Efficient simulation of vibrations in an electric motor

Abstract

A method (300) for simulating vibrations in an electric motor including a rotatable shaft which extends horizontally and is secured, via at least two intermediary elements of finite stiffness on each side of a rotor, to a stator. The method comprises: providing (310) a 1-degree-of-freedom, 1-DOF, model to represent the electric motor in terms of a horizontal component of a transverse displacement of the shaft; simulating (320) vibrations in the shaft by solving a second-order differential equation for said displacement, which differential equation includes an oscillating drive term; and performing a technical manipulation (330) relating to the electric motor on the basis of the vibration simulation.

Description

EFFICIENT SIMULATION OF VIBRATIONS IN AN ELECTRIC MOTOR TECHNICAL FIELD

[0001] The present disclosure relates to the field of computer simulation of vibrating mechanical systems and in particular to efficient methods for simulating vibrations in an electric motor, especially an alternating-current (AC) motor. Example uses of the simulation results may include: control of the electric motor, control of an adaptive vibration-attenuation system for the electric motor, design of the electric motor, and early discovery of mechanical degradation in the electric motor. BACKGROUND

[0002] The importance of computer simulation in research and development of complex machinery, including the design phase as well as subsequent improvements, is uncontested. Model-based computer simulations also play an increasing role in real-time control of the machinery, where simulated data may supplement measured data for thereby improving the responsiveness and / or accuracy of conventional feedback loops. Diagnostics, including runtime diagnostics, is a further use case, where regular comparison with computer simulations – or regular re-identification of model parameters – may help discover mechanical degradation (e.g., structural faults) before they cause serious damage to the machinery.

[0003] CN103746630A relates to real-time control of heavy electric machinery by conventional methods. A large electric motor that rests on multiple ground supports is considered. Each support is provided with an electromagnetic actuator for generating an anti-phase vibration that attenuates a vertical vibration of the electric motor relative to the ground. The electromagnetic actuator is driven by an AC current which is controlled in real time. To bring about this real-time control, the authors of CN103746630A use an adapted version of so-called global sliding mode control, a robust control technique that is suitable for controlling partially unknown systems. In operation, a series of test runs with different candidate values assigned to certain coefficients ^^^, ^^of a control law ansatz are executed and evaluated with respect to an objective function. The best performing candidate values are selected for onward use.

[0004] JP2005020949A discloses a simulation setup having as its central component a frequency-characteristic computation device which includes a storage unit configured to store a frequency characteristic equation constituted by a plurality of vibration models, each with one degree of freedom. It is understood that a vibration model is a second-order ordinary differential equation, and each vibration model represents one machine element. The outputs from all vibration models are combined.

[0005] US5502650A discloses an apparatus for predicting vibration conditions in a rotating electric system with speed-dependent dynamical properties. The prediction is based on shaft vibrations sensed by displacement sensors, which are input to a resonance equation. US5502650A is concerned with vertical rotors.

[0006] These and other disclosures of the state of the art fail to address the problem of simulating vibrations in an electric motor with a rotatable shaft which extends horizontally and is secured, via at least two intermediary elements of finite stiffness on each side of a rotor, to a stator. In general terms, a good computer simulation should be physically accurate at the same time as it is computationally tractable, particularly if online or real-time simulation is intended. Careful and computationally efficient modeling of the electric motor is therefore of utmost importance. SUMMARY

[0007] One objective of the present disclosure is to is to make available a physically accurate yet computationally efficient method for simulating vibrations in an electric motor, especially an AC motor, with the structure outlined above. A further objective is to propose a simple but still faithful model of such an electric motor, for use in computer simulations. A further objective is to provide simulation- based tools for troubleshooting, diagnostics, runtime control and / or design of such an electric motor.

[0008] At least some of these objectives are achieved by the invention as defined by the independent claims. The dependent claims relate to advantageous embodiments of the invention.

[0009] In a first aspect of the present disclosure, there is provided a method for simulating vibrations in an electric motor. The electric motor includes a rotatableshaft that extends horizontally and is secured, via at least two intermediary elements of finite stiffness on each side of a rotor, to a stator. The method comprises: providing a 1-degree-of-freedom (1-DOF) model to represent the electric motor in terms of a horizontal component of a transverse displacement ^ of the shaft; simulating vibrations in the shaft by solving a second-order differential equation for said displacement, which differential equation includes an oscillating drive term; and performing a technical manipulation relating to the electric motor on the basis of the vibration simulation.

[0010] In other words, the inventor has discovered a well-performing 1-DOF model, where the electric motor is represented in terms of a horizontal component of a transverse displacement ^ of the shaft, which allows the vibrations to be simulated by (numerically) solving a second-order differential equation for said displacement ^. The physical accuracy of the 1-DOF model has been validated empirically by way of reference simulations and experiments to be presented below.

[0011] In different embodiments, the 1-DOF model can be used for simulating vibrations in an electric motor with a more complex structure. For example, the intermediary elements may be a bearing journal, an end-plate arrangement and / or a bearing housing. Further, 1-DOF model is used for simulating an electric motor that is secured to a foundation via at least one foot. The accuracy of the results remains satisfactory.

[0012] In some embodiments, the providing of the 1-DOF model includes estimating an amplitude and / or a phase of a vibratory response of the electric motor when the electric motor is subjected to an oscillating excitation force, and assigning a value to a parameter of the 1-DOF model based on the estimated vibratory response. The estimation may include peak-to-peak estimation, an inverse root-mean-square calculation, a Hilbert transform, and it may be based on direct or indirect measurements on the electric motor.

[0013] In some embodiments, the simulation method is performed online and / or in real time. For example, a finite impulse response (FIR) filter configured to approximate a Hilbert transform may be used to estimate an amplitude, ultimately to identify parameters of the 1-DOF model.

[0014] Different embodiments provide various technical uses of the results of simulating the vibrations in an electric motor. The technical uses can be described as technical manipulations relating to an existing electric motor or an electric motor to be manufactured.

[0015] For one thing, the results may be used to generate a control signal to be applied to the electric motor.

[0016] Further, the results of simulating the vibrations in the electric motor may form the basis of controlling an auxiliary system associated with the electric motor, especially an adaptive vibration attenuation system.

[0017] Further, one or more design parameters of the electric motor may be set on the basis of the results of simulating the vibrations in the electric motor. Optionally, an electric motor can be manufactured in accordance with the thus assigned value of the design parameter.

[0018] Further still, the 1-DOF model may be used for detecting an evolution over time of the electric motor, namely, by repeatedly comparing measured vibration data with simulated vibration data. If the measured vibration data begins to differ from the simulated vibration data, it may be concluded that a once adequate parameter assignment to the model does not fits the electric motor any longer, which may be an early sign of mechanical wear, deterioration and the like in the electric motor. The simulated vibration data may for example be obtained while varying a parameter of the 1-DOF model and comparing the measured vibration data with the simulated vibration data for different values of the parameter. (This is essentially a parameter estimation. The detection of the evolution over time of the electric motor is thus, in other words, achieved by repeatedly performing parameter estimation with respect to the 1-DOF model.) Such an approach can be used in order to diagnose the electric motor very accurately, so that subtle degradations in the electric motor can be captured. On the one hand, the 1-DOF model has been shown to have a high sensitivity to variations in its parameters. On the other hand, the diagnosis can be based on a much larger dataset than just single measurements.

[0019] In a second aspect of the present disclosure, there is provided a computer program comprising instructions which, when the instructions are executed by a computer, cause the computer to carry out the method of the first aspect.

[0020] In a third aspect, there is provided a computer program comprising instructions which, when the instructions are loaded into a memory of a computer, provide a computer-executable 1-DOF model to represent an electric motor in terms of a horizontal component of a transverse displacement ^ of the shaft.

[0021] Each of the computer programs may be stored or distributed on a data carrier. As used herein, a “data carrier” may be a transitory data carrier, such as modulated electromagnetic or optical waves, or a non-transitory data carrier. Non- transitory data carriers include volatile and non-volatile memories, such as permanent and non-permanent storage media of magnetic, optical or solid-state type. Still within the scope of “data carrier”, such memories may be fixedly mounted or portable.

[0022] Generally, all terms used in the claims are to be interpreted according to their ordinary meaning in the technical field, unless explicitly defined otherwise herein. All references to “a / an / the element, apparatus, component, means, step, etc.” are to be interpreted openly as referring to at least one instance of the element, apparatus, component, means, step, etc., unless explicitly stated otherwise. The steps of any method disclosed herein do not have to be performed in the exact order described, unless this is explicitly stated. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Aspects and embodiments are now described, by way of example, with reference to the accompanying drawings, on which: figure 1 contains an axial elevation (left panel) and a side elevation (right panel) of an electric motor secured to a foundation; figure 2 illustrates a 1-DOF system model representing the electric motor; figure 3 is a flowchart of a vibration simulation method according to embodiments herein; figure 4 is a plot of offline parameter identification by a maximum-likelihood (ML) method; figure 5 is a comparison of ML and root-mean-square (RMS) estimates of velocity close to a critical frequency;figure 6 is a plot of numerical derivatives with respect to excitation frequency, which have been computed based on ML and RMS estimates of the amplitudes; and figure 7 is a plot of amplitude as a function of angular speed, when computed by a detailed motor model and a 1-DOF model. DETAILED DESCRIPTION

[0024] The aspects of the present disclosure will now be described more fully hereinafter with reference to the accompanying drawings, on which certain embodiments of the invention are shown. These aspects may, however, be embodied in many different forms and should not be construed as limiting; rather, these embodiments are provided by way of example so that this disclosure will be thorough and complete, and to fully convey the scope of all aspects of invention to those skilled in the art. Like numbers refer to like elements throughout the description.

[0025] Figure 1 includes two elevations of an electric motor 100 secured to a foundation (or substrate) 170 by two feet 170 with a transverse spacing of 2^ units. An axial direction ^ of the electric motor 100 is defined by its shaft, which is horizontally oriented in figure 1 at a uniform height ℎ above the foundation 170. A left end surface of the motor 100 is depicted in the left-hand elevation, in which the axial direction extends perpendicular to the plane of figure 1. Further axes ^ (transverse, horizontal) and ^ (vertical, downward) are indicated which form, together with the axial direction ^, a reference frame for describing the geometry of the electric motor 100, as well as forces, torques and vibratory movement therein. The right-hand elevation is a side view of the electric motor.

[0026] An axis of rotation Ω of the shaft 110 is parallel with the axial direction ^. The shaft 110 is connected to a rotor 152 to allow transfer of at least torque along the axis Ω. The shaft 110 may allow a limited axial movement of the rotor 152, so as to absorb axial shocks and / or attenuate vibrations in the axial direction. The rotor 152 is rotatably arranged with an airgap inside a stator 151 of the motor 100 for thereby receiving an electromagnetically induced torque in a per se known manner. Constructively, the shaft 110 is secured to the stator 151 via a plurality of intermediary elements of finite stiffness on each side of the rotor 152. The stator 151 may include a portion acting as a protective housing of the electric motor 100. The intermediary elements may include one or more of: a bearing journal (or roller bearing) 120, anend-plate arrangement 130, and a bearing housing 140. The shaft 100 and the stator 151 form an elastic structure, notably because of the finite stiffness of the intermediary elements (i.e., they are not modeled as rigid bodies), which is susceptible to mechanical vibrations. Facilitating the understanding of such vibrations is one of the aims of the present disclosure.

[0027] A detailed model of electric motors of the general type under consideration is proposed in Werner, U. (2017), “Mathematical multibody model of a soft mounted induction motor regarding forced vibrations due to dynamic rotor eccentricities considering electromagnetic field damping”. Journal of Applied Mathematics and Physics, 5, 346-364. doi:10.4236 / jamp.2017.52032 The model envisages large, fast-running AC motors, and induction motors in particular. In summary, the detailed model considers the electric motor as being composed of four subsystems: rotor, stator, bearing journal and end-plates system, and bearing housing system. The shaft journals system is connected to each bearing housing by the stiffness matrix for the bearing, ^^, and the damping matrix ^^. The two bearing housings are connected to the stator by the stiffness matrix ^^and the damping matrix ^^. The electromagnetism is considered by the magnetic stiffness and damping matrices represented by ^^and ^^, respectively. The foundations stiffnesses and damping are represented byand ^^, with suffices ^, ^, ^, ^ representing horizontal direction, vertical direction, left side, and right side, respectively.

[0028] The model according to Werner (2017) predicts the displacement of major points in the electric motor as a function of time under given conditions. It can be used to monitor and diagnose several critical components of the system including foundation, bearings and housing parameters. Another important aspect of the model is the consideration of the airgap torque as the excitation and the reactive torque to counterbalance it. However, there are nine degrees of freedoms, six masses, ten stiffness coefficients and nine damping constants to identify and tune. This level of complexity might not be feasible in practical applications, both due to computational constraints and to the massive system identification effort needed.

[0029] The inventor has realized, and demonstrated empirically, that a considerably simpler model of the electric motor 100 may be relied upon for the purpose of computer simulation of the vibratory behavior. More precisely, a careful modal analysis supported by experiments has led the inventor to treat the problem as an unbalance vibration of a rotating shaft, notably since the vibrations are usually excited by unbalance in the rotor 151 and / or the shaft 110. The single-degree-of- freedom or one-degree-of-freedom (1-DOF), system model illustrated in figure 2 will be considered. Here, a shaft 110 with a disk-like cross section is seen rotating at the angular velocity of Ω and is shown at a momentary angle Ω^ in the (^, ^)-plane at time ^. The disk has a center of gravity ^ located at a distance ^ from the centroid ^, and it is acted upon by forces ^^, ^^. The momentary orientation of the resultant force on the disk is indicated by the unit vector ^^^; the resultant force is normally rotating in the case of imbalances. The equation of the motion in horizontal direction ^ is given by:where the coefficients ^, ^ represent damping and stiffness, respectively. This has been found to be the dominating vibration mode. Equation (1) is equivalent to ^̈ + 2^^ ^ ^^̇ + ^^^ = ^Ω^cos Ωt ,(2)where (3)and ^^= is a resonance frequency in the absence of damping. In the cases of main interest, the vibratory response of such a system in steady state will be syn- chronized with the angular velocity Ω of the shaft 110 and has a phase lag of ^ behind the rotating imbalance direction. The response can be factorized as ^(^) = |^(Ω)| cos(Ω^ − ^) . (4) with the frequency-dependent amplitudewhereThe phase angle in (4) is given by

[0030] In a more general system description, the independent variable in equations (4)–(7) is denoted by ^, which may or may not be equal to the angular velocity Ω of the shaft 110. The shaft 110 may for example experience mechanical vibrations induced by a frequency converter although it does not rotate, Ω = 0.

[0031] The proposed 1-DOF model has been validated with respect to the detailed model according to Werner (2017), and more precisely by checking the agreement of normalized amplitude values provided by each of the models. The amplitude is known to depend on eccentricity, mass and angular speed. Figure 7 illustrates, for a series of numerical simulations, the amplitude’s dependence on angular speed for constant eccentricity and mass, wherein the dashed curve corresponds to the values obtained by the model according to Werner (2017) and the solid curve to values for the 1-DOF model. As seen in figure 7, the amplitude values were in good agreement ^ for angular speeds in the range 0.6 ≤ ^^≤ 1.4. Further, the location of the critical frequency, where the amplitude is maximal, differed by approximately 1% from the location predicted by the model according to Werner (2017).

[0032] Figure 3 is a flowchart of a method 300 for simulating vibrations in an electric motor 100. As explained above, the electric motor 100 is assumed to have a rotatable shaft 110 which extends horizontally and is secured, via at least two intermediary elements 120, 130 of finite stiffness on each side of a rotor 152, to a stator 151.

[0033] The method 300 can be executed on a general-purpose processing resource, such as a local or centralized processor, or a network of processors in distributed implementations. Depending on the way the 1-DOF model has been setup and on the nature of the technical manipulation, as foreseen by different embodiments, the processing resource may further be connected to suitable sensors and actuators (not shown) to enable an exchange of sensor data and / or control data with these. In general terms, sensor data provides information about the physical reality and control data is used to bring about desired changes in the physical reality. In relation to such exchange of sensor data and control data, the method 300 may be performed in an online or real-time fashion, wherein the executing processing resource processes an incoming stream of sensor data at the same pace as the stream is received and / or the processing resource generates an outgoing stream of control data at a pace sufficient to control the connected actuators during operation. Alternatively, in other embodiments, the method 300 is executed offline, in the sense that it processes a static set of input data and / or it generates a static set of output data, which may optionally be stored temporarily before it is used in a technical application.

[0034] In a first main step 310 of the method 300, a 1-DOF model suitable for representing the electric motor 100 is provided. The 1-DOF model relates to a horizontal component of a transverse displacement of the rotatable shaft 110, denoted ^. The example model defined by one or more of equations (1)–(7) can be used for this purpose. Here, the drive term on the right-hand side of equations (1) and (2) oscillates at a frequency corresponding to the angular speed of the shaft 110, e.g., they have an equal number of periods per second.

[0035] Providing the 1-DOF model may include various system-identification activities carried out on an actual electric motor 100. In particular, an amplitude and / or a phase of a vibratory response of the electric motor 100 to an oscillating excitation force may be estimated (step 312). On this basis, values may be assigned (step 314) to one or more of the parameters ^, ^, ^, ^ appearing in the model equations. The assigned value is computed from the estimated amplitude and / or phase of the vibratory response. The estimation may be based on direct or indirect measurements, and statistical methods may be applied.

[0036] The amplitude estimation may be based on peak-to-peak estimation (step 312.2), inverse RMS calculation (step 312.4), calculations including a Hilbert- type transform (step 312.6) or a combination of one or more of these. In particular, afinite impulse response filter, which is configured to approximate a Hilbert-type transform, may be used (step 312.8) in order to estimate the amplitude.

[0037] In some embodiments, which are suitable for offline implementations, the system is excited by harmonic forces with several excitation frequencies, and data is collected after a stabilization period has elapsed at each of the excitation frequency points. The frequency range should be selected such that the expected critical fre- quency is approximately midway between the lowest and highest excitation- frequency point. A two-stage approach may be followed: - Estimate the amplitudes and phases of the sinusoidal responses to the harmonic excitation forces at the steady state operation conditions. - Use the estimated amplitudes and equations (3)–(7) together with a nonlinear least-squares method to estimate stiffness, damping and oscillation amplitude. For purposes of the analysis, the harmonic forces and signals are written as a Fourier series in the form that all the frequencies in the model are expressed by the main cyclic period ^^with harmonics ^^= ^^^ for ^ = 1, 2, 3 … , ^. The following signal model may be used: 1,2, … , ^.(8)where ^ is the number of the samples in the sequence, ^ the number of harmonics, and ^[^] is Gaussian noise.

[0038] In other embodiments, the amplitudes of the sinusoids are estimated using the Hilbert transform. More precisely, according to a theoretical result, every real- valued vibration process ^(^) measured by a sensor can be imagined to be one of the possible projections on the real axis of some analytical signal ^(^)∈ ℂ. The quadrature part (on the imaginary axis) of the analytical signal ^(^) is the Hilbert transform of the vibration processes, that is, ^^,(9)where the integral is to be understood as a Cauchy principal value. The analytical signal can thus be reconstructed as ^(^)= ^(^)+ ^^̅(^) (10)with amplitude ^(^)=|^(^)|. In this disclosure, the term Hilbert-type transform meant to cover no only (9) but also various rescaled versions thereof, such as by removing the factor 1 / ^, which can easily be compensated by adapting the later calculations.

[0039] With respect to online or real-time implementations, peak-to-peak estimation (step 312.2) and variants thereof may be suitable options, as is the use of a FIR filter configured to approximate a Hilbert transform. Further, peak-to-peak estimation may as well be feasible on limited processing resources, such as a SandBox™ module in a frequency converter, and without access to a dedicated matrix processing library, with operations such as numerical matrix inversion. In a straightforward implementation, the peak-to-peak value is the difference between a vibration period’s maximum and minimum value divided by two. A statistically better estimate is based on the fact that for pure sinusoids the root-mean-square (RMS) value of the full cycle is 1 / √2 of the amplitude. Accordingly, the amplitude can be obtained by an inverse RMS calculation, i.e., multiplying a sensed RMS value by √2. If the amplitudes of the higher harmonics are not exactly zero, low-pass filtering of the signals with a cut-off frequency selected between the first and the second harmonic could improve the amplitude estimation and thus the accuracy of the estimated coefficients of the 1-DOF model.

[0040] With continued reference to implementations of peak-to-peak estimation, it has been found that the damping coefficient ^ in electric motors 100 of main inte- rest is negligible in comparison with the stiffness coefficient ^. Therefore, it is pos- sible to ignore the damping, meaning that the peak amplitude is attained at ^ =^ ^^(critical frequency) rather than the exact theoretical result^^ −so that the stiffness can be computed approximately as ^ ≈ ^^^for the critical frequency. On this basis, the frequency with the peak amplitude can be found by searching for the zero point of the numerical derivative of equation (6), namely,^^^^^^^^^2^ ≈ −^^^^^^^+ 16^^^^^^^^+ ^^^^^^− 16^^^^^^^^+ 2^^^^^^^^^^ −24Δ^^(11)where ^^^^, ^^^^, ^^, ^^^^, ^^^^are the frequency components of the exciting harmonic force, Δ^^is the separation of two consecutive frequency components, and ^^is as in equation (8). When applied to a sampled signal, the numerical derivative (11) tends to behave like an absolutely stable FIR filter. In equation (11), it is computed by a finite-difference approximation.

[0041] In practice, the zero-point may be located between two excitation frequency points. A more accurate estimate is obtained by using linear or nonlinear interpolation between the points where the derivative changes sign from positive to negative and by calculating the zero point using the obtained interpolation curve.

[0042] The real phase difference between measured velocity and measured torque is an important input parameter for controlling a vibration-attenuation system associated with the electric motor 100. To be precise, the phase difference of interest is the difference between the phases of the first harmonics of the measured velocity and torque signals. It is beneficial to calculate the difference using a maximum- likelihood sinusoidal estimation for the first harmonics and defining the length the modelling period exactly integer multiple of the fundamental frequency, i.e. to use a sequence with ^^= ^^× 2^^^ / ^^samples, where ^^is the number of full cycles and ^^is the sampling frequency. This will cause the diagonal and off-diagonal elements of the likelihood function to have a special form, whereby the phases of the first harmonics of velocity ^ and torque ^ signals can be estimated as per: ^^,^= arctan^^,^= arctanThe sought-for phase difference is obtained as Δ^ = ^^,^− ^^,^. To summarize, the theoretical results reviewed above enable an online implementation of a parameter estimation algorithm which provides the critical frequency and the phase difference between measured velocity and measured torque.

[0043] Practical considerations relating to the numerical estimation of the critical frequency are further illustrated by figures 4 to 6.

[0044] Figure 4 is a plot of offline parameter identification by a maximum- likelihood (ML) method, wherein a smooth waveform, consistent with amplitude in a neighborhood of resonance according to equation (6), has been fitted to the datapoints. The exact location of the critical frequency can thus be determined with reference to the smooth waveform rather than having to select one of the datapoints.

[0045] Figure 5 for its part is a comparison of ML estimates (×) and RMS estimates (+) of velocity close to a critical frequency. The horizonal scale is different from that of figure 4. It is seen that the ML and RMS approaches are generally consistent with each other.

[0046] Figure 6 finally is a plot of numerical derivatives with respect to excitation frequency according to equation (11). The derivatives have been computed based on ML estimates (×) and RMS estimates (+), respectively, and the best approximation of the critical frequency 32.16 Hz is indicated by a vertical dash-dotted line.

[0047] In a next step 320 of the method 300, vibrations in the shaft are simulated by solving a second-order differential equation for said displacement. The differential equation includes an oscillating drive term. In the equations (1) and (2), the drive term is the term proportional to cos Ω^ on the right-hand side, which is in units of force or acceleration.

[0048] Next, within step 330, one or more technical manipulations relating to the electric motor are performed on the basis of the vibration simulation in step 320. The technical manipulations generally include an interaction with the physical reality, e.g., they create or modify an electric motor in the real world, influence the operation of such an electric motor, or provide an indication of a condition prevailing in the electric motor. A number of examples will be reviewed.

[0049] First, the technical manipulation may include generating a control signal on the basis of the vibration simulation and applying (step 332) the control signal to a physical entity. For example, the control signal may be suitable for an electric motor 100 of the type simulated. Another example is where the control signal is suitable for – and applied to – an adaptive vibration attenuation system which is associated with the electric motor 100; reference is made to the above-described estimation of thecritical frequency and the phase difference between measured velocity and measured torque, which are key inputs for tuning the vibration attenuation system to the operating electric motor 100. The vibration attenuation system can then be controlled (step 334) based on the generated control signal.

[0050] Secondly, the technical manipulation may include determining (step 336) one or more design parameters which describe the structural (dimensions, materials etc.) or electric properties of an electric motor 100 of the simulated type. The design parameters may be determined, for example, in order to prevent undesired resonance, to make the electric motor 100 compatible with a foundation 170 in an intended worksite, or to achieve similar benefits. The second type of technical manipulation may optionally include manufacturing an electric motor 100 in accordance with the design parameters thus determined.

[0051] Thirdly, the efficient computer simulations made possible by the teachings in the present disclosure can be used for diagnostic purposes, including trouble- shooting. More precisely, the technical manipulation may include detecting an evolution over time of the electric motor 100 by repeatedly comparing (step 338) measured vibration data with simulated vibration data. In other words, parameter identification applied to one or more parameters of the 1-DOF model can be utilized to detect degradation of the electric motor 100 and / or the environment in which it is installed. The degradation, including early signs thereof, may include structural faults, such as: i) change in stiffness of a foot 160 through which the electric motor 100 is secured to the foundation 170, including various cases of “soft foot”, e.g., especially rocking soft foot, angled soft foot, induced soft foot, squishy soft foot; ii) bearing faults; iii) misalignment of shaft and bearings; iv) unbalance, e.g. due to uneven erosion or debris buildup on rotating elements; v) mechanical looseness; vi) stator problems, e.g., eccentricity, looseness, insulation faults; and vii) rotor problems, e.g., broken or cracked rotor bars, short-circuited rotor laminations, loose rotor bars.

[0052] The promising outlook of this detection approach is supported by a numerical sensitivity study and a laboratory experiment that the inventor has carried out. In a first part of the numerical sensitivity study, the effect on horizontal fre- quency and amplitude of a change of the horizontal stiffness of the left foot by –50%, –25% and +25% relative to a reference value was investigated using the detailed model according to Werner (2017). In a second part, the effect of changing the stiffness of both feet – by the same amounts – was explored. In third and fourth parts, concurrent changes in vertical stiffnesses were added. Each time, there was a significant and thus easily discoverable change in the frequency and amplitude of the horizontal component of the vibration. Because of the strong impact on the horizontal component, it may be concluded that the 1-DOF model proposed herein may, despite its simplicity, be used as an accurate tool for diagnosing an electric motor 100 of the type under consideration. In mathematical terms, the 1-DOF model has a high sensitivity to structural faults affecting one of the feet 160 of electric motor 100, so that a parameter identification process applied to the 1-DOF model is likely to return distinctly different values of its parameters, and of the stiffness ^ in particular.

[0053] A purpose of the above-mentioned experiment was to further investigate what impact a change in stiffness may have on the vibratory response of the electric motor 100. In the laboratory experiment, a trial installation with an electric motor was modified by loosening one of the bolts to the foundation. The bolt was unscrewed by half a turn from the tight position, resulting in a change in the stiffness of the foundation and thus a shift of the critical frequency for which the peak amplitude is reached. More precisely, the loosened bolt was found to have decreased the stiffness by about 12% and to have reduced the critical frequency by 6%. There was also slight increase in the vibration velocity magnitude with the loosened bolt case. In the state of the art, soft foot conditions have usually been diagnosed and identified by vibration analysis and alignment measurements (see for example EP0920953A2), which include significantly more work-intensive handling and causes downtime.

[0054] The aspects of the present disclosure have mainly been described above with reference to a few embodiments. However, as is readily appreciated by a person skilled in the art, other embodiments than the ones disclosed above are equally possible within the scope of the invention, as defined by the appended patent claims.

Claims

CLAIMS 1. A method (300) for simulating vibrations in an electric motor (100), the electric motor (100) including a rotatable shaft (110) which extends horizontally and is secured, via at least two intermediary elements (120, 130) of finite stiffness on each side of a rotor (152), to a stator (151), the method comprising: providing (310) a 1-degree-of-freedom, 1-DOF, model to represent the electric motor in terms of a horizontal component of a transverse displacement (^) of the shaft; simulating (320) vibrations in the shaft by solving a second-order differential equation for said displacement, which differential equation includes an oscillating drive term; and performing a technical manipulation (330) relating to the electric motor on the basis of the vibration simulation.

2. The method (300) of claim 1, wherein said intermediary elements include a bearing journal (120).

3. The method (300) of claim 1 or 2, wherein said intermediary elements include an end-plate arrangement (130).

4. The method (300) of any of claims 1 to 3, wherein said intermediary elements include a bearing housing (140).

5. The method (300) of any of the preceding claims, wherein the electric motor (100) is secured to a foundation (170) via at least one foot (160).

6. The method (300) of any of the preceding claims, wherein the drive term oscillates at a frequency corresponding to the angular speed of the shaft (110).

7. The method (300) of any of the preceding claims, wherein the providing (310) of the 1-DOF model comprises: estimating (312) an amplitude and / or a phase of a vibratory response of the electric motor (100) to an oscillating excitation force; and assigning (314) a value to a parameter of the 1-DOF model.

8. The method (300) of claim 7, wherein the providing (310) of the 1-DOF model comprises estimating (312) an amplitude based on at least one of:peak-to-peak estimation (312.2); an inverse root-mean-square calculation (312.4); a Hilbert-type transform (312.6); a finite impulse response filter (312.8) configured to approximate a Hilbert transform.

9. The method (300) of any of the preceding claims, which is performed online and / or in real time.

10. The method (300) of claim 9, wherein the providing (310) of the 1-DOF model comprises estimating (312) an amplitude using a finite impulse response filter (312.8) configured to approximate a Hilbert transform.

11. The method (300) of any of the preceding claims, wherein the technical manipulation (330) comprises applying (332) a control signal generated on the basis of the vibration simulation of the electric motor (100).

12. The method (300) of any of the preceding claims, wherein the technical manipulation (330) comprises controlling (334) an adaptive vibration attenuation system, which is associated with the electric motor (100), on the basis of the vibration simulation.

13. The method (300) of any of the preceding claims, wherein the technical manipulation (330) comprises assigning (336) a value to a design parameter of the electric motor (100).

14. The method (300) of claim 13, wherein the technical manipulation (330) further comprises manufacturing an electric motor (100) in accordance with the assigned value of the design parameters.

15. The method (300) of any of the preceding claims, wherein the technical manipulation (330) comprises detecting an evolution over time of the electric motor (100) by repeatedly comparing (338) measured vibration data with simulated vibration data.

16. A computer program comprising instructions which, when the instructions are executed by a computer, cause the computer to carry out the method (300) of any of the preceding claims.

17. A computer program comprising instructions which, when the instructions are loaded into a memory of a computer, provide a computer-executable 1-DOF model to represent an electric motor in terms of a horizontal component of a transverse displacement (^) of the shaft, wherein the electric motor (100) includes a rotatable shaft (110) which extends horizontally and is secured, via at least two intermediary elements (120, 130) of finite stiffness at each end of the shaft, to a stator (150).

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