In order to simplify the calculation, the vibration system (the piezoelectric elements and the stator) can be characterized as a 2-degree-of-freedom spring-mass-damping system. The modal coordinates (qA, qB) and the dynamic functions can be depicted as 4q=qA(t)qB(t)=ξcos⁡(wt)ξsin⁡(wt),5moq¨A+doq˙A+koqA=kcUA+εkcUB+Fd1,moq¨B+doq˙B+koqB=kcUB+εkcUA+Fd2, where w=2πFs denotes the angular driving frequency, mo, do and ko represent modal mass, modal damping and modal stiffness, respectively, and ε is the imbalance coefficient between two-phase signals. Fd1 and Fd2 are the respective modal forces in both the tangential direction and the vertical direction. Besides, if λ is the wavelength of the traveling wave and k=2π/λ, cos⁡kx⌢ and sin⁡kx⌢ are the eigenmode functions. When Φ(r) is the radial variance of this mode shape, the orthogonal modal shape function can be represented as 6Φw=ΦwA(r,x⌢)ΦwB(r,x⌢)=Φrcos⁡kx⌢Φrsin⁡kx⌢. Thus, the vibration response of the stator stuck with the piezoceramics can be obtained by multiplying Eqs. (4) and (6); the result is depicted as 7w0(r,x⌢,t)=ΦwTq=Φ(r)ξcos⁡(kx⌢-ωt). Finally, the tangential speed of the surface point can be derived. If h is the half-thickness of the stator and the middle radius of the stator is rp, whose radial variance is defined as ψav=Φ(rp), then the stator speed can be expressed as 8vs(rp,x⌢,t)=-hddt∂w0(x⌢,t)∂x⌢=-hξkψavcos⁡(kx⌢-wt).

Figure 4 illustrates the contact schematics of the motor with teeth discretization, and the stator–rotor contact model is built on the assumption that the rotor is rigid over all the contact areas, whereas the friction layer can be modeled as a series of linear springs. The yellow points mean the contact borders with the zone [-x0,x0], and the demarcation points between the traction zone and the braking zone are -x1 and x1.

According to Fig. 4 and Eq. (9), the coordinates of contact border x0 and stick point x1 yield 9x0=1karccos⁡h-z(t)ψavξ,10x1=1karccos⁡θ˙rR02khψavξω, where z(t) is the real-time vertical displacement of the rotor and θ˙r means the rotor speed. When the joint stiffness of the dispersed springs is Kf, the unit pressure distributing from the contact interface to the stator can be given by 11f(x)=KfΔx=Kfψavξ[cos⁡(kx)-cos⁡(kx0)]. As shown in Fig. 5, every tooth can be simplified as a cuboid whose length and width are Rc and f, respectively. Since USR60 operates in ninth mode with 90 teeth, 10 teeth scatter in every wavelength. The coordinate plane OYZ is built on the central axis of the middle tooth, and the x axis is assigned along the traveling wave direction.

In succession, the left teeth edges are recorded as Aj, while the right ones are assigned as Bj. As shown in Fig. 5, the code numbers are named for every tooth, and only half of No. 1 and No. 11 are included in one wavelength. We define the gap width as d. Therefore every tooth position can be determined by its edge, whose horizontal coordinate and vertical coordinate are depicted as 12X⌢Aj=(2n-13)f2N+(n-6)dN,X⌢Bj=(2n-11)f2N+(n-6)dN,13ZAj=h+ψavξhkcos⁡(wt-kX⌢Aj),ZBj=h+ψavξhkcos⁡(wt-kX⌢Bj). Therefore, there are four cases when comparing the vertical coordinates with the stator half-thickness h. For convenience, the real contact boundaries are symbolized as s1, s2 with the following four cases. a.

If ZAj<h and ZBj<h, both edges of the tooth are apart from the contact area, and this tooth makes no contributions to the force generation.

In terms of the whole TWUSM, the vertical displacement and the rotational velocity of the rotor can be described in the third and fourth functions in Eq. (27), where dz represents the vertical damping and dr means the rotational damping. Finally, the final analytical model can be depicted as 26moq¨A+doq˙A+koqA=kcUA+εkcUB+Fdt1+Fdn1,moq¨B+doq˙B+koqB=kcUB+εkcUA+Fdt2+Fdn2,mrW¨r+dzW˙r=FZ-FN,Jrθ¨r+drθ˙r=Tr-Tload. It should be emphasized that the bridges between the circuit model and the motor model are the equivalent electrical voltages (φva,φvb), which can be expressed as Eq. (28). The efficiency ηe from the input power to the output power can be depicted as Eq. (29), which is used to evaluate the energy utilization factors with variable loads. 27φva=Fd1=Fdn1+Fdt1φvb=Fd2=Fdn2+Fdt228ηe=PoutPin=2Trθ˙rPA+PB

As shown in Fig. 9, when the amplitude Um is 1.3 V and the frequency Fs is 41.5 kHz, the startup velocity curves with or without teeth discretization are compared. As can be seen, the results with the teeth discretization method achieve better fitness. The fitness is 55 % if the teeth structure is ignored, and the fitness increases to 89 % when the teeth discretization is implemented. Moreover, the 2 ms delay which occurs in the results without the teeth discretization model disappears with the proposed model. This can be explained by eliminating the driving effect of the tooth gap by refining the interaction forces and the contact status. Furthermore, the contact area which has been limited to the teeth space rather than the whole ring is closer to reality. Furthermore, the startup speed responses actuated by various frequencies, and a certain amplitude, are displayed in Fig. 10a. To gain more detailed observations, the speed curves before 5 ms are displayed in Fig. 10b. All results under different frequencies show good agreements between the numerical results and experimental ones and the differences of the response process. Figure 10b indicates that the motor starts more quickly and obtains less overshoot with the increasing frequency.

Due to the complexity of piezoelectric ceramics, the vibration conditions are different in diverse regions before or after the resonant points. In order to obtain the ideal working range, especially the frequency interval of the proposed integrated model, the rotor velocities of different amplitudes and frequencies are displayed in Fig. 11a and b.

It can be seen from Fig. 11a that the linearity between the driving amplitudes and the rotor speed is evident on the whole. And Fig. 11b demonstrates that the speed gradually increases and then drops to a lower value when the frequency is changed from 40 to 44 kHz. Moreover, the frequencies located on the peak velocity decrease as the amplitude increases. This is due to the variation of the natural frequency of the stator caused by the softening nonlinearity. We can conclude that when the frequency comes close to the resonant frequency, the simulation model cannot describe the speed adequately, which may result from the simplification of the stator modal model, as the frequency higher than the resonant peak is usually chosen as the working range, where the fitness becomes better especially from 41 and 44 kHz. The effective fitting results further prove the feasibility of the proposed model.

In all, the above test of speed performances with two driving parameters verifies the validity of the model located in a specific working area, where the amplitude is higher than 0.7 V and the frequency is higher than 41 kHz.

In order to verify the model in the functions of external load, Fig. 12 illustrates the comparison of mechanical characteristics from the simulations and the experiments. As can be seen, the velocity–torque curve exists in good agreement within the frequencies from 42 to 43 kHz, while the results occur with a little error with the lower frequency. Also, we can observe that the efficiency becomes higher when the frequency is near the resonant value, and the optimal torque we should impose on the motor will be less than 0.5 N m.