{"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":"2026-02-15T20:17:22","modified_gmt":"2026-02-15T20:17:22","slug":"equivalent-spring-stiffness","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/bn\/calculators\/equivalent-spring-stiffness\/","title":{"rendered":"\u09b8\u09ae\u09a4\u09c1\u09b2\u09cd\u09af \u09b8\u09cd\u09aa\u09cd\u09b0\u09bf\u0982 \u0995\u09a0\u09cb\u09b0\u09a4\u09be \u0995\u09cd\u09af\u09be\u09b2\u0995\u09c1\u09b2\u09c7\u099f\u09b0 (\u09b8\u09bf\u09b0\u09bf\u099c &amp; \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2) | \u09ac\u09bf\u09a8\u09be\u09ae\u09c2\u09b2\u09cd\u09af\u09c7 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u099f\u09c1\u09b2"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Equivalent Spring Stiffness Calculator\",\"alternateName\":[\"Combined Spring Rate Calculator\",\"Series Parallel Springs Calculator\"],\"description\":\"Free online engineering calculator for equivalent spring stiffness. Supports series and parallel configurations with up to 5 springs and optional natural frequency calculation.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/equivalent-spring-stiffness\/\",\"applicationCategory\":\"Engineering Calculator\",\"operatingSystem\":\"Any (Web Browser)\",\"offers\":{\"@type\":\"Offer\",\"price\":\"0\",\"priceCurrency\":\"EUR\"},\"creator\":{\"@type\":\"Organization\",\"name\":\"Vibromera\",\"url\":\"https:\/\/vibromera.eu\/\"},\"datePublished\":\"2025-01-01\",\"dateModified\":\"2025-06-14\",\"inLanguage\":\"en\",\"isAccessibleForFree\":true,\"featureList\":[\"Series and parallel spring configurations\",\"Up to 5 springs\",\"Natural frequency calculation\",\"Metric and Imperial units\",\"Auto-calculation on input change\",\"URL parameter sharing\",\"Calculation history\"],\"keywords\":\"equivalent spring stiffness, series springs, parallel springs, combined spring rate, natural frequency\"}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[\n{\"@type\":\"Question\",\"name\":\"When should I use series vs parallel spring connections?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Use parallel springs when you need to increase the overall stiffness \u2014 for example, mounting multiple springs side by side under a machine. The equivalent stiffness is the sum of all individual stiffnesses. Use series springs when you need to decrease the overall stiffness or increase flexibility \u2014 for example, stacking springs end-to-end.\"}},\n{\"@type\":\"Question\",\"name\":\"How do parallel springs increase stiffness?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"In a parallel arrangement, all springs share the same deflection but each carries a portion of the load. 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}\n    .vc-copy-btn, .vc-section-chevron, .vc-faq-chevron, .vc-presets, .vc-promo { display:none!important; }\n}\n<\/style>\n\n<div class=\"vc-calculator\" id=\"vc-spring-stiffness\">\n\n<!-- Header -->\n<header class=\"vc-header\">\n    <p class=\"vc-header-eyebrow\">Free Engineering Tool<\/p>\n    <h1 class=\"vc-header-title\">Equivalent Spring Stiffness Calculator<\/h1>\n    <p class=\"vc-header-subtitle\">Calculate equivalent spring stiffness for springs connected in series or parallel. Enter up to 5 individual spring rates and optionally compute natural frequency.<\/p>\n    <div class=\"vc-badges\">\n        <span class=\"vc-badge\">Series &amp; Parallel<\/span>\n        <span class=\"vc-badge\">Up to 5 Springs<\/span>\n        <span class=\"vc-badge\">Natural Frequency<\/span>\n    <\/div>\n<\/header>\n\n<!-- Calculator Card -->\n<div class=\"vc-card\">\n    <div class=\"vc-unit-bar\">\n        <button type=\"button\" class=\"vc-unit-btn vc-active\" data-system=\"metric\">Metric (N\/mm, kg)<\/button>\n        <button type=\"button\" class=\"vc-unit-btn\" data-system=\"imperial\">Imperial (lbf\/in, lb)<\/button>\n    <\/div>\n\n    <form class=\"vc-form\" id=\"vc-form\" autocomplete=\"off\">\n        <div class=\"vc-form-grid\">\n            <div class=\"vc-field\">\n                <label class=\"vc-label\" for=\"vc-conn\">Connection Type<\/label>\n                <select class=\"vc-select\" id=\"vc-conn\">\n                    <option value=\"parallel\" selected>Parallel \u2014 springs share deflection<\/option>\n                    <option value=\"series\">Series \u2014 springs share force<\/option>\n                <\/select>\n            <\/div>\n            <div class=\"vc-field\">\n                <label class=\"vc-label\" for=\"vc-num\">Number of Springs<\/label>\n                <select class=\"vc-select\" id=\"vc-num\">\n                    <option value=\"2\">2<\/option>\n                    <option value=\"3\">3<\/option>\n                    <option value=\"4\" selected>4<\/option>\n                    <option value=\"5\">5<\/option>\n                <\/select>\n            <\/div>\n            <div class=\"vc-field\" id=\"vc-field-k1\">\n                <label class=\"vc-label\" for=\"vc-k1\">Spring k\u2081 <span class=\"vc-label-hint vc-stiff-unit\">(N\/mm)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-k1\" inputmode=\"decimal\" step=\"any\" min=\"0.001\" placeholder=\"e.g. 50\">\n            <\/div>\n            <div class=\"vc-field\" id=\"vc-field-k2\">\n                <label class=\"vc-label\" for=\"vc-k2\">Spring k\u2082 <span class=\"vc-label-hint vc-stiff-unit\">(N\/mm)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-k2\" inputmode=\"decimal\" step=\"any\" min=\"0.001\" placeholder=\"e.g. 50\">\n            <\/div>\n            <div class=\"vc-field\" id=\"vc-field-k3\">\n                <label class=\"vc-label\" for=\"vc-k3\">Spring k\u2083 <span class=\"vc-label-hint vc-stiff-unit\">(N\/mm)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-k3\" inputmode=\"decimal\" step=\"any\" min=\"0.001\" placeholder=\"e.g. 50\">\n            <\/div>\n            <div class=\"vc-field\" id=\"vc-field-k4\">\n                <label class=\"vc-label\" for=\"vc-k4\">Spring k\u2084 <span class=\"vc-label-hint vc-stiff-unit\">(N\/mm)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-k4\" inputmode=\"decimal\" step=\"any\" min=\"0.001\" placeholder=\"e.g. 50\">\n            <\/div>\n            <div class=\"vc-field\" id=\"vc-field-k5\" style=\"display:none\">\n                <label class=\"vc-label\" for=\"vc-k5\">Spring k\u2085 <span class=\"vc-label-hint vc-stiff-unit\">(N\/mm)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-k5\" inputmode=\"decimal\" step=\"any\" min=\"0.001\" placeholder=\"e.g. 50\">\n            <\/div>\n            <div class=\"vc-field vc-field-full\">\n                <label class=\"vc-label\" for=\"vc-mass\">Mass <span class=\"vc-label-hint\">\u2014 optional, for natural frequency<\/span> <span class=\"vc-label-hint\" id=\"vc-mass-unit-hint\">(kg)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-mass\" inputmode=\"decimal\" step=\"any\" min=\"0.001\" placeholder=\"e.g. 200\">\n            <\/div>\n        <\/div>\n\n        <!-- Presets -->\n        <div class=\"vc-presets\">\n            <div class=\"vc-presets-label\">Quick presets<\/div>\n            <button type=\"button\" class=\"vc-preset-btn\" data-conn=\"parallel\" data-n=\"4\" data-k=\"50,50,50,50\">4\u00d750 N\/mm Parallel<\/button>\n            <button type=\"button\" class=\"vc-preset-btn\" data-conn=\"series\" data-n=\"2\" data-k=\"100,200\">100+200 N\/mm Series<\/button>\n            <button type=\"button\" class=\"vc-preset-btn\" data-conn=\"parallel\" data-n=\"3\" data-k=\"80,80,80\">3\u00d780 N\/mm Parallel<\/button>\n        <\/div>\n    <\/form>\n\n    <!-- Results -->\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\">\n                <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>\n                Copy\n            <\/button>\n        <\/div>\n        <div class=\"vc-result-grid\">\n            <div class=\"vc-rcard vc-rcard-primary\">\n                <div class=\"vc-rcard-label\">Equivalent Stiffness (k_eq)<\/div>\n                <div class=\"vc-rcard-value\" id=\"vc-r-keq\">\u2014<\/div>\n            <\/div>\n            <div class=\"vc-rcard\">\n                <div class=\"vc-rcard-label\">Connection Type<\/div>\n                <div class=\"vc-rcard-value\" id=\"vc-r-conn\">\u2014<\/div>\n            <\/div>\n            <div class=\"vc-rcard\">\n                <div class=\"vc-rcard-label\">Number of Springs<\/div>\n                <div class=\"vc-rcard-value\" id=\"vc-r-num\">\u2014<\/div>\n            <\/div>\n            <div class=\"vc-rcard\" id=\"vc-rcard-identical\" style=\"display:none\">\n                <div class=\"vc-rcard-label\">Simplified Formula<\/div>\n                <div class=\"vc-rcard-value\" id=\"vc-r-identical\">\u2014<\/div>\n            <\/div>\n            <div class=\"vc-rcard\" id=\"vc-rcard-freq\" style=\"display:none\">\n                <div class=\"vc-rcard-label\">Natural Frequency (f_n)<\/div>\n                <div class=\"vc-rcard-value\" id=\"vc-r-freq\">\u2014<\/div>\n            <\/div>\n            <div class=\"vc-rcard\" id=\"vc-rcard-omega\" style=\"display:none\">\n                <div class=\"vc-rcard-label\">Angular Frequency (\u03c9_n)<\/div>\n                <div class=\"vc-rcard-value\" id=\"vc-r-omega\">\u2014<\/div>\n            <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n\n<!-- Theory & Formulas -->\n<div class=\"vc-section vc-open\" id=\"vc-sec-theory\">\n    <button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"true\">\n        <span class=\"vc-section-toggle-text\">\n            <span class=\"vc-section-icon\">\ud83d\udcd8<\/span>\n            <span class=\"vc-section-title\">Theory &amp; Formulas<\/span>\n        <\/span>\n        <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>\n    <\/button>\n    <div class=\"vc-section-body\">\n        <div class=\"vc-section-inner vc-theory\">\n            <h3>Parallel Connection<\/h3>\n            <p>Springs share the same deflection; forces add up:<\/p>\n            <div class=\"vc-formula-box\" id=\"vc-katex-1\"><\/div>\n\n            <h3>Series Connection<\/h3>\n            <p>Springs share the same force; deflections add up:<\/p>\n            <div class=\"vc-formula-box\" id=\"vc-katex-2\"><\/div>\n\n            <h3>Identical Springs Simplification<\/h3>\n            <p>For <em>n<\/em> identical springs of stiffness <em>k<\/em>:<\/p>\n            <div class=\"vc-formula-box\" id=\"vc-katex-3\"><\/div>\n\n            <h3>Natural Frequency<\/h3>\n            <p>With a known mass mounted on the spring system:<\/p>\n            <div class=\"vc-formula-box\" id=\"vc-katex-4\"><\/div>\n            <p>Where <strong>k<sub>eq<\/sub><\/strong> is in N\/m (multiply N\/mm by 1000) and <strong>m<\/strong> is in kg.<\/p>\n\n            <h3>Practical Example<\/h3>\n            <div class=\"vc-example-box\">\n                <div class=\"vc-example-title\">Example \u2014 Machine on 4 Parallel Springs<\/div>\n                <p><strong>Given:<\/strong> Mass m = 200 kg, 4 identical springs in parallel, each k = 45 N\/mm.<\/p>\n                <p>k<sub>eq<\/sub> = 4 \u00d7 45 = <strong>180 N\/mm<\/strong> = 180,000 N\/m<\/p>\n                <p>\u03c9<sub>n<\/sub> = \u221a(180,000 \/ 200) = \u221a900 = <strong>30 rad\/s<\/strong><\/p>\n                <p>f<sub>n<\/sub> = 30 \/ (2\u03c0) = <strong>4.77 Hz<\/strong><\/p>\n            <\/div>\n\n            <div class=\"vc-info-box\">\n                <p><strong>\ud83d\udca1 Tip:<\/strong> Parallel springs increase stiffness (stiffer system, higher natural frequency). Series springs decrease stiffness (softer system, lower natural frequency).<\/p>\n            <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n\n<!-- Related Calculators -->\n<div class=\"vc-section\" id=\"vc-sec-related\">\n    <button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\">\n        <span class=\"vc-section-toggle-text\">\n            <span class=\"vc-section-icon\">\ud83d\udd17<\/span>\n            <span class=\"vc-section-title\">Related Calculators<\/span>\n        <\/span>\n        <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>\n    <\/button>\n    <div class=\"vc-section-body\">\n        <div class=\"vc-section-inner\">\n            <div class=\"vc-related\">\n                <a class=\"vc-related-link\" href=\"\/calculators\/natural-frequency\/\">Natural Frequency (Mass-Spring)<\/a>\n                <a class=\"vc-related-link\" href=\"\/calculators\/spring-selection\/\">Spring Selection by Frequency<\/a>\n                <a class=\"vc-related-link\" href=\"\/calculators\/critical-speed\/\">Spring Critical Speed<\/a>\n                <a class=\"vc-related-link\" href=\"\/calculators\/balancing-platform\/\">Balancing Platform Design<\/a>\n            <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n\n<!-- FAQ -->\n<div class=\"vc-section\" id=\"vc-sec-faq\">\n    <button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\">\n        <span class=\"vc-section-toggle-text\">\n            <span class=\"vc-section-icon\">\u2753<\/span>\n            <span class=\"vc-section-title\">Frequently Asked Questions<\/span>\n        <\/span>\n        <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>\n    <\/button>\n    <div class=\"vc-section-body\">\n        <div class=\"vc-section-inner\">\n            <div class=\"vc-faq-list\" id=\"vc-faq-list\"><\/div>\n        <\/div>\n    <\/div>\n<\/div>\n\n<!-- Promo -->\n<div class=\"vc-promo\">\n    <div class=\"vc-promo-text\">\n        <strong>Vibromera \u2014 Portable Balancing &amp; Vibration Analysis<\/strong><br>\n        Professional portable balancers, vibration analyzers, and condition monitoring systems. Optimize spring selection and verify isolation performance on-site. Used in 50+ countries.\n    <\/div>\n    <a class=\"vc-promo-link\" href=\"https:\/\/vibromera.eu\/\" target=\"_blank\" rel=\"noopener\">Learn More<\/a>\n<\/div>\n\n<!-- Footer -->\n<footer class=\"vc-footer\">\n    <p>\u00a9 2024\u20132025 <a href=\"https:\/\/vibromera.eu\/\">Vibromera<\/a> \u2014 Engineering Calculators<\/p>\n    <p style=\"font-size:12px;margin-top:4px;\">Last updated: June 2025<\/p>\n    <div class=\"vc-footer-links\">\n        <a href=\"\/calculators\/engineering-calculators\/\">All Calculators<\/a>\n        <a href=\"https:\/\/vibromera.eu\/contacts\/\">Contact<\/a>\n    <\/div>\n<\/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\n    var SLUG = 'equivalent-spring-stiffness';\n    var unitSystem = 'metric';\n    var KG2LB = 2.20462;\n    var NMM2LBFIN = 5.71015;\n\n    function $(id) { return document.getElementById(id); }\n    function fmt(n, d) {\n        if (isNaN(n)) return '\u2014';\n        if (n === 0) return '0';\n        if (Math.abs(n) < 0.001) return n.toExponential(2);\n        if (d !== undefined) { var f = n.toFixed(d); return f.replace(\/(\\.\\d*?)0+$\/, '$1').replace(\/\\.$\/, ''); }\n        if (Math.abs(n) < 1) return n.toFixed(4);\n        if (Math.abs(n) < 10) return n.toFixed(3);\n        if (Math.abs(n) < 100) return n.toFixed(2);\n        if (Math.abs(n) < 10000) return n.toFixed(1);\n        return Math.round(n).toLocaleString('en-US');\n    }\n\n    \/* \u2500\u2500 UPDATE VISIBLE SPRING FIELDS \u2500\u2500 *\/\n    function updateFields() {\n        var n = parseInt($('vc-num').value) || 4;\n        for (var i = 1; i <= 5; i++) {\n            var el = $('vc-field-k' + i);\n            if (el) el.style.display = (i <= n) ? 'flex' : 'none';\n        }\n    }\n\n    \/* \u2500\u2500 CALCULATION \u2500\u2500 *\/\n    function calculate() {\n        var n = parseInt($('vc-num').value) || 4;\n        var conn = $('vc-conn').value;\n        var kValues = [];\n        var valid = true;\n\n        for (var i = 1; i <= n; i++) {\n            var val = parseFloat($('vc-k' + i).value);\n            if (!val || val <= 0) { valid = false; break; }\n            kValues.push(val);\n        }\n\n        if (!valid || kValues.length === 0) {\n            $('vc-results').classList.remove('vc-visible');\n            return;\n        }\n\n        \/* Convert to metric (N\/mm) *\/\n        var kMetric = kValues.map(function(k) { return unitSystem === 'imperial' ? k \/ NMM2LBFIN : k; });\n\n        \/* Compute equivalent stiffness in N\/mm *\/\n        var keqNmm;\n        if (conn === 'parallel') {\n            keqNmm = 0;\n            for (var j = 0; j < kMetric.length; j++) keqNmm += kMetric[j];\n        } else {\n            var invSum = 0;\n            for (var j = 0; j < kMetric.length; j++) invSum += 1 \/ kMetric[j];\n            keqNmm = 1 \/ invSum;\n        }\n\n        \/* Check if all springs identical *\/\n        var allIdentical = true;\n        for (var j = 1; j < kMetric.length; j++) {\n            if (Math.abs(kMetric[j] - kMetric[0]) > 0.001) { allIdentical = false; break; }\n        }\n\n        \/* Display units *\/\n        var stiffUnit = unitSystem === 'imperial' ? 'lbf\/in' : 'N\/mm';\n        var keqDisplay = unitSystem === 'imperial' ? keqNmm * NMM2LBFIN : keqNmm;\n\n        \/* Primary result *\/\n        $('vc-r-keq').innerHTML = fmt(keqDisplay) + ' <span class=\"vc-rcard-unit\">' + stiffUnit + '<\/span>';\n        $('vc-r-conn').textContent = conn === 'parallel' ? 'Parallel' : 'Series';\n        $('vc-r-num').textContent = n;\n\n        \/* Identical springs info *\/\n        var identCard = $('vc-rcard-identical');\n        if (allIdentical) {\n            identCard.style.display = '';\n            var kSingle = kValues[0];\n            if (conn === 'parallel') {\n                $('vc-r-identical').innerHTML = n + ' \u00d7 ' + fmt(kSingle) + ' = ' + fmt(keqDisplay) + ' <span class=\"vc-rcard-unit\">' + stiffUnit + '<\/span>';\n            } else {\n                $('vc-r-identical').innerHTML = fmt(kSingle) + ' \/ ' + n + ' = ' + fmt(keqDisplay) + ' <span class=\"vc-rcard-unit\">' + stiffUnit + '<\/span>';\n            }\n        } else {\n            identCard.style.display = 'none';\n        }\n\n        \/* Natural frequency (optional mass) *\/\n        var massVal = parseFloat($('vc-mass').value);\n        var freqCard = $('vc-rcard-freq');\n        var omegaCard = $('vc-rcard-omega');\n        if (massVal && massVal > 0) {\n            var massKg = unitSystem === 'imperial' ? massVal \/ KG2LB : massVal;\n            var keqNm = keqNmm * 1000;\n            var omega = Math.sqrt(keqNm \/ massKg);\n            var fn = omega \/ (2 * Math.PI);\n            freqCard.style.display = '';\n            omegaCard.style.display = '';\n            $('vc-r-freq').innerHTML = fmt(fn) + ' <span class=\"vc-rcard-unit\">Hz<\/span>';\n            $('vc-r-omega').innerHTML = fmt(omega) + ' <span class=\"vc-rcard-unit\">rad\/s<\/span>';\n        } else {\n            freqCard.style.display = 'none';\n            omegaCard.style.display = 'none';\n        }\n\n        $('vc-results').classList.add('vc-visible');\n\n        \/* URL params *\/\n        var u = new URL(window.location);\n        u.searchParams.set('conn', conn);\n        u.searchParams.set('n', n);\n        for (var i = 1; i <= n; i++) u.searchParams.set('k' + i, $('vc-k' + i).value);\n        if (massVal && massVal > 0) u.searchParams.set('mass', $('vc-mass').value);\n        else u.searchParams.delete('mass');\n        history.replaceState(null, '', u.toString());\n\n        \/* History *\/\n        var summary = conn + ', n=' + n + ' \u2192 k_eq=' + fmt(keqDisplay) + ' ' + stiffUnit;\n        var histEntry = { conn: conn, n: n };\n        for (var i = 1; i <= n; i++) histEntry['k' + i] = $('vc-k' + i).value;\n        if (massVal && massVal > 0) histEntry.mass = $('vc-mass').value;\n        histEntry._summary = summary;\n        saveHistory(histEntry);\n    }\n\n    \/* \u2500\u2500 UNIT TOGGLE \u2500\u2500 *\/\n    var unitBtns = document.querySelectorAll('.vc-unit-btn');\n    unitBtns.forEach(function(btn) {\n        btn.addEventListener('click', function() {\n            unitBtns.forEach(function(b) { b.classList.remove('vc-active'); });\n            this.classList.add('vc-active');\n            unitSystem = this.getAttribute('data-system');\n            var stiffUnit = unitSystem === 'imperial' ? '(lbf\/in)' : '(N\/mm)';\n            document.querySelectorAll('.vc-stiff-unit').forEach(function(el) { el.textContent = stiffUnit; });\n            $('vc-mass-unit-hint').textContent = unitSystem === 'imperial' ? '(lb)' : '(kg)';\n            try { localStorage.setItem('vc_units', unitSystem); } catch(e) {}\n            calculate();\n        });\n    });\n\n    \/* \u2500\u2500 PRESETS \u2500\u2500 *\/\n    document.querySelectorAll('.vc-preset-btn').forEach(function(btn) {\n        btn.addEventListener('click', function(e) {\n            e.preventDefault();\n            var conn = this.getAttribute('data-conn');\n            var n = parseInt(this.getAttribute('data-n'));\n            var kArr = this.getAttribute('data-k').split(',');\n\n            $('vc-conn').value = conn;\n            $('vc-num').value = n;\n            updateFields();\n\n            for (var i = 0; i < 5; i++) {\n                var inp = $('vc-k' + (i + 1));\n                if (inp) {\n                    if (i < kArr.length) {\n                        var kVal = parseFloat(kArr[i]);\n                        inp.value = unitSystem === 'imperial' ? fmt(kVal * NMM2LBFIN, 1) : kVal;\n                    } else {\n                        inp.value = '';\n                    }\n                }\n            }\n            $('vc-mass').value = '';\n            calculate();\n        });\n    });\n\n    \/* \u2500\u2500 AUTO-CALCULATE \u2500\u2500 *\/\n    $('vc-form').addEventListener('input', function() { updateFields(); calculate(); });\n    $('vc-form').addEventListener('change', function() { updateFields(); calculate(); });\n\n    \/* \u2500\u2500 COPY \u2500\u2500 *\/\n    $('vc-copy-btn').addEventListener('click', function() {\n        var stiffUnit = unitSystem === 'imperial' ? 'lbf\/in' : 'N\/mm';\n        var massUnit = unitSystem === 'imperial' ? 'lb' : 'kg';\n        var n = parseInt($('vc-num').value) || 4;\n        var conn = $('vc-conn').value;\n        var text = 'Equivalent Spring Stiffness Calculation\\n' +\n            '\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\\n' +\n            'Connection: ' + (conn === 'parallel' ? 'Parallel' : 'Series') + '\\n' +\n            'Springs: ' + n + '\\n';\n        for (var i = 1; i <= n; i++) text += 'k' + i + ': ' + $('vc-k' + i).value + ' ' + stiffUnit + '\\n';\n        text += 'k_eq: ' + $('vc-r-keq').textContent + '\\n';\n        var massVal = parseFloat($('vc-mass').value);\n        if (massVal && massVal > 0) {\n            text += 'Mass: ' + $('vc-mass').value + ' ' + massUnit + '\\n';\n            text += 'f_n: ' + $('vc-r-freq').textContent + '\\n';\n            text += '\u03c9_n: ' + $('vc-r-omega').textContent + '\\n';\n        }\n        text += '\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\\n' +\n            'vibromera.eu\/calculators\/equivalent-spring-stiffness\/';\n\n        if (navigator.clipboard) {\n            navigator.clipboard.writeText(text).then(function() {\n                var btn = $('vc-copy-btn');\n                btn.classList.add('vc-copied');\n                btn.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!';\n                setTimeout(function() {\n                    btn.classList.remove('vc-copied');\n                    btn.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';\n                }, 2000);\n            });\n        }\n    });\n\n    \/* \u2500\u2500 COLLAPSIBLE SECTIONS \u2500\u2500 *\/\n    document.querySelectorAll('.vc-section-toggle').forEach(function(btn) {\n        btn.addEventListener('click', function() {\n            var section = this.closest('.vc-section');\n            var isOpen = section.classList.contains('vc-open');\n            section.classList.toggle('vc-open');\n            this.setAttribute('aria-expanded', !isOpen);\n        });\n    });\n\n    \/* \u2500\u2500 FAQ \u2500\u2500 *\/\n    var faqData = [\n        { q: 'When should I use series vs parallel spring connections?', a: '<p>Use parallel springs when you need to increase the overall stiffness \u2014 for example, mounting multiple springs side by side under a machine. The equivalent stiffness is the sum of all individual stiffnesses.<\/p><p>Use series springs when you need to decrease the overall stiffness or increase flexibility \u2014 for example, stacking springs end-to-end. The equivalent stiffness is always less than the weakest individual spring.<\/p>' },\n        { q: 'How do parallel springs increase stiffness?', a: '<p>In a parallel arrangement, all springs share the same deflection but each carries a portion of the load. The total force is the sum of the forces in each spring, so the effective stiffness is k<sub>eq<\/sub> = k\u2081 + k\u2082 + \u2026 + k\u2099.<\/p><p>For n identical springs of stiffness k, the equivalent stiffness is simply n \u00d7 k.<\/p>' },\n        { q: 'What are real-world examples of springs in series?', a: '<p>Series spring configurations appear in many engineering systems:<\/p><ul><li>Multi-stage shock absorbers with stacked springs<\/li><li>Automotive valve springs in some configurations<\/li><li>Rubber isolation pads stacked vertically<\/li><li>Multi-layer leaf spring systems<\/li><li>Drill string assemblies<\/li><\/ul>' },\n        { q: 'Does adding springs change the natural frequency of a system?', a: '<p>Yes. Natural frequency depends on both mass and stiffness: f = (1\/2\u03c0)\u221a(k\/m). Adding springs in parallel increases k<sub>eq<\/sub>, raising the natural frequency. Adding springs in series decreases k<sub>eq<\/sub>, lowering it.<\/p><p>Doubling identical parallel springs doubles k<sub>eq<\/sub> and increases frequency by \u221a2 \u2248 1.41.<\/p>' },\n        { q: 'How can I test or measure spring stiffness?', a: '<p>Spring stiffness can be measured several ways:<\/p><ul><li><strong>Static test:<\/strong> Apply a known force and measure deflection. k = F\/\u03b4.<\/li><li><strong>Compression testing machine:<\/strong> Record the full force-deflection curve.<\/li><li><strong>Dynamic test:<\/strong> Mount a known mass, measure free vibration frequency. k = m \u00d7 (2\u03c0f)\u00b2.<\/li><li><strong>Manufacturer datasheet:<\/strong> Most commercial springs have a rated spring rate.<\/li><\/ul>' }\n    ];\n\n    var faqList = $('vc-faq-list');\n    faqData.forEach(function(faq) {\n        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    });\n    faqList.addEventListener('click', function(e) {\n        var btn = e.target.closest('.vc-faq-q');\n        if (!btn) return;\n        btn.closest('.vc-faq-item').classList.toggle('vc-open');\n    });\n\n    \/* \u2500\u2500 HISTORY \u2500\u2500 *\/\n    function saveHistory(entry) {\n        var key = 'vc_history_' + SLUG;\n        var arr = [];\n        try { arr = JSON.parse(localStorage.getItem(key)) || []; 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