{"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":"2026-07-12T00:55:36","modified_gmt":"2026-07-12T00:55:36","slug":"damping-ratio-calculator","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/ro\/calculators\/damping-ratio-calculator\/","title":{"rendered":"Calculator raport amortizare | Decrement logaritmic \u0219i l\u0103\u021bime de band\u0103 la jum\u0103tate de putere | Instrument gratuit #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        $('vc-results').classList.add('vc-visible');\n        drawViz(zeta);\n\n        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Calculator\\n\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\\nMethod: '+$('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>Calculator online gratuit al raportului de amortizare. Determina\u021bi raportul de amortizare \u03b6 prin decrementare logaritmic\u0103 (amplitudini de v\u00e2rf succesive) sau prin metoda l\u0103\u021bimii de band\u0103 la jum\u0103tate de putere. Calcul automat cu formule.<\/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\/ro\/wp-json\/wp\/v2\/calculator\/100073","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/ro\/wp-json\/wp\/v2\/calculator"}],"about":[{"href":"https:\/\/vibromera.eu\/ro\/wp-json\/wp\/v2\/types\/calculator"}],"version-history":[{"count":3,"href":"https:\/\/vibromera.eu\/ro\/wp-json\/wp\/v2\/calculator\/100073\/revisions"}],"predecessor-version":[{"id":102364,"href":"https:\/\/vibromera.eu\/ro\/wp-json\/wp\/v2\/calculator\/100073\/revisions\/102364"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/ro\/wp-json\/wp\/v2\/media?parent=100073"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/ro\/wp-json\/wp\/v2\/categories?post=100073"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/ro\/wp-json\/wp\/v2\/tags?post=100073"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}