Contact Force Numerical Analysis of a Wedge-Wave Ultrasonic Motor

Known as piezoelectric motors, ultrasonic motors (USMs) consist of a piezoelectric ceramic attached to an elastic structure. The resulting piezoelectric effects are used to stimulate particular structural vibration modes through the application of a high-frequency AC voltage. As particles move along the surface of the elastic structure, elliptical trajectories are formed, and friction drives either the rotation of the rotor or the linear motion of the slider. USMs are ideal for quiet environments due to their vibration frequency exceeding 20 kHz, which is beyond the range of human hearing. Because of their low-amplitude displacement, USMs also provide high-precision positioning control.

In certain applications, USMs can replace conventional electromagnetic motors because of their high torque, low revolution speed, precise positioning, steady output control, resistance to magnetic interference, structural simplicity, ease of fabrication, and lack of requirement for a reduction drive. The use of USM is common in medical devices such as magnetic resonance imaging machines and other devices that require protection from magnetic interference. The use of USMs in recent years has expanded to include cameras, cellphones, medical devices, aerospace industries, high-precision positioning devices, robots, and a variety of machines. In comparison to electromagnetic motors, USMs can achieve higher torque with a smaller size, lighter weight, and simpler structure. Numerous studies have focused on developing piezoelectric motors [1–14].

An analysis of the vibration mode of a motor’s stator was carried out by Maeno et al. [15]. In regard to the elliptical trajectory motion of the stator’s surface teeth, they reported good consistency between the calculated and experimental results. They also integrated a penalty function formulation into contact analysis to simulate the dynamic contact behavior between the stator and rotor. In order to obtain the torque-revolution speed of the stator and rotor, motor efficiency, and energy loss, the researchers used the displacement and speed of the stator and rotor as well as the distribution of the stick and slip areas. Hirata and Ueha [16] proposed a method for calculating the load characteristics of a traveling-wave USM and estimated the motor’s performance using an equivalent circuit model that emphasized the vibrator’s resonance frequency and ignored the relationship between the equivalent mass and circuit stiffness. They estimated the contact pressure of the rotor vibrator using a two-dimensional (2D) elastic contact model and reported that the simulation and experimental results were consistent.

Using the Rayleigh-Ritz assumed-mode energy method and dynamic equation including strain and electric potential, Hagood and McFarland [17] obtained an approximate solution for dynamic simulation. They assumed the stator to be a rigid body and the rotor to be a linear spring. A compressed spring system was also defined for the area where the stator and rotor overlapped. For the steady-state, they ignored the inertial forces of the rotor and defined the input parameters of the USM, such as the driving voltage, axial preload, and additional torque, and determined the effects of the stator teeth and tooth height on the performance of the motor.

Storck et al. [18,19] reported that the stick-and-slip process of a motor’s stator and rotor was primarily influenced by the elasticity of the friction layer material. In their experiments, they assumed the friction layer to be an elastic material and the spring damper contact system to be oriented in the axial and tangential directions. They applied the equations of motion for a single mass element in the friction layer to deduce the normal and frictional forces while accounting for the effects of the inertial forces. They determined the motor’s output torque, revolution speed, and efficiency from the equivalent spring and damper, and their results indicated that the efficiency loss of the motor was the smallest in the presence of stick friction in the stator and rotor.

Vasiljev et al. [20] conducted a finite element simulation of a USM dynamic contact using ansys finite element software (ANSYS, Inc., Canonsburg, PA). To account for the contact behavior between the stator and the slider, they included a friction layer on the slider, along with surface-to-surface contact elements. They established two sets of contact pair parameters for the stator and slider-friction layers and ignored the damping effects. They used the parameters to create a displacement-time curve of the stator contact points along the x- and y-axes within the first three actuating periods of the slider. The results indicated that the motion trajectory of the contact point was similar to that in the time-harmonic analysis. In order to measure the amplitude of the contact points, the researchers used a laser vibrometer and obtained an error of 5% to 8%.

Shigematsu and Kurosawa [21–25] performed a set of studies utilizing Herzian contact theory to examine the contact force of the friction layer in a surface acoustic wave (SAW) motor. The research encompassed measurement, analysis, and modeling, and involved establishing the physical attributes and design principles associated with this phenomenon. In terms of contact between objects of different sizes and materials, they discovered that Hertzian contact theory could be used to obtain analytical expressions of normal forces and that Coulomb’s law of friction could be used to obtain frictional forces. However, they discovered that Hertzian contact theory could not describe the nonlinear characteristics of frictional forces and that the defined model could not accurately predict the motion of a SAW motor.

In 2010, Shi et al. [26] introduced an innovative linear USM that incorporated a wheel-shaped stator and employed two perpendicular fourth-bending modes. In the same year, Radi et al. [27] put forward a mathematical model that elucidated the dynamic interaction between an elastic object and an obstacle. In 2011, Shi et al [14]. proposed a new type of standing-wave-based linear USM that combined first-longitudinal and second-bending (L1B2) modes. Additionally, in the same year, Shi et al. [28] identified the optimal control strategy for a USM, aiming to maximize its efficiency.

A standing-wave brass-lead zirconate titanate (PZT) tubular USM was introduced by Park et al. [29]. In a separate study, Liu et al. [30] presented a high-power linear USM that utilized a bending vibration transducer to mitigate modal degeneration issues often encountered in USM designs incorporating composite vibration modes. Zhou et al. [31] proposed a modal decoupling method for linear USMs to simplify their design and improve their performance. In the same year, Mashimo [32] proposed one of the smallest USMs to date: a micro-USM that employed a stator with a volume of approximately 1 mm³. Subsequently, Peled et al. [33] conducted a comprehensive analysis, examining high-precision motion solutions utilizing L1B2 USMs. Their review encompassed various aspects, including the design of motor structures, complete motion solution architectures, material considerations, motor drive, and control mechanisms, as well as performance envelopes.

More studies were conducted in 2018. For example, Izuhara et al. [34] proposed a linear micro-USM regarded as one of the smallest linear actuators that could generate a practical amount of force. A spherical USM capable of operating at multiple degrees-of-freedom (DOF) with a single stator was designed by Mizuno et al. [35], providing both high-holding and low-speed torque. Liu et al. [36] conducted experiments on a two-DOF USM that enabled linear motion using a single longitudinal-bending hybrid sandwich transducer. Wang et al. [37] utilized a linear USM to construct a motion system for a probe station, delivering high resolution, rapid response, and extensive travel range. Similarly, in the same year, Lu et al. [38] developed a single-mode linear motor utilizing two types of PZT ceramics. Yu [39] proposed suitable assumption to simplify a model of a surface acoustic wave (SAW) motor, incorporating the concept of an elastic friction layer to analyze the two-dimensional contact problem.

A novel noncontact USM utilizing near-field acoustic levitation was introduced by Shi et al. [40]. This motor design incorporated four flexure pivot-tilting pads driven by high-frequency vibration generated by piezoelectric actuators. In a parallel effort during the same year, Sun et al. [41] devised a technique to determine the contact interface variables that are challenging to directly measure. They built a USM energy output efficiency model that substantially reduced the prediction errors under different preloads compared with the conventional efficiency model. A proposal was put forward by Sun et al. [42] to investigate the coefficient of friction signals during the running-in process by conducting a sliding experiment involving a ring and a stationary disk.

A new two-stage degradation model was introduced by Wang et al. [43] to forecast the variation in preload of a linear motion ball guide, taking into account machining errors. In a similar vein, Li et al. [44] proposed a novel semi-empirical model for the friction coefficient, which modified the classical elastic friction theory, and established the relationship between the friction coefficient and parameters such as normal pressure and velocity. Deng et al. [45], on the other hand, employed a finite element method to assess the transient rolling/slipping contact behaviors between wheels and rails. Their objective was to investigate the characteristics of wheel slips and develop more precise slip protection methods.

In this study, we analyzed the dynamic displacement response of an elastic friction layer under load by simplifying the wedge-wave ultrasonic motor (WW-USM) (shown in Fig. 1) to a two-dimensional (2D) contact problem through suitable assumptions, based on References [46] and [47]. To establish a model for the contact friction between the stator and rotor of the motor and simulate transient contact mechanics, we performed a finite element simulation using ansys. Using actual displacement and external boundary conditions, we tested the optimal convergence and obtained reasonable calculation results by selecting appropriate contact parameter values. We used these results to estimate the characteristic rotational speed versus torque curve of the motor [48].

In the provided equation, T represents the stress tensor matrix, while S represents the strain tensor matrix. The matrix $cE$ signifies the elastic stiffness under a fixed electric field, whereas e represents the matrix of piezoelectric constants. The transport matrix of piezoelectric constants is denoted by $eT$⁠. The electrical displacement vector matrix represented by D, and E denotes the electric field vector. Lastly, the matrix $εS$ represents the dielectric constants under the condition of constant strain.

where the damping matrix is set to proportional damping (i.e., $C=αM+βKuu$⁠), F represents the contact force, encompassing both normal and tangential components, between the motor’s stator and rotor. On the other hand, Q represents the total electric charge induced in the piezoelectric material as a result of the applied electric field.

Equation (5) is an eigenvalue determinant. The n eigenvalues of Eq. (5) are the natural frequencies $ω$⁠.

In this study, we used ansys to calculate structural dynamic response. Furthermore, an analysis was conducted to examine the motion trajectory of the wedge waves on the stator as well as the contact interaction between the stator and the rotor. In addition, we used the Newmark time integration method to perform a transient response analysis. In this section, we discuss the theory underlying this method.

where M, C, and K represent mass, damping, and stiffness matrices of the system, respectively. The displacement vector is denoted as u, and the vector F represents the external forces acting on the system.

where $K¯=K+1αΔt2M+δαΔtC$ and the displacement of $un+1$ can be obtained from the results obtained at time $t$⁠, according to Eq. (11). Because the Newmark time integration method is unconditionally stable, a larger time increment $Δt$ can be used in the calculations, although a smaller time increment is still recommended for congruence and accurate results.

The corresponding internal element force $Fn+1$ is calculated using $un+1$⁠, and the difference $ΔR$ is modified until it is less than or equal to the convergence criterion $Rcri$⁠. The displacement of the $i$th iteration $un+i$ is the approximate solution of the nonlinear equation. In the illustration of this iterative process shown in Fig. 2, the curve represents the nonlinear changes in the structure and the corresponding displacement due to the forces acting on the structure. The matrices $Fn$ and $un$ represent the internal element force and displacement at time $tn$⁠, respectively. In the Newton–Raphson approximation method, $P0$ is defined as the starting point of the first iterative step. Point $P1$ is determined by defining a tangent to the gradient $KT1$ and its corresponding vertical line. The next iteration is performed when the difference between the external and internal forces $ΔR$ exceeds the convergence criterion. Point $Pi$ is determined at the $i$th iteration with the corresponding displacement $un+i$ and internal element force $Fn+i$⁠. If $ΔR$ is smaller than the convergence criterion, then its corresponding displacement $un+i$ is regarded as the desired approximate solution, indicating the convergence of the iterative process. However, if the curve is too complex or the defined value of the starting point $P0$ is inappropriate, divergence may occur during the iterative process.

ansys contact analysis is used to simulate contact between two objects [49,50]. However, the contact pair on the contact surfaces of the two objects must first be defined, including the target surface and contact surface, which include the respective target and contact elements. The calculated penetration between the elements is then used to obtain the contact force distributed across the elements and nodes. This nonlinear solution process yields an approximate solution for node displacement.

where $Fnormal$ is the contact force, $knormal$ is the contact stiffness, and $xp$ is the penetration depth. Figure 3 illustrates the contact between two objects in a finite element model. A contact spring is present between the objects, and its constant is regarded as the contact stiffness. By adjusting the contact stiffness, appropriate convergence criteria and reasonable calculation results can be obtained. A higher contact stiffness indicates weaker penetration between objects. Thus, the contact stiffness determines the penetration depth. However, increased contact stiffness makes calculations using the Newton–Raphson iterative method more complex, which in turn increases the difficulty of achieving convergence in nonlinear problems.

In this study, we used the Coulomb friction model in ansys to simulate the contact and friction between objects. In general, the relative speed of two objects in their contact area is zero when stick friction, or static friction, occurs, where $μs$ is the coefficient of friction. Similarly, the relative speed of two objects in their contact area is nonzero when sliding friction, or kinetic friction, occurs, where $μd$ is the coefficient of friction. In our contact analysis, the relationship between the coefficients of static and kinetic friction was defined as $μs=FACT×μd$⁠, where $FACT$ is the ratio of static to kinetic friction coefficients.

To avoid unreasonably excessive shear stress, the largest shear stress value of the contact area can be specified. For two objects in contact, sliding occurs when the shear stress of the friction area exceeds the largest allowable shear stress. At the friction area, the maximum shear stress is $τmax=Sy$/$3$⁠, where $Sy$ is the tensile and yield strength of the material and the maximum shear stress $τmax$ is the von Mises stress.

In the illustrated cylindrical configuration of the WWUSM (refer to Fig. 1), a PZT-4 piezoelectric tube sourced from Eleceram Technology in Taoyuan City, Taiwan, was securely bonded to a stainless-steel wedge. Additionally, the piezoelectric tube was compactly bonded to the stainless-steel base, forming a cohesive assembly. Using a preloaded compression spring, the inclined face of the wedge was then set in close contact with a naval brass rotor. Table 1 details the material properties of the rotor. The PZT-4 piezoelectric tube operated in a radial poling direction and the inner tube wall was uniformly coated with an electrode to produce traveling waves of four wavelengths to drive the rotor. To facilitate measurement, a modal sensor along with two sets of comb electrodes, namely, A and B, were screen-printed onto the outer surface of the PZT-4 piezoelectric tube. The material properties of the PZT-4 piezoelectric tube, as provided in Table 2, are outlined [51].

The cylindrical wedge motor stator was discretized using 3D ANSYS, resulting in a mesh with 55,858 elements and 75,167 nodes. Figure 4(a) illustrates that all 3840 nodes situated at the base’s bottom were assigned with zero degrees-of-freedom (DOFs). To accelerate the ansys analytical process and save memory, none of the four holes at the bottom of the stainless-steel base were tapped. Figure 4(b) shows the mode shape of stator F(m, n) obtained from the modal analysis, where m = 1 is the axial mode number and n = 4 is the circumferential mode number. The corresponding resonance frequency of 36.065 kHz was set as the target excitation mode. The outer wall of the PZT tube was coated with electrodes capable of exciting four petal-like traveling waves (i.e., dual-phased electrodes A and B). The cylindrical wedge and base of the motor’s stator were made of stainless steel. Their material properties are detailed in Table 3.

The wedge stator exhibits a monotonic increase in twist angle along the axial position z in terms of its structure. As a consequence, the torque on the inner slope of the wedge remains consistent. Furthermore, as the contact point gradually shifts toward the wedge’s tip, the rotational speed of the rotor experiences a corresponding increase. However, in this study, the wedge tip was too flexible and could not generate sufficient friction to drive the rotor. In other words, the USM had a low torque output at high revolution speeds and vice versa. By utilizing the contact point between the rotor and the inner slope of the wedge, it is possible to eliminate the tradeoff between torque and revolution speed.

Let us consider a case in which two transducers A and B are excited in accordance with sinusoidal functions sin(ωt) and cos(ωt), respectively, and let us assume that the motor’s stator has a damping factor ζ = 0.2%. In this case, the F(1, 4) excitation mode is generated using a 400 V_p-p alternating voltage and by conducting a harmonic analysis in ansys. As a result of damping, the particle motion in each cross section undergoes phase changes. In addition, the particle motion along the circumferential direction generates amplitude changes with respect to the corresponding electrode positions.

For the purpose of this investigation, the structure of the rotor was specifically designed, taking into account the contact point between the stator and rotor of the motor. As an example, in a stator configuration with a wedge angle set to 15 deg, the contact point was positioned 1.26 mm below the tip of the wedge. To enhance both thrust and friction characteristics, various components were incorporated with the contact area between the stator and rotor. The creation of the rotor’s mesh was carried out in ansys, utilizing Solid92 tetrahedron cone element and TAEGE170 contact area elements, as depicted in Fig. 5. Figure 6 shows that the mesh of the stator's wedge was produced using the Solid5 hexahedron elements and CONTA173 contact area elements.

The meshing of the cylindrical wedge motor’s stator and rotor in ansys comprised a total of 63,840 elements and 80,640 nodes. With this mesh, it was assumed that the 3840 nodes located at the base’s bottom had zero DOFs. Figure 7 illustrates the meshes used for the motor's stator and rotor.

The analysis of transient response was utilized to calculate the contact mechanics between the stator and rotor of the motor, as well as to predict the rotational speed and torque of the USM. Due to the substantial nonlinearity involved in dynamic contact, attaining convergence holds crucial importance in the iterative process. The determination of the number of iterative steps, convergence, and the validity of results relied on the specific real parameters and essential options assigned to the contact elements.

In ansys, contact analysis requires contact elements to simulate the contact patterns in the contact area of two objects. When these elements are specified, the contact conditions of the two objects are considered in the solution process. However, before the solution process begins, an appropriate range should be specified for the contact region. An excessively small range may result in penetration in undefined contact areas, whereas an excessively large range increases the calculation time. The focus of this study was solely on the contact mechanics analysis of the USM, specifically limited to the finite element models of the stator structure and rotor. The contact point of the stator was considered as a plane, with the assumption that the contact involved a rigid body interacting with an elastic body. The rotor was also considered to be a rigid body, consisting of a 10-node three-dimensional (3D) SOLID92 tetrahedron element. A contact pair was established on the stator’s wedge and the rotor’s surface such that the rotor and stator made surface-to-surface contact in 3D space. The contact surface of the rotor was associated with the target element TARGE170, while the wedge’s surface was linked to the contact element CONTA173. Next, the element coordinate systems (ESYSs) of the contact and target elements were tested. The normal vector of the contact ESYS was directed toward the target element, and the normal vector of the target ESYS was directed toward the contact element. Finally, system calculations were performed based on the established contact pair.

In contact pairs, the actual constants of target and contact elements are used to determine the nonpenetration contact conditions. The following are the parameters that enhance model convergence and considerably affect the analysis results:

1. Contact stiffness

   During the establishment of surface-to-surface contact elements, the contact stiffness of the elements must be specified. Greater contact stiffness leads to enhanced simulation accuracy and reduced penetration between the stator and rotor of the motor. However, in the context of Newton–Raphson method, augmenting the contact stiffness gradient results in an increased number of iterative steps. Consequently, this elevation in iterative steps, along with the associated computational complexity, poses challenges for achieving convergence in nonlinear solution processes. Contact stiffness can therefore be incremented based on specified contact stiffness values, after which the convergence and reasonableness of the results can be determined.

2. Allowable penetration

   In contact area calculations, the penetration of two objects must be less than the maximum allowable penetration. During an iterative process, convergence is unfeasible if the actual penetration is greater than the allowable penetration, and the system continues iteration until the resulting penetration qualifies. The specifications of allowable penetration are similar to those of contact stiffness in which poor allowable penetration is close to the actual contact conditions. However, this requires a long calculation time and may be associated with nonconvergence.

3. Pinball region

   When a target element is present in the defined detection range of a contact element, the system performs calculations on the basis of the contact force between the elements. An excessively small specified detection range may cause the system to issue warnings because it is less likely to detect contact, whereas an excessively large range may slow the solving process because more items are searched. Hence, to confirm whether the specified range is suitable for detecting the occurrence of contact, the appropriate contact range should first be assessed.

4. Maximum shear stress

   Unrealistic results resulting from excessive contact stress can be avoided by specifying the maximum shear stress of the friction area, defined as $τmax=Sy$/$3$⁠, where $Sy$ is the tensile and yield strength of the material.

5. Coefficient of friction

   Given that the friction between the rotor and the stator encompasses both stick and sliding frictions, the designated static and kinetic coefficients of friction play a significant role in determining the frictional stress of the contact element, the extent of sliding contact, and the overall motor output performance. Hence, it is of utmost importance to adjust specific specifications related to the contact and target elements. In this particular study, careful consideration was given to selecting appropriate specifications based on general principles and the nature of contact between the stator and the rotor. Some of the key options are listed as follows:

6. Contact algorithm

   In this study, a penalty method algorithm was used. In this method, high precision can be achieved only by increasing the number of equilibrium iterations. However, additional iterations require more temporal and spatial calculations.

7. Detection criteria of contact elements

   The detection of target elements by contact elements can be either Gaussian-point detection or nodal detection. In this study, because the stator was considered a rigid body, nodal detection was used, preventing contact pair penetration.

8. Contact behavior

   In this study, the motor’s stator and rotor were considered to be inseparable, i.e., their surfaces adhered to each other, even under sliding conditions.

9. Update of contact stiffness

   When the motor’s stator and rotor are in contact, continuous changes in the contact region result in continuous changes in the system’s stiffness. Therefore, during each iteration, the system automatically updates the contact and tangential stiffnesses to converge to a solution.

10. Initial specifications of contact area

    If even a tiny gap is present before the objects come into contact, convergence difficulties may arise at the moment of initial contact. Therefore, after the finite element model is established, correlation specifications can be established for initial contact points. For example, the activation options for gap closure allow for constant contact between the stator and rotor, thereby minimizing errors in model and mesh generation.

Figure 8 exhibits the continuous-mode diagrams portraying the transient response of contact mechanics. By the 240th-step (approximately 6.6 ms), the system reaches a steady-state in which all deformations on the stator are focused on the rotor. Table 4 provides a detailed list of the specified actual constants for the contact friction model used in the WWUSM.

Figure 7 demonstrates the generation of a circular rotor model positioned at the upper region of the contact point within the stator structure. To initiate the desired motion, a sinusoidal voltage with a peak-to-peak amplitude of 400 volts (V_p–p) was applied to the surface electrodes of phase actuators A and B. The voltages of the actuators were 90 deg out of phase. Subsequently, an examination was carried out to determine the downward preload force exerted on the top of the rotor. Before commencing the solution process, the coordinates of the rotor were converted from Cartesian to cylindrical representation. Upon activation of the stator, the friction forces arising from the contact caused the rotor to undergo rotation. It required a minimum of 240 cycles (approximately 6.6 ms) to achieve a steady-state displacement and speed, as depicted in Fig. 9. As the radial displacement U_r of the rotor approached zero, the radial displacement U_θ exhibited a progressive increase over time, inducing a counterclockwise rotation of the rotor (attributable to the positive displacement). Figure 10 provides a comparison analysis of the simulation and measurement results for the rotational speed and torque, considering preload forces of 0.98 N and 4.116 N.

Both the mechanical composition and driving method affect the performance of USMs, with the mechanical composition having the greater effect. The mechanical considerations encompass various factors, such as the parallelism of relative displacement between the stator and rotor, the perpendicularity of the piezoelectric ultrasonic units concerning the applied preload force, and the bonding technique employed for securing the rotor and load cell. The mechanical components include the springs that support the rotations of the rotor. Although the properties of an optimal spring do not change regardless of its position, the homogeneity of force distribution is difficult to maintain with such a spring. Therefore, in this study, springs were replaced by a piezoelectric disk because applying a steady DC voltage can help achieve force distribution homogeneity. Typically, it is desirable for the coefficient of friction between the rotor and the stator to remain constant. Moreover, in order to avoid any significant obstruction caused by substantial instantaneous displacement during rotor startup, it is advisable to keep the maximum static coefficient of friction between the two components within reasonable limits.

The following criteria should be considered in the selection of the friction layer:

1. It is preferable to minimize the coefficients of friction between the rotor and stator components. Conventional rotors have a high seal resistance, which increases the overall resistance and renders them unsuitable for use in WWUSMs that require low driving forces.

2. The rotor should be as light as possible. A lighter rotor mitigates the effects of inertia on the performance of the motor and increases its response speed.

3. The rotations should be highly precise. If the precision of the motor’s working direction is insufficient, it can lead to inaccuracies in rotations.

Among the driving parameters of a WWUSM are the preload force, driving voltage, and number of burst periods. According to the preliminary empirical results, driving voltage is the most critical parameter, and preload forces have no clear effects (within the scope of our study). The step behavior of a WWUSM can be easily observed with additional burst periods and a suitable voltage. With few burst periods and a high voltage, the process becomes limited by the performance of the measuring instrument; only a long-term cumulative displacement can be obtained, and the step displacement behavior cannot be observed. The most vital part of a WWUSM system is the contact between the rotor and the stator.

According to the literature on contact friction theory, the characteristics of contact friction can be mathematically expressed, and various contact friction characteristics, such as the ratio of friction and normal forces and the relative speed of two objects, can be iteratively integrated into coefficient of friction functions. During the calculation of relative speed, the type of contact (area or point contact) between the stator and rotor must be clearly specified before analysis, in addition to the fixed value of the revolution speed.

In this research, the objective was to investigate the dynamic displacement response of the elastic friction layer under load by simplifying the USM to a 2D contact problem through suitable assumptions. Typically, the displacement field of the friction layer surface can be determined using numerical wave integration methods. According to the dispersion curve of the friction layer, the propagation of wedge-shaped guided waves can occur on the friction layer only when the layer is thick or when the load frequency is high. The friction layer of USM rotors is often a few micrometers thick, and the effects of the propagation poles of the wedge-shaped guided waves are negligible in wave number integration.

According to preliminary calculations, the displacement fields generated by different loads have mutual effects. During the calculation of load distribution under displacement conditions, the specified preload force and other conditions can facilitate convergence. The displacement field resulting from the loading forces at the contact friction layer subsurface can then be determined using numerical integration in ansys. Although the obtained measurement results cannot be used for verification, the simulation results can still be verified by observing physical phenomena.

In this study, wedge deformations were first specified to analyze the contact shear between the friction layer and the wedge as well as the final displacement of the friction layer. The opposing rotation of the rotor was facilitated by the frictional force produced by the friction layer. However, the specified displacement conditions did not represent the actual displacement of the friction layer, resulting in failure to obtain precise contact stress values. Therefore, to determine the contact stress more accurately, additional comprehensive conditions must be specified. Furthermore, in order to accurately determine the effective friction force, it is essential to account for the deformation velocity of the friction layer when calculating relative velocity between the friction layer and stator.

Finite element simulation was performed in this study using the ansys finite element software to establish contact friction models for a motor’s stator and rotor and simulate the transient contact mechanics involved. The finding from the ansys simulation demonstrates the significance of the contact between the stator and rotor in the contact friction model when determining the actual parameters. The selection of appropriate contact friction model parameters, based on optimal criteria, served as a reference for analyzing the transient contact mechanics response during the design process.

Examining the dynamic behavior of the friction layer in WWUSM enables a comprehensive exploration of the isotropic material properties of both the friction layer and the wedge, fostering in-depth discussions. This analytical approach can be extended to conventional USMs as well. Furthermore, it allows for the determination of the relationship between the surface displacement of the friction layer and the external forces exerted upon it. In addition, parameters such as the displacement field and motor characteristics can be analyzed when the conditions for either displacement or contact stress are specified. However, actual displacement conditions are difficult to determine because several conditions are required for iteration.

• Ministry of Science and Technology in Taiwan (Grant No. MOST 112-2221-E-239-002; Funder ID: 10.13039/501100004663).

• National United University (Grant No. 112-NUUPRJ-05).