Mechanical engineering

Mechanical engineering studies and designs machines and physical systems using mechanics, thermodynamics, fluid flow and materials. A pump, engine or robotic joint links forces and motion to energy conversion and dissipation. Design analysis must distinguish ideal kinematics from the loads and losses required to realize it. For example, a linkage may permit a desired motion while its bearings overheat or its members deflect excessively. Measurement and testing connect such component-level predictions to actual system behaviour.

Condition-based maintenance uses observations of a machine's condition to inform inspection, repair or replacement decisions. The chain runs from a physical measurement, through an interpreted condition indicator, to an action rule. A signal change alone is not that complete decision process. For a gearbox, useful observations can include a gear vibration spectrum, lubricant debris and direct inspection of gear surface pitting.

A motivating primary experiment monitored24 pinion/face-gear sets during endurance testing. Accelerometers and shaft-timing sensors supplied vibration records, while an in-line oil-debris sensor provided another physical channel. Inspection observations supplied evidence about actual damage. The study compared indicator responses and their detection/false-alarm tradeoffs, showing why an indicator needs validation against component condition.^[ref]

For an illustrative decision process, compare an operating-state-matched baseline with repeated new measurements, investigate a persistent change, and select the response using evidence about the affected component. An alarm threshold that flags8 of10 damaged test cases and1 of90 healthy cases has observed detection fraction0.8 and false-alarm fraction about0.0111. These hypothetical counts describe one test dataset; choosing an action also depends on failure consequences, inspection cost and uncertainty. Receiver operating characteristic analysis helps expose the threshold tradeoff.

Detection and prediction answer different questions. Estimating remaining useful life additionally requires a model connecting future operation and degradation to a stated end condition. A detector that recognizes existing pits does not automatically supply that model. Changes in load, speed or sensor mounting can also change vibration, so a maintenance decision should state the measurement conditions and the interpretation being tested.

Remaining useful life is the future time, cycle count or other usage until a component reaches a specified end-of-life criterion, conditional on its current condition and assumed future operation. The endpoint must be stated: crossing a performance limit, exhausting a discharge and reaching a long-term degradation threshold are different events. For an increasing damage variable $D$, threshold $D_c$ and present time $t$, a mathematical definition is the first-passage timeA prediction of this quantity supports condition-based maintenance, while requiring more than detection of existing damage.

For an illustrative deterministic model, let current normalized damage be0.60, the end threshold1.00 and future damage rate0.02 per hour. Linear extrapolation gives $R=(1-0.60)/0.02=20\mathrm{h}$. If the plausible future rate instead ranges from0.015 to0.025 per hour, the corresponding times range from about26.67 down to16h. Remaining useful life uncertainty therefore includes future operating conditions and degradation-model uncertainty, as well as measurement error.

A primary NASA battery-aging dataset makes the endpoint concrete. A custom testbed cycled electric batteries under controlled load and temperature, recording quantities including electric potential difference, electric current and temperature. The reported aging endpoint was30% loss of rated capacity, from2Ah to1.4Ah. That long-term endpoint is distinct from the terminal-potential threshold used to end an individual discharge.^[ref]

A useful validation compares predictions made before the endpoint with subsequently observed crossing times under documented operation. Reporting one predicted number without its endpoint, usage assumptions or uncertainty can conceal the main sources of disagreement between a model and a real endurance test.

An end-of-life criterion specifies the condition at which a component is treated as having exhausted its useful operation for a stated purpose. It supplies the endpoint needed to define remaining useful life. The criterion may be a performance threshold or an observed failure condition; useful operation can end before complete physical destruction. Its units, measurement procedure and operating conditions belong in the specification.

A primary NASA battery-aging test used30% loss of rated capacity, from2Ah to1.4Ah, as the aging endpoint. The apparatus separately used terminal-potential limits to finish individual discharges. Those two endpoints describe different processes and timescales.^[ref] A prediction stated only as “time until the battery ends” would therefore omit an essential physical definition.

For an illustrative linear capacity model, suppose present capacity is1.8Ah and future fade is0.01Ah per cycle. A1.4Ah end criterion gives40 remaining cycles, whereas a1.2Ah criterion gives60. The assumed degradation path is unchanged; the endpoint definition changes the reported lifetime. The example thresholds and rate are hypothetical, rather than a recommendation or a fit to the NASA data.

Competing end-of-life criteria also need a logical rule. If operation ends when either limit is first reached, the lifetime is $R=\min (R_1,R_2)$. Threshold crossings at12 and20h then give12h. A rule requiring both limits to be exceeded would define another event. Recording the criterion before evaluating a prediction avoids changing the target after observing the result, and supports an interpretable remaining-life validation error.

Competing end-of-life criteria are two or more conditions, any one of which can terminate useful operation under the stated specification. They define a remaining useful life by the earliest relevant crossing. If the prospective crossing times, measured from the present, are $T_1,\dots ,T_n$, thenThis is the event structure studied by competing risks. A condition ending one operating cycle must not inadvertently be treated as the component's aging endpoint: NASA's battery experiment separately defines a terminal-potential discharge cutoff and a30% rated-capacity-loss aging criterion.^[ref]

For an original mechanical example, suppose a component can reach either an unacceptable vibration level in12h or an unacceptable temperature in20h under a specified operating scenario. If either condition stops operation, its predicted useful life is12h. Requiring both conditions would instead define another stopping rule. An observed early crossing also does not establish when the other criterion would have been reached if the test had continued.

For uncertain crossing times, the survival function of useful life isOnly under independence does this equal the product of the individual survival functions. For an explicitly hypothetical independent pair of exponential distributions with rates $0.01{\mathrm{h}}^{-1}$ and $0.02{\mathrm{h}}^{-1}$, the product is $e^{-0.03t}$ when $t$ is measured in hours. The expected first crossing is $1/0.03\approx 33.33\mathrm{h}$, smaller than either individual mean,100h or50h.

The independence assumption is substantive. Shared temperature, load or damage can correlate crossing times, so multiplying marginal survival functions can give the wrong answer. Moreover, an intervention triggered by the first crossing can change the subsequent trajectory. The physical end-of-life criterion, dependence model and assumed future operation therefore belong with any numerical lifetime prediction.

Remaining useful life uncertainty describes uncertainty about the future crossing time of a stated end-of-life criterion. It can arise from current-condition measurements, degradation parameters, model inadequacy and future operation. The same cycle count need not imply the same condition. NASA's battery-aging dataset explicitly documents variation associated with discharge depth, rest periods and intrinsic differences, and distinguishes discharge-time prediction from long-term aging prediction.^[ref]

For an original two-scenario example, let normalized current damage be0.60 and the threshold1.00. Suppose future damage rate is0.01 per hour with probability0.25 and0.02 per hour with probability0.75. The corresponding remaining useful life values are40 and20h. Their expected lifetime isThe mean rate is0.0175 per hour, but extrapolating with that rate gives $0.40/0.0175\approx 22.86\mathrm{h}$. Averaging the inputs before a nonlinear inverse calculation therefore gives a different answer from averaging the predicted lifetimes.

A forecast can report a probability distribution or a prediction interval together with the assumed future-use scenarios. A narrow interval for the estimated mean is not automatically a narrow interval for a future component's crossing time. The reported uncertainty must identify the random quantity it covers and the model that supplies its probabilities.

Remaining-life validation error compares a forecast made at a specified prediction time with the endpoint subsequently observed. Testing predictions on independent endurance records helps assess whether stated intervals cover observed outcomes at the claimed frequency. Such evaluation must preserve the original endpoint and future-use assumptions; otherwise a numerical disagreement can mix model error with a changed prediction question.

A remaining-life validation error compares a forecast of remaining useful life made at a specified time with the subsequently observed time to the same end-of-life criterion. If the criterion is reached at $t_f$ and a prediction is issued at $t<t_f$, the reference life is $R(t)=t_f-t$. Define the signed error asWith this convention, positive error means predicting more useful life than was observed. A forecast must retain its issue time and endpoint definition for this difference to have a physical meaning.

For an original cycle-count example, suppose the specified endpoint is observed at cycle100. At cycle60, a forecast of50 remaining cycles has reference life40 and error $+10$. At cycle90, a forecast of20 has reference life10 and the same error $+10$. Their absolute relative errors are25% and100%, respectively. Equal cycle errors therefore need not have equal significance near the endpoint. A relative error divided by the reference life is undefined at the endpoint itself.

Errors should also be distinguished from aggregate scores. Two forecasts with errors $+10$ and $-10$ have zero mean signed error, but mean absolute error10 and mean squared error100 in squared cycle units. Cancellation can hide large individual discrepancies. Reporting errors by prediction time preserves information that a single average discards.

NASA's controlled battery tests supply a concrete physical setting: measured current, terminal potential and temperature accompany charge/discharge records, with a defined capacity-loss aging endpoint. The records can support testing forecasts against later observations while retaining the endpoint definition.^[ref] Their different operating histories also make indiscriminate pooling at equal cycle count inappropriate.

The remaining-life validation cutoff separates information available when issuing the forecast from information revealed later for scoring it. Using the eventual crossing time to choose the forecast, or tuning a model on the same future record used to claim validation, changes the experiment. For a prediction interval, one can additionally count how often the subsequently observed life lies inside the interval, while retaining its stated probability level and assumptions about future operation.

The remaining-life validation cutoff is the information boundary used when reconstructing a forecast of remaining useful life at a particular issue time. Observations available by that time can enter the forecast; subsequent observations can establish the eventual end-of-life criterion crossing for scoring it. Keeping these roles separate prevents data leakage from turning a retrospective calculation into an apparently successful prediction.

For an original example, suppose a test reaches its specified endpoint at cycle100 and a forecast is evaluated as if issued at cycle60. The reference remaining life is40 cycles, known only after observing the later endpoint. Constructing a predictor that returns $100-t$ because the complete test record already reveals100 produces zero remaining-life validation error by construction. That is not evidence that the endpoint was predictable from the first60 cycles.

Preprocessing has the same information constraint. A trailing three-sample average $z_t=(x_{t-2}+x_{t-1}+x_t)/3$ uses observations available at $t$. A centered average $z_t^{\prime }=(x_{t-1}+x_t+x_{t+1})/3$ needs a future observation. The centered average can be legitimate with a declared one-sample reporting delay, but then its forecast issue time is $t+1$, not $t$. Parameter fitting, normalization and selection among candidate models also belong inside the information boundary.

NASA's battery-aging records provide a physical example of the required distinction: they contain measured discharge current, terminal potential and temperature, together with a capacity-loss aging endpoint and run-to-failure histories.^[ref] A validation experiment can reveal only the prefix through its chosen cutoff to a prediction method, save the forecast, and subsequently compare it with the endpoint in the withheld suffix.

A fixed model trained on other complete histories can be evaluated on a new component's prefix. A method that updates online must instead document which new observations have become available before each forecast. In either case, preserving the cutoff, endpoint definition and assumptions about future operation makes the resulting remaining-life validation error interpretable. Pooling all prediction times into one score can conceal a violation at a particular cutoff.

A rotating accelerometer is an accelerometer whose sensing axes rotate with a moving component, such as a gear shaft. It can measure motion close to its source, but its output combines the point's actual acceleration with a changing projection direction. Treating its channels as fixed laboratory axes can therefore change the interpretation of a gear vibration spectrum. A NASA transmission experiment mounted a microelectromechanical system sensor package on a pinion and compared it with stationary housing measurements.^[ref]

For a rigid shaft rotating in a plane, let its center have inertial acceleration ${\mathbf{a}}_O$. An attached point at radius $r$ haswhere ${\mathbf{e}}_r,{\mathbf{e}}_t$ rotate with the shaft. The radial contribution is the rotating-accelerometer centripetal term. At constant $\omega =100rad/s$ and $r=0.020\mathrm{m}$, its magnitude is $200m/{\mathrm{s}}^2$ even without vibration of the center. For the ideal specific force convention, the measured vector is ${\mathbf{a}}_P-\mathbf{g}$ projected onto each sensor axis, so gravitational acceleration can also produce a changing projection.

Suppose a laboratory acceleration $A\cos (2\pi f_{m}t)$ projects onto a sensor direction rotating at $f_r$. The product contains components at $f_m-f_r$ and $f_m+f_r$:This measurement geometry can produce modulation sidebands without a damaged tooth. The NASA radial channel showed1854 and2060Hz around a1957Hz mesh frequency with103Hz rotation. Paired tangential channels also allowed different motion contributions to be separated by sums and differences.^[ref] Correct interpretation therefore specifies mounting radius, orientation, measurement calibration and rotation timing, as well as sensor bandwidth.

The rotating-accelerometer centripetal term is the inward acceleration contribution $-r{\omega }^2$ at a sensor rigidly attached a distance $r$ from a rotation axis. With outward radial unit vector ${\mathbf{e}}_r$ and tangential vector ${\mathbf{e}}_t$, differentiating the rotating position givesIt is a real kinematic acceleration, rather than an electronic zero error. A rotating accelerometer measures the relevant projection of specific force, so center motion and gravitational acceleration must also be included when interpreting an actual channel. A primary rotating-sensor experiment explicitly separates radial and tangential measurement contributions.^[ref]

At $r=0.020\mathrm{m}$ and $\omega =100rad/s$, the radial contribution is $-200m/{\mathrm{s}}^2$. It is steady in the rotating radial coordinate even though its direction continuously turns in a laboratory frame. Reversing rotation preserves this term because $\omega$ is squared; the tangential term instead follows angular acceleration.

A rotating-accelerometer speed-ripple term remains when speed is not constant. If $\omega (t)={\omega }_0+\epsilon \cos \Omega t$, thenFor the preceding radius, ${\omega }_0=100rad/s$ and ripple amplitude $\epsilon =1rad/s$ give mean $-200.01m/{\mathrm{s}}^2$, fundamental ripple amplitude4m/s² and second-ripple amplitude0.01m/s². Subtracting a record's mean removes only the steady part. A spectral component can therefore arise from speed variation in this kinematic term, which should be checked against the measured rotation history before attributing it to another vibration source.

The rotating-accelerometer speed-ripple term is the changing part of centripetal acceleration caused by variation in shaft speed. For a radial sensor at fixed radius $r$, the contribution is $a_r=-r\omega (t{)}^2$. Writing $\omega ={\omega }_0+\delta \omega$ givesThe term linear in speed variation can be appreciable even when the ripple is small relative to mean speed. Interpreting a rotating accelerometer also requires its measurement axes and their changing projection of other acceleration components.^[ref]

If $\delta \omega =\epsilon \cos \Omega t$, the mean is $-r({\omega }_0^2+{\epsilon }^2/2)$, the ripple at angular frequency $\Omega$ has amplitude $2r|{\omega }_0\epsilon |$, and the component at $2\Omega$ has amplitude $r{\epsilon }^2/2$. With $r=0.020\mathrm{m}$, ${\omega }_0=100rad/s$ and $\epsilon =1rad/s$, these are mean $-200.01m/{\mathrm{s}}^2$ and amplitudes4 and0.01m/s². Removing the mean retains both changing terms.

A synchronized shaft-speed estimate permits a model-based correction using the measured square of speed. Its accuracy is limited by centripetal subtraction error. Linearizing $-r{\omega }^2$ gives error magnitude approximately $|{\omega }_0^2\delta r+2r_0{\omega }_0\delta \omega |$. At the preceding operating point, a radius error10µm contributes0.10m/s² and a speed error0.1rad/s contributes0.40m/s² in magnitude. The signs and timing of those errors matter when subtracting signals.

This correction removes the modeled centripetal contribution, while retaining the need to account for center motion, angular acceleration, specific force and sensor orientation. In particular, speed ripple is a physical kinematic input to the channel; it need not indicate a separate structural vibration mode.

Centripetal subtraction error is the residual error left when a modeled centripetal contribution is removed from a rotating accelerometer signal. For the positive magnitude $C=r{\omega }^2$, small errors in radius and angular speed giveThe residual's sign depends on whether the correction is overestimated or underestimated. A correct model also needs synchronized speed information and the appropriate sensor-axis projection; a primary rotating-sensor experiment documents the importance of radial/tangential directions and rotating-frame effects.^[ref]

Using first-order propagation of uncertainty, the standard uncertainty satisfiesThe covariance term must be included when the estimates are correlated.^[ref] For $r=0.020\mathrm{m}$, $\omega =100rad/s$, $u_r=10\mu \mathrm{m}$ and $u_{\omega }=0.1rad/s$, the two uncorrelated contributions have magnitudes0.10 and0.40m/s², giving $u_C\approx 0.4123m/{\mathrm{s}}^2$.

A centripetal timing error can remain even with accurate individual channels. If the speed used for correction lags by $\Delta t$, then $\Delta \omega \approx -\dot{\omega }\Delta t$. At the same radius and speed, acceleration rate $\dot{\omega }=10rad/{\mathrm{s}}^2$ and lag5ms produce correction error approximately $-0.20m/{\mathrm{s}}^2$. Channel timing therefore belongs in the measurement model. These linearized formulas describe small errors; they do not replace accounting for other motion or specific force contributions.

A centripetal timing error occurs when the speed used to correct a rotating accelerometer sample refers to another time. For centripetal magnitude $C(t)=r\omega (t{)}^2$, let the correction use $\hat{C}(t)=r\omega (t-\tau {)}^2$, with positive lag $\tau$. The correction error isfor small lag. It is one contribution to centripetal subtraction error. A primary rotating-sensor experiment shows why channel direction, rotation reference and phase matter when interpreting motion spectra.^[ref]

With $r=0.020\mathrm{m}$, $\omega =100rad/s$, $\dot{\omega }=10rad/{\mathrm{s}}^2$ and lag5ms, the first-order error is $-0.20m/{\mathrm{s}}^2$. Both channels can be individually accurate while their combination is mistimed.

For periodic speed ripple $\omega ={\omega }_0+\epsilon \cos \Omega t$, keep terms first order in $\epsilon$. The correction-error amplitude isThis is a centripetal phase error with phase offset $\Omega \tau$. For $r=0.020\mathrm{m}$, ${\omega }_0=100rad/s$, $\epsilon =1rad/s$, $\Omega =20rad/s$ and $\tau =0.005\mathrm{s}$, the amplitude is about0.3998m/s². Its small-lag approximation is0.40m/s².

The relevant timing is relative between the channels and any filtering stages. A common shift of both timestamps leaves that relative delay unchanged. Rate estimates formed from a time window also need their effective time stated, rather than assuming the newest recorded timestamp represents an instantaneous speed at the accelerometer sample.

A centripetal phase error is a mismatch in the phase of a time-varying centripetal correction relative to the signal being corrected. It is a frequency-specific form of centripetal timing error. A constant centripetal magnitude has no changing phase to misalign, but speed ripple gives an oscillating contribution. For angular velocity $\omega (t)={\omega }_0+\epsilon \cos \Omega t$, expansion of $r\omega (t{)}^2$ to first order in $\epsilon$ gives an oscillation of amplitude $A=2r{\omega }_0\epsilon$ when $r,{\omega }_0,\epsilon$ are positive.

Let the true oscillating correction be $A\cos \theta$ and the applied correction $A\cos (\theta -\varphi )$. Their difference isso its amplitude is $2A|\sin (\varphi /2)|$. Equivalently, phasor subtraction gives magnitude $|A(e^{-i\varphi }-1)|$. Even perfectly matched amplitudes do not guarantee cancellation when their phases differ.

For an original example, $r=0.020\mathrm{m}$, ${\omega }_0=100rad/s$ and $\epsilon =1rad/s$ give $A=4m/{\mathrm{s}}^2$. A phase lag0.10rad leaves amplitude $8\sin (0.05)\approx 0.3998m/{\mathrm{s}}^2$. The centripetal phase-error tolerance for residual amplitude at most $0.10m/{\mathrm{s}}^2$, choosing the small phase-error range around zero, isFor a pure delay $\varphi =\Omega \tau$ at $\Omega =20rad/s$, this corresponds to about1.25ms. The same delay gives a larger phase error at a higher frequency.

A primary rotating-sensor experiment compared internal rotating MEMS sensors with stationary transmission-housing accelerometers; its analysis distinguishes rotating-frame spectral components and reconstruction of a rotation reference.^[ref] The calculation above is an illustrative correction model, not a measured error attributed to that apparatus. A whole-period delay cancels one ideal periodic component, but does not generally align transients or other frequencies.

A centripetal phase-error tolerance bounds the relative phase mismatch allowed when subtracting an oscillating centripetal correction. It translates an acceptable residual acceleration into a phase requirement and, for a known frequency, a timing requirement. With equal true and correction amplitudes $A>0$, a centripetal phase error $\varphi$ leaves residual amplitude $2A|\sin (\varphi /2)|$. For a budget $0\le E_{\max }<2A$, the permitted principal mismatch around zero isFor a pure time lag at angular frequency $\Omega >0$, divide this bound by $\Omega$ to obtain the lag tolerance. An allowable phase at one frequency is not a universal allowable delay for every spectral component.

For an original example, $A=4m/{\mathrm{s}}^2$ and $E_{\max }=0.10m/{\mathrm{s}}^2$ give a bound of about0.02500rad. At $\Omega =20rad/s$ this is approximately1.25ms; at $200rad/s$ it is approximately0.125ms. The tenfold change follows from the frequency dependence of phase delay, without changing the acceleration budget.

A centripetal amplitude-error budget is needed if the applied correction amplitude $\hat{A}$ differs from $A$. Phasor subtraction givesEven at zero phase error, the residual is at least $|\hat{A}-A|$. If that exceeds $E_{\max }$, timing improvement alone cannot meet the requirement. Otherwise, for positive amplitudes and a budget smaller than their sum, the allowed principal phase isWith $A=4$, $\hat{A}=3.96$ and budget0.10 in consistent acceleration units, the phase bound falls to about0.02303rad, or1.15ms at20rad/s.

The tolerance is a property of the correction model and the chosen operating range. A primary rotating MEMS-sensor study analyzes rotating-frame frequency components and reconstruction of a rotation reference; these are practical reasons to retain channel timing and phase information.^[ref] The numerical tolerances here are illustrative design calculations, not measured limits asserted for that apparatus.

A centripetal amplitude-error budget allocates the permitted error in an oscillating centripetal correction between amplitude mismatch and phase mismatch. If the true correction amplitude is $A>0$, the applied amplitude is $\hat{A}>0$ and the relative phase is $\varphi$, phasor subtraction gives residual amplitudeThe first term remains even with perfect timing. The second consumes part of the same acceleration-error budget; their separate limits cannot both be set to the entire allowed residual.

Write $\hat{A}=A(1+ϵ)$. For small $ϵ$ and $\varphi$, the leading approximation isThis provides an approximate joint budget, rather than permission to allow the maximum phase error and maximum gain error simultaneously. For an original example, $A=4m/{\mathrm{s}}^2$ and residual limit $0.10m/{\mathrm{s}}^2$ give relative budget0.025. Assigning relative amplitude error0.015 leaves an approximate phase budget0.020rad. The exact formula, with $\hat{A}=4.06m/{\mathrm{s}}^2$, gives a slightly smaller bound of about0.01985rad.

A centripetal speed-gain error illustrates how the mismatch can arise. For $C=r{\omega }^2$, using radius $r+\delta r$ and a speed reading scaled by $g$ producesWith correct radius and $g=1.01$, the correction is2.01% high. In a small-ripple model whose constant speed and ripple amplitude are both scaled by the same sensor gain, the oscillating correction amplitude has this same gain factor. For $A=4m/{\mathrm{s}}^2$ that alone gives0.0804m/s² amplitude mismatch. The constant centripetal term also needs its own error check; budgeting only the ripple does not control it.

Rotating-frame components and the rotation reference are experimentally consequential: a primary MEMS-sensor study compares rotating transmission measurements with stationary housing accelerometers and reconstructs a timing reference.^[ref] The budgets above are original illustrative calculations. Actual limits require the relevant operating speeds, radii, frequency range and measurement calibration, alongside the centripetal phase-error tolerance.

A centripetal speed-gain error is an error in a calculated centripetal correction caused by a multiplicative scale error in the measured angular velocity. For true centripetal magnitude $C=r{\omega }^2$ and indicated speed $\hat{\omega }=g\omega$, the computed magnitude with correct radius isWriting $g=1+ϵ$ gives exact relative error $2ϵ+{ϵ}^2$. The leading relative error is therefore twice the speed-scale error, rather than equal to it.

For an original example, let $r=0.020\mathrm{m}$ and $\omega =100rad/s$, giving $C=200m/{\mathrm{s}}^2$. A1% high speed indication gives $\hat{C}=204.02m/{\mathrm{s}}^2$, an overestimate4.02m/s². Subtracting that value from a radial reading would leave an error of the opposite sign if the true motion were otherwise modeled correctly. The example illustrates sensitivity; it is not a measured calibration error of a particular sensor.

Radius error combines with speed gain. If the indicated radius is $r+\delta r$, thenTo first order, relative error is $\delta r/r+2ϵ$. Apparent cancellation between these terms at one condition does not separately identify either calibration parameter. An independent radius measurement or speed reference helps distinguish their contributions.

An additive speed offset $b$ behaves differently:Measurements at several speeds can distinguish the quadratic scale contribution from the linear and constant offset contributions, subject to the remaining model and measurement errors. Near zero true speed, a relative-error description becomes unsuitable even though the absolute expression remains meaningful.

The same gain factor applies to the first-order ripple correction $2r{\omega }_0\epsilon$ when both mean speed and ripple amplitude share that speed scale. The resulting mismatch enters the centripetal amplitude-error budget, while relative channel timing enters the centripetal phase-error tolerance. A primary rotating MEMS-sensor experiment identifies rotating-frame components and reconstructs a rotation reference, making the physical need for consistent speed and timing information concrete.^[ref]

At a specified operating point, the sensitivity coefficient to speed is $\partial C/\partial \omega =2r\omega$. A speed standard uncertainty therefore contributes approximately $2r|\omega |u(\omega )$ to the correction's standard uncertainty, with radius uncertainty and any covariance included separately.^[ref]

A mechanical bearing supports relative motion while transmitting constraint forces between machine components. A shaft support can permit rotation while constraining translation, with the permitted motions depending on its construction and arrangement. A ball bearing uses rolling contacts; a journal bearing supports a shaft through a sliding or fluid-film interface. The support reaction is distinct from resistance to the permitted motion: an ideal support can transmit a large transverse force with zero resisting shaft torque.^[ref]

For a simple ideal shaft supported at axial positions0 and $L$, an upward load $P$ applied at $a$ requires reactionsWith $P=200\mathrm{N}$, $L=100\mathrm{mm}$ and $a=40\mathrm{mm}$, the downward support loads are120N and80N. These reactions exist even if the shaft rotates without an applied resisting torque. In a real mechanical bearing, an illustrative frictional resisting moment $M_f=0.020\mathrm{N}\mathrm{m}$ at angular speed $100rad/s$ dissipates $M_f|\omega |=2\mathrm{W}$; force equilibrium alone does not determine that loss.

A locating bearing arrangement fixes the shaft's axial position relative to the housing, while a non-locating bearing support permits the required axial displacement, for example from thermal expansion. The manufacturer distinguishes this arrangement from adjusted and floating arrangements.^[ref] Radial support does not by itself establish which support carries axial thrust.

Selection must therefore consider permitted motion, combined loads, alignment and operating conditions. An idler support reaction determines a shaft-level load; it does not directly give each rolling contact's Hertzian contact pressure or a service-life prediction. Those require the bearing's internal geometry and a suitable load/life model.

A non-locating bearing support carries the required radial load while allowing the shaft's intended axial displacement relative to the housing. It complements a locating bearing, which establishes axial position. The movement may occur within the bearing or through a suitable ring-to-seat arrangement; its purpose is to avoid imposing an unwanted axial constraint when dimensions change.^[ref]

An important cause of relative movement is thermal expansion. For an illustrative shaft with constant linear thermal expansion coefficient $\alpha$, length $L$ and uniform temperature increase $\Delta T$, free extension isUsing hypothetical $\alpha =12\times 10^{-6}{\mathrm{K}}^{-1}$, $L=0.50\mathrm{m}$ and $\Delta T=50\mathrm{K}$ gives0.30mm. If the housing expands differently, the required bearing axial displacement allowance depends on the shaft's extension relative to that housing, as well as assembly tolerances.

For contrast within the same mechanical model, constrain both ends of an elastic bar against extension. Axial strain balance gives a compressive bearing thermal-expansion constraint load with magnitude $F=EA\alpha \Delta T$. With hypothetical Young's modulus $E=200\mathrm{GPa}$ and area $A=100\mathrm{m}{\mathrm{m}}^2$, the preceding temperature change gives12kN. Real supports and housings are compliant, so this rigid-constraint calculation illustrates the mechanism rather than predicting a universal bearing load.

A non-locating bearing is not generally a support with no useful constraint: it can still transmit substantial transverse forces. Its permitted axial movement must remain compatible with the shaft's other functional requirements, including gear positioning. The actual support arrangement determines whether thermal movement is accommodated without losing the required location.

Bearing axial displacement allowance is the axial movement available in a stated shaft/support arrangement before an unintended restraint is encountered. A non-locating bearing support accommodates relative shaft/housing displacement while a locating bearing establishes axial position. The required movement includes thermal changes and assembly variation, rather than shaft expansion alone.^[ref]

For a simple uniformly heated shaft and housing, estimate relative free displacement asusing the applicable linear thermal expansion coefficients and distances from their common axial reference. A shaft extension0.30mm and housing extension0.12mm give relative movement0.18mm. Assuming the housing stationary would overstate that thermal contribution in this example.

The complete bearing axial travel interval can be asymmetric. Suppose the operating states require relative motion between $-0.12$ and $+0.18\mathrm{mm}$, with an independent assembly offset between $-0.05$ and $+0.05\mathrm{mm}$. Adding the intervals gives required travel from $-0.17$ to $+0.23\mathrm{mm}$ about the nominal reference. Its total span is0.40mm, but centering a0.40mm travel range symmetrically at zero would miss the positive endpoint by0.03mm.

Once available movement is exhausted, additional expansion can generate a bearing thermal end-stop load. Movement may be accommodated inside a bearing or at a suitable seat, depending on the actual design. This interval model is bookkeeping for a specified support arrangement; the relevant travel limits must correspond to that physical mechanism and its permitted direction.

A bearing axial travel interval states the range of relative shaft/housing positions that must be accommodated, or the range physically available, using an explicit axial reference. It is more informative than one unsigned clearance number when heating, cooling and assembly offsets move the shaft in different directions. A non-locating bearing support accommodates such movement while a locating bearing determines the reference position.^[ref]

Suppose thermal operating states give displacement $d\in [-0.12,0.18]\mathrm{mm}$ and an independent assembly offset lies in $[-0.05,0.05]\mathrm{mm}$. Adding the endpoint bounds gives required displacement $d\in [-0.17,0.23]\mathrm{mm}$. This is a bound over the permitted combinations, not a standard deviation.

Let available physical travel be $[a,b]$ and choose bearing nominal axial setting $x_0$. Requiring every displacement in $[d_{\min },d_{\max }]$ to remain within the travel givesorIf physical travel is $[-0.20,0.20]\mathrm{mm}$ for the preceding displacement range, both endpoints force $x_0=-0.03\mathrm{mm}$. Setting $x_0=0$ would exceed the positive limit by0.03mm despite equal total required and available spans.

The example has no spare endpoint margin: requiring an additional positive margin at both ends makes the nominal-setting interval empty. Correlated thermal inputs can also make a naive independent-range bound conservative; a tighter result requires the permitted joint operating states. The interval must refer to the actual movement mechanism, because reaching its end can introduce a bearing thermal end-stop load.

A bearing nominal axial setting is the chosen reference position of a shaft/support assembly within its available bearing axial travel interval. It determines how much travel remains in each direction before thermal or assembly changes encounter a limit. The locating support establishes the axial reference, while the non-locating bearing arrangement accommodates the permitted movement.^[ref]

Let physical travel be $[a,b]$, nominal setting $x_0$ and all required changes be $d\in [d_{\min },d_{\max }]$. Feasibility requiresThe interval is nonempty exactly when the available span is at least the required span. This follows directly by requiring $a\le x_0+d_{\min }$ and $x_0+d_{\max }\le b$.

For an original example, physical travel $[-0.30,0.30]\mathrm{mm}$ and required changes $[-0.17,0.23]\mathrm{mm}$ permit settings from $-0.13$ to $+0.07\mathrm{mm}$. Setting $x_0=-0.03\mathrm{mm}$ gives operating endpoints $-0.20$ and $+0.20\mathrm{mm}$, leaving0.10mm at each limit. Setting $x_0=0$ instead leaves0.13mm on one side and0.07mm on the other.

The worst endpoint bearing axial travel margin isIn this symmetric-margin objective, its maximum occurs at $x_0=(a+b-d_{\min }-d_{\max })/2$, with $m_{\max }=[(b-a)-(d_{\max }-d_{\min })]/2$. The example gives0.10mm. A real setting can have additional positioning requirements; balancing these geometric margins alone does not determine every functional constraint.

The bearing axial travel margin is the smallest remaining distance between the required axial operating range and either limit of the available bearing axial travel interval. It measures how much extra displacement can be accommodated before the assembly reaches a limit. The physical purpose of the non-locating support is to accommodate relative axial displacement, including thermal changes and accumulated dimensional variation.^[ref]

Let physical limits be $[a,b]$, bearing nominal axial setting be $x_0$, and required changes range from $d_{\min }$ to $d_{\max }$. The two end margins and their worst value areA negative margin means that at least one required position is outside the available interval. Zero means touching a limit, with no reserve in that direction.

For an original example, $[a,b]=[-0.30,0.30]\mathrm{mm}$, changes $[-0.17,0.23]\mathrm{mm}$ and setting $x_0=-0.03\mathrm{mm}$ give $m_{-}=m_{+}=0.10\mathrm{mm}$. A bearing axial setting tolerance of $\eta \in [-0.04,0.04]\mathrm{mm}$ expands the possible operating range to $[-0.24,0.24]\mathrm{mm}$. The guaranteed margin is then0.06mm. Choosing $x_0=0$ would instead give a guaranteed margin of only0.03mm under the same uncertainty bound.

In general, a bounded setting error $|\eta |\le ϵ$ changes the guaranteed margin to $m-ϵ$. This follows by reducing the lower end margin with $\eta =-ϵ$ and the upper end margin with $\eta =+ϵ$. A required reserve $m_{\mathrm{req}}$ therefore needs $m\ge m_{\mathrm{req}}+ϵ$. The example can retain a0.05mm reserve with the centered setting and a0.04mm error bound.

A deterministic tolerance bound and a standard uncertainty are different specifications. Replacing the bound by one standard deviation would not preserve the guarantee without additional distributional assumptions. The margin calculation also concerns geometric travel: it does not by itself establish acceptable contact stress, friction or bearing thermal end-stop load after a limit is reached.

A bearing axial setting tolerance specifies the permitted difference between the realized axial reference position of an assembly and its bearing nominal axial setting. It consumes some of the available bearing axial travel margin. The setting tolerance must therefore be included when deciding whether all required thermal and operating changes remain inside the physical travel limits. The bearing manufacturer's locating/non-locating arrangement discussion explicitly includes relative displacement and accumulated dimensional variation.^[ref]

Let physical travel be $[a,b]$, required displacement changes be $[d_{\min },d_{\max }]$, and setting error be $\eta \in [{\eta }_{\min },{\eta }_{\max }]$. To retain a prescribed margin $m_{\mathrm{req}}\ge 0$ for every allowed combination, the nominal setting must satisfyThe lower bound follows from the smallest possible realized position $x_0+d_{\min }+{\eta }_{\min }$; the upper bound follows from the largest. This is a worst-case guarantee over the stated ranges and needs no assumption that the errors are statistically independent.

For an original example, take $[a,b]=[-0.30,0.30]\mathrm{mm}$, required changes $[-0.17,0.23]\mathrm{mm}$, setting error $[-0.02,0.04]\mathrm{mm}$ and required margin0.05mm. The admissible nominal-setting interval is $[-0.06,-0.02]\mathrm{mm}$. Choosing $x_0=-0.04\mathrm{mm}$ gives worst-case operating endpoints $-0.23$ and $+0.23\mathrm{mm}$, leaving0.07mm at each end. The asymmetric error range has shifted the balanced nominal setting relative to a symmetric tolerance assumption.

A bearing axial tolerance stack combines the contributing dimensions and their signs before applying this range calculation. Two separate bounded errors of $\pm 0.01$ and $\pm 0.02\mathrm{mm}$ can add to $\pm 0.03\mathrm{mm}$ in the worst case. Taking a square root of summed squares would answer a statistical uncertainty question under additional assumptions; it would not reproduce that deterministic bound. The distinction between a bound and standard uncertainty matters when a guaranteed clearance or reserve is specified.

The resulting interval can be empty. That outcome identifies incompatible travel, operating-change, tolerance and reserve requirements; adjusting the nominal setting alone cannot resolve it. A nonempty interval addresses axial geometry but does not establish acceptable friction, elastic loading or contact conditions after an end stop engages.

A bearing axial tolerance stack expresses the effect of contributing dimensional deviations on the realized axial position of a bearing assembly. Each contribution enters with the sign and sensitivity imposed by the geometry. The resulting setting variation consumes bearing axial travel margin. The manufacturer's locating/non-locating bearing discussion identifies relative axial displacement and accumulated dimensional variation as design considerations.^[ref]

For an original linear stack, let shaft, housing and shim deviations be $\delta L_s$, $\delta L_h$ and $\delta t$, with setting errorIf their independent admissible ranges are respectively $\pm 0.02$, $\pm 0.03$ and $\pm 0.01\mathrm{mm}$, the worst-case setting range is $\pm 0.06\mathrm{mm}$. The positive extreme uses the upper shaft and shim deviations with the lower housing deviation. Here “independent admissible ranges” means that all combinations are allowed, rather than a probabilistic independence assertion.

A design with nominal margin0.10mm and required reserve0.05mm can accommodate at most0.05mm of bounded setting error. This stack is therefore too wide by0.01mm. Reducing the housing deviation bound from0.03 to0.02mm makes the worst-case sum0.05mm and meets that geometric reserve exactly. A bearing nominal axial setting can shift the range but cannot shrink its width.

Statistical propagation answers another question. With coefficient vector $a=(1,-1,1)$ and covariance matrix $\Sigma$, the setting variance isFor independent errors with standard uncertainties0.010,0.010 and0.005mm, this gives $u(\eta )=0.015\mathrm{mm}$. If the shaft and housing errors instead have correlation0.9 while the shim remains independent, the subtraction cancels much of their shared variation:giving about0.00671mm. Reversing a geometric sign or changing correlation changes that result. The GUM requires the covariance contribution when inputs are correlated.^[ref]

Neither statistical value replaces the0.06mm deterministic bound without a justified model relating probabilities and allowed ranges. Shared reference measurements or common thermal conditions can introduce dependence. Keeping the dimensional signs, bounds and covariance assumptions in the stack makes the resulting bearing axial setting tolerance reproducible.

A bearing thermal-expansion constraint load is an axial reaction generated when supports prevent the relative thermal expansion that a shaft/housing system would otherwise undergo. For a uniform elastic shaft of length $L$, area $A$, Young's modulus $E$ and free extension ${\delta }_{\mathrm{th}}=\alpha L\Delta T$, completely rigid axial restraint gives compressive force magnitudeA locating/non-locating support arrangement is intended to accommodate the required displacement rather than impose this unwanted extra axial constraint.^[ref]

Actual supports have finite mechanical compliance. Let shaft axial compliance be $C_b=L/(EA)$ and the combined restraining support/housing compliance be $C_s$. Compatibility for a fully engaged restraint givesFor hypothetical $L=0.50\mathrm{m}$, $E=200\mathrm{GPa}$ and $A=100\mathrm{m}{\mathrm{m}}^2$, $C_b=2.5\times 10^{-8}m/N$. Free extension0.30mm gives12kN against rigid supports. If $C_s=10^{-7}m/N$, the force falls to2.4kN because part of the mismatch is taken up by support displacement.

A gap changes the onset of restraint. If the system permits positive free travel $g$ before reaching an axial stop, a simple one-direction model gives bearing thermal end-stop loadWith the preceding compliant supports and $g=0.10\mathrm{mm}$, the result is1.6kN. Below the gap-closing displacement, this modeled stop contributes no force. Applied gear axial force, bearing preload, temperature gradients and other constraints require their own contributions; the simple expression represents one stated thermal compatibility problem.

A bearing thermal end-stop load is the axial reaction that appears when relative thermal expansion exhausts the free travel of a support arrangement. In a quasistatic one-direction model, let free thermal mismatch be ${\delta }_{\mathrm{th}}$, initial travel to the stop be $g$, shaft compliance be $C_b$ and restraining support compliance be $C_s$. The engaged reaction magnitude isThe gap first closes; additional mismatch is taken up by elastic deformation of the shaft and support. This explains why the bearing axial displacement allowance matters. The locating/non-locating arrangement is intended to accommodate relative motion rather than create an unintended thermal restraint.^[ref]

For hypothetical $C_b=2.5\times 10^{-8}m/N$, $C_s=10^{-7}m/N$, $g=0.10\mathrm{mm}$ and ${\delta }_{\mathrm{th}}=0.30\mathrm{mm}$, the effective stiffness is $1/(C_b+C_s)=8MN/m$ and the load is1.6kN. The stored elastic energy in this combined linear compliance isThis is stored energy; the frictionless elastic model does not itself predict heat loss during a complete reversible cycle.

If the housing is fixed and free mismatch is $\alpha L\Delta T$, the bearing end-stop engagement temperature increment is $\Delta T_{\ast }=g/(\alpha L)$. With hypothetical $\alpha =12\times 10^{-6}{\mathrm{K}}^{-1}$ and $L=0.50\mathrm{m}$, the0.10mm gap closes at about16.67K temperature increase. Further heating raises the load according to the engaged stiffness.

The positive-part relation is a simple instance of unilateral contact complementarity: the stop supplies a reaction only when the corresponding clearance is exhausted. Shaft inertia, impact, friction or bearing preload require a more complete model. In particular, this static compatibility calculation does not determine a collision force when a moving assembly first hits the stop.

The bearing end-stop engagement temperature is the temperature state at which free relative expansion first exhausts the clearance to an axial stop. It must specify the initial setting and how both shaft and housing temperatures change. A non-locating bearing arrangement normally accommodates relative expansion; an exhausted travel limit changes the axial constraint.^[ref]

For an illustrative common temperature increase $\Delta T$, write free relative movement aswhere ${\delta }_0$ is the assembly offset toward the stop. With positive remaining gap and $\beta >0$, engagement occurs atFor hypothetical ${\alpha }_s=12\times 10^{-6}{\mathrm{K}}^{-1}$, ${\alpha }_h=8\times 10^{-6}{\mathrm{K}}^{-1}$ and $L_s=L_h=0.50\mathrm{m}$, $\beta =2\times 10^{-6}m/K$. A0.10mm gap with no initial offset closes after50K common heating. Treating the housing as fixed would instead give16.67K, a different boundary-condition model.

An initial offset ${\delta }_0=0.02\mathrm{mm}$ reduces the engagement increase to40K. With combined axial compliance $C_b+C_s=1.25\times 10^{-7}m/N$, the engaged bearing thermal reaction slope isAt60K increase, the excess20K beyond engagement therefore gives320N of bearing thermal end-stop load in this linear quasistatic model.

If $\beta =0$, common uniform heating does not change the relative gap in this model. If $\beta <0$, heating moves away from the stated stop; cooling or another operating change can be relevant instead. Temperature gradients or independently heated components require their actual relative-displacement model rather than a universal scalar engagement temperature.

The bearing thermal reaction slope is the local rate of change of a bearing assembly's thermal constraint force with temperature along a specified thermal path. For constant combined axial mechanical compliance $C=C_b+C_s$, relative free-expansion rate $\beta$ and an engaged end stop, the linear model givesIts units are force per temperature, such asN/K. It combines thermal mismatch with elastic restraint; it is not a stiffness measured inN/m. The need to accommodate differential expansion motivates locating/non-locating arrangements in the bearing manufacturer's design guidance.^[ref]

For an original one-stop model with initial relative position ${\delta }_0$, gap $g$, uniform temperature change $\Delta T$ and $\beta >0$,The slope is zero while the gap is open and $\beta /C$ after engagement. At the ideal switching point, the two one-sided slopes differ. The stated slope therefore needs the contact state; it should not be applied throughout the open-gap regime.

Take $\beta =2\mu m/K$ and $C=0.125\mu m/N$. The engaged slope is16N/K. If the bearing end-stop engagement temperature corresponds to a40K increase, increases of60K and70K give forces320N and480N. Conversely, those two force measurements would imply a16N/K slope, and a separately established $\beta$ would imply $C=\beta /(dF/dT)=0.125\mu m/N$. This is a possible bearing thermal stiffness identification measurement, not a reported manufacturer's test.

A bench assessment could measure temperature and axial force at successive equilibrated states after engagement. Measuring both shaft and housing temperatures helps distinguish their actual thermal path from an assumed common temperature. The inferred combined mechanical compliance does not separately identify $C_b$ and $C_s$. Temperature-dependent elasticity, friction, changing contact or temperature gradients can make the observed slope vary, while transient heating can produce a path different from the equilibrated one.

As a local sensitivity coefficient, the engaged slope also propagates a temperature contribution to force standard uncertainty: for a known16N/K slope and $u(T)=0.5\mathrm{K}$, that contribution is8N. Uncertainty in the thermal coefficient, compliance and gap requires its own terms, with covariance included when inputs are correlated.^[ref]

Bearing thermal stiffness identification uses measured force changes along a specified thermal path, together with independently known expansion behavior, to infer the axial restraint of an assembly. The observed bearing thermal reaction slope is $q=dF/dT$ inN/K. Mechanical stiffness is force per displacement, so converting $q$ into stiffness requires the free relative-expansion rate $\beta$ in displacement per temperature. In the constant-compliance engaged model,Here $C_b$ and $C_s$ represent the bearing and supporting structure's axial mechanical compliance contributions.

For an original identification example, suppose equilibrated engaged measurements give forces160,320 and480N at temperature increases50,60 and70K. A linear regression gives slope16N/K and, within this model, an extrapolated zero-force engagement increase of40K. If an independent free-expansion measurement gives $\beta =2\mu m/K$, the total compliance is $0.125\mu m/N$ and total stiffness is $8N/\mu \mathrm{m}=8\times 10^{6}N/m$. A16N/K thermal slope is therefore not itself a16N/m spring stiffness.