{"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\/tr\/calculators\/damping-ratio-calculator\/","title":{"rendered":"SDOF Damping Ratio | Log Decrement &#038; Low-Damping FRF"},"content":{"rendered":"\r\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"SDOF Damping-Ratio Extraction\",\"description\":\"Calculate exact viscous damping ratio from free-decay logarithmic decrement or a low-damping half-power FRF estimate, with the method assumptions and measured-frequency definitions enforced.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/damping-ratio-calculator\/\",\"applicationCategory\":\"Engineering 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.vc-faq-chevron{transform:rotate(180deg)}.vc-faq-a{max-height:0;overflow:hidden;transition:max-height .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}.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 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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}}\r\n<\/style>\r\n<div class=\"vc-calculator\" id=\"vc-damping-ratio\"><header class=\"vc-header\"><p class=\"vc-header-eyebrow\">Linear viscous SDOF methods<\/p><h1 class=\"vc-header-title\">SDOF Damping-Ratio Extraction<\/h1><p class=\"vc-header-subtitle\">Use exact logarithmic decrement for an unforced exponential free decay, or the conventional half-power FRF estimate only for a stationary, isolated, lightly damped mode.<\/p><div class=\"vc-badges\"><span class=\"vc-badge\">Log decrement: exact SDOF<\/span><span class=\"vc-badge\">FRF bandwidth: \u03b6 &lt; 0.05<\/span><span class=\"vc-badge\">Source required<\/span><\/div><\/header>\r\n<div class=\"vc-card\"><form class=\"vc-form\" id=\"vc-form\" autocomplete=\"off\"><div class=\"vc-form-grid\"><div class=\"vc-field vc-field-full\"><label class=\"vc-label\" for=\"vc-method\">Extraction method<\/label><select class=\"vc-select\" id=\"vc-method\"><option value=\"logdec\">Logarithmic decrement: free-decay peaks<\/option><option value=\"bandwidth\">Half-power bandwidth: measured FRF estimate<\/option><\/select><\/div><div class=\"vc-field vc-log\"><label class=\"vc-label\" for=\"vc-a1\">Earlier same-signed peak magnitude A\u2081 <span class=\"vc-label-hint\">(&gt;0)<\/span><\/label><input class=\"vc-input\" id=\"vc-a1\" type=\"text\" inputmode=\"decimal\"><\/div><div class=\"vc-field vc-log\"><label class=\"vc-label\" for=\"vc-a2\">Later same-signed peak magnitude A\u2082 <span class=\"vc-label-hint\">(0&lt;A\u2082&lt;A\u2081)<\/span><\/label><input class=\"vc-input\" id=\"vc-a2\" type=\"text\" inputmode=\"decimal\"><\/div><div class=\"vc-field vc-log\"><label class=\"vc-label\" for=\"vc-cycles\">Damped periods N between peaks <span class=\"vc-label-hint\">(integer \u22651)<\/span><\/label><input class=\"vc-input\" id=\"vc-cycles\" type=\"text\" inputmode=\"numeric\"><\/div><div class=\"vc-field vc-bw\" style=\"display:none\"><label class=\"vc-label\" for=\"vc-fp\">Measured response peak frequency f<sub>p<\/sub> <span class=\"vc-label-hint\">(Hz; &gt;0)<\/span><\/label><input class=\"vc-input\" id=\"vc-fp\" type=\"text\" inputmode=\"decimal\"><\/div><div class=\"vc-field vc-bw\" style=\"display:none\"><label class=\"vc-label\" for=\"vc-fl\">Lower 1\/\u221a2-amplitude frequency f<sub>l<\/sub> <span class=\"vc-label-hint\">(Hz)<\/span><\/label><input class=\"vc-input\" id=\"vc-fl\" type=\"text\" inputmode=\"decimal\"><\/div><div class=\"vc-field vc-bw\" style=\"display:none\"><label class=\"vc-label\" for=\"vc-fu\">Upper 1\/\u221a2-amplitude frequency f<sub>u<\/sub> <span class=\"vc-label-hint\">(Hz)<\/span><\/label><input class=\"vc-input\" id=\"vc-fu\" type=\"text\" inputmode=\"decimal\"><\/div><div class=\"vc-field vc-field-full\"><label class=\"vc-label\" for=\"vc-source\">Data source, response type, mode isolation and model\/measurement basis<\/label><input class=\"vc-input\" id=\"vc-source\" type=\"text\" maxlength=\"300\"><\/div><\/div><p id=\"vc-error\" role=\"alert\" style=\"margin:.75rem 0 0;color:#b42318\"><\/p><div class=\"vc-warning-box\"><p style=\"margin:0\" id=\"vc-gate\"><strong>Log-decrement gate:<\/strong> subtract equilibrium\/offset; use magnitudes of same-signed peaks separated by exactly N damped periods in an unforced, linear exponential SDOF decay. Beating, multiple modes, nonlinear damping or forced response invalidate the direct result.<\/p><\/div><\/form><div class=\"vc-results\" id=\"vc-results\"><div class=\"vc-results-head\"><h2 class=\"vc-results-title\">Method result<\/h2><button type=\"button\" class=\"vc-copy-btn\" id=\"vc-copy-btn\">Copy<\/button><\/div><div class=\"vc-result-grid\"><div class=\"vc-rcard vc-rcard-primary\"><div class=\"vc-rcard-label\">Viscous damping ratio \u03b6<\/div><div class=\"vc-rcard-value\" id=\"vc-r-zeta\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Percent critical damping<\/div><div class=\"vc-rcard-value\" id=\"vc-r-pct\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\" id=\"vc-r-q-label\">Logarithmic decrement \u03b4<\/div><div class=\"vc-rcard-value\" id=\"vc-r-quantity\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\">SDOF quality factor Q=1\/(2\u03b6)<\/div><div class=\"vc-rcard-value\" id=\"vc-r-Q\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Method check<\/div><div class=\"vc-rcard-value\" id=\"vc-r-check\">\u2014<\/div><\/div><\/div><p id=\"vc-r-note\" style=\"margin:0;color:var(--vc-ink-secondary)\"><\/p><\/div><\/div>\r\n<div class=\"vc-section vc-open\"><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\">Equations, exactness and measurement boundary<\/span><\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner vc-theory\"><div class=\"vc-formula-box\">\u03b4=(1\/N)ln(A\u2081\/A\u2082); &nbsp; \u03b6=\u03b4\/\u221a((2\u03c0)\u00b2+\u03b4\u00b2) &nbsp; [exact for stated unforced SDOF decay]<\/div><div class=\"vc-formula-box\">\u03b6 \u2248 (f<sub>u<\/sub>\u2212f<sub>l<\/sub>)\/(2f<sub>p<\/sub>) &nbsp; only for \u03b6&lt;0.05 using a measured magnitude FRF<\/div><div class=\"vc-formula-box\">Q=1\/(2\u03b6) &nbsp; for the stated SDOF quality-factor relationship<\/div><p><a href=\"https:\/\/ntrs.nasa.gov\/citations\/20170005173\" target=\"_blank\" rel=\"noopener\">NASA\/TM-2016-218227, Extracting Damping Ratio from Dynamic Data and Numerical Solutions<\/a>, Table 1 and sections 1.1\/A.2, identifies the common SDOF assumption. It marks logarithmic decrement exact for an unforced SDOF exponential decay and the measured-spectrum half-power formula approximate for \u03b6&lt;0.05.<\/p><p>For an FFT magnitude FRF, f<sub>l<\/sub> and f<sub>u<\/sub> are selected at 1\/\u221a2 of peak amplitude. For a PSD\/quadratic quantity, use one-half of the peak. The denominator is the measured peak-center frequency f<sub>p<\/sub> in this approximation\u2014not an automatically known undamped natural frequency. NASA distinguishes the exact power-curve relationship from the commonly measured FRF approximation.<\/p><p><a href=\"https:\/\/ocw.mit.edu\/courses\/2-003sc-engineering-dynamics-fall-2011\/resources\/estimation-of-natural-frequencies-and-damping-ratios-from-measured-response-the-logarithmic-decrement\/\" target=\"_blank\" rel=\"noopener\">MIT OpenCourseWare<\/a> separately teaches damping estimation from measured free response for an SDOF system.<\/p><div class=\"vc-warning-box\"><p style=\"margin:0\"><strong>No universal modal result:<\/strong> closely spaced modes, leakage, inadequate frequency resolution, noise, non-proportional\/non-viscous or nonlinear damping, changing boundary conditions and non-stationarity require curve fitting or another validated modal-identification method. This page does not claim ISO 1940\/21940 compliance.<\/p><\/div><\/div><\/div><\/div><footer class=\"vc-footer\"><p>\u00a9 2024\u20132026 <a href=\"https:\/\/vibromera.eu\/\">Vibromera<\/a><\/p><p>NASA-sourced SDOF extraction formulas with enforced method limits. Scientific review: July 2026.<\/p><\/footer><\/div>\r\n<script>(function(){'use strict';function $(i){return document.getElementById(i)}function raw(v){var s=String(v).trim().replace(\/\\s+\/g,'');if(!\/^[+]?(?:\\d+(?:[.,]\\d*)?|[.,]\\d+)$\/.test(s))return NaN;return Number(s.replace(',','.'))}function pos(v){var x=raw(v);return Number.isFinite(x)&&x>0?x:NaN}function f(x){return x.toFixed(9).replace(\/(\\.\\d*?)0+$\/,'$1').replace(\/\\.$\/,'')}function toggle(){var log=$('vc-method').value==='logdec';document.querySelectorAll('.vc-log').forEach(function(x){x.style.display=log?'flex':'none'});document.querySelectorAll('.vc-bw').forEach(function(x){x.style.display=log?'none':'flex'});$('vc-gate').innerHTML=log?'<strong>Log-decrement gate:<\/strong> subtract equilibrium\/offset; use magnitudes of same-signed peaks separated by exactly N damped periods in an unforced, linear exponential SDOF decay. Beating, multiple modes, nonlinear damping or forced response invalidate the direct result.':'<strong>Bandwidth gate:<\/strong> use a stationary measured magnitude FRF of an isolated SDOF-like mode; f_l and f_u are the 1\/\u221a2-amplitude points around measured peak f_p. 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