{"id":100210,"date":"2026-02-15T20:27:34","date_gmt":"2026-02-15T20:27:34","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100210"},"modified":"2026-07-13T23:19:47","modified_gmt":"2026-07-13T23:19:47","slug":"rotor-critical-speed","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/sr\/calculators\/rotor-critical-speed\/","title":{"rendered":"Documented Point-Mass Lateral-Frequency Worksheet"},"content":{"rendered":"\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Documented Point-Mass Lateral-Frequency Worksheet\",\"description\":\"Calculate an equivalent undamped point-mass lateral natural-frequency marker from documented effective mass and lateral stiffness, with an optional ideal Euler-Bernoulli solid-shaft helper and explicit rotor-dynamic limits.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/rotor-critical-speed\/\",\"applicationCategory\":\"EngineeringApplication\",\"operatingSystem\":\"Any\",\"offers\":{\"@type\":\"Offer\",\"price\":\"0\"},\"creator\":{\"@type\":\"Organization\",\"name\":\"Vibromera\",\"url\":\"https:\/\/vibromera.eu\/\"},\"dateModified\":\"2026-07-14\",\"inLanguage\":\"en\",\"isAccessibleForFree\":true}<\/script>\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Does the worksheet calculate the first critical speed of a real rotor?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"No. It calculates the undamped natural-frequency marker of one documented effective point mass and one linear lateral stiffness. A real rotor-bearing-seal train can have multiple speed-dependent modes and requires a rotor-dynamic model and response evidence.\"}},{\"@type\":\"Question\",\"name\":\"Is the result an ISO or API acceptance verdict?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"No. ISO 21940-11 and ISO 21940-12 concern rotor balancing behaviour and tolerances; ISO 21940-12 states that structural resonances are outside its scope. Exact API equipment criteria depend on the applicable licensed edition, equipment scope and clauses.\"}},{\"@type\":\"Question\",\"name\":\"When may the solid-shaft helper be used?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Only for the displayed ideal Euler-Bernoulli case: a uniform solid circular massless beam, constant documented Young modulus, one point mass at the stated location, and ideal support boundary conditions. It does not include shaft mass, real bearings, damping, gyroscopic effects, seals or housings.\"}},{\"@type\":\"Question\",\"name\":\"Why is there no safe or danger zone?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"A universal percentage band would ignore the applicable machine standard, mode, damping, excitation order, operating range, transient passage and response analysis. The worksheet reports arithmetic comparison only and does not issue an engineering verdict.\"}}]}<\/script>\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\":\"Point-mass lateral frequency\",\"item\":\"https:\/\/vibromera.eu\/calculators\/rotor-critical-speed\/\"}]}<\/script>\n<style>\n:root{--vc-surface:#fff;--vc-alt:#f8f6f2;--vc-ink:#1a1a1a;--vc-secondary:#5a5650;--vc-muted:#807b73;--vc-accent:#b84f22;--vc-accent-light:#fdf0ea;--vc-yellow:#825f00;--vc-yellow-light:#fff8dc;--vc-red:#9d2b24;--vc-red-light:#fff0ee;--vc-border:#d9d4cc;--vc-border-light:#e8e4dd;--vc-shadow:0 1px 3px rgba(26,26,26,.06),0 4px 12px 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12px;color:var(--vc-muted);font-size:12px}.vc-footer a{color:var(--vc-accent)}@media(max-width:820px){.vc-grid,.vc-mode-panel.vc-active{grid-template-columns:1fr 1fr}.vc-result-grid{grid-template-columns:1fr 1fr}}@media(max-width:560px){.vc-grid,.vc-mode-panel.vc-active,.vc-result-grid{grid-template-columns:1fr}.vc-pair{grid-template-columns:minmax(0,1fr) 112px}.vc-form,.vc-results{padding:18px}.vc-results-head{align-items:flex-start;flex-direction:column}}@media print{.vc-section-body,.vc-results{display:block!important}.vc-copy,.vc-chevron,.vc-actions{display:none}}\n<\/style>\n<div class=\"vc-calculator\">\n<header class=\"vc-header\"><p class=\"vc-eyebrow\">Documented one-DOF linear model \u00b7 no equipment verdict<\/p><h1 class=\"vc-title\">Documented Point-Mass Lateral-Frequency Worksheet<\/h1><p class=\"vc-subtitle\">Calculate the equivalent undamped natural-frequency marker of one documented effective point mass and one linear lateral stiffness at the same degree of freedom. The result is not automatically the first critical speed of a real rotor and is not an ISO or API acceptance calculation.<\/p><div class=\"vc-badges\"><span class=\"vc-badge\">\u03c9<sub>n<\/sub> = \u221a(k<sub>eff<\/sub>\/m<sub>eff<\/sub>)<\/span><span class=\"vc-badge\">Direct stiffness or ideal beam helper<\/span><span class=\"vc-badge\">No universal separation band<\/span><span class=\"vc-badge\">No pass\/fail<\/span><\/div><\/header>\n<div class=\"vc-card\"><form class=\"vc-form\" id=\"vc-form\" novalidate><div class=\"vc-grid\">\n<div class=\"vc-subhead\">Documented model inputs<\/div>\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-mode\">Stiffness input method<\/label><select class=\"vc-select\" id=\"vc-mode\"><option value=\"direct\">Documented equivalent lateral stiffness<\/option><option value=\"beam\">Ideal solid circular beam helper<\/option><\/select><span class=\"vc-hint\">Changing method clears method-specific values and prior results<\/span><\/div>\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-mass\">Effective point mass m<sub>eff<\/sub><\/label><div class=\"vc-pair\"><input class=\"vc-input\" id=\"vc-mass\" inputmode=\"decimal\" autocomplete=\"off\"><select class=\"vc-select\" id=\"vc-mass-unit\" aria-label=\"Mass unit\"><option value=\"kg\">kg<\/option><option value=\"lbm\">lbm<\/option><\/select><\/div><span class=\"vc-hint\">At the selected lateral degree of freedom; not automatically total rotor mass<\/span><\/div>\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-forcing-speed\">Optional documented synchronous forcing speed<\/label><input class=\"vc-input\" id=\"vc-forcing-speed\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"rpm; leave blank if not compared\"><span class=\"vc-hint\">Non-negative magnitude; output is arithmetic only<\/span><\/div>\n<div class=\"vc-mode-panel vc-active\" id=\"vc-direct-panel\">\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-stiffness\">Equivalent linear lateral stiffness k<sub>eff<\/sub><\/label><div class=\"vc-pair\"><input class=\"vc-input\" id=\"vc-stiffness\" inputmode=\"decimal\" autocomplete=\"off\"><select class=\"vc-select\" id=\"vc-stiffness-unit\" aria-label=\"Stiffness unit\"><option value=\"Nm\">N\/m<\/option><option value=\"Nmm\">N\/mm<\/option><option value=\"lbfin\">lbf\/in<\/option><\/select><\/div><span class=\"vc-hint\">At the same degree of freedom and direction as m<sub>eff<\/sub><\/span><\/div>\n<\/div>\n<div class=\"vc-mode-panel\" id=\"vc-beam-panel\">\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-support\">Ideal boundary and point-mass location<\/label><select class=\"vc-select\" id=\"vc-support\"><option value=\"simply_mid\">Pin\u2013pin, point mass at midspan (C = 48)<\/option><option value=\"fixed_mid\">Clamped\u2013clamped, point mass at midspan (C = 192)<\/option><option value=\"cantilever_end\">Clamped\u2013free, point mass at free end (C = 3)<\/option><\/select><span class=\"vc-hint\">Support choice also fixes the assumed mass\/load location<\/span><\/div>\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-length\">Ideal beam span\/length L<\/label><div class=\"vc-pair\"><input class=\"vc-input\" id=\"vc-length\" inputmode=\"decimal\" autocomplete=\"off\"><select class=\"vc-select\" id=\"vc-length-unit\" aria-label=\"Length unit\"><option value=\"mm\">mm<\/option><option value=\"in\">in<\/option><\/select><\/div><span class=\"vc-hint\">Between ideal supports, or clamp to free end<\/span><\/div>\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-diameter\">Uniform solid diameter d<\/label><input class=\"vc-input\" id=\"vc-diameter\" inputmode=\"decimal\" autocomplete=\"off\"><span class=\"vc-hint\">Uses the same length unit as L<\/span><\/div>\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-modulus\">Documented Young modulus E<\/label><div class=\"vc-pair\"><input class=\"vc-input\" id=\"vc-modulus\" inputmode=\"decimal\" autocomplete=\"off\"><select class=\"vc-select\" id=\"vc-modulus-unit\" aria-label=\"Young modulus unit\"><option value=\"GPa\">GPa<\/option><option value=\"ksi\">ksi<\/option><\/select><\/div><span class=\"vc-hint\">Use alloy-, temperature- and state-specific controlled data<\/span><\/div>\n<\/div>\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-system\">Machine, lateral degree of freedom and direction<\/label><input class=\"vc-input\" id=\"vc-system\" autocomplete=\"off\" placeholder=\"Equipment ID; station\/DOF; x or y direction\"><\/div>\n<div class=\"vc-field vc-wide\"><label class=\"vc-label\" for=\"vc-mass-record\">Effective-mass source and reduction record<\/label><input class=\"vc-input\" id=\"vc-mass-record\" autocomplete=\"off\" placeholder=\"Drawing, modal model or test; revision\/date; why this mass belongs at this DOF\"><\/div>\n<div class=\"vc-field vc-wide\"><label class=\"vc-label\" for=\"vc-stiffness-record\">Stiffness or beam-property source and idealization record<\/label><input class=\"vc-input\" id=\"vc-stiffness-record\" autocomplete=\"off\" placeholder=\"Static\/modal test or model; or drawing\/material source plus support and point-mass idealization\"><\/div>\n<div class=\"vc-field vc-wide\"><label class=\"vc-label\" for=\"vc-speed-record\">Optional forcing-speed source and applicable operating\/excitation record<\/label><input class=\"vc-input\" id=\"vc-speed-record\" autocomplete=\"off\" placeholder=\"Required only when a forcing speed is entered; source\/revision, excitation order and operating state\"><\/div>\n<label class=\"vc-check vc-wide\" for=\"vc-confirm\"><input id=\"vc-confirm\" type=\"checkbox\"><span>I confirm that m<sub>eff<\/sub> and k<sub>eff<\/sub> describe the same linear lateral degree of freedom and direction; both are positive and applicable to the documented state. If the beam helper is used, I accept its uniform solid massless Euler\u2013Bernoulli beam, constant E, ideal support and single point-mass assumptions. I will treat N<sub>eq<\/sub> only as an undamped one-DOF speed-equivalent marker\u2014not as a certified critical speed, safe operating speed, ISO\/API verdict or substitute for rotor-dynamic response analysis.<\/span><\/label>\n<div class=\"vc-actions vc-wide\"><button class=\"vc-calc-btn\" type=\"submit\">Calculate documented frequency marker<\/button><\/div>\n<\/div><\/form><div class=\"vc-error\" id=\"vc-error\" role=\"alert\"><\/div>\n<div class=\"vc-results\" id=\"vc-results\" aria-live=\"polite\"><div class=\"vc-results-head\"><div><h2 class=\"vc-results-title\">Equivalent undamped point-mass result<\/h2><div class=\"vc-results-basis\" id=\"vc-results-basis\">\u2014<\/div><\/div><button type=\"button\" class=\"vc-copy\" id=\"vc-copy\">Copy record<\/button><\/div><div class=\"vc-result-grid\">\n<div class=\"vc-result vc-primary\"><div class=\"vc-result-label\">Speed-equivalent marker N<sub>eq<\/sub><\/div><div class=\"vc-result-value\" id=\"vc-speed-result\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Natural frequency f<sub>n<\/sub><\/div><div class=\"vc-result-value\" id=\"vc-frequency-result\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Angular natural frequency \u03c9<sub>n<\/sub><\/div><div class=\"vc-result-value\" id=\"vc-omega-result\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Normalized effective mass<\/div><div class=\"vc-result-value\" id=\"vc-mass-result\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Normalized effective stiffness<\/div><div class=\"vc-result-value\" id=\"vc-stiffness-result\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Compliance 1\/k<sub>eff<\/sub><\/div><div class=\"vc-result-value\" id=\"vc-compliance-result\">\u2014<\/div><\/div>\n<div class=\"vc-result\" id=\"vc-beam-result-box\"><div class=\"vc-result-label\">Ideal beam I and L\/d<\/div><div class=\"vc-result-value\" id=\"vc-beam-result\">Not used<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Optional forcing-speed comparison<\/div><div class=\"vc-result-value\" id=\"vc-comparison-result\">Not entered<\/div><\/div>\n<\/div><div class=\"vc-danger\"><strong>Model boundary:<\/strong> this is one linear, undamped, stationary-coordinate point-mass\/stiffness model. It omits distributed shaft mass, multiple discs and modes, bearing\/support\/housing flexibility and damping, cross-coupling and anisotropy, seals and fluids, gyroscopic and rotary-inertia effects, shear deformation, coupling\/train interaction, temperature\/state changes, nonlinearities and speed-dependent coefficients. Do not use its marker as a machine clearance, run\/avoid, or acceptance verdict.<\/div><\/div><\/div>\n\n<section class=\"vc-section vc-open\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"true\"><span class=\"vc-section-title\">Implemented equations and dimensional check<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\">\n<h3>Direct documented-stiffness model<\/h3><div class=\"vc-formula\">m<sub>eff<\/sub>x\u0308 + k<sub>eff<\/sub>x = 0<br>\u03c9<sub>n<\/sub> = \u221a(k<sub>eff<\/sub>\/m<sub>eff<\/sub>) rad\/s<br>f<sub>n<\/sub> = \u03c9<sub>n<\/sub>\/(2\u03c0) Hz<br>N<sub>eq<\/sub> = 60f<sub>n<\/sub> rpm<\/div>\n<p>The ratio k\/m has units (N\/m)\/kg = s<sup>\u22122<\/sup>; its square root is rad\/s, with the radian dimensionless. N<sub>eq<\/sub> is only the rotational speed whose 1\u00d7 frequency equals f<sub>n<\/sub>. It is not a claim that an actual rotor will have its first response peak at that speed.<\/p>\n<h3>Optional ideal solid circular beam helper<\/h3><div class=\"vc-formula\">I = \u03c0d<sup>4<\/sup>\/64<br>k<sub>beam<\/sub> = CEI\/L<sup>3<\/sup><br>C = 48: pin\u2013pin, central point mass\/load<br>C = 192: clamped\u2013clamped, central point mass\/load<br>C = 3: clamped\u2013free, point mass\/load at free end<\/div>\n<p>The helper derives static point-load stiffness from Euler\u2013Bernoulli deflection formulas and then uses the same one-DOF frequency equation. The shaft itself is treated as massless; E, I and L are constant; and the support and point-mass location are ideal. The arithmetic does not become a distributed-beam eigenfrequency calculation.<\/p>\n<\/div><\/div><\/section>\n\n<section class=\"vc-section\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-title\">Unit normalization<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\"><div class=\"vc-table-wrap\"><table class=\"vc-table\"><thead><tr><th>Entered quantity<\/th><th>Internal SI conversion<\/th><th>Boundary<\/th><\/tr><\/thead><tbody>\n<tr><td>lbm<\/td><td>1 lbm = 0.45359237 kg exactly<\/td><td>Mass, not pound-force<\/td><\/tr>\n<tr><td>N\/mm<\/td><td>1 N\/mm = 1000 N\/m<\/td><td>Linear stiffness<\/td><\/tr>\n<tr><td>lbf\/in<\/td><td>1 lbf\/in = 4.4482216152605\/0.0254 = 175.126835246&#8230; N\/m<\/td><td>Uses the standard pound-force and international inch<\/td><\/tr>\n<tr><td>in<\/td><td>1 in = 0.0254 m exactly<\/td><td>L and d must use the same selected unit<\/td><\/tr>\n<tr><td>GPa<\/td><td>1 GPa = 10<sup>9<\/sup> Pa<\/td><td>E is not selected from a universal material preset<\/td><\/tr>\n<tr><td>ksi<\/td><td>1 ksi = 1000 lbf\/in\u00b2 = 6,894,757.293168&#8230; Pa<\/td><td>Pressure\/stress unit for E<\/td><\/tr>\n<\/tbody><\/table><\/div><p>The exact pound and inch basis follows NIST <a href=\"https:\/\/www.nist.gov\/pml\/special-publication-811\/nist-guide-si-appendix-b-conversion-factors\/nist-guide-si-appendix-b8\" target=\"_blank\" rel=\"noopener\">SP 811 Appendix B.8<\/a>. Changing a unit clears the affected numeric field and prior result rather than silently reinterpreting a number.<\/p><\/div><\/div><\/section>\n\n<section class=\"vc-section\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-title\">Primary evidence and published numerical check<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\">\n<h3>NASA one-DOF rotor boundary<\/h3><p>NASA\/CR\u20142004-213069, <a href=\"https:\/\/ntrs.nasa.gov\/api\/citations\/20040073438\/downloads\/20040073438.pdf?attachment=true\" target=\"_blank\" rel=\"noopener\">Disk Crack Detection for Seeded Fault Engine Test<\/a>, presents a simplified one-degree-of-freedom Jeffcott rotor with disk mass M and shaft stiffness k<sub>s<\/sub>. It gives \u03c9<sub>cr<\/sub> = \u221a(k<sub>s<\/sub>\/M) and limits that representation to single-disk assemblies under relatively rigid bearings at relatively low speeds near or below the first bending critical. The report also shows that response depends on speed ratio and damping; it does not support the former page\u2019s statement that vibration grows \u201cexponentially.\u201d<\/p>\n<h3>Published NPTEL arithmetic reproduced<\/h3><p>NPTEL\/IIT Guwahati, <a href=\"https:\/\/archive.nptel.ac.in\/content\/storage2\/courses\/112103024\/module2\/lec1\/5.html\" target=\"_blank\" rel=\"noopener\">Single-DOF Damped Rotor Model, examples 2.1\u20132.2<\/a>, uses m = 10 kg and k = 100 kN\/m to obtain \u03c9<sub>n<\/sub> = 100 rad\/s. Direct mode reproduces 100 rad\/s, 15.915494309&#8230; Hz and 954.929658551&#8230; rpm. This is a public university worked example of the one-DOF relation, not a universal machine acceptance criterion.<\/p>\n<h3>Beam-helper provenance<\/h3><p>MIT OpenCourseWare 2.080, <a href=\"https:\/\/ocw.mit.edu\/courses\/2-080j-structural-mechanics-fall-2013\/3533c046dafcc488f1432e92d05ec210_MIT2_080JF13_Lecture5.pdf\" target=\"_blank\" rel=\"noopener\">Structural Mechanics Lecture 5<\/a>, gives central point-load deflections PL\u00b3\/(48EI) for pin\u2013pin support and, by its clamped\u2013clamped point-load expression, PL\u00b3\/(192EI) at midspan. MIT 1.050 <a href=\"https:\/\/ocw.mit.edu\/courses\/1-050-solid-mechanics-fall-2004\/fd4eff39aec922b8c07660006f40686e_pset04_11.pdf\" target=\"_blank\" rel=\"noopener\">Problem Set 11 reference sheet<\/a> gives PL\u00b3\/(3EI) for an end-loaded cantilever. Inverting deflection\/load yields the three helper stiffnesses.<\/p>\n<\/div><\/div><\/section>\n\n<section class=\"vc-section\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-title\">ISO and API scope\u2014what this worksheet does not claim<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\">\n<div class=\"vc-table-wrap\"><table class=\"vc-table\"><thead><tr><th>Reference<\/th><th>Verified public status\/scope on 13 July 2026<\/th><th>Treatment here<\/th><\/tr><\/thead><tbody>\n<tr><td><a href=\"https:\/\/www.iso.org\/standard\/54074.html\" target=\"_blank\" rel=\"noopener\">ISO 21940-11:2016<\/a>, Edition 1; Amendment 1:2022<\/td><td>Published; under systematic review at stage 90.20. It establishes balancing procedures and unbalance tolerances for rotors with rigid behaviour. ISO 1940-1:2003 is shown as withdrawn and replaced.<\/td><td>No structural critical-speed formula or \u201cbelow 70%\u201d rigid-rotor rule is attributed to it.<\/td><\/tr>\n<tr><td><a href=\"https:\/\/www.iso.org\/standard\/50429.html\" target=\"_blank\" rel=\"noopener\">ISO 21940-12:2016<\/a>, Edition 1<\/td><td>Published and confirmed in 2025 at stage 90.93. It concerns balancing rotors with flexible behaviour and explicitly places structural resonances and their modification outside its scope.<\/td><td>No claim that this worksheet classifies rigid\/flexible behaviour or demonstrates balancing conformity.<\/td><\/tr>\n<tr><td>API Std 610, 612 and 617<\/td><td>The official <a href=\"https:\/\/www.api.org\/products-and-services\/standards\/standards-plan\" target=\"_blank\" rel=\"noopener\">API Standards Plan<\/a> lists Std 610 Edition 13 (29 June 2026), Std 612 Edition 8 (1 November 2020), and Std 617 Edition 9 (1 April 2022). API TR 684-1 Edition 1 was published in 2019 and Edition 2 is under development.<\/td><td>Former universal 115%\/120% and \u00b120% claims are removed. Exact applicability and separation\/response clauses remain <strong>NEEDS_LICENSED_SOURCE<\/strong> for the selected equipment, edition and contract.<\/td><\/tr>\n<\/tbody><\/table><\/div><div class=\"vc-warning\"><strong>No substitute standard:<\/strong> an owner, purchaser, equipment standard or project specification may require a particular lateral analysis, separation margin, damping assumption, unbalance response, train model or acceptance evidence. Identify and apply the licensed requirement; do not infer it from this worksheet.<\/div>\n<\/div><\/div><\/section>\n\n<section class=\"vc-section\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-title\">Why real rotor critical speeds require a fuller model<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\">\n<p>A real rotor-bearing system can have several lateral modes. Natural frequencies and response may change with rotational speed because of gyroscopic effects, bearing and seal coefficients, temperature and operating state. Critical speed is tied to an excitation intersection and response, not merely a static solid-shaft stiffness divided by a disk mass.<\/p>\n<ul><li>Use a controlled mass\/stiffness or finite-element rotor model with actual stations, distributed shaft mass, discs, couplings and overhangs.<\/li><li>Represent bearing, support, housing, seal and fluid coefficients with their applicable speed\/load\/state dependence.<\/li><li>Include rotary inertia, gyroscopic moments, shear deformation and anisotropy when material to the model.<\/li><li>Evaluate the applicable excitation orders on a Campbell diagram and calculate damped unbalance\/forced response as required.<\/li><li>Validate with run-up\/coast-down, modal or other controlled evidence where appropriate, and apply the licensed equipment\/project acceptance criteria.<\/li><\/ul>\n<p>The optional forcing-speed output reports only r = N<sub>forcing<\/sub>\/N<sub>eq<\/sub> and \u0394N = N<sub>forcing<\/sub> \u2212 N<sub>eq<\/sub>. It deliberately does not convert either number into \u201cOK,\u201d \u201ccaution,\u201d \u201cdanger,\u201d \u201csafe\u201d or \u201cunsafe.\u201d<\/p>\n<\/div><\/div><\/section>\n\n<section class=\"vc-section\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-title\">Confirmed defects removed from the former calculator<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\"><div class=\"vc-table-wrap\"><table class=\"vc-table\"><thead><tr><th>Former content or behaviour<\/th><th>Problem and correction<\/th><\/tr><\/thead><tbody>\n<tr><td>\u201cRayleigh method\u201d and distributed-mass capability<\/td><td>The code used only \u221a(k\/m) with a point mass and three static stiffness formulas; no distributed shaft mass or Rayleigh quotient was implemented. The replacement names the actual point-mass model.<\/td><\/tr>\n<tr><td>Result labelled first critical speed<\/td><td>The model cannot establish the first critical of a real rotor-bearing-seal system. It now reports an equivalent undamped speed-frequency marker.<\/td><\/tr>\n<tr><td>Universal \u00b120%\/\u00b140% danger, caution and OK zones<\/td><td>No controlled source or machine scope supported those verdicts. They are removed; optional comparison is arithmetic only.<\/td><\/tr>\n<tr><td>Universal API 610\/612\/617 percentage claims<\/td><td>Edition, equipment scope, clause and operating definition were not controlled. Exact criteria are not guessed and are marked NEEDS_LICENSED_SOURCE.<\/td><\/tr>\n<tr><td>\u201cRigid rotor typically below 70% of critical\u201d attributed around ISO balancing<\/td><td>ISO 21940-11\/-12 address balancing behaviour; the public ISO 21940-12 scope excludes structural resonances. The unsupported shortcut and withdrawn ISO 1940 label are removed.<\/td><\/tr>\n<tr><td>Steel\/stainless\/aluminium modulus presets<\/td><td>Young modulus depends on the controlled material, alloy, condition and temperature. The replacement requires a documented value and has no authoritative-looking preset.<\/td><\/tr>\n<tr><td>Text example 6029 N\/mm, 448.2 rad\/s, 4280 rpm<\/td><td>The former code\u2019s displayed inputs give 6040.049&#8230; N\/mm, 448.703&#8230; rad\/s and 4284.804&#8230; rpm, so prose and code disagreed. The new published example is generated by the same audited model and independently tested.<\/td><\/tr>\n<tr><td>\u201cVibration amplifies exponentially\u201d<\/td><td>The standard damped one-DOF response is a rational function of frequency ratio and damping, not exponential growth. The misleading wording is removed.<\/td><\/tr>\n<tr><td>Defaults, presets, auto-calculation, partial parsing and persisted state<\/td><td>The page could issue an apparently authoritative answer without a controlled model record. It now starts blank, validates complete finite inputs, requires provenance and confirmation, and calculates only on explicit submit.<\/td><\/tr>\n<\/tbody><\/table><\/div><\/div><\/div><\/section>\n\n<section class=\"vc-section\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-title\">FAQ<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\">\n<div class=\"vc-faq\"><button type=\"button\">Can I enter total rotor mass?<\/button><div>Only if the controlled one-DOF reduction shows that total mass is the correct effective mass at the selected lateral coordinate. For a real mode, effective\/modal mass depends on the coordinate and mode shape.<\/div><\/div>\n<div class=\"vc-faq\"><button type=\"button\">Does a fixed\u2013fixed selection describe ordinary bearings?<\/button><div>No. It means an ideal beam with zero translation and zero slope at both ends. Real bearing, housing and pedestal flexibility must be represented in the actual rotor model or in a documented equivalent stiffness.<\/div><\/div>\n<div class=\"vc-faq\"><button type=\"button\">Does resonance amplitude become infinite?<\/button><div>Only the ideal undamped linear steady-state model has an unbounded mathematical response exactly at resonance. Real response depends on damping, forcing, nonlinear limits and the complete system; this worksheet calculates no amplitude.<\/div><\/div>\n<div class=\"vc-faq\"><button type=\"button\">May I use the result to choose an operating speed?<\/button><div>No. Identify the applicable licensed standard\/specification and excitation orders, build the required rotor-dynamic model, evaluate damped response and transients, and obtain the responsible engineering review.<\/div><\/div>\n<\/div><\/div><\/section>\n\n<section class=\"vc-section\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-title\">Related calculators<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\"><div class=\"vc-related\"><a href=\"\/calculators\/shaft-deflection\/\">Shaft-deflection worksheet<\/a><a href=\"\/calculators\/natural-frequency\/\">Natural-frequency worksheet<\/a><a href=\"\/calculators\/bearing-stiffness\/\">Bearing-stiffness worksheet<\/a><\/div><\/div><\/div><\/section>\n<footer class=\"vc-footer\"><p><a href=\"https:\/\/vibromera.eu\/\">Vibromera<\/a> engineering reference worksheet \u00b7 revised 14 July 2026<\/p><\/footer>\n<\/div>\n<script>\n(function(){'use strict';\nvar LB_TO_KG=0.45359237,LBF_TO_N=4.4482216152605,IN_TO_M=0.0254,G0=9.80665,NMM_TO_NM=1000,LBFIN_TO_NM=LBF_TO_N\/IN_TO_M,PSI_TO_PA=LBF_TO_N\/(IN_TO_M*IN_TO_M),KSI_TO_PA=1000*PSI_TO_PA,SUPPORT_C={simply_mid:48,fixed_mid:192,cantilever_end:3};\nfunction parseNumber(value){var s=String(value==null?'':value).trim().replace(',','.');if(!\/^[+-]?(?:(?:\\d+(?:\\.\\d*)?)|(?:\\.\\d+))(?:[eE][+-]?\\d+)?$\/.test(s))return null;var n=Number(s);return Number.isFinite(n)?n:null}\nfunction normalizeMass(value,unit){if(!Number.isFinite(value)||value<=0||!['kg','lbm'].includes(unit))throw new Error('mass');var m=value*(unit==='kg'?1:LB_TO_KG);if(!Number.isFinite(m)||m<=0)throw new 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