{"id":100084,"date":"2026-02-15T20:17:22","date_gmt":"2026-02-15T20:17:22","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100084"},"modified":"2026-07-12T02:34:45","modified_gmt":"2026-07-12T02:34:45","slug":"equivalent-spring-stiffness","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/ro\/calculators\/equivalent-spring-stiffness\/","title":{"rendered":"Equivalent Linear Spring Stiffness | Series, Parallel &#038; SDOF"},"content":{"rendered":"\r\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Equivalent Linear Spring Stiffness\",\"description\":\"Calculate equivalent stiffness for ideal linear springs in one-dimensional series or parallel arrangements and optional SDOF natural frequency.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/equivalent-spring-stiffness\/\",\"applicationCategory\":\"Engineering Calculator\",\"operatingSystem\":\"Any\",\"offers\":{\"@type\":\"Offer\",\"price\":\"0\",\"priceCurrency\":\"EUR\"},\"creator\":{\"@type\":\"Organization\",\"name\":\"Vibromera\",\"url\":\"https:\/\/vibromera.eu\/\"},\"dateModified\":\"2026-07-12\",\"inLanguage\":\"en\",\"isAccessibleForFree\":true}<\/script>\r\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"BreadcrumbList\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/vibromera.eu\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Calculators\",\"item\":\"https:\/\/vibromera.eu\/calculators\/\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Equivalent linear spring stiffness\",\"item\":\"https:\/\/vibromera.eu\/calculators\/equivalent-spring-stiffness\/\"}]}<\/script>\r\n<link rel=\"preconnect\" href=\"https:\/\/fonts.googleapis.com\">\r\n<style>\r\n:root {\r\n    --vc-bg: #f4f1ec;\r\n    --vc-surface: #ffffff;\r\n    --vc-surface-alt: #f8f6f2;\r\n    --vc-ink: #1a1a1a;\r\n    --vc-ink-secondary: #5a5650;\r\n    --vc-ink-muted: #8a857e;\r\n    --vc-accent: #c85a2a;\r\n    --vc-accent-hover: #b04d22;\r\n    --vc-accent-light: #fdf0ea;\r\n    --vc-blue: #2a5c8c;\r\n    --vc-blue-light: #eaf1f8;\r\n    --vc-green: #2a7a4b;\r\n    --vc-green-light: #eaf8ef;\r\n    --vc-yellow: #a67c00;\r\n    --vc-yellow-light: #fef9e8;\r\n    --vc-red: #b91c1c;\r\n    --vc-red-light: #fef2f2;\r\n    --vc-border: #d9d4cc;\r\n    --vc-border-light: #e8e4dd;\r\n    --vc-shadow: 0 1px 3px rgba(26,26,26,0.06), 0 4px 12px rgba(26,26,26,0.04);\r\n    --vc-shadow-lg: 0 4px 12px rgba(26,26,26,0.08), 0 16px 40px rgba(26,26,26,0.06);\r\n    --vc-radius: 8px;\r\n    --vc-radius-lg: 12px;\r\n    --vc-font: 'DM Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', sans-serif;\r\n    --vc-mono: 'JetBrains Mono', 'Consolas', 'Monaco', monospace;\r\n    --vc-display: 'Fraunces', Georgia, serif;\r\n}\r\n\r\n.vc-calculator {\r\n    font-family: var(--vc-font);\r\n    font-size: 15px;\r\n    line-height: 1.65;\r\n    color: var(--vc-ink);\r\n    max-width: 960px;\r\n    margin: 0 auto;\r\n    padding: 20px 16px 40px;\r\n    -webkit-font-smoothing: antialiased;\r\n}\r\n.vc-calculator *, .vc-calculator *::before, .vc-calculator *::after { box-sizing: border-box; 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}\r\n    .vc-faq-a { max-height:none!important; }\r\n    .vc-results { max-height:none!important; padding:24px!important; }\r\n    .vc-copy-btn, .vc-section-chevron, .vc-faq-chevron, .vc-presets, .vc-promo { display:none!important; }\r\n}\r\n<\/style>\r\n<div class=\"vc-calculator\" id=\"vc-spring-stiffness\"><header class=\"vc-header\"><p class=\"vc-header-eyebrow\">Ideal one-dimensional linear model<\/p><h1 class=\"vc-header-title\">Equivalent Linear Spring Stiffness<\/h1><p class=\"vc-header-subtitle\">Combine two to five positive tangent spring rates in a true one-dimensional series or parallel arrangement. Optional natural frequency uses an ideal undamped single-degree-of-freedom model.<\/p><div class=\"vc-badges\"><span class=\"vc-badge\">Series<\/span><span class=\"vc-badge\">Parallel<\/span><span class=\"vc-badge\">Strict SI input<\/span><span class=\"vc-badge\">Optional SDOF<\/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\"><label class=\"vc-label\" for=\"vc-conn\">Arrangement<\/label><select class=\"vc-select\" id=\"vc-conn\"><option value=\"parallel\">Parallel \u2014 common deflection<\/option><option value=\"series\">Series \u2014 common force<\/option><\/select><\/div><div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-num\">Active springs<\/label><select class=\"vc-select\" id=\"vc-num\"><option value=\"2\">2<\/option><option value=\"3\">3<\/option><option value=\"4\">4<\/option><option value=\"5\">5<\/option><\/select><\/div><div class=\"vc-field\" id=\"vc-field-k1\"><label class=\"vc-label\" for=\"vc-k1\">Spring k1 <span class=\"vc-label-hint\">(N\/mm; &gt;0)<\/span><\/label><input class=\"vc-input\" id=\"vc-k1\" type=\"text\" inputmode=\"decimal\"><\/div><div class=\"vc-field\" id=\"vc-field-k2\"><label class=\"vc-label\" for=\"vc-k2\">Spring k2 <span class=\"vc-label-hint\">(N\/mm; &gt;0)<\/span><\/label><input class=\"vc-input\" id=\"vc-k2\" type=\"text\" inputmode=\"decimal\"><\/div><div class=\"vc-field\" id=\"vc-field-k3\"><label class=\"vc-label\" for=\"vc-k3\">Spring k3 <span class=\"vc-label-hint\">(N\/mm; &gt;0)<\/span><\/label><input class=\"vc-input\" id=\"vc-k3\" type=\"text\" inputmode=\"decimal\"><\/div><div class=\"vc-field\" id=\"vc-field-k4\"><label class=\"vc-label\" for=\"vc-k4\">Spring k4 <span class=\"vc-label-hint\">(N\/mm; &gt;0)<\/span><\/label><input class=\"vc-input\" id=\"vc-k4\" type=\"text\" inputmode=\"decimal\"><\/div><div class=\"vc-field\" id=\"vc-field-k5\"><label class=\"vc-label\" for=\"vc-k5\">Spring k5 <span class=\"vc-label-hint\">(N\/mm; &gt;0)<\/span><\/label><input class=\"vc-input\" id=\"vc-k5\" type=\"text\" inputmode=\"decimal\"><\/div><div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-mass\">Total effective moving mass <span class=\"vc-label-hint\">(kg; optional, &gt;0)<\/span><\/label><input class=\"vc-input\" id=\"vc-mass\" type=\"text\" inputmode=\"decimal\"><\/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\"><strong>Model boundary:<\/strong> all active elements must be linear about the operating point and act along the same generalized coordinate. Series requires the same force through every element; parallel requires the same deflection. Preload, geometric leverage, bending, contact changes, damping and spring\/mount mass are not included.<\/p><\/div><\/form><div class=\"vc-results\" id=\"vc-results\"><div class=\"vc-results-head\"><h2 class=\"vc-results-title\">Equivalent ideal model<\/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\">Equivalent stiffness keq<\/div><div class=\"vc-rcard-value\" id=\"vc-r-keq\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\">SI stiffness<\/div><div class=\"vc-rcard-value\" id=\"vc-r-nm\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Imperial conversion<\/div><div class=\"vc-rcard-value\" id=\"vc-r-lbfin\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Compliance 1\/keq<\/div><div class=\"vc-rcard-value\" id=\"vc-r-comp\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Natural frequency fn<\/div><div class=\"vc-rcard-value\" id=\"vc-r-fn\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Angular frequency \u03c9n<\/div><div class=\"vc-rcard-value\" id=\"vc-r-omega\">\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\udcd0<\/span><span class=\"vc-section-title\">Derivation and references<\/span><\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner vc-theory\"><div class=\"vc-formula-box\">Parallel: keq = \u03a3ki &nbsp; (common deflection, forces add)<\/div><div class=\"vc-formula-box\">Series: 1\/keq = \u03a3(1\/ki) &nbsp; (common force, deflections add)<\/div><p><a href=\"https:\/\/ocw.mit.edu\/courses\/2-76-multi-scale-system-design-fall-2004\/3aa5862a1724b75c3e4aa7a6fee6c511_reading_l3.pdf\" target=\"_blank\" rel=\"noopener\">MIT OpenCourseWare Precision Engineering Principles<\/a> states these compliance\/stiffness addition rules. A separate <a href=\"https:\/\/ocw.mit.edu\/courses\/1-105-solid-mechanics-laboratory-fall-2003\/0f895868598a52276cafb4875f9c277b_exp3_03.pdf\" target=\"_blank\" rel=\"noopener\">MIT solid-mechanics laboratory note<\/a> derives the two-spring series result from equilibrium and compatibility.<\/p><div class=\"vc-formula-box\">\u03c9n = \u221a(keq\/m), &nbsp; fn = \u03c9n\/(2\u03c0)<\/div><p>The frequency result applies only to a rigid effective mass on massless linear springs with one degree of freedom and no damping. <a href=\"https:\/\/ocw.mit.edu\/courses\/2-002-mechanics-and-materials-ii-spring-2004\/4432b26565dc8e422abaa65123d9b3ea_hw1_sol.pdf\" target=\"_blank\" rel=\"noopener\">MIT Mechanics and Materials<\/a> states this boundary explicitly. For real isolators, include the effective mass of mounts\/springs, geometry, damping and all coupled modes.<\/p><div class=\"vc-warning-box\"><p style=\"margin:0\"><strong>Unit correction:<\/strong> input is always N\/mm and kg. Results also show N\/m and lbf\/in, using exact SI definitions: 1 lbf=4.4482216152605 N and 1 in=25.4 mm. The old toggle reinterpreted unchanged numbers and could alter physical stiffness by 5.71\u00d7.<\/p><\/div><\/div><\/div><\/div><footer class=\"vc-footer\"><p>\u00a9 2024\u20132026 <a href=\"https:\/\/vibromera.eu\/\">Vibromera<\/a><\/p><p>Reference idealization only; validate the actual force\u2013deflection curve and effective dynamic mass. Scientific review: July 2026.<\/p><\/footer><\/div>\r\n<script>(function(){'use strict';var LBF_IN_PER_N_MM=25.4\/4.4482216152605;function $(i){return document.getElementById(i)}function pos(v){var s=String(v).trim().replace(\/\\s+\/g,'');if(!\/^[+]?(?:\\d+(?:[.,]\\d*)?|[.,]\\d+)$\/.test(s))return NaN;var x=Number(s.replace(',','.'));return Number.isFinite(x)&&x>0?x:NaN}function fmt(x){var y=Math.round(x*1e9)\/1e9;if(Math.abs(x-y)<=2e-12*Math.max(1,Math.abs(x)))x=y;return Number(x.toPrecision(12)).toString()}function clearResult(){['vc-r-keq','vc-r-nm','vc-r-lbfin','vc-r-comp','vc-r-fn','vc-r-omega'].forEach(function(id){$(id).textContent='\u2014'});$('vc-r-note').textContent='';$('vc-results').classList.remove('vc-visible')}function updateFields(){var n=Number($('vc-num').value);for(var i=1;i<=5;i++)$('vc-field-k'+i).style.display=i<=n?'flex':'none'}function calc(){var n=Number($('vc-num').value),conn=$('vc-conn').value,ks=[],any=false;for(var i=1;i<=n;i++){any=any||$('vc-k'+i).value.trim();ks.push(pos($('vc-k'+i).value))}var massRaw=$('vc-mass').value.trim(),mass=massRaw?pos(massRaw):null,bad=!['series','parallel'].includes(conn)||![2,3,4,5].includes(n)||!ks.every(Number.isFinite)||(massRaw&&!Number.isFinite(mass));if(bad){clearResult();$('vc-error').textContent=any||massRaw?'Enter every active spring as a positive unsuffixed N\/mm value;mass may be blank or a positive unsuffixed kg value.':'';return}var keq;if(conn==='parallel')keq=ks.reduce(function(a,b){return a+b},0);else keq=1\/ks.reduce(function(a,b){return a+1\/b},0);var omega=mass?Math.sqrt(keq*1000\/mass):null,fn=omega?omega\/(2*Math.PI):null;$('vc-error').textContent='';$('vc-r-keq').textContent=fmt(keq)+' N\/mm';$('vc-r-nm').textContent=fmt(keq*1000)+' N\/m';$('vc-r-lbfin').textContent=fmt(keq*LBF_IN_PER_N_MM)+' lbf\/in';$('vc-r-comp').textContent=fmt(1\/keq)+' mm\/N';$('vc-r-fn').textContent=fn?fmt(fn)+' Hz':'Not calculated';$('vc-r-omega').textContent=omega?fmt(omega)+' rad\/s':'Not calculated';$('vc-r-note').textContent=(conn==='parallel'?'Common deflection;forces summed.':'Common force;deflections summed.')+(mass?' 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