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Modeling and analysis of stick-slip motion in a linear piezoelectric ultrasonic motor considering ultrasonic oscillation effect
International Journal of Mechanical Sciences 107 (2016) 215–224
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Modeling and analysis of stick-slip motion in a linear piezoelectric ultrasonic motor considering ultrasonic oscillation effect Xiang Li a,n, Zhiyuan Yao a, Ranchao Wu b a b
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China School of Mathematics, Anhui University, Hefei 230039, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 27 September 2015 Received in revised form 13 January 2016 Accepted 14 January 2016 Available online 21 January 2016
Piezoelectric ultrasonic motor is an electromechanical coupling system, which is driven by the interface friction between the stator and the rotor/slider. A novel friction model is developed to study the stick–slip motion in a linear piezoelectric ultrasonic motor using longitudinal and bending modes. In this model, the influence of the ultrasonic oscillation on the dynamic friction coefficient is taken into account, which is described by a friction law related to the vibration amplitude at the interface. In addition, the friction contact model involves intermittent separation non-linearity induced by the variable normal force, which can result in complex stick–slip–separation motion at the contact interface. Analytical transition criteria between stick and slip are established by using the switch model. Based on the developed model, numerical simulations are performed to analyze the force transmission between the stator and the slider involving stick–slip motion. A transformed phase plane is proposed to study various states of the contact interface (stick, slip and separation), on which slip–separation, stick–slip, stick–slip–separation, and pure stick motions are intuitively characterized. The influence of some input parameters on the stick–slip motion is analyzed. Furthermore, the low-voltage dead-zone behavior is clarified by stick motion. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Friction Stick–slip Vibration Switch model Piezoelectric ultrasonic motor
1. Introduction Piezoelectric ultrasonic motor is a newly developed motor, which has received much attention due to their unique characteristics such as high power density, high torque, fast response and compact size [1,2]. In particular, the piezoelectric motor has minute control characteristics because it utilizes large mechanical friction and needs no additional braking system. Linear piezoelectric ultrasonic motor is one branch of piezoelectric ultrasonic motors, which can move in straight line and has much freedom design and high-thrust force [3,4]. As an electromechanical coupling system, a common characteristic of all piezoelectric ultrasonic motors is their energy conversion [1]. In the first stage, piezoelectric components convert electrical energy into high-frequency (ultrasonic frequency) mechanical oscillations. Depending on the geometry of the stator (vibrator) and the piezoelectric excitation, generally two orthogonal vibration modes (mainly longitudinal and bending, but also torsion) are superposed on an elliptical motion of the surface/driving tip. In the second stage, the high-frequency vibration is rectified into macroscopic unidirectional rotary or linear motion of a rotor/slider (driven component) n
Corresponding author. Tel.: þ 86 02584895853. E-mail address: [email protected] (X. Li).
http://dx.doi.org/10.1016/j.ijmecsci.2016.01.016 0020-7403/& 2016 Elsevier Ltd. All rights reserved.
based on the contact mechanisms occurring at the interface between the stator and the rotor/slider. The stick–slip motion between the stator and the slider of a linear ultrasonic piezoelectric motor occurring at the second stage of the energy conversion will be discussed in detail in the following chapters of this paper. Stick–slip motion exists in many machineries with the relative motion of contact surfaces, and a considerable number of theoretical and experimental studies have been made to investigate stick–slip motion [5–10]. Because the piezoelectric ultrasonic motors are driven by the friction resulting from the relative motion of the stator and the rotor/slider, stick–slip motion in piezoelectric motors has significant influence on the dynamics of the motors. It is known that the slipping can cause the motor to consume more energy, which will reduce the efficiency and shorten the life. On the other hand, the sticking can cause the output speed of the motor to decline, and even lead to deadzone behavior. However, stick–slip motion is very difficult to measure in experiment due to the characteristic of high-frequency vibration of piezoelectric ultrasonic motor. As a part of the contact mechanisms of piezoelectric motors, stick–slip motion has been studied by virtue of various contact models [11–18]. In 1992, Maeno et al. [11] studied the stator/rotor contact in a traveling wave rotary piezoelectric motor by finite element method. They indicated that the distribution of the stick/slip exists in the contact area. In 1995, Cao and Wallaschek [12] proposed a visco-elastic contact model for traveling wave piezoelectric
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motor. They studied stick–slip behavior in detail and indicated that three different slip and two different stick zones exist. In 2000, Moal et al. [15] studied the mechanical energy transductions in a standing wave piezoelectric motor based on an analytical model in which the stick and slip phases are characterized analytically. In 2006, Qu et al. [18] investigated the influence of contact mechanics on the performance of a traveling wave rotary piezoelectric motor based on a viscoelastic contact model, in which the stick–slip motion was considered. They introduced a new judge function to judge whether the conversion between stick zone and slip zone. Given the majority of the publications in this field are devoted to the study of rotary piezoelectric ultrasonic motors and the velocity of the rotor is assumed to be constant, this paper focuses on the stick–slip motion of a linear piezoelectric ultrasonic motor based on an analytical model in which the slider velocity is time dependent. Therefore, not only this model can be used to investigate the stick–slip motion at the steady stage but also it can be used to study the stick–slip motion at the transient stage. Different from the previous work, the influence of the ultrasonic oscillation on the dynamic friction is considered in this paper, which can further affect the stick–slip motion. This paper develops an analytical model to study the stick–slip motion in a typical linear piezoelectric ultrasonic motor utilizing longitudinal vibration and bending vibration modes. Based on the dynamic model proposed in [19], a friction model subjected to both friction non-linearity and intermittent separation non-linearity is established taking into account the influence of ultrasonic vibration. In this model, the dynamic friction coefficient is modeled by an amplitude-dependent law. The transition criteria between stick and slip are deduced via the switch model, which are applied effectively in numerical simulations. The force transmission between the stator and the slider involving complex stick–slip–separation motion is analyzed numerically, which is helpful to further understand the operation mechanism of the motor. Various friction states including slip–separation, stick–slip, stick–slip–separation and pure stick are characterized by steady-state trajectories on a transformed phase plane. Numerical simulations are performed to study the influence of the possible parameters on the stick–slip motion based on the developed model, and the results indicate that the preload denotes the easiness for the occurrence of stick–slip motion and the input voltage amplitude acts to suppress the occurrence. Furthermore, the dead-zone behavior caused by low input voltage can be predicted by the developed stick–slip model and clarified by pure stick motion.
Linear motion
y
Slider
x
j k
Contact tip
Linear guide 1
i
2
V sin ωt
Stator V sin ωt
3
4
Preload spring
Fig. 1. Physical model of the linear piezoelectric ultrasonic motor.
Longitudinal vibration
Bending vibration
Linear motion Slider Elliptic motion
Fig. 2. Drive mechanism of the linear piezoelectric ultrasonic motor utilizing longitudinal vibration and bending vibration modes.
Furthermore, the direction of motion of the slider can be changed by applying the excitation signals to the electrodes 2–3.
3. Modeling of stick–slip motion 2. Description of the physical model The physical model shown in Fig. 1 is a typical bimodal linear piezoelectric ultrasonic motor utilizing the first longitudinal and second bending modes reported in [20]. It consists of a stator with a contact tip, a preload spring, a slider and a linear guide. The stator is a rectangular piezoelectric plate of dimensions 30 mm 7:5 mm 3 mm (length (l), width (bs ) and thickness (hs )), which is served as a piezoelectric vibrator in the system. To excite both longitudinal and bending modes of the stator, the polarization direction of the stator is in the thickness direction (k-direction in Fig. 1). The electrode is divided into four sections denoted by 1, 2, 3 and 4, as shown in Fig. 1. Fig. 2 illustrates the drive mechanism of the linear piezoelectric ultrasonic motor. When one pair of sinusoidal signals with a single driving frequency close to the natural frequencies of the first longitudinal mode (L1-mode) and the second bending mode (B2mode) of the stator are applied to the electrodes 1–4, the stator is excited in a longitudinal vibration and a bending vibration simultaneously, again generating an elliptic motion at the contact tip of the stator. As a result, a friction force is produced at the contact interface under a preload provided by the preload spring, which can drive the slider to perform a macroscopic linear motion.
In [19], Tsai et al. have developed an integral dynamic model for a linear piezoelectric motor utilizing longitudinal vibration and bending vibration modes, in which the dynamics of the electrical driving circuit was considered. This paper concentrates on the mechanical energy conversion rather than on the electrical energy conversion. Here, the driving voltage is directly applied to the electrodes of the stator as shown in Fig. 1, and the electrical driving circuit equations are ignored in the following modeling process. 3.1. Analytical model For simplification, the short stubby stator is assumed to be a Timoshenko beam and the contact tip on the stator is assumed to be rigid. A cross section of the Timoshenko beam (the stator) is shown in Fig. 3, and the displacement field ðu ¼ u1 i þ u2 j þ u3 kÞ of a material point in the beam can be expressed as: 8 u1 ðy; tÞ ¼ wðy; tÞ; > > > > < u2 ðx; y; tÞ ¼ uðy; tÞ xψ ðy; tÞ ð1Þ ; ¼ uðy; tÞ xð∂w > ∂y þ β ðy; tÞÞ; > > > : u ðy; tÞ ¼ 0; 3
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type in which the contact area between the stator and the rotor/ slider moves along with a traveling wave on the stator. The second is an “intermittent” contact type, in which the two bodies are in contact intermittently and the contact area of the stator remains the same. As a standing wave piezoelectric motor, the linear piezoelectric motor discussed in this paper belongs to the latter contact type motors. The contact mechanism at the interface is described by an elastic spring existing in between the contact tip and the slider [19]. The normal force acting on the interface is given as: ( F P þ kðud ust Þ; F P þ kðud ust Þ 4 0 FN ¼ : ð7Þ 0; F P þ kðud ust Þ r 0 Fig. 3. The cross section of Timoshenko beam (the stator).
where uðy; tÞ is the axial displacement of the beam at center point O. wðy; tÞ is the transverse displacement of the beam. ψ ðy; tÞ is the rotation of the line element along the centerline, and β ðy; tÞ is the shear angle. It is noted that the rotation ψ ðy; tÞ can be removed in the dynamics equations of the stator, because the natural frequency that corresponds to the mode of rotation is much higher than the driving frequency [19]. By assumed mode method, the displacement fields can be expressed as: uðy; tÞ ¼ ϕL ðyÞul ðtÞ; ð2Þ wðy; tÞ ¼ ϕB ðyÞwb ðtÞ; ; where ul ðtÞ and wb ðtÞ are the generalized coordinates. ϕL ðyÞ and ϕB ðyÞ are the corresponding assumed mode shape functions. Here ϕL ðyÞ and ϕB ðyÞ are the eigenfunctions of a Bernoulli–Euler beam corresponding to L1-mode and B2-mode vibration, respectively. With a free-free boundary condition, the two mode shape functions [21] are given as: ( ϕL ðyÞ ¼ cos πly; ð3Þ ϕB ðyÞ ¼ α2 ð cos β2 y þ cosh β2 yÞ þ ð sin β2 y þ sinh β2 yÞ; where β2 ¼ 2:499753π =l; α2 ¼ ð sin β2 l sinh β2 lÞ=ðcosh β2 l cos β2 lÞ: Utilizing Hamilton’s principle and the assumed mode method [2,19], the dynamic equations for the L1-mode and B2-mode of the stator are given by M l u€ l ðtÞ þ C l u_ l ðtÞ þ K l ul ðtÞ ¼ Al V 0 sin ωt þ ηN F N ;
ð4Þ
€ b ðtÞ þ C b w _ b ðtÞ þ K b wb ðtÞ ¼ Ab V 0 sin ωt þ ηT F T ; Mb w
ð5Þ
where M l ; C l ; K l and M b ; C b ; K b are the mass, damping and stiffness for the L1-mode and B-2 mode of the stator, respectively. V 0 is the amplitude of the exciting voltage. ω is the circular frequency of the exciting voltage. F N and F T represent normal force and tangential force acting on the interface between the stator and the slider, respectively. Al and Ab are the force factors related to the electromechanical coupling effects. ηN and ηT are the load coefficients responding to the normal force and tangential force, respectively. The explicit expresses of the parameters in (4) and (5) are given in Appendix A. Through force equilibrium, the motion equation of the slider is given by M x€ ¼ F T þ F f ; where F f ¼ signð x_ Þμsd ðF N þMgÞ:
ð6Þ
Here M is the mass of the slider, x_ and x€ are the velocity and acceleration of the slider in the x direction, respectively. μsd is the dynamic friction coefficient of the linear guide, g is the gravitational acceleration. sign() represents the sign function. From the contact viewpoint, piezoelectric ultrasonic motors can be divided into two groups [22]. One is called a “traveling” contact
Here F P is the preload and k is the stiffness of the equivalent spring. ud is the total dynamic longitudinal elongation of the stator and is equal to ðϕL ðlÞ ϕL ð0ÞÞ Uul . ust is the static elongation of the stator under the preload only and is equal to ηN F P ðϕL ðlÞ ϕL ð0ÞÞ=K l . It is noted that if the condition of F P þ kðud ust Þ r0 occurs, the contact tip will separate from the slider and the interface normal force is vanished. The tangential force resulting from the friction force at the interface is related to the relative velocity of the contact tip and the slider in the tangential direction. The relative velocity is defined as: vrel ¼ x_ tip x_ ;
ð8Þ
where x_ tip is the tangential velocity of the contact tip and is equal to
ϕB ðlÞw_ b , which can be obtained from the stator dynamics. Depending on the variable normal force, the contact situation between the contact tip and the slider in the normal direction can be divided into two states: the contact state (F N 40) and the separation state (F N ¼ 0). There are two types of friction forces in the contact state: static friction force F s corresponding to the stick phase and the dynamic friction force F d corresponding to the slip phase. The friction force is not present during the separation state owing to the normal force equal to zero. In light of Coulomb friction model, the tangential force acting on the interface is expressed as: 8 ) > F A μs F N ; μs F N ; F N 40 and vrel ¼ 0; ðstickÞ > < s contact F T ¼ F d ¼ signðvrel Þμd F N ; F N 4 0 and vrel a 0; ðslip Þ > > :0 F ¼ 0; separation N
ð9Þ where μs and μd are the static and dynamic friction coefficients of the interface, respectively. The static friction force in the stick phase is limited by the maximum static friction force, i.e. jF s j r μs F N . During the contact state, the transition point from stick to slip is reached when jF s j 4 μs F N . If the contact tip is sticking to the slider then x€ tip ¼ x€ ; and one can obtain the sticking force F s by using Eqs. (5) and (8) Fs ¼
ϕB ðlÞ Mb
_ b þ K b wb Ab V 0 sin ωtÞ þ ðC b w
Ff ϕB ðlÞηT 1 = : M Mb M
ð10Þ
It is worth noting that the ultrasonic oscillation has influence on the dynamic friction. It is known that dynamic friction is related to the vibration amplitude at the contact surface [23]: The dynamic friction coefficient decreases with increasing vibration amplitude. The dynamic friction coefficient in this paper is modeled by the following amplitude-dependent law:
μd ðAN Þ ¼ aμ þ ðμd aμ Þe bμ AN ;
ð11Þ
which exponentially decays to a residual value of friction aμ , as shown in Fig. 4. AN is the normal vibration amplitude at the contact tip, which can be used to represent the vibration amplitude at the contact interface. μd represents the dynamic friction coefficient without considering the influence of the ultrasonic oscillation. Parameter bμ in Eq. (11) is an attenuation factor that is used
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4. Numerical simulation and discussion
to characterize the decay rate of the dynamic friction coefficient μd with increasing vibration amplitude AN .
In this section, numerical simulation was conducted with the governing equations derived in Section 3. The Runge-Kutta method was used for the numerical integral of the ordinary differential equations in the model. In this paper, the piezoelectric stator is made of mainly piezoelectric ceramic PZT-8. The physical parameters of the piezoelectric ultrasonic motor are shown in Table 1. By using ANSYS simulation, the natural frequencies of L1-mode and B-2 mode of the stator can be obtained which are 39841 Hz and 40257 Hz respectively, and the excitation frequency f d was chosen to be 40 kHz throughout the simulations. The calculation time step length in numerical simulation is taken for 1e 7 s considering the excitation frequency of 40 kHz. In addition, the value of parameter ε in the switch model is taken for 1e 10 in the simulations.
3.2. Switch model for simulating stick–slip motion
4.1. Determination of the model parameters
The switch model [24] is an effective method to simulate the stick–slip motion. It treats the system as three different sets of ordinary differential equations: one for the stick phase, a second for the transition from stick to slip and a third for the slip phase. At each time step the state vector is inspected to determine whether the system is in the slip mode, in the transition mode or in the stick mode. The conditions for changing to the slip mode or the stick mode operate as switches between the systems. In the switch model, a region of small relative velocity called stick band is defined as vrel r ε; where ε{1. This finite region is necessary for digital computation because an exact value of zero is rarely computed. The system is considered to be in the stick mode if the relative velocity lies within the stick band, and it is considered to be the slip mode if the relative velocity lies outside the stick band. Especially, the system is considered to be in transition from stick to slip, if the relative velocity lies within the stick band and if the static friction force in the stick phase exceeds the breakaway friction force μs F N . In the contact state (F N 4 0), the dynamic equations governing the switch model can be expressed as: Stick mode: vrel r ε and jF s j r μs F N
Here, we describe in detail the determination of the model parameters used in the simulation. Based on the physical parameters of the stator in Table 1, the parameters used in Eqs. (4) and (5) except modal damping C l and C b can be firstly calculated using the formulas in Appendix A. A laser vibrometer (PSV-300, Polytec GmbH, Germany) is used to measure the longitudinal and transversal vibration amplitudes at the contact tip of the stator without applying preload under different voltage amplitudes when the exciting frequency is 40 kHz. The results are shown in Figs. 5 and 6, respectively. In Figs. 5 and 6, using Eqs. (4) and (5) the modal damping C l and C b can be identified by data fitting to let the calculated vibration amplitude agree with the measured one. The dynamic friction coefficient of the linear guide μsd , the static and dynamic friction coefficients μs and μd can be measured by experiment. Considering the vertical vibration amplitude of the contact tip is difficult to measure after applying a preload in experiment, since the contact tip will contact with the slider when a preload is applied. As a result, the friction parameters aμ and bμ in (11) are very difficult to be obtained by experiment. The values of the parameters aμ and bμ in this paper are determined as follows. It is reported that the reduction rate of dynamic friction coefficient in real ultrasonic motors can reach over 50% due to ultrasonic vibration [25], and the parameter aμ was chosen to be 0.15 in the simulation. In order to estimate the attenuation factor bμ , the normal vibration amplitude AN for various combinations of voltage amplitude and preload is calculated by using Eq. (4), as shown in Fig. 7. Based on the magnitude of the normal vibration amplitude, the parameter bμ was chosen to be 1 106 in the simulation. Of course this is strictly speaking not very correct. Nevertheless, it can depict the physical characteristics that the dynamic friction coefficient decreases with increasing vertical vibration amplitude at the
Fig. 4. The dynamic friction coefficient related to the vibration amplitude at the interface.
€ b ðtÞ þ C b w _ b ðtÞ þ K b wb ðtÞ ¼ Ab V 0 sin ωt þ ηT F s ; Mb w
ð12Þ
M x€ ¼ F s þ F f :
ð13Þ
Stick to slip transition mode: vrel r ε and jF s j 4 μs F N
€ b ðtÞ þ C b w _ b ðtÞ þ K b wb ðtÞ ¼ Ab V 0 sin ωt þ ηT signðF s Þμs F N ; Mb w
ð14Þ
M x€ ¼ signðF s Þμs F N þ F f :
ð15Þ
Slip mode: vrel 4 ε € b ðtÞ þ C b w _ b ðtÞ þ K b wb ðtÞ ¼ Ab V 0 sin ωt þ ηT signðvrel Þμd F N ; Mb w ð16Þ M x€ ¼ signðvrel Þμd F N þF f :
ð17Þ
Where the static friction force F s is given by Eq. (10). The thickness parameter ε of the stick band is often chosen to be as small as possible without affecting the calculation speed. On the other hand, the dynamics equations governing separation mode (F N ¼ 0) can be expressed as: € b ðtÞ þ C b w _ b ðtÞ þ K b wb ðtÞ ¼ Ab V 0 sin ωt; Mb w
ð18Þ
M x€ ¼ signð x_ Þμsd Mg:
ð19Þ
As a result, the complex stick–slip–separation motion in the linear piezoelectric motor can be analyzed based on the secondorder ordinary differential Eqs. (4), (12–19).
Table 1 Physical parameters of the motor. Parameter
Value
Length of the stator l Width of the stator bs Thickness of the stator hs Density of the stator ρ
30 mm 7.5 mm 3 mm
Mechanical compliance of PZT-8 cE11
12:6 1010 N=m2
Mechanical compliance of PZT-8 cE66 Piezoelectric parameter of PZT-8 e31 Mass of the slider M Static friction coefficient μs Dynamic friction coefficient μd Dynamic friction coefficient of the linear guide μsd Stiffness of the preload spring
3:46 1010 N=m2 5:2 C=m2 0:2 kg 0:5 0.35 0:04 4000 N=m
7600 kg=m3
X. Li et al. / International Journal of Mechanical Sciences 107 (2016) 215–224
219
1.2 -6
x 10 1.2
0.9
1 N
(m)
1
0.8
Normal amplitude A
Vibration amplitude (µm)
1.1
0.7 0.6 0.5 0.4
0.8 0.6 0.4 0.2 0 25
0.3 0.2
Pre 50
100
150
200
250
20 loa dF
P (N )
Voltage amplitude (V)
400 300 200
15 100 10 0
Fig. 5. Experiment and fitting results of the longitudinal vibration amplitude at the contact tip of the stator without applying preload. Measured vibration amplitude (asterisk); Calculated vibration amplitude by Eq. (4) (solid line).
eV0
g Volta
(V)
Fig. 7. The normal vibration amplitude AN for various combinations of voltage amplitude and preload. Table 2 Parameters in numerical simulation.
0.8
Parameter
Value
Mass for L1-mode of the stator M l Damping for L1-mode of the stator C l Stiffness for L1-mode of the stator K l Mass for B2-mode of the stator M b Damping for B2-mode of the stator C b Stiffness for B2-mode of the stator K b Force factor corresponding to L1-mode Al Force factor corresponding to B2-mode Ab Equivalent stiffness k
0:0029 kg 173 Ns/m 1.888 108 N/m 0:005 kg 178 Ns/m 3.147 108 N/m 0:1086 0:1246 1.7378 107 N/m
Vibration amplitude (µm)
0.7
0.6
0.5
0.4
0.3
0.2
0.1 50
100
150
200
250
Voltage amplitude (V)
Fig. 6. Experiment and fitting results of the transversal vibration amplitude at the contact tip of the stator without applying preload. Measured vibration amplitude (asterisk); Calculated vibration amplitude by Eq. (5)(solid line).
contact surface in ultrasonic motors [25,26]. The parameters employed in the simulation are shown in Table 2. 4.2. The force transmission between the stator and slider considering stick–slip motion In order to validate the developed model, the force transmission between the stator and the slider considering stick–slip motion is investigated numerically. Furthermore, the analysis of the transmission behavior is helpful to further understand the drive mechanism of the motor. The flow chart for calculation of the time-domain responses of the motor velocity and interface friction is shown in Fig. 8. The time-domain response of the velocity of the slider is shown in Fig. 9, in which two observations marked A and B are used to analyze the force transmission between the stator and the slider during transient and steady-state stages, respectively. Figs. 10 and 11 show the time-domain behaviors of the tangential force F T , the limiting static friction force 7 μs F N and the relative velocity vrel at the time span A and time span B, respectively. As shown in Fig. 10, stick– slip motion can be observed and two transition points from stick to
slip are marked, which indicate the developed model can be used to analyze stick–slip motion effectively in the linear piezoelectric ultrasonic motor. As shown in Fig. 11, the time span B from t ¼ 46353 46378 μs is further divided into five periods P 1 P 5 that represent an overall driving period (1/40,000 s). As the relative velocity is positive at the period P 1 ðfrom t 1 to t 2 Þ, the tangential force is a positive driving slip friction force such that the slider is pushed forward. As the relative velocity is equal to zero at the period P 2 ðfrom t 2 to t 3 Þ, the contact tip sticks to the slider and the tangential force is a negative static friction force that opposes the motion of the slider. The stick motion ends at time t 3 when the static friction force reaches to its limit μs F N , then the slider slips with respect to the contact tip but is pulled backward due to negative braking slip friction force at the periods P 3 ðfrom t 3 to t 4 Þ and P 5 ðfrom t 5 to t 6 Þ. At the period P 4 ðfrom t 4 to t 5 Þ, the contact tip separates from the slider and the tangential force is zero. The corresponding hysteresis loop experiencing the complex stick–slip–separation motion is shown in Fig. 12, in which various states of the interface between the stator and slider are characterized. By comparing Figs. 10 and 11, it is found that the contact tip spends more time within a cycle at the period A than that at the period B sticking to the slider. This is because the velocity of the slider at the starting stage is slower than that at the steady stage. 4.3. Steady-state trajectories on the transformed phase plane It is known that the second-order system dynamics can be intuitively characterized by means of the trajectories of the system on the phase plane. Here, we propose a transformed phase plane to analyze the stick–slip motion at the steady state, on which various friction states including slip–separation, stick–slip,
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Fig. 10. The tangential force and the relative velocity at the time period A.
Fig. 8. The flow chart for calculation of the time-domain responses of the motor velocity and interface friction. 0.04 0.035
B
Slider velocity (m/s)
0.03 0.025 0.02 0.015 0.01 A
0.005
Fig. 11. The tangential force and the relative velocity at the time period B.
0
10 -0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Time (s)
T
Tangential force F (N)
5
Fig. 9. The time-domain response of the slider velocity under V 0 ¼ 200 V; F P ¼ 14 N.
stick–slip–separation and pure stick are intuitively characterized by steady-state trajectories. Fig. 13 shows the transformed phase plane where the abscissa ytip is the displacement of the contact tip in the normal direction and the ordinate vrel is the relative velocity in the tangential direction. The stick phase is expressed by a segment PQ on the abscissa axis (vrel ¼ 0), the driving slip phase (DSP) is by a semi-infinite plane lower than the abscissa axis, and the braking slip phase (BSP) is by a semi-infinite plane upper than the abscissa axis. In addition, the separation phase is defined by a region S between the two lines L1 and L2 n o c ð20Þ S ¼ ðytip ; vrel Þ A R2 ymin tip r ytip r ytip ; where ymin tip is the minimum value of the contact tip displacement in the normal direction, which can be obtained from the stator dynamics. yctip is a critical displacement of the contact tip under a certain preload in the normal direction, which can be obtained from the separating critical condition of F P þ kðud ust Þ ¼ 0, and is
Driving slip
0 Separation -5 Braking slip -10 Stick -15 -5
-4
-3
-2
-1
0
1
2
Transversal displacement at the tip x tip (m)
3
4 x 10
Fig. 12. The hysteresis loop of the interface experiencing stick–slip–separation motion.
given by yctip ¼
F P ηN F P : 2k Kl
ð21Þ
X. Li et al. / International Journal of Mechanical Sciences 107 (2016) 215–224
221
vrel DSP
L2
c ytip
min ytip
Q
P
ytip
O
Stick phase
rel
S
0.05
Relative velocity v (m/s)
L1
BSP
Fig. 13. The transformed phase plane and four states of the interface; The region DSP denotes driving slip phase; The region BSP denotes braking slip phase; The segment PQ denotes stick phase; The region S denotes separation phase.
0
-0.05
-0.1 L1
-0.15
L2 vrel=0
-0.2
Trajectories -0.25 -8
-6
0.04 DSP
0.02
-2
0
2
-0.02
4 x 10
Fig. 15. The trajectories experiencing slip–separation motion under V0 ¼ 200 V, FP ¼10 N.
0 Stick phase
BSP
rel
0.02
-0.04 -0.06
0.01
-0.08 L1
-0.1
0
-0.12
vrel=0
-0.14
Trajectories
-0.16 -6
-5
-4
-3
-2
-1
0
1
Longitudinal displacement y tip (m)
2
-0.01
rel
L2
Relative velocity v (m/s)
Relative velocity v (m/s)
-4
Longitudinal displacement y tip (m)
3 x 10
Fig. 14. A typical example of the trajectories experiencing stick–slip–separation motion under V0 ¼200 V, FP ¼ 14 N.
-0.02 -0.03 L1 -0.04
L2 vrel=0
-0.05
It is noting that the separating region S is not present when (the line L2 is on the left side of L1 ), which means the contact tip cannot separate from the slider. Fig. 14 shows a representative example of the steady-state trajectories experiencing stick–slip–separation motion under The trajectories staying on the line vrel ¼ 0 correspond to the stick phase and the trajectories lying inside the region between the lines L1 and L2 correspond to the separation phase. The trajectories upper than the line vrel ¼ 0 correspond to the driving slip phase and the trajectories lower than the vrel ¼ 0 excluding those lying within the region between the lines L1 and L2 correspond to the braking slip phase. Fig. 15 shows the trajectories experiencing slip-separation motion under , in which the trajectories cross the line vrel ¼ 0 but does not stay on it. Fig. 16 shows the trajectories experiencing stick–slip motion without separation under V 0 ¼ 200 V; F P ¼ 18 N, in which the line L2 is on the left side of L1 . However, the trajectories degenerate to a straight line that stays on the line vrel ¼ 0 when the preload is large in a certain degree, as shown in Fig. 17, meaning that the contact tip remains stuck to the slider in an overall period which is called pure stick motion. yctip o ymin tip
4.4. Influence of some parameters on the stick–slip motion Based on the steady-state trajectories on the transformed phase plane, numerical simulation was conducted for the various combinations of the voltage amplitude and the preload in order to investigate the influence of the two input parameters on the stick–slip motion in the linear piezoelectric motor. Figs. 18 and 19 show the
-0.06 -8
Trajectories
-6
-4
-2
0
Longitudinal displacement y tip (m)
2
4 x 10
-7
Fig. 16. The trajectories experiencing stick–slip motion without separation under V0 ¼200 V, FP ¼ 18 N.
numerical results for various combinations of voltage amplitude and preload including and excluding the influence of ultrasonic oscillation on the dynamic friction, respectively. It is found that stick–slip motion occurs with larger value of preload for a constant value of voltage amplitude, and with smaller value of voltage amplitude for a constant value of preload, which indicates the preload denotes the easiness for the occurrence of stick–slip motion and the voltage amplitude acts to suppress the occurrence. Furthermore, pure stick motion is found to occur with lower voltage and larger preload, which means the contact tip can stick to the slider all the time under relative low input voltage and larger preload. By comparing Figs. 18 and 19, slip–separation motion is found to occur more easily considering the influence of ultrasonic oscillation on the dynamic friction, while stick–slip motion occurs more easily without considering the influence of ultrasonic oscillation on the dynamic friction. This is because the ultrasonic oscillation at the interface can reduce dynamic friction coefficient, and the contact tip can move more freely. It is found that dynamic friction coefficient decreases with increasing voltage amplitude, and preload has little influence on the dynamic friction coefficient, as shown in Figs. 20 and 21, respectively.
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0.1
500 450
0.05
350
0
Voltage amplitude V (V)
rel
Relative velocity v (m/s)
400 0
-0.05
-0.1 L1
-0.15
300 250 200 150
L2
100
vrel=0
-0.2
Trajectories -0.25 -10
-5
0
50 0
5
10
12
14
16
x 10
Longitudinal displacement y tip (m)
18
20
22
24
26
Preload FP (N)
Fig. 17. The trajectories experiencing pure stick motion under V 0 ¼ 200 V; F P ¼ 25 N. 500 450
Fig. 19. Stick–slip map for various combinations of voltage amplitude and preload without considering the ultrasonic vibration effect; open circles: slip–separation motion, solid circles: stick–slip motion; asterisk: pure stick motion; the dynamic friction coefficient is given by μ in numerical simulation.
400
0
Voltage amplitude V (V)
0.36
350 0.34
300
Dynamic friction coefficient µ
d
250 200 150 100 50 0 10
12
14
16
18
20
22
24
0.32 0.3 0.28 0.26 0.24
26
Preload FP (N)
0.22
Fig. 18. Stick–slip map for various combinations of voltage amplitude and preload considering the ultrasonic vibration effect; open circles: slip–separation motion, solid circles: stick–slip motion; asterisk: pure stick motion; the dynamic friction coefficient is given by Eq. (11) in numerical simulation.
50
100
150
200
250
300
350
400
450
Fig. 20. Relationship between the dynamic friction coefficient and the voltage amplitude under Fp ¼ 14 N. 0.25 0.245 0.24 d
It is known that compensation of the dead-zone behavior is very important for precision control of the piezoelectric ultrasonic motor [2]. However, the majority of the previous dynamic models of the piezoelectric motor cannot explain the dead-zone behavior reasonably. Here, we provide an explanation for the dead-zone behavior by the viewpoint of stick motion. To observe the real dead zone of the linear ultrasonic motor (LUSM) under low voltage, an experimental setup is constructed to measure the velocity of the LUSM under different input voltages as shown in Fig. 22. It consists of the LUSM prototype, a signal generator (AFG 3022B, Tektronix, USA), a power amplifier (HFVA-42, Nanjing Funeng Technology Co. Ltd, China), a laser sensor (LK-H150, Keyence, Japan) and a computer. Fig. 23 shows the photo of the LUSM prototype used in experiment. The stator of the LUSM is assembled by using a holding box. It is noted that the preload can be changed by adjusting the blot connected with the preload spring, and the preload is fixed at the value of 14 N in the experiment. The voltage amplitude can be adjusted by the power amplifier. Fig. 24 shows the simulation and experimental results for the steady-state velocity of the slider under different values of input voltage and a fixed excitation frequency of
0
Voltage amplitude V 0 (V)
Dynamic friction coefficient µ
5. Low-voltage dead-zone behavior related to stick motion
0.2
0.235 0.23 0.225 0.22 0.215 0.21 0.205 0.2 10
15
20
25
Preload FP (N)
Fig. 21. Relationship between the dynamic friction coefficient and the preload under V0 ¼ 400 V.
X. Li et al. / International Journal of Mechanical Sciences 107 (2016) 215–224
low voltages are simulated as shown in Fig. 25. It is found that the values of these relative velocity fall within the given stick band ½ 1e 10; 1e 10, which means the contact tip isn’t able anymore to ‘escape’ the stick mode, and the slider won’t move with respect to the contact tip. This phenomenon causes the well-known dead-zone behavior in the voltage–velocity characteristic of the motor.
PC
Signal generator
LUSM
Power amplifier
223
Laser sensor
Fig. 22. Experimental setup for measuring velocity of the LUSM.
Slider 6. Conclusions
Contact tip
In this work, a novel friction model is proposed to study the stick– slip motion in a well-known linear piezoelectric ultrasonic motor. Different from the previous work, the influence of ultrasonic oscillation on the dynamic friction is taken into account in this model, in which the dynamic friction coefficient is modeled by an amplitude-dependent friction law. By using the switch model, the stick–slip motion is effectively simulated. The force transmission between the stator and the slider involving stick–slip motion is analyzed numerically, which provides more physical insights for the stick–slip motion in the linear piezoelectric motor. Various friction states of the interface including slip–separation, stick–slip, stick–slip–separation and pure stick are intuitively characterized by steady-state trajectories on a transformed phase plane. The influences of input voltage amplitude and external preload on stick–slip motion are investigated with the developed model. The results show that the input voltage amplitude acts to suppress the occurrence and the external preload denotes the easiness for the occurrence of stick–slip motion. Meanwhile, the pure stick motion of the interface is found to occur when the input voltage is low and the preload is relatively larger. It is worth noting that the pure stick motion can cause nonlinear dead-zone behavior of the motor, meaning that the pure stick motion should be avoided in the motor control. Furthermore, the stick–slip motion in the piezoelectric motor is contrasted with and without considering the influence of ultrasonic oscillation on the dynamic friction, and the result shows that ultrasonic oscillation can suppress the occurrence of stick-slip motion.
Stator
Friction bar
Holding box Preload spring Blot Fig. 23. Photo of experimental prototype.
Steady state velocity of the slider (m/s)
0.25 Simulation with stick-slip model Experiment
0.2
0.15
0.1
0.05
0 0
50
100
150
200
250
300
350
400
Acknowledgments
Voltage amplitude (V)
Fig. 24. Steady-state velocity of the slider under different input voltage amplitudes. 3
x 10 V0=25 V
2.5
Appendix A. Parameter expressions in dynamic equations of the stator
V0=50 V V0=75 V
rel
Relative velocity v (m/s)
This work is supported by the Program of National Natural Science Foundation of China (No. 51275229).
2
Z
V0=100 V
Ml ¼
l 0
ρAðϕL ðyÞÞ2 dy
ðA:1Þ
ρAðϕB ðyÞÞ2 dy
ðA:2Þ
1.5
Z Mb ¼
1
0
Z
0.5
l
Kl ¼
l 0
cE11 AðϕL ðyÞÞ2 dy
0
ðA:3Þ
0
ðA:4Þ
0
Z -0.5
Kb ¼ 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0
0.05
Time (s)
Fig. 25. The time-domain responses of the relative velocity under low voltages.
40 kHz, and a dead-zone about 120 V is observed from the simulation curve, which is in good agreement with the experimental result (around 100 V). With the developed stick–slip model, the timedomain responses of the relative velocity in 50 ms under several
Z Al ¼
Ab ¼
l
l
0
1 2
cE66 AðϕB ðyÞÞ2 dy
0
e31 AE3 ϕL ðyÞdy
Z
l 0
00
e31 AE3 ϕB ðyÞdy
ðA:5Þ
ðA:6Þ
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ηN ¼ ϕL ð0Þ ϕL ðlÞ
ðA:7Þ
ηT ¼ ϕB ð0Þ ϕB ðlÞ
ðA:8Þ
Here ρ is the density of the stator, A is the area of the cross-section area that is equal to bs hs . cE11 and cE66 are the mechanical compliance of the piezoelectric ceramic, e31 is the parameter of the piezoelectric ceramic. E3 denotes the amplitude of average electric field intensity of the stator due to the excitation voltage that is equal to V 0 =hs .
References [1] Ueha S. Ultrasonic motors theory and applications. Oxford: Clarendon; 1993. [2] Zhao Cs. Ultrasonic motors technologies and applications. Beijing: Science; 2010. [3] Vyshnevskyy O, Kovalev S, Wischnewskiy W. A novel, single-mode piezoceramic plate actuator for ultrasonic linear motors. IEEE Trans Ultrason Ferroelectr Freq Control 2005;52:2047–53. [4] Zhou Sl, Yao Zy. Design and optimization of a modal independent linear ultrasonic motor. IEEE Trans Ultrason Ferroelectr Freq Control 2014;61:535–46. [5] Yang Bd, Chu Ml, Menq Ch. Stick–slip–separation analysis and non-linear stiffness and damping characterization of friction contacts having variable normal load. J Sound Vib 1998;210:461–81. [6] Porter M, Hills Da. A flat punch pressed against an elastic interlayer under conditions of slip and separation. Int J Mech Sci 2002;44:465–74. [7] Nakano K. Two dimensionless parameters controlling the occurrence of stickslip motion in a 1-DOF system with Coulomb friction. Tribol Lett 2006;24:91–8. [8] Behrendt J, Weiss C, Hoffmann Np. A numerical study on stick-slip motion of a brake pad in steady sliding. J Sound Vib 2011;330:636–51. [9] Fang H, Xu J. Stick-slip effect in a vibration-driven system with dry friction: sliding bifurcations and optimization. J Appl Mech Trans ASME 2014;81:051001. [10] Atkins T. Prediction of sticking and sliding lengths on the rake faces of tools using cutting forces. Int J Mech Sci 2015;91:33–45. [11] Maeno T, Tsuukimoto T, Miyake A. Finite-element analysis of the rotor/stator contact in a ring-type ultrasonic motor. IEEE Trans Ultrason Ferroelectr Freq Control 1992;39:668–74.
[12] Cao X, Wallaschek J. Modelling and analysis of dynamic contact problems in travelling wave ultrasonic motors. in: Proceedings of the 3th international congress on industrial and applied mathematics. Hamburg; 1995. [13] Hagedorn P, Schmidt Jp, Bingqi M. A note on the contact problem in an ultrasonic travelling wave motor. Int J Non-Linear Mech 1996;31:915–24. [14] Le Pm, Minotti P. 2-D analytical approach of the stator-rotor contact problem including rotor bending effects for high torque piezomotor design. Eur J Mech A: Solids 1997;16:1067–103. [15] Moal Pl, Joseph E, Ferniot Jc. Mechanical energy transductions in standing wave ultrasonic motors: Analytical modelling and experimental investigations. Eur J Mech A: Solids 2000;19:849–71. [16] Xu X, Liang Yc, Lee Hp, Lin Wz, Lim Sp, Lee Kh, Shi Xh. Mechanical modeling of longitudinal oscillation ultrasonic motors and temperature effect analysis. Smart Mater Struct 2003;12:514–23. [17] Storck H, Wallaschek J. The effect of tangential elasticity of the contact layer between stator and rotor in travelling wave ultrasonic motors. Int J Non-Linear Mech 2003;38:143–59. [18] Qu J, Sun F, Zhao C. Performance evaluation of travelling wave ultrasonic motor based on a model with visco-elastic friction layer on stator. Ultrasonics 2006;45:22–31. [19] Tsai Ms, Lee Ch, Hwang Sh. Dynamic modeling and analysis of a bimodal ultrasonic motor. IEEE Trans Ultrason Ferroelectr Freq Control 2003;50:245–56. [20] Tomikawa Y, Takano T, Ueda H. Thin rotary and linear ultrasonic motors using a double-mode piezoelectric vibrator of the first longitudinal and second bending modes. Jpn J Appl Phys 1992;31:3073–6. [21] Rao S. Mechanical vibrations. Reading, MA: Addison-Wesley; 1995. [22] Wallaschek J. Contact mechanics of piezoelectric ultrasonic motors. Smart Mater Struct 1998;7:369–81. [23] Chowdhury Mohammad Asaduzzaman, Maksud Helali Md. The effect of amplitude of vibration on the coefficient of friction for different materials. Tribol Int 2008;41:307–14. [24] Leine Ri, Van Campen Dh, De Kraker A. Stick-slip vibrations induced by alternate friction models. Nonlinear Dyn 1998;16:41–54. [25] Qu J, Tian X, Sun F. Experiment study on friction reduction of ultrasonic vibration based on travelling wave ultrasonic motor. Tribology 2007;27:73–7. [26] Mashimo T, Terashima K. Experimental verification of elliptical motion model in travelling wave ultrasonic motors. IEEE/ASME Trans Mechatron 2015;20:2699–707.
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Modeling and analysis of stick-slip motion in a linear piezoelectric ultrasonic motor considering ultrasonic oscillation effect Xiang Li a,n, Zhiyuan Yao a, Ranchao Wu b a b
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China School of Mathematics, Anhui University, Hefei 230039, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 27 September 2015 Received in revised form 13 January 2016 Accepted 14 January 2016 Available online 21 January 2016
Piezoelectric ultrasonic motor is an electromechanical coupling system, which is driven by the interface friction between the stator and the rotor/slider. A novel friction model is developed to study the stick–slip motion in a linear piezoelectric ultrasonic motor using longitudinal and bending modes. In this model, the influence of the ultrasonic oscillation on the dynamic friction coefficient is taken into account, which is described by a friction law related to the vibration amplitude at the interface. In addition, the friction contact model involves intermittent separation non-linearity induced by the variable normal force, which can result in complex stick–slip–separation motion at the contact interface. Analytical transition criteria between stick and slip are established by using the switch model. Based on the developed model, numerical simulations are performed to analyze the force transmission between the stator and the slider involving stick–slip motion. A transformed phase plane is proposed to study various states of the contact interface (stick, slip and separation), on which slip–separation, stick–slip, stick–slip–separation, and pure stick motions are intuitively characterized. The influence of some input parameters on the stick–slip motion is analyzed. Furthermore, the low-voltage dead-zone behavior is clarified by stick motion. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Friction Stick–slip Vibration Switch model Piezoelectric ultrasonic motor
1. Introduction Piezoelectric ultrasonic motor is a newly developed motor, which has received much attention due to their unique characteristics such as high power density, high torque, fast response and compact size [1,2]. In particular, the piezoelectric motor has minute control characteristics because it utilizes large mechanical friction and needs no additional braking system. Linear piezoelectric ultrasonic motor is one branch of piezoelectric ultrasonic motors, which can move in straight line and has much freedom design and high-thrust force [3,4]. As an electromechanical coupling system, a common characteristic of all piezoelectric ultrasonic motors is their energy conversion [1]. In the first stage, piezoelectric components convert electrical energy into high-frequency (ultrasonic frequency) mechanical oscillations. Depending on the geometry of the stator (vibrator) and the piezoelectric excitation, generally two orthogonal vibration modes (mainly longitudinal and bending, but also torsion) are superposed on an elliptical motion of the surface/driving tip. In the second stage, the high-frequency vibration is rectified into macroscopic unidirectional rotary or linear motion of a rotor/slider (driven component) n
Corresponding author. Tel.: þ 86 02584895853. E-mail address: [email protected] (X. Li).
http://dx.doi.org/10.1016/j.ijmecsci.2016.01.016 0020-7403/& 2016 Elsevier Ltd. All rights reserved.
based on the contact mechanisms occurring at the interface between the stator and the rotor/slider. The stick–slip motion between the stator and the slider of a linear ultrasonic piezoelectric motor occurring at the second stage of the energy conversion will be discussed in detail in the following chapters of this paper. Stick–slip motion exists in many machineries with the relative motion of contact surfaces, and a considerable number of theoretical and experimental studies have been made to investigate stick–slip motion [5–10]. Because the piezoelectric ultrasonic motors are driven by the friction resulting from the relative motion of the stator and the rotor/slider, stick–slip motion in piezoelectric motors has significant influence on the dynamics of the motors. It is known that the slipping can cause the motor to consume more energy, which will reduce the efficiency and shorten the life. On the other hand, the sticking can cause the output speed of the motor to decline, and even lead to deadzone behavior. However, stick–slip motion is very difficult to measure in experiment due to the characteristic of high-frequency vibration of piezoelectric ultrasonic motor. As a part of the contact mechanisms of piezoelectric motors, stick–slip motion has been studied by virtue of various contact models [11–18]. In 1992, Maeno et al. [11] studied the stator/rotor contact in a traveling wave rotary piezoelectric motor by finite element method. They indicated that the distribution of the stick/slip exists in the contact area. In 1995, Cao and Wallaschek [12] proposed a visco-elastic contact model for traveling wave piezoelectric
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motor. They studied stick–slip behavior in detail and indicated that three different slip and two different stick zones exist. In 2000, Moal et al. [15] studied the mechanical energy transductions in a standing wave piezoelectric motor based on an analytical model in which the stick and slip phases are characterized analytically. In 2006, Qu et al. [18] investigated the influence of contact mechanics on the performance of a traveling wave rotary piezoelectric motor based on a viscoelastic contact model, in which the stick–slip motion was considered. They introduced a new judge function to judge whether the conversion between stick zone and slip zone. Given the majority of the publications in this field are devoted to the study of rotary piezoelectric ultrasonic motors and the velocity of the rotor is assumed to be constant, this paper focuses on the stick–slip motion of a linear piezoelectric ultrasonic motor based on an analytical model in which the slider velocity is time dependent. Therefore, not only this model can be used to investigate the stick–slip motion at the steady stage but also it can be used to study the stick–slip motion at the transient stage. Different from the previous work, the influence of the ultrasonic oscillation on the dynamic friction is considered in this paper, which can further affect the stick–slip motion. This paper develops an analytical model to study the stick–slip motion in a typical linear piezoelectric ultrasonic motor utilizing longitudinal vibration and bending vibration modes. Based on the dynamic model proposed in [19], a friction model subjected to both friction non-linearity and intermittent separation non-linearity is established taking into account the influence of ultrasonic vibration. In this model, the dynamic friction coefficient is modeled by an amplitude-dependent law. The transition criteria between stick and slip are deduced via the switch model, which are applied effectively in numerical simulations. The force transmission between the stator and the slider involving complex stick–slip–separation motion is analyzed numerically, which is helpful to further understand the operation mechanism of the motor. Various friction states including slip–separation, stick–slip, stick–slip–separation and pure stick are characterized by steady-state trajectories on a transformed phase plane. Numerical simulations are performed to study the influence of the possible parameters on the stick–slip motion based on the developed model, and the results indicate that the preload denotes the easiness for the occurrence of stick–slip motion and the input voltage amplitude acts to suppress the occurrence. Furthermore, the dead-zone behavior caused by low input voltage can be predicted by the developed stick–slip model and clarified by pure stick motion.
Linear motion
y
Slider
x
j k
Contact tip
Linear guide 1
i
2
V sin ωt
Stator V sin ωt
3
4
Preload spring
Fig. 1. Physical model of the linear piezoelectric ultrasonic motor.
Longitudinal vibration
Bending vibration
Linear motion Slider Elliptic motion
Fig. 2. Drive mechanism of the linear piezoelectric ultrasonic motor utilizing longitudinal vibration and bending vibration modes.
Furthermore, the direction of motion of the slider can be changed by applying the excitation signals to the electrodes 2–3.
3. Modeling of stick–slip motion 2. Description of the physical model The physical model shown in Fig. 1 is a typical bimodal linear piezoelectric ultrasonic motor utilizing the first longitudinal and second bending modes reported in [20]. It consists of a stator with a contact tip, a preload spring, a slider and a linear guide. The stator is a rectangular piezoelectric plate of dimensions 30 mm 7:5 mm 3 mm (length (l), width (bs ) and thickness (hs )), which is served as a piezoelectric vibrator in the system. To excite both longitudinal and bending modes of the stator, the polarization direction of the stator is in the thickness direction (k-direction in Fig. 1). The electrode is divided into four sections denoted by 1, 2, 3 and 4, as shown in Fig. 1. Fig. 2 illustrates the drive mechanism of the linear piezoelectric ultrasonic motor. When one pair of sinusoidal signals with a single driving frequency close to the natural frequencies of the first longitudinal mode (L1-mode) and the second bending mode (B2mode) of the stator are applied to the electrodes 1–4, the stator is excited in a longitudinal vibration and a bending vibration simultaneously, again generating an elliptic motion at the contact tip of the stator. As a result, a friction force is produced at the contact interface under a preload provided by the preload spring, which can drive the slider to perform a macroscopic linear motion.
In [19], Tsai et al. have developed an integral dynamic model for a linear piezoelectric motor utilizing longitudinal vibration and bending vibration modes, in which the dynamics of the electrical driving circuit was considered. This paper concentrates on the mechanical energy conversion rather than on the electrical energy conversion. Here, the driving voltage is directly applied to the electrodes of the stator as shown in Fig. 1, and the electrical driving circuit equations are ignored in the following modeling process. 3.1. Analytical model For simplification, the short stubby stator is assumed to be a Timoshenko beam and the contact tip on the stator is assumed to be rigid. A cross section of the Timoshenko beam (the stator) is shown in Fig. 3, and the displacement field ðu ¼ u1 i þ u2 j þ u3 kÞ of a material point in the beam can be expressed as: 8 u1 ðy; tÞ ¼ wðy; tÞ; > > > > < u2 ðx; y; tÞ ¼ uðy; tÞ xψ ðy; tÞ ð1Þ ; ¼ uðy; tÞ xð∂w > ∂y þ β ðy; tÞÞ; > > > : u ðy; tÞ ¼ 0; 3
X. Li et al. / International Journal of Mechanical Sciences 107 (2016) 215–224
217
type in which the contact area between the stator and the rotor/ slider moves along with a traveling wave on the stator. The second is an “intermittent” contact type, in which the two bodies are in contact intermittently and the contact area of the stator remains the same. As a standing wave piezoelectric motor, the linear piezoelectric motor discussed in this paper belongs to the latter contact type motors. The contact mechanism at the interface is described by an elastic spring existing in between the contact tip and the slider [19]. The normal force acting on the interface is given as: ( F P þ kðud ust Þ; F P þ kðud ust Þ 4 0 FN ¼ : ð7Þ 0; F P þ kðud ust Þ r 0 Fig. 3. The cross section of Timoshenko beam (the stator).
where uðy; tÞ is the axial displacement of the beam at center point O. wðy; tÞ is the transverse displacement of the beam. ψ ðy; tÞ is the rotation of the line element along the centerline, and β ðy; tÞ is the shear angle. It is noted that the rotation ψ ðy; tÞ can be removed in the dynamics equations of the stator, because the natural frequency that corresponds to the mode of rotation is much higher than the driving frequency [19]. By assumed mode method, the displacement fields can be expressed as: uðy; tÞ ¼ ϕL ðyÞul ðtÞ; ð2Þ wðy; tÞ ¼ ϕB ðyÞwb ðtÞ; ; where ul ðtÞ and wb ðtÞ are the generalized coordinates. ϕL ðyÞ and ϕB ðyÞ are the corresponding assumed mode shape functions. Here ϕL ðyÞ and ϕB ðyÞ are the eigenfunctions of a Bernoulli–Euler beam corresponding to L1-mode and B2-mode vibration, respectively. With a free-free boundary condition, the two mode shape functions [21] are given as: ( ϕL ðyÞ ¼ cos πly; ð3Þ ϕB ðyÞ ¼ α2 ð cos β2 y þ cosh β2 yÞ þ ð sin β2 y þ sinh β2 yÞ; where β2 ¼ 2:499753π =l; α2 ¼ ð sin β2 l sinh β2 lÞ=ðcosh β2 l cos β2 lÞ: Utilizing Hamilton’s principle and the assumed mode method [2,19], the dynamic equations for the L1-mode and B2-mode of the stator are given by M l u€ l ðtÞ þ C l u_ l ðtÞ þ K l ul ðtÞ ¼ Al V 0 sin ωt þ ηN F N ;
ð4Þ
€ b ðtÞ þ C b w _ b ðtÞ þ K b wb ðtÞ ¼ Ab V 0 sin ωt þ ηT F T ; Mb w
ð5Þ
where M l ; C l ; K l and M b ; C b ; K b are the mass, damping and stiffness for the L1-mode and B-2 mode of the stator, respectively. V 0 is the amplitude of the exciting voltage. ω is the circular frequency of the exciting voltage. F N and F T represent normal force and tangential force acting on the interface between the stator and the slider, respectively. Al and Ab are the force factors related to the electromechanical coupling effects. ηN and ηT are the load coefficients responding to the normal force and tangential force, respectively. The explicit expresses of the parameters in (4) and (5) are given in Appendix A. Through force equilibrium, the motion equation of the slider is given by M x€ ¼ F T þ F f ; where F f ¼ signð x_ Þμsd ðF N þMgÞ:
ð6Þ
Here M is the mass of the slider, x_ and x€ are the velocity and acceleration of the slider in the x direction, respectively. μsd is the dynamic friction coefficient of the linear guide, g is the gravitational acceleration. sign() represents the sign function. From the contact viewpoint, piezoelectric ultrasonic motors can be divided into two groups [22]. One is called a “traveling” contact
Here F P is the preload and k is the stiffness of the equivalent spring. ud is the total dynamic longitudinal elongation of the stator and is equal to ðϕL ðlÞ ϕL ð0ÞÞ Uul . ust is the static elongation of the stator under the preload only and is equal to ηN F P ðϕL ðlÞ ϕL ð0ÞÞ=K l . It is noted that if the condition of F P þ kðud ust Þ r0 occurs, the contact tip will separate from the slider and the interface normal force is vanished. The tangential force resulting from the friction force at the interface is related to the relative velocity of the contact tip and the slider in the tangential direction. The relative velocity is defined as: vrel ¼ x_ tip x_ ;
ð8Þ
where x_ tip is the tangential velocity of the contact tip and is equal to
ϕB ðlÞw_ b , which can be obtained from the stator dynamics. Depending on the variable normal force, the contact situation between the contact tip and the slider in the normal direction can be divided into two states: the contact state (F N 40) and the separation state (F N ¼ 0). There are two types of friction forces in the contact state: static friction force F s corresponding to the stick phase and the dynamic friction force F d corresponding to the slip phase. The friction force is not present during the separation state owing to the normal force equal to zero. In light of Coulomb friction model, the tangential force acting on the interface is expressed as: 8 ) > F A μs F N ; μs F N ; F N 40 and vrel ¼ 0; ðstickÞ > < s contact F T ¼ F d ¼ signðvrel Þμd F N ; F N 4 0 and vrel a 0; ðslip Þ > > :0 F ¼ 0; separation N
ð9Þ where μs and μd are the static and dynamic friction coefficients of the interface, respectively. The static friction force in the stick phase is limited by the maximum static friction force, i.e. jF s j r μs F N . During the contact state, the transition point from stick to slip is reached when jF s j 4 μs F N . If the contact tip is sticking to the slider then x€ tip ¼ x€ ; and one can obtain the sticking force F s by using Eqs. (5) and (8) Fs ¼
ϕB ðlÞ Mb
_ b þ K b wb Ab V 0 sin ωtÞ þ ðC b w
Ff ϕB ðlÞηT 1 = : M Mb M
ð10Þ
It is worth noting that the ultrasonic oscillation has influence on the dynamic friction. It is known that dynamic friction is related to the vibration amplitude at the contact surface [23]: The dynamic friction coefficient decreases with increasing vibration amplitude. The dynamic friction coefficient in this paper is modeled by the following amplitude-dependent law:
μd ðAN Þ ¼ aμ þ ðμd aμ Þe bμ AN ;
ð11Þ
which exponentially decays to a residual value of friction aμ , as shown in Fig. 4. AN is the normal vibration amplitude at the contact tip, which can be used to represent the vibration amplitude at the contact interface. μd represents the dynamic friction coefficient without considering the influence of the ultrasonic oscillation. Parameter bμ in Eq. (11) is an attenuation factor that is used
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X. Li et al. / International Journal of Mechanical Sciences 107 (2016) 215–224
4. Numerical simulation and discussion
to characterize the decay rate of the dynamic friction coefficient μd with increasing vibration amplitude AN .
In this section, numerical simulation was conducted with the governing equations derived in Section 3. The Runge-Kutta method was used for the numerical integral of the ordinary differential equations in the model. In this paper, the piezoelectric stator is made of mainly piezoelectric ceramic PZT-8. The physical parameters of the piezoelectric ultrasonic motor are shown in Table 1. By using ANSYS simulation, the natural frequencies of L1-mode and B-2 mode of the stator can be obtained which are 39841 Hz and 40257 Hz respectively, and the excitation frequency f d was chosen to be 40 kHz throughout the simulations. The calculation time step length in numerical simulation is taken for 1e 7 s considering the excitation frequency of 40 kHz. In addition, the value of parameter ε in the switch model is taken for 1e 10 in the simulations.
3.2. Switch model for simulating stick–slip motion
4.1. Determination of the model parameters
The switch model [24] is an effective method to simulate the stick–slip motion. It treats the system as three different sets of ordinary differential equations: one for the stick phase, a second for the transition from stick to slip and a third for the slip phase. At each time step the state vector is inspected to determine whether the system is in the slip mode, in the transition mode or in the stick mode. The conditions for changing to the slip mode or the stick mode operate as switches between the systems. In the switch model, a region of small relative velocity called stick band is defined as vrel r ε; where ε{1. This finite region is necessary for digital computation because an exact value of zero is rarely computed. The system is considered to be in the stick mode if the relative velocity lies within the stick band, and it is considered to be the slip mode if the relative velocity lies outside the stick band. Especially, the system is considered to be in transition from stick to slip, if the relative velocity lies within the stick band and if the static friction force in the stick phase exceeds the breakaway friction force μs F N . In the contact state (F N 4 0), the dynamic equations governing the switch model can be expressed as: Stick mode: vrel r ε and jF s j r μs F N
Here, we describe in detail the determination of the model parameters used in the simulation. Based on the physical parameters of the stator in Table 1, the parameters used in Eqs. (4) and (5) except modal damping C l and C b can be firstly calculated using the formulas in Appendix A. A laser vibrometer (PSV-300, Polytec GmbH, Germany) is used to measure the longitudinal and transversal vibration amplitudes at the contact tip of the stator without applying preload under different voltage amplitudes when the exciting frequency is 40 kHz. The results are shown in Figs. 5 and 6, respectively. In Figs. 5 and 6, using Eqs. (4) and (5) the modal damping C l and C b can be identified by data fitting to let the calculated vibration amplitude agree with the measured one. The dynamic friction coefficient of the linear guide μsd , the static and dynamic friction coefficients μs and μd can be measured by experiment. Considering the vertical vibration amplitude of the contact tip is difficult to measure after applying a preload in experiment, since the contact tip will contact with the slider when a preload is applied. As a result, the friction parameters aμ and bμ in (11) are very difficult to be obtained by experiment. The values of the parameters aμ and bμ in this paper are determined as follows. It is reported that the reduction rate of dynamic friction coefficient in real ultrasonic motors can reach over 50% due to ultrasonic vibration [25], and the parameter aμ was chosen to be 0.15 in the simulation. In order to estimate the attenuation factor bμ , the normal vibration amplitude AN for various combinations of voltage amplitude and preload is calculated by using Eq. (4), as shown in Fig. 7. Based on the magnitude of the normal vibration amplitude, the parameter bμ was chosen to be 1 106 in the simulation. Of course this is strictly speaking not very correct. Nevertheless, it can depict the physical characteristics that the dynamic friction coefficient decreases with increasing vertical vibration amplitude at the
Fig. 4. The dynamic friction coefficient related to the vibration amplitude at the interface.
€ b ðtÞ þ C b w _ b ðtÞ þ K b wb ðtÞ ¼ Ab V 0 sin ωt þ ηT F s ; Mb w
ð12Þ
M x€ ¼ F s þ F f :
ð13Þ
Stick to slip transition mode: vrel r ε and jF s j 4 μs F N
€ b ðtÞ þ C b w _ b ðtÞ þ K b wb ðtÞ ¼ Ab V 0 sin ωt þ ηT signðF s Þμs F N ; Mb w
ð14Þ
M x€ ¼ signðF s Þμs F N þ F f :
ð15Þ
Slip mode: vrel 4 ε € b ðtÞ þ C b w _ b ðtÞ þ K b wb ðtÞ ¼ Ab V 0 sin ωt þ ηT signðvrel Þμd F N ; Mb w ð16Þ M x€ ¼ signðvrel Þμd F N þF f :
ð17Þ
Where the static friction force F s is given by Eq. (10). The thickness parameter ε of the stick band is often chosen to be as small as possible without affecting the calculation speed. On the other hand, the dynamics equations governing separation mode (F N ¼ 0) can be expressed as: € b ðtÞ þ C b w _ b ðtÞ þ K b wb ðtÞ ¼ Ab V 0 sin ωt; Mb w
ð18Þ
M x€ ¼ signð x_ Þμsd Mg:
ð19Þ
As a result, the complex stick–slip–separation motion in the linear piezoelectric motor can be analyzed based on the secondorder ordinary differential Eqs. (4), (12–19).
Table 1 Physical parameters of the motor. Parameter
Value
Length of the stator l Width of the stator bs Thickness of the stator hs Density of the stator ρ
30 mm 7.5 mm 3 mm
Mechanical compliance of PZT-8 cE11
12:6 1010 N=m2
Mechanical compliance of PZT-8 cE66 Piezoelectric parameter of PZT-8 e31 Mass of the slider M Static friction coefficient μs Dynamic friction coefficient μd Dynamic friction coefficient of the linear guide μsd Stiffness of the preload spring
3:46 1010 N=m2 5:2 C=m2 0:2 kg 0:5 0.35 0:04 4000 N=m
7600 kg=m3
X. Li et al. / International Journal of Mechanical Sciences 107 (2016) 215–224
219
1.2 -6
x 10 1.2
0.9
1 N
(m)
1
0.8
Normal amplitude A
Vibration amplitude (µm)
1.1
0.7 0.6 0.5 0.4
0.8 0.6 0.4 0.2 0 25
0.3 0.2
Pre 50
100
150
200
250
20 loa dF
P (N )
Voltage amplitude (V)
400 300 200
15 100 10 0
Fig. 5. Experiment and fitting results of the longitudinal vibration amplitude at the contact tip of the stator without applying preload. Measured vibration amplitude (asterisk); Calculated vibration amplitude by Eq. (4) (solid line).
eV0
g Volta
(V)
Fig. 7. The normal vibration amplitude AN for various combinations of voltage amplitude and preload. Table 2 Parameters in numerical simulation.
0.8
Parameter
Value
Mass for L1-mode of the stator M l Damping for L1-mode of the stator C l Stiffness for L1-mode of the stator K l Mass for B2-mode of the stator M b Damping for B2-mode of the stator C b Stiffness for B2-mode of the stator K b Force factor corresponding to L1-mode Al Force factor corresponding to B2-mode Ab Equivalent stiffness k
0:0029 kg 173 Ns/m 1.888 108 N/m 0:005 kg 178 Ns/m 3.147 108 N/m 0:1086 0:1246 1.7378 107 N/m
Vibration amplitude (µm)
0.7
0.6
0.5
0.4
0.3
0.2
0.1 50
100
150
200
250
Voltage amplitude (V)
Fig. 6. Experiment and fitting results of the transversal vibration amplitude at the contact tip of the stator without applying preload. Measured vibration amplitude (asterisk); Calculated vibration amplitude by Eq. (5)(solid line).
contact surface in ultrasonic motors [25,26]. The parameters employed in the simulation are shown in Table 2. 4.2. The force transmission between the stator and slider considering stick–slip motion In order to validate the developed model, the force transmission between the stator and the slider considering stick–slip motion is investigated numerically. Furthermore, the analysis of the transmission behavior is helpful to further understand the drive mechanism of the motor. The flow chart for calculation of the time-domain responses of the motor velocity and interface friction is shown in Fig. 8. The time-domain response of the velocity of the slider is shown in Fig. 9, in which two observations marked A and B are used to analyze the force transmission between the stator and the slider during transient and steady-state stages, respectively. Figs. 10 and 11 show the time-domain behaviors of the tangential force F T , the limiting static friction force 7 μs F N and the relative velocity vrel at the time span A and time span B, respectively. As shown in Fig. 10, stick– slip motion can be observed and two transition points from stick to
slip are marked, which indicate the developed model can be used to analyze stick–slip motion effectively in the linear piezoelectric ultrasonic motor. As shown in Fig. 11, the time span B from t ¼ 46353 46378 μs is further divided into five periods P 1 P 5 that represent an overall driving period (1/40,000 s). As the relative velocity is positive at the period P 1 ðfrom t 1 to t 2 Þ, the tangential force is a positive driving slip friction force such that the slider is pushed forward. As the relative velocity is equal to zero at the period P 2 ðfrom t 2 to t 3 Þ, the contact tip sticks to the slider and the tangential force is a negative static friction force that opposes the motion of the slider. The stick motion ends at time t 3 when the static friction force reaches to its limit μs F N , then the slider slips with respect to the contact tip but is pulled backward due to negative braking slip friction force at the periods P 3 ðfrom t 3 to t 4 Þ and P 5 ðfrom t 5 to t 6 Þ. At the period P 4 ðfrom t 4 to t 5 Þ, the contact tip separates from the slider and the tangential force is zero. The corresponding hysteresis loop experiencing the complex stick–slip–separation motion is shown in Fig. 12, in which various states of the interface between the stator and slider are characterized. By comparing Figs. 10 and 11, it is found that the contact tip spends more time within a cycle at the period A than that at the period B sticking to the slider. This is because the velocity of the slider at the starting stage is slower than that at the steady stage. 4.3. Steady-state trajectories on the transformed phase plane It is known that the second-order system dynamics can be intuitively characterized by means of the trajectories of the system on the phase plane. Here, we propose a transformed phase plane to analyze the stick–slip motion at the steady state, on which various friction states including slip–separation, stick–slip,
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X. Li et al. / International Journal of Mechanical Sciences 107 (2016) 215–224
Fig. 10. The tangential force and the relative velocity at the time period A.
Fig. 8. The flow chart for calculation of the time-domain responses of the motor velocity and interface friction. 0.04 0.035
B
Slider velocity (m/s)
0.03 0.025 0.02 0.015 0.01 A
0.005
Fig. 11. The tangential force and the relative velocity at the time period B.
0
10 -0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Time (s)
T
Tangential force F (N)
5
Fig. 9. The time-domain response of the slider velocity under V 0 ¼ 200 V; F P ¼ 14 N.
stick–slip–separation and pure stick are intuitively characterized by steady-state trajectories. Fig. 13 shows the transformed phase plane where the abscissa ytip is the displacement of the contact tip in the normal direction and the ordinate vrel is the relative velocity in the tangential direction. The stick phase is expressed by a segment PQ on the abscissa axis (vrel ¼ 0), the driving slip phase (DSP) is by a semi-infinite plane lower than the abscissa axis, and the braking slip phase (BSP) is by a semi-infinite plane upper than the abscissa axis. In addition, the separation phase is defined by a region S between the two lines L1 and L2 n o c ð20Þ S ¼ ðytip ; vrel Þ A R2 ymin tip r ytip r ytip ; where ymin tip is the minimum value of the contact tip displacement in the normal direction, which can be obtained from the stator dynamics. yctip is a critical displacement of the contact tip under a certain preload in the normal direction, which can be obtained from the separating critical condition of F P þ kðud ust Þ ¼ 0, and is
Driving slip
0 Separation -5 Braking slip -10 Stick -15 -5
-4
-3
-2
-1
0
1
2
Transversal displacement at the tip x tip (m)
3
4 x 10
Fig. 12. The hysteresis loop of the interface experiencing stick–slip–separation motion.
given by yctip ¼
F P ηN F P : 2k Kl
ð21Þ
X. Li et al. / International Journal of Mechanical Sciences 107 (2016) 215–224
221
vrel DSP
L2
c ytip
min ytip
Q
P
ytip
O
Stick phase
rel
S
0.05
Relative velocity v (m/s)
L1
BSP
Fig. 13. The transformed phase plane and four states of the interface; The region DSP denotes driving slip phase; The region BSP denotes braking slip phase; The segment PQ denotes stick phase; The region S denotes separation phase.
0
-0.05
-0.1 L1
-0.15
L2 vrel=0
-0.2
Trajectories -0.25 -8
-6
0.04 DSP
0.02
-2
0
2
-0.02
4 x 10
Fig. 15. The trajectories experiencing slip–separation motion under V0 ¼ 200 V, FP ¼10 N.
0 Stick phase
BSP
rel
0.02
-0.04 -0.06
0.01
-0.08 L1
-0.1
0
-0.12
vrel=0
-0.14
Trajectories
-0.16 -6
-5
-4
-3
-2
-1
0
1
Longitudinal displacement y tip (m)
2
-0.01
rel
L2
Relative velocity v (m/s)
Relative velocity v (m/s)
-4
Longitudinal displacement y tip (m)
3 x 10
Fig. 14. A typical example of the trajectories experiencing stick–slip–separation motion under V0 ¼200 V, FP ¼ 14 N.
-0.02 -0.03 L1 -0.04
L2 vrel=0
-0.05
It is noting that the separating region S is not present when (the line L2 is on the left side of L1 ), which means the contact tip cannot separate from the slider. Fig. 14 shows a representative example of the steady-state trajectories experiencing stick–slip–separation motion under The trajectories staying on the line vrel ¼ 0 correspond to the stick phase and the trajectories lying inside the region between the lines L1 and L2 correspond to the separation phase. The trajectories upper than the line vrel ¼ 0 correspond to the driving slip phase and the trajectories lower than the vrel ¼ 0 excluding those lying within the region between the lines L1 and L2 correspond to the braking slip phase. Fig. 15 shows the trajectories experiencing slip-separation motion under , in which the trajectories cross the line vrel ¼ 0 but does not stay on it. Fig. 16 shows the trajectories experiencing stick–slip motion without separation under V 0 ¼ 200 V; F P ¼ 18 N, in which the line L2 is on the left side of L1 . However, the trajectories degenerate to a straight line that stays on the line vrel ¼ 0 when the preload is large in a certain degree, as shown in Fig. 17, meaning that the contact tip remains stuck to the slider in an overall period which is called pure stick motion. yctip o ymin tip
4.4. Influence of some parameters on the stick–slip motion Based on the steady-state trajectories on the transformed phase plane, numerical simulation was conducted for the various combinations of the voltage amplitude and the preload in order to investigate the influence of the two input parameters on the stick–slip motion in the linear piezoelectric motor. Figs. 18 and 19 show the
-0.06 -8
Trajectories
-6
-4
-2
0
Longitudinal displacement y tip (m)
2
4 x 10
-7
Fig. 16. The trajectories experiencing stick–slip motion without separation under V0 ¼200 V, FP ¼ 18 N.
numerical results for various combinations of voltage amplitude and preload including and excluding the influence of ultrasonic oscillation on the dynamic friction, respectively. It is found that stick–slip motion occurs with larger value of preload for a constant value of voltage amplitude, and with smaller value of voltage amplitude for a constant value of preload, which indicates the preload denotes the easiness for the occurrence of stick–slip motion and the voltage amplitude acts to suppress the occurrence. Furthermore, pure stick motion is found to occur with lower voltage and larger preload, which means the contact tip can stick to the slider all the time under relative low input voltage and larger preload. By comparing Figs. 18 and 19, slip–separation motion is found to occur more easily considering the influence of ultrasonic oscillation on the dynamic friction, while stick–slip motion occurs more easily without considering the influence of ultrasonic oscillation on the dynamic friction. This is because the ultrasonic oscillation at the interface can reduce dynamic friction coefficient, and the contact tip can move more freely. It is found that dynamic friction coefficient decreases with increasing voltage amplitude, and preload has little influence on the dynamic friction coefficient, as shown in Figs. 20 and 21, respectively.
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X. Li et al. / International Journal of Mechanical Sciences 107 (2016) 215–224
0.1
500 450
0.05
350
0
Voltage amplitude V (V)
rel
Relative velocity v (m/s)
400 0
-0.05
-0.1 L1
-0.15
300 250 200 150
L2
100
vrel=0
-0.2
Trajectories -0.25 -10
-5
0
50 0
5
10
12
14
16
x 10
Longitudinal displacement y tip (m)
18
20
22
24
26
Preload FP (N)
Fig. 17. The trajectories experiencing pure stick motion under V 0 ¼ 200 V; F P ¼ 25 N. 500 450
Fig. 19. Stick–slip map for various combinations of voltage amplitude and preload without considering the ultrasonic vibration effect; open circles: slip–separation motion, solid circles: stick–slip motion; asterisk: pure stick motion; the dynamic friction coefficient is given by μ in numerical simulation.
400
0
Voltage amplitude V (V)
0.36
350 0.34
300
Dynamic friction coefficient µ
d
250 200 150 100 50 0 10
12
14
16
18
20
22
24
0.32 0.3 0.28 0.26 0.24
26
Preload FP (N)
0.22
Fig. 18. Stick–slip map for various combinations of voltage amplitude and preload considering the ultrasonic vibration effect; open circles: slip–separation motion, solid circles: stick–slip motion; asterisk: pure stick motion; the dynamic friction coefficient is given by Eq. (11) in numerical simulation.
50
100
150
200
250
300
350
400
450
Fig. 20. Relationship between the dynamic friction coefficient and the voltage amplitude under Fp ¼ 14 N. 0.25 0.245 0.24 d
It is known that compensation of the dead-zone behavior is very important for precision control of the piezoelectric ultrasonic motor [2]. However, the majority of the previous dynamic models of the piezoelectric motor cannot explain the dead-zone behavior reasonably. Here, we provide an explanation for the dead-zone behavior by the viewpoint of stick motion. To observe the real dead zone of the linear ultrasonic motor (LUSM) under low voltage, an experimental setup is constructed to measure the velocity of the LUSM under different input voltages as shown in Fig. 22. It consists of the LUSM prototype, a signal generator (AFG 3022B, Tektronix, USA), a power amplifier (HFVA-42, Nanjing Funeng Technology Co. Ltd, China), a laser sensor (LK-H150, Keyence, Japan) and a computer. Fig. 23 shows the photo of the LUSM prototype used in experiment. The stator of the LUSM is assembled by using a holding box. It is noted that the preload can be changed by adjusting the blot connected with the preload spring, and the preload is fixed at the value of 14 N in the experiment. The voltage amplitude can be adjusted by the power amplifier. Fig. 24 shows the simulation and experimental results for the steady-state velocity of the slider under different values of input voltage and a fixed excitation frequency of
0
Voltage amplitude V 0 (V)
Dynamic friction coefficient µ
5. Low-voltage dead-zone behavior related to stick motion
0.2
0.235 0.23 0.225 0.22 0.215 0.21 0.205 0.2 10
15
20
25
Preload FP (N)
Fig. 21. Relationship between the dynamic friction coefficient and the preload under V0 ¼ 400 V.
X. Li et al. / International Journal of Mechanical Sciences 107 (2016) 215–224
low voltages are simulated as shown in Fig. 25. It is found that the values of these relative velocity fall within the given stick band ½ 1e 10; 1e 10, which means the contact tip isn’t able anymore to ‘escape’ the stick mode, and the slider won’t move with respect to the contact tip. This phenomenon causes the well-known dead-zone behavior in the voltage–velocity characteristic of the motor.
PC
Signal generator
LUSM
Power amplifier
223
Laser sensor
Fig. 22. Experimental setup for measuring velocity of the LUSM.
Slider 6. Conclusions
Contact tip
In this work, a novel friction model is proposed to study the stick– slip motion in a well-known linear piezoelectric ultrasonic motor. Different from the previous work, the influence of ultrasonic oscillation on the dynamic friction is taken into account in this model, in which the dynamic friction coefficient is modeled by an amplitude-dependent friction law. By using the switch model, the stick–slip motion is effectively simulated. The force transmission between the stator and the slider involving stick–slip motion is analyzed numerically, which provides more physical insights for the stick–slip motion in the linear piezoelectric motor. Various friction states of the interface including slip–separation, stick–slip, stick–slip–separation and pure stick are intuitively characterized by steady-state trajectories on a transformed phase plane. The influences of input voltage amplitude and external preload on stick–slip motion are investigated with the developed model. The results show that the input voltage amplitude acts to suppress the occurrence and the external preload denotes the easiness for the occurrence of stick–slip motion. Meanwhile, the pure stick motion of the interface is found to occur when the input voltage is low and the preload is relatively larger. It is worth noting that the pure stick motion can cause nonlinear dead-zone behavior of the motor, meaning that the pure stick motion should be avoided in the motor control. Furthermore, the stick–slip motion in the piezoelectric motor is contrasted with and without considering the influence of ultrasonic oscillation on the dynamic friction, and the result shows that ultrasonic oscillation can suppress the occurrence of stick-slip motion.
Stator
Friction bar
Holding box Preload spring Blot Fig. 23. Photo of experimental prototype.
Steady state velocity of the slider (m/s)
0.25 Simulation with stick-slip model Experiment
0.2
0.15
0.1
0.05
0 0
50
100
150
200
250
300
350
400
Acknowledgments
Voltage amplitude (V)
Fig. 24. Steady-state velocity of the slider under different input voltage amplitudes. 3
x 10 V0=25 V
2.5
Appendix A. Parameter expressions in dynamic equations of the stator
V0=50 V V0=75 V
rel
Relative velocity v (m/s)
This work is supported by the Program of National Natural Science Foundation of China (No. 51275229).
2
Z
V0=100 V
Ml ¼
l 0
ρAðϕL ðyÞÞ2 dy
ðA:1Þ
ρAðϕB ðyÞÞ2 dy
ðA:2Þ
1.5
Z Mb ¼
1
0
Z
0.5
l
Kl ¼
l 0
cE11 AðϕL ðyÞÞ2 dy
0
ðA:3Þ
0
ðA:4Þ
0
Z -0.5
Kb ¼ 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0
0.05
Time (s)
Fig. 25. The time-domain responses of the relative velocity under low voltages.
40 kHz, and a dead-zone about 120 V is observed from the simulation curve, which is in good agreement with the experimental result (around 100 V). With the developed stick–slip model, the timedomain responses of the relative velocity in 50 ms under several
Z Al ¼
Ab ¼
l
l
0
1 2
cE66 AðϕB ðyÞÞ2 dy
0
e31 AE3 ϕL ðyÞdy
Z
l 0
00
e31 AE3 ϕB ðyÞdy
ðA:5Þ
ðA:6Þ
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X. Li et al. / International Journal of Mechanical Sciences 107 (2016) 215–224
ηN ¼ ϕL ð0Þ ϕL ðlÞ
ðA:7Þ
ηT ¼ ϕB ð0Þ ϕB ðlÞ
ðA:8Þ
Here ρ is the density of the stator, A is the area of the cross-section area that is equal to bs hs . cE11 and cE66 are the mechanical compliance of the piezoelectric ceramic, e31 is the parameter of the piezoelectric ceramic. E3 denotes the amplitude of average electric field intensity of the stator due to the excitation voltage that is equal to V 0 =hs .
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