Impact of road surface roughness and magnetic force on the in-wheel motor magnet gap

2.1. The structure of the IWM

The IWM used in this paper is the permanent magnet synchronous motor (PMSM). The basic structure of the IWM is shown in Fig. 2. The insulation level of the IWM is $F$. The IWM parameters are listed in Table 1. The stator winding is a three-phase Y-type, with 6 coils in series per phase, 31 turns per coil. The coils are in signal-layer chain connection.

Fig. 2The structure of the IWM

2.2. Mathematical model of the UMF

When the impact of the core reluctance and the saturation effect are not taken into consideration in a sine-wave powered PMSM, the flux density can be expressed as [18, 19]:

where $F\left(\theta ,t\right)$ is the magnetomotive force (MMF); $\mathrm{\Lambda }\left(\theta ,t\right)$ is the magnet gap permeance.

In order to obtain the flux density B, the following magnet gap permeance $\mathrm{\Lambda }$ and the MMF $F$ must firstly be derived, as considered below.

2.2.1. Relative magnet gap permeance

For ideal motor operating conditions, the geometric centre of the rotor magnetic pole and the stator core should be the same, meaning that the length of the magnet gap is uniformly distributed along the circumference. However, in actual motor operations, the relative positions of the stator and rotor deviate from the ideal state due to factors such as manufacture installation and processing errors, shaft bending, relative rotor run-out, and bearing wear, which causes magnet gap deformation along the circumferential and axial directions. The IWM is a three-dimension object. Fig. 3 shows the circumference distribution of the length of the magnet gap in any one of the cross-sections ($XY$ plane) along the axial direction $Z$ after the MGD is produced. As shown, the motor magnet gap is no longer distributed uniformly along the circumferential direction.

The function in Fig. 3 shows that the length of the even/uneven magnet gap of the motor at time $t$ can be specifically expressed as:

2

$g\left(\theta ,z,t\right)=g_0\left(1-\epsilon \left(z\right)\mathrm{c}\mathrm{o}\mathrm{s}(\theta -\omega t-\gamma )\right),$

3

$\epsilon \left(z\right)=\frac{e}{g_0},$

where $g_0$ is the nominal gap thickness; $\epsilon$ is the gap eccentricity in axial direction $z$ cross-section; $\omega$ is the motor mechanical rotating speed; $\gamma$ is the rotor initial position.

The Fourier series of the magnet gap permeance during eccentricity is expressed as:

4

$\Lambda \left(\theta ,z,t\right)=\frac{{\mu }_0}{g_e\left(\theta ,z,t\right)}=\sum _{\lambda =0}^{\infty }{\mathrm{\Lambda }}_{\lambda }\mathrm{c}\mathrm{o}\mathrm{s}[\lambda \left(\theta -\omega t-\gamma \right)],$

where ${\mu }_0$ is the air permeability; ${\mathrm{\Lambda }}_{\lambda }$ is the Fourier coefficient; $\lambda$ is the Fourier series.

The Fourier coefficient in Eq. (4) is calculated by:

5

${\Lambda }_{\lambda }=\left\{\begin{array}{l}\frac{{\mu }_0{\mathrm{\Lambda }}_0^{\text{'}}}{g_0}=\frac{{\mu }_0}{g_0\sqrt{1-{\epsilon }^2}},\lambda =0,\\ \frac{{\mu }_0{\mathrm{\Lambda }}_{\lambda }^{\text{'}}}{g_0}=\frac{{2\mu }_0}{g_0\sqrt{1-{\epsilon }^2}{\left(\frac{1-\sqrt{1-{\epsilon }^2}}{\epsilon }\right)}^{\lambda }},\lambda \ge 1.\end{array}\right.$

Then the relation curve between $\epsilon$ and ${\mathrm{\Lambda }}_{\lambda }^{\text{'}}$ can be obtained according to the above equations as shown in Fig. 4.

Fig. 3Motor MGD

Fig. 4Relation between ε and Λλ'

The above figure shows the change of each magnet gap permeance Fourier coefficient with the gap eccentricity in Eq. (5). As shown, all the coefficients increase with the motor eccentricity. And the coefficients decrease with the order under the same motor eccentricity $\epsilon$. When $\epsilon$ is relatively small, the Fourier coefficient in each order can be expressed using the following equation:

6

${\Lambda }_{\lambda }=\left\{\begin{array}{l}\frac{{\mu }_0{\mathrm{\Lambda }}_0^{\text{'}}}{g_0}=\frac{{\mu }_0}{g_0},\lambda =0,\\ \frac{{\mu }_0{\mathrm{\Lambda }}_{\lambda }^{\text{'}}}{g_0}=\frac{{\mu }_0{\epsilon }^{\lambda }}{g_0{2}^{\lambda -1}},\lambda \ge 1.\end{array}\right.$

After substitution of Eq. (6) into Eq. (4), the magnet gap permeance is expressed as:

7

$\mathrm{\Lambda }\left(\theta ,z,t\right)={\mu }_0\frac{\left[1+\frac{\sum _{\lambda =1}^{\infty }{(\epsilon }^{\lambda }\mathrm{c}\mathrm{o}\mathrm{s}\left(\theta -\omega t-\gamma \right))}{2^{\lambda -1}}\right]}{g_0}.$

3.1. Test platform

An electric-wheel-based vibration performance test platform, as shown in Fig. 5, was built to simulate RSR and composite excitations to test the IWM MGD. An in-depth analysis of the rules describing the effect these excitations have on the magnet gap is investigated by using this platform.

3.2. Vibration model of the test system

Ignoring the influence of the bearing free gap and the lubricant membrane, and the bearing is taken as an equivalent spring-damper system in the vertical direction. Then the vibration model of the test system is constructed as shown in Fig. 6.

Fig. 5Test platform

a) Schematic diagram of experiment set-up

b) Experimental set-up

Fig. 6The system model

In Fig. 6, $m_1$ is the unsprung mass of the wheel; $m_{31}$ is the total mass of the knighthead and the hub; $m_{32}$ is the mass of stator and shell. $m_{33}$ is the mass of the rotor; $m_a$ is the mass of the wheel support disk; $y_0$ is the excitation single. $y_i$ is the corresponding vertical displacement of the masses, $i=$ 1, 31, 33; $k_i$ is the stiffness of the tyre and the bearing respectively, $i=$ 1, 4$j$ ($j=$ 1, 2); $c_i$ is the damping of the tyre and the bearing respectively, $i=$ 1, 4$j$ ($j=$ 1, 2); $k_a$ and $k_b$ are the stiffness of the bench springs.

The vibration equation of the test system is:

where $M$, $C_m$, $K_m$ and $F$ are the mass matrix, mechanical damping matrix, mechanical stiffness matrix and external excitation force matrix.

The test system parameters are shown in Table 2.

4.1. Impact of road excitation on the motor MGD

The vibration exciter was set at different frequencies with the same amplitude to perform sinusoidal excitations on the IWM drive system. Taking into account that the frequency range of the actual road excitation is generally stays within 35 Hz, the excitation frequency was set to be 2 Hz, 3 Hz, 4 Hz, 5 Hz, 6 Hz, 7 Hz, 8Hz, 12 Hz, 16 Hz, 20 Hz, 24 Hz, 28 Hz, 30 Hz, 35 Hz and 45 Hz to test the motor MGD and the vibration acceleration of the test bench, wheel rim, and stator [23]. Fig. 7 shows the collected signal from the MGD (relative displacement of the stator and rotor), the test bench vibration acceleration (TBVA), the rim vibration acceleration (RVA) and the stator vibration acceleration (SVA) in the time domain when the excitation frequency was 8 Hz (for the sake of brevity, the vibration signals from each component generated from the other excitation frequencies are not presented here). To clearly display each vibration signal, the time-domain waveform displays the results only for a section of the test period, with the actual test period lasting 30 s.

The time-domain signals in Fig. 7 show that the vibration acceleration from the test bench produces a large shock. As seen in Fig. 6, this is mainly due to the test bench is contacted with the exciter head. Here, the bench spring is always in contact with the test bench, and the tire is always in contact with the test bench. This is because of the preloaded bench spring. Only the exciter head simulating the road excitation may be separated from the test bench. This situation (where the exciter head separates from the test bench) simulates how the tire loses contact with the ground during driving. The difference is that the tire is flexible, but the test bench is rigid. Therefore, a large shock is produced when the exciter head contacts the test bench. The impact energy is absorbed by the tire once the vibration energy has been transmitted through the tire to the rim, with the vibration at the rim forming a regular sine wave. We used the amount of displacement at the rim as the input and the motor MGD $e$ as the output to examine the transmission characteristics of the system.

A Fourier transform was applied to each vibration response signal to obtain the amplitude-frequency characteristics of each vibration at the different excitation frequencies. In addition, during measurements of the IWM drive system vibrations, a non-contact displacement sensor was used to directly measure the relative displacement response of the stator and rotor of the motor, whereas a contact acceleration sensor was used to measure the vibration response of the test bench, wheel rim, and motor stator. The two frequency transformation relationships are:

where $d_f$ is the displacement response, $a_f$ is the acceleration response, and ${\omega }_r$ is the response frequency.

Fig. 7The test result in time domain

a) Magnet gap deformation

b) Test bench acceleration

c) Rim vibration acceleration

d) Stator vibration acceleration

Since the signal obtained in the experiment inevitably contains noise, it is necessary to correct the discrete frequency spectrum obtained after the Fourier transformation. We used a four-point energy centrobaric correction method to correct the frequency and amplitude of each vibration response [24-25]. The obtained discrete frequency spectrum of the motor MGD $e$ and the rim displacement (RD) are shown in Figs. 8 and 9.

Fig. 8The discrete spectra of e

Fig. 9The discrete spectra of RD

The actual detected excitation amplitude and the set value are not completely consistent in most cases, as there are various uncertainties in the experiment. Therefore, normalization of the motor MGD $e$ and the RD is performed. Simultaneously, to test the accuracy of the theory, the excitation signals were substituted into the system vibration equation in Section 2.2 to produce a simulated calculation. Normalized results from a comparison between the obtained from the simulation and the experiment are shown in Table 3 and Fig. 10.

From Table 3 and Fig. 10, some conclusions can be ascertained: 1) Although uncertainties in the experimental process led to some disagreement between the theoretical and experimental values, similar trends were observed in the theoretical and experimental results for the normalized values within the frequency range examined by the experiment, indicating that the theory is valid. 2) The normalized value increases with the excitation frequency, indicating that a higher excitation frequency at same amplitude causes a larger motor MGD. 3) At 6 Hz, the normalization trends upward in a similar way as the adjacent frequencies. We can preliminarily conclude that the excitation frequency is close to the natural frequency of the experimental IWM drive system. To verify this inference, the natural frequency of the IWM drive system at each order was calculated and found to be 5.9, 71.4, 132 and 169.1 Hz, respectively, meaning that frequency resonance does occur in the system for first-order frequencies.

Fig. 10Theoretical and experimental results

4.2. Impact of composite excitation on the motor MGD

The actual vehicle uses an IWM for driving. As seen in the derivation in Section 1.2, the magnetic force of the IWM is related not only to factors such as its structure and rotation speed but also to the shape of the IWM system and its installation method, which can cause motor MGD. When a UMF caused by the MGD acts on the stator, rotor and associated components, it further increases the degree of the MGD, resulting in adverse effects on vehicle dynamics. Based on the first experiment, we examine the change seen in an IWM magnet gap for RSR and UMF excitations. The goal of this effort is to conduct a preliminary study on the impact that a composite excitation has on the magnet gap and simultaneously verify the results obtained by using the theory.

In this experiment, the IWM electric wheel was rotated, as the road excitation cannot be exerted directly on the tire due to experimental limitations. However, previous studies have shown that the maximum support shaft displacement can still reach as much as 0.0307 m for a B class RSR excitation when the vehicle speed is 60 km/h. Moreover, the excitation magnitude was 1 mm in this experiment. Therefore, our study simulated the loading signals from road was applied directly to the motor support shaft using the platform to lift the IWM drive system. In addition, there was no load applied on the IWM in this experiment. In reality, the IWM operates under a certain load. For a larger motor load, a stronger magnetic field is produced by the stator current to maintain a constant motor speed. Put another way, for the same MGD, the UMF of the motor load is larger than without according to Eqs. (1)-(22). Therefore, the impact with/without the load should be the same, even if the amplitude may be different. For the purposes of this experiment, this assumption is reasonable.

The control excitation frequencies for the vibration exciter were set to 2 Hz, 4 Hz, 8 Hz, 12 Hz, 16 Hz, 24 Hz, 28 Hz, 35 Hz and 45 Hz, respectively. For each excitation frequency, the motor was allowed to operate at 200 rpm and 800 rpm to measure the MGD, support shaft vibration acceleration (SSVA), SVA and exciter vibration acceleration (EVA). Figs. 11-14 show the measured signals when the excitation frequency was 8 Hz (for the sake of brevity, the vibration signals from each component generated by the other excitation frequencies are not presented here). To clearly present each vibration signal, the time-domain waveform shows only a section of the test period, with the actual test period lasting 30 s.

Fig. 11Test result of the magnet gap deformation

a) 200 rpm

b) 800 rpm

Fig. 12Test result of the support shaft vibration acceleration

a) 200 rpm

b) 800 rpm

Results from the 8 Hz excitation frequency test for various IWM rotation speeds are shown in Figs. 11-14 and indicate the followings: 1) The collected signals from the motor MGD $e$ show that an uneven magnet gap exists when the IWM is running, and the degree of unevenness in the magnet gap increases with the motor speed. This result shows that electromagnetic excitation created during operation of the IWM leads to the deterioration of the MGD. 2) Observations of various vibration signals from the system for the same excitation frequency show the followings: (i) When the IWM speed increases, the amount of vibration from the system components also increases, verifying that the electromagnetic excitation from the IWM is related to its rotation frequency. A greater rotation frequency causes an increased MGD. (ii) Under these two rotation speeds, the exciter acceleration signal contains many glitches, with the degree of vibration acceleration at 800 rpm being significantly larger than that at 200 rpm, indicating that the magnetic excitation is transmitted via the motor-supporting components and the wheel-rim transfer to the vibration exciter. This shows that for a vehicle operating in a real-world situation, the RSR will not only cause IWM MGD, but the UMF caused by the MGD will be transmitted to the wheel rim in turn, resulting in a negative impact on vehicle dynamics.

Fig. 13Test result of the stator vibration acceleration

a) 200 rpm

b) 800 rpm

Fig. 14Test result of the exciter vibration acceleration

a) 200 rpm

b) 800 rpm

In order to obtain additional visually intuitive characteristic from the time-domain waveform, a similar spectral analysis was performed on the time domain signals collected from the various composite excitations. The corrected discrete amplitude spectrums from the MGD $e$ for various motor rotation speeds, and from the supporting shaft displacement (SSD), are shown in Figs. 15 and 16 for each excitation frequency of the IWM drive system.

Fig. 15The discrete spectra of e

Fig. 16The discrete spectra of SSD

Normalization of the motor MGD $e$ and SSD are performed for comparisons. Simultaneously, the excitation signals are substituted into the system vibration equation in Section 2.2 to provide a simulated calculation. The normalized results from a comparison between the obtained transmission characteristics from the simulation and those from the experiment are shown in Table 4. The normalized experimental results for the different IWM speeds are shown in Fig. 17.

As seen in Fig. 17, the normalized value increases with the motor speed for the same RSR excitation frequency, indicating that a higher motor speed causes a larger motor MGD. The results not only confirm the existence of the UMF but also indicate the size of the UMF excitation is related to the rotation frequency of the IWM. Based on Eqs. (1)-(22), the UMF for the different motor speeds are calculated to be time-variable. However, the mean value of the UMF is proportional to its rotation speed, which has been verified by many studies [26].

Fig. 17Experimental results with different IWM speed

From the figure, we can also see that the normalized value increases at 24 Hz and after 35 Hz. The reasons for this, which are shown in the corrected discrete amplitude spectrum of the MGD in Fig. 15, are that the amplitude of the MGD has the same order of magnitude. From Fig. 16, we see that the SSD amplitude is relatively small at 24 and 35 Hz compared to the other amplitudes, causing an increase in the normalized value. The main reason for the decrease in the SSD is that there may be two excitations with the same frequency and different phases acting together on the supporting shaft. There is no doubt that the RSR excitation is one of the excitation sources. The other excitations should come from the UMF, which may have the same frequency harmonic as the RSR. The two excitation signals containing the same frequency superpose on the supporting shaft together, causing the supporting shaft to oscillate at the same frequency. However, the vibration amplitude decreases, as determined by the phase and amplitude of the two excitation signals. This analysis will be verified later using specialized testing.