Dynamic modeling and characteristics analysis of a modal-independent linear ultrasonic motor

Ultrasonics 72 (2016) 117–127

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Ultrasonics journal homepage: www.elsevier.com/locate/ultras

Dynamic modeling and characteristics analysis of a modal-independent linear ultrasonic motor Xiang Li ⇑, Zhiyuan Yao, Shengli Zhou, Qibao Lv, Zhen Liu State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

a r t i c l e

i n f o

Article history: Received 6 May 2015 Received in revised form 22 July 2016 Accepted 29 July 2016 Available online 30 July 2016 Keywords: Linear ultrasonic motor Whole machine model Energy method Dynamics

a b s t r a c t In this paper, an integrated model is developed to analyze the fundamental characteristics of a modalindependent linear ultrasonic motor with double piezoelectric vibrators. The energy method is used to model the dynamics of the two piezoelectric vibrators. The interface forces are coupled into the dynamic equations of the two vibrators and the moving platform, forming a whole machine model of the motor. The behavior of the force transmission of the motor is analyzed via the resulting model to understand the drive mechanism. In particular, the relative contact length is proposed to describe the intermittent contact characteristic between the stator and the mover, and its role in evaluating motor performance is discussed. The relations between the output speed and various inputs to the motor and the start-stop transients of the motor are analyzed by numerical simulations, which are validated by experiments. Furthermore, the dead-zone behavior is predicted and clarified analytically using the proposed model, which is also observed in experiments. These results are useful for designing servo control scheme for the motor. Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction Ultrasonic motors (USMs) are an attractive alternative to conventional electromagnetic drives for many applications in precision positioning due to their advantages such as high torque at low speed, quick response, accurate position, no electromagnetic and auto-locking on power outage [1,2]. The operation of the USMs is based on excitation of acoustic waves in a solid using the inverse piezoelectric effect. Functionally, USMs can be divided into two categories: traveling-wave ultrasonic motors (TWUM) and standing-wave ultrasonic motors (SWUM). One well-known design of the TWUMs is a ring-shape rotary motor proposed by Sashida in 1983, which is applied on focusing system of camera by Canon Co. in 1987 [3]. In contrast to the TWUM, the SWUMs are simpler to design and fabricate, among which a well-known design is a bimodal ultrasonic motor with longitudinal and bending (flexural) mode developed by the Israeli Company Nanomotion in 1995 [4]. The linear ultrasonic motor (LUSM) is an important branch of USMs, which can directly produce linear motion and direct output thrust. Since 1980s, various LUSMs have been designed and investigated [2,4–9]. The working principle of most LUSMs is to excite the stator of the motor to produce two orthogonal vibration modes.

⇑ Corresponding author. E-mail address: [email protected] (X. Li). http://dx.doi.org/10.1016/j.ultras.2016.07.018 0041-624X/Ó 2016 Elsevier B.V. All rights reserved.

Superposition of the two vibration modes generates an elliptical motion at the driving tip that can drive the slider to create a linear motion under a proper preload. In 2014, Zhou and Yao [10] proposed a novel modal-independent LUSM with double piezoelectric vibrators, which operates based on the kick-feet principle. The motor has good output performances because of superior frequency consistency that attributes to the decoupled vibration modes of the two vibrators. Note that the dynamic model not only can be used to predict and evaluate the performance of the USM, but also can serve as references for controller design [1,11–14]. However, it is not easy to develop mathematical models for USMs due to their complex contact mechanism. There are some in-depth analyses for the contact mechanism of TWUM (traveling wave contact type) [15,16]. However, the linear equivalent-spring model [17,18] is still the most common contact model describing the impact-like contact characteristic of SWUM (intermittent contact type). In this paper, a nonlinear spring is adopted to represent the normal interface force. The finite element method is an attractive approach for developing dynamic model [19,20], in which the Lagrange multiplier method is usually adopted to treat the contact problem between the stator and the rotor (slider). Energy method is found to be the most appropriate choice for addressing USM modeling. In 1995, Hagood and Mcfarland [21] initiated a general model for the TWUM using Hamilton’s principle, which lays the foundation for a general

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framework capable of representing numerous USMs and severing as a useful design tool for optimizing the configuration of the motors later. In [17], Tsai et al. developed an integrated dynamic model for a bi-modal standing-wave LUSM based on the energy method, and they investigated the force transmission between the stator and slider and the nonlinear deadzone of the motor which provided physical insights to the dynamics of the standing-wave LUSM. However, the effects of various inputs to the motor such as preload and excitation frequency on the motor performance are not discussed. Recently, an analytical model [22] was proposed to evaluate the performance of a plate SWUM using dual-frequency drive, and the initial load torque was determined by using probability theory to identify dead zone. However, the analytical model failed to capture the stator dynamics of the SWUM. The LUSM studied in this paper is a modal-independent LUSM reported in [10]. Although a mathematical model was developed to explain some basic characteristics of the motor, the model is too simple to analyze the dynamics of the motor effectively and an integrated model is still lacking. The purpose of this work is to develop an integrated model to understand the operation principle of the motor and analyze the dynamic behaviors of the motor. Several coupled dynamic equations governing the dynamics of the LUSM are deduced, into which the interface forces are incorporated. The behavior of the force transmission between the stator and the moving platform is investigated via the resulting model, which can shed more physical light on how the motor operates. The contact state of the stator and mover is discussed by a dimensionless parameter. Finally, the start-stop transients of the motor and the relationships between the output speed and various inputs are investigated numerically and experimentally, which provide valuable guidelines for controller design of the LUSM. 2. LUSM characterizations The motor investigated in this research is a modal-independent LUSM previously developed in [10], as shown in Fig. 1. The motor is mainly composed of a stator with a driving tip, a mover-A with a friction bar and a linear platform. The stator is fixed on the holding box by the flexible champing parts and the mover-A is fixed on the linear platform by fixing parts. The distance between the holding box and the mover-A can be adjusted by turning the bolt and the preload is applied and regulated in this way. The stator and mover-A compose two piezoelectric vibrators of the motor which vibrate in the x direction and the y direction, respectively, and the resonance frequencies of the two vibrators are very close to each other. The second longitudinal mode (L2-mode) of the stator and the first longitudinal mode (L1-mode) of the mover-A are utilized in this LUSM, and the frequency of L2-mode for the stator is 50; 298 Hz and that of L-1 mode for the mover-A is 50; 316 Hz. The operating mechanism of the motor is to excite the stator and the mover-A by applying two sinusoidal signals with the same frequency but a phase difference of p=2 to them such that the two vibrators vibrate at equal frequencies, perpendicularly to each other, with a phase difference of p=2. Therefore, under a preload the platform is driven by the friction in the x direction. The schematics of the two piezoelectric vibrators are shown in Fig. 2. Under the assumptions of modeling the stator as a bar with an varying cross-section and the mover-A as a plate, the two piezoelectric vibrators are made by sticking piezoelectric ceramics (PZT-8) onto the metal substrate made of 60Si2Mn steel. The polarization and wiring of the piezoelectric ceramics of the stator and the mover-A are also illustrated in Fig. 2. The champing parts of the stator and the mover-A are placed near the nodal planes of the longitudinal vibrations, and the effect of the champing parts on the longitudinal-mode vibrations of the two vibrators is ignored in modeling.

3. LUSM modeling Energy method and Newton’s law are adopted to construct the dynamic models of the stator and the mover-A in this section. The interface model is modeled in Section 3.1 and then the forced vibration equations of the two vibrators (the stator and mover-A) are derived in Section 3.2. Finally, the motion equation of the platform is deduced in Section 3.3. 3.1. The interface model To establish the interface model, one assumption that the driving plane made by Al2 O3 ceramic is rigid body and the driving tip is elastic body [10]. Two resultant forces act on the driving plane: a normal force F N and a tangential force F T . The relationship between the driving tip’s static deformation y (in meters) and the y direction force f ðyÞ (in N) is given by [10]

f ðyÞ ¼ a3 y3 þ a2 y2 þ a1 y;

ð1Þ

where a1 ¼ 44:2  106 , a2 ¼ 0:123  1012 and a3 ¼ 42:3  1018 . The schematic of the contact of the stator and mover-A in the y direction is illustrated in Fig. 3. The driving tip is simplified to an equivalent nonlinear spring, and the flexible champing parts of the stator are equivalent to a linear spring and the equivalent stiffness k is equal to 2:2  104 [10]. The vibration of the stator in the y direction is also considered in the contact model. The picture (a) in Fig. 3 represents the original state of the system when the preload is not applied. The picture (b) in Fig. 3 represents the static deformation of the driving tip after applying the preload F p only, which is denoted by dc . The normal direction contact dynamics of the stator and the mover-A under the excitation voltage is given by the picture (c) in Fig. 3, in which the vibration displacements of the stator and driving plane are y1 and y2 in the y direction, respectively. From the figure, it is not difficult to obtain that the driving tip will contact with the driving plane when the condition dc þ y1  y2 > 0 occurs, and a positive normal force acts on the driving plane. If the condition dc þ y1  y2 6 0 occurs, the driving tip will separate from the driving plane and the normal force is equal to zero. Thus, the normal force F N acting on the driving plane can be expressed as


FN ¼

f ðdc þ y1  y2 Þ; dc þ y1  y2 > 0 0;

dc þ y1  y2 6 0

;

ð2Þ

where f ðÞ expresses the cubic function denoted by Eq. (1), and dc can be calculated by Eq. (1) when the preload is known. y1 and y2 are determined by several coupled dynamic equations that will be developed later. The tangential force is the dynamic friction force caused by the relative motion of the driving tip and the driving plane in the horizontal direction. Using the Coulomb friction law, the tangential force F T can be expressed as

F T ¼ signðx_ 1  x_ 2 Þld F N ;

ð3Þ

where sign() is the sign function, and ld is the dynamic friction coefficient of the interface. x_ 1 and x_ 2 are the tangential velocities of the driving tip and the platform, respectively, which can be obtained from the dynamics of the stator and the platform developed later. 3.2. The vibrations of the two vibrators For simplification, the stator is considered to be a variable cross-section bar with champed boundary conditions at the L-2 mode nodal planes and free boundary conditions at the either end. The mover-A is considered to be a uniform-section plate with

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y

Mover-A

Linear platform

Fixing part

x PZT

Cross roller guide

Friction bar

Stator

Driving tip

PZT

PZT

Flexible clamping part

Preload bolt

Holding box

Fig. 1. Schematic of the LUSM.

(a)

(b)

Fig. 2. Schematics of the two piezoelectric vibrators. (a) The stator and (b) the mover-A.

y

Original state

Apply the preload only

Mover-A

Mover-A

Apply the voltage

x Mover-A y2

Driving plane

Stator

c

k

y2

Nonlinear spring δc

FP

Stator

FP

(a)

(b)

FN

Stator

y1 c

k

y1

c

k

FP

(c)

Fig. 3. The schematic of the contact of the stator and mover-A in the y direction.

champed boundary conditions at the L-1 mode nodal plane and free boundary conditions at the either end. Here two types of vibrations are considered: the first is the forced longitudinal vibration of the stator and the mover-A, and the other is the forced vibration of the stator in the y direction.

The forced longitudinal vibration equation of the stator is derived first. Based on the structure of the stator as shown in Fig. 2(a), four piezoelectric ceramic patches (PZTs) are attached to the stator symmetrically, and x0 is the distance of the PZT from the driving tip. The thickness effect of the PZT is ignored because of

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the thickness hp ¼ 0:5 mm of the PZT being very small, and the effect of a single PZT on the stator takes into account firstly. Hence, when the input voltage V 0 is applied to PZT, the average field intensity of the PZT is

E3 ðx; tÞ ¼

V 0 eixt hp

ð4Þ

The strain of the PZT in the x direction can be expressed as

e1 ðx; tÞ ¼

@uðx; tÞ ; @x

ð5Þ

where uðx; tÞ is the displacement of the longitudinal vibration of the stator. According to the second class of piezoelectric equations [2] one can obtain




E ¼ ½ 0 0 E3 

r1 ðx; tÞ ¼ cE11 e1 ðx; tÞ  e31 E3 ðx; tÞ

;

ð6Þ

where r1 ðx; tÞ is the stress of the PZT in the x direction, and cE11 , e31 are the piezoelectric parameter and the dielectric constant, respectively. By the assumed mode method, the displacement field is expressed as

uðx; tÞ ¼

1 X /j ðxÞqj ðtÞ;

ð7Þ

j¼1

where /j ðxÞ is the jth mode shape function for the longitudinal vibration of the stator and qj ðtÞ is the corresponding generalized coordinate. Utilizing the energy method, the longitudinal vibration equation of the stator under excitation of the distribute PZTs can be obtained [2]

€j ðtÞ þ C j q_ j ðtÞ þ K j qj ðtÞ ¼ F j ðtÞ; Mj q

ð8Þ

where Mj , C j , K j and F j ðtÞ are order j modal mass of the stator, modal damping, modal stiffness and modal force respectively. The modal force is given by

F j ðtÞ ¼

eixt 2hp

Z

ls =2

ls =2

e31 V 0 Sp /0j ðxÞdx;

ð9Þ

where ls is the length of the stator and Sp is the cross sectional area of the PZT on it. By using Duhamel integral, the modal force of a single PZT is

F j ðtÞ ¼

eixt 2hp

Z

ls =2

ls =2

e31 V 0 Sp /0j ðxÞdðx  x0 Þdx;

ð10Þ

where dðÞ expresses Dirac’s delta function. Note that the four PZTs on the stator are symmetrically, modal force is linearly addible and the total modal force under excitation of all PZTs is

2eixt F j ðtÞ ¼ hp

Z

ls =2

ls =2

e31 V 0 Sp /0j ðxÞdðx

 x0 Þdx

ð11Þ

Takes the dynamic friction force acting on the stator into account, the dynamic equation for the longitudinal mode of the stator is modified by addition of dynamic friction force

€j ðtÞ þ C j q_ j ðtÞ þ K j qj ðtÞ ¼ F j ðtÞ  /j ð0ÞF T Mj q

ð12Þ

Since only L-2 mode of the stator is considered, uðx; tÞ can be replaced by

uðx; tÞ ¼ /2 ðxÞq2 ðtÞ;

ð13Þ

where /2 ðxÞ is the mode shape function for the L-2 mode of the stator, q2 ðtÞ is the corresponding generalized coordinate.

The stator is considered to be a variable cross-section bar, which is decomposed into five subsections with the length li ði ¼ 1; 2; . . . ; 5Þ as shown in Fig. 2(a). Using mechanical vibration theory [23], the mode shape function for the L-2 mode of the stator without considering the PZTs can be determined by the boundary conditions and the compatibility conditions at contact areas of the five subsections 8 a1 cos aðx þ 0:058Þ; > > > > cx cx > > < a2 e cos bðx þ 0:017Þ þ a3 e sinbðx þ 0:017Þ; /2 ðxÞ ¼ a4 cos aðx þ 0:002Þ þ a5 sin aðx þ 0:002Þ; > > > a4 ecx cosbðx  0:002Þ þ a3 ecx sin bðx  0:002Þ; > > > : a6 cos aðx  0:017Þ þ a7 sin aðx  0:017Þ;

x 2 ðls =2; l2  l3 =2Þ x 2 ðl2  l3 =2;l3 =2Þ x 2 ðl3 =2; l3 =2Þ x 2 ðl3 =2; l3 =2 þ l4 Þ

;

x 2 ðl3 =2 þ l4 ;ls =2Þ ð14Þ

where the coefficients a1 ¼ 1:02, a2 ¼ 0:79, a3 ¼ 0:17, a4 ¼ 1:21, a5 ¼ 1:15, a6 ¼ 0:19, a7 ¼ 0:16, a ¼ 61:53, b ¼ 49:45, and c ¼ 36:63. The modal Eq. (12) is further modified as

€2 ðtÞ þ C 2 q_ 2 ðtÞ þ K 2 q2 ðtÞ ¼ F 2 ðtÞ  /2 ð0ÞF T M2 q

ð15Þ

Here M 2 , K 2 and C 2 correspond to the modal mass, modal stiffness and damping for the L-2 mode of the stator, respectively. The first term on the right side of Eq. (15) is the generalized force resulting from the excitation of the four PZTs on the stator which can be obtained from Eq. (11), and the second term is the generalized force associated with the dynamic friction force which can be obtained by using Dirac’s delta function and mode shape orthogonality. The vibration displacement of the driving tip x1 in the x direction can be expressed as

x1 ¼ uð0; tÞ ¼ /2 ð0Þq2 ðtÞ

ð16Þ

Based on the contact model shown in Fig. 3 and Newton’s second law, the vibration equation of the stator in the y direction is given as

€1 þ cy_ 1 þ ky1 ¼ F p  F N ; my

ð17Þ

where m, c and k correspond to the mass of the stator, equivalent damping and equivalent stiffness of the stator’s champing parts. Like full the derivation of the dynamic equation for the longitudinal mode of the stator, the modal force resulting from the excitation of the two PZTs on the mover-A is given as

F j ðtÞ ¼

eixt hp

Z

lm

0

 0 ðyÞdðy  y Þdy; e31 V 0 Sp / 0 j

ð18Þ

where lm is the length of the mover-A without the friction bar, Sp is the cross sectional area of a single PZT on it. Since only L-1 mode of the mover-A is utilized, the displacement field v ðy; tÞ can be expressed as

v ðy; tÞ ¼ / 1 ðyÞq1 ðtÞ;

ð19Þ

 1 ðyÞ is the mode shape function for the L-1 mode of the where / mover-A, q1 ðtÞ is the corresponding generalized coordinate. The longitudinal vibration of the plate can be approximately considered as the longitudinal vibration of a rod. So, with a freefree boundary condition, the mode shape function for the L-1 mode of the mover-A without considering the friction bar and the PZTs is given as

 1 ðyÞ ¼ cos p y / lm

ð20Þ

Takes the normal force acting on the mover-A into account, the dynamic equation for the L-1 mode of the move-A is given as

 1 ð0ÞF N ; 1 ðtÞ ¼ F 1 ðtÞ þ / €1 ðtÞ þ C 1 q _ 1 ðtÞ þ K 1 q M1 q

ð21Þ

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X. Li et al. / Ultrasonics 72 (2016) 117–127

where M 1 , K 1 and C 1 correspond to the modal mass, modal stiffness and damping for the L-1 mode of the mover-A, respectively. The first term on the right side of Eq. (21) is the generalized force resulting from the excitation of the two PZTs on the mover-A and can be obtained from (18), and the second term is the generalized force associated with the normal force which can be obtained by using Dirac’s delta function and mode shape orthogonality. The vibration displacement of the driving plane y2 in the y direction can be expressed as

 1 ð0Þq1 ðtÞ y2 ¼ v ð0; tÞ ¼ /

ð22Þ

3.3. The motion of the platform Through force equilibrium the motion equation of the linear moving platform is obtained

M€x2 þ C x_ 2 ¼ F T þ F f  F load ;

ð23Þ

Here M is the sum of the mass of the platform and the mover-A with fixing parts, since the mover-A is fixed on the platform by the fixing parts. C is the viscous damping coefficient and is chosen arbitrarily but kept small so to not significantly detract from the thrust output. F f is the friction force existing in the linear guide of the platform. F load is the applied load which can be used to measure the load capability of the LUSM, and the non-loaded velocity is obtained when F load ¼ 0. It should be note that the friction force F f includes the effect of the normal force F N and that of the gravity forces of the platform and the mover-A with fixing parts. The friction force existing in the linear guide is given as [17]


Ff ¼

F T ;

jF T j < jF f max j

F f max ; jF T j P jF f max j

;

ð24Þ

with

F f max ¼ signðx_ 2 Þls ðF N þ 9:81MÞ

ð25Þ

Here ls is friction coefficient of the linear guide. If the tangential force is smaller than F f max , the tangential force cannot overcome the friction and the platform remains still. A dead-zone area caused by this characteristic will be analyzed next. 4. Simulation and experimental validations In this section, numerical simulations are performed to study the transient and steady-state dynamics of the LUSM via the developed model. By using MATLAB software (The MathWorks, Natick, MA), a Runge–Kutta integration scheme is utilized to solve numerically the derived coupled dynamic equations, and the calculation step length h is taken for 1e  7 s because of the high working frequency. The structural parameters of the motor are listed in Table 1, and the material parameters can be found in [10], which are not listed here. The modal parameters including M 2 , K 2 , M 1 and K 1 are determined by the formulas listed in Appendix A, and the damping parameters are modified according to the experimental results. The friction parameters including ld and ls can be measured by experiments. The parameters used for the simulation are listed in Table 2 unless stated otherwise. The photograph of the prototype LUSM as well as the test devices used in the experiments is shown in Fig. 4. The force sensor (BK-2F, China Academy of Aerospace Aerodynamics, Beijing, PR China) between the preload bolt and the holding box is used to measure the preload. A laser sensor, a Renishaw Laser XL-80 (Renishaw Inc., Gloucestershire, UK) is used to measure the output speed of the LUSM, and the linear mirror of the laser sensor is fixed on the fixing part of the mover-A. Furthermore, the loaded speed can be measured when the rope holds weights through a fixed pulley.

Table 1 Structural parameters of the motor (mm). l1

l2

l3

l4

l5

bs

hs

hd

lm

bm

hm

x0

y0

41

15

4

15

41

16

4.5

1.5

50

30

10

32.5

26

Table 2 Simulation parameters. Parameter

Value

Excitation frequency f d Voltage amplitude V 0 Applied preload F p Mass for L2-mode of the stator M 2 Damping for L2-mode of the stator C 2 Stiffness for L2-mode of the stator K 2

50:3 kHz 200 V 40 N 0:0202 kg 39:75 Ns=m

Damping for L1-mode of the mover-A C 1

2:021  109 N=m 0:0572 kg 17:8 Ns=m

Stiffness for L1-mode of the mover-A K 1 Mass of the stator m Viscous damping coefficient c Total mass of the platform M Viscous damping coefficient C Friction coefficient of the interface ld Static friction coefficient of the linear guide

5:7204  109 N=m 0:15 kg 1:15 Ns=m 0:32 kg 1 Ns=m 0.3 0.04

Mass for L1-mode of the mover-A M 1

ls

4.1. The steady-state responses of the stator and mover-A To obtain a proper operating frequency, the steady-state vibration amplitudes of the driving tip and driving plane under the preload equal to 40 N and an excitation voltage peak-peak value of 200 V within the frequency range from 48 to 52 kHz are first calculated based on the vibration equations of the two vibrators, as shown in Fig. 5. It is found that the driving plane has the greatest normal vibration at the resonance frequency of 50:5 kHz, and the driving tip has the greatest tangential vibration at the resonance frequency of 50:3 kHz. In this study, the driving frequency is chosen to be 50:3 kHz, unless stated otherwise. The vibration amplitude of the driving plane in y direction at the steady state is compared with that of the stator in y direction shown in Fig. 6. It is observed that the amplitude of the stator in y direction (red1 heavy solid line) is much smaller than that of the driving plane (blue fine solid line), suggesting that the micro-amplitude oscillation of the stator in y direction can be ignored. The operating principle of the LUSM can be understood via the trajectory of the driving tip. It is worthwhile mentioning that the trajectory of the driving tip is an ellipse relative to the longitudinal vibration of the driving plane which is called the ‘relative elliptical motion’ in this research. The relative elliptical motion of the driving tip with a p=2 phase difference is compared with that of the driving tip with a p=2 phase difference shown in Fig. 7. It is found that the relative elliptical motions under the two different phase differences have opposite directions, which implies the motion direction of the platform can be reversed by changing the phase difference from p=2 to p=2. 4.2. The force transmission during the transient and steady states To understand the drive mechanism of the LUSM from the viewpoint of mechanics, numerical simulations are performed to analyze the force transmission between the stator and the linear platform in this section using the numerical observation method proposed in [17]. The time-domain responses of the velocity and displacement of the moving platform are shown in Fig. 8. It is found that the time 1 For interpretation of color in Fig. 6, the reader is referred to the web version of this article.

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X. Li et al. / Ultrasonics 72 (2016) 117–127

Mover-A

Linear platform

Fixed pulley

Rope Stator Holding box

Base

Preload bolt Force sensor

x 10

6

Wires

Fig. 4. The prototype of the LUSM and the test devices.

Displacement at the driving tip relative to the driving plane (m)

Linear mirror

-6

π/2 -π/2

4

2

0

-2

-4 -4

-3

-2

-1

0

1

2

Displacement at the driving tip in x-direction (m) x 10

Driving plane Driving tip

Vibration amplitude (m)

7 6 5 4 3 2 1 0 48

48.5

49

49.5

50

50.5

51

51.5

52

Excitation frequency (kHz) Fig. 5. The vibration amplitudes of the driving tip and driving plane versus excitation frequency.

8 6

x 10

-6

Driving plane vibration amplitude in y-direction Stator vibratiion amplitude in y-direction

4

Amplitude (m)

4 x 10

-6

Fig. 7. The relative elliptical motions of the driving tip under the phase difference equal to p=2 and  p=2.

-6

8

3

2 0 -2 -4 -6 9000 9100 9200 9300 9400 9500 9600 9700 9800 9900 10000

Time (μs) Fig. 6. The vibration amplitudes of the driving plane and the stator in y direction.

period at the starting stage is very short and steady-stage is reached in roughly 40 ms. Furthermore, the displacement of the moving platform, which is obtained by integration of the velocity, increases linearly at the steady-state stage, as expected. Two time spans, A and B marked in Fig. 8, represent the time periods at the starting and steady-state stages. The simulation results of the force transmission between the stator and the moving platform at the time spans A and B are shown in Figs. 9 and 10, respectively. From Fig. 9, it is observed that the positive work done by the tangential friction force F T on the platform is much more than the negative work within a cycle T, which can explain the velocity of platform increases quickly at the starting stage. As shown in Fig. 10, the time span B is further divided into four periods P1 —P 4 . The overall period P 1 —P4 (around 20 ls) represents a cycle for the driving tip to perform a complete relative elliptical motion. As the driving tip velocity x_ 1 is higher than the platform velocity x_ 2 at the period P1 (from t 1 to t 2 ), the tangential force F T is a positive driving force such that the platform is pushed forward. However, at the periods of P 2 (t2 to t3 ) and P 4 (t4 to t5 ), the velocities of the driving tip are lower than the velocities of the platform. Thus, the tangential force F T is a negative resistance such that the platform is pulled backward. The driving tip is separated from the driving plane at the period P3 (t 3 to t 4 ), and the tangential force is zero because the normal force F N vanishes at this period. 4.3. Analysis of the contact state between driving tip and driving plane Because the performances of USM depend considerably on contact state between the stator and rotor (slider) [18,19,24], it is necessary to discuss the effects of contact state on the dynamics of the LUSM. Although it has been reported that the contact state in a traveling wave USM can be measured by using an electric contact method [24], the test is not only complex but also costly. Here, a simple numerical method is proposed to analyze contact state between the stator and the driving plane based on the timedomain response of the normal force F N at the steady state. Fig. 11 shows the time-domain steady-state response of the normal force. It is found that every contact period is a repeat of the previous one, and thus the contact states at the steady state can be represented by the contact state at a single cycle. It should be noted that this contact characteristic is not true for the dualfrequency drive [22].

X. Li et al. / Ultrasonics 72 (2016) 117–127

0.6 B

0.025

0.4

0.02

0.3

0.015

0.2

0.01

A

0.1

Displacement (m)

0.5

Velocity (m/s)

In order to describe the contact state of the stator and the driving plane, the relative contact length (RCL) k is introduced and defined as

0.03 Velocity Displacement

0.005

0

0 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Time (s) Fig. 8. The time-domain responses of the velocity and displacement of the moving platform.

80

N, m/s

0

-20 -40 T

-60

2030

2035

2040

2045

2050

2055

2060

Time (μs) Fig. 9. The driving tip/platform velocities and the tangential force at the time period A.

80

Tangential force Platform velocity x 100 Tip velocity x 100

60 40

N, m/s

ð26Þ

where t c and t s represent the contacting and separating time of the stator and the driving tip within a single cycle T, respectively. It is clear that the value of the relative contact length k varies between 0 and 1, and the driving tip has a period separating from the driving plane when k < 1 while the driving tip is not able to separate from the driving plane throughout the cycle when k ¼ 1. Fig. 12 illustrates the numerical method for calculating the RCL at the steady state. As shown in Fig. 12, a single cycle T at the steady state is divided into a series of discrete time nodes with evenly spaced time step h, which are denoted by t n1 ; tn2 ; . . . ; tnN and the corresponding discrete normal force nodes are denoted by F n1 ; F n2 ; . . . ; F nN . The subscript n represents the number of the contact periods. Note that the normal force F N is positive when the driving tip contacts with the driving tip and the number of positive normal force nodes can be obtained using MATLAB function. Accordingly, the relative contact length k can be calculated approximately by

    T 1 ¼ int N ¼ int h fd  h

20

20 P3

P2

0

P1

P4

-20 -40 -60 t1

-80 4.716

tc ts ¼1 ; T T

ð27Þ

with

40

-80 2025

k¼

 numðfjF nj > 0; j ¼ 1; 2; :::; N gÞ k¼ N

Tangential force platform velocity x 100 Tip velocity x 100

60

123

4.7165

t2

t3

4.717

t4

4.7175

4.718

Time (μs)

t5

4.7185

4.719

4.7195

x 10

4

Fig. 10. The driving tip/platform velocities and the tangential force at the time period B.

ð28Þ

Here num() represents the count function that counts the number of elements in a set, and int() represents the integral function. N is the number of time nodes in a single cycle T, and f d is the driving frequency. To analyze how the contact states between the driving tip and driving plane affect the motor performance, we propose two observations (RCL curve and steady-state velocity curve) in a single figure based on the simulation results. Fig. 13 shows the RCL and the moving platform velocity at the steady state under different values of voltage. Three different regions marked with A–C in Fig. 13 are used to demonstrate the relationship between the contact states and the platform velocity when the voltage varies. In region A ðV 0 ¼ 20—40 VÞ, which can be described as the dead-zone area in which the tangential force cannot overcome the static friction force, thus the platform remains still. In region B ðV 0 ¼ 40—80 VÞ, the tangential force is able to overcome the static friction force as the voltage increases, and the platform starts to move. However, the excitation voltage is not larger enough to produce the separation of the driving tip from the driving plane due to the RCL equal to 1 in region B. The driving tip starts to have a period separating from the driving plane when the input voltage increases beyond 80 V, where the RCL is lower than 1, as shown in region C in Fig. 13. Similar results are obtained when the preload varies as shown in Fig. 14 in which three different regions marked with A–C are also used to demonstrate the relationship between the contact states and the platform velocity under different values of preload. In region A ðF p ¼ 40—140 NÞ, the driving tip has a period separating from the driving plane due to the RCL lower than 1. However, the driving tip will not separate from the driving plane when the preload increases beyond 140 N, which is mainly because the static deformation of the driving tip increases with the increase of preload. As expected, the platform velocity will decline to zero as the preload goes up to some extent as shown in region C ðF p ¼ 160—200 NÞ in Fig. 14, which can be described as the deadzone area. This is because the maximum static friction force F f max is dramatically influenced by the normal force F N that will change

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X. Li et al. / Ultrasonics 72 (2016) 117–127

4.4. Analysis of the relations between the output speed and various inputs to the motor

250

150

N

100

50

0 Separation stage

-50 4.739

4.74

4.741

4.742

4.743

4.744

4.745

4.746

Time (μs)

4.747

x 10

4

Fig. 11. The time-domain response of the normal force at the steady state.

FN Fnj +1 Fnj

0

tc

ts

nT h t n1 tnj tnj +1

(n+1)T t nN

t

Fig. 12. Numerical method for calculating the RCL at the steady state.

with the applied preload. In looking into Figs. 13 and 14, it can be concluded that the non-separation behavior (RCL = 1) between the driving tip and driving plane is unfavorable for the output speed of the motor.

In this section, the relationships between the output speed of the motor and various inputs including voltage amplitude, preload, driving frequency and applied load are discussed by virtue of numerical and experimental methods. Fig. 15 shows the relationship between non-loaded speed and voltage amplitude under the preload F p ¼ 40 N and driving frequency f d ¼ 50:3 kHz. The output speed is found to increase with the increasing voltage amplitude. A nonlinear speed saturation phenomenon is observed in experiments when the input voltage is high (more than 300 V), which cannot be fully captured in the presented model. This issue could be addressed by taking into account the nonlinearity of piezoelectric element in modeling, which is not discussed in the present work. In addition, a deadzone is observed both in the simulations and experiments at low voltage, which is explained in previous section. Another desirable property found in Fig. 15 is the output speed is linearly dependent on the amplitude of the voltage signals in a certain region ð100—300 VÞ, which is favorable for the speed servo control design. Fig. 16 shows the relationship between non-loaded speed and preload under the voltage V 0 ¼ 200 V and driving frequency f d ¼ 50:3 kHz. From Fig. 16, it is observed that the non-loaded speed of the motor increases firstly with the preload, then gradually decreases with the increase of the preload. The physical reasons for the effects of the preload on the non-loaded speed are explained as follows: the interface friction force increases as the preload increases, which is favorable for the acceleration of the mover-A in some extent. On the other hand, the tangential amplitude of the driving tip associated with the non-loaded speed decreases with the increasing preload, which is unfavorable for the non-loaded speed. In addition, the RCL dramatically increases when the preload is too high as shown in Fig. 14, which might result in increased duration in a cycle for braking force on the mover-A, causing a decrease in non-loaded speed of the motor. Therefore, an optimal applied preload (about 30 N) exists for maximizing the non-loaded speed as shown in Fig. 16. Fig. 17 shows the relationship between non-loaded speed and driving frequency under the voltage V 0 ¼ 200 V and preload F p ¼ 40 N. From the figure, it is observed that the LUSM has dead

1.2

0.55

1.1 1

Relative contact length

0.5

RCL Velocity

0.45

0.9

0.4

0.8

0.35

0.7 0.6

0.3 C

B

A

0.25

0.5

0.2

0.4

0.15

0.3

0.1

0.2

0.05

0.1

0

0 20

40

60

80

100

120

140

160

180

-0.05 200

Voltage (V) Fig. 13. The RCL and the steady-state velocity of the platform under different voltage.

Steady-state velocity of the platform (m/s)

Normal force F (N)

200

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X. Li et al. / Ultrasonics 72 (2016) 117–127

1.1

0.9 0.8

RCL Velocity

0.9

0.7

0.8

0.6

0.7

0.5

A

0.6

C

B

0.4

0.5

0.3

0.4

0.2

0.3

0.1

0.2

0

0.1 40

60

80

100

120

140

160

Steady-state velocity of the platform (m/s)

Relative contact length

1

-0.1 200

180

Preload (N) Fig. 14. The RCL and the steady-state velocity of the platform under different preload.

0.7

Non-load speed of the platform (m/s)

Non-load speed of the platform (m/s)

1.2 Simulation Experiment

1 0.8 0.6 0.4 0.2 0 -0.2

0

50

100

150

200

250

300

350

Simulation Experiment

0.6 0.5 0.4 0.3 0.2 0.1 0 45

400

46

47

Fig. 15. Averaged steady-state speed of the platform versus voltage amplitude.

49

50

51

52

53

54

55

Fig. 17. Averaged steady-state speed of the platform versus excitation frequency.

0.7

0.5 Simulation Experiment

0.45

Simulation Experiment

0.6

0.4

0.5

0.35

Speed (m/s)

Non-load speed of the platform (m/s)

48

Excitation frequency (kHz)

Voltage amplitude (V)

0.4 0.3

0.3 0.25 0.2 0.15

0.2

0.1 0.1

0.05 0

0

20

40

60

80

100

120

140

160

180

200

Preload (N) Fig. 16. Averaged steady-state speed of the platform versus applied preload.

0 0

5

10

15

20

25

30

35

40

Load (N) Fig. 18. Averaged steady-state speed of the platform versus applied load.

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X. Li et al. / Ultrasonics 72 (2016) 117–127

150

speed drops with the increasing applied load, and when the load is 40 N the motor will stop. This implies that the maximum thrust of the motor is about 40 N under this initial input. However, it should be noted that the load capability of the LUSM can be improved by selecting a proper applied preload [10].

3

100

2

50

1

0

0

-50 0.005

0.01

0.015

0.02

4.5. On/off transients of the LUSM

-1 0.025

Time (s) Fig. 19. The responses of the platform displacement and velocity to turning the motor on for only 20 ms (simulation results).

Functional Generator (Tectronix, AFG3022)

PC

Power Amplifier (Foneng, HFVP-153)

Laser sensor (Renishaw, XL-80)

LUSM

Fig. 20. Experimental setup for measuring the start-stop transients of the LUSM.

zones when the driving frequency is considerably off the resonance frequency, and the serious nonlinearity is observed when the excitation frequency is close to the resonance frequency of 50:3 kHz that may be associated with the contact nonlinearity. This indicates that the excitation frequency is not a preferred control input. Finally, the relation between output speed and applied load, under the voltage V 0 ¼ 200 V, preload F p ¼ 40 N and driving frequency f d ¼ 50:3 kHz, is shown in Fig. 18. As expected, the output

Considering the characteristic of the quick response of USM, the transients to turning the motor on and off are investigated numerically and experimentally in this section. With the developed model, the responses of the moving platform displacement and velocity to turning the motor on for only 20 ms are simulated under the voltage V 0 ¼ 100 V, preload F p ¼ 40 N and driving frequency f d ¼ 50:3 kHz, and the results are shown in Fig. 19. It is found that the steady-stage of the velocity is reached in roughly 15 ms, and the velocity decays linearly over time after turning the motor off at time 20 ms and reduces to zero at time 22:5 ms which conforms to Coulomb friction model. Furthermore, the motor is left on for twenty milliseconds and the platform moves about 2:3 mm, and the resolution could be increased if the input voltage envelope is modulated properly. Using the experimental setup shown in Fig. 20, the transients to turning the motor on and off can be measured, and the sampling frequency of the laser sensor is set to be 100 kHz and the experimental conditions are same as the simulation. Fig. 21 shows the responses of the moving platform displacement and velocity to turning the motor on for only 20 ms. It is found that the experimental results agree basically with the simulation results presented in Fig. 19. However, the velocity is found to fluctuate largely at a period of time ð0:006—0:014 sÞ, which could be caused by wear behavior at the contact interface and instability of the platform in the experiments. 5. Conclusion In this paper, a whole-machine dynamic model including a contact model and four coupled governing equations is developed to investigate the dynamical characteristics of a modal-independent LUSM. The force transmission between the stator and the moving platform is analyzed to provide more physical understandings for the operating principle of the motor. One contribution of this work

150

3

Velocity (mm/s)

Displacement Velocity 100

2

50

1

0

0

-50

0

Displacement (mm)

0

Displacement (mm)

Velocity (mm/s)

Displacement Velocity

-1 0.0025 0.005 0.0075 0.01 0.0125 0.015 0.0175 0.02 0.0225 0.025

Time (s) Fig. 21. The responses of the platform displacement and velocity to turning the motor on for only 20 ms (experiment results).

X. Li et al. / Ultrasonics 72 (2016) 117–127

is to investigate the contact state between stator and driving plane via the defined RCL, and its role in evaluating motor speed is discussed. It is shown that non-separation behavior (RCL = 1) between the stator and driving plane is unfavorable for output speed of the motor. Differing from structural analysis for the LUSM presented in [10], the present work focuses on model development for the LUSM and some complex dynamical behaviors such as deadzone behavior and start-stop transient dynamics of the motor are discussed, which cannot be analyzed by the mathematical model proposed in the previous work. Furthermore, the relationships between the output speed and several possible control inputs including amplitude/frequency of the excitation voltage, preload and applied load are established via the developed model, which are useful for designing servo control scheme for the motor. Acknowledgments This work is supported by the Program of National Natural Science Foundation of China (No. 51275229), and by the 973 Program of National Basic Research (No. 2011CB707602). Appendix A After taking out of the piezoelectric elements on the stator and mover-A, the explicit expresses of modal mass and modal stiffness for L-2 mode of the stator and L-1 mode of the mover-A are given by

Z M2 ¼

ls =2

Z K2 ¼

ls =2

ls =2

Z

lm

0

Z K1 ¼ 0

lm

2

ðA:1Þ

EAs ðxÞð/02 ðxÞÞ dx;

ðA:2Þ

qAm ð/ 1 ðyÞÞ dy;

ðA:3Þ

 0 ðyÞÞ2 dy; EAm ð/ 1

ðA:4Þ

ls =2

M1 ¼

qAs ðxÞð/2 ðxÞÞ2 dx;

2

where q and E are the density and Young’s modulus of 60Si2Mn steel respectively considering that both the stator and mover-A are made of 60Si2Mn steel. Am ¼ bm hm is the cross sectional area of the mover-A. As ðxÞ is the cross sectional area of the varying cross-section stator, which is a piecewise function of longitudinal coordinate x and is given by

8 bs hs ; > > > > > > < bs hs x=l4 ; As ðxÞ ¼ bs hd ; > > > > bs hs x=l4 ; > > : bs hs ;

ls =2 < x < l2  l3 =2 l2  l3 =2 < x < l3 =2 : l3 =2 < x < l3 =2 l3 =2 < x < l3 =2 þ l4 l3 =2 þ l4 < x < ls =2

ðA:5Þ

127

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