 Original Article
 Open Access
 Published:
Novel Traveling Wave Sandwich Piezoelectric Transducer with Single Phase Drive: Theoretical Modeling, Experimental Validation, and Application Investigation
Chinese Journal of Mechanical Engineering volume 34, Article number: 101 (2021)
Abstract
Most of traditional traveling wave piezoelectric transducers are driven by two phase different excitation signals, leading to a complex control system and seriously limiting their applications in industry. To overcome these issues, a novel traveling wave sandwich piezoelectric transducer with a singlephase drive is proposed in this study. Traveling waves are produced in two driving rings of the transducer while the longitudinal vibration is excited in its sandwich composite beam, due to the coupling property of the combined structure. This results in the production of elliptical motions in the two driving rings to achieve the drive function. An analytical model is firstly developed using the transfer matrix method to analyze the dynamic behavior of the proposed transducer. Its vibration characteristics are measured and compared with computational results to validate the effectiveness of the proposed analytical model. Besides, the driving concept of the transducer is investigated by computing the motion trajectory of surface points of the driving ring and the quality of traveling wave of the driving ring. Additionally, application example investigations on the driving effect of the proposed transducer are carried out by constructing and assembling a tracked mobile system. Experimental results indicated that 1) the assembled tracked mobile system moved in the driving frequency of 19410 Hz corresponding to its maximum mean velocity through frequency sensitivity experiments; 2) motion characteristic and traction performance measurements of the system prototype presented its maximum mean velocity with 59 mm/s and its maximum stalling traction force with 1.65 N, at the excitation voltage of 500 V_{RMS}. These experimental results demonstrate the feasibility of the proposed traveling wave sandwich piezoelectric transducer.
Introduction
Piezoelectric ceramic material utilizes its positive/converse piezoelectric effect to achieve the energy conversion between electric energy and mechanical energy, being one of commonly used smart materials in industry and research [1]. Piezoelectric transducer is the typical application of piezoelectric ceramic materials and can be employed as sensors or actuators, being widely applied to aerospace, optics, robotics, and biomedical engineering [2]. Some piezoelectric transducers utilizing the converse piezoelectric effect can convert electrical energy to microamplitude mechanical vibration energy, thereby they can be designed as the stator of ultrasonic motors [3]. Vibration characteristics of the piezoelectric transducers directly influence the mechanical output performance of ultrasonic motors. This is because the microamplitude vibration of the piezoelectric transducer is transmitted to the macro motion of slider/rotor of ultrasonic motors by friction. Elliptical motion is one of the significant evaluation indexes for the vibration characteristics of the piezoelectric transducers. Therefore, how to construct efficient elliptical motion in the driving tip of the piezoelectric transducer is the key step of ultrasonic motor design.
In general, to generate the effective elliptical motion of the driving tip, the piezoelectric transducers are expected to operate in the coupling vibration mode [4]. This is because the elliptical motion of the driving tip is used to produce the friction between the stator and the rotor/slider of ultrasonic motors for achieving macro rotary/linear motion. Thus, the generation of the elliptical motion is significant for piezoelectric transducers. However, for producing the elliptical motion, two vibration modes are generally required to be stimulated and coupled in the piezoelectric transducers, increasing the difficulties of design, manufacturing, and assembling of transducer [5, 6]. Additionally, how to achieve the resonant frequency consistency of the two selected vibration modes is also a difficult issue for the control system of the piezoelectric transducers. To avoid these problems, singlephase excited piezoelectric transducer is the preferred option [7]. This is because this type of piezoelectric transducer only needs a phase drive signal to achieve the elliptical motion of its driving tip, eliminating the problem of adjusting resonant frequencies and simplifying the control system [8].
Researchers have made some efforts in the research of singlephase excited piezoelectric transducers [9,10,11,12,13,14,15,16,17,18,19]. Kurosawa et al. [9] used a small mass to attach in a ring type piezoelectric transducer, realizing two orthogonal vibration modes with a slightly different resonant frequency excited in the ring transducer under a single power supply. Therefore, a circular ultrasonic motor was constructed. Utilization of the structural property of rectangular piezoelectric plate and the asymmetrical resonant excitation, Vyshnevskyy et al. [10] proposed a singlemode linear ultrasonic motor. For this motor operating principle, a twodimensional extensional standing wave is stimulated in the piezoelectric plate to achieve the slopeline elliptical motion of its driving tip. Based on the internal coupling effect caused by the crystal anisotropy, Tamura et al. [11] designed a thinplate piezoelectric transducer to be the stator of an ultrasonic motor. A coupling vibration mode was excited in a LiNbO_{3} rectangular plate vibrator to achieve the miniature singlephase drive ultrasonic motor. He et al. [12] presented a tubular ultrasonic motor that only uses a single vibration bending mode of a piezoelectric tube to produce rotation motion. The piezoelectric tube is only excited by one driving signal, and no vibration mode coupling is required. Yokoyama et al. [13] developed a linear ultrasonic motor using a single resonance mode driven by a single excitation signal, in which two square plates linked by Vshaped beams are employed as the motor stator. Liu et al. [14] designed a sandwich plated piezoelectric transducer with single vibration mode, producing slash elliptical motion at its driving tip to push a slider for constructing a linear ultrasonic motor. Chang et al. [15] proposed a screwthreadtype ultrasonic motor with singlephase excitation by designing an asymmetrical piezoelectric transducer to generate two orthogonal bending vibration modes. Based on the structural properties of the transducer, Ma et al. [16,17,18,19] developed standing wave piezoelectric transducers with single phase excitation for pushing a rotor/slider to achieve ultrasonic motors, respectively. Most aforementioned piezoelectric transducers are designed as the standing wave vibrator for ultrasonic motors. However, the wear problem is hard to avoid for standing wave ultrasonic motors due to the limited contact area between their stator and slider/rotor. It has been demonstrated that traveling wave ultrasonic motors are the costefficient, durable and stable actuator during industrial applications due to the large contact area between the stator and the rotor during operation.
Therefore, inspired by the transducer design method in Refs. [9,10,11,12,13,14,15,16,17,18,19], a novel traveling wave sandwich piezoelectric transducer with single phase excitation is proposed in this study. Only one excitation signal is required to apply to the transducer, and traveling waves with the same rotational direction are produced in two driving rings utilizing the structural coupling property of a beamring combined element. This is done in order to achieve elliptical motion of surface points of the two driving rings, largely simplifying the control system. A pretensioning mechanism is designed to mount four groups of complete lead zirconate titanate piezoelectric ceramic (PZT) elements by one bolt, avoiding the unequal prestresses issue of all PZT elements.
This paper is organized as follows. Initially, the configuration and operating principle of the proposed transducer are explained in detailed. Subsequently, an analytical model is developed utilizing the transfer matrix method (TMM) to study the dynamic behavior of the proposed transducer. In addition, experimental validation is conducted in order to validate the feasibility of the developed transfer matrix model and to confirm the operating principle and design of the proposed transducer. Moreover, motion trajectory of surface points and quality of the driving ring are calculated based on the developed model to study the driving concept of the transducer. Application example of the proposed transducer is then investigated to evaluate the driving performance of the proposed transducer prototype. Finally, a conclusion is presented.
Structure Design and Operating Principle
Structure Design
The threedimensional view of the proposed traveling wave sandwich piezoelectric transducer excited by a singlephase signal is illustrated in Figure 1. It is composed of a sandwich composite beam and two driving rings. The sandwich composite beam is comprised of a variable crosssection beam, a pretensioning mechanism, and four groups of PZT elements. The pretensioning mechanism consists of two isosceles trapezoid blocks, four right trapezoid blocks, and a bolt. Each PZT element is polarized along its thickness direction, and two pieces of complete and rectangular PZT elements with opposite polarization directions form a group of excitation unit. One isosceles trapezoid block and two right trapezoid blocks form a rectangular block. Therefore, the pretensioning mechanism is comprised of two rectangular blocks and a bolt. The middle of the variable crosssection beam is designed with two rectangular grooves at its upper and lower surfaces. A rectangular block and two groups of PZT elements placed at its both ends are combined together to fill into one rectangular groove. Therefore, the two isosceles trapezoid blocks are connected by the bolt, ensuring the pretensioning mechanism and the four groups of PZT elements are mounted in the variable crosssection beam. This ensures that all the PZT elements are applied with equal prestress. The two driving rings are placed at both ends of the sandwich composite beam.
Operating Principle
In order to achieve the drive function, elliptical motion is required to be produced at surface points of the two driving rings. As an electrical signal is simultaneously applied to all the PZT elements, the sandwich composite beam is excited to vibrate at the first order longitudinal vibration, and two driving rings are simultaneously stimulated to vibrate at a third order inplane radial bending vibration, as shown in Figure 2(a). The aforementioned two vibrations are the natural vibration mode of the driving ring and sandwich composite beam. Once the resonant frequencies of the two vibrations are close or the same, they will be stimulated in the ringbeam composite structure under the external forced excitation. This is the reason why the first order longitudinal vibration and the third order inplane radial bending vibration can be simultaneously excited in the transducer under a singlephase signal. Therefore, a periodical movement of each surface point of the driving ring along its radial direction is achieved. It should be pointed out that the two third order inplane bending vibrations of the two driving rings have a spatial phase difference of π. Due to the structural coupling between the sandwich composite beam and two driving rings, the sixth order bending vibration is induced in the sandwich composite beam, and another third order inplane bending vibration is produced in the driving ring along the tangential direction as well, as shown in Figure 2(b). Since the longitudinal vibration of the sandwich composite beam has two mechanical parameters (longitudinal force F and longitudinal velocity v), whereas the inplane bending vibration of the ring holds six mechanical parameters (radial force Q_{R}, tensile force N, bending moment M, radial displacement u, tangential displacement Ω, and rotational slope ψ), it cannot meet the rules of force balance and equal velocity at the connecting interface between the ring and beam. Therefore, it will be an unbalance state at the connection interface. In order to keep a balance state, another four mechanical parameters (shear velocity w, angular velocity \(\dot{\psi }\), bending moment M, and shear force Q) should be included in the sandwich composite beam. This means that the bending vibration is induced in the sandwich composite beam. Meanwhile, due to the existence of the four shear mechanical parameters (tensile force N, bending moment M, radial displacement u, tangential displacement Ω, and rotational slope ψ), the inplane tangential bending vibration is also induced in the driving rings. The two third order inplane bending vibrations produced in each driving ring along the radial and tangential directions, respectively, have a spatial phase difference of π/2, leading to the generation of traveling wave in each driving ring. For each surface point, its movements are occurred in the radial and tangential directions of the driving ring in the same time. This results in the generation of elliptical motion of the driving rings. As the two induced third order inplane bending vibrations of the two driving rings along the tangential direction have a spatial phase difference of π, the traveling waves produced in the two driving rings rotate the same direction. Therefore, surface points of the driving rings move in elliptical motion with the same direction, as shown in Figure 2(c).
Analytical Modelling
To further understand the operating principle and analyze the dynamic behavior of the proposed transducer, an analytical model is developed using the TMM in Section 3. Compared to other analytical solutions, the TMM minimizes the computational effort and provides a precise solution. Using the TMM, a complex system can be divided into many simple and continuous elements, such as beams, rods, or plates [20,21,22]. Then the independent transfer matrix model can be described for each element according to the basic wave motion equations and the arbitrary harmonic boundary conditions [23]. Through transfer conditions, the overall transfer matrix model of the entire system can be calculated by combining the independent transfer matrices of all discrete elements. Applying boundary conditions of the system, a linear equation set can be obtained and solved to describe the dynamic behavior of the complex system. For the PZT element, the mechanical and electrical properties can be simultaneously considered in its transfer matrix model [24], making the TMM suitable for the analysis of piezoelectric transducers.
The discrete model of the proposed sandwich piezoelectric transducer is shown in Figure 3. The transducer is divided into eleven elements, including two curved elastic beams, seven elastic beams, and two sandwich beams. The sandwich beam is composed of an elastic beam and two groups of PZT elements located on the upper and lower surfaces of the elastic beam, respectively. As the four groups of PZT elements are excited to vibrate at the longitudinal vibration, the seven elastic beams and two sandwich beams are subjected to this vibration and the bending vibration induced by the structural coupling of the sandwich composite beam and driving ring. Therefore, the longitudinal and bending vibration transfer matrix models should be created for these beam elements. As the two curved beams are excited at the inplane bending vibration, the corresponding transfer matrix model should be carried out. In the following sections, the aforementioned transfer matrix model of each discrete element will be given and explained in detail.
Longitudinal Vibration Modelling
Longitudinal vibration is the major vibration stimulated in the sandwich composite beam, therefore, the transfer matrix model of each discrete element in this beam must first be developed. In Refs. [23, 25, 26], the longitudinal vibration transfer matrix model of the elastic beam has been presented as
where v_{il} and v_{ir} are the left and right velocities of the elastic beam i, respectively; F_{il} and F_{ir} are the left and right forces of the elastic beam i, respectively, and T_{iL} is the longitudinal vibration transfer matrix with following components: \(T_{iL,11} = \,\cos (k_{i} l_{i} )\), \(T_{iL,12} = \frac{j}{{z_{i} }}\sin (k_{i} l_{i} )\), \(T_{iL,21} = jz_{i} \sin (k_{i} l_{i} )\), and \(T_{iL,22} = \cos (k_{i} l_{i} )\), in which \(z_{i} = A_{i} \sqrt {\rho_{i} E_{i} }\) is the impedance, k_{i} = ω/c_{i} is the wave number; \(c_{i} = \sqrt {E_{i} /\rho_{i} }\) is the wave velocity, ω is the angular frequency; A_{i} is the crosssection area; E_{i} and ρ_{i} are the Young’s modulus and density of the elastic beam i, respectively; and l_{i} is the length of the elastic beam i (The subscript i is the number of the discrete element, in which i \(\in\)[1 11] (i.e., i belongs to the interval 1 to 11)).
Due to the fact that each group of PZT elements operates at their d_{33} vibration mode to produce the longitudinal vibration, its transfer matrix model (see in Figure 4) must simultaneously include the mechanical and electrical coupling properties [23,24,25,26], as written:
where ()_{pl} and ()_{pr} are the parameters (force and velocity) of the two ends of the group of PZT elements, respectively; U and I are the input voltage and induced current of the group of PZT elements, respectively, and T_{pL} denotes the transfer matrix of the group of PZT elements operating in the d_{33} vibration mode, in which its matrix components are expressed as follows: \(T_{pL,11} = \cos (k_{p} l_{p} )\), \(T_{pL,12} = \frac{j}{{A_{p} z_{p} }}\sin (k_{p} l_{p} )\), \(T_{pL,13} = \frac{{j\alpha_{p} }}{{A_{p} z_{p} }}\sin (k_{p} l_{p} )\), \(T_{pL,21} = jA_{p} z_{p} \sin (k_{p} l_{p} )\), \(T_{pL,22} = T_{pL,11}\), \(T_{pL,23} = \alpha_{p} (\cos (k_{p} l_{p} )  1)\), \(T_{pL,31} = T_{pL,23}\), \(T_{{pL,{32}}} = T_{{pL,1{3}}}\), and \(T_{{pL,{33}}} = j\omega C_{0p} + \frac{{j\alpha_{p}^{2} }}{{A_{p} z_{p} }}\sin (k_{p} l_{p} )\), where k_{p}=ω/c_{i} is the wave number, \(z_{p} = 1/(c_{p} s_{33}^{E} )\) the impedance, \(c_{p} = 1/\sqrt {\rho_{p} s_{33}^{E} }\) the wave velocity, \(C_{0p} = \frac{{4A_{p} }}{{l_{p} }}(\varepsilon_{33}^{T}  \frac{{d_{33}^{2} }}{{s_{33}^{E} }})\), and \(\alpha_{p} = \frac{{2A_{p} d_{33} }}{{s_{33}^{E} l_{p} }}\) the electromechanical translation factor of the PZT element, in which \(s_{33}^{E}\) represents the compliance parameter, \(\varepsilon_{33}^{E}\) denotes the dielectric coefficients, d_{33} denotes the piezoelectric strain coefficients, ρ_{p} denotes the density, l_{p} is the total length of a group of PZT elements, and A_{p} is the crosssection area of the PZT element.
For the sandwich beam, its transfer matrix model in longitudinal vibration can be described by combining the transfer matrices of the elastic beam and the two group of PZT elements and considering the boundary conditions, as shown in Figure 5. In Ref. [26], the coupling longitudinal vibration transfer matrix model was developed and expressed as:
where ()_{sl} and ()_{sr} are the parameters (force and velocity) of the left and right ends of the sandwich beam, respectively; U_{sL} and I_{sL} are the input voltage and induced current of the sandwich beam, respectively; T_{sL} is the longitudinal vibration transfer matrix of the sandwich beam and the detailed expression of its matrix components can be found in Appendix A.
Bending Vibration Modelling
Due to the structural coupling of the sandwich composite beam and driving rings, all discrete elements in the sandwich composite beam are induced to present bending vibration. According to the geometrical property of the elastic beam, its bending vibration can be formulated using the Timoshenko beam theory by considering the rotary inertia and shear deformation, which has been reported by Pestel et al. [20], written as
where shear velocity w, angular velocity \(\dot{\psi }\), bending moment M, and shear force Q are the mechanical parameters to describe the bending vibration of the elastic beam, and T_{iB} is the bending vibration transfer matrix of the elastic beam, in which its matrix components can be found in the Appendix B.
For the sandwich beam, the two groups of PZT elements are subjected with the same bending vibration as elastic beams, meaning that the electric parameters do not involve in the bending vibration. Therefore, the expression of the bending vibration transfer matrix model of the sandwich beam should be described the same as that of the elastic beam:
where T_{sB} is the bending vibration transfer matrix of the sandwich beam with the substitutions of \(\sigma_{s} = \frac{{\mu_{s} \omega^{2} }}{{(GA)_{s} }}l_{p}^{2}\), \(\tau_{s} = \frac{{\left( {\rho I} \right)_{s} \omega^{2} }}{{(EI)_{s} }}l_{p}^{2}\), \(\beta_{s}^{4} = \frac{{\mu_{s} \omega^{2} }}{{(EI)_{s} }}l_{p}^{4}\), \(a_{s} = \frac{{l_{p}^{2} }}{{\left( {EI} \right)_{s} }}\), and \(\left\{ {\begin{array}{*{20}c} {(GA)_{s} = \kappa^{2} W\left( {h_{e} G_{e} + \frac{{2h_{p} }}{{s_{55}^{E} }}} \right)} \\ {(EI)_{s} = \frac{{h_{e}^{3} WE_{e} }}{12} + \frac{2W}{{3s_{33}^{E} }}\left( {\left( {\frac{{h_{e} }}{2} + h_{p} } \right)^{3}  \left( {\frac{{h_{e} }}{2}} \right)^{3} } \right)} \\ {(\rho I)_{s} = \frac{{\rho_{e} Wh_{e}^{3} }}{12} + \frac{2}{3}\rho_{p} W\left( {\left( {\frac{{h_{e} }}{2} + h_{p} } \right)^{3}  \left( {\frac{{h_{e} }}{2}} \right)^{3} } \right)} \\ {\mu_{s} = \rho_{e} Wh_{e} + 2\rho_{p} Wh_{p} } \\ \end{array} } \right.\), in which e is the element number 5 or 7, h_{p} is the height of the PZT element, and \(s_{55}^{E}\) is the shear compliance parameter of the PZT element.
The curved beams are excited to vibrate at the third order inplane bending vibration. Therefore, their transfer matrix models should consider six mechanical parameters, including radial force Q_{R}, tensile force N, bending moment M, radial displacement u, tangential displacement Ω, and rotational slope ψ. By solving the governing equations of the inplane bending vibration, Irie et al. [27] calculated the coefficient matrix H, written as:
where c is the curved beam number 1 or 11; R = (R_{1}+R_{2})/2 is the radius in the neutral axis of the curved beam while its outer and inner radii are denoted as R_{1} and R_{2}, respectively; the quantities k^{2}, \(k_{1}^{2}\) and \(k_{2}^{2}\) are the dimensionless parameters, and \(n = \rho_{c} A_{c} R\omega^{2}\).
Therefore, the transfer matrix model of the curved beam with the inplane bending vibration can be formulated as:
where Z_{a} = [u_{a} Q_{Ra} Ω_{a} ψ_{a} M_{a} N_{a}]^{T} and Z_{b} = [u_{b} Q_{Rb} Ω_{b} ψ_{b} M_{b} N_{b}]^{T} are the state vectors of both ends of the curved beam, respectively, and T_{c}(θ) = exp(Hθ) is the transfer matrix with the dimension of 6×6 as the curved beam is with the angle θ.
The state vector of the curved beam includes the force and displacement parameters, while that of its neighbouring elastic beam contains the force and velocity parameters. Therefore, to achieve the matching of the state vectors between the elastic beam and curved beam, the displacement parameters of the curved beam should be converted as the velocity parameters according to the formulation of u = jωv. The new expression of the transfer matrix model of the curved beam can be written as
where \(\varvec{Z}{^{\prime}_{{\varvec{a}}}} = [v_{a} \quad Q_{Ra} \quad w_{a} \quad \dot{\psi }_{a} \quad M_{a} \quad N_{a} ]^{{\text{T}}}\) and \(\varvec{Z}{^{\prime}_{{\varvec{b}}}} = [v_{b} \quad Q_{Rb} \quad w_{b} \quad \dot{\psi }_{b} \quad M_{b} \quad N_{b} ]^{{\text{T}}}\) are the input and output state vectors of the curved beam containing the force and velocity parameters, and \(\varvec{T}{^{\prime}_{{\varvec{c}}}} \left( {\varvec{\theta}} \right)\) is the new expression of the inplane beading vibration transfer matrix of the curved beam.
Damping Characteristic
The damping effect of the real material has a substantial influence on the output performance of the proposed transducer and should therefore be considered in the transfer matrix model developed here. As reported in Ref. [28], the damping effect can be described by the complex Young’s modulus \(\underline {E}_{i} = E_{i} (1 + j\frac{1}{{\zeta_{i} }})\) for general materials, and the complex compliance coefficient \(\underline {S}_{33}^{E} = S_{33}^{E} /(1 + j\frac{1}{{\zeta_{p} }})\) for the PZT element, in which ζ_{i} and ζ_{p} are the quality factors of the general material and the PZT element, respectively. Due to the specific material properties of the PZT element, its damping effect still includes the dielectric and piezoelectric losses. These two losses can be presented by a complex dielectric constant \(\underline {\varepsilon }_{33}^{T}\) and a complex piezoelectric constant \(\underline {d}_{33}\), as formulated
with the dissipation factors tan(δ) and tan(η) for the dielectric loss and piezoelectric loss, respectively.
Transfer Matrix Model of Combined Vibrations
As the longitudinal and bending vibrations are simultaneously generated in the sandwich composite beam, a total transfer matrix model used to describe these two vibrations together should be developed to simplify the final analytical model. The combined transfer matrix model for the elastic beam can be formulated as
where \({\varvec{Z}}_{{{\varvec{il}}}} = \left[ {\begin{array}{*{20}c} {{v_{il}} } & {{F}_{{{il}}} } & {{w}_{{{il}}} } & {{\dot{\psi }}_{{{il}}} } & {{M}_{{{il}}} } & {{Q}_{{{il}}} } \\ \end{array} } \right]^{{\text{T}}}\) and \({\varvec{Z}}_{{{\varvec{ir}}}} = \left[ {\begin{array}{*{20}c} {{{v}}_{{{ir}}} } & {{F}_{{{ir}}} } & {{w}_{{{ir}}} } & {{\dot{\psi }}_{{{ir}}} } & {{M}_{{{ir}}} } & {{Q}_{{{ir}}} } \\ \end{array} } \right]^{{\text{T}}}\) are the input and output state vectors of the elastic beam i, including all mechanical parameters for describing the longitudinal and bending vibrations, respectively, and T_{i} is the combined transfer matrix with the dimension of 6 × 6 for the elastic beam i.
Combining Eqs. (3) and (5), the total transfer matrix model of the sandwich beam is expressed as
where \({\varvec{Z}}_{{{\varvec{sl}}}} { = }\left[ {\begin{array}{*{20}c} {{{v}}_{{{{sl}}}} } & {{{F}}_{{{{sl}}}} } & {{{U}}_{{{{sL}}}} } & {{{w}}_{{{{sl}}}} } & {{\dot{\psi }}_{{{{sl}}}} } & {{{M}}_{{{{sl}}}} } & {{{Q}}_{{{{sl}}}} } \\ \end{array} } \right]^{{{T}}}\) and \({\varvec{Z}}_{{{\varvec{sr}}}} { = }\left[ {\begin{array}{*{20}c} {{{v}}_{{{{sr}}}} } & {{{F}}_{{{{sr}}}} } & {{{I}}_{{{{sL}}}} } & {{{w}}_{{{{sr}}}} } & {{\dot{\psi }}_{{{{sr}}}} } & {{{M}}_{{{{sr}}}} } & {{{Q}}_{{{{sr}}}} } \\ \end{array} } \right]^{{{T}}}\) are input and output state vectors of the sandwich beam containing mechanical and electrical parameters of the two vibrations, respectively, and T_{s} is the total transfer matrix with the dimension of 7 × 7 for the sandwich beam.
Transfer Conditions
In order to combine all the discrete elements, transfer conditions between two neighbouring elements should be analyzed by following the rules of force balance and equal velocity at the connecting interface. In our proposed transducer, five kinds of transfer conditions exist. The first transfer condition is defined that the vibration is transferred from the left curved beam 1 to the neighbouring elastic beam 2, which can be formulated as
where C_{1}, C_{2}, and C_{3} are the transfer condition matrices with the dimension of 6×6; Z_{1}=[v_{1} Q_{R1} w_{1} \(\dot{\psi }_{1}\) M_{1} N_{1}]^{T} and Z_{2}=[v_{2} Q_{R2} w_{2} \(\dot{\psi }_{2}\) M_{2} N_{2}]^{T} are the input and output state vectors of the curved beam 1, respectively; \({\varvec{Z}}_{{\user2{3}}} = \left[ {\begin{array}{*{20}c} {v_{3} } & {F_{3} } & {w_{3} } & {\dot{\psi }_{3} } & {M_{3} } & {Q_{3} } \\ \end{array} } \right]^{{\text{T}}}\) is the input state vector of the elastic beam 2.
The second transfer condition is used to describe the connection between two neighbouring elastic beams i and i+1, which can be written as:
where the transfer condition matrix C_{u} is a unit matrix with the dimension of 6×6, \({\varvec{Z}}_{{{\user2{2i}}}} = \left[ {\begin{array}{*{20}c} {v_{2i} } & {F_{2i} } & {w_{2i} } & {\dot{\psi }_{2i} } & {M_{2i} } & {Q_{2i} } \\ \end{array} } \right]^{{\text{T}}}\) and \({\varvec{Z}}_{{{\user2{2i}} + {\user2{1}}}} = \left[ {\begin{array}{*{20}c} {v_{2i + 1} } & {F_{2i + 1} } & {w_{2i + 1} } & {\dot{\psi }_{2i + 1} } & {M_{2i + 1} } & {Q_{2i + 1} } \\ \end{array} } \right]^{{\text{T}}}\) are the output and input state vectors of the elastic beam i and i+1, respectively.
The third kind of transfer condition is that the transmission of the vibration is from the elastic beam i to the sandwich beam. Due to the unequal dimension of their state vectors, this transfer condition should be formulated as:
where C_{4} is the transfer condition matrix with the dimension of 7×7.
The fourth kind of transfer condition is defined as the vibration that is transferred from the sandwich beam to the neighboring elastic beam i. In this case, the transfer condition should be described as
where \({\varvec{Z}}_{{{\user2{2i}}  1}} = \left[ {\begin{array}{*{20}c} {v_{2i  1} } & {F_{2i  1} } & {w_{2i  1} } & {\dot{\psi }_{2i  1} } & {M_{2i  1} } & {Q_{2i  1} } \\ \end{array} } \right]^{\text{T}}\) is the input state vector of the elastic beam i.
The last transfer condition is that the vibration transfers from the elastic beam 10 to the right curved beam, which is written as:
where C_{5} and C_{6} are the transfer condition matrices with the dimension of 6×6, \({\varvec{Z}}_{{{\user2{20}}}} = \left[ {\begin{array}{*{20}c} {{v}_{{{20}}} } & {{F}_{{{20}}} } & {{w}_{{{20}}} } & {{\dot{\psi }}_{{{20}}} } & {M}_{{{20}} } & {{Q}_{{{20}}} } \\ \end{array} } \right]^{\text{T}}\) is the output state vector of the elastic beam 10, Z_{21} = [v_{21} Q_{R21} w_{21} \(\dot{\psi }_{21}\) M_{21} N_{21}]^{T} and Z_{22} = [v_{22} Q_{R22} w_{22} \(\dot{\psi }_{22}\) M_{22} N_{22}]^{T} are the input and output state vectors of the curved beam 11, respectively.
Using the aforementioned five kinds of transfer conditions, all the discrete elements can be physically combined together.
Overall Transfer Matrix Model
Combining all independent transfer matrices of the divided elements, transfer conditions, and boundary conditions, the overall transfer matrix model of the proposed transducer is formulated as Eq. (18).
In this overall transfer matrix model, S_{m} is the systematic matrix, including total physical transfer matrix T_{pm}, total transfer condition matrix C_{m}, and total boundary condition extraction matrix B_{m}, Z_{m} is the state vector containing all input and output parameters, and A_{m} is the total electrical excitation conditions, as the entire transducer is a closed system without any mechanical boundary conditions. In the total boundary condition extraction matrix B_{m}, the matrix B with the dimension of 1×7 is used to extract the excitation voltage U_{sL} applied to the sandwich beam. Therefore, the corresponding value of the extracted excitation voltage is expressed by the matrix A with the dimension of 1×1. It can be noted that the total transfer matrix model the proposed transducer is a linear equation set, which can be easily solved using the Matlab software. Thus, the dynamic behavior of the transducer can be calculated by solving the developed transfer matrix model.
Experimental Validation
To validate the effectiveness of the developed transfer matrix model and to confirm the feasibility of the design and operating principle of the proposed transducer, we first manufactured and assembled the prototype of the transducer and then conducted vibration measurements. The prototype of the proposed transducer with a weight of 225 g is depicted in Figure 6, and its geometrical sizes are listed in Table 1. All elastic beams and curved elastic beams were made of stainless steel, and the PZT element was made of PIC181 type piezoelectric ceramic (PI Company, Germany). The material parameters of the transducer prototype are listed in Table 2.
An experimental system was built up to measure threedimensional vibration characteristics of the transducer prototype, as shown in Figure 7. The major component of the experimental system is a 3D laser Doppler vibrometer (MSA1003D, Polytec, Germany), which is composed of a control system, a XY positioning stage, and a laser head. The control system is used to analyze and process the measured data. The laser head is adopted to emit three laser beams, and it can move up and down in the Z direction. Additionally, the transducer prototype was placed on the XY positioning stage, and a foam was located between the transducer prototype and the XY positioning stage in order to keep the free boundary conditions of the transducer prototype. Therefore, the experimental system enabled us to measure the threedimensional vibration characteristics of the transducer prototype.
Frequency Response Characteristic
The frequency response characteristic of the transducer prototype was initially measured to obtain the resonant frequency of the operating vibration mode. The measured frequency swept from 19 kHz to 20 kHz with an interval of 1 Hz while the excitation voltage was 1 V. The connection position between the sandwich composite beam and left driving ring was tested to obtain the frequency response characteristic of the transducer prototype, and the result was plotted in Figure 8, leaving a resonant frequency of 19392 Hz. Under the same excitation conditions, the frequency response characteristic was calculated by the developed transfer matrix model (Eq. (18)), as also illustrated in Figure 8. The amplitudes (u_{left} and u_{right}) of both ends of the sandwich composite beam were computed to exhibit the frequency response characteristic of the transducer. It can be noted that the calculated resonant frequency of the operating vibration mode is 19821 Hz for the transducer, showing a frequency difference of 429 Hz compared to the experimental results. The difference between the calculated and measured resonant frequencies is mainly due to the fact that our developed analytical model does not consider the influence of prestress applied on the PZT elements during assembly. The comparison still demonstrated the feasibility of the created transfer matrix model and confirmed the validation of the operating principle of the proposed transducer.
Vibration Shapes
At the measured resonant frequency of 19392 Hz, the vibration shapes of the left driving ring and the sandwich composite beam were measured with the excitation voltage of 2 V using the aforementioned experimental system, as illustrated in Figure 9. To further validate the developed transfer matrix model, the vibration shapes of the transducer were computed and compared to the experimental results in Figure 9. The calculated vibration shapes were computed at the calculated resonant frequency of 19821 Hz. The measured and computed first order longitudinal vibration shapes of the sandwich composite beam presented in Figure 9(a) showed good agreement with each other. Due to the structural coupling between the sandwich composite beam and the left driving ring, the sixth order bending vibration was induced in the composited beam, as illustrated in Figure 9(b). It can be seen that the computed bending vibration shape matches well with the experimental result. In Figure 9(c), the radial and tangential displacements of each point in the left driving ring are calculated and exhibit the inplane bending vibration shapes of the left driving ring in the radial and tangential directions. The third order inplane bending vibration shapes in radial and tangential directions were observed, respectively, in which the 0° position is the connection place between the sandwich composite beam and left driving ring. The measured third order bending vibration shapes in tangential and radial directions matched well with the calculation results. The measured and calculated results indicated that the two third order inplane bending vibration shapes in tangential and radial directions have a spatial phase difference of π/2. This is the reason why the traveling wave can be generated in each driving ring at a singlephase excitation. Certainly, the maximum amplitude of the left driving ring in the radial direction is larger than that in the tangential direction, indicating that the standing wave ratio (SWR) is larger than traveling wave. Therefore, energy dissipation is inevitable as the metal track is driven by the transducer prototype. Comparisons between the experimental and simulated results adequately proved that the operating principle of the proposed transducer is feasible and the developed transfer matrix model is effective.
Investigation on the Driving Concept
Since the proposed transducer is excited to produce the traveling wave in each driving ring by a singlephase signal, therefore, to investigate its driving concept is very important for further understanding of the operating principle of the transducer. Two aspects can be used to describe the driving concept clearly. One is the motion trajectory of surface points of the driving ring, and another one is the quality of the traveling wave of the driving rings. In this section, the investigations on the driving concept of the transducer are carried out.
Motion Trajectory
At first, we experimentally and theoretically investigated the motion trajectory of surface points of the driving ring. Three points (points A, B, and C) on the central line of the left driving ring were selected to study their motion trajectories, as shown in Figure 10(a), in which the surface region from point A to point C is used to drive the metal track and point B is located on the middle of this region. The excitation conditions were the same as in the last section (The corresponding frequency was 19392 Hz for the test and 19821 Hz for the calculation, and the excitation voltage is 2 V), and the motion trajectories of the selected three points are measured and plotted in Figure 10(b). Due to the fact that the central line of the left driving ring was used to analyze and describe the vibration characteristic of the transducer in our developed analytical model, the motion trajectories of the selected points were computed and compared to the experimental results, as illustrated in Figure 10(b). The measured and computed motion trajectories shown that the selected points moved elliptically, demonstrating that the proposed transducer with a singlephase excitation can achieve the effective drive on the metal track. Since the induced vibration mode is used to couple and generate traveling waves in the driving rings, the not pure traveling wave is produced in each driving ring, leading to that the standing wave ratio of each driving ring is higher than 1. This results in the trajectory difference between the points A and C.
Quality of Traveling Wave
To study the influence of geometrical dimensions on the quality of the traveling wave in the driving rings, the modelbased investigation was conducted. The quality of the traveling wave in each driving ring directly reflects the driving effect of the transducer. To quantitatively investigate the quality of the traveling wave in the driving ring, the SWR is utilized as the figure of merit. This is because the vibration of the driving ring is composed of standing wave and traveling wave at the singlephase excitation. The SWR is defined as the maximum amplitude divided by the minimum amplitude. A pure traveling wave occurs when the SWR of the driving ring is equal to 1, while a pure standing wave exists as the SWR of the driving ring is infinite. Once the SWR of the driving ring is greater than 1, it indicates that the generated traveling wave is not pure. To calculate the SWRs of the driving ring in the radial and tangential directions, the whole driving ring is first divided into 360 elements and then amplitudes of all element interfaces in both directions are computed, respectively.
Due to the fact that the tangential displacement of surface points of the driving ring is induced by the structural coupling between the sandwich composite beam and driving ring, it means that the generated traveling wave would be disturbed by standing wave. To investigate the influence of geometrical dimensions on the SWR of the driving ring, we conducted calculations. At first, we studied the influence of the inner radius (R_{1}) and outer radius (R_{2}) of the driving ring individually. When R_{1} was varied from 15 mm to 16 mm in steps of 0.1 mm or R_{2} was changed from 19 mm to 20 mm in steps of 0.1 mm, the minimum SWR of the left driving ring is more than 100 in both directions, indicating that these two geometrical dimensions have a small influence on the SWR. Then we investigated the influence of the length (l_{2}) of the elastic beam element connecting with the driving ring. The computed SWRs of the left driving ring in the radial and tangential directions are plotted in Figure 11(a), when l_{2} varied from 20 mm to 50 mm with a step of 0.5 mm. It can be observed that a specific length range existed as l_{2} varied from 33 mm to 40 mm, due to the fact that the SWRs of the left driving ring in both directions at this length range are relatively smaller than other lengths. The minimum SWRs for the radial and tangential directions are 2.282 and 1.914, respectively, both corresponding to the value of l_{2} of 33.5 mm, meaning that the quality of the traveling wave in the driving ring is highest at this length. It also can be seen that the resonant frequency of the operating vibration mode of the transducer linearly decreased with the increase of l_{2}. Selected the length of l_{2} of 33.5 mm and its corresponding resonant frequency of 19490 Hz as the driving frequency, the amplitudes of the left driving ring in both directions are computed and plotted in Figure 11(b). It can be found that the amplitude of the driving ring in the radial direction is larger than that in the tangential direction. This indicates that the output force of the metal track driven by the transducer is better than its output velocity. In a word, the length of l_{2} can be used as the key geometrical dimension to improve the quality of the traveling wave in the driving rings under a singlephase excitation.
Application Example Investigation
To verify the driving effect of the proposed traveling wave sandwich piezoelectric transducer, an application example is presented in this section. In Refs. [29, 30], tracked mobile systems driven by traveling wave piezoelectric transducers with two phases excitation signals have been investigated. A metal track is adopted as the rotor to be driven by a piezoelectric transducer via friction. Compared to traditional driving forms utilizing electromagnetic motors and retarding mechanisms, the piezoelectric actuated method greatly simplifies the configuration of the tracked mechanism, eliminating the problem of lubricant volatilization and deflation within operating in vacuum environments, and is free from electromagnetic interference. Therefore, the approach utilizing piezoelectric transducers to drive track presents potential application for planetary explorations. Referring to our previous designs, the application example of this proposed transducer is still to build a tracked mobile system, as shown in Figure 12. A metal track with smooth inner surface is tensioned along the outer surfaces of the two driving rings of the proposed traveling wave sandwich piezoelectric transducer.
The prototype of the tracked mobile system with the weight of 395 g was assembled, as shown in Figure 13. The dimensions of each track block are 2.76 mm in height, 13 mm in width, and 7.52 mm in length. In this section, we conducted experimental investigations on the tracked mobile system prototype to verify the driving performance of the transducer. To measure motion characteristics of the tracked mobile system, an experimental setup was built, as shown in Figure 14. It includes a power amplifier, a signal generator, an oscilloscope, and a current probe. The prototype of the tracked mobile system with a singlephase excitation moved on a vibration isolation platform, and its motion situations were recorded by a camera. Thus, the motion characteristics of the assembled tracked mobile system were obtained by evaluating the videos.
Frequency Sensitivity Measurement
Although the resonant frequency of the operating principle of the transducer prototype has been determined, the assembly of the metal track changes its boundary conditions. Therefore, we conducted the frequency sensitivity measurement to find an optimal driving frequency with the maximum mean velocity for the tracked mobile system prototype. The experimental results are plotted in Figure 15. The excitation voltage applied to the system prototype was 300 V_{RMS}. The driving frequency varied from 19360 Hz to 19460 Hz with a step of 10 Hz. The results showed that the mean velocity of the system prototype first increased and then decreased when the driving frequency increased, leaving an optimal driving frequency of 19410 Hz corresponding to the maximum mean velocity of the system prototype of 45.6 mm/s. Compared to the measured resonant frequency of the transducer prototype in the last section, the obtained driving frequency of the assembled system was shifted by 18 Hz. This indicates that the installation of the metal track changed the resonant frequency of the transducer prototype, due to the fact that its vibration property was primarily influenced by the contact stiffness between the transducer and the metal track.
Motion Characteristic
Based on the measured optimal driving frequency of 19410 Hz, motion characteristics of the system prototype were tested under different voltages by using the aforementioned experimental setup (Figure 14), as illustrated in Figure 16. The motion characteristic curve showed that the mean velocity of the system prototype rose with the increase of the excitation voltage. It can be found that a specific mean velocity of the system prototype occurred in the excitation voltage of 250 V_{RMS}, making that the motion characteristic curve varied nonlinear. This is caused by the accumulation of experimental errors due to the fact that each measured mean velocity was calculated in terms of the experimental results of 5 times. The maximum mean velocity of 59 mm/s was obtained when the excitation voltage was 500 V_{RMS}. The results demonstrated that the transducer prototype can drive the metal track while it was excited with a singlephase voltage varied from 150 V_{RMS} to 500 V_{RMS}.
Traction Performance
The traction force is generally adopted as an important index for evaluating the performance of the mobile system. For our assembled tracked mobile system, we built an experimental setup to test its traction performance, as illustrated in Figure 17(a). To avoid slide phenomenon during measurements of the assembled system, the surface of the vibration isolation platform was covered with abrasive papers with the roughness of P100 (Germany standard). The measured optimal frequency of 19410 Hz was still used as the driving frequency for traction performance measurements of the prototype.
First, the maximum stalling traction performance was measured under different excitation voltages, and the results were plotted in Figure 17(b). The maximum stalling traction force increased linearly when the excitation voltage increased. When the voltage was 500 V_{RMS}, the maximum stalling traction force reached up to 1.65 N. For real applications, the traction performance of the tracked mobile system should be improved.
Additionally, the loading characteristic of the system prototype was measured using the same experimental setup when the excitation voltage was 300 V_{RMS}, as shown in Figure 17(c). With the loading weight increased, the mean velocity of the system prototype decreased. At each loading weight, the mean velocity of the system prototype was calculated according to the experimental results of 5 times. These results preliminarily present the noteworthy driving performances of the proposed traveling wave sandwich piezoelectric transducer with a singlephase excitation.
Comparisons
The measured performances of the single phase excited tracked mobile system are compared to some developed traveling wave tracked mobile systems under the two excitation voltages, as listed in Table 3. The maximum mean velocity and loading weight of the tracked mobile system driven by a surfacebonded type piezoelectric transducer with four driving rings [29] are 57 mm/s and 240 g under two excitation voltages of 460 V_{RMS}, respectively. The tracked mobile system driven by a sandwich piezoelectric transducer with two driving rings [30] achieved a maximum mean velocity of 72 mm/s under two excitation voltages of 500 V_{RMS} and a maximum loading weight of 1440 g under two excitation voltages of 300 V_{RMS}. Although the maximum mean velocity of the proposed tracked mobile system is a little lower than the aforementioned two tracked mobile systems, it only required a singlephase excitation voltage, simplifying the control system. It should be pointed out that the loading weight for the tracked mobile system in Refs. [29, 30] indicates the actual loading while our measured loading weight in the last section represents the traction force for the assembled system prototype, thus, the comparison is not given for these three tracked mobile systems.
Conclusions

(1)
A novel traveling wave sandwich piezoelectric transducer with single phase excitation was proposed, theoretically analyzed, experimentally validated and investigated in this study. Using the structural coupling property between the sandwich composite beam and driving rings, the proposed piezoelectric transducer can produce traveling waves in the driving rings under a singlephase excitation signal, greatly simplifying the control system.

(2)
An overall transfer matrix model was first created to describe the dynamic behavior of the proposed traveling wave sandwich piezoelectric transducer.

(3)
Then the vibration measurements were carried out on the transducer prototype and compared to the calculation results. The comparisons showed that the computed vibration characteristics of the transducer were in good agreement with the measured results, demonstrating the effectiveness of the developed transfer matrix model and confirming the operating principle of the proposed transducer.

(4)
Additionally, the elliptical trajectories of the surface points of the driving ring were obtained, and the influence of geometrical dimensions on the quality of traveling wave of the driving ring was investigated, showing that the computed minimum SWRs of the driving ring in the radial and tangential directions are 2.282 and 1.914, respectively, at the value of l_{2} of 33.5 mm.

(5)
Finally, a case study on evaluating the driving performance of the proposed traveling wave sandwich piezoelectric transducer was carried out by constructing and assembling a tracked mobile system. The frequency sensitivity results indicated that the driving frequency of the system prototype was 19410 Hz corresponding to the maximum mean velocity and shifting 18 Hz compared to the measured resonant frequency of the transducer prototype. The maximum mean velocity and maximum stalling traction force of the system prototype were respectively 59 mm/s and 1.65 N when the excitation voltage was 500 V_{RMS}. The mean velocity of the system prototype decreased when the loading weight increased. These experimental results demonstrated that the driving effect of the proposed traveling wave sandwich piezoelectric transducer with a singlephase excitation deserves attention.
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Acknowledgements
Not applicable.
Funding
Supported by the National Science Foundation of China (Grants Nos. 51905262 and U2037603), the Natural Science Foundation of Jiangsu Province (Grant No. BK20190398), and the State Key Laboratory of Mechanical System and Vibration (Grant No. MSV202011).
Author information
Affiliations
Contributions
LW proposed the novel design of the piezoelectric transducer and developed its dynamical analytical model. FB guided in analyzing the operating principle of the piezoelectric transducer. VH conducted the dynamical analytical modeling of the piezoelectric transducer and experimental verification. LW and JJ carried out the experimental investigation on the case study of assembled tracked mobile system. LW and JT were in charge of the whole structure of the manuscript. All authors contributed to the preparation of the manuscript. All authors read and approved the final manuscript.
Authors’ Informations
Liang Wang is currently a lecturer in Mechanical Engineering at Nanjing University of Aeronautics and Astronautics, China. He received his doctoral degree from Nanjing University of Aeronautics and Astronautics, China, in 2018. His research interest includes piezoelectric actuators, mobile robotic systems, and ultrasonic motors.
Fushi Bai is currently an associate professor in Marine Science and Technology at Northwestern Polytechnical University, China. He received his doctoral degree in mechanical engineering from Leibniz University Hannover, Germany, in 2019. He mainly focuses on ultrasonic cavitation and peening process.
Viktor Hofmann received the Diploma degree in mechanical engineering from Leibniz University Hannover, Germany, in 2013. He is currently working toward the Ph.D. degree in the Institute of Dynamics and Vibration Research, Leibniz University Hannover, Germany, researching in piezoelectric actuators, and tactile display.
Jiamei Jin is currently a Professor with the State Key Laboratory of Mechanics and Control of Mechanical Structures at Nanjing University of Aeronautics and Astronautics, China. He received the Ph.D. degree in mechatronics engineering from Nanjing University of Aeronautics and Astronautics, China, in 2007. His research interests include piezoelectric actuators, ultrasonic motors, ultrasonic machining, and robotics.
Jens Twiefel received the Diploma degree in mechanical engineering from University of Paderborn, Germany, in 2004, and the Ph.D. degree in mechanical engineering from the Institute of Dynamics and Vibration Research, Leibniz University Hannover, Germany, in 2010. From 2004 to 2007, he was a research assistant at Mechatronics and Dynamics of Heinz Nixdorf Institute, University of Paderborn. Since 2007, he is a research assistant in the Institute of Dynamics and Vibration Research, Leibniz University Hannover, Germany, in which, he is promoted as the head of Piezoelectric and Ultrasonic Technology Research Group in the Institute of Dynamics and Vibration Research since 2008. His research interests include piezoelectric transducers, ultrasonic motors, ultrasonic machining, and vibration energy harvesting.
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Appendices
Appendices
Appendix A
The components of the longitudinal vibration transfer matrix T_{sL} of the sandwich beam are expressed as follows: \(T_{sL,11} = \frac{{T_{pL,12} T_{iL,11} + 2T_{iL,12} T_{pL,11} }}{{2T_{iL,12} + T_{pL,12} }}\), \(T_{sL,12} = \frac{{T_{iL,12} T_{pL,12} }}{{2T_{iL,12} + T_{pL,12} }}\), \(T_{sL,13} = \frac{{2T_{iL,12} T_{pL,13} }}{{2T_{iL,12} + T_{pL,12} }}\), \(T_{sL,21} = 2T_{pL,21} + T_{iL,21} + \frac{{2(T_{pL,22}  T_{iL,22} )\left( {T_{iL,11}  T_{pL,11} } \right)}}{{2T_{iL,12} + T_{pL,12} }}\), \(T_{sL,22} = \frac{{T_{pL,12} T_{iL,22} + 2T_{iL,12} T_{pL,22} }}{{2T_{iL,12} + T_{pL,12} }}\), \(T_{sL,23} = 2T_{pL,23} + \frac{{2T_{pL,13} (T_{iL,22}  T_{pL,22} )}}{{2T_{iL,12} + T_{pL,12} }}\), \(T_{sL,31} = 2T_{pL,31} + 2T_{pL,32} \frac{{T_{iL,11}  T_{pL,11} }}{{2T_{iL,12} + T_{pL,12} }}\), \(T_{sL,32} = \frac{{2T_{iL,12} T_{pL,32} }}{{2T_{iL,12} + T_{pL,12} }}\), and \(T_{sL,33} = 2T_{pL,33}  \frac{{2T_{pL,13} T_{pL,32} }}{{2T_{iL,12} + T_{pL,12} }}\).
Appendix B
The components of the bending vibration transfer matrix T_{iB} of the elastic beam i are given as follows: \(T_{iB,11} = c_{0i}  \sigma_{i} c_{2i}\), \(T_{iB,12} =  l_{i} \left[ {c_{1i}  (\sigma_{i} + \tau_{i} )c_{3i} } \right]\), \(T_{iB,13} =  j\omega a_{i} c_{2i}\), \(T_{iB,14} = \frac{{  j\omega a_{i} l_{i} \left[ {(\beta_{i}^{4} + \sigma_{i}^{2} )c_{3i}  \sigma_{i} c_{1i} } \right]}}{{\beta_{i}^{4} }}\), \(T_{iB,21} =  \frac{{\beta_{i}^{4} c_{3i} }}{{l_{i} }}\), \(T_{iB,22} = c_{0i}  \tau_{i} c_{2i}\), \(T_{iB,23} = \frac{{j\omega a_{i} (c_{1i}  \tau_{i} c_{3i} )}}{{l_{i} }}\), \(T_{iB,24} =  T_{ib,13}\), \(T_{iB,31} =  \frac{{\beta_{i}^{4} c_{2i} }}{{j\omega a_{i} }}\), \(T_{iB,32} = \frac{{l_{i} \left[ {(\beta_{i}^{4} + \tau_{i}^{2} )c_{3i}  \tau_{i} c_{1i} } \right]}}{{j\omega a_{i} }}\), \(T_{iB,33} = T_{iB,22}\), \(T_{iB,34} =  T_{iB,12}\), \(T_{iB,41} =  \frac{{\beta_{i}^{4} (c_{1i}  \sigma_{i} c_{3i} )}}{{j\omega a_{i} l_{i} }}\), \(T_{iB,42} =  T_{iB,31}\), \(T_{iB,43} =  T_{iB,21}\), and \(T_{iB,44} = T_{iB,11}\), in which \(\left\{ {\begin{array}{*{20}l} {c_{0i} = \frac{{\lambda_{2i}^{2} \cosh \lambda_{1i} + \lambda_{1i}^{2} \cos \lambda_{2i} }}{{\lambda_{1i}^{2} + \lambda_{2i}^{2} }}} \\ {c_{1i} = \frac{{\frac{{\lambda_{2i}^{2} }}{{\lambda_{1i} }}\sinh \lambda_{1i} + \frac{{\lambda_{1i}^{2} }}{{\lambda_{2i} }}\sin \lambda_{2i} }}{{\lambda_{i1}^{2} + \lambda_{i2}^{2} }}} \\ {c_{2i} = \frac{{\cosh \lambda_{1i}  \cos \lambda_{2i} }}{{\lambda_{1i}^{2} + \lambda_{2i}^{2} }}} \\ {c_{3i} = \frac{{\frac{1}{{\lambda_{1i} }}\sinh \lambda_{1i}  \frac{1}{{\lambda_{2i} }}\sin \lambda_{2i} }}{{\lambda_{1i}^{2} + \lambda_{2i}^{2} }}} \\ {\lambda_{1i,2i} = \sqrt {\sqrt {\beta_{i}^{4} + \frac{{(\sigma_{i}  \tau_{i} )^{2} }}{4}} \mp \frac{{\sigma_{i} + \tau_{i} }}{2}} } \\ \end{array} } \right.\) with \(\sigma_{i} = \frac{{\mu_{i} \omega^{2} }}{{(GA)_{i} }}l_{i}^{2}\), \(\tau_{i} = \frac{{\left( {\rho I} \right)_{i} \omega^{2} }}{{(EI)_{i} }}l_{i}^{2}\), \(\beta_{i}^{4} = \frac{{\mu_{i} \omega^{2} }}{{(EI)_{i} }}l_{i}^{4}\), and \(a_{i} = \frac{{l_{i}^{2} }}{{\left( {EI} \right)_{i} }}\), where \((GA)_{i} = \kappa^{2} Wh_{i} G_{i}\) is the shear stiffness, \((EI)_{i} = Wh_{i}^{3} E_{i} /12\) is the bending stiffness, \((\rho I)_{i} = Wh_{i}^{3} \rho_{i} /12\) is the moment of inertia, \(\mu_{i} = Wh_{i} \rho_{i}\) is the mass, G_{i} is the shear modulus, and κ is the correction factor depending on the elastic beam crosssection.
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Wang, L., Bai, F., Hofmann, V. et al. Novel Traveling Wave Sandwich Piezoelectric Transducer with Single Phase Drive: Theoretical Modeling, Experimental Validation, and Application Investigation. Chin. J. Mech. Eng. 34, 101 (2021). https://doi.org/10.1186/s1003302100623x
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Keywords
 Traveling wave
 Sandwich piezoelectric transducer
 Single phase excitation
 Transfer matrix method
 Ultrasonic motor