{"id":100073,"date":"2026-02-15T20:15:07","date_gmt":"2026-02-15T20:15:07","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100073"},"modified":"2026-02-15T20:15:07","modified_gmt":"2026-02-15T20:15:07","slug":"damping-ratio-calculator","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/uz\/calculators\/damping-ratio-calculator\/","title":{"rendered":"Damping Ratio Calculator | Log Decrement &amp; Half-Power Bandwidth | Free Tool #066"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Damping Ratio Calculator\",\"alternateName\":[\"Log Decrement Calculator\",\"Half-Power Bandwidth Calculator\",\"Zeta Calculator\"],\"description\":\"Free online calculator for damping ratio from logarithmic decrement or half-power bandwidth method.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/damping-ratio-calculator\/\",\"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\":[\"Logarithmic decrement method\",\"Half-power bandwidth method\",\"Multiple cycle averaging\",\"Auto-calculation\",\"Quality factor Q\"],\"keywords\":\"damping ratio, log decrement, half-power bandwidth, zeta, vibration, modal analysis\"}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[\n{\"@type\":\"Question\",\"name\":\"What is damping ratio \u03b6?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Damping ratio \u03b6 (zeta) is the ratio of actual damping to critical damping. \u03b6 = 0 means undamped, \u03b6 = 1 is critically damped, 0 < \u03b6 < 1 is underdamped (oscillatory). Most mechanical systems have \u03b6 between 0.01 and 0.1.\"}},\n{\"@type\":\"Question\",\"name\":\"How does the logarithmic decrement method work?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Measure successive peak amplitudes x\u2081 and x\u2082 in free vibration. The log decrement \u03b4 = ln(x\u2081\/x\u2082). For n cycles: \u03b4 = (1\/n)ln(x\u2081\/x\u2099\u208a\u2081). Then \u03b6 = \u03b4\/\u221a(4\u03c0\u00b2 + \u03b4\u00b2) \u2248 \u03b4\/(2\u03c0) for small damping.\"}},\n{\"@type\":\"Question\",\"name\":\"How does the half-power bandwidth method work?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"From a frequency response function, measure the natural frequency fn and the two frequencies f\u2081, f\u2082 where amplitude drops to 1\/\u221a2 of peak (\u22123 dB). Then \u03b6 = (f\u2082 \u2212 f\u2081)\/(2fn) = \u0394f\/(2fn).\"}},\n{\"@type\":\"Question\",\"name\":\"Which method is more accurate?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Log decrement is better for lightly damped systems (\u03b6 < 0.1) with clear free vibration decay. Half-power bandwidth works well for forced vibration measurements and moderate damping. For very light damping, averaging over many cycles improves log decrement accuracy.\"}},\n{\"@type\":\"Question\",\"name\":\"What is the quality factor Q?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Quality factor Q = 1\/(2\u03b6). A high Q indicates low damping and a sharp resonance peak. Q = 50 means \u03b6 = 0.01. Q is commonly used in acoustics, electronics, and structural dynamics.\"}}\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\":\"Damping Ratio Calculator\",\"item\":\"https:\/\/vibromera.eu\/calculators\/damping-ratio-calculator\/\"}\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 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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 .15s}.vc-promo-link:hover{background:var(--vc-accent-hover)}.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}@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-damping-ratio\">\n\n<header class=\"vc-header\">\n    <p class=\"vc-header-eyebrow\">Free Engineering Tool #066<\/p>\n    <h1 class=\"vc-header-title\">Damping Ratio Calculator<\/h1>\n    <p class=\"vc-header-subtitle\">Determine damping ratio \u03b6 from logarithmic decrement (successive peak amplitudes) or half-power bandwidth method. Supports multi-cycle averaging.<\/p>\n    <div class=\"vc-badges\"><span class=\"vc-badge\">Log Decrement<\/span><span class=\"vc-badge\">Half-Power Bandwidth<\/span><span class=\"vc-badge\">Auto-Calc<\/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 vc-field-full\">\n                <label class=\"vc-label\" for=\"vc-method\">Method<\/label>\n                <select class=\"vc-select\" id=\"vc-method\">\n                    <option value=\"logdec\" selected>Logarithmic Decrement (free vibration peaks)<\/option>\n                    <option value=\"bandwidth\">Half-Power Bandwidth (frequency response)<\/option>\n                <\/select>\n            <\/div>\n\n            <!-- Log decrement inputs -->\n            <div class=\"vc-field\" id=\"vc-f-x1\">\n                <label class=\"vc-label\" for=\"vc-x1\">Peak Amplitude x\u2081<\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-x1\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 10\" value=\"10\">\n            <\/div>\n            <div class=\"vc-field\" id=\"vc-f-x2\">\n                <label class=\"vc-label\" for=\"vc-x2\">Peak Amplitude x\u2082 <span class=\"vc-label-hint\">(after n cycles)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-x2\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 6\" value=\"6\">\n            <\/div>\n            <div class=\"vc-field\" id=\"vc-f-ncyc\">\n                <label class=\"vc-label\" for=\"vc-ncyc\">Number of Cycles n <span class=\"vc-label-hint\">(between x\u2081 and x\u2082)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-ncyc\" inputmode=\"decimal\" step=\"1\" min=\"1\" placeholder=\"e.g. 1\" value=\"1\">\n            <\/div>\n\n            <!-- Bandwidth inputs -->\n            <div class=\"vc-field\" id=\"vc-f-fn\" style=\"display:none\">\n                <label class=\"vc-label\" for=\"vc-fn\">Natural Frequency f<sub>n<\/sub> <span class=\"vc-label-hint\">(Hz)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-fn\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 100\" value=\"100\">\n            <\/div>\n            <div class=\"vc-field\" id=\"vc-f-f1\" style=\"display:none\">\n                <label class=\"vc-label\" for=\"vc-f1\">Lower \u22123 dB Frequency f\u2081 <span class=\"vc-label-hint\">(Hz)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-f1\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 97.5\" value=\"97.5\">\n            <\/div>\n            <div class=\"vc-field\" id=\"vc-f-f2\" style=\"display:none\">\n                <label class=\"vc-label\" for=\"vc-f2\">Upper \u22123 dB Frequency f\u2082 <span class=\"vc-label-hint\">(Hz)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-f2\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 102.5\" value=\"102.5\">\n            <\/div>\n        <\/div>\n\n        <div class=\"vc-presets\">\n            <div class=\"vc-presets-label\">Quick presets<\/div>\n            <button type=\"button\" class=\"vc-preset-btn\" data-method=\"logdec\" data-x1=\"10\" data-x2=\"6\" data-n=\"1\">Log dec: 10\u21926 (1 cycle)<\/button>\n            <button type=\"button\" class=\"vc-preset-btn\" data-method=\"logdec\" data-x1=\"5\" data-x2=\"1.5\" data-n=\"5\">Log dec: 5\u21921.5 (5 cycles)<\/button>\n            <button type=\"button\" class=\"vc-preset-btn\" data-method=\"bandwidth\" data-fn=\"100\" data-f1=\"97.5\" data-f2=\"102.5\">Bandwidth: fn=100, \u0394f=5 Hz<\/button>\n        <\/div>\n    <\/form>\n\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\"><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>\n        <\/div>\n        <div class=\"vc-result-grid\">\n            <div class=\"vc-rcard vc-rcard-primary\">\n                <div class=\"vc-rcard-label\">Damping Ratio \u03b6<\/div>\n                <div class=\"vc-rcard-value\" id=\"vc-r-zeta\">\u2014<\/div>\n            <\/div>\n            <div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Damping (%)<\/div><div class=\"vc-rcard-value\" id=\"vc-r-pct\">\u2014<\/div><\/div>\n            <div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Log Decrement \u03b4 \/ Bandwidth \u0394f<\/div><div class=\"vc-rcard-value\" id=\"vc-r-delta\">\u2014<\/div><\/div>\n            <div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Quality Factor Q<\/div><div class=\"vc-rcard-value\" id=\"vc-r-Q\">\u2014<\/div><\/div>\n            <div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Damping Classification<\/div><div class=\"vc-rcard-value\" id=\"vc-r-class\">\u2014<\/div><\/div>\n        <\/div>\n        <div class=\"vc-viz\" id=\"vc-viz\" style=\"text-align:center;margin:16px 0 8px\"><\/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\"><div class=\"vc-section-inner vc-theory\">\n        <h3>Logarithmic Decrement Method<\/h3>\n        <div class=\"vc-formula-box\" id=\"vc-katex-1\"><\/div>\n        <div class=\"vc-formula-box\" id=\"vc-katex-2\"><\/div>\n        <h3>Half-Power Bandwidth Method<\/h3>\n        <div class=\"vc-formula-box\" id=\"vc-katex-3\"><\/div>\n        <h3>Quality Factor<\/h3>\n        <div class=\"vc-formula-box\" id=\"vc-katex-4\"><\/div>\n\n        <div class=\"vc-example-box\">\n            <div class=\"vc-example-title\">Example \u2014 Log Decrement<\/div>\n            <p><strong>Given:<\/strong> x\u2081 = 10 mm, x\u2082 = 6 mm, n = 1 cycle<\/p>\n            <p>\u03b4 = (1\/1) \u00d7 ln(10\/6) = ln(1.667) = <strong>0.5108<\/strong><\/p>\n            <p>\u03b6 = 0.5108 \/ \u221a(4\u03c0\u00b2 + 0.5108\u00b2) = 0.5108 \/ 6.304 = <strong>0.0811<\/strong> (8.11%)<\/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 log decrement approximation \u03b6 \u2248 \u03b4\/(2\u03c0) is accurate to within 1% for \u03b6 < 0.1. For higher damping, use the exact formula.<\/p><\/div>\n    <\/div><\/div>\n<\/div>\n\n<div class=\"vc-section\" id=\"vc-sec-related\"><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><div class=\"vc-section-body\"><div class=\"vc-section-inner\"><div class=\"vc-related\">\n    <a class=\"vc-related-link\" href=\"\/calculators\/vibration-mount-selection\/\">Vibration Mount Selection<\/a>\n    <a class=\"vc-related-link\" href=\"\/calculators\/machine-vibration-isolation\/\">Machine Vibration Isolation<\/a>\n    <a class=\"vc-related-link\" href=\"\/calculators\/rubber-mount-calculator\/\">Rubber Mount Calculator<\/a>\n    <a class=\"vc-related-link\" href=\"\/calculators\/residual-unbalance-iso1940\/\">Residual Unbalance (ISO 1940)<\/a>\n<\/div><\/div><\/div><\/div>\n\n<div class=\"vc-section\" id=\"vc-sec-faq\"><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><div class=\"vc-section-body\"><div class=\"vc-section-inner\"><div class=\"vc-faq-list\" id=\"vc-faq-list\"><\/div><\/div><\/div><\/div>\n\n<div class=\"vc-promo\"><div class=\"vc-promo-text\"><strong>Vibromera \u2014 Portable Balancing &amp; Vibration Analysis<\/strong><br>Professional field balancing instruments and vibration analysis software. Used in 50+ countries.<\/div><a class=\"vc-promo-link\" href=\"https:\/\/vibromera.eu\/\" target=\"_blank\" rel=\"noopener\">Learn More<\/a><\/div>\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;\">Damping ratio from log decrement and half-power bandwidth. 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\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='damping-ratio-calculator';\n    function $(id){return document.getElementById(id)}\n    function fmt(n,d){if(isNaN(n)||!isFinite(n))return'\u2014';var f=n.toFixed(d===undefined?4:d);return f.replace(\/(\\.\\d*?)0+$\/,'$1').replace(\/\\.$\/,'')}\n\n    function toggleMethod(){\n        var m=$('vc-method').value;\n        var isLog=m==='logdec';\n        $('vc-f-x1').style.display=isLog?'flex':'none';\n        $('vc-f-x2').style.display=isLog?'flex':'none';\n        $('vc-f-ncyc').style.display=isLog?'flex':'none';\n        $('vc-f-fn').style.display=isLog?'none':'flex';\n        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class=\"vc-rcard-unit\">(log dec \u03b4)<\/span>';\n        } else {\n            var fn=parseFloat($('vc-fn').value);\n            var f1=parseFloat($('vc-f1').value);\n            var f2=parseFloat($('vc-f2').value);\n            if(!fn||fn<=0||!f1||f1<=0||!f2||f2<=0||f1>=f2||f1>=fn||f2<=fn){$('vc-results').classList.remove('vc-visible');return}\n            var df=f2-f1;\n            zeta=df\/(2*fn);\n            delta=2*Math.PI*zeta\/Math.sqrt(1-zeta*zeta);\n            $('vc-r-delta').innerHTML=fmt(df,2)+' <span class=\"vc-rcard-unit\">Hz (\u0394f bandwidth)<\/span>';\n        }\n\n        var Q=1\/(2*zeta);\n        $('vc-r-zeta').innerHTML=fmt(zeta,4);\n        $('vc-r-pct').innerHTML=fmt(zeta*100,2)+' <span class=\"vc-rcard-unit\">%<\/span>';\n        $('vc-r-Q').innerHTML=fmt(Q,1);\n        $('vc-r-class').innerHTML='<span style=\"font-size:16px\">'+classify(zeta)+'<\/span>';\n        $('vc-results').classList.add('vc-visible');\n        drawViz(zeta);\n\n        saveHistory({method:method,zeta:fmt(zeta,4),_summary:'\u03b6='+fmt(zeta,4)+' ('+method+')'});\n    }\n\n    $('vc-method').addEventListener('change',function(){toggleMethod();calculate()});\n\n    document.querySelectorAll('.vc-preset-btn').forEach(function(btn){btn.addEventListener('click',function(e){e.preventDefault();\n        var m=this.getAttribute('data-method');$('vc-method').value=m;toggleMethod();\n        if(m==='logdec'){$('vc-x1').value=this.getAttribute('data-x1');$('vc-x2').value=this.getAttribute('data-x2');$('vc-ncyc').value=this.getAttribute('data-n')}\n        else{$('vc-fn').value=this.getAttribute('data-fn');$('vc-f1').value=this.getAttribute('data-f1');$('vc-f2').value=this.getAttribute('data-f2')}\n        calculate();\n    })});\n\n    $('vc-form').addEventListener('input',calculate);$('vc-form').addEventListener('change',calculate);\n\n    $('vc-copy-btn').addEventListener('click',function(){var text='Damping Ratio Calculator\\n\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\\nMethod: '+$('vc-method').value+'\\n\u03b6: '+$('vc-r-zeta').textContent+'\\nDamping: '+$('vc-r-pct').textContent+'\\nQ: '+$('vc-r-Q').textContent+'\\n\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\\nvibromera.eu\/calculators\/'+SLUG+'\/';if(navigator.clipboard){navigator.clipboard.writeText(text).then(function(){var b=$('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 damping ratio \u03b6?',a:'<p>Damping ratio \u03b6 (zeta) is the ratio of actual damping to critical damping. \u03b6 = 0 is undamped, \u03b6 = 1 is critically damped, 0 &lt; \u03b6 &lt; 1 is underdamped. Most structures have \u03b6 = 0.01\u20130.10.<\/p>'},\n        {q:'How does the logarithmic decrement method work?',a:'<p>Measure successive peak amplitudes x\u2081 and x\u2082 in free vibration decay. \u03b4 = (1\/n)ln(x\u2081\/x<sub>n+1<\/sub>). Then \u03b6 = \u03b4\/\u221a(4\u03c0\u00b2 + \u03b4\u00b2).<\/p>'},\n        {q:'How does the half-power bandwidth method work?',a:'<p>From FRF: measure f<sub>n<\/sub> and the \u22123 dB frequencies f\u2081, f\u2082. Then \u03b6 = (f\u2082 \u2212 f\u2081)\/(2f<sub>n<\/sub>).<\/p>'},\n        {q:'Which method is more accurate?',a:'<p>Log decrement for lightly damped systems with clear free decay. Bandwidth for forced vibration with moderate damping.<\/p>'},\n        {q:'What is the quality factor Q?',a:'<p>Q = 1\/(2\u03b6). High Q = low damping, sharp resonance. Q = 50 \u2192 \u03b6 = 0.01. Used in acoustics, electronics, and structural dynamics.<\/p>'}\n    ];\n    var faqList=$('vc-faq-list');faqData.forEach(function(faq){faqList.innerHTML+='<div class=\"vc-faq-item\"><button type=\"button\" class=\"vc-faq-q\"><span>'+faq.q+'<\/span><svg class=\"vc-faq-chevron\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><path d=\"M6 9l6 6 6-6\"\/><\/svg><\/button><div class=\"vc-faq-a\"><div class=\"vc-faq-a-inner\">'+faq.a+'<\/div><\/div><\/div>'});\n    faqList.addEventListener('click',function(e){var b=e.target.closest('.vc-faq-q');if(b)b.closest('.vc-faq-item').classList.toggle('vc-open')});\n\n    function drawViz(zeta){\n        var c=$('vc-viz'),W=380,H=160,pad=30,top=20,bot=16;\n        var svg='<svg width=\"'+W+'\" height=\"'+H+'\" viewBox=\"0 0 '+W+' '+H+'\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">';\n        var cx=W\/2,cy=H\/2;\n        var chartW=W-2*pad,chartH=H-top-bot;\n        var baseY=top+chartH\/2;\n        \/\/ Axis\n        svg+='<line x1=\"'+pad+'\" y1=\"'+baseY+'\" x2=\"'+(W-pad)+'\" y2=\"'+baseY+'\" stroke=\"#d9d4cc\" stroke-width=\"1\"\/>';\n        svg+='<line x1=\"'+pad+'\" y1=\"'+top+'\" x2=\"'+pad+'\" y2=\"'+(top+chartH)+'\" stroke=\"#d9d4cc\" stroke-width=\"1\"\/>';\n        \/\/ Decaying sine wave\n        var wd=Math.sqrt(1-Math.min(zeta*zeta,0.999));\/\/ damped frequency ratio\n        var nCycles=6,pts=[],envU=[],envL=[];\n        var tMax=nCycles*2*Math.PI\/wd;\n        for(var i=0;i<=200;i++){\n            var t=i\/200*tMax;\n            var env=Math.exp(-zeta*t);\n            var y=env*Math.cos(wd*t);\n            var px=pad+i\/200*chartW;\n            var py=baseY-y*chartH\/2*0.85;\n            pts.push(px.toFixed(1)+','+py.toFixed(1));\n            envU.push(px.toFixed(1)+','+(baseY-env*chartH\/2*0.85).toFixed(1));\n            envL.push(px.toFixed(1)+','+(baseY+env*chartH\/2*0.85).toFixed(1));\n        }\n        \/\/ Envelope (dashed)\n        svg+='<polyline points=\"'+envU.join(' ')+'\" fill=\"none\" stroke=\"#e74c3c\" stroke-width=\"1\" stroke-dasharray=\"4,3\" opacity=\"0.5\"\/>';\n        svg+='<polyline points=\"'+envL.join(' ')+'\" fill=\"none\" stroke=\"#e74c3c\" stroke-width=\"1\" stroke-dasharray=\"4,3\" opacity=\"0.5\"\/>';\n        \/\/ Wave\n        svg+='<polyline points=\"'+pts.join(' ')+'\" fill=\"none\" stroke=\"#3578b2\" stroke-width=\"2\"\/>';\n        \/\/ Labels\n        svg+='<text x=\"'+cx+'\" y=\"12\" text-anchor=\"middle\" fill=\"#2c3e50\" font-size=\"11\" font-weight=\"600\">Decaying Response (\\u03B6 = '+zeta.toFixed(3)+')<\/text>';\n        svg+='<text x=\"'+(W-pad)+'\" y=\"'+(baseY+14)+'\" text-anchor=\"end\" fill=\"#8a857e\" font-size=\"9\">time<\/text>';\n        svg+='<text x=\"'+(pad-4)+'\" y=\"'+(top+8)+'\" text-anchor=\"end\" fill=\"#8a857e\" font-size=\"9\">x<\/text>';\n        svg+='<text x=\"'+(pad+chartW*0.15)+'\" y=\"'+(baseY-chartH*0.35)+'\" fill=\"#e74c3c\" font-size=\"9\">e\\u207B\\u1DBF\\u1D57<\/text>';\n        svg+='<\/svg>';\n        c.innerHTML=svg;\n    }\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('\\\\delta = \\\\frac{1}{n} \\\\ln\\\\frac{x_1}{x_{n+1}}',$('vc-katex-1'),{displayMode:true,throwOnError:false});\n        katex.render('\\\\zeta = \\\\frac{\\\\delta}{\\\\sqrt{4\\\\pi^2 + \\\\delta^2}}',$('vc-katex-2'),{displayMode:true,throwOnError:false});\n        katex.render('\\\\zeta = \\\\frac{\\\\Delta f}{2 f_n} = \\\\frac{f_2 - f_1}{2 f_n}',$('vc-katex-3'),{displayMode:true,throwOnError:false});\n        katex.render('Q = \\\\frac{1}{2\\\\zeta}',$('vc-katex-4'),{displayMode:true,throwOnError:false});\n    }catch(e){}}\n\n    function init(){toggleMethod();renderFormulas();calculate()}\n    if(document.readyState==='loading')document.addEventListener('DOMContentLoaded',init);else init();\n})();\n<\/script>\n","protected":false},"excerpt":{"rendered":"<p>Free online damping ratio calculator. Determine damping ratio \u03b6 from logarithmic decrement (successive peak amplitudes) or half-power bandwidth method. Auto-calculation with formulas.<\/p>","protected":false},"featured_media":0,"template":"","meta":{"ai_generated_summary":"","footnotes":""},"categories":[],"tags":[],"class_list":["post-100073","calculator","type-calculator","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/vibromera.eu\/uz\/wp-json\/wp\/v2\/calculator\/100073","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/uz\/wp-json\/wp\/v2\/calculator"}],"about":[{"href":"https:\/\/vibromera.eu\/uz\/wp-json\/wp\/v2\/types\/calculator"}],"version-history":[{"count":1,"href":"https:\/\/vibromera.eu\/uz\/wp-json\/wp\/v2\/calculator\/100073\/revisions"}],"predecessor-version":[{"id":100388,"href":"https:\/\/vibromera.eu\/uz\/wp-json\/wp\/v2\/calculator\/100073\/revisions\/100388"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/uz\/wp-json\/wp\/v2\/media?parent=100073"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/uz\/wp-json\/wp\/v2\/categories?post=100073"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/uz\/wp-json\/wp\/v2\/tags?post=100073"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}