Magnetostriction Vibration and Acoustic Noise in Motor Stator Cores

This study focuses on the impact of magnetostriction on vibration and acoustic noise emitted from motor stator cores. Typically, motor vibration and acoustic noise are attributed to radial electromagnetic forces, torque ripple, and pulse-width modulation switching. However, it is important to consider the influence of magnetostriction in iron core materials with high magnetostriction. In this study, an analytical model was developed to derive the equivalent magnetostrictive force in a global cylindrical coordinate system to understand the effect of magnetostriction on motor iron cores. Three core materials with different magnetostriction characteristics were used to fabricate three individual stator cores for comparative experiments. To isolate the effect of magnetostriction, an additional toroidal winding was added to excite the stator yoke, avoiding the generation of electromagnetic forces and inducing vibration solely by magnetostriction. The magnetostrictive strain, vibration, and sound pressure level of the three stator cores were measured and compared. The results clearly indicate that magnetostriction has a significant impact on vibration and acoustic noise in motor cores, particularly in high-magnetostriction core materials.

has been extensively researched [3], [4], [5].Various techniques have been proposed to mitigate NVH issues stemming from radial force ripple, including the elimination of specific air-gap flux density harmonics, magnet skewing, winding optimization, and selective force component reduction [6], [7], [8], [9].Although torque ripple has a less direct influence on the radial vibration of housing components or the stator back yoke [3], it can induce vibrations in other system parts, such as the gearbox of an electric vehicle's powertrain [10].Fractional-slot topologies and magnet skewing have been proposed to reduce torque ripple [11], [12].Regarding the reduction of the NVH resulting from PWM switching, numerous control strategies have been proposed [13].
In contrast to three NVH sources that were previously mentioned, research focusing on the impact of MS on the vibroacoustic performance of motors remains limited, and a definitive conclusion is yet to be reached.In power transformers, the MS of iron core materials is known to cause deformation and serve as a primary NVH source [14], [15].Similarly, the iron core materials in rotating electric machines exhibit MS-borne deformation when magnetized.Several studies have highlighted the influence of MS on motor core deformation.For example, the MS of conventional silicon steel was reported to contribute to approximately 50% of the total deformation observed in the stator teeth of an induction machine [16].Moreover, the MS of an amorphous iron material was found to contribute as much as 80% of the deformation in the stator teeth of a switched reluctance motor (SRM) [17].
In [18], finite element analysis (FEA) revealed that the sound pressure level (SPL) generated solely by MS in an SRM constructed from amorphous iron 2605SA1 was comparable to that produced by pure electromagnetic forces.This finding underscores the importance of considering MS in the NVH assessment of motors with high-MS core materials such as amorphous iron and cobalt-iron alloys [19].In contrast, for motors made of conventional 3% silicon steel, the MS-induced SPL is lower than that caused by electromagnetic forces.However, MS should not be ignored in scenarios that require precise NVH evaluations [18].
As previously mentioned, the contribution of MS to vibration and acoustic noise can be simulated using FEA.Based on the FEA method and results presented in [20], MS mitigates the peak vibration harmonics in a permanent-magnet motor.In addition, other FEA methods suggested that MS deformation accumulates with that caused by electromagnetic force, leading to higher total vibrations [21].Experimental studies have been conducted to evaluate the contribution of MS to the acoustic noise of motors.For instance, an experimental study showed that an SRM made of amorphous iron, which has a high MS, exhibits higher overall SPLs than a motor made of a core material with a lower MS [18].Similarly, in [22], it was reported that significantly higher acoustic noise emissions were observed from a motor made of amorphous iron with a high MS.Moreover, the interaction between the electromagnetic forces and MS at the switching harmonics of PWM control was evaluated in [23].It was deduced that MS could either intensify or mitigate the deformations caused by electromagnetic forces depending on the harmonic order.
The contribution of MS to motor vibration and acoustic noise remains ambiguous, primarily because of the scarcity of experimental studies and the following challenging factors: 1) MS and electromagnetic forces occur simultaneously when motor cores are magnetized [16], [17].To the best of our knowledge, it is impossible to experimentally measure the contribution of MS alone during normal motor operation.2) Different iron core materials exhibit markedly different MS characteristics.Although MS could be negligible in some materials, it is the dominant source of deformation in others [18].3) An equivalent MS force is distributed over the entire iron core body, in contrast to electromagnetic forces that primarily occur on the surface of the iron core.This difference results in a more complicated relationship between the MS and core deformation [20].With the increasing demand for superior motor NVH performance, it is necessary to understand all the contributing factors to NVH, including MS.
Owing to the numerous unknown aspects of MS contributions in motor NVH, the authors recognize the considerable complexity in analyzing the combined effects of electromagnetic forces and MS.Furthermore, research on the impact of MS in motor core deformation is still limited.Therefore, a study that includes a comprehensive analysis and experimental evaluation of MS alone would be highly beneficial, which is currently lacking in the existing literature.The study would be a fundamental reference for future research in motor NVH evaluations considering both electromagnetic forces and MS.
The primary objective of this study is to present the fundamental theory, FEA results, and experimental evidence showing, explaining, and predicting the influence of MS on motor vibration and acoustic noise.
In Section II, an analytical model is presented to calculate the equivalent MS forces within the global cylindrical coordinate system.The model provides an intuitive understanding on the tendency of MS to deform the motor core.The derived equivalent MS forces are verified in a ring-shaped core using FEA.In Section III, the strain, vibration, and acoustic noise of the three stator cores are measured experimentally and compared.The three stator cores were made of three different iron core materials with significantly different MS characteristics.In the experiment, these stator cores were excited by toroidal windings added to the stator yokes; hence, the vibration originated exclusively from MS.In the conference version of this article [24], the measured strain, vibration, and SPL results were presented and compared in a preliminary manner.In this journal version, first, the theory and analytical model of the equivalent MS forces are introduced and verified.Second, a comparison between the measured and FEA MS strains is added.Furthermore, the spectral analyses of the measured vibrations are presented.

II. THEORETICAL ANALYSIS ON MAGNETOSTRICTION
In this section, to understand the tendency of MS to deform the motor core, an analytical model is introduced to calculate the equivalent MS force f .The procedure of obtaining the equivalent MS force f is illustrated in Fig. 1.First, MS strain ε and stress σ tensors, which are both essential for deriving the equivalent MS force f , are developed from the distribution of the flux density B. Subsequently, the equivalent MS force f is derived in the global cylindrical coordinate system, which is preferable for motor cores with cylindrical structures.Finally, the derived equivalent MS force f is validated using a ring-core example in FEA.
In the equations presented in this section, vectors and matrices are distinctly denoted using bold font.Specifically, vectors are enclosed within curly brackets, as in {B}, while matrices are enclosed within square brackets, as in [σ].

A. Magnetostriction Strain Tensor in Cartesian Coordinate Systems
Let B x , B y , and B z be the three components of flux density B in the global Cartesian xyz system as: The superscript (xyz) indicates that the vector is defined in the global Cartesian xyz system, and "T " represents the transpose of a matrix or vector.Given the arbitrary flux density B at a point inside a magnetic material, the expression for the corresponding MS deformation at that point is not trivial.However, a trivial expression for an isotropic material exists when B aligns with one of the axes of the coordinate system [25].Therefore, we rotate the coordinate system to align the x-axis of a new local Cartesian xỹz coordinate system with the vector B as shown in Fig. 2(a).Let R be the rotation matrix that aligns the x-axis with B. Consequently, the components of B in the new local xỹz system can be calculated as: Assuming isotropic MS characteristics, the deformation caused by an x-axis-aligned flux density is illustrated in Fig. 2 [25].A cubic element with an initial edge length d is shown in Fig. 2. The gray and blue bodies represent the undeformed and MS-deformed states, respectively.The deformation is exaggerated to enhance clarity.As shown in Fig. 2(b), the MS deformation causes tensile strain λ in the x-direction, that is, the direction of the flux density B. Simultaneously, MS also causes compressive strains −κλ in two perpendicular directions, ỹ and z.Here, κ represents the magnetostrictive Poisson's ratio, which is the ratio of the compressive strain to the tensile strain caused by MS.In the side and front views shown in Fig. 2(b), all the edges remain perpendicular to each other under the MS deformation.Hence, the shear strains in all the directions are zero.In most materials, the actual anisotropic MS behavior leads to non-zero shear strains.However, measurements have shown that the shear strain in non-oriented steels is negligible [26], [27].
Considering the aforementioned tensile and compressive strains and zero shear strains, the MS strain tensor ε in the local Cartesian xỹz system can be expressed as: The superscript (xỹz) indicates that the MS strain tensor ε is given by the new local xỹz system.In (3), the diagonal elements are the normal strains in each direction.The off-diagonal elements are zero because of the zero MS shear strains mentioned previously.Specifically, λ denotes the MS tensile strain in the direction of B. The value of λ depends on the flux density magnitude |B| and initial stress in the material.

B. Derivation of Magnetostriction Strain Tensor in Global Cylindrical Coordinate System
In this subsection, the MS strain tensor, given by (3) in the local Cartesian xỹz system, is transformed into the global cylindrical rϕz system.R, C, and P are the matrices for the coordinate transformation among these four systems and are also shown in Fig. 3.For instance, R is the matrix that converts the global Cartesian xyz system into the local Cartesian xỹz system, as previously presented in (2).Based on the coordinate transformation rule of vectors and tensors, the flux density vector B and strain tensor ε in xỹz and rϕz systems satisfy the following transformation relationships: Let us define: then, ( 4) and ( 5) are simplified as: Here, Q directly transforms the xỹz system to the rϕz system.
In the case of a 3-dimensional problem, P , C, and Q are all 3 by 3 orthonormal matrices.
Calculating each element of the strain tensor ε in the rϕz system from (8) yields: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where δ kl is the Kronecker delta.i and j are the row and column indices of a matrix, respectively.In ( 9), 3 k=1 Q ik Q jk = δ ij holds true because of the orthonormality of Q mentioned previously.To relate the strain tensor ε obtained in ( 9) with the flux density vector B, it is necessary to express the coefficients Q 11 , Q 21 , and Q 31 in terms of the three components B r , B ϕ , and B z of B. Therefore, ( 2) is substituted into (7), and the following is obtained: Finally, substituting (10) into ( 9) yields the MS strain tensor ε given in the global cylindrical coordinate system rϕz:

C. Magnetostriction Stress Tensor
Assuming that the MS strain ε in ( 11) is caused by certain forces (which are the equivalent MS forces discussed in the next subsection), the corresponding stress on the element satisfies Hooke's law in the theory of elasticity [28]: where {σ} (rϕz) and {ε} (rϕz) are the MS stress and strain tensors in the global cylindrical rϕz coordinate system, respectively, and are written in Voigt's notation; D denotes the stiffness tensor.
For an isotropic material, D is given by .
where E and ν are Young's modulus and Poisson's ratio, respectively, used in Hooke's law.κ is the MS Poisson's ratio, which indicates the ratio of the MS strain in the magnetization direction and those perpendicular to it.
It is important to note that ε and σ in (12) do not represent the final strain and stress of an actual body undergoing MS deformation.This is because the constraints from the boundary conditions and strain compatibility further limit the deformation after MS is induced in the body.In other words, the MS strain ε and MS stress σ can be regarded as the inputs, and the constraints on the body form the system for that input.The output of the system is the actual deformation of the body.In summary, ε and σ in (12) are not the final states of an actual body; however, analyzing ε and σ provides insight into the tendency of MS to deform the body.

D. Equivalent Magnetostriction Force
In this subsection, the equivalent MS force is derived using the virtual work principle.The derivation should be performed in a cylindrical rϕz coordinate system; however, it is overly extensive in that format.Consequently, the derivation process is performed in a Cartesian coordinate system instead.Finally, the equivalent MS force in a cylindrical coordinate system is provided in the end.To simplify the presentation, superscripts indicating the coordinate system, such as (xyz) and (rϕz), are omitted.Additionally, the Einstein notation is employed to further simplify the presentation.
Fig. 4(a) illustrates the model used for deriving the equivalent MS force.The body is assumed to be magnetized and allowed to freely deform.The corresponding MS deformation can be realized by imposing an equivalent MS body force f b within volume V and an equivalent MS surface force f s on surface S. In the presence of f b and f s , a stress distribution denoted by σ emerges and causes the entire body to attain a force equilibrium state.This model permits all the body points to deform freely, ignoring both the constraining boundary conditions and strain compatibility conditions.Although such an assumption does not align with real-world scenarios, it presents precise insight into the tendency of MS to deform the body, namely, the equivalent MS force.
Assuming a virtual displacement field δu on the body, the virtual work δW conducted by the MS body force f b and MS surface force f s is given by [28]: where f s i , f b i , and δu i indicate the i-th component of the vectors f s , f b , and δu, respectively.Simultaneously, the virtual displacement δu induces a change in the elastic potential energy δU of the body, which is expressed as [28]: where δε ij is the virtual strain derived from the virtual displacement δu based on the relationships given by the following strain-displacement relationship [28], where u i is the i-th component of the displacement vector u, and x i is the i-th coordinate of a coordinate system.From (15), Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
the virtual strain is obtained as [28] δε ij = 1 2 Considering the symmetry of the stress tensor, that is σ ij = σ ji , the following equality is obtained from ( 16): Substituting ( 17) into ( 14), and applying partial integration and the divergence theorem in order, the following form of δU is obtained: where n j is the j-th component of the normal vector n on the body surface S as shown in Fig. 4(a).According to the principle of virtual work, the virtual work δW done by the external forces f s and f b is equal to the change in the elastic potential energy δU of the body for a system in equilibrium.Therefore, δU = δW .By substituting ( 13) and ( 18) into δU = δW , the following equation can be derived: Using the fundamental lemma of the calculus of variation, the summation within the parentheses of each integral must be zero for (19) to hold universally true for any arbitrary δu.Consequently, the two subsequent equalities related to the equivalent MS forces is obtained as: Equation ( 20) is the element form of the equivalent MS forces.These can also be expressed in vector form as: The two equations presented in (21) correspond to the stress vector defined by the Cauchy stress tensor and equation of equilibrium in the theory of elasticity [28].
As previously stated, ( 21) is derived in a Cartesian coordinate system.If the derivation is carried out in a cylindrical coordinate system, the same vector form given by ( 21) will be obtained.The derivation procedure is consistent with that in the Cartesian coordinate system.However, it is more extensive in format because the Einstein notation is not applicable to energy calculations involving strain and stress tensors written in cylindrical coordinate systems.For the element form, f s is identical in the Cartesian and cylindrical coordinate systems, which is given in (20).However, the component form of f b has a different expression from that in (20) because the divergence of the stress tensor σ is different under the two coordinate systems.Expanding the divergence of the stress tensor σ in a cylindrical coordinate system yields the component form of f b as:

E. Verification of Equivalent MS Force Equations
A ring-core model, as shown in Fig. 4(b), is used to verify the equivalent MS forces in (21) and (22).First, the equivalent MS force densities f b and f s on the element with a bold black frame in Fig. 4(b) are calculated based on ( 21) and (22).Subsequently, the acquired force densities are set as the inputs for the structural FEA to validate (21) and (22).The validation is carried out by comparing the MS strain value λ assigned in the theoretical equations with that obtained from the FEA.The following calculations are carried out in the global cylindrical coordinate system rϕz.
Inside the ring shown in Fig. 4(b), a uniform tangential flux density is assumed as: Subsequently, the magnetostriction strain written in the Voigt notation is determined from (11) as: Then, substituting (24) into (12) yields: (25) Given the MS stress σ in (25), the equivalent MS forces can be obtained from ( 21) and (22).The surface force on the outer surface denoted by the vector {n r+ } = {1 0 0} T and the top surface denoted by {n z+ } = {0 0 1} T , are derived as follows: For the surface force f s,r− on the inner surface denoted by {n r− } = {−1 0 0} T and f s,z− on the bottom surface denoted by {n z− } = {0 0 − 1} T , they are the opposite of f s,r+ and f s,z− , respectively.
Next, the equivalent MS body force f b is calculated using (22).First, all the derivative terms in (22) are zero because the stress elements are independent of the space coordinates r, ϕ and z, as shown in (25).Moreover, given that the shear components in (25) are also zero, f b ϕ and f b z in ( 22) are both zero.In contrasts, f b r is non-zero and can be calculated as follows Because all the parameters in (30) are positive, f b is directed outward, leading to the expansion of the ring core.
To validate the equivalent MS forces theoretically obtained in ( 26)-( 30), these MS forces are applied to the meshes on a ring core in the ANSYS static structural FEA.The ring core shown in Fig. 5(a) has a mean radius of 100 mm and a thickness of 10 mm.In this validation, the Young's modulus E, MS Poisson's ratio κ, and mechanical Poisson's ratio ν of the core are set to 200 GPa, 0.5, and 0.3, respectively.The MS strain value λ is set to 10 ppm as an example.Substituting these values into ( 26  that the theoretical equivalent MS forces, derived from ( 21) and ( 22), successfully produce the assigned MS strain within the ring.Hence, ( 21) and ( 22) are validated.

III. MEASUREMENT OF MAGNETOSTRICTIVE STRAIN, VIBRATION, AND ACOUSTIC NOISE
In the previous section, the influence of MS on object deformation was analytically demonstrated.This section presents an experimental validation of the occurrence of MS on stator cores through the measurement of strain, vibration, and acoustic noise attributable to MS.Furthermore, the pronounced effect of MS on the vibration and acoustic noise emissions from stator cores is shown.

A. Magnetostriction of Core Materials
In this subsection, the MS values with respect to the flux density of the three core materials used in this study are compared.The three core materials are 6.5% high-silicon steel 10JNEX900, amorphous iron 2605SA1, and 3% conventional silicon steel 20JNEH1200.6.5% high-silicon steel is employed in dc/dc reactors in some solar power systems owing to its advantages of low acoustic noise and reduced core loss.In addition, amorphous iron is utilized in power transformer applications because of its low core loss.Although neither material has been widely adopted in motor applications, their pronounced low core loss characteristics may provide significant efficiency enhancements in future motor applications [29], [30].
Fig. 6 shows the B-H curves of the three studied materials.The Young's modulus was measured using stress-strain tests on the laminated core samples.The measured values of the highsilicon steel, amorphous iron, and conventional silicon steel were determined as 172 GPa, 120 GPa, and 174 GPa, respectively.
The MS characteristics of these materials were measured using strain gauges installed on closed-loop laminated cores.During the measurements, the cores were excited by a 10-Hz sinusoidal voltage.The measured MS profiles with respect to the flux density are shown in Fig. 7.The differences in the measured MS values of the three materials were significant.For instance, at a flux density of 1 T, the amorphous iron exhibits a high MS value of 11.0 ppm.In contrast, the conventional and high-silicon steels show MS values of 3.5 ppm and 0.3 ppm, respectively.These differences in the MS values cause significantly different behaviors in terms of deformation, vibration, and acoustic noise, which will be discussed in the following subsections.

B. Stator Yoke Excitation
In motors, MS and electromagnetic forces occur simultaneously, making it challenging to evaluate the individual contribution of MS to the vibroacoustic performance.In this study, an additional toroidal winding is introduced to the stator cores to address this challenge.
Fig. 8(a) and (b) show a cross-sectional view of the stator core model and a photograph of the fabricated stator core, respectively.The stator core has an additional toroidal winding that generates flux inside the stator yoke, thus avoiding the generation of any electromagnetic force.Consequently, deformation, vibration, and acoustic noise originate only from MS in this core model.The outer diameter of the stator cores is 190 mm, and the stack length is 50 mm.This stator was designed for a switched reluctance generator installed in hybrid electric vehicles.During the experiment, the stator was suspended using a thread through the top ear using a crane.A search coil was wound around the stator yoke for flux density detection, as shown in Fig. 8. Let v s (t) be the induced voltage measured by the search coil, and N and A be the number of turns of the search coil and effective cross-sectional area of the stator yoke, respectively.The flux density B Y (t) inside the stator yoke is then calculated by The strain, vibration, and SPL of the excited stators were measured and compared under two different excitations.The flux density B Y (t) inside the stator yoke under these two excitations is shown in Fig. 9.In the first excitation method, the stators were excited by a sinusoidally varying flux density in the stator yoke as shown in Fig. 9(a).The sinusoidal flux density was achieved  by applying a sinusoidal voltage source to the toroidal winding using a precision power amplifier, NF4520 A. This sinusoidal flux excitation was used to measure the MS resulting from the fundamental component of the flux density.In contrast, in Fig. 9(b), a square-shaped flux density was provided to the yoke to measure the MS caused by the high-order harmonics.The square-shaped flux density was achieved using an inverter given a square current reference.For both excitations, the frequency range spanned from 25 Hz to 1 kHz, with increments of 25 Hz.

C. Measurement of Magnetostrictive Strain
The strains on the outer surface of the stator core were measured using strain gauges.These strain gauges convert local mechanical deformations into measurable electrical signals.The positions of the strain gauges are shown in Fig. 10(a).Fig. 10(b) shows a photograph of the strain gauges installed on the outer surface of the stator.
As shown in Fig. 10, the strain gauges were installed at two locations: (A) directly behind the stator tooth and (B) directly behind the slot.To avoid redundancy, this discussion focuses on the measurement at position B that exhibits the highest strain across the entire circumference.The results of the measurements at position A were discussed in [24].
As shown in Fig. 10(b), two strain gauges were installed at two different axial positions at B. Because the strain outputs were mostly identical at these two positions, the average value was adopted to increase the signal-to-noise ratio.
A non-inductive strain gauge, KFNB-2-350 (KYOWA), was used for this measurement.The gauge has a nominal output error of 2%.The signal from the gauge was amplified using the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.show the strains measured on the three stator cores under the sinusoidal and square flux excitations at 100 Hz, respectively.Only positive strains were observed under both excitations, indicating that the stators expanded regardless of the magnetization direction.The shapes of the strain waveform in Fig. 11 correspond to those of the flux density waveforms previously shown in Fig. 9.In both Fig. 11(a) and (b), the amorphous iron stator core exhibits the highest strain peak among the three materials, which is 2.3 times higher than that of the conventional silicon steel stator core.In contrast, the high-silicon steel stator core exhibits the lowest strain peak, which is 1/10 of that measured in the conventional silicon steel stator core.
In Fig. 11(a), the peak strain is 7.0 ppm for the amorphous iron stator.This value is lower than the measured MS value of 11.0 ppm at 1 T, as previously shown in Fig. 7.This reduction was verified using the FEA MS simulation in JMAG.Fig. 12(a) In FEA, the stator yoke was excited with a flux density of 1 T, as shown in Fig. 12(a), which is consistent with the peak flux density in the experiment.Fig. 12(c) shows the resulting tangential normal strain ε ϕϕ on the outer surfaces of the three stator cores.In the case of the amorphous iron stator, the FEA strain value at 70 • , which is the same location of the strain gauge in the experiment, was calculated to be 7.1 ppm.This FEA result agrees with the experimental value of 7.0 ppm.Regarding the high-silicon steel and conventional silicon steel stator cores, the FEA strain values are 0.2 ppm and 2.4 ppm at 70 • , respectively.These two FEA values are lower than the experimentally measured values of 0.3 ppm and 3.0 ppm, respectively.This discrepancy may be due to the MS anisotropy in silicon steel.It was reported that MS anisotropy is significant in non-oriented electrical steel materials [31].For these materials, the MS values are the lowest in the rolling direction and highest in the transverse direction.In the FEA simulation shown in Fig. 12, the adopted MS values were those that were experimentally measured in the rolling direction.Consequently, the MS deformation was underestimated in the FEA in the high-silicon steel and conventional silicon steel stator cores.In contrast, amorphous iron does not exhibit MS anisotropy owing to its non-crystalline nature; hence, its measured and FEA strains have a good correspondence.

D. Measurement of Magnetostrictive Vibration
Fig. 13 shows the FFT analyzer used in this study and the accelerometers installed on the outer surface of the stator core.The accelerometers were evenly distributed at 20-degree intervals, except for the six positions near the ears.The output signals of the twelve accelerometers were synchronized using the FFT analyzer, DS-5000.Both the amplitudes and phases of the acceleration at the twelve points were measured; thus, information on the vibration mode shapes could be obtained.
Fig. 14(a) compares the acceleration spectra measured by sensor 1 on the three stator cores under 250-Hz sinusoidal flux excitation.The most significant acceleration component is the second at 500 Hz, which is twice the excitation frequency of 250 Hz, indicating that the MS is not affected by the polarity of the excitation.Furthermore, the multiples of the second component are also significant in the spectra because the strain waveforms are not perfectly sinusoidal, as shown in Fig. 11(a).The comparison of the acceleration magnitudes among the three  materials reveals that the amorphous iron stator exhibits the highest acceleration, whereas the high-silicon steel stator core exhibits the lowest acceleration.Fig. 14(b) and (c) show the amplitudes of the second and fourth acceleration components with respect to the excitation frequency, respectively.The amorphous iron stator consistently exhibits the highest acceleration for both components across all excitation frequencies from 25 Hz to 1000 Hz.The amplitudes of these measured accelerations are in accordance with those of the measured strains presented in Fig. 11.
After comparing the acceleration amplitudes, the focus is shifted to the acceleration phase to determine the vibration mode.To further specify the mode shapes excited in the stator cores, Fig. 16 shows the phases of the second harmonic of the accelerations measured at the twelve points.The phase of sensor 1 is set as the reference and fixed to −π/2 in all the figures.The phases measured at six different excitation frequencies in the three stator cores are presented.Except for 200 Hz, the phases measured at all the twelve points are mostly identical.Thus, these accelerations are synchronous in phase, indicating mode-0 or breathing-mode vibrations.These synchronous vibrations are due to the uniform and concurrent flux density distribution throughout the stator yoke.
To verify the asynchronous phases observed under the 200-Hz excitation, the modal characteristics of the cores were measured by modal hammering tests.In the modal hammering test, sensor 1 in Fig. 13 was removed for the hammer hit, and the remaining eleven accelerometers were used for measuring the accelerations excited by the hammer hit.Through the modal hammering test results shown in Fig. 17, the resonance frequencies of mode 2 are approximately 400 Hz in the three stator cores.Consequently, in Fig. 16, these mode-2 resonances are excited under 200-Hz excitation in the high-silicon steel and conventional silicon steel stator cores.In contrast, the mode-2 vibration cannot be observed in the amorphous stator core owing to its isotropic MS characteristic previously mentioned.Similarly, slight mode-3 resonances can be observed in the stator cores under 600-Hz excitation in Fig. 16.In conclusion, MS excites the corresponding resonances when its frequency matches any resonance frequency of the stator core.This phenomenon is similar to that between electromagnetic forces and vibration.

E. Measurement of Sound Pressure Level
Fig. 18 shows the setup used to measure the SPL of the acoustic noise generated by MS.In this setup, the microphone was positioned 1 m from the stator yoke in the radial direction.
The Campbell diagrams shown in Figs.19 and 20 summarize the measured SPLs under the sinusoidal and square flux excitations, respectively.The SPL trajectories can be clearly observed at the second, fourth, sixth, and eighth harmonics, as indicated by the white arrows.In these diagrams, only even-order harmonics exist because MS is independent of the magnetization direction.
In Fig. 19(a)-(c), the highest SPL point for each material is highlighted in red, which are 56 dB, 83 dB, and 76 dB in the high-silicon steel, amorphous iron, and conventional silicon steel stator cores, respectively.These highest SPL points are all observed at the trajectory of the second harmonic owing to the limited harmonics in the sinusoidal flux excitation.However,  trajectories of harmonics with higher orders also exist because of the nonlinearity between the MS value and flux density, as previously shown in Fig. 7.
Fig. 20(a)-(c) show the Campbell diagrams under the square flux excitation.These diagrams show significantly higher SPLs than those in Fig. 19.In contrast to Fig. 19, high-order harmonic trajectories, such as the sixth and eighth, are also clearly shown in Fig. 20.Comparing the SPLs on the second-harmonic trajectory in Figs.19 and 20, they have similar values owing to the similar fundamental components of the flux densities in the two excitation methods.However, the SPL of the fourth, sixth, and eighth harmonics are significantly higher in Fig. 20 owing to the higher harmonic components in the square flux excitation.For example, the peak SPLs on the fourth harmonic trajectories increased from 29 dB to 54 dB in high-silicon steel, 76 dB to 82 dB in amorphous iron, and 56 dB to 78 dB in conventional silicon steel stator cores.Furthermore, the highest SPL in each Campbell diagram is no longer on the second harmonic trajectory as those highlighted in Fig. 19.The highest SPLs are 65 dB at the eighth harmonic trajectory in the high-silicon steel, 98 dB at the sixth harmonic trajectory in the amorphous iron, and 79 dB at the fourth harmonic trajectory in the conventional silicon steel stator core, as indicated by the red fonts.The reason for these high SPLs is that the respective trajectory coincides with the mode-0 resonances of these stator cores, as shown in Fig. 17.
Fig. 21 compares the A-weighted SPLs measured under the two types of excitation from 25 Hz to 1000 Hz.The solid and dashed curves indicate the overall SPLs under the sinusoidal and square flux excitations, respectively.First, the overall SPLs of the high-silicon steel and amorphous iron stator cores are the lowest and highest, respectively, at most excitation frequencies.In other words, materials with higher MS show higher overall SPLs, provided that there is no electromagnetic force.Second, the dashed curves are consistently higher than the solid ones for all the three materials, indicating that the square flux density results in higher SPLs than the sinusoidal flux density.This difference suggests that a reduction in the flux density harmonics in the stator yoke can reduce the MS-borne overall SPLs.It should be noted that the amorphous iron stator core has two extremely high overall SPL values under the square flux density excitation.The SPL reaches 101.4 dB and 101.6 dB at 700 Hz and 925 Hz, respectively, when the mode-0 resonance experiences significant excitation.The peak overall SPLs in the conventional silicon steel and high-silicon steel are 80.6 dB and 65.6 dB, respectively, under the square flux excitation.
The experimental results presented in this section, obtained when only MS affects the stator cores, indicate that materials with a higher MS tend to produce higher vibration and acoustic noise.However, this does not imply that MS always increases the vibration and acoustic noise during actual motor operation.It is essential to consider the combined effects of the electromagnetic force and MS.The overall vibration and acoustic noise may increase or decrease depending on their respective vibration phases.
IV. CONCLUSION In this study, the vibration and acoustic noise caused by the magnetostriction (MS) of three iron core materials were evaluated.First, an analytical model for deriving the MS strain, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.stress, and equivalent force under the global cylindrical coordinate system was introduced and verified using a ring-core example in finite element analysis.Using this analytical model, the tendency of MS to deform the motor stator core can be quickly computed.To experimentally evaluate the effect of MS on the stator core vibration and acoustic noise, three stator cores were fabricated using three iron materials with significantly different MS characteristics.These stator cores were excited by an additional toroidal winding to avoid the generation of electromagnetic forces.The MS-borne strain, vibration, and acoustic noise were measured and compared.The results show that stator cores expand because of MS, and those with greater MS levels tend to have higher vibration and acoustic noise, provided that no electromagnetic force exists.Furthermore, when compared to the sinusoidal flux excitation, square flux excitation intensifies the vibration and acoustic noise for all the three materials.This distinction suggests that minimizing the harmonics of the flux density in the stator cores can help reduce the vibration and acoustic noise caused by MS.
The analytical model and experimental results from this study offer valuable insights into the effect of MS on the vibration and acoustic noise of stator cores.However, considering MS alone is insufficient to provide direct suggestions for motor noise, vibration, and harshness.It is crucial to emphasize that in real-world motor operation, both electromagnetic forces and MS impacts should be considered.Their combined influence may either intensify or mitigate the overall vibrations and acoustic noise.Further research is necessary to delve deeper into the interplay between electromagnetic forces and MS during real motor operation.

Fig. 3
Fig. 3 illustrates the components of the flux density vector B in four coordinate systems: global Cartesian xyz, local Cartesian xỹz, global cylindrical rϕz, and local cylindrical r φz systems.R, C, and P are the matrices for the coordinate transformation among these four systems and are also shown in Fig. 3.For instance, R is the matrix that converts the global Cartesian xyz system into the local Cartesian xỹz system, as previously presented in (2).Based on the coordinate transformation rule of vectors and tensors, the flux density vector B and strain tensor ε in xỹz and rϕz systems satisfy the following transformation relationships:
)-(30) yields |f s,r+ | = |f s,r− | = |f s,z+ | = |f s,z− | = 0.77 × 10 6 Pa and |f b | = 46 × 10 6 N/m 3 .Subsequently, these forces are mapped onto the ring meshes to calculate the strain distribution in the FEA.The normal strain in the tangential direction ε ϕϕ is shown in Fig. 5(a).In Fig. 5(a), the strain values on the outer and inner surfaces are 9.2 ppm and 10.8 ppm, respectively.Fig. 5(b) shows the distribution of ε ϕϕ along the radial direction of the ring.The average value of ε ϕϕ within the ring is approximately 10.0 ppm, which is identical with the MS value λ = 10 ppm assigned in (26)-(30).This consistency indicates

Fig. 6 .
Fig. 6.B-H curves of the three studied core materials.

Fig. 12 .
Fig. 12.(a) Flux density and (b) tangential normal strain ε ϕϕ of the amorphous iron stator in FEA.(c) Tangential normal strain ε ϕϕ distributions on the outer surfaces of the three stators.

Fig. 14 .
Fig. 14.(a) Acceleration spectra measured by sensor 1 under 250-Hz sinusoidal flux excitation.(b) Amplitude of second component and (c) fourth component with respect to the excitation frequency.
Fig. 15(a) and (b) show the measured acceleration waveforms in the conventional silicon steel under 100-Hz and 200-Hz sinusoidal flux excitations, respectively.In each figure, the upper row depicts the measured acceleration waveforms, whereas the lower row highlights their second frequency harmonics, which are the most significant components.Under 100-Hz excitation, as shown in Fig. 15(a), the second harmonics measured from the twelve sensors are mostly in phase, indicating a radial mode-0 vibration, or breathing mode.Conversely, Fig. 15(b) shows notable phase differences in the second harmonics under 200-Hz excitation, indicating the occurrence of other vibration modes instead of the radial mode-0 vibration.This distinction indicates that MS excites different vibration modes at different frequencies.

Fig. 16 .
Fig. 16.Relative phases of second acceleration component at the twelve measurement points in the three stator cores under sinusoidal flux excitation at six excitation frequencies.

Fig. 18 .
Fig. 18.Setup for measurement of sound pressure level.

Fig. 21 .
Fig. 21.Overall sound pressure level measured from three stator cores under sinusoidal and square flux excitations.