{"id":100298,"date":"2026-02-15T20:32:31","date_gmt":"2026-02-15T20:32:31","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100298"},"modified":"2026-07-17T09:05:11","modified_gmt":"2026-07-17T09:05:11","slug":"vibration-transmissibility","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/bg\/calculators\/vibration-transmissibility\/","title":{"rendered":"Linear SDOF Transmissibility Worksheet | Magnitude &#038; Phase"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Linear SDOF Transmissibility Worksheet\",\"description\":\"Calculates magnitude and phase from the same complex transfer function for one declared linear viscously damped SDOF model. It does not establish ISO compliance, isolator suitability, acceptance or operating action.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/vibration-transmissibility\/\",\"applicationCategory\":\"EngineeringApplication\",\"operatingSystem\":\"Any web browser\",\"inLanguage\":\"en\",\"isAccessibleForFree\":true,\"featureList\":[\"Base-motion or force-transmission model explicitly selected\",\"Magnitude and phase from the same complex transfer function\",\"No isolation-efficiency, ISO-compliance or approval claim\",\"Evidence and applicability record\"]}\n<\/script>\n<script type=\"application\/ld+json\">\n{\"@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\":\"Linear SDOF Transmissibility 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.tr-table-wrap{width:100%;max-width:100%;min-width:0;overflow-x:auto}#vbm-tr table{width:100%;border-collapse:collapse;min-width:840px;font-size:13px}#vbm-tr th,#vbm-tr td{padding:10px;border:1px solid #cbd5e1;text-align:left;vertical-align:top}#vbm-tr th{background:#eaf0f8}#vbm-tr .tr-status{font-weight:800}#vbm-tr details{margin-top:9px;border:1px solid var(--line);border-radius:8px;background:#fff}#vbm-tr summary{padding:11px 13px;cursor:pointer;font-weight:800}#vbm-tr details>div{padding:0 13px 13px;color:#425066}#vbm-tr .tr-small{font-size:13px;color:#536075}#vbm-tr .tr-foot{margin-top:22px;padding:16px;border-top:1px solid var(--line);color:#536075;font-size:13px}\n@media(max-width:720px){#vbm-tr .tr-grid,#vbm-tr .tr-checks,#vbm-tr .tr-out-grid,#vbm-tr .tr-equations{grid-template-columns:minmax(0,1fr)}#vbm-tr .tr-full,#vbm-tr fieldset{grid-column:1}#vbm-tr .tr-card{padding:15px}#vbm-tr table{min-width:720px}}\n<\/style>\n<main id=\"vbm-tr\">\n  <section class=\"tr-hero\" aria-labelledby=\"tr-title\">\n    <p class=\"tr-kicker\">Controlled linear-dynamics worksheet<\/p>\n    <h1 id=\"tr-title\">Linear SDOF Transmissibility Worksheet<\/h1>\n    <p class=\"tr-lead\">Calculate magnitude and phase for one declared, steady-state, harmonically excited, linear single-degree-of-freedom system with viscous damping. The magnitude and phase now come from the same complex transfer function.<\/p>\n    <div class=\"tr-badges\"><span class=\"tr-badge\">Base motion or force transmission<\/span><span class=\"tr-badge\">Correct complex phase<\/span><span class=\"tr-badge\">No ISO compliance claim<\/span><span class=\"tr-badge\">No isolator approval<\/span><\/div>\n    <div class=\"tr-alert\"><strong>Boundary:<\/strong> this is general linear-dynamics arithmetic, not a formula issued by ISO 2017, ISO 2041 or ISO 10846. A result does not establish that a real mount is linear, SDOF, viscously damped, stable, safe, correctly installed or suitable for service.<\/div>\n  <\/section>\n\n  <section class=\"tr-card\" aria-labelledby=\"tr-model-heading\">\n    <h2 id=\"tr-model-heading\">Declared model and evidence<\/h2>\n    <form id=\"tr-form\" novalidate>\n      <div class=\"tr-grid\">\n        <div class=\"tr-field\"><label for=\"tr-case\">Case \/ calculation ID<\/label><input id=\"tr-case\" type=\"text\" maxlength=\"120\" autocomplete=\"off\"><span class=\"tr-hint\">Traceable identifier; no default case is assumed.<\/span><\/div>\n        <div class=\"tr-field\"><label for=\"tr-asset\">Vibration source \/ asset<\/label><input id=\"tr-asset\" type=\"text\" maxlength=\"180\" autocomplete=\"off\"><\/div>\n        <div class=\"tr-field\"><label for=\"tr-receiver\">Receiver \/ supported item<\/label><input id=\"tr-receiver\" type=\"text\" maxlength=\"180\" autocomplete=\"off\"><\/div>\n        <div class=\"tr-field\"><label for=\"tr-axis\">Axis, point and positive-direction convention<\/label><input id=\"tr-axis\" type=\"text\" maxlength=\"220\" autocomplete=\"off\"><span class=\"tr-hint\">For force work, state whether the recorded support reaction has been polarity-reversed.<\/span><\/div>\n        <div class=\"tr-field tr-full\"><label for=\"tr-condition\">Operating and measurement condition<\/label><textarea id=\"tr-condition\" maxlength=\"600\"><\/textarea><\/div>\n        <div class=\"tr-field\"><label for=\"tr-model\">Transmissibility model<\/label><select id=\"tr-model\"><option value=\"\">Select the physical ratio<\/option><option value=\"motion\">Base motion: response X \/ base Y<\/option><option value=\"force\">Force transmission: transmitted F \/ applied F<\/option><\/select><\/div>\n        <div class=\"tr-field\"><label for=\"tr-quantity\">Matched input\/output quantity<\/label><select id=\"tr-quantity\"><option value=\"\">Select quantity<\/option><option value=\"displacement\">Displacement amplitude<\/option><option value=\"velocity\">Velocity amplitude<\/option><option value=\"acceleration\">Acceleration amplitude<\/option><option value=\"force\">Force amplitude<\/option><\/select><span class=\"tr-hint\">Motion ratios require the same derivative quantity; the force model requires force.<\/span><\/div>\n        <div class=\"tr-field\"><label for=\"tr-frequency\">Harmonic excitation frequency f (Hz)<\/label><input id=\"tr-frequency\" type=\"text\" inputmode=\"decimal\" autocomplete=\"off\"><span class=\"tr-hint\">Finite and greater than zero.<\/span><\/div>\n        <div class=\"tr-field\"><label for=\"tr-natural\">Undamped natural frequency f<sub>n<\/sub> (Hz)<\/label><input id=\"tr-natural\" type=\"text\" inputmode=\"decimal\" autocomplete=\"off\"><span class=\"tr-hint\">f<sub>n<\/sub> = (1 \/ 2\u03c0) \u221a(k \/ m); do not substitute a damped resonance peak without derivation.<\/span><\/div>\n        <div class=\"tr-field\"><label for=\"tr-zeta\">Viscous damping ratio \u03b6<\/label><input id=\"tr-zeta\" type=\"text\" inputmode=\"decimal\" autocomplete=\"off\"><span class=\"tr-hint\">Dimensionless, finite and non-negative. The numerical ceiling is only a software guard.<\/span><\/div>\n        <div class=\"tr-field\"><label for=\"tr-fn-basis\">Natural-frequency basis<\/label><input id=\"tr-fn-basis\" type=\"text\" maxlength=\"260\" autocomplete=\"off\"><span class=\"tr-hint\">Model, test, fit, date and source.<\/span><\/div>\n        <div class=\"tr-field\"><label for=\"tr-zeta-basis\">Damping-ratio basis<\/label><input id=\"tr-zeta-basis\" type=\"text\" maxlength=\"260\" autocomplete=\"off\"><span class=\"tr-hint\">Method, condition, uncertainty and source.<\/span><\/div>\n        <div class=\"tr-field\"><label for=\"tr-reference\">Controlled calculation \/ model reference<\/label><input id=\"tr-reference\" type=\"text\" maxlength=\"320\" autocomplete=\"off\"><\/div>\n        <div class=\"tr-field\"><label for=\"tr-reviewer\">Reviewer and review date<\/label><input id=\"tr-reviewer\" type=\"text\" maxlength=\"180\" autocomplete=\"off\"><\/div>\n        <fieldset>\n          <legend>Eight applicability and decision gates<\/legend>\n          <div class=\"tr-checks\">\n            <label class=\"tr-check\" for=\"tr-ev-linear\"><input id=\"tr-ev-linear\" type=\"checkbox\">Linearity and time invariance are adequate over the stated amplitude and condition.<\/label>\n            <label class=\"tr-check\" for=\"tr-ev-sdof\"><input id=\"tr-ev-sdof\" type=\"checkbox\">One mode\/DOF is an adequate representation in the evaluated frequency region.<\/label>\n            <label class=\"tr-check\" for=\"tr-ev-harmonic\"><input id=\"tr-ev-harmonic\" type=\"checkbox\">The input is steady-state harmonic and transients\/non-harmonic content are out of scope.<\/label>\n            <label class=\"tr-check\" for=\"tr-ev-viscous\"><input id=\"tr-ev-viscous\" type=\"checkbox\">A linear viscous damper model is adequate; friction, hysteresis and nonlinear mounts are addressed separately.<\/label>\n            <label class=\"tr-check\" for=\"tr-ev-frequency\"><input id=\"tr-ev-frequency\" type=\"checkbox\">The undamped natural frequency and excitation frequency refer to the same axis\/configuration.<\/label>\n            <label class=\"tr-check\" for=\"tr-ev-damping\"><input id=\"tr-ev-damping\" type=\"checkbox\">The damping estimate is applicable to this amplitude, temperature, preload and frequency region.<\/label>\n            <label class=\"tr-check\" for=\"tr-ev-quantity\"><input id=\"tr-ev-quantity\" type=\"checkbox\">Input\/output quantity, amplitude convention, axis and polarity are matched.<\/label>\n            <label class=\"tr-check\" for=\"tr-ev-decision\"><input id=\"tr-ev-decision\" type=\"checkbox\">Clearance, travel, load, stability, durability and acceptance decisions use a separate controlled procedure.<\/label>\n          <\/div>\n        <\/fieldset>\n      <\/div>\n      <div class=\"tr-actions\"><button type=\"submit\">Calculate declared model<\/button><button type=\"button\" class=\"tr-secondary\" id=\"tr-clear\">Clear<\/button><\/div>\n      <div id=\"tr-errors\" class=\"tr-errors\" role=\"alert\" aria-live=\"assertive\"><\/div>\n    <\/form>\n  <\/section>\n\n  <section id=\"tr-result\" class=\"tr-result\" data-state=\"empty\" aria-live=\"polite\" aria-atomic=\"true\">\n    <div id=\"tr-result-title\" class=\"tr-result-title\">No result<\/div>\n    <div id=\"tr-result-summary\" class=\"tr-result-summary\">Enter a traceable case and submit explicitly.<\/div>\n    <div class=\"tr-out-grid\">\n      <div class=\"tr-out\"><div class=\"tr-out-label\">Frequency ratio r = f \/ f<sub>n<\/sub><\/div><div id=\"tr-out-r\" class=\"tr-out-value\">\u2014<\/div><\/div>\n      <div class=\"tr-out\"><div class=\"tr-out-label\">Magnitude T = |H|<\/div><div id=\"tr-out-t\" class=\"tr-out-value\">\u2014<\/div><\/div>\n      <div class=\"tr-out\"><div class=\"tr-out-label\">Output phase relative to input<\/div><div id=\"tr-out-phase\" class=\"tr-out-value\">\u2014<\/div><\/div>\n      <div class=\"tr-out\"><div class=\"tr-out-label\">Signed amplitude change (T \u2212 1) \u00d7 100%<\/div><div id=\"tr-out-change\" class=\"tr-out-value\">\u2014<\/div><\/div>\n      <div class=\"tr-out\"><div class=\"tr-out-label\">Non-zero crossover comparison<\/div><div id=\"tr-out-crossover\" class=\"tr-out-value\">\u2014<\/div><\/div>\n      <div class=\"tr-out\"><div class=\"tr-out-label\">Evidence gates recorded<\/div><div id=\"tr-out-evidence\" class=\"tr-out-value\">0 of 8<\/div><\/div>\n    <\/div>\n    <div id=\"tr-model-note\" class=\"tr-model-note\">Select the physical input\/output ratio. A dimensionless number without that definition is ambiguous.<\/div>\n  <\/section>\n\n  <section class=\"tr-card\" aria-labelledby=\"tr-equations-heading\">\n    <h2 id=\"tr-equations-heading\">Equations, variables and scope<\/h2>\n    <p>For the declared base-motion model, H = X\/Y. For the declared force-transmission model, H = F<sub>transmitted<\/sub>\/F<sub>applied<\/sub> under the stated scalar positive-direction convention. In the same ideal spring\u2013mass\u2013viscous-damper arrangement they have the same complex form:<\/p>\n    <div class=\"tr-equations\">\n      <div class=\"tr-eq\"><code>r = f \/ f<sub>n<\/sub><\/code><span class=\"tr-small\">f and f<sub>n<\/sub> are in the same frequency unit; r is dimensionless.<\/span><\/div>\n      <div class=\"tr-eq\"><code>H(i r) = (1 + i 2 \u03b6 r) \/ (1 \u2212 r\u00b2 + i 2 \u03b6 r)<\/code><span class=\"tr-small\">This numerator is essential to both magnitude and phase.<\/span><\/div>\n      <div class=\"tr-eq\"><code>T = |H| = \u221a[1 + (2 \u03b6 r)\u00b2] \/ \u221a[(1 \u2212 r\u00b2)\u00b2 + (2 \u03b6 r)\u00b2]<\/code><span class=\"tr-small\">T is an amplitude ratio, not energy efficiency.<\/span><\/div>\n      <div class=\"tr-eq\"><code>\u03c6 = atan2(2 \u03b6 r, 1) \u2212 atan2(2 \u03b6 r, 1 \u2212 r\u00b2)<\/code><span class=\"tr-small\">Wrapped to \u2212180\u00b0\u2026+180\u00b0; output is relative to input under the stated polarity.<\/span><\/div>\n      <div class=\"tr-eq\"><code>amplitude change = (T \u2212 1) \u00d7 100%<\/code><span class=\"tr-small\">Negative means modelled amplitude reduction; positive means amplification.<\/span><\/div>\n      <div class=\"tr-eq\"><code>T = 1 at r = \u221a2 (non-zero crossover)<\/code><span class=\"tr-small\">This follows algebraically from this H and does not shift with \u03b6. At \u03b6 = 0 and r = 1 the ideal model is singular.<\/span><\/div>\n    <\/div>\n    <p class=\"tr-small\" style=\"margin-top:12px\"><strong>General engineering model, not an ISO formula.<\/strong> NASA sources below provide the spring\u2013mass\u2013damper equation, transfer function and magnitude relationship. The ISO records establish vocabulary, information-exchange and laboratory-measurement scope; public abstracts do not establish this calculator or an approval criterion.<\/p>\n  <\/section>\n\n  <section class=\"tr-card\" aria-labelledby=\"tr-sources-heading\">\n    <h2 id=\"tr-sources-heading\">Source and lifecycle register<\/h2>\n    <div class=\"tr-table-wrap\"><table><thead><tr><th>ID<\/th><th>Source \/ exact status<\/th><th>Used for<\/th><th>Not used for<\/th><\/tr><\/thead><tbody>\n      <tr id=\"source-S1\"><td>S1<\/td><td><a href=\"https:\/\/gipoc.grc.nasa.gov\/pims\/MMAP\/PIMS_ORIG\/MEIT\/MEIT_pdfs\/meit2004\/Section_17.pdf\" target=\"_blank\" rel=\"noopener\">NASA, Fundamentals of Microgravity Vibration Isolation, MEIT-2004, Section 17<\/a>, pp. 10 and 15<\/td><td>Base-excited SDOF equation and P(s) = (d s + k)\/(m s\u00b2 + d s + k). The displayed magnitude and phase are derived from the same P(i\u03c9).<\/td><td>No real-isolator acceptance, clearance, load, stability or durability criterion.<\/td><\/tr>\n      <tr id=\"source-S2\"><td>S2<\/td><td><a href=\"https:\/\/ntrs.nasa.gov\/citations\/20110024049\" target=\"_blank\" rel=\"noopener\">Niebuhr &amp; Hagen, Development of the Vibration Isolation System for the Advanced Resistive Exercise Device<\/a>, NASA NTRS 20110024049, publication record 2011, proceedings 2012, p. 4<\/td><td>Published transmissibility magnitude equation and a rounded application table. The Z-axis 0.11\/0.09 Hz, \u03b6=0.10 case gives model T\u22481.868 versus reported 1.9.<\/td><td>No universal damping range or approval threshold.<\/td><\/tr>\n      <tr id=\"source-S3\"><td>S3<\/td><td><a href=\"https:\/\/www.iso.org\/standard\/28919.html\" target=\"_blank\" rel=\"noopener\">ISO 2017-1:2005, Edition 1<\/a> \u2014 Published; last confirmed 2019; currently stage 90.60 (under review)<\/td><td>Public scope: information exchange among users, manufacturers and suppliers for isolation-system applications.<\/td><td>The public record does not issue this SDOF formula or prove suitability. Exact requirements: <span class=\"tr-status\">NEEDS_LICENSED_SOURCE<\/span>.<\/td><\/tr>\n      <tr id=\"source-S4\"><td>S4<\/td><td><a href=\"https:\/\/www.iso.org\/standard\/68734.html\" target=\"_blank\" rel=\"noopener\">ISO 2041:2018, Edition 4<\/a> \u2014 Published; confirmed 2024; stage 90.93<\/td><td>Current mechanical-vibration vocabulary lifecycle and scope.<\/td><td>Protected definitions\/clauses are not reproduced or inferred: <span class=\"tr-status\">NEEDS_LICENSED_SOURCE<\/span>.<\/td><\/tr>\n      <tr id=\"source-S5\"><td>S5<\/td><td><a href=\"https:\/\/www.iso.org\/standard\/38936.html\" target=\"_blank\" rel=\"noopener\">ISO 10846-1:2008, Edition 2<\/a> \u2014 Published; confirmed 2022; stage 90.93<\/td><td>Public scope: principles\/guidance for laboratory determination of resilient-element transfer properties and selection of the relevant series part.<\/td><td>No claim that this ideal analytical model is an ISO 10846 laboratory result. Exact procedure: <span class=\"tr-status\">NEEDS_LICENSED_SOURCE<\/span>.<\/td><\/tr>\n      <tr id=\"source-S6\"><td>S6<\/td><td><a href=\"https:\/\/www.iso.org\/standard\/34560.html\" target=\"_blank\" rel=\"noopener\">ISO 10846-3:2002, Edition 1<\/a> \u2014 Published; confirmed 2022; stage 90.93<\/td><td>Public scope: indirect laboratory method for dynamic transfer stiffness of resilient supports, including measurement of vibration transmissibility under stated test conditions.<\/td><td>No protected apparatus, validity-band or uncertainty detail is embedded: <span class=\"tr-status\">NEEDS_LICENSED_SOURCE<\/span>.<\/td><\/tr>\n    <\/tbody><\/table><\/div>\n    <p class=\"tr-small\">Source access and technical review: 17 July 2026. Recheck ISO 2017-1 before controlled future use because its official record is at review stage 90.60.<\/p>\n  <\/section>\n\n  <section class=\"tr-card\" aria-labelledby=\"tr-corrections-heading\">\n    <h2 id=\"tr-corrections-heading\">What was corrected<\/h2>\n    <details><summary>1. Phase now belongs to the displayed transfer function<\/summary><div>The former page used the numerator-inclusive magnitude but \u03c6 = atan2(2\u03b6r, 1\u2212r\u00b2), which is the phase of a different denominator-only response. The worksheet uses the complex numerator and denominator for both outputs. For f=50 Hz, f<sub>n<\/sub>=15 Hz and \u03b6=0.05. The former display was about +178.1\u00b0; the declared H gives \u2212159.68\u00b0.<\/div><\/details>\n    <details><summary>2. Force and motion ratios are no longer an unused switch<\/summary><div>The former selector never changed the calculation or defined the numerator and denominator. The selected model now changes the physical definition and enforces a compatible quantity. Relative displacement, insertion loss, power efficiency and multi-DOF transfer paths are not silently substituted.<\/div><\/details>\n    <details><summary>3. The crossover contradiction was removed<\/summary><div>For this H, the non-zero solution of T=1 is exactly r=\u221a2 for any finite \u03b6. The former FAQ said damping shifts it while the theory section said it is independent of damping.<\/div><\/details>\n    <details><summary>4. r=1 is not labelled as the exact damped peak<\/summary><div>The former plot labelled T(r=1) as \u201cPeak\u201d. For numerator-inclusive transmissibility and \u03b6&gt;0 the exact maximum is generally below r=1. The worksheet reports the requested point and does not invent a peak claim.<\/div><\/details>\n    <details><summary>5. \u201cIsolation efficiency\u201d was replaced by an amplitude statement<\/summary><div>(1\u2212T)\u00d7100% is an amplitude reduction under the declared model, not thermodynamic, power or installation efficiency. The worksheet reports signed amplitude change and keeps engineering approval separate.<\/div><\/details>\n    <details><summary>6. Defaults, auto-calculation and detached result state were removed<\/summary><div>No presets, implicit defaults, URL parameters, local storage, clipboard export, dynamic markup insertion, external formula CDN or automatic calculation remains. Full-string decimal point\/comma parsing, explicit submit, stale-state invalidation and error clearing are tested.<\/div><\/details>\n  <\/section>\n\n  <footer class=\"tr-foot\">This worksheet does not replace modal testing, an isolator supplier\u2019s controlled data, clearance\/travel checks, load and stability analysis, fatigue\/durability review, installation verification or an applicable licensed standard\/procedure.<\/footer>\n<\/main>\n<script>\n(function(){'use strict';\n  function byId(id){return document.getElementById(id)}\n  var form=byId('tr-form'),result=byId('tr-result'),errorsBox=byId('tr-errors');\n  var required=['tr-case','tr-asset','tr-receiver','tr-axis','tr-condition','tr-model','tr-quantity','tr-frequency','tr-natural','tr-zeta','tr-fn-basis','tr-zeta-basis','tr-reference','tr-reviewer'];\n  var gates=['tr-ev-linear','tr-ev-sdof','tr-ev-harmonic','tr-ev-viscous','tr-ev-frequency','tr-ev-damping','tr-ev-quantity','tr-ev-decision'];\n  var outputs=['tr-out-r','tr-out-t','tr-out-phase','tr-out-change','tr-out-crossover'];\n  function parseStrict(raw){var s=String(raw).trim();if(!\/^[+]?(?:\\d+(?:[.,]\\d*)?|[.,]\\d+)$\/.test(s)||(\/[.]\/.test(s)&&\/,\/.test(s)))return null;var n=Number(s.replace(',','.'));return Number.isFinite(n)?n:null}\n  function cleanText(id){return byId(id).value.trim()}\n  function fmt(n,d){if(!Number.isFinite(n))return '\u2014';if(n===0)return '0';var digits=Math.abs(n)>=10000?3:(d===undefined?6:d),s=n.toPrecision(digits);if(Math.abs(n)>=1e-4&&Math.abs(n)<1e7)s=Number(s).toString();return s}\n  function clearOutputs(){for(var i=0;i<outputs.length;i++)byId(outputs[i]).textContent='\u2014';byId('tr-out-evidence').textContent='0 of 8';byId('tr-model-note').textContent='Select the physical input\/output ratio. A dimensionless number without that definition is ambiguous.'}\n  function setErrors(items){errorsBox.textContent='';if(!items.length)return;var ul=document.createElement('ul');for(var i=0;i<items.length;i++){var li=document.createElement('li');li.textContent=items[i];ul.appendChild(li)}errorsBox.appendChild(ul)}\n  function modelText(model,quantity){if(model==='motion')return 'H = X\/Y for matched harmonic '+quantity+' amplitudes. Output phase is response X relative to base input Y.';return 'H = F_transmitted\/F_applied for the declared scalar positive direction. If instrumentation reports the opposite support-reaction polarity, correct that convention before comparing phase.'}\n  function setState(state,title,summary){result.dataset.state=state;byId('tr-result-title').textContent=title;byId('tr-result-summary').textContent=summary}\n  function markStale(){if(result.dataset.state==='ready'||result.dataset.state==='incomplete'||result.dataset.state==='singular'){setState('stale','Result is stale','The model, frequencies, damping, evidence or traceability record changed. Submit again before using the arithmetic.')}}\n  form.addEventListener('input',markStale);form.addEventListener('change',markStale);\n  form.addEventListener('submit',function(event){event.preventDefault();var issues=[];\n    for(var i=0;i<required.length;i++){var el=byId(required[i]);if(!String(el.value).trim())issues.push((el.labels&&el.labels[0]?el.labels[0].textContent:required[i])+' is required.')}\n    var rawF=byId('tr-frequency').value.trim(),rawFn=byId('tr-natural').value.trim(),rawZeta=byId('tr-zeta').value.trim(),f=parseStrict(rawF),fn=parseStrict(rawFn),zeta=parseStrict(rawZeta),model=byId('tr-model').value,quantity=byId('tr-quantity').value;\n    if(rawF&&(f===null||f<=0||f>1e12))issues.push('Excitation frequency must be finite, greater than 0 and no greater than 1e12 Hz.');\n    if(rawFn&&(fn===null||fn<=0||fn>1e12))issues.push('Undamped natural frequency must be finite, greater than 0 and no greater than 1e12 Hz.');\n    if(rawZeta&&(zeta===null||zeta<0||zeta>1e6))issues.push('Damping ratio must be finite, non-negative and no greater than the 1e6 software guard.');\n    if(model==='motion'&&quantity==='force')issues.push('Base-motion transmissibility requires displacement, velocity or acceleration on both sides, not force.');\n    if(model==='force'&&quantity!=='force')issues.push('Force transmissibility requires force for both numerator and denominator.');\n    if(issues.length){clearOutputs();setErrors(issues);setState('error','Cannot calculate','Correct the listed fields. Previous outputs were cleared.');return}\n    var r=f\/fn,a=2*zeta*r,dr=1-r*r,den=dr*dr+a*a,count=0;for(var g=0;g<gates.length;g++)if(byId(gates[g]).checked)count++;\n    setErrors([]);byId('tr-out-r').textContent=fmt(r,8);byId('tr-out-evidence').textContent=count+' of 8';byId('tr-model-note').textContent=modelText(model,quantity);\n    if(den===0){byId('tr-out-t').textContent='unbounded (ideal model)';byId('tr-out-phase').textContent='undefined at singularity';byId('tr-out-change').textContent='not finite';byId('tr-out-crossover').textContent='r < \u221a2 but \u03b6 = 0, r = 1 is singular';setState('singular','Ideal undamped resonance is singular','The ideal linear model has no finite steady-state amplitude at \u03b6 = 0 and r = 1. Real behaviour requires damping, nonlinearity, limits and transient analysis.');return}\n    var hRe=(dr+a*a)\/den,hIm=(-a*r*r)\/den,T=Math.hypot(hRe,hIm),phase=Math.atan2(hIm,hRe)*180\/Math.PI,change=(T-1)*100,sq2=Math.SQRT2,tol=64*Number.EPSILON*Math.max(1,Math.abs(r),sq2),cross;\n    if(![r,hRe,hIm,T,phase,change].every(Number.isFinite)){clearOutputs();setState('error','Cannot calculate','The calculation exceeded finite numerical range. Previous outputs were cleared.');return}\n    if(Math.abs(r-sq2)<=tol){T=1;change=0;cross='At r = \u221a2: T = 1 crossover'}else if(r<sq2)cross='Below \u221a2: amplitude amplification (T > 1)';else cross='Above \u221a2: amplitude attenuation (T < 1)';\n    byId('tr-out-t').textContent=fmt(T,8);byId('tr-out-phase').textContent=fmt(phase,8)+'\u00b0';byId('tr-out-change').textContent=(change>0?'+':'')+fmt(change,8)+'%';byId('tr-out-crossover').textContent=cross;\n    if(count===8)setState('ready','Model result \u2014 evidence record complete','All eight gates are recorded. This remains an ideal SDOF model result, not ISO compliance or isolator approval.');else setState('incomplete','Model result \u2014 evidence incomplete','The arithmetic is shown, but only '+count+' of 8 applicability\/decision gates are recorded. Do not use it as an approval.');\n  });\n  byId('tr-clear').addEventListener('click',function(){form.reset();setErrors([]);clearOutputs();setState('empty','No result','Enter a traceable case and submit explicitly.');byId('tr-case').focus()});\n})();\n<\/script>\n\n","protected":false},"excerpt":{"rendered":"<p>Calculate base-motion or force-transmission magnitude and phase for a declared linear viscously damped SDOF model. No ISO compliance or isolator approval.<\/p>","protected":false},"featured_media":0,"template":"","meta":{"ai_generated_summary":"","footnotes":""},"categories":[],"tags":[],"class_list":["post-100298","calculator","type-calculator","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/vibromera.eu\/bg\/wp-json\/wp\/v2\/calculator\/100298","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/bg\/wp-json\/wp\/v2\/calculator"}],"about":[{"href":"https:\/\/vibromera.eu\/bg\/wp-json\/wp\/v2\/types\/calculator"}],"version-history":[{"count":4,"href":"https:\/\/vibromera.eu\/bg\/wp-json\/wp\/v2\/calculator\/100298\/revisions"}],"predecessor-version":[{"id":102647,"href":"https:\/\/vibromera.eu\/bg\/wp-json\/wp\/v2\/calculator\/100298\/revisions\/102647"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/bg\/wp-json\/wp\/v2\/media?parent=100298"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/bg\/wp-json\/wp\/v2\/categories?post=100298"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/bg\/wp-json\/wp\/v2\/tags?post=100298"}],"curies":[{"name":"\u0440\u0430\u0431\u043e\u0442\u043d\u0430 \u0441\u0440\u0435\u0449\u0430","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}