# Numerical Analysis on Chaotic Vibration of Drive System for a Movable Tooth Piezoelectric Motor

Authors: Chong Li; Jichun Xing; Jiwen Fang; Zhong Zhao
Journal: Shock and Vibration
 (2017)

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## Abstract

The nonlinear dynamic equations of the drive system for movable tooth piezoelectric motor are established. Using these equations, the chaotic vibrations of the system are investigated. The results show that chaotic vibrations occur in the movable tooth drive system under some parameters. The average mesh stiffness, theoretical radius, and wave generator offset significantly influence the nonlinear chaotic vibrations of the drive system of the movable tooth piezoelectric motor. The ranges for the system parameters that lead to a motor with bad dynamics are shown. The results can be used to predict the dynamic load and optimize power density of the proposed piezoelectric motor.


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## Body

## 1. Introduction

Piezoelectric motors have been widely utilized in fields such as electromechanical actuating, mechanical energy harvesting, aeronautics, and astronautics [1–3]. In 1988, Kurosawa et al. proposed a cylinder-shaped piezoelectric motor in which the bending modes were used to increase its mechanical output [4]. In 2002, Koc et al. proposed a piezoelectric motor with two orthogonal bending modes of a hollow cylinder; later the motor was used in camera of phones by Samsung Corp [5]. What is more, in 2014, Mashimo designed a micropiezoelectric motor in which the size of stator was only one cubic millimeter [6].Meanwhile, inchworm piezoelectric actuator is the special friction type piezoelectric motor. In 2003, Mrad et al. developed a motor that was designed based on the classical inchworm principle, and a hybrid control method was used for the motor [7]. In 2010, A ultracompact, zero-power magnetic latching piezoelectric inchworm motor was proposed. With a 50 kHz driving frequency, precision of 5 μm at a scalable full stroke of 5 mm and a speed of 10 mm/s were achieved [8]. Besides, Li et al. designed an inchworm type rotary actuator in 2014. In the motor, six piezostacks and flexure hinges were used to realize large rotation ranges with high accuracy both in the forward and in backward motions. The test results indicate that the maximum output torque was 19.6 Nmm [9]. What is more, Gu and Xing proposed a rotary inchworm piezoelectric motor. The motor has advantages such as high positioning accuracy, large output torque, and large travel [10].Although the piezoelectric motors have been widely used in modern life. However, the friction between the stator and the rotor in piezoelectric motors limits its output torque and operating life.Hence, a noncontact piezoelectric motor in which the fluid between the stator and rotor was proposed by Yamayoshi and Hirose in 1992 [11]. In 2009, a noncontact piezoelectric motor with two flexural standing wave vibration disks was proposed by Yamayoshi et al., and the rotating speed of which is up to 2000 rpm at a very low driving voltage of 3 V [12]. In 2012, a noncontact piezoelectric actuator with spherical rotors was designed by Chen et al., and a maximum rotational speed of 521 r/min was obtained under excitation voltage of 300 V [13]. Two years later, Qiu et al. proposed a noncontact piezoelectric rotary motor that was modulated by giant electrorheological fluid [14]. In short, the noncontact piezoelectric motors eliminated the friction between the stator and rotor and increased the efficiency and lifetime. However, the output torque of which is too small to satisfy the practical need.Therefore, the authors proposed a movable tooth piezoelectric motor, and, hence, a larger transmission ratio and a larger output torque of which can be obtained [15]. The structure and working principle of the proposed motor are shown in Figures 1 and 2. The motor consists of driving system and transmission system. The driving system includes piezoelectric actuator (1), z-shaped rod (2), swaying rod (3) and adjusting spring (8), the transmission system contains movable tooth (4), rigid cog (5), rotor (6), and wave generator (7).Figure 1
Movable tooth piezoelectric motor.(1) Piezoelectric actuator; (2) z-shaped rod; (3) swaying rod; (4) movable tooth; (5) rigid cog; (6) rotor; (7) wave generator; (8) spring.Figure 2
Working principle of movable tooth piezoelectric motor.The driving source of the motor consists of two piezoelectric actuators with a phase difference of 90 degrees. When the piezoelectric actuators are subject to cosine signals with a positive bias and 90 degrees phase difference, they generate an axial elongation. With z-shaped rod, the swaying rod is pushed toward the side of the springs. When the voltage of the exciting signals returns to zero, the swaying rod returns to the initial position under the elastic force of the springs. Then, the swaying rod swings back and forth in two directions performing a continuous harmonic wave on wave generator. Here, 30 movable teeth and 29 rigid cog teeth are used. With the effect of movable tooth drive, the rotation can achieve a transmission ratio of 30. Here, the movable teeth are steel balls and its diameter is 1 mm. When working, the movable teeth locate in the groove of the rotor. The rigid cog is formed by tooth profile equation, and the equation can be expressed by(1)X=bcos⁡θ-arcsin⁡asin⁡icp-1θb+acos⁡icpθ±rpcos⁡ψ,Y=bsin⁡θ-arcsin⁡asin⁡icp-1θb+asin⁡icpθ±rpsin⁡ψ,where θ is rotational angle of the rotor; ψ is the angle between x-axis and tangent of central trajectory of movable tooth; b is coefficient, b=rs+rp; rs and rp are radius of wave generator and movable tooth.The proposed movable tooth piezoelectric motor is a nonlinear coupled system, and the nonlinear coupled dynamics have an important influence on operating behavior of the motor. In [16], the nonlinear free vibration of driving system of the movable tooth piezoelectric motor was studied. The chaotic vibrations of the drive system influence the dynamics of the motor, which will decrease the load carrying capacity. In [17], the chaotic vibrations of driving system for the piezoelectric motor were investigated. The results show that the chaotic vibrations of the driving system for the motor occur under some parameters. However, the chaotic vibrations characteristics of the transmission system for the piezoelectric motor are unknown.So, the purpose of this paper is to reveal the influence of chaotic vibrations of the transmission system on dynamics of the proposed motor. Using Runge-Kutta numerical method, the nonlinear dynamic performance and the effect of average mesh stiffness, theoretical radius, and wave generator offset on chaotic vibrations characteristics are investigated. The results can be used to predict the dynamic load and optimize power density of the motor.


## 2. Nonlinear Dynamic Equations

Dynamic model of drive system of movable tooth piezoelectric motor is shown in Figure3. Figures 3(a) and 3(b) are the relative displacement between movable teeth and central elements (rotor, rigid cog, and wave generator) and forces of movable tooth system, respectively. In Figure 3, the coordinate system OXY is attached to the foundation, oxy to rotor 6, and oixiyi to the ith movable tooth. Subscripts s, c, r, p represent, respectively, wave generator, rigid cog, rotor, and movable tooth. Besides, xj, yj, uj represent x, y direction and circumferential direction linear displacement (j=s,c,r,p1,…,pz). Wave generator offset a is the distance between rotor center O and wave generator center O′.Figure 3
Dynamic model of drive system of movable tooth piezoelectric motor.
(a)
Relative displacement (b)
Forces of movable tooth systemFrom Figure3(a), the projection of relative displacement from central elements to movable teeth along meshing line can be written as(2)δsi=xs-xpicos⁡ϕ1i+ys-ypisin⁡ϕ1i+ussin⁡ϕ3i,δci=xc-xpicos⁡θi+yc-ypisin⁡θi-ucsin⁡ϕθi,δri=xpi-xrsin⁡ϕi+yr-ypicos⁡ϕi-ur,where ϕ1i=ϕi+ϕ3i, ϕθi=ϕi-θi. θi is angular displacement of elements, ϕi is the angle between line ooi and coordinate axis x direction, oon is the line from center of the nth movable tooth and rotor center, ϕ3i is the angle between line oo3i and coordinate axis x direction, and oo3i is the line from center of the nth movable tooth and vibrator center.From Figure3(b), the linear dynamic equations of drive system for movable tooth piezoelectric motor can be expressed by(3)msx¨s+∑i=1Zksδsicos⁡ϕ1i+kszxs=0,msy¨s+∑i=1Zksδsisin⁡ϕ1i+kszys=0,Isrs2u¨s-∑i=1Zksδsisin⁡ϕ3i+kstus=Tsrs,mcx¨c+∑i=1Zkcδcicos⁡θi+kczxc=0,mcy¨c+∑i=1Zkcδcisin⁡θi+kczyc=0,Icrc2u¨c-∑i=1Zkcδcisin⁡ϕθi+kctuc=0,mrx¨r-∑i=1Zkrδrisin⁡ϕi+krzxr=0,mry¨r+∑i=1Zkrδricos⁡ϕi+krzyr=0,Irrr2u¨r-∑i=1Zkrδri+krtur=0,mpx¨pi-kcδcicos⁡θi+krδrisin⁡ϕi-ksδsicos⁡ϕ1i=0,mpy¨pi-kcδcisin⁡θi-krδricos⁡ϕi-ksδsisin⁡ϕ1i=0,Iprp2u¨i+u¨j=0,where mj, Ij are mass and relative mass of each element, kj, kjz, kjt represent the mesh stiffness and radial and tangent supporting stiffness between the movable teeth and central elements, ri is theoretical radius of elements, and Ts is output torque of wave generator.The output torque of movable tooth piezoelectric motor changes with rotational corner of rotor is shown in Figure4. In Figure 4(a), the output torque jump at θ=n·π/435. And in Figure 4(b), the output torque equation in single-period is complex, applying polynomial curves to fit torque equation, and yields(4)T1=τ1θ2+τ2θ+τ30≤θ<π435,where τ1, τ2, τ3 are fitting coefficients.Figure 4
Output torque changes with rotational corner of rotor.
(a)
Multiperiod (b)
Single-periodApplying Fourier transform, (3) can be written as(5)Tθ=a02+∑i=1nancos⁡2nπθτ+bnsin⁡2nπθτ=a02+∑i=1nan2+bn2sin⁡2nπθτ+ϕT,where an and bn are Fourier coefficients, (6)ϕT=arctan⁡anbn,a0=23τ2τ1+ττ2+2τ3,an=2τ∫0τT1θcos⁡2nπθτdθ=τ2τ1π2n2,bn=2τ∫0τT1θsin⁡2nπθτdθ=-τ2τ1+ττ2πn.Assume that rotational angle increment and output torque increment of rotor areδθ and δTr, respectively. So, the Taylor series of Tr at θ=θ0 can be expressed by(7)Tr=Tr0+δTr=Tθ0+T′θ0δθ+12T′′θ0δθ2+⋯=a02+∑i=1nan2+bn2sin⁡2nπθ0τ+ϕT+∑i=1n2nπτan2+bn2cos⁡2nπθ0τ+ϕTδθ-∑i=1n4n2π2τ2an2+bn2sin⁡2nπθ0τ+ϕTδθ2+⋯.Output torque increment of rotor is(8)δTr=∑i=1n2nπτan2+bn2cos⁡2nπθ0τ+ϕTδθ-∑i=1n4n2π2τ2an2+bn2sin⁡2nπθ0τ+ϕTδθ2+⋯.Hence, the relationship between output torqueT and meshing force between movable tooth and rotor Frj can be written as(9)T=∑j=1nFrjSj=∑j=1nkrSj2δθ.From (9), the nonlinear meshing stiffness between movable tooth and rotor can be expressed by(10)kr=an2+bn2Sj2∑i=1n2nπτcos⁡2nπθ0τ+ϕT-4n2π2τ2sin⁡2nπθ0τ+ϕTδθ+⋯=an2+bn2Sj2·∑i=1nAk-2nπAkτtan⁡2nπθ0τ+ϕTδθ+⋯=k-r+Δkr,where (11)Δkr=-2nπAkan2+bn2τSj2∑i=1ntan⁡2nπθ0τ+ϕTδθ,k-r=an2+bn2Sj2∑i=1nAk,Ak=2nπτcos⁡2nπθ0τ+ϕT.Following the same approach, the nonlinear meshing stiffness between movable tooth and wave generator, rigid cog can be written as(12)ks=k-s-2nπτtan⁡2nπθ0τ+ϕTδθk-s=k-s+Δks,kc=k-c-2nπτtan⁡2nπθ0τ+ϕTδθk-c=k-c+Δkc.Further, substituting (10) and (12) into (3), the nonlinear dynamic equations of movable tooth drive system can be expressed by(13)Mq¨+Kq=F+ΔF,where M, K, q are mass matrix, stiffness matrix, and generalized coordinates array of the system; F and ΔF are outer forces array and nonlinear meshing forces increment array, ΔF=BkuiεΛ; and Λ is array that contains only sine and cosine element.(14)Bk=2nπAkan2+bn2τSj2∑i=1ntan⁡2nπθ0τ+ϕT.


## 3. Results and Discussion

Equation (13) can be resolved by using a Matlab software tool that involves the fourth-order Runge-Kutta method. The time step is T/100, and the total time is 400T. Because the beginning of the calculation yields unstable responses, we discard the response of the first 300 periods and use only the response of the final 100T, where T is the period of the system vibration. Torsional vibration behaves obviously in movable tooth drive system, so the torsional vibration response of central elements is selected to discuss. The parameters of the numerical example are given in Table 1.Table 1
Parameters of example system.
Parameters Rotor Rigid cog Wave generator Movable tooth Mass (kg) 1.31 × 10−2 5.64 × 10−2 2.59 × 10−2 3.30 × 10−2 Equivalent mass (kg) 9.34 × 10−2 8.41 × 10−2 1.30 × 10−2 1.32 × 10−2 Diameter (mm) 10~50 10~50 10~50 2 Radial support stiffness (N/m) 5 × 108 5 × 108 5 × 108 0 Tangent support stiffness (N/m) 0 1 × 109 0 0 Mesh stiffness (N/m) 0~1 × 108 0~3 × 108 0~1 × 108 0


### 3.1. Influence of Average Mesh Stiffness on Chaotic Vibration

Figure5 gives bifurcation diagrams of torsional vibration of movable tooth drive system for average mesh stiffness (ks-av,kc-av,kp-av) varying from (0,0,0) to (1×108,3×108,1×108). Here, the bifurcation diagrams can show sudden changes of the vibration stage when some parameters changes. Moreover, the Poincare section method is utilized to obtain the bifurcation diagrams. From Figure 5, the following are known.Figure 5
Bifurcation with average mesh stiffness.
(a)
k s - a v = 0~1 × 108 N/m (b)
k s - a v = 1 × 10 7  ~  7 × 10 7 N/m (c)
k c - a v = 0~1 × 108 N/m (d)
k c - a v = 1 × 10 8  ~  3 × 10 7 N/m (e)
k p - a v = 0~1 × 108 N/m (f)
k p - a v = 0~4 × 107 N/m(1) Whenks-av=1×107 N/m, 1.7 × 107 N/m, 3.8 × 107 N/m, 9.2 × 107 N/m, the bifurcation diagram of wave generator shows sudden changes of vibration state. When ks-av<1×107 N/m, the response amplitude is at stable state and the system tends to do quasiperiodic motion.(2) Similarly, whenkc-av=6×103 N/m, 3.3 × 107 N/m, 6.7 × 107 N/m, 8 × 107 N/m and kp-av=2×103 N/m, 1.2 × 107 N/m, 4 × 107 N/m, 7 × 107 N/m, the bifurcation diagrams of rigid cog and rotor behave in sudden changes of vibration state.(3) Changes of bifurcation diagram with average mesh stiffness of elements are similar. However, the starting point of sudden changes is different for each central element. It is because the dimension and natural frequency of each element are different, and the vibration intensity varies from one element to another. So this is the reason why the starting point of sudden changes is different.Figure6 shows Poincare map of torsional vibration response for different average mesh stiffness. Poincare map uses simple discrete mapping points to show continuous motions. Taking one mapping point each exciting period, Poincare map of the torsional vibration of movable tooth drive system can be obtained. Figure 7 shows power spectrum of torsional vibration response for different average mesh stiffness. Power spectrum shows changes of the signal power along with exciting frequency from which chaotic motion can be determined. Here, periodogram method is used to calculate power spectrum of the vibration. From Figures 6 and 7, the following are known.Figure 6
Poincare map of torsional vibration response for different average mesh stiffness.
(a)
k s - a v = 1 × 10 7 N/m (b)
k s - a v = 3.8 × 10 7 N/m (c)
k c - a v = 6 × 10 3 N/m (d)
k c - a v = 6.7 × 10 7 N/m (e)
k p - a v = 2 × 10 3 N/m (f)
k p - a v = 4 × 10 7 N/mFigure 7
Power spectrum of torsional vibration response for different average mesh stiffness.
(a)
k s - a v = 1 × 10 7 N/m (b)
k s - a v = 3.8 × 10 7 N/m (c)
k c - a v = 6 × 10 3 N/m (d)
k c - a v = 6.7 × 10 7 N/m (e)
k p - a v = 2 × 10 3 N/m (f)
k p - a v = 4 × 10 7 N/m(1) As the average mesh stiffness changes, the vibration displacements of central elements change. Atks-av=1×107 N/m, 3.8 × 107 N/m, kc-av=6×103 N/m, 6.7×107 N/m, and kp-av=2×103 N/m, 4×107 N/m, jumping of the vibration displacement of central elements occurs. Here, a small change of the average mesh stiffness can cause significant change of the vibration displacements. It causes unstable vibration of the movable tooth drive system.(2) Atks-av=1×107 N/m, 3.8×107 N/m, kc-av=6×103 N/m, 6.7×107 N/m, and kp-av=2×103 N/m, 4×107 N/m, a distributed point set with a certain geometry appears on the Poincare map, the power spectra become continuous. It shows that chaotic vibration occurs in piezoelectric actuator. Meanwhile, At above average mesh stiffness points, the vibration displacements of the central elements are larger than those for other average mesh stiffness. Among them, the vibration displacement of the movable tooth drive system is the larger at ks-av=3.8×107 N/m and kp-av=2×103 N/m.(3) The chaotic vibration of movable tooth drive system occurs when average mesh stiffnesskj-av exceeds a specific value. Moreover, the starting point of chaotic vibration is different for each central element. When kp-av is very small, the chaotic vibration of rotor occurs, but the chaotic vibration of wave generator occurs only when kp-av increases larger.In general, from chaotic vibration theory, it is known that break of quasiperiodic torus is the main reason leading to the chaotic vibration of the drive system. In addition, the chaotic vibration is sensitive for the small change of main parameters. Sometimes, the chaotic vibration only occurs in a moment. Hence, the average mesh stiffness is the primary parameter that affects the chaotic vibration of the piezoelectric motor. To obtain a good dynamic performance, the average mesh stiffness should be selected properly.


### 3.2. Influence of Theoretical Radius on Chaotic Vibration

Figure8 gives bifurcation diagrams of torsional vibration of movable tooth drive system for theoretical radius rj varying from 10 mm to 50 mm. Here, the Poincare section method is utilized to obtain the bifurcation diagrams. Changing radius rj from 10 mm to 50 mm, one Poincare map is calculated for each theoretical radius. Based on it, the bifurcation diagram of torsional vibration for theoretical radius is given. From Figure 8, the following are known.Figure 8
Bifurcation with theoretical radius.
(a)
r s = 10~50 mm (b)
r s = 2~10 mm (c)
r c = 10~50 mm (d)
r c = 2~10 mm (e)
r r = 10~50 mm (f)
r r = 2~10 mm(1) In bifurcation diagram of wave generator, sudden change of vibration displacement occurs atrs=3.8 mm, and intermittent sudden changes happen during rs = 3.8~10 mm. However, the maximum vibration displacement increases steadily when rs>10 mm.(2) In bifurcation diagram of rigid cog, whenrc=14 mm, major jump change of vibration displacement appears, and small jump change occurs at rc=6 mm. Besides, the vibration displacement increases stability when rc>14 mm.(3) In bifurcation diagram of rotor, it is similar to rigid cog. The difference is that the major sudden change of vibration displacement of rotor occurs atrr=8 mm, earlier than rc=14 mm; moreover, the number of small jump changes increases when rr<8 mm.(4) Compared with rigid cog and rotor, the displacement sudden change of wave generator happens when theoretical radius is quite small. Hence, the chaotic vibration of wave generator tends to take place earlier than rigid cog and rotor. The reason is that the external forces apply on the wave generator. The wave generator is easier to generate vibration. So, in order to obtain a good performance of the piezoelectric motor, the vibration of wave generator should be suppressed.Figures9 and 10, Figures 11 and 12, and Figures 13 and 14 show Poincare map and power spectrum of vibration response of wave generator, rigid cog, and rotor for different theoretical radius. From Figures 9–14, the following are known.Figure 9
Poincare map of vibration response of wave generator for different theoretical radius.
(a)
r s = 3.7 mm (b)
r s = 3.8 mm (c)
r s = 9 mm (d)
r s = 20 mmFigure 10
Power spectrum of vibration response of wave generator for different theoretical radius.
(a)
r s = 3.7 mm (b)
r s = 3.8 mm (c)
r s = 9 mm (d)
r s = 20 mmFigure 11
Poincare map of vibration response of rigid cog for different theoretical radius.
(a)
r c = 5.5 mm (b)
r c = 6 mm (c)
r c = 13 mm (d)
r c = 14 mmFigure 12
Power spectrum of vibration response of rigid cog for different theoretical radius.
(a)
r c = 5.5 mm (b)
r c = 6 mm (c)
r c = 13 mm (d)
r c = 14 mmFigure 13
Poincare map of vibration response of rotor for different theoretical radius.
(a)
r r = 5.2 mm (b)
r r = 5.4 mm (c)
r r = 7.8 mm (d)
r r = 8 mmFigure 14
Power spectrum of vibration response of rotor for different theoretical radius.
(a)
r r = 5.2 mm (b)
r r = 5.4 mm (c)
r r = 7.8 mm (d)
r r = 8 mm(1) Atrs=3.7 mm, a distributed point set with a ring structure appears on the Poincare map of wave generator, and the power spectra becomes discrete. This shows that the wave generator exhibits a quasiperiod vibration. However, at rs=3.8 mm, the power spectra become continuous. It is shown that chaotic vibration occurs in wave generator, and rs=3.8 mm becomes a demarcation point from quasiperiod vibration to chaotic vibration. Similarly, at rs=9 mm or 20 mm, the wave generator shows quasiperiod vibrations.(2) Atrc=5.5 mm, a distributed point set with a certain geometry appears on the Poincare map of rigid cog; the power spectra appear to be continuous. It is shown that chaotic vibration occurs in rigid cog. However, at rc=6 mm, the power spectra turn into continuous, so the rigid cog changes into quasiperiod vibration. By the same token, at rc=13 mm and 14 mm, the rigid cog appears to be chaotic vibration and quasiperiod vibration, respectively. Besides, the power spectra at rc>14 mm and rc=6 mm and 14 mm are similar. This shows that the chaotic vibration disappears when rc>14 mm.(3) The law of vibration response of rotor is similar to rigid cog. Atrr=5.2 mm and 7.8 mm, the system exhibits chaotic vibration. And at rr=5.4 mm and 8 mm, the quasiperiod vibration is shown in rotor. Similarly, the chaotic vibration disappears when rr>8 mm.In general, the chaotic vibration sudden change occurs when theoretical radius takes some certain value. From chaotic vibration theory, the phenomenon that chaos attractors suddenly appear or disappear is interior crisis or boundary crisis. The reason for generating of the two-type crisis is that the collision between chaos attractors and unstable period occurs at interior and boundary of attraction basin. Therefore, when theoretical radius changes, the way to chaotic vibration is paroxysmal.


### 3.3. Influence of Wave Generator Offset on Chaotic Vibration

Figure15 gives bifurcation diagrams of torsional vibration of movable tooth drive system for wave generator offset a varying from 0 to 1 mm. Figures 16–18 show power spectrum and time response of wave generator, rigid cog, and rotor for different wave generator offset. From Figures 15–18, the following are known.Figure 15
Bifurcation with wave generator offset.
(a)
Wave generator (a = 0~1 mm) (b)
Wave generator (a = 0~0.4 mm) (c)
Rigid cog (a = 0~1 mm) (d)
Rotor (a = 0~1 mm)Figure 16
Power spectrum and time response of wave generator for different wave generator offset.
(a)
Power spectrum fora=0 (b)
Time response fora=0 (c)
Power spectrum fora=0.04 mm (d)
Time response fora=0.04 mm (e)
Power spectrum fora=0.2 mm (f)
Time response fora=0.2 mmFigure 17
Power spectrum and time response of rigid cog for different wave generator offset.
(a)
Power spectrum fora=0 (b)
Time response fora=0 (c)
Power spectrum fora=0.06 mm (d)
Time response fora=0.06 mm (e)
Power spectrum fora=0.16 mm (f)
Time response fora=0.16 mmFigure 18
Power spectrum and time response of rotor for different wave generator offset.
(a)
Power spectrum fora=0 (b)
Time response fora=0 (c)
Power spectrum fora=0.1 mm (d)
Time response fora=0.1 mm(1) Ata=0, the vibration displacement of wave generator is very small from bifurcation diagram and time response graph, and a single discrete peak appears in power spectrum. This shows that wave generator exhibits a period-one vibration. At a=0.04 mm, a sudden change appears in bifurcation diagram, aperiodic amplitude jumping vibrations occur on the time response of wave generator, and power spectrum becomes continuous. It is shown that the chaotic vibration takes place on wave generator. However, displacement of chaotic vibration behaves invariably when a>0.3 mm.(2) Ata=0, 0.06 mm and 0.16 mm, sudden changes occur in bifurcation diagram of rigid cog. The power spectrum behaves continuously, and the time response appears aperiodic. It shows that initial stage chaotic vibration occurs on rigid cog. For the length ranges of [(0,0.06),(0.06,0.16)], the stable quasiperiod vibration of rigid cog occurs.(3) From Figure15(d), vibration displacement of rotor does not change with a. At a=0 and 0.1 mm, the rotor exhibits quasiperiod vibration accompanied with chaotic vibration.In short, from chaotic vibration theory, the main way to chaotic vibration is paroxysmal. But for the rotor, break of quasiperiodic torus is another reason for the chaotic vibration. Hence, wave generator offseta has also a significant influence on the chaotic vibration of the movable tooth drive system. To obtain a good performance of the piezoelectric motor, the wave generator offset should be chosen properly as well.


## 3.1. Influence of Average Mesh Stiffness on Chaotic Vibration

Figure5 gives bifurcation diagrams of torsional vibration of movable tooth drive system for average mesh stiffness (ks-av,kc-av,kp-av) varying from (0,0,0) to (1×108,3×108,1×108). Here, the bifurcation diagrams can show sudden changes of the vibration stage when some parameters changes. Moreover, the Poincare section method is utilized to obtain the bifurcation diagrams. From Figure 5, the following are known.Figure 5
Bifurcation with average mesh stiffness.
(a)
k s - a v = 0~1 × 108 N/m (b)
k s - a v = 1 × 10 7  ~  7 × 10 7 N/m (c)
k c - a v = 0~1 × 108 N/m (d)
k c - a v = 1 × 10 8  ~  3 × 10 7 N/m (e)
k p - a v = 0~1 × 108 N/m (f)
k p - a v = 0~4 × 107 N/m(1) Whenks-av=1×107 N/m, 1.7 × 107 N/m, 3.8 × 107 N/m, 9.2 × 107 N/m, the bifurcation diagram of wave generator shows sudden changes of vibration state. When ks-av<1×107 N/m, the response amplitude is at stable state and the system tends to do quasiperiodic motion.(2) Similarly, whenkc-av=6×103 N/m, 3.3 × 107 N/m, 6.7 × 107 N/m, 8 × 107 N/m and kp-av=2×103 N/m, 1.2 × 107 N/m, 4 × 107 N/m, 7 × 107 N/m, the bifurcation diagrams of rigid cog and rotor behave in sudden changes of vibration state.(3) Changes of bifurcation diagram with average mesh stiffness of elements are similar. However, the starting point of sudden changes is different for each central element. It is because the dimension and natural frequency of each element are different, and the vibration intensity varies from one element to another. So this is the reason why the starting point of sudden changes is different.Figure6 shows Poincare map of torsional vibration response for different average mesh stiffness. Poincare map uses simple discrete mapping points to show continuous motions. Taking one mapping point each exciting period, Poincare map of the torsional vibration of movable tooth drive system can be obtained. Figure 7 shows power spectrum of torsional vibration response for different average mesh stiffness. Power spectrum shows changes of the signal power along with exciting frequency from which chaotic motion can be determined. Here, periodogram method is used to calculate power spectrum of the vibration. From Figures 6 and 7, the following are known.Figure 6
Poincare map of torsional vibration response for different average mesh stiffness.
(a)
k s - a v = 1 × 10 7 N/m (b)
k s - a v = 3.8 × 10 7 N/m (c)
k c - a v = 6 × 10 3 N/m (d)
k c - a v = 6.7 × 10 7 N/m (e)
k p - a v = 2 × 10 3 N/m (f)
k p - a v = 4 × 10 7 N/mFigure 7
Power spectrum of torsional vibration response for different average mesh stiffness.
(a)
k s - a v = 1 × 10 7 N/m (b)
k s - a v = 3.8 × 10 7 N/m (c)
k c - a v = 6 × 10 3 N/m (d)
k c - a v = 6.7 × 10 7 N/m (e)
k p - a v = 2 × 10 3 N/m (f)
k p - a v = 4 × 10 7 N/m(1) As the average mesh stiffness changes, the vibration displacements of central elements change. Atks-av=1×107 N/m, 3.8 × 107 N/m, kc-av=6×103 N/m, 6.7×107 N/m, and kp-av=2×103 N/m, 4×107 N/m, jumping of the vibration displacement of central elements occurs. Here, a small change of the average mesh stiffness can cause significant change of the vibration displacements. It causes unstable vibration of the movable tooth drive system.(2) Atks-av=1×107 N/m, 3.8×107 N/m, kc-av=6×103 N/m, 6.7×107 N/m, and kp-av=2×103 N/m, 4×107 N/m, a distributed point set with a certain geometry appears on the Poincare map, the power spectra become continuous. It shows that chaotic vibration occurs in piezoelectric actuator. Meanwhile, At above average mesh stiffness points, the vibration displacements of the central elements are larger than those for other average mesh stiffness. Among them, the vibration displacement of the movable tooth drive system is the larger at ks-av=3.8×107 N/m and kp-av=2×103 N/m.(3) The chaotic vibration of movable tooth drive system occurs when average mesh stiffnesskj-av exceeds a specific value. Moreover, the starting point of chaotic vibration is different for each central element. When kp-av is very small, the chaotic vibration of rotor occurs, but the chaotic vibration of wave generator occurs only when kp-av increases larger.In general, from chaotic vibration theory, it is known that break of quasiperiodic torus is the main reason leading to the chaotic vibration of the drive system. In addition, the chaotic vibration is sensitive for the small change of main parameters. Sometimes, the chaotic vibration only occurs in a moment. Hence, the average mesh stiffness is the primary parameter that affects the chaotic vibration of the piezoelectric motor. To obtain a good dynamic performance, the average mesh stiffness should be selected properly.


## 3.2. Influence of Theoretical Radius on Chaotic Vibration

Figure8 gives bifurcation diagrams of torsional vibration of movable tooth drive system for theoretical radius rj varying from 10 mm to 50 mm. Here, the Poincare section method is utilized to obtain the bifurcation diagrams. Changing radius rj from 10 mm to 50 mm, one Poincare map is calculated for each theoretical radius. Based on it, the bifurcation diagram of torsional vibration for theoretical radius is given. From Figure 8, the following are known.Figure 8
Bifurcation with theoretical radius.
(a)
r s = 10~50 mm (b)
r s = 2~10 mm (c)
r c = 10~50 mm (d)
r c = 2~10 mm (e)
r r = 10~50 mm (f)
r r = 2~10 mm(1) In bifurcation diagram of wave generator, sudden change of vibration displacement occurs atrs=3.8 mm, and intermittent sudden changes happen during rs = 3.8~10 mm. However, the maximum vibration displacement increases steadily when rs>10 mm.(2) In bifurcation diagram of rigid cog, whenrc=14 mm, major jump change of vibration displacement appears, and small jump change occurs at rc=6 mm. Besides, the vibration displacement increases stability when rc>14 mm.(3) In bifurcation diagram of rotor, it is similar to rigid cog. The difference is that the major sudden change of vibration displacement of rotor occurs atrr=8 mm, earlier than rc=14 mm; moreover, the number of small jump changes increases when rr<8 mm.(4) Compared with rigid cog and rotor, the displacement sudden change of wave generator happens when theoretical radius is quite small. Hence, the chaotic vibration of wave generator tends to take place earlier than rigid cog and rotor. The reason is that the external forces apply on the wave generator. The wave generator is easier to generate vibration. So, in order to obtain a good performance of the piezoelectric motor, the vibration of wave generator should be suppressed.Figures9 and 10, Figures 11 and 12, and Figures 13 and 14 show Poincare map and power spectrum of vibration response of wave generator, rigid cog, and rotor for different theoretical radius. From Figures 9–14, the following are known.Figure 9
Poincare map of vibration response of wave generator for different theoretical radius.
(a)
r s = 3.7 mm (b)
r s = 3.8 mm (c)
r s = 9 mm (d)
r s = 20 mmFigure 10
Power spectrum of vibration response of wave generator for different theoretical radius.
(a)
r s = 3.7 mm (b)
r s = 3.8 mm (c)
r s = 9 mm (d)
r s = 20 mmFigure 11
Poincare map of vibration response of rigid cog for different theoretical radius.
(a)
r c = 5.5 mm (b)
r c = 6 mm (c)
r c = 13 mm (d)
r c = 14 mmFigure 12
Power spectrum of vibration response of rigid cog for different theoretical radius.
(a)
r c = 5.5 mm (b)
r c = 6 mm (c)
r c = 13 mm (d)
r c = 14 mmFigure 13
Poincare map of vibration response of rotor for different theoretical radius.
(a)
r r = 5.2 mm (b)
r r = 5.4 mm (c)
r r = 7.8 mm (d)
r r = 8 mmFigure 14
Power spectrum of vibration response of rotor for different theoretical radius.
(a)
r r = 5.2 mm (b)
r r = 5.4 mm (c)
r r = 7.8 mm (d)
r r = 8 mm(1) Atrs=3.7 mm, a distributed point set with a ring structure appears on the Poincare map of wave generator, and the power spectra becomes discrete. This shows that the wave generator exhibits a quasiperiod vibration. However, at rs=3.8 mm, the power spectra become continuous. It is shown that chaotic vibration occurs in wave generator, and rs=3.8 mm becomes a demarcation point from quasiperiod vibration to chaotic vibration. Similarly, at rs=9 mm or 20 mm, the wave generator shows quasiperiod vibrations.(2) Atrc=5.5 mm, a distributed point set with a certain geometry appears on the Poincare map of rigid cog; the power spectra appear to be continuous. It is shown that chaotic vibration occurs in rigid cog. However, at rc=6 mm, the power spectra turn into continuous, so the rigid cog changes into quasiperiod vibration. By the same token, at rc=13 mm and 14 mm, the rigid cog appears to be chaotic vibration and quasiperiod vibration, respectively. Besides, the power spectra at rc>14 mm and rc=6 mm and 14 mm are similar. This shows that the chaotic vibration disappears when rc>14 mm.(3) The law of vibration response of rotor is similar to rigid cog. Atrr=5.2 mm and 7.8 mm, the system exhibits chaotic vibration. And at rr=5.4 mm and 8 mm, the quasiperiod vibration is shown in rotor. Similarly, the chaotic vibration disappears when rr>8 mm.In general, the chaotic vibration sudden change occurs when theoretical radius takes some certain value. From chaotic vibration theory, the phenomenon that chaos attractors suddenly appear or disappear is interior crisis or boundary crisis. The reason for generating of the two-type crisis is that the collision between chaos attractors and unstable period occurs at interior and boundary of attraction basin. Therefore, when theoretical radius changes, the way to chaotic vibration is paroxysmal.


## 3.3. Influence of Wave Generator Offset on Chaotic Vibration

Figure15 gives bifurcation diagrams of torsional vibration of movable tooth drive system for wave generator offset a varying from 0 to 1 mm. Figures 16–18 show power spectrum and time response of wave generator, rigid cog, and rotor for different wave generator offset. From Figures 15–18, the following are known.Figure 15
Bifurcation with wave generator offset.
(a)
Wave generator (a = 0~1 mm) (b)
Wave generator (a = 0~0.4 mm) (c)
Rigid cog (a = 0~1 mm) (d)
Rotor (a = 0~1 mm)Figure 16
Power spectrum and time response of wave generator for different wave generator offset.
(a)
Power spectrum fora=0 (b)
Time response fora=0 (c)
Power spectrum fora=0.04 mm (d)
Time response fora=0.04 mm (e)
Power spectrum fora=0.2 mm (f)
Time response fora=0.2 mmFigure 17
Power spectrum and time response of rigid cog for different wave generator offset.
(a)
Power spectrum fora=0 (b)
Time response fora=0 (c)
Power spectrum fora=0.06 mm (d)
Time response fora=0.06 mm (e)
Power spectrum fora=0.16 mm (f)
Time response fora=0.16 mmFigure 18
Power spectrum and time response of rotor for different wave generator offset.
(a)
Power spectrum fora=0 (b)
Time response fora=0 (c)
Power spectrum fora=0.1 mm (d)
Time response fora=0.1 mm(1) Ata=0, the vibration displacement of wave generator is very small from bifurcation diagram and time response graph, and a single discrete peak appears in power spectrum. This shows that wave generator exhibits a period-one vibration. At a=0.04 mm, a sudden change appears in bifurcation diagram, aperiodic amplitude jumping vibrations occur on the time response of wave generator, and power spectrum becomes continuous. It is shown that the chaotic vibration takes place on wave generator. However, displacement of chaotic vibration behaves invariably when a>0.3 mm.(2) Ata=0, 0.06 mm and 0.16 mm, sudden changes occur in bifurcation diagram of rigid cog. The power spectrum behaves continuously, and the time response appears aperiodic. It shows that initial stage chaotic vibration occurs on rigid cog. For the length ranges of [(0,0.06),(0.06,0.16)], the stable quasiperiod vibration of rigid cog occurs.(3) From Figure15(d), vibration displacement of rotor does not change with a. At a=0 and 0.1 mm, the rotor exhibits quasiperiod vibration accompanied with chaotic vibration.In short, from chaotic vibration theory, the main way to chaotic vibration is paroxysmal. But for the rotor, break of quasiperiodic torus is another reason for the chaotic vibration. Hence, wave generator offseta has also a significant influence on the chaotic vibration of the movable tooth drive system. To obtain a good performance of the piezoelectric motor, the wave generator offset should be chosen properly as well.


## 4. Conclusions

In this paper, the nonlinear dynamic performance and chaotic vibration of transmission system for movable tooth piezoelectric motor are investigated. The results show the following.(1) The average mesh stiffness has an significant influence on the chaotic vibration of the transmission system. Atks-av=1×107 N/m, 3.8×107 N/m, kc-av=6×103 N/m, 6.7×107 N/m, and kp-av=2×103 N/m, 4×107 N/m, the chaotic vibration occurs in the transmission system of the motor.(2) As theoretical radius changes, the wave generator exhibits chaotic vibration whenrs is quite small.(3) The influence of wave generator offseta on chaotic vibration of wave generator is most obvious.



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*Source: 3216010-2017-10-12.xml*