed among the rigid body mode (low-speed) and each flexible mode. A common engineering approach:<\/p>\n            <div class=\"vc-formula-box\" id=\"vc-katex-2\"><\/div>\n            <p>Where N is the number of flexible modes + 1 (rigid body mode). Each mode receives an equal share of the total unbalance budget.<\/p>\n            <h3>Correction Planes<\/h3>\n            <p>For N<sub>flex<\/sub> flexible modes, the minimum number of correction planes is:<\/p>\n            <div class=\"vc-formula-box\" id=\"vc-katex-3\"><\/div>\n            <div class=\"vc-example-box\">\n                <div class=\"vc-example-title\">Example \u2014 Gas Turbine Rotor<\/div>\n                <p><strong>Given:<\/strong> m = 500 kg, n = 8000 RPM, G2.5, 2 flexible modes<\/p>\n                <p>\u03c9 = 2\u03c0 \u00d7 8000\/60 = 837.76 rad\/s<\/p>\n                <p>e<sub>per<\/sub> = 2.5 \u00d7 1000 \/ 837.76 = <strong>2.984 \u03bcm<\/strong><\/p>\n                <p>U<sub>total<\/sub> = 2.984 \u00d7 500 = <strong>1492 g\u00b7mm<\/strong><\/p>\n                <p>Per-mode = 1492 \/ 3 = <strong>497 g\u00b7mm<\/strong> (3 modes: 1 rigid + 2 flexible)<\/p>\n                <p>Planes needed: 2 + 2 = <strong>4 correction planes<\/strong><\/p>\n            <\/div>\n            <div class=\"vc-warning-box\"><p style=\"margin:0;font-size:14px;color:var(--vc-ink-secondary);\"><strong>\u26a0\ufe0f Note:<\/strong> The equal-distribution model is a simplified approach. In practice, modal mass participation may differ, and actual tolerances should be verified by rotor dynamics analysis.<\/p><\/div>\n        <\/div>\n    <\/div>\n<\/div>\n\n<div class=\"vc-section\" id=\"vc-sec-related\">\n    <button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-toggle-text\"><span class=\"vc-section-icon\">\ud83d\udd17<\/span><span class=\"vc-section-title\">Related Calculators<\/span><\/span><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><\/button>\n    <div class=\"vc-section-body\"><div class=\"vc-section-inner\"><div class=\"vc-related\">\n        <a class=\"vc-related-link\" href=\"\/calculators\/residual-unbalance-iso1940\/\">Residual Unbalance (ISO 21940-11)<\/a>\n        <a class=\"vc-related-link\" href=\"\/calculators\/coupling-unbalance-tolerance\/\">Coupling Unbalance Tolerance<\/a>\n        <a class=\"vc-related-link\" href=\"\/calculators\/campbell-diagram-calculator\/\">Campbell Diagram<\/a>\n        <a class=\"vc-related-link\" href=\"\/calculators\/trial-weight-calculator\/\">Trial Weight Calculator<\/a>\n    <\/div><\/div><\/div>\n<\/div>\n\n<div class=\"vc-section\" id=\"vc-sec-faq\">\n    <button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-toggle-text\"><span class=\"vc-section-icon\">\u2753<\/span><span class=\"vc-section-title\">Frequently Asked Questions<\/span><\/span><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><\/button>\n    <div class=\"vc-section-body\"><div class=\"vc-section-inner\"><div class=\"vc-faq-list\" id=\"vc-faq-list\"><\/div><\/div><\/div>\n<\/div>\n\n<div class=\"vc-promo\"><div class=\"vc-promo-text\"><strong>Vibromera \u2014 Portable Balancing &amp; Vibration Analysis<\/strong><br>Balance flexible rotors in the field. Multi-plane balancing support with professional instruments. Used in 50+ countries.<\/div><a class=\"vc-promo-link\" href=\"https:\/\/vibromera.eu\/\" target=\"_blank\" rel=\"noopener\">Learn More<\/a><\/div>\n\n<footer class=\"vc-footer\"><p>\u00a9 2024\u20132025 <a href=\"https:\/\/vibromera.eu\/\">Vibromera<\/a> \u2014 Engineering Calculators<\/p><p style=\"font-size:12px;margin-top:4px;\">Based on ISO 21940-14. Last updated: February 2025<\/p><div class=\"vc-footer-links\"><a href=\"\/calculators\/engineering-calculators\/\">All Calculators<\/a><a href=\"https:\/\/vibromera.eu\/contacts\/\">Contact<\/a><\/div><\/footer>\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    var SLUG='flexible-rotor-balancing';\n    function $(id){return document.getElementById(id)}\n    function fmt(n,d){if(isNaN(n)||!isFinite(n))return '\u2014';var f=n.toFixed(d===undefined?1:d);return f.replace(\/(\\.\\d*?)0+$\/,'$1').replace(\/\\.$\/,'')}\n\n    function calculate(){\n        var mass=parseFloat($('vc-mass').value);\n        var speed=parseFloat($('vc-speed').value);\n        var grade=parseFloat($('vc-grade').value);\n        var modes=parseInt($('vc-modes').value);\n        if(!mass||mass<=0||!speed||speed<=0){$('vc-results').classList.remove('vc-visible');return}\n\n        var omega=2*Math.PI*speed\/60;\n        var eper=grade*1000\/omega;\n        var uTotal=eper*mass;\n        var nModes=modes+1; \/\/ rigid + flexible\n        var uMode=uTotal\/nModes;\n        var planes=modes+2;\n\n        $('vc-r-total').innerHTML=(uTotal>=1000?fmt(uTotal,0):fmt(uTotal,1))+' <span class=\"vc-rcard-unit\">g\u00b7mm<\/span>';\n        $('vc-r-mode').innerHTML=(uMode>=1000?fmt(uMode,0):fmt(uMode,1))+' <span class=\"vc-rcard-unit\">g\u00b7mm per mode<\/span>';\n        $('vc-r-ecc').innerHTML=fmt(eper,2)+' <span class=\"vc-rcard-unit\">\u03bcm<\/span>';\n        $('vc-r-planes').innerHTML=planes+' <span class=\"vc-rcard-unit\">planes<\/span>';\n        $('vc-r-omega').innerHTML=fmt(omega,1)+' <span class=\"vc-rcard-unit\">rad\/s<\/span>';\n        $('vc-results').classList.add('vc-visible');\n        saveHistory({mass:mass,speed:speed,grade:grade,modes:modes,uTotal:fmt(uTotal,1),_summary:mass+'kg, '+speed+'RPM, G'+grade+', '+modes+' modes \u2192 '+fmt(uMode,1)+' g\u00b7mm\/mode'});\n    }\n\n    document.querySelectorAll('.vc-preset-btn').forEach(function(btn){btn.addEventListener('click',function(e){e.preventDefault();$('vc-mass').value=this.getAttribute('data-m');$('vc-speed').value=this.getAttribute('data-s');$('vc-grade').value=this.getAttribute('data-g');$('vc-modes').value=this.getAttribute('data-md');calculate()})});\n    $('vc-form').addEventListener('input',calculate);$('vc-form').addEventListener('change',calculate);\n\n    $('vc-copy-btn').addEventListener('click',function(){var text='Flexible Rotor Balancing Tolerance\\n\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\\nMass: '+$('vc-mass').value+' kg\\nSpeed: '+$('vc-speed').value+' RPM\\nGrade: G'+$('vc-grade').value+'\\nModes: '+$('vc-modes').value+'\\nTotal: '+$('vc-r-total').textContent+'\\nPer-Mode: '+$('vc-r-mode').textContent+'\\nPlanes: '+$('vc-r-planes').textContent+'\\n\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\\nvibromera.eu\/calculators\/flexible-rotor-balancing\/';if(navigator.clipboard){navigator.clipboard.writeText(text).then(function(){var b=$('vc-copy-btn');b.classList.add('vc-copied');b.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!';setTimeout(function(){b.classList.remove('vc-copied');b.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'},2000)})}});\n\n    document.querySelectorAll('.vc-section-toggle').forEach(function(btn){btn.addEventListener('click',function(){var s=this.closest('.vc-section');s.classList.toggle('vc-open');this.setAttribute('aria-expanded',s.classList.contains('vc-open'))})});\n\n    var faqData=[\n        {q:'What is a flexible rotor?',a:'<p>A flexible rotor operates above one or more critical speeds, so it deforms during rotation. The unbalance distribution changes with speed as different bending modes are excited. Examples: turbine shafts, generators, high-speed compressor rotors.<\/p>'},\n        {q:'What is ISO 21940-14?',a:'<p>ISO 21940-14 addresses balance quality of flexible rotors. It extends ISO 21940-11 (rigid rotor) concepts to account for modal unbalance and speed-dependent deformation.<\/p>'},\n        {q:'How does flexible rotor balancing differ from rigid?',a:'<p>Rigid: 2-plane low-speed balancing. Flexible: N+2 planes for N modes with speed-dependent correction. Each mode has its own tolerance based on modal mass and the mode\\'s critical speed.<\/p>'},\n        {q:'What is modal unbalance?',a:'<p>Modal unbalance is the unbalance component exciting a specific bending mode. Each mode acts as an independent SDOF system. Total balance requires addressing each significant mode separately.<\/p>'},\n        {q:'How many correction planes do I need?',a:'<p>N<sub>flex<\/sub> + 2 planes. 1 mode \u2192 3 planes, 2 modes \u2192 4 planes, 3 modes \u2192 5 planes. Locations must effectively influence each mode shape.<\/p>'}\n    ];\n    var fl=$('vc-faq-list');faqData.forEach(function(f){fl.innerHTML+='<div class=\"vc-faq-item\"><button type=\"button\" class=\"vc-faq-q\"><span>'+f.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\">'+f.a+'<\/div><\/div><\/div>'});\n    fl.addEventListener('click',function(e){var b=e.target.closest('.vc-faq-q');if(!b)return;b.closest('.vc-faq-item').classList.toggle('vc-open')});\n\n    function saveHistory(entry){var key='vc_history_'+SLUG;var arr=[];try{arr=JSON.parse(localStorage.getItem(key))||[]}catch(e){}entry._ts=new Date().toISOString();arr.unshift(entry);if(arr.length>10)arr=arr.slice(0,10);try{localStorage.setItem(key,JSON.stringify(arr))}catch(e){}}\n\n    function renderFormulas(){try{\n        katex.render('U_{total}=\\\\frac{G\\\\times 1000}{\\\\omega}\\\\times m\\\\;[g\\\\cdot mm]',$('vc-katex-1'),{displayMode:true,throwOnError:false});\n        katex.render('U_{mode}=\\\\frac{U_{total}}{N_{flex}+1}',$('vc-katex-2'),{displayMode:true,throwOnError:false});\n        katex.render('N_{planes}=N_{flex}+2',$('vc-katex-3'),{displayMode:true,throwOnError:false});\n    }catch(e){}}\n\n    function init(){renderFormulas();calculate()}\n    if(document.readyState==='loading'){document.addEventListener('DOMContentLoaded',init)}else{init()}\n})();\n<\/script>\n","protected":false},"excerpt":{"rendered":"<p>ISO 21940-14 \u09aa\u09cd\u09b0\u09a4\u09bf \u09a8\u09ae\u09a8\u09c0\u09af\u09bc \u09b0\u09cb\u099f\u09b0 \u09ad\u09be\u09b0\u09b8\u09be\u09ae\u09cd\u09af \u09b8\u09b9\u09a8\u09b6\u09c0\u09b2\u09a4\u09be \u099c\u09a8\u09cd\u09af \u09ac\u09bf\u09a8\u09be\u09ae\u09c2\u09b2\u09cd\u09af\u09c7 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u0995\u09cd\u09af\u09be\u09b2\u0995\u09c1\u09b2\u09c7\u099f\u09b0\u0964 \u09b0\u09cb\u099f\u09b0 \u09ad\u09b0, \u09b8\u09b0\u09cd\u09ac\u09be\u09a7\u09bf\u0995 \u0997\u09a4\u09bf \u098f\u09ac\u0982 \u09a8\u09ae\u09a8\u09c0\u09af\u09bc \u09ae\u09cb\u09a1\u09c7\u09b0 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u0989\u09aa\u09b0 \u09ad\u09bf\u09a4\u09cd\u09a4\u09bf \u0995\u09b0\u09c7 \u09aa\u09cd\u09b0\u09a4\u09bf-\u09ae\u09cb\u09a1 \u09ad\u09be\u09b0\u09b8\u09be\u09ae\u09cd\u09af\u09b9\u09c0\u09a8\u09a4\u09be \u09b8\u09b9\u09a8\u09b6\u09c0\u09b2\u09a4\u09be \u0997\u09a3\u09a8\u09be \u0995\u09b0\u09c1\u09a8\u0964<\/p>","protected":false},"featured_media":0,"template":"","meta":{"ai_generated_summary":"","footnotes":""},"categories":[],"tags":[],"class_list":["post-100100","calculator","type-calculator","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/calculator\/100100","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/calculator"}],"about":[{"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/types\/calculator"}],"version-history":[{"count":1,"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/calculator\/100100\/revisions"}],"predecessor-version":[{"id":100402,"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/calculator\/100100\/revisions\/100402"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/media?parent=100100"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/categories?post=100100"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/tags?post=100100"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}