## 1 Introduction  
Traveling wave actuators are one of the most popular piezoelectric devices that are used in numerous industrial, medical, and life-science fields [1, 2]. This type of actuator has features like high accuracy, microscale resolution, self-locking, high torque, and quick response [3, 4, 5]. Usually, the operating principle of traveling wave actuators is based on the excitation of traveling wave vibrations in a ring-type stator [6, 7, 8]. However, the amplitudes of the contact surface vibrations of such stators are low. Therefore, todring-shaped stators are used. The configuration of the tooth, including height, cross section, and shape, is based on the mechanical output requirements of the motor. However, the tooth-based stators have height limitations because of buckling phenomena [9, 10, 11]. Buckling occurs when the preload force between the stator and rotor increases and exceeds the critical load. In this case, uncontrolled slipping between the stator and rotor appears, energy losses and wear at the contact area increase, and, as a result, the performance and reliability of the actuator decreases [12]. Moreover, these disadvantages limit the design flexibility of the stator and, as a result, possible application areas of the traveling wave actuators. Therefore, to overcome these disadvantages of toothed stators, toothless traveling wave actuators should be introduced. [PERSON] et al. reported a cylindrical traveling wave actuator based on a micromachined bulk vibrator and glass case [13]. The operating principle of this actuator is based on the first or second out-of-plane bending vibration modes of the step-shaped cylinder. The excitation of the actuator is performed using four signals which have a phase difference of \(\pi\)/2. On the basis of numerical and experimental investigations, the authors concluded that the proposed motor can achieve a rotational velocity of 1000 RPM with excitation signal frequency and amplitude of 67 kHz and 40 V\({}_{\text{P-P}}\), respectively. The authors also performed numerical investigations of when the motor was driven at higher vibration mode and concluded that usage of a higher vibration mode could increase the performance of the motor. [PERSON] et al. reported a rotary traveling wave actuator with design based on a toothless stator and rotor pair with irregular cross sections [14]. The cross sections of the stator and rotor were modified by cylindrical cavities. The modifications were made in order to increase the displacement amplitudes of the actuator and overcome buckling phenomena. This actuator's operation is based on synchronous traveling wave vibrations in the modified stator and rotor. The authors concluded that the proposed approach could increase the mechanical performance and design flexibility of simple traveling wave actuators, as well as overcome the disadvantages of tooth-based traveling wave actuators. [PERSON] and [PERSON] reported a toothless traveling wave rotary motor based on a ring-shaped stator, bar-shaped transducer, and two-sided rotors [15]. The simple design of the stator and separated heat source and friction interface make the motor more attractive compared to the conventional design of traveling wave motors. Employment of the proposed toothless stator eliminates the disadvantages related to tooth buckling phenomena and makes the design more flexible and easily customizable to special requirements. The authors performed numerical and experimental investigations of the motor and concluded that the proposed motor can operate in both longitudinal and bending vibration modes. According to the results, the motor is able to provide 0.25 Nm and 50 r/min at a frequency of 48.625 kHz when an excitation voltage of 300 V\({}_{\text{p-p}}\) is applied. [PERSON] et al. reported a 2-degree-of-freedom (DOF) piezoelectric toothless traveling wave rotary motor [16]. The motor design is based on a hemispherical stator with two sets of piezoelectric patches.