{"id":100209,"date":"2026-02-15T20:27:32","date_gmt":"2026-02-15T20:27:32","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100209"},"modified":"2026-07-13T07:59:15","modified_gmt":"2026-07-13T07:59:15","slug":"rotor-acceleration-time","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/fr\/calculators\/rotor-acceleration-time\/","title":{"rendered":"Documented Constant-Net-Torque Speed-Transition Worksheet"},"content":{"rendered":"\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Documented Constant-Net-Torque Speed-Transition Worksheet\",\"description\":\"Calculate a signed single-axis speed-transition time from documented effective inertia and constant net torque, with exact unit normalization and explicit model limits.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/rotor-acceleration-time\/\",\"applicationCategory\":\"EngineeringApplication\",\"operatingSystem\":\"Any\",\"offers\":{\"@type\":\"Offer\",\"price\":\"0\"},\"creator\":{\"@type\":\"Organization\",\"name\":\"Vibromera\",\"url\":\"https:\/\/vibromera.eu\/\"},\"dateModified\":\"2026-07-13\",\"inLanguage\":\"en\",\"isAccessibleForFree\":true}<\/script>\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Is this an ISO motor-sizing calculation?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"No. It is a general-mechanics constant-net-torque calculation based on net torque equals effective inertia times angular acceleration. It does not select a motor, drive, brake, coupling, gearbox or protection system and does not claim ISO conformity.\"}},{\"@type\":\"Question\",\"name\":\"What torque belongs in the worksheet?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Use the signed net torque at the same shaft and about the same positive axis as the entered speeds and effective inertia. It must already include the applicable driving and resisting torques and must remain constant over the whole transition.\"}},{\"@type\":\"Question\",\"name\":\"May I use average motor torque when torque varies with speed?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Not in general. For a monotonic transition with constant inertia and speed-dependent net torque, time is the integral of constant J divided by net torque(omega) with respect to omega. Time-, state- or configuration-dependent systems require the governing differential equations and drive limits.\"}},{\"@type\":\"Question\",\"name\":\"How is load inertia reflected through a gearbox?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Reflect every inertia to the calculation shaft by kinetic-energy equivalence: J reflected equals J source times the square of source angular speed divided by calculation-shaft angular speed. If i equals motor speed divided by load speed, load inertia reflected to the motor is J load divided by i squared.\"}}]}<\/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\":\"Constant-net-torque speed transition\",\"item\":\"https:\/\/vibromera.eu\/calculators\/rotor-acceleration-time\/\"}]}<\/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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.vc-chevron{transform:rotate(180deg)}.vc-section-body{display:none}.vc-section.vc-open .vc-section-body{display:block}.vc-section-inner{padding:0 22px 22px;border-top:1px solid var(--vc-border-light);color:var(--vc-secondary)}.vc-section-inner h3{font:700 17px var(--vc-display);color:var(--vc-ink);margin:22px 0 9px}.vc-section-inner p,.vc-section-inner li{font-size:14px}.vc-formula{font:500 13px\/1.75 var(--vc-mono);padding:13px 15px;border:1px solid var(--vc-border);border-radius:6px;background:var(--vc-alt);overflow-x:auto}.vc-warning{padding:13px 15px;border-left:4px solid var(--vc-yellow);background:var(--vc-yellow-light);color:#5d4708;border-radius:4px;margin:14px 0}.vc-danger{padding:13px 15px;border-left:4px solid var(--vc-red);background:var(--vc-red-light);color:var(--vc-red);border-radius:4px;margin:14px 0}.vc-table-wrap{overflow-x:auto}.vc-table{width:100%;border-collapse:collapse;margin:14px 0;font-size:12px}.vc-table th,.vc-table td{padding:8px 9px;border:1px solid var(--vc-border-light);text-align:left;vertical-align:top}.vc-table th{background:var(--vc-alt);color:var(--vc-ink)}.vc-faq{border:1px solid var(--vc-border-light);border-radius:6px;margin-top:8px}.vc-faq button{width:100%;padding:13px 14px;border:0;background:var(--vc-alt);font-weight:700;text-align:left;cursor:pointer}.vc-faq div{display:none;padding:13px 14px;border-top:1px solid var(--vc-border-light)}.vc-faq.vc-open div{display:block}.vc-related{display:flex;gap:9px;flex-wrap:wrap;margin-top:14px}.vc-related a{padding:7px 12px;border:1px solid var(--vc-border);border-radius:6px;text-decoration:none;color:var(--vc-secondary)}.vc-footer{text-align:center;padding:28px 12px;color:var(--vc-muted);font-size:12px}.vc-footer a{color:var(--vc-accent)}@media(max-width:820px){.vc-grid{grid-template-columns:1fr 1fr}.vc-result-grid{grid-template-columns:1fr 1fr}}@media(max-width:560px){.vc-grid,.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\">Single-axis general mechanics \u00b7 constant net torque<\/p><h1 class=\"vc-title\">Documented Constant-Net-Torque Speed-Transition Worksheet<\/h1><p class=\"vc-subtitle\">Calculate the time for one signed angular-speed transition from a documented total effective inertia and a constant signed net torque at the same shaft. The result is not a motor, drive or brake selection and is not an ISO conformity calculation.<\/p><div class=\"vc-badges\"><span class=\"vc-badge\">\u03c4<sub>net<\/sub> = J\u03b1<\/span><span class=\"vc-badge\">t = J(\u03c9<sub>2<\/sub> \u2212 \u03c9<sub>1<\/sub>)\/\u03c4<sub>net<\/sub><\/span><span class=\"vc-badge\">Signed speeds and power<\/span><span class=\"vc-badge\">No equipment verdict<\/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-inertia\">Total effective\/reflected inertia J<\/label><div class=\"vc-pair\"><input class=\"vc-input\" id=\"vc-inertia\" inputmode=\"decimal\" autocomplete=\"off\"><select class=\"vc-select\" id=\"vc-inertia-unit\" aria-label=\"Inertia unit\"><option value=\"kgm2\">kg\u00b7m\u00b2<\/option><option value=\"lbmft2\">lbm\u00b7ft\u00b2<\/option><option value=\"slugft2\">slug\u00b7ft\u00b2<\/option><\/select><\/div><span class=\"vc-hint\">About the calculation shaft and axis<\/span><\/div>\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-torque\">Constant signed net torque \u03c4<sub>net<\/sub><\/label><div class=\"vc-pair\"><input class=\"vc-input\" id=\"vc-torque\" inputmode=\"decimal\" autocomplete=\"off\"><select class=\"vc-select\" id=\"vc-torque-unit\" aria-label=\"Torque unit\"><option value=\"Nm\">N\u00b7m<\/option><option value=\"lbfft\">lbf\u00b7ft<\/option><\/select><\/div><span class=\"vc-hint\">Driving minus all resisting torque at the same shaft<\/span><\/div>\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-speed-unit\">Angular-speed unit<\/label><select class=\"vc-select\" id=\"vc-speed-unit\"><option value=\"rpm\">rpm<\/option><option value=\"Hz\">Hz (rev\/s)<\/option><option value=\"rads\">rad\/s<\/option><\/select><span class=\"vc-hint\">Changing a unit clears its affected numeric field(s)<\/span><\/div>\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-speed-initial\">Initial signed speed \u03c9<sub>1<\/sub><\/label><input class=\"vc-input\" id=\"vc-speed-initial\" inputmode=\"decimal\" autocomplete=\"off\"><span class=\"vc-hint\">Negative values mean the opposite chosen rotation<\/span><\/div>\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-speed-final\">Final signed speed \u03c9<sub>2<\/sub><\/label><input class=\"vc-input\" id=\"vc-speed-final\" inputmode=\"decimal\" autocomplete=\"off\"><span class=\"vc-hint\">May cross zero if the same constant torque remains valid<\/span><\/div>\n<div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-system\">Machine, shaft and positive-axis convention<\/label><input class=\"vc-input\" id=\"vc-system\" autocomplete=\"off\" placeholder=\"Equipment ID; calculation shaft; positive rotation\"><\/div>\n<div class=\"vc-field vc-wide\"><label class=\"vc-label\" for=\"vc-inertia-record\">Inertia source and reflection record<\/label><input class=\"vc-input\" id=\"vc-inertia-record\" autocomplete=\"off\" placeholder=\"Drawing\/test\/catalogue, revision\/date; included components; speed-ratio convention and reflection calculation\"><\/div>\n<div class=\"vc-field vc-wide\"><label class=\"vc-label\" for=\"vc-torque-record\">Net-torque source, operating range and load record<\/label><input class=\"vc-input\" id=\"vc-torque-record\" autocomplete=\"off\" placeholder=\"Torque-speed\/drive data or controlled analysis; driving, load, friction, windage and transmission effects included\"><\/div>\n<label class=\"vc-check vc-wide\" for=\"vc-confirm\"><input id=\"vc-confirm\" type=\"checkbox\"><span>I confirm that J and \u03c4<sub>net<\/sub> are expressed at the same calculation shaft and about the same signed axis; J is positive and constant; \u03c4<sub>net<\/sub> is the signed sum after all applicable driving and resisting torques and remains constant from \u03c9<sub>1<\/sub> to \u03c9<sub>2<\/sub>; the entered torque has the sign needed for the transition; and I will not treat the result as proof that a motor, drive, brake, coupling, gearbox, thermal duty or protection system is adequate.<\/span><\/label>\n<div class=\"vc-actions vc-wide\"><button class=\"vc-calc-btn\" type=\"submit\">Calculate documented transition<\/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\">Constant-net-torque 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\">Transition time t<\/div><div class=\"vc-result-value\" id=\"vc-time-result\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Signed angular acceleration \u03b1<\/div><div class=\"vc-result-value\" id=\"vc-alpha-result\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Signed speed change \u0394\u03c9<\/div><div class=\"vc-result-value\" id=\"vc-delta-result\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Signed rotational-energy change \u0394E<sub>rot<\/sub><\/div><div class=\"vc-result-value\" id=\"vc-energy-result\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Average signed net mechanical power<\/div><div class=\"vc-result-value\" id=\"vc-average-power-result\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Endpoint net mechanical powers P<sub>1<\/sub> \/ P<sub>2<\/sub><\/div><div class=\"vc-result-value\" id=\"vc-endpoint-power-result\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Normalized inertia and torque<\/div><div class=\"vc-result-value\" id=\"vc-normalized-result\">\u2014<\/div><\/div>\n<div class=\"vc-result\"><div class=\"vc-result-label\">Normalized signed speeds<\/div><div class=\"vc-result-value\" id=\"vc-speeds-result\">\u2014<\/div><\/div>\n<\/div><div class=\"vc-danger\"><strong>Model boundary:<\/strong> this is the rigid, single-axis, constant-J, constant-net-torque solution only. It does not model a torque-speed curve, current\/voltage limits, field weakening, slip, compliance\/backlash, resonances, traction loss, thermal limits, regenerative capacity, brake heat, control ramps or transient load events. Signed mechanical energy and power are not electrical input power or dissipated heat.<\/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\">Equations, signs and dimensional check<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\">\n<h3>Equations implemented<\/h3><div class=\"vc-formula\">\u03c4<sub>net<\/sub> = J\u03b1<br>\u0394\u03c9 = \u03c9<sub>2<\/sub> \u2212 \u03c9<sub>1<\/sub><br>t = \u0394\u03c9\/\u03b1 = J\u0394\u03c9\/\u03c4<sub>net<\/sub> &gt; 0<br>\u0394E<sub>rot<\/sub> = \u00bdJ(\u03c9<sub>2<\/sub>\u00b2 \u2212 \u03c9<sub>1<\/sub>\u00b2)<br>P\u0304<sub>net<\/sub> = \u0394E<sub>rot<\/sub>\/t = \u03c4<sub>net<\/sub>(\u03c9<sub>1<\/sub> + \u03c9<sub>2<\/sub>)\/2<br>P<sub>1<\/sub> = \u03c4<sub>net<\/sub>\u03c9<sub>1<\/sub>; P<sub>2<\/sub> = \u03c4<sub>net<\/sub>\u03c9<sub>2<\/sub><\/div>\n<p>The positive axis is chosen by the user. Angular speeds, angular acceleration, torque, energy change and power retain their signs. The worksheet accepts the transition only when non-zero \u0394\u03c9 and non-zero \u03c4<sub>net<\/sub> have the same sign, so calculated time is positive. A negative energy change is ordinary mechanical-energy removal, not an error.<\/p>\n<p>Dimensions close: (kg\u00b7m\u00b2)(rad\/s)\/(N\u00b7m) reduces to seconds because the radian is dimensionless and N\u00b7m = kg\u00b7m\u00b2\/s\u00b2. Likewise \u00bdJ\u03c9\u00b2 is joules and \u03c4\u03c9 is watts.<\/p>\n<div class=\"vc-warning\"><strong>Reversal through zero:<\/strong> the arithmetic can represent reversal if the same signed net torque is valid throughout. Mechanical power changes sign when \u03c9 crosses zero. Verify the drive, brake, transmission and control behavior separately.<\/div>\n<\/div><\/div><\/section>\n<section class=\"vc-section\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-title\">Exact 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 unit<\/th><th>Conversion to the internal SI quantity<\/th><th>Important distinction<\/th><\/tr><\/thead><tbody>\n<tr><td>lbm\u00b7ft\u00b2<\/td><td>1 lbm\u00b7ft\u00b2 = 0.45359237 \u00d7 0.3048\u00b2 = 0.0421401100938048 kg\u00b7m\u00b2<\/td><td>Mass moment using the avoirdupois pound mass<\/td><\/tr>\n<tr><td>slug\u00b7ft\u00b2<\/td><td>1 slug\u00b7ft\u00b2 = 1.3558179483314004 kg\u00b7m\u00b2<\/td><td>Not interchangeable with lbm\u00b7ft\u00b2; it is 32.174048556&#8230; times larger<\/td><\/tr>\n<tr><td>lbf\u00b7ft<\/td><td>1 lbf\u00b7ft = 4.4482216152605 \u00d7 0.3048 = 1.3558179483314004 N\u00b7m<\/td><td>Torque conversion; the same number as slug\u00b7ft\u00b2 conversion has a different dimension<\/td><\/tr>\n<tr><td>rpm<\/td><td>\u03c9 = rpm \u00d7 2\u03c0\/60 rad\/s<\/td><td>Signed revolutions per minute<\/td><\/tr>\n<tr><td>Hz (rev\/s)<\/td><td>\u03c9 = Hz \u00d7 2\u03c0 rad\/s<\/td><td>Here Hz explicitly means signed revolutions per second<\/td><\/tr>\n<tr><td>rad\/s<\/td><td>No scale change<\/td><td>Internal angular-speed unit<\/td><\/tr>\n<\/tbody><\/table><\/div><p>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> lists pound-foot-squared, pound-force-foot, slug and rpm conversion factors. The prior page used approximately 1.35582 for an input labelled lb\u00b7ft\u00b2; that is the slug\u00b7ft\u00b2 scale, not the lbm\u00b7ft\u00b2 scale.<\/p><\/div><\/div><\/section>\n<section class=\"vc-section\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-title\">Effective inertia and torque boundary<\/span><span class=\"vc-chevron\">\u2304<\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\">\n<h3>Reflect all inertias to one shaft<\/h3><p>Total J is the sum of rotor, coupling, gear and load inertias after every component has been expressed as kinetic-energy-equivalent inertia at the calculation shaft. For a component at angular speed \u03c9<sub>source<\/sub> and a calculation shaft at \u03c9<sub>calc<\/sub>:<\/p><div class=\"vc-formula\">J<sub>ref,calc<\/sub> = J<sub>source<\/sub>(\u03c9<sub>source<\/sub>\/\u03c9<sub>calc<\/sub>)\u00b2<\/div>\n<p>If i = \u03c9<sub>motor<\/sub>\/\u03c9<sub>load<\/sub>, load inertia reflected to the motor is J<sub>load<\/sub>\/i\u00b2. State the ratio convention explicitly and include the inertias of relevant gears, pulleys and couplings. Oriental Motor\u2019s official engineering guidance likewise notes that gearing reduces load inertia reflected to the motor by the gear ratio squared. Compliance, backlash and losses require a fuller model.<\/p>\n<h3>Net torque means the signed remainder at that same shaft<\/h3><p>\u03c4<sub>net<\/sub> is not merely motor nameplate or stall torque. It is the signed sum after the applicable motor\/prime-mover torque, process\/load torque, friction, windage, transmission losses and other torques have been brought to the same shaft and sign convention. The torque must remain constant over the entire speed interval for this closed-form time to apply.<\/p>\n<div class=\"vc-warning\"><strong>Variable torque:<\/strong> when J remains constant and \u03c4<sub>net<\/sub> varies only with speed over a monotonic interval, use t = \u222b[J\/\u03c4<sub>net<\/sub>(\u03c9)]d\u03c9. A simple arithmetic or time average of torque is not generally equivalent. If torque depends on time, temperature, current, voltage, controller state or another dynamic state\u2014or if effective inertia changes\u2014solve the governing differential equations and apply the real drive limits.<\/div>\n<\/div><\/div><\/section>\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>General mechanics, not an ISO-prescribed formula<\/h3><p>MIT OpenCourseWare, <a href=\"https:\/\/ocw.mit.edu\/courses\/8-01sc-classical-mechanics-fall-2016\/mit8_01scs22_chapter17.pdf\" target=\"_blank\" rel=\"noopener\">Chapter 17: Two Dimensional Rotational Dynamics<\/a>, equation 17.3.23, states the fixed-axis relationship between total external torque, moment of inertia and angular acceleration. This worksheet applies that general-mechanics relationship to a documented constant-J and constant-net-torque interval. No ISO clause, acceptance limit or equipment-selection rule is claimed.<\/p>\n<h3>Published university example reproduced<\/h3><p>The University of Maryland Physics 121 <a href=\"https:\/\/physics.umd.edu\/studinfo\/courses\/Phys121\/Brill\/2005\/hw8s.html\" target=\"_blank\" rel=\"noopener\">wire-spool worked solution<\/a> uses J = 2.25 kg\u00b7m\u00b2 and \u03c4 = 9 N\u00b7m, obtains \u03b1 = 4 rad\/s\u00b2, and after 2 s obtains \u0394\u03c9 = 8 rad\/s. Entering 0 to 8 rad\/s reproduces t = 2 s. The same documented model gives \u0394E<sub>rot<\/sub> = 72 J, average net mechanical power 36 W, and endpoint powers 0 W and 72 W.<\/p>\n<p>The University of Tennessee physics module independently explains that torque is directional and that net torque is the vector sum of torques. NIST SP 811 supplies the unit conversion basis. These are public sources; no closed normative coefficient or table is embedded.<\/p>\n<\/div><\/div><\/section>\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 behavior<\/th><th>Problem and correction<\/th><\/tr><\/thead><tbody>\n<tr><td>Input labelled lb\u00b7ft\u00b2 multiplied by 1.35582<\/td><td>That scale corresponds to slug\u00b7ft\u00b2, not lbm\u00b7ft\u00b2. The former conversion overstated literal pound-mass-foot-squared inertia by about 32.174. Separate explicit lbm\u00b7ft\u00b2 and slug\u00b7ft\u00b2 units now use distinct NIST-based factors.<\/td><\/tr>\n<tr><td>Metric\/imperial toggle reinterpreted existing numbers<\/td><td>Changing a unit could silently change the physical case. Each unit is now explicit, and changing it clears the affected numeric field(s) and prior result.<\/td><\/tr>\n<tr><td>Only non-negative speeds and absolute speed change<\/td><td>Direction, braking and reversal were lost. Initial\/final speeds, torque, acceleration, energy and power now retain a documented sign convention.<\/td><\/tr>\n<tr><td>Absolute energy and power outputs<\/td><td>Absolute values hid whether rotational energy was added or removed. The replacement reports signed \u0394E and signed net mechanical power and does not relabel them as electrical input or brake heat.<\/td><\/tr>\n<tr><td>\u201cNet motor torque minus load torque; include friction\u201d<\/td><td>The shaft, direction and transmission boundary were not controlled. The replacement requires a signed same-shaft net-torque record including all applicable driving and resisting contributions.<\/td><\/tr>\n<tr><td>Variable torque: \u201cintegrate numerically or use time-averaged torque\u201d<\/td><td>A generic time-average torque is not a valid substitute. The replacement shows the speed-domain integral and directs state-dependent systems to their differential equations and real drive limits.<\/td><\/tr>\n<tr><td>Ambiguous reflected-inertia ratio<\/td><td>The old N-motor\/N-load notation could mean speeds or tooth counts and reverse the result. The replacement states energy-equivalent reflection and defines i = motor speed\/load speed before giving J<sub>load,ref<\/sub> = J<sub>load<\/sub>\/i\u00b2.<\/td><\/tr>\n<tr><td>Defaults, presets, auto-calculation, partial parsing, persistence and dynamic innerHTML<\/td><td>The page could display an apparently authoritative result without a controlled load record. It now starts blank, requires provenance and confirmation, validates complete finite input, uses explicit submit and writes results with textContent.<\/td><\/tr>\n<\/tbody><\/table><\/div><\/div><\/div><\/section>\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\">Does the calculated time prove the motor or brake is adequate?<\/button><div>No. Check the complete torque-speed envelope, current and voltage limits, controller ramps, duty cycle, temperatures, regenerative or dissipation capacity, transmission ratings, couplings, protection and the governing equipment\/application requirements.<\/div><\/div>\n<div class=\"vc-faq\"><button type=\"button\">What if torque falls with speed?<\/button><div>Do not enter a guessed average. With constant J, integrate J\/\u03c4<sub>net<\/sub>(\u03c9) over a monotonic interval using controlled data, or solve the state equations when torque depends on time, current, voltage, temperature, controller state or changing configuration.<\/div><\/div>\n<div class=\"vc-faq\"><button type=\"button\">Can the worksheet calculate deceleration?<\/button><div>Yes. For positive rotation slowing down, \u0394\u03c9 is negative and net torque must also be negative. The energy and average net mechanical power will normally be negative. This still does not calculate brake temperature or electrical regeneration.<\/div><\/div>\n<div class=\"vc-faq\"><button type=\"button\">Why distinguish lbm\u00b7ft\u00b2 and slug\u00b7ft\u00b2?<\/button><div>They are different inertia units by a factor of about 32.174. A field labelled only lb\u00b7ft\u00b2 is unsafe unless the mass convention is explicit.<\/div><\/div>\n<\/div><\/div><\/section>\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\/angular-acceleration-calculator\/\">Angular-acceleration worksheet<\/a><a href=\"\/calculators\/rotational-kinetic-energy\/\">Rotational kinetic-energy worksheet<\/a><a href=\"\/calculators\/motor-torque-calculator\/\">Motor-torque worksheet<\/a><\/div><\/div><\/div><\/section>\n<footer class=\"vc-footer\"><p><a href=\"https:\/\/vibromera.eu\/\">Vibromera<\/a> engineering reference worksheet \u00b7 revised 13 July 2026<\/p><\/footer>\n<\/div>\n<script>\n(function(){'use strict';\nvar LB_TO_KG=0.45359237,FT_TO_M=0.3048,LBF_TO_N=4.4482216152605,LBMFT2_TO_KGM2=LB_TO_KG*FT_TO_M*FT_TO_M,SLUGFT2_TO_KGM2=LBF_TO_N*FT_TO_M,LBFFT_TO_NM=LBF_TO_N*FT_TO_M;\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 normalizeInputs(x){if(!x||!['kgm2','lbmft2','slugft2'].includes(x.inertiaUnit)||!['Nm','lbfft'].includes(x.torqueUnit)||!['rpm','Hz','rads'].includes(x.speedUnit))throw new Error('units');if(!Number.isFinite(x.inertia)||x.inertia<=0||!Number.isFinite(x.netTorque)||x.netTorque===0||!Number.isFinite(x.initialSpeed)||!Number.isFinite(x.finalSpeed))throw new Error('domain');var inertiaFactor=x.inertiaUnit==='kgm2'?1:(x.inertiaUnit==='lbmft2'?LBMFT2_TO_KGM2:SLUGFT2_TO_KGM2),torqueFactor=x.torqueUnit==='Nm'?1:LBFFT_TO_NM,speedFactor=x.speedUnit==='rpm'?2*Math.PI\/60:(x.speedUnit==='Hz'?2*Math.PI:1),y={inertiaKgm2:x.inertia*inertiaFactor,netTorqueNm:x.netTorque*torqueFactor,omega1:x.initialSpeed*speedFactor,omega2:x.finalSpeed*speedFactor};if(!Number.isFinite(y.inertiaKgm2)||y.inertiaKgm2<=0||!Number.isFinite(y.netTorqueNm)||y.netTorqueNm===0||!Number.isFinite(y.omega1)||!Number.isFinite(y.omega2))throw new Error('range');return y}\nfunction rotorSpeedTransitionModel(x){var y=normalizeInputs(x),deltaOmega=y.omega2-y.omega1;if(!Number.isFinite(deltaOmega)||deltaOmega===0)throw new Error('transition');if(Math.sign(deltaOmega)!==Math.sign(y.netTorqueNm))throw new Error('sign');var alpha=y.netTorqueNm\/y.inertiaKgm2,time=deltaOmega\/alpha,energyChange=.5*y.inertiaKgm2*deltaOmega*(y.omega2+y.omega1),averagePower=energyChange\/time,power1=y.netTorqueNm*y.omega1,power2=y.netTorqueNm*y.omega2;if(![alpha,time,energyChange,averagePower,power1,power2].every(Number.isFinite)||time<=0)throw new Error('range');return{normalized:y,deltaOmega:deltaOmega,alpha:alpha,time:time,energyChange:energyChange,averagePower:averagePower,power1:power1,power2:power2}}\nwindow.vbmRotorSpeedTransitionModel={parseNumber:parseNumber,normalizeInputs:normalizeInputs,rotorSpeedTransitionModel:rotorSpeedTransitionModel,constants:{LB_TO_KG:LB_TO_KG,FT_TO_M:FT_TO_M,LBF_TO_N:LBF_TO_N,LBMFT2_TO_KGM2:LBMFT2_TO_KGM2,SLUGFT2_TO_KGM2:SLUGFT2_TO_KGM2,LBFFT_TO_NM:LBFFT_TO_NM}};\nvar form=document.getElementById('vc-form');if(!form)return;var error=document.getElementById('vc-error'),results=document.getElementById('vc-results'),lastRecord='';\nfunction displayNumber(n,d){return n.toLocaleString('en-US',{maximumFractionDigits:d,minimumFractionDigits:0})}\nfunction quantityText(n,unit){return displayNumber(n,12)+' '+unit}\nfunction showError(message){error.textContent=message;error.classList.add('vc-show');results.classList.remove('vc-visible')}\nfunction clearError(){error.textContent='';error.classList.remove('vc-show')}\nfunction clearFields(ids){ids.forEach(function(id){document.getElementById(id).value=''});lastRecord='';results.classList.remove('vc-visible');clearError()}\ndocument.getElementById('vc-inertia-unit').addEventListener('change',function(){clearFields(['vc-inertia'])});document.getElementById('vc-torque-unit').addEventListener('change',function(){clearFields(['vc-torque'])});document.getElementById('vc-speed-unit').addEventListener('change',function(){clearFields(['vc-speed-initial','vc-speed-final'])});\nform.addEventListener('submit',function(event){event.preventDefault();clearError();var inertia=parseNumber(document.getElementById('vc-inertia').value),torque=parseNumber(document.getElementById('vc-torque').value),speed1=parseNumber(document.getElementById('vc-speed-initial').value),speed2=parseNumber(document.getElementById('vc-speed-final').value);if([inertia,torque,speed1,speed2].some(function(v){return v===null})){showError('Enter a complete finite number in every numeric field. Decimal point and decimal comma are accepted.');return}var system=document.getElementById('vc-system').value.trim(),inertiaRecord=document.getElementById('vc-inertia-record').value.trim(),torqueRecord=document.getElementById('vc-torque-record').value.trim();if(!system){showError('Record the machine, calculation shaft and positive-axis convention.');return}if(!inertiaRecord){showError('Record the source, components and any reflection calculation used for total effective inertia.');return}if(!torqueRecord){showError('Record the signed net-torque source, included loads and operating speed range.');return}if(!document.getElementById('vc-confirm').checked){showError('Confirm the same-shaft sign convention, constant-model assumptions and equipment-selection limitations before calculating.');return}var input={inertia:inertia,inertiaUnit:document.getElementById('vc-inertia-unit').value,netTorque:torque,torqueUnit:document.getElementById('vc-torque-unit').value,initialSpeed:speed1,finalSpeed:speed2,speedUnit:document.getElementById('vc-speed-unit').value},r;try{r=rotorSpeedTransitionModel(input)}catch(ex){if(ex&&ex.message==='sign'){showError('The signed net torque must have the same sign as final speed minus initial speed, so the requested transition has positive time.');return}if(ex&&ex.message==='transition'){showError('Initial and final speed must be different.');return}showError('Inertia must be positive; net torque must be non-zero; all converted values and results must remain finite.');return}var y=r.normalized;document.getElementById('vc-time-result').textContent=quantityText(r.time,'s');document.getElementById('vc-alpha-result').textContent=quantityText(r.alpha,'rad\/s\u00b2');document.getElementById('vc-delta-result').textContent=quantityText(r.deltaOmega,'rad\/s');document.getElementById('vc-energy-result').textContent=quantityText(r.energyChange,'J');document.getElementById('vc-average-power-result').textContent=quantityText(r.averagePower,'W');document.getElementById('vc-endpoint-power-result').textContent=quantityText(r.power1,'W')+' \/ '+quantityText(r.power2,'W');document.getElementById('vc-normalized-result').textContent=quantityText(y.inertiaKgm2,'kg\u00b7m\u00b2')+' \u00b7 '+quantityText(y.netTorqueNm,'N\u00b7m');document.getElementById('vc-speeds-result').textContent=quantityText(y.omega1,'rad\/s')+' \u2192 '+quantityText(y.omega2,'rad\/s');document.getElementById('vc-results-basis').textContent=system+' \u00b7 '+inertiaRecord+' \u00b7 '+torqueRecord;lastRecord='Constant-net-torque speed-transition record\\nSystem\/axis: '+system+'\\nInertia source\/reflection: '+inertiaRecord+'\\nNet-torque source\/load record: '+torqueRecord+'\\nJ: '+quantityText(y.inertiaKgm2,'kg\u00b7m\u00b2')+'\\nNet torque: '+quantityText(y.netTorqueNm,'N\u00b7m')+'\\nSpeed: '+quantityText(y.omega1,'rad\/s')+' to '+quantityText(y.omega2,'rad\/s')+'\\nTime: '+quantityText(r.time,'s')+'\\nAngular acceleration: '+quantityText(r.alpha,'rad\/s\u00b2')+'\\nEnergy change: '+quantityText(r.energyChange,'J')+'\\nAverage net mechanical power: '+quantityText(r.averagePower,'W')+'\\nEndpoint net powers: '+quantityText(r.power1,'W')+' \/ '+quantityText(r.power2,'W')+'\\nGeneral-mechanics constant-J and constant-net-torque result only; not equipment selection or ISO conformity.';results.classList.add('vc-visible')});\ndocument.getElementById('vc-copy').addEventListener('click',function(){if(lastRecord&&navigator.clipboard&&navigator.clipboard.writeText)navigator.clipboard.writeText(lastRecord)});document.querySelectorAll('.vc-section-toggle').forEach(function(button){button.addEventListener('click',function(){var section=button.closest('.vc-section'),open=section.classList.toggle('vc-open');button.setAttribute('aria-expanded',open?'true':'false')})});document.querySelectorAll('.vc-faq button').forEach(function(button){button.addEventListener('click',function(){button.parentElement.classList.toggle('vc-open')})});\n})();\n<\/script>\n\n","protected":false},"excerpt":{"rendered":"<p>Calculate a signed single-axis speed-transition time from documented effective inertia and constant net torque, with exact lbm\/slug unit handling and explicit limits.<\/p>","protected":false},"featured_media":0,"template":"","meta":{"ai_generated_summary":"","footnotes":""},"categories":[],"tags":[],"class_list":["post-100209","calculator","type-calculator","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/vibromera.eu\/fr\/wp-json\/wp\/v2\/calculator\/100209","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/fr\/wp-json\/wp\/v2\/calculator"}],"about":[{"href":"https:\/\/vibromera.eu\/fr\/wp-json\/wp\/v2\/types\/calculator"}],"version-history":[{"count":2,"href":"https:\/\/vibromera.eu\/fr\/wp-json\/wp\/v2\/calculator\/100209\/revisions"}],"predecessor-version":[{"id":102548,"href":"https:\/\/vibromera.eu\/fr\/wp-json\/wp\/v2\/calculator\/100209\/revisions\/102548"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/fr\/wp-json\/wp\/v2\/media?parent=100209"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/fr\/wp-json\/wp\/v2\/categories?post=100209"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/fr\/wp-json\/wp\/v2\/tags?post=100209"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}