A method for real-time simulation of the noise and vibration of electric machines

Abstract

Driver-in-the-loop vehicle simulation platforms are utilised to improve the vehicle design at an early stage, in particular with regard to noise, vibration and harshness. Here, a novel, real time simulation model applied to automotive electric motors is described, generating a capability to demonstrate noise and vibration from initial design decisions, avoiding experimental or emulated recordings. Thus, a rapid method for modelling sound and vibration of a brushed DC permanent magnet motor under various loads and speeds is shown. The motor can be controlled with a PWM signal or through direct current. To obtain the sound and vibration response at a specified load condition, which will be used for quick motor design/integration, a rapid and widely applied model is needed. Hence, in this work, a simplified model has been established for analysing the permanent-magnet machine in MATLAB/Simulink, achieving a real-time/quasi-real-time estimation of the sound radiation due to the internal magnetic force. The real-time internal magnetic force is calculated under both loading and unloading conditions in a simplified polar coordinate system with consideration of the stator slotting effect. Then, a simplified structural model is created using design data appropriate to an early motor design iteration, and the real-time sound radiation profile is estimated by the modal superposition model. This method is validated with the experiment for a DC motor but can be easily extended to other types of machines such as permanent magnet synchronous motors found in automotive electrical vehicles.

Keywords

Automotive components driving modelling/simulation electric vehicles engine noise transportation noise vehicle comfort vehicle noise/vibration vehicle simulation/modelling

Introduction

The automotive sector is facing significant challenges following the successful integration of electric powertrains. The traditional industrial design process using physical prototypes is too expensive, causes focus in too narrow a geographical area and creates issues of delay during reworking,¹ the cycle of iterative design not being suitable for the fast introduction of new models. Instead, simulation and emulation methods are favoured through the use of digital twin models of the vehicle,² which also allow greater product development through software after the vehicle is launched. The emphasis on real time simulation is emphasised for condition monitoring, both before launch and during service. In the specific field of Noise, Vibration and Harshness (NVH), the digital twin needs to represent the narrow band tonal components of the electric motor into the body-in-white structure so that development targets may be set.^3–5 Mehrgou et al.³ show examples of modern design practice for NVH, incorporating multi-objective optimisation with AI tools to highlight patterns and features.

There are a range of fidelity levels⁶ from analytical models to full multi-physics simulations covering the full range of non-linear electromagnetic, thermal designs, but often soft coupling⁷ between lower fidelity models are preferable given the lack of higher detail validation models (especially the difficulty measuring dynamic electromagnetic fields) at high rotations per minute. A key interest is in the integration of driver-in-the loop simulators,^1,6 where design engineers, managers, potential customers may explore the design space in order to objectively set targets. Xue et al. note that the NVH process is particularly unsuitable to narrow band performance targets due to the influence of human perception.¹ The driver-in-the loop simulations typically rely on either experimental measurements or simulation methods to obtain the acoustics for the vehicle. The former provides a fastest path to obtain a data set of a (prototype) vehicle, including wind and tyre noise. However, the vision for electric vehicles is a virtual vehicle based exclusively on simulation models.

In this paper, the focus is on producing both a method and workflow by which a virtual electric motor can be incorporated into a driver-in-the-loop simulator, for NVH performance. A novel approach, required for the real-time implementation, results in a simulation model which can replace the pre-recorded acoustic generation, thus allowing design alterations and more responsive target setting. An engineer would be able to obtain driver feedback on sound and vibration more rapidly than currently possible. In addition, the workflow is a direct substitute for the existing frequency response function excitation to the body-in-white that most manufacturers rely on. This is an important realistic, industrial advantage, as authors have shown that changes to audio soundscapes and visual details can influence the impression of the vehicle under development. Given the pressure to avoid manufacturing a test vehicle to record and test, it is important to be able to set realistic targets by simulation which agree with visual and driving dynamics,⁸ so this paper addresses this need through a real-time implementation of an electric motor powertrain.

High fidelity simulations are available, usually based on the finite element modelling (FEM) of the structure, see for example the review by Ho and Fu⁹ covering both two and three dimensional (2D, 3D) approaches, with rotation and different linear or simplified⁵/non-linear¹⁰ approaches to the electromagnetic fields, but these are nowhere near real-time. Indeed, the computational cost for a typical permanent magnet synchronous motor (PMSM) has led to different NVH approaches, including the use of the boundary element method to discretise surfaces rather than volumes for acoustic radiation,⁴ but still replying on finite element modelling for both the electromagnetic and structural response. Nevertheless, even these are unsuitable for real-time application, the indirect method needing to run per individual frequency rather than for a time domain solution. In this paper, analytical expressions for the electromagnetic forcing avoid these computational restrictions.

The typical real-time driver-in-the-loop simulators need powerful computers to run graphics and motion platforms, therefore the audio is often limited to pre-recorded sound clips taken from a database^8–12 and merged together through a complex mixing algorithm (taking into account vehicle type and state). The powertrain, wind, road and background noise are pitch blended before the sound is played through speakers or headphones. Blommer and Greenberg¹¹ show the scale of the challenge building a database using real pre-recorded sounds, where the vehicle needs to be driven on a track or located in a wind tunnel and the engine switched off in 10 mph increments, but also every 500 RPM of the engine (this would have been an internal combustion engine, but the same principle applies to permanent magnet synchronous motors (PMSM)). Genuit and Fiebig¹² note that the noise and comfort evaluations with acoustics were frequently and subconsciously influenced by vibrations through the seat. As such, there exists sufficient evidence that a digital design would benefit from both simulated audio but also the simultaneous and accurate provision of vibration information, without the need to first build a vehicle to record experimental data. This paper shows such an approach as a distinct advantage available to run on a single computer and transfer information to a motion simulator through a UDP network protocol with no effect of latency.

In this paper, the novel contribution includes,

A workflow producing radial vibration and acoustic radiation pattern from an automotive electric motor.

An analytical modelling approach solving the electromagnetic force equations for a high rotational speed and small integration timestep.

Methodology to support exploration of the design space for appropriate NVH target setting in an automotive industrial environment.

Provision of a coupled structural model which provides natural frequencies, mode shapes and shows coupling mechanisms to the high speed rotation (hence links to wave speeds and annular Doppler shifted modes).

An end to end solution which is robust and fits the criteria of a real-time operation for a driver-in-the loop simulator.

Provision of validated vibration and acoustic data which can produce real-time audio files suitable for transfer over a standard network connection.

Provision of a real-time digital-twin electric motor sound without pre-recording or low fidelity emulation.

Through the elimination of existing state of the art pre-recorded sound clips, the actual real time sound will be generated which is linked to the design. This is a far more powerful approach allowing the design to be changed in simulation and presented both to supporting engineers and managers.

The real-time methodology is demonstrated through the use of Simulink, which negates the typical approach of modelling the whole motor (electromagnetic, thermodynamic, cooling and structural response) using finite elements, an approach which is impractical at an early design stage when knowledge of coil size, shape, windings, magnet sizes or cooling boundary conditions is not available, or when the parameters of such complex models are not yet identified. Thus, the novel real-time model for use on a whole vehicle simulator based on initial design data is clearly advantageous. In a discussion of validation criteria, it is shown that variability in unknown boundary conditions can significantly influence the measured sound in an experiment, therefore this analytical approach provides stability in the early design phase.

Following a short background, the methodology is demonstrated in section ‘Analytical formulations’, comprising the electromagnetic flux density conversion to the radial and tangential forces on the motor shell, and a modal superposition approach to produce the acceleration of the motor shell. In section ‘Model validation and verification’, a validation excise is undertaken followed by conclusions.

Background

To demonstrate the methodology, the problem is described using a brushed DC motor, however, this is equally applicable to a 3-phased AC PMSM motor, with highlighted modifications to the electromagnetic equations.

Permanent-magnet direct current (PMDC) motors are widely used in both industrial and domestic applications for their simplicity, low cost and reliability. The noise and vibration behaviours were extensively studied for the permanent magnet brushless motor,^13–18 however, there are few publications which join all elements of the methodology, from user demand through to the noise and vibration of the brushed PMDC motor. He et al.¹⁹ derived an analytical model for a two-dimensional electromagnetic field in polar coordinates and validated this against an FE model and experiment. Furlan et al.,²⁰ developed a numerical coupled electromagnetic-mechanical-acoustic model to predict sound radiation of the DC motor, although it included boundary element models and as such, did not run in real time. This provides evidence of a need, starting from simple design data to be able to predict noise and vibration characteristics in real-time, for simulator integration, that this paper provides as a novel contribution.

Finite element analysis is a common method to predict the noise and vibration of the E-motor,^21,22 this includes electromagnetic analysis to estimate the magnetic force, mechanical analysis to predict modes and displacement and then using an acoustic model to calculate the noise level. This method can have a high level of accuracy, subject to the supply of the detailed parameters and fine meshes, at the expense of high computational running time and the need for a high level of detail of part specification, including material properties and geometry. In real world validation, variability is often overlooked.

This research provides an analytical model, which can be an alternative method of calculating the acoustic radiation from the motor with high fidelity. In this model, as the flowchart in Figure 1 shows, a simplified motor model is established based on a given motor constant and winding resistance. The user demand is a pulse-width-modulated (PWM) voltage and the current in the coils and the torque generated by the motor can be estimated in real-time.

Figure 1.

Noise and vibration flowchart.

The simplified motor model simulates the propulsion side, where such models come in the form of a system of ordinary differential equations with lumped parameters (motor constant, winding resistance/inductance, rotor inertia or as look-up tables/torque maps). They are generally easier to parametrise and use at an early stage of development compared to the complete electromagnetic alternative. Critically, they are much easier to link to the electric circuit including the DC bus, battery, inverter etc., for a full system simulation. Therefore, one is included in this work to generate the high level current/speed states and the associated torque as a function of driver input and operating conditions. The current and speed are then fed to the detailed electromagnetic model that calculates the relevant excitation on the motor casing. The torque calculated by the detailed electromagnetic model compares well with that calculated by the simplified model, offering an opportunity for model verification/validation. This closed validation loop between the simplified and the electromagnetic model is clearly shown in Figure 1.

Using the high level inputs from the simplified model, the radial and tangential flux densities are calculated by the electromagnetic model, then through the Maxwell tensor function, the flux density is converted into radial and tangential forces acting on the motor shell. This utilises simplified data available at the early design stage for example, motor size, number of slots, magnets and approximate coil windings. Crucially, at this early design stage, the fine details of the motor assembly, boundary conditions such as adhesive joint mating, coil winding overlapping surface area, variable friction mechanisms and thermal/cooling changes are unlikely to be known. So, a recognition that industrial information cannot fully satisfy the needs of the highest fidelity, non-linear, multi-physics, offline finite element solver is essential. This is the electrical analogy of mechanical design, where machine elements use of empirical equations provide an early design option²³ where finite elements cannot possibly solve problems. This limitation of information must also be recognised in validation, where comparisons to real world measurements will be different to that early design knowledge.

The rotor torque can be calculated by integration of the tangential force and validated with the torque generated by the simplified motor model (a useful verification). In order to obtain the structural response for the motor casing through the body-in-white vehicle structure two options are available: a finite element model (FEM) or a modal superposition using just natural frequencies and mode shapes. The former is computationally expensive for real time application involving a driving simulator, hence FEM is only relied on for obtaining the necessary mode shapes and eigenvalues for the modal superposition approach, valid through the imposition of a loss factor in the equations.

A modal superposition model is created, with the radial electromagnetic force applied to the motor shell, therefore, the real-time displacement/velocity/acceleration of each node can be calculated in real-time.

The assumptions of this model are: (1) The motor shell is physically homogeneous, (2) The coupled change of the magnetic field due to a radial structural deformation is neglected, (3) The permeability of iron is infinite and the permanent magnets are magnetised in the radial direction with constant recoil permeability and (4) The excitation caused by the frictional forces, for example, brush and bearing, is simplified as white noise acting on the motor shell.

Available design data

The motor used for validation of the model is a Parvalux M63QN-0011 (Figure 2), with a regulated 12 V supply voltage and 250 W power rating, the no-load output speed is 3000 RPM with a maximum torque of 0.8 Nm. Two ferrite magnets are adhesively bonded to the rolled steel casing, which also holds two carbon brushes, as Figure 2(b) shows. The rotor has 16 slots with 93 mm effective length (Figure 2(c)) held with ball bearings in aluminium caps. The parameters of the motor are shown in Table 1.

Figure 2.

The exterior and interior of the DC motor for measurement: (a) motor shell and shaft, (b) motor magnet, brushes and commutator and (c) rotor and caps.

Table 1.

Parameters of motor design data.

Parameters	Value	Unit
No pole pairs	$p$ = 1	–
No rotor slots	$Q_{s}$ = 16	–
Magnetic arc per pole pitch ratio	$α_{p}$ = 125/180	–
Radius rotor surface	$R_{r}$ = 0.032	m
Radius stator surface	$R_{s}$ = 0.042	m
Radius permanent magnets surface	$R_{m}$ = 0.034	m
Length motor shell	$L_{shell}$ = 175	mm
Magnet remanence *	$B_{r}$ = 0.2	T
Relative recoil perm.	$μ_{r}$ = 1.02	–
Angle single slot opening	$α_{0}$ = 0.07	rad
No of series turns * armature winding	$N_{ph}$ = 10	–
Motor input voltage	12	Volt
Motor max speed	3000	RPM
Total resistance winding	$R_{resis}$ = 0.573	Ohms
Winding inductance*	$L$ = 0.00001	H
Inertia of rotor	0.000283	kg m²
Friction torque per	0.00001	Nms
rad/s (linear)*

^*Estimated value.

Analytical formulations

Electric magnetic force generation

In a typical brushed PMDC motor, as shown in Figure 3, the permanent magnets are bonded on the inner surface of the motor housing in pairs. In the air gap between the permanent magnets and the rotor, the magnetic field is comprised of two components: a field from the permanent magnets, and the armature reactions.¹⁹ A field solution of air gap magnetic flux density is given by Zhu et al.,¹⁷ based on the 2-D Laplacian and quasi-Poissonian equation, whereas the flux density from the armature reaction can be derived from the formulation of a single coil electromagnetic field.^19,24 Whilst this is a 2D formulation, it is noted that this is a fully accepted approach in the literature, covering both analytical and finite element approaches, see Król et al.²⁵ The rotor slot effect can be taken into account by introducing a complex relative air gap permeance based on conformal transformation, which can be found in Zarko et al.²⁶ Hence the radial and tangential direction of flux density in the air gap can be expressed as:

\begin{matrix} B_{sr} = B_{r} λ_{a} + B_{t} λ_{b}, B_{st} = B_{t} λ_{a} - B_{r} λ_{b} \end{matrix}

(1)

where $B_{r} = B_{r_mag} + B_{r_arm}$ is the sum of radial magnetic and armature flux density, $B_{t} = B_{t_mag} + B_{t_arm}$ is the sum of tangential magnetic and armature flux density, $λ_{a}$ and $λ_{b}$ are real and imaginary parts of the air gap permeance.

Figure 3.

Schematic diagram of a PMDC motor.

According to the Maxwell stress tensor, the radial and tangential force densities in the air gap can be expressed in terms of $μ_{0}$ , the magnetic constant as:

\begin{matrix} f_{r} & = \frac{1}{2 μ_{0}} (B_{r}^{2} - B_{t}^{2}), f_{t} = \frac{1}{2 μ_{0}} 2 B_{r} B_{t} \end{matrix}

(2)

Slotless flux density from the permanent magnets

The first of the two flux densities comes from the movement of the rotor material (with slots) passing through the magnetic field of the permanent magnets. According to Zhu et al.,^17,27 the magnetic flux density in the air space can be expressed as:

\begin{matrix} B_{r_mag} (r, θ) = \sum_{n = 1, 3, 5 . . .}^{\infty} K_{B} (n) \cdot f_{Br} (r) \cdot \cos (np θ) \\ B_{t_mag} (r, θ) = \sum_{n = 1, 3, 5 . . .}^{\infty} K_{B} (n) \cdot f_{Bt} (r) \cdot \sin (np θ) \end{matrix}

(3)

where the sum over angular order ( $n$ ) transforms the angular field around the stator into an equivalent periodic mode shape of flux distribution. The term $f_{Br} (r)$ is dimensionless and provides an amplitude scaling in the air gap for each angular order, while $K_{B} (n)$ provides the specific flux density based on the permeability and the remanence of the magnets. These are provided for completeness in the Appendix.

Flux density from the armature reaction

The second of the two flux densities comes from the current flow in the coils. The slotless flux density from the armature reaction can be simplified as an integration of magnetic fields induced by a set of thin coils placed on a smooth rotor surface.^19,24 The slotless armature reaction field of a PMDC motor can be expressed as:

\begin{matrix} B_{r_arm} (r, θ) = \sum_{m = 1}^{\infty} C_{m} (r) \cos (m θ - \frac{m π}{2 p}) \\ B_{t_arm} (r, θ) = \sum_{m = 1}^{\infty} C_{m} (r) \sin (m θ - \frac{m π}{2 p}) \end{matrix}

(4)

where

\begin{matrix} C_{m} (r) = \frac{4 I μ_{0}}{π r} \frac{R_{r}}{m b_{0}} {(\frac{R_{r}}{r})}^{m} \frac{R_{s}^{2 m} + r^{2 m}}{R_{s}^{2 m} - R_{r}^{2 m}} \cdot \\ [\sum_{k = 1}^{Q_{s} / 2} \sin (m (2 k - 1) \frac{π}{Q_{s}})] \sin (m \frac{α_{0}}{2}) . \end{matrix}

(5)

The current $I$ is in the armature coils, $α_{0}$ is the angle of slot openings, $b_{0} = α_{0} R_{r}$ is the length of slot openings and $Q_{s}$ the number of slots.

Slot effect

The effect of rotor slotting, shown in Figure 3, is accounted for by introducing a complex, relative air gap permeance ( $λ$ ) based on the slot shape transformation, which can be found in Zarko et al.²⁶ The permeance is calculated numerically for a wide range of angles for a single slot, however to replicate this for every slot a quicker and simpler method is through an analytical Fourier series, split into a waveform of real and imaginary components, hence $λ_{a}$ and $λ_{b}$ in equation (1) can be expressed as:

\begin{matrix} λ_{a} = λ_{0} + \sum_{n = 1}^{N_{λ}} λ_{an} \cos (n Q_{s} θ), λ_{b} = \sum_{n = 1}^{N_{λ}} λ_{bn} \sin (n Q_{s} θ), \end{matrix}

(6)

where $N_{λ}$ is the maximum order of the Fourier series (eight as the upper limit in Matlab), $λ_{0}$ , $λ_{an}$ and $λ_{bn}$ are the Fourier coefficients that are calculated from the $λ$ waveform. The equations for $λ$ are taken from Zarko et al.²⁶ which also provides a validation source for the waveform (and thus the Fourier amplitude in this paper).

Modal superposition model

For the electric motor, the magnetic force in the air gap of the machine is acting on the internal components of the shell, for example, permanent magnets (DC motor shown in Figure 3) or slot tooth (PMSM motor). These periodic excitation forces lead to vibration of the inner surface and then are transmitted to the outer surface via compressive and flexural waves. This vibration is the body-in-white structural excitation.

As the mathematical method must be employed in real-time, this transmission path is represented using a modal superposition model, which relies solely on natural frequencies and eigenvectors. The reference structural model can be from finite element predictions (as in this paper), empirical estimations or experimental measurement by laser vibrometers. The modal superposition method is widely used due to the computational speed²⁸ and high accuracy, subject to the correct natural frequencies, mode shapes and damping. It has been used for electric motor structural design^7,25,28 and to understand coupling between components and geometry, such as winding geometry,²⁹ therefore the use in this paper as part of the wholistic methodology is endorsed.

By simplifying the motor stator and rotor as a decoupled linear system where the vibration of the housing does not lead to a change in the force, the excitation can be amplified by a transfer impedance characterised by the modes of the system.³⁰ The resonances occur when the frequencies of the excitation force coincide with the system eigenvalues. The list of reference natural frequencies and mass normalised mode shapes are calculated by FE package ABAQUS, using an FE representation of the motor casing in its entirety, including the shell, magnets, adhesive glue and space for the motor brushes.

The implementation is summarised here, for a thorough derivation refer to Wan et al.³¹ and Tsai.³² For a forced response the equation of motion can be expressed as

[M] {\overset{\cdot\cdot}{u}} + [C] {\overset{\cdot}{u}} + [K] {u} = {F (t)}

(7)

where $[M]$ , $[C]$ and $[K]$ , are mass, damping and stiffness matrices respectively, ${\overset{\cdot\cdot}{u}}$ , ${\overset{\cdot}{u}}$ and ${u}$ are nodal acceleration, velocity and displacement, respectively. ${F (t)}$ are time domain nodal forces (the applied forces from the electromagnetic field).

A structural spatial FE model described by equation (7) with $m$ degrees of freedom will provide $m$ eigenvalues and the associated $n$ eigenvectors. A subset of $m < n$ eigenvalues with their corresponding mass-normalised eigenvectors can then be selected to represent the structural response of the system. For simplicity, the system is considered initially without damping and by pre- and post- multiplying with the transpose eigenvector and eigenvector matrices respectively, it is converted to an $m \times n$ system of decoupled equations in modal coordinates, $Z$ , as shown in equation (8).

[M_{r}] \overset{\cdot\cdot}{Z} (t) + [K_{r}] Z (t) = [Φ]^{T} F (t)

(8)

where,

\begin{array}{l} [M_{r}] = [\begin{matrix} 1 & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & 1 \end{matrix}], [K_{r}] = [\begin{matrix} ω_{1}^{2} & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & ω_{m}^{2} \end{matrix}] \\ [Φ] = [\begin{matrix} α_{11} & α_{21} & \dots & α_{m 1} \\ α_{12} & α_{22} & \dots & α_{m 2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ α_{1 n} & α_{2 n} & \dots & α_{m n} \end{matrix}] \end{array}

Damping is then added via the inclusion of a damping term in each decoupled equation, as shown in equation (9), with $ζ_{m}$ as the damping ratio at the $m$ th mode.

[M_{r}] \overset{\cdot\cdot}{Z} (t) + [C_{r}] \overset{\cdot}{Z} (t) + [K_{r}] Z (t) = [Φ]^{T} F (t)

(9)

where,

[C_{r}] = [\begin{matrix} ζ_{1} ω_{1} & \cdot \cdot \cdot & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \cdot \cdot \cdot & ζ_{m} ω_{m} \end{matrix}],

Since $[M]$ , $[C]$ and $[Ω]$ are diagonal, for each mode the problem can be solved as decoupled equations ${\overset{\cdot}{Z}}_{m} + ω {\overset{\cdot}{Z}}_{m} + ω^{2} Z_{m} = F_{m} (t)$ , which is computationally inexpensive and so contributes to the novel real-time implementation. Then as $u_{m} (t) = Φ_{m} Z_{m} (t)$ , the spatial response is obtained by superposition of the response of each decoupled equation as indicated by equation (10).

u_{n} (t) = \sum^{m} Φ_{mn} Z_{m} (t) .

(10)

Hence, for any force acting on the nodes, the displacement for each node can be calculated by the modal superposition model.

Model validation and verification

The validation for the real-time Simulink model is considered with reference to its constituent elements shown in Figure 1. In this section, the magnetic and armature fluxes are compared to the results in the literature. A comparison is undertaken of the structural frequency response function between the modal superposition method and the response generated by ABAQUS. Finally, the model is established against the experiment measurements of a DC motor in a vibration laboratory.

Magnetic flux validation

As the rotor turns, the interaction of the iron core with the permanent magnets generate a localised radial and tangential magnetic force. Using the motor parameters shown in Table A1 (parameters from Zarko et al.²⁶), predictions of the radial magnetic flux density against rotor angle are shown in Figure 4(a), as periodic wave forms with a significant change of gradient for small changes of rotor angle. It is this square wave which causes the unsteady forces on the motor shell (at frequencies similar to a Sinc function), thus a vibration or acoustic pitch with harmonics provides an audible feedback to a vehicle driver. The analytical match (blue lines in Figure 4) can be compared to the waveform given in Li et al.³³ (red line) where the underline shape is a good match with differences provided by the slot effect. The slot effect generates a higher narrow band frequency content dependent on the number of slots and magnet pairs. The shape of the slot effect (yellow lines in Figure 4) agrees well with the result from Li et al.³³

Figure 4.

Flux density in (a) radial and (b) tangential directions from permanent magnets.

Similarly the prediction of tangential flux densities is shown in Figure 4(b), with positive and negative flux producing impulse forces on the shell. This contributes to the excitation frequencies and harmonics of torque ripple. Again, the simulated waveform compares well with Li et al.³³ and it is noted that the ratio of radial to tangential peak flux density is also similar between Li (3.4) and this paper’s simulation (4.6).

Armature flux validation

As current flows in the coil windings located in the armature, a magnetic field is generated with radial and tangential flux density components. The radial component is validated using parameters taken from He et al.¹⁹ and included in Table A2. The radial flux density as seen in Figure 5, is plotted against the angle where the decomposition of the waveform into a sum of Sine waves is clear. Once again, this contributes to the acoustic pitch variation with speed, however, the amplitude of this component significantly depends on the user demand (through the current loading).

Figure 5.

Radial flux density from the armature reaction.

Through a comparison with Tian et al.³⁴ the shape of armature flux density is broadly correct, subject to scaling of motor power.

The tangential flux density from the armature can be aggregated for all of the coils around the circumference, this is the net torque on the rotor which provides the useful work. For completeness, the equations for the simplified PMDC motor used in this work are included in the Appendix. The torque generated for the same input current by both the simplified and electromagnetic model is shown in Figure 6. The torque ripple due to the high frequency switching of the current is also visible in the figure.

Figure 6.

Comparison of the torque generated from the simplified motor model with that calculated by the electromagnetic forces. Blue is simplified model and red is electromagnetic force. The red overlays the blue almost exactly and the reader’s attention is drawn to the excellent match between peaks of duration approximately 0.2 s. The curves match as close as is possible to show visually.

The red dash line is the torque calculated by summing all of tangential element moments, which matches a high level of accuracy in both long term trend and instantaneous output.

In the example shown, the PWM signal is at 5 kHz and does not change with motor speed. Thus, it causes characteristic narrow-band whine.

Structural response validation

The finite element method can be used to generate natural frequencies and mode shapes, however, this is computationally expensive and does not meet the requirement for real-time simulation. It does allow the generation of frequency response functions (structural radial surface acceleration for an impulse input). A faster method to get this FRF is though the modal superposition method.

The outer shell of the DC motor is generated in the finite element program ABAQUS. As shown in Figure 7, the steel shell comprises 11220 hexahedral elements (type C3D8R), the magnets are formed of 6720 elements while the glue interface is represented as a tie constraint. A virtual hammer test is conducted to provide the accelerance of the surface against frequency. This is shown as a blue line in Figure 8. An illustration of mode shapes is provided in Figure 7(b).

Figure 7.

(a) ABAQUS model showing shell and magnets and (b) example of generated mode shapes at different frequencies.

Figure 8.

Modal superposition validated with FEA.

A list of natural frequencies with corresponding eigenvectors representing $Φ$ in equation (9), is exported from ABAQUS to an rpt file and then loaded into Matlab. A virtual hammer test is then created in Simulink to ensure the same response can be replicated, this uses the natural frequencies and mode shapes from simulation with the appropriate response damping implemented as a constant viscous damping value.

A virtual hammer test has been applied in both ABAQUS steady-state dynamic and the modal superposition model (as shown in Figure 9). The frequency response function obtained through real-time Simulink is shown on the red line in Figure 8. The match is highly accurate showing that the analytical model in Simulink can provide a robust prediction subject to a suitable FEA model.

Figure 9.

Flow chart of simulation for the virtual hammer test.

Whilst validation against finite element models is shown to be excellent, validation against real world experiments is also important and whilst these are outside of the scope of this paper, several important points will be made. Capturing accurate acoustics and vibration data at higher frequencies seen in electric motors is challenging due to the short wavelength in comparison to boundary condition changes. Typical automotive industry practices¹¹ include looking at frequency response functions with band averaging,³⁵ due to modal overlap and damping. The variability in experimental measurements is often unsuitable for narrow band frequency identification,³⁶ see for example real vehicle measurements with statistical distributions,³⁷ or where repeats of experimental measurements are averaged to provide a mean result with confidence levels. Hence the focus of this paper is providing the acoustic experience for a driver-in-the loop simulator at an early stage of the design process, where parameters aren’t well known.

Probabilistic methods should be promoted over single, narrow band metrics such as accuracy of natural frequencies. Durand et al. show that the automotive finite element models should built taking into account both model and data uncertainties,³⁵ leading to standard deviations and stochastic responses. Here the problem then becomes the determination of the uncertainty bounds in both material properties, geometrical tolerances and attachment boundary conditions. In this particular case of a DC motor, it has been shown that the uncertainty in boundary condition attachment of the adhesive bonding can create significant alterations in natural frequencies and mode shape dominance.³⁸ In Gao et al.³⁹ moving from a ferrite magnet adhesively bonded to a hard steel shell with a variation in adhesive thickness from 0.5 to 0.8 mm caused a movement of resonant frequencies by 10% or greater, but the amplitude was affected more than the location of the frequency. Similarly, in validating the simulation model of an electric motor to experimental measurements, Wu et al. looked for overall vibration levels being close, and peak values being ‘relatively matched’,²⁹ while Lin et al. looked for similar trends²⁸ as validation quality. In order to fully validate finite element simulations against experimentations, the uncertainty from manufacturing must be quantified, for example stamping, fabricating, as detailed by Beltrán-Pulido et al.⁴⁰ with ranges allowing sensitivity studies to be performed. All of this is before operating variability is taken into account, such as thermal⁷ or non-linear friction models in bearings and contact components, mechanical noise and rotor eccentricity altering sound pressure levels by up to 4 dB(A).²⁸

Experiment validation

A series of experiments provide additional validation of the motor model by uncoupling the radial force from the magnetic flux from the similar radial force from the armature response. Subsequently, it is possible to measure the vibration from the overall motor.

As Figure 10 shows, there are two DC motors connected to the shaft with jaw couplings: a larger, more powerful DC motor matching both Figure 2 and the structural model in the simulation. In the first test, which replicates the radial force from the permanent magnets in equation (3), the main motor is driven by the ancillary motor as Figure 11(a) shows. As the main motor is in an open circuit, there is no induced current in the armature winding. Hence in this test the electromagnetic force is generated from only the permanent magnets. The brushes on the main motor are removed to reduce the effect of frictional force.

Figure 10.

Experiment setup.

Figure 11.

Schematic diagram of the experimental setup: (a) testing radial permanent magnet force and (b) overall motor response.

For the second test, the main DC motor (at the top of the rig) is excited with a high current PWM signal, which drives the ancillary motor. Without any load on the ancillary motor, it would rotationally accelerate to a high speed limited by friction in the bearings or saturation of the main motor at peak torque. Thus an electrical resistor is connected in a closed circuit, as Figure 11(b) shows. By switching the resistance value, the ancillary motor limits the peak rotational speed. The aim of this test is validation of the overall motor response.

The acceleration responses normal to the surface were measured by two shear ceramic PCB accelerometers (model: 352C33) attached via wax and powered with internal charge amplifiers. To acquire the data, two National Instruments chassis units were connected to a laptop; A NI-9162 chassis with a NI-9215 16bit, 10 V ADC input, acquired the laser tachometer signal and PWM signal from the signal generator. The accelerometers were connected to a NI-9234 card with IEPE provision, all signals being acquired at a sampling rate of 51 kHz. The rotational speed is coupled to the pitch from the motor and so is measured through a non-contact laser tachometer (VLS/DA1) with analogue output (0–6 volt represents 50–6000 RPM).

The accelerometer positions differ by 90° from each other as Figure 11 shows, also reflected in the label $A$ and $B$ in Figure 7(a). In order to minimise the effect of frame resonance, both motors were fixed on plates with damping sheets as shown in Figure 10. The motors were connected with a flexible shaft connector to reduce the effect of shaft misalignment.

Validation of magnetic response

In the first test, the ancillary motor was powered by a 12 V DC signal, rotationally accelerating from 300 RPM to its maximum speed of approximately 5000 RPM, according to Figure 13. Figure 12(a) shows the surface acceleration of the motor, processed using a short-time Fourier transform, the main axes being time against frequency, where the amplitude of the acceleration is shown as colour bars in the form of power spectrum density (on the dB scale).

Figure 12.

Experimental validation for ancillary motor drives main motor (without brushes): (a) experiment acceleration spectrogram, (b) speed of the motor, (c) simulation radial flux density spectrogram and (d) simulation acceleration spectrogram.

Figure 13.

Flowchart of simulation for the first test.

There are three main artefacts to highlight: (i) the orders of the magnetic force which change with rotational speed, (ii) constant horizontal lines representing excited structural responses (small forces can lead to a large resonant vibrations) and (iii) constant broadband horizontal frequencies which are independent of the motor speed and come from excitation of the bearings. Where any two artefacts coincide a large amplitude can be audibly heard.

For validation, the experimental motor speed is measured using the non-contact laser tachometer, shown in Figure 12(b). That motor acceleration is then imported as the simulation input, to provide real-time prediction of the radial flux density in Figure 12(c). The gradient and number of the lines is perfect match to the experiment.

Through application of the modal superposition method, the surface acceleration can be predicted, as shown in Figure 12. The amplitude of the predicted orders is good with the structural damping reducing the amplitude for higher orders, however, the prediction does not include the bearing vibration (which appears in the experimental measurement as the horizontal lines). Nevertheless, where the radial order coincide with the structural modes the amplitude is greatly increased.

Validation of armature response

The main motor is driven by a 12 V PWM signal with 5 kHz frequency as detailed in the flowchart in Figure 14. By increasing the PWM duty cycle from 20% to 80%, the speed of the motor is accelerated from 500 to 3000 RPM in 18 s, which approximates the experimental measurements. Spikes in the signal are instantaneous losses in the non-contact measurement and could easily be smoothed with a low-pass filter.

Figure 14.

Flowchart of simulation for the second test.

A $3 Ω$ , high power resistor is installed as an electrical load. The surface acceleration of the main motor was measured and the frequency spectrum graph was plotted in Figure 15(a). Two additional factors are present in this result, a far higher broadband excitation due to significantly more frictional excitation from the brushes and the PWM signal with harmonics at 5 kHz. The latter exhibit a Doppler shift with speed where circumferential waves travel faster with the rotor and against. At high frequency there are many structural modes excited by this PWM signal.

Figure 15.

Experimental validation for main motor drives ancillary motor: (a) experiment acceleration spectrogram, (b) time against motor speed (RPM; experimental measured compared with simulation), (c) simulation radial flux density spectrogram and(d) simulation acceleration spectrogram.

The PWM signal that was recorded from the experiment was input to the simplified motor model and the time domain motor speed (shown in Figure 15(b)) and winding current were generated. The simulated radial flux are shown in Figure 15(c), where the electromagnetic orders are shown increasing frequency with speed (small variations are due to the difficulty of replicating the exact motor speed). The fixed horizontal lines from the PWM are also evident.

With the speed and current input to the electromagnetic force model, the radial force for each element can be output to the modal superposition model and the structural response can be simulated. A white noise was also added to the modal superposition model to represent the frictional excitation of the motor from sources such as the brushes and bearings. A spectrum graph for the point acceleration at which the exact accelerometer attachment position was plotted in Figure 15(d).

The simulated acceleration results are a good match with the experiment, containing all relevant major frequencies at the right amplitude including the Doppler shift. This result also highlights the central importance of an accurate finite element model, which not trivial given the uncertainty in manufacturing boundary conditions, modal overlap and uncertain material damping where friction is involved.

Model performance

This motor noise and vibration model is implemented in Simulink using the Bogacki-Shampine solver (ode3), as shown in the Appendix. The step size of the Simulink model is fixed at $1 \times 10^{- 5}$ s based on the maximum speed of the motor and being able to resolve the slot effect (this provides a rotor resolution of 0.36 degrees per time step at 6000 RPM).

The model input can either be the speed of the motor (first experiment) or be the PWM signal that measured in the experiment (second test). The complex relative air gap permeance ( $λ$ ) in equation (6) was calculated in the initialised MATLAB function with 7200 points in the range of 0 to $2 π$ and then a look-up table is used during the run to find the corresponding rotating angle ( $ϕ$ ) at each time-step. Compared with calculating $λ$ at each time step, this dramatically increases the simulation speed. The increment of angular position in the lookup table is therefore 50 millidegrees, based on nearest neighbour sorting. In offline testing, this was found to be practically indistinguishable from spline and curve fitting solutions as the rotational angle increment was so small.

The number of $θ$ points is a main factor that can influence the speed of simulation, as is the number of modes ( $m$ in equation (8)). Table 2 shows the simulation time in Simulink for a prediction of 5 s dynamic response of the motor with different settings for $θ$ and $m$ , using the rapid accelerator mode on a desktop PC with 6 cores CPU. This time will obviously be reduced by a more advanced PC.

Table 2.

Model running time for 5 s simulation with different length of $θ$ and number of modes.

No. of m	20 (s)	40 (s)	60 (s)
No. of θ	20 (s)	40 (s)	60 (s)
length ( $θ$ ) = 24	1.9	2.4	4.7
length ( $θ$ ) = 48	4.7	5.0	7.2
length ( $θ$ ) = 90	5.5	6.2	9.6
length ( $θ$ ) = 180	7.5	8.7	11.5
length ( $θ$ ) = 360	14.3	15.3	23.3
length ( $θ$ ) = 720	24.4	26.7	43.0

It is obvious that the simulation is faster than real time when the number of $θ$ is below 48 and the number of modes is smaller than 40. However, a smaller number of $θ$ means the mesh of the model is coarser and some details of the electromagnetic force will be missing as a consequence. A comparison of the flux density generated with different lengths of $θ$ are shown in Figure 16. The figure plotted with length( $θ$ ) $= 24$ only can observe 4 harmonic curves, whereas the figure plotted with length( $θ$ ) $= 360$ has all orders resolved. For simulating the motor in this paper, the critical length of $θ$ that could provide considerable precision of the result is length( $θ$ ) $= 90$ .

Figure 16.

Comparison of the radial flux density generated by the simulation model with different length of $θ$ : (a) length ( $θ$ ) = 24 and (b) length ( $θ$ ) = 360.

Conclusion

A novel analytical methodology for real time implementation of the noise and vibration of an electric motor for use in a driver-in-the-loop motion simulator has been provided. This model combines the modal superposition model with the analytical electromagnetic force model. A set of experiment tests are applied to validate this model. This model can predict the high fidelity simulation results with a minimum requirement of motor parameters and can achieve real-time simulation in Simulink rapid accelerator mode. By replacement of the DC electromagnetic force model to the PMSM analytical model, this model can be extended to predict the noise and vibration behaviour of a PMSM motor. The methodology for generation of noise and vibration data for use in a driver-in-the-loop simulator when design data is relatively uncertain, has been produced, allowing managers, engineers and potential customers to explore the design space for NVH target setting in an objective fashion.

Footnotes

Appendix

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to acknowledge the funding support from Innovate UK and the Advanced Propulsion Centre (APC) for carrying out this work.

ORCID iDs

Dan J O’Boy

Georgios Mavros

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