Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation

Mechanical Systems and Signal Processing xxx (xxxx) xxx

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Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation Xiang Li a,⇑, Chaohao Kan a, Yuan Cheng a, Zhiwei Chen a, Taian Ren b a b

School of Electrical Engineering and Automation, Hefei University of Technology, Hefei 230039, China Industrial Training Center, Hefei University of Technology, Hefei 230039, China

a r t i c l e

i n f o

Article history: Received 9 July 2019 Received in revised form 8 October 2019 Accepted 24 October 2019 Available online xxxx Keywords: Ultrasonic motor Piezoelectric vibrator Nonlinear electroelasticity Dynamical modeling Performance evaluation

a b s t r a c t This paper proposes a nonlinear whole-machine dynamic model for a typical bimodal standing wave ultrasonic motor considering nonlinear electroelasticity of piezoelectric vibrator under high excitation voltage. To capture the inherent nonlinearities of the piezoelectric element, nonlinear constitutive relations of piezoelectric element are adopted to derive the nonlinear electromechanical coupling model of the stator. The contact mechanism between the stator and the mover including stick-slip-separation dynamics and asperity contact mechanics is modeled based on a physical-based equivalent spring model and a modified Coulomb friction model. The measured nonlinear relationships between the excitation voltage and the amplitude response of the stator near resonant frequency are used along with analytical approximations for the nonlinear vibration responses by harmonic balance method to identify the critical nonlinear parameters. Finally, experiments are conducted to validate the proposed model, and the results show that the simulations agree well with the experimental results. In particular, nonlinear speed saturation phenomenon and the resonant frequency drift of the motor can be predicted and analyzed by the developed model when the motor is driven at a higher voltage, which are useful for the performance evaluation and working range choice of such motors under highpower input conditions. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Due to some unique merits distinguishing from traditional electromagnetic motors, such as high accuracy, rapid dynamic response, compactness in sizes and no electromagnetic interference [1], ultrasonic motors have been applied in precision positioning [2], machining system [3], robotics [4] and medical device [5]. In recent years, two types of ultrasonic motors including traveling wave ultrasonic motor (TWUM) and standing wave ultrasonic motor (SWUM) have attracted a wide interest in the field of piezoelectric actuators [6–9], and their working principle is determined by two-stage energy conversion. Based on the inverse piezoelectric effect of the piezoelectric elements, the electrical energy is converted into ultrasonicfrequency mechanical vibration energy of the stator (piezoelectric vibrator) in the first stage. Through vibro-impact frictional coupling, the high-frequency mechanical oscillation is transformed into macroscopic unidirectional rotary or linear motion

⇑ Corresponding author. E-mail address: [email protected] (X. Li). https://doi.org/10.1016/j.ymssp.2019.106475 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: X. Li, C. Kan, Y. Cheng et al., Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j. ymssp.2019.106475

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of a driven component in the second stage. As a result, ultrasonic motor has highly nonlinear characteristics due to some complex nonlinear factors such as the piezoelectric element nonlinearities and contact nonlinearities between the stator and the rotor (mover), which poses a challenge in obtaining its accurate dynamic model. An accurate dynamic model not only can be utilized to predict and evaluate the motor performance, but also can be served as the basis for motor optimization and controller design. Ultrasonic motor is a complex electromechanical coupling system, energy methods seem to be an appropriate choice for modeling its dynamics. A pioneering framework on dynamic modeling of the ultrasonic motor was proposed by Hagood and McFarland [10] in 1995, where they developed an electromechanical coupling model for a TWUM by using Rayleigh-Ritz assumed mode energy method. Later in 2003, Tsai et al. [11] adopted a similar method to develop an integral dynamic model for a well-known SWUM produced by Nanomotion Company, in which they investigated the force transfer in detail between the stator and slider and clarified the nonlinear deadzone behavior. Based on the energy principle, Li et al. [12] developed a thermal-mechanical-electric coupling dynamic model for a SWUM and investigated the coupling interaction between the temperature rise and the motor characteristics. Finite volume method was also employed to study the dynamics of the TWUM by Marquez et al. [13]. Recently, He et al. [14] investigated the dynamics of a linear SWUM with V-configuration stator by using a hybrid simulation model that combined with the finite element method and Rayleigh-Ritz assumed mode method. However, nonlinear electroelasticity of piezoelectric vibrator is not included in all above-mentioned models, and some well-known nonlinear behaviors of piezoelectric vibrators excited by high voltage such as amplitude saturation effect and softening behavior [15,16] cannot be predicted and analyzed. Nakagawa et al. [17] proposed a nonlinear dynamical model for a TWUM, in which the dynamic model of the motor was modeled as a second-order nonlinear oscillator and the softening spring-type nonlinearity was represented by a cubic equation. However, the stator was modeled as a three DOF oscillator and several empirical formulas were adopted in the model, which failed to model the inherent nonlinearities of the piezoelectric vibrator. Furthermore, the SWUMs are often driven at a higher voltage in order to improve their mechanical characteristics [18– 20], whereas some nonlinear characteristics of the motor including speed saturation [21] and resonant frequency drift [12] that were observed in the experiments under such excitation conditions have not been further clarified and analyzed. This work will focus on providing more physical insight into some important nonlinear characteristics of the SWUM under high voltage excitation by taking nonlinear electroelasticity of the stator into account. Meanwhile, it is expected that the speed saturation phenomenon and the resonant frequency drift of the SWUM under high voltage excitation can be predicted with the model considering nonlinear electroelasticity of the stator, which cannot be predicted by a linear model based on the linear theory [12,21]. On the other hand, the model-based performance evaluation for linear SWUM is significant to the motor optimization and control [14,22], but the accuracy is greatly affected by some complicated nonlinear contact mechanics such as asperity contact mechanics [21] and stick-slip-separation dynamics [22]. To the knowledge of the authors, none of literatures has modeled the dynamics of linear SWUMs taking into account nonlinear electroelasticity of the piezoelectric vibrator as well as the nonlinearities caused by the stator/mover contact and friction. The purpose of this study is to develop a physics-based nonlinear whole-machine model to analyze and evaluate the nonlinear characteristics of a typical bimodal linear SWUM under high-voltage excitation, and the primary novelty of this work is the consideration of the inherent nonlinearities of the stator. The paper is organized as follows. Firstly, the structure and the operating principle of the motor are introduced briefly. Secondly, a nonlinear whole-machine dynamic model of the motor considering nonlinear electroelasticity is established, where the nonlinear electro-mechanical coupling model of the stator is derived by nonlinear constitutive relations of piezoelectric material, and the contact mechanism between the stator and mover is modeled via an asperity contact based equivalent spring model and a modified Coulomb friction model in which both the stick-slip-separation dynamics and asperity contact mechanics are included. Meanwhile, several critical nonlinear parameters in the model are identified by the measured results for the stator vibration responses and the analytical approximations for nonlinear vibration responses based on the harmonic balance method. Finally, simulation results are compared with the experimental results. It is shown that the speed saturation phenomenon and the resonant frequency drift of the motor can be accurately predicted and analyzed with the developed model when the motor is driven at a higher voltage.

2. SWUM prototype and operating principle Fig. 1 shows the schematic of the bimodal linear SWUM studied in this paper, which consists of a stator with a contact tip, a mover with a friction bar, a preload spring and a housing box. The stator is symmetrically clamped by four support nylon blots at the nodes of the second bending vibration mode. Eight PZT-8 piezoelectric patches are bonded symmetrically onto a metal substrate to form the stator, and its geometrical properties and material properties can be found in the literature [12]. The first-order longitudinal mode (L1-mode) and the second-order bending mode (B2-mode) of the stator are excited when two phase sinusoidal excitation voltage signals with the same frequency that are close to the natural frequencies for the two modes (f1 = 39516 Hz and f2 = 39469 Hz, respectively) but a phase difference of p=2 are diagonally applied to the electrodes on the piezoelectric patches, then a micro elliptic motion is formed at the contact tip that produces a friction force for driving the mover to perform a macro linear motion, as illustrated in Fig. 2. Please cite this article as: X. Li, C. Kan, Y. Cheng et al., Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j. ymssp.2019.106475

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Friction bar Preload spring

Contact tip

Stator

3

Lin ear mo tio Mover n

A phase exciting signal B phase exciting signal

Clamping blot

Fig. 1. Schematic of the SWUM.

3. Nonlinear modeling In this section, a nonlinear whole-machine dynamic model of the SWUM is developed, where the stator is modeled as a piezoelectric vibrator considering nonlinear electroelasticity, and the complex interface mechanism between the stator and mover is modeled considering stick-slip-separation dynamics and asperity contact mechanics. 3.1. Nonlinear stator model Since the shear mode of the stator cannot be excited within the operating frequency of the motor [12], the stator herein is modeled as an Euler-Bernoulli beam with symmetric clamped conditions at the edge, as shown in Fig. 3. Furthermore, the space between the piezoelectric patches on the stator is neglected in modeling. Fig. 4 shows the mechanical models of the simplified stator and the contact tip, where F N and F T represent the normal force and the tangential force at the interface, respectively. The clamping effect on the stator is modeled as well, where L1 is the distance between the clamping and the longitudinal side edges of the beam and L2 is the distance between the clamping at the transverse side edges. The reaction forces caused by the clamping parts are denoted by F 1 and F 2 that are given by

F1 ¼

L1 FT ; L2

F2 ¼


 L1 FT : 1þ L2

ð1Þ

With the assumption of the Euler-Bernoulli beam, the strain-displacement relations for the L1-mode and B2-mode of the beam can be respectively expressed as:

eL ¼

@uðx;tÞ ; @x

eB ¼ 12 y @

2

ð2Þ

wðx;tÞ ; @x2

where eL ; eB are the normal strains for the L1-mode and B2-mode of the beam, respectively, and u ¼ uðx; tÞ and w ¼ wðx; tÞ are the axial displacement and deflection of the beam, respectively.

Linear motion

B2-mode vibration

L1-mode vibration

Mover Elliptic motion

+

=

Fig. 2. Illustration of operating principle.

Please cite this article as: X. Li, C. Kan, Y. Cheng et al., Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j. ymssp.2019.106475

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L2

L1

y

L1 hs

O

x

hs

O

z

ts t p Fig. 3. Simplified stator with clamped conditions at the edge.

Since the piezoelectric 31-effect of the piezoceramic is utilized in the stator, the constitutive relations within the piezoelectric element considering nonlinear elastic and coupling behavior of the piezoceramic are described by nonlinear constitutive equations [23]

T p1 ¼ cE11 S1 þ cE111 S21 þ cE1111 S31  e31 E3  e311 S1 E3  e3111 S21 E3 ;

ð3Þ

D3 ¼ e31 S1 þ e311 S1 E3 þ e3111 S21 E3 þ eS33 E3 ;

where T p1 and S1 represent the stress and strain along the length of the beam, respectively. D3 and E3 indicate the electric displacement and electric field that develops through the thickness of the piezoelectric patches. The material coefficients cE11 and cE66 , cE111 and cE1111 and e31 ; e311 ; and e3111 are the second, third, and fourth order elastic and electroelastic tensor components, respectively. eS33 is the electric permittivity at zero strain. It should be noted that the strain S1 ¼ eL for L1-mode, while S1 ¼ eB for B2-mode. As a result, the stress T p1 and the electric displacement D3 of the piezoelectric element are different for L1-mode and B2-mode of the stator vibration. To prevent confusion, we use T p1 and D3 to represent the stress and the p ^ 3 to represent the stress and the electric displacement for B2-mode electric displacement for L1-mode, while use T^ and D 1

hereafter. When two excitation voltage signals with V A ¼ V 0 sinxt and V B ¼ V 0 cosxt are applied to the electrodes of the piezoelectric elements, across which the electrical field can be approximately expressed as:

E3 ðtÞ ¼

8 <

VA tp

; ðx; y; zÞ 2 X1 [ X4 [ X5 [ X8

ð4Þ

:  V B ; ðx; y; zÞ 2 X2 [ X3 [ X6 [ X7 tp

where the detailed expresses for zones Xi ði ¼ 1; 2; :::; 8Þ can be found in the literature [12]. The kinetic energy within the stator is given by

 2  2 RL T ¼ 12 0 ½qs As @u þ qs As @w  dx @t @t  2 @w2 RL þ2 0 ½qp Ap @u þ q A  dx; p p @t @t

ð5Þ

in which As ¼ 2hs ts and Ap ¼ hs tp are the cross-sectional areas of the substrate and a single piezoelectric patch, respectively. The total potential energy of the stator is given by

R L  2 R L  2 2 U ¼ 12 Es As 0 @u dx þ 12 Es Is 0 @@xw2 dx @x R p ^ 3 E3 ÞdV p þ 12 V p ðT p1 eL þ T^ 1 eB  D3 E3  D

ð6Þ

3

where Is ¼ 2t s hs =3 is the moment of the inertia of the substrate. V p is the total volumes of the piezoelectric elements on the stator. In addition, the variational work caused by external forces can be expressed as:

dW F ¼ F N  dujx¼0  F N  dujx¼L þ F T  dwjx¼L  F 1 L2  dhjx¼L1 þ F T L1  dhjx¼L ;

ð7Þ

where h ¼ @w is the rotation angle of the beam. Again, it is noted that the stator is clamped at the nodes of the B2-mode of the @x stator, thus the rotation angle at the clamping points can be ignored. As a result, Eq. (7) can be further reduced to

F2 u

FT FN FN

FN

FT

FN

FT F1

Stator

Contact tip

Fig. 4. Mechanical models of the stator and contact tip.

Please cite this article as: X. Li, C. Kan, Y. Cheng et al., Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j. ymssp.2019.106475

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dW F ¼ F N  dujx¼0  F N  dujx¼L þ F T  dwjx¼L þ F T L1  dhjx¼L ;

5

ð8Þ

Then a modified Hamilton’s principle [24] is adopted to derive the governing equations for the dynamics of the stator considering clamping and contact-based operating conditions

Z

t2

d

Z ðT  UÞdt þ

t1

t2

dW F dt ¼ 0

ð9Þ

t1

Furthermore, Rayleigh-Ritz assumed mode method is used to obtain the ordinary differential dynamic equations of the stator instead of the partial differential dynamic equations derived from (9). Considering that the motor operates with the L1-mode and B2-mode of the stator, the displacement fields of the stator are expressed as:

uðx; tÞ ¼ /L ðxÞq1 ðtÞ; wðx; tÞ ¼ /B ðxÞq2 ðtÞ;

ð10Þ

where q1 ðtÞ; q2 ðtÞ are the generalized coordinates, /L ðxÞ; /B ðxÞ are the assumed mode shape functions corresponding to uðx; tÞ and wðx; tÞ, respectively. Here the functions /L ðxÞ and /B ðxÞ are chosen to be the eigenfunctions of an EulerBernoulli beam for L1-mode and B2-mode of the stator vibration, respectively. With a free-free boundary condition, the shape functions are given by

/L ðxÞ ¼ cos pLx ; /B ðxÞ ¼ a2 ðcosb2 x þ coshb2 xÞ þ ðsinb2 x þ sinhb2 xÞ;

ð11Þ

in which b2 ¼ 2:499753p=L; a2 ¼ ðsinb2 L  sinhb2 LÞ=ðcoshb2 L  cosb2 LÞ: Substituting the displacement fields in (10) into the energy and virtual work terms in (9), and then integrating the domain, one can derive the dynamic equations of the stator excluding its structural damping

€1 ðtÞ þ K 1 q1 ðtÞ þ cE111 a1 q21 ðtÞ þ cE1111 b1 q31 ðtÞ M1 q ¼ a0 ðtÞ þ gN F N þ e311 a1 ðtÞ þ e311 a2 ðtÞq1 ðtÞ þ e3111 a3 ðtÞq1 ðtÞ þ e3111 a4 ðtÞq1 ðtÞ2 ; €2 ðtÞ þ K 2 q2 ðtÞ þ cE1111 b2 q32 ðtÞ M2 q ¼ b0 ðtÞ þ gT F T þ e311 b1 ðtÞ þ e311 b2 ðtÞq2 ðtÞ þ e3111 b3 ðtÞq2 ðtÞ þ e3111 b4 ðtÞq2 ðtÞ2 ;

ð12Þ

ð13Þ

in which the nonlinear parameters cE111 ; cE1111 ; e311 and e3111 will be identified later, and the definitions of the rest coefficients are given in Appendix A. By adding two linear modal damping terms to the equations, the dynamic equations are modified as:

€1 ðtÞ þ C 1 q_ 1 ðtÞ þ K 1 q1 ðtÞ þ cE111 a1 q21 ðtÞ þ cE1111 b1 q31 ðtÞ M1 q ¼ a0 ðtÞ þ gN F N þ e311 a1 ðtÞ þ e311 a2 ðtÞq1 ðtÞ þ e3111 a3 ðtÞq1 ðtÞ þ e3111 a4 ðtÞq1 ðtÞ2 ; €2 ðtÞ þ C 2 q_ 2 ðtÞ þ K 2 q2 ðtÞ þ cE1111 b2 q32 ðtÞ M2 q ¼ b0 ðtÞ þ gT F T þ e311 b1 ðtÞ þ e311 b2 ðtÞq2 ðtÞ þ e3111 b3 ðtÞq2 ðtÞ þ e3111 b4 ðtÞq2 ðtÞ2 :

ð14Þ

ð15Þ

Here the damping coefficients C 1 and C 2 can be determined by half-power bandwidth method based on the frequency scanning results of the stator at a low applied voltage [12]. Moreover, experimental investigations shown that a linear damping law can provide a sufficiently accurate description of the damping behavior for piezo-beam systems [25]. 3.2. Interface model The contact type between the stator and the mover of SWUM is characterized by an impact-like intermittent contact mode [21]. For simplicity, an equivalent spring model based on asperity contact is utilized to describe the normal contact between the stator and the mover, where both the interface roughness and intermittent separation are considered, and the normal interface force is given by

 FN ¼

F P þ ke utip ; 0

F P þ ke utip > 0; ðcontactÞ F P þ ke utip 6 0; ðseparationÞ

ð16Þ

where F P is the applied preload. utip is the dynamic displacement of the contact tip which is associated with the deformation of the interface asperities and the longitudinal vibration of the stator. The contact tip separates from the mover and the interface normal vanishes, when the condition of F P þ utip 6 0 is satisfied. The parameter ke represents the stiffness of the equivalent spring. During the real contact process between the contact tip and friction bar, the contact of the two surfaces takes place on asperities, as shown Fig. 5. In this paper, the assumption of the elastically deformed asperities is adopted because of a relative low applied preload, and the contact of the asperities is described by GW elastic contact model [26]. In GW model, it is assumed that the heights of the asperities follow a certain probability distribution function that is often described by a GausPlease cite this article as: X. Li, C. Kan, Y. Cheng et al., Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j. ymssp.2019.106475

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Nominal flat surface (Friction bar) Asperities

Summit radius

Asperity heights

Mean of asperity heights Nominal flat surface (Contact tip) Fig. 5. Contacts of asperities between the contact tip and friction bar.

sian distribution in many engineering rough surfaces [26]. By an approximate method, the nonlinear contact force between the two roughness surfaces for an exponential distribution and elastic contact can be expressed as [27]

PðdÞ ¼

c

pffiffiffiffi


1=2

pbE An Rq k5=2

ekd=Rq ;

Rs

ð17Þ

where E denotes the combined Young’s modulus of two materials that equals to 1=E ¼ ð1  t21 Þ=E1 þ ð1  t22 Þ=E2 . An is the nominal contact area. c ¼ 17 and k ¼ 3 are curved-fitted constants that cover most practical ranges [27]. Rq and Rs are the standard deviation of surface roughness and the average radius of asperities, respectively. b ¼ gRq Rs is the roughness parameter, g is the areal density of asperities. d represents the normal distance between the two roughness surfaces, which is associated with the dynamic normal displacement of the contact tip and the static deformation of the asperities at the interface. It is given by [21]

d ¼ dc  ðutip þ r0 Þ;

ð18Þ

where dc is the critical mean normal separation between the rough surfaces that is determined by the surface roughness. r0 is the initial contact deformation of the asperities under the preload only, which can be calculated by (17). The actual measured values of these roughness parameters in [21] will be adopted approximately to calculate the nonlinear contact force in terms of (17). Under the assumption of the elastically deformed asperities, the parameter d is periodic and time-varied due to the periodic longitudinal vibration of stator, then the interface contact force PðdÞ changes periodically. According to the equivalent principle of the impulse during one contact period, the relationship between the equivalent spring model described by (16) and the roughness contact model described by (17) can be determined by

Z

tþT c

Z ðF p þ ke utip Þdt ¼

t

tþT c

ð19Þ

PðdÞdt: t

Then the stiffness parameter ke is solved as

R tþT c ke ¼

t

PðdÞdt  F p T c ; R tþT c utip dt t

ð20Þ

in which T c is the contact duration between the contact tip and the friction bar in one overall driving period, which can be estimated by an electrical contact test [21]. Meanwhile, Eq. (20) implies that the stiffness of the equivalent spring is associated with the roughness of the contact surfaces, which is more close to the real physical situation. On the other hand, the interface tangential force between the contact tip and friction bar is modeled based on Coulomb friction model. Moreover, in order to include the stick-slip-separation dynamics that could occur between the contact tip and friction bar, the interface tangential force is defined as [14]:

( FT ¼

lc F N Erfð3:6  10a v rel Þ; F N –0; ðslip and stickÞ 0

F N ¼ 0; ðseparationÞ

;

ð21Þ

with

v rel ¼ v tip  v m :

ð22Þ

where Erf is the Gauss error function, lc is the Coulomb friction coefficient at the interface. a is the sliding constant factor that determines the accuracy level in the Coulomb friction and controls the transition speed between slip and stick [28]. v tip and v m are the tangential velocity of the contact tip and the mover velocity, respectively. Please cite this article as: X. Li, C. Kan, Y. Cheng et al., Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j. ymssp.2019.106475

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3.3. Mover kinetic model By force equilibrium, the kinetic equation of the mover can be expressed as:

mv_ m þ cv m ¼ F T þ F f  F load :

ð23Þ

Here m is the mass of the mover, c is the damping coefficient that can be identified in terms of the transient response of the motor velocity. F f and F load are the friction force existing in the linear guide and the applied load, respectively. Moreover, in order to characterize the dead-zone behavior, the friction force F f is defined as follows:

 Ff ¼

F load  F T ; jF T j < jf max j þ F load ; F f max ; jF T j P jf max j þ F load

ð24Þ

with

f max ¼ signðv m Þls ðF N þ 9:81mÞ; where

ð25Þ

ls is the maximum static friction coefficient of the linear guide.

4. Determination of nonlinear parameters Excluding the interface forces in (14) and (15), the vibration equations of the stator removing the mover are reduced to

€1 ðtÞ þ C 1 q_ 1 ðtÞ þ K 1 q1 ðtÞ þ cE111 a1 q21 ðtÞ þ cE1111 b1 q31 ðtÞ ¼ a0 ðtÞ þ e311 a1 ðtÞ þ e311 a2 ðtÞq1 ðtÞ þ e3111 a3 ðtÞq1 ðtÞ M1 q þ e3111 a4 ðtÞq1 ðtÞ2 ;

ð26Þ

€2 ðtÞ þ C 2 q_ 2 ðtÞ þ K 2 q2 ðtÞ þ cE1111 b2 q32 ðtÞ ¼ b0 ðtÞ þ e311 b1 ðtÞ þ e311 b2 ðtÞq2 ðtÞ þ e3111 b3 ðtÞq2 ðtÞ M2 q þ e3111 b4 ðtÞq2 ðtÞ2 :

ð27Þ

A first approximate solution of the two degree-of-freedom nonlinear vibration equations above is found by harmonic balance method [29]. For this purpose, the steady state solutions for the modal displacements are assumed as:

q1 ðtÞ ¼ A cosðxt þ u1 Þ;

ð28Þ

q2 ðtÞ ¼ B cosðxt þ u2 Þ;

ð29Þ

where u1 and u2 is the respective phase offset. For harmonic excitation near a fundamental frequency, this approximation is sufficiently accurate for describing steady state oscillations [29]. By the shape functions described in (11), the normal and tangential displacements of the contact tip on the stator removing the mover can be approximately expressed as:

utip ¼ /L ðLÞq1 ðtÞ ¼ /L ðLÞA cosðxt þ u1 Þ; wtip ¼ /B ðLÞq2 ðtÞ ¼ /B ðLÞB cosðxt þ u2 Þ;

ð30Þ

Substituting the assumed solutions into modal Eqs. (26) and (27) and neglecting all but the first harmonic, one can yield the following two tenth order polynomial equations for the amplitudes A and B respectively, after tedious trigonometric operations

a10 A10 þ a8 A8 þ a7 A7 þ a6 A6 þ a5 A5 þ a4 A4 þ a2 A2 ¼ 1;

ð31Þ

b10 B10 þ b8 B8 þ b7 B7 þ b6 B6 þ b4 B4 þ b3 B3 þ b2 B2 ¼ 1;

ð32Þ

where the coefficients are listed in Appendix B. The nonlinear relationships between the amplitudes and the excitation voltage at natural frequencies can be obtained by numerically resolving the roots of the nonlinear algebraic expressions above, which are used to identify the nonlinear parameters with the help of the measured amplitude responses of the stator later. The longitudinal and bending vibration responses of the stator removing the mover are measured by a 3D laser Doppler vibrometer system (PSV-500F-B, Polytec, Inc., Waldbronn, Germany), as shown Fig. 6, where the stator is clamped at its nodes of B2 vibration mode and is excited by two-phase sinusoidal voltage signals generated and amplified by a signal generator and two power amplifiers. The normal and tangential vibration amplitudes of the contact tip at the stator can be simultaneously measured via the 3D Doppler vibrometer system. Figs. 7 and 8 show the measured normal and tangential amplitude responses of the contact tip obtained by exciting the stator at different applied voltages, where the exciting frequency is fixed at f e ¼ 39:5 kHz that is very close to the two natural frequencies for L1-mode and B2-mode of the stator. It can be seen that relationship between excitation voltage and amplitude response is approximately linear at lower voltages, whereas a nonlinear saturation amplitude behavior is clearly shown under a higher voltage (higher than 250 V). The nonlinear relationships are used for the determination of the four non-linear parameters cE111 ; cE1111 ; e311 and e3111 by (30)–(32) along with a nonlinear least-squares optimization algorithm implemented in MATLAB. Please cite this article as: X. Li, C. Kan, Y. Cheng et al., Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j. ymssp.2019.106475

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3D Doppler Vibrometer System

Stator

Front view

Signal generator

Power Amplifier

Fig. 6. Experimental setup for measuring the vibration responses of the stator.

Given the material parameters listed in Table 1, all modal parameters and coefficients except for non-linear parameters in the vibration Eqs. (26) and (27) can be obtained based on Appendix A, by which the optimal fitting curves to the experimental data are shown in Figs. 7 and 8, and the corresponding identified nonlinear parameters are listed in Table 2. For comparison, the tangential response amplitude is measured at the resonance frequency near the natural frequency for the B2-mode of the stator under different excitation voltages, and the measured and calculated results with the identified nonlinear parameters are shown in Fig. 9, which exhibits a good agreement between the theoretical and experimental results, and an obvious softening behavior caused by nonlinear electroelasticity of the piezoelectric vibrator (stator) is observed in the figure. 5. Simulation and experimental validation In this section, the comparisons between the simulation and experimental investigations will be performed to validate the nonlinear whole-machine model of the motor developed in Section 3, and the performance evaluation and some nonlinear characteristics of the motor under high excitation voltage will be discussed next. 5.1. The motion trajectory of the contact tip As described in Section 2, the micro elliptical motion at the contact tip resulting from the superposition of the L1 and B2 modes of the stator is significantly important to the force transmission between the stator and mover as well as the motor performance. The steady-state trajectory of the contact tip removing the mover can be simulated by using (30), and the actual trajectory of the contact tip can be obtained by extracting the normal and tangential displacements of the contact tip measured by the experiment in Fig. 6. Figs. 10 and 11 show the comparisons of the simulation and experimental results for the motion trajectories of the contact tip with different excitation conditions. It is found that the simulation and exper-

Fig. 7. Normal amplitude responses of the contact tip with different excitation voltages.

Please cite this article as: X. Li, C. Kan, Y. Cheng et al., Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j. ymssp.2019.106475

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Fig. 8. Tangential amplitude responses of the contact tip with different excitation voltages.

imental results with different excitation conditions are basically coincident, which further verifies the effectiveness of the nonlinear model of the stator developed in Section 3. Fig. 10 shows the comparison of the simulation and experimental results for the motion trajectories of the contact tip under different excitation voltages, where the excitation frequency is fixed at 39.5 kHz. As expected, the magnitude of the elliptical trajectory increases with the excitation voltage, while the increasing magnitude exhibits a saturation phenomenon due to the amplitude saturation effect of the stator under higher excitation voltage as described in Figs. 7 and 8. Fig. 11 shows the comparison of the simulation and experimental results for the motion trajectories of the contact tip under different excitation frequencies, where the excitation voltage peakpeak value is fixed at 400 V, and the results show that the magnitude of the elliptical trajectory is greater when the excitation frequency is 39 kHz or 39.2 kHz than the magnitude when the excitation frequency is 39.5 kHz which is more closer to the natural frequencies for the L1 and B2 modes of the stator. The reason behind this phenomenon is the softening behavior due to nonlinearities under high excitation voltage as described in Fig. 9. As a result, for the case of high excitation voltage, the choice of the excitation frequency should be slightly lower than the natural frequencies for the L1 and B2 modes of the stator in order to enlarge the magnitude of the elliptical trajectory at the contact tip. 5.2. Performance evaluation of the motor To simulate the motor dynamics, a Runge-Kutta integration scheme is employed to solve the nonlinear differential dynamic equations by using MATLAB software (The MathWorks, Natick, MA). The parameters used in the simulation are listed in Table 3 unless stated otherwise. The experimental setup for measuring the motor performance is shown in Fig. 12. A signal generator (AFG 3022B, Tektronix, Inc. USA) is used to generate the two-phase excitation signals with 900 phase shift, which are amplified by two power amplifiers (HFVA-42, Nanjing Funeng Technology Corp., China). The frequency and amplitude of the excitation voltage V A and V B can be adjusted by the signal generator and power amplifier, respectively, and the preload is fixed at the value of 24.8 N which follows the simulation value. A laser sensor (LK-H150, Keyence, Japan) linked with a computer is utilized to measure the speed of the mover with or without a load. Fig. 13 shows the simulation and experimental results for the no-load speed of the motor with different excitation voltages at a fixed frequency of 40 kHz. A low-voltage (lower than 50 V) deadzone behavior is observed both in simulation and experimental results. The motor speed is found to increase with the increasing excitation voltage within the range from 50 V to 400 V, and an obvious nonlinear speed saturation phenomenon is observed when the excitation voltage is higher than 250

Table 1 Material parameters of the stator. Parameter

Value

Density of 45 steel qs Young modulus of 45 steel Es

7850 kg=m3

Poisson ratio of 45 steel Density of PZT-8 qp

t

2:09  1011 Pa 0.269 7600 kg=m3

Mechanical compliance of PZT-8 cE11

12:23  1010 N=m2

Mechanical compliance of PZT-8 cE66 Piezoelectric parameter of PZT-8 e31 Damping coefficient C 1 Damping coefficient C 2

3:73  1010 N=m2 5:2 C=m2 39 Ns=m 48 Ns=m

Please cite this article as: X. Li, C. Kan, Y. Cheng et al., Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j. ymssp.2019.106475

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Table 2 Identified non-linear parameters. Parameter Value

C E111 ðN=m2 Þ 1:4583  10

C E1111 ðN=m2 Þ 17

12

2:8856  10

e311 ðm=VÞ 3:3142  10

e3111 ðm=VÞ 6

6:4235  103

Fig. 9. Back-bone curve (tangential amplitude at the contact tip and resonant frequency near the natural frequency for B2-mode of the stator under different excitation voltages).

V, which can be attributed to the amplitude saturation effect of the stator under high excitation voltage as described in the previous section. Furthermore, the simulation results based on the linear model excluding nonlinear electroelasticity (i.e. e311 ¼ e3111 ¼ C E111 ¼ C E1111 ¼ 0) and by the nonlinear model of the motor considering nonlinear electroelasticity are compared in the figure, which presents the advantages of the proposed nonlinear model in evaluating the motor performance under high-power input conditions. It should be noted that, for ultrasonic motors, the resonant frequency of the motor is not the resonant frequency of the stator due to the influences of the preload and contact stiffness [30], and the actual resonant frequency of the motor is found to be higher than the resonant frequency of the stator [31]. Hence, the driving frequency for the motor is selected as 40 kHz that is higher than the natural frequencies of the stator in Fig. 13. Fig. 14 shows the simulation and experimental results for the no-load speed of the motor versus the driving frequency under different excitation voltages, in which a good agreement between the simulations and experiments is observed. In the case of 100 V, the maximum speed of the motor occurs at the driving frequency about 40. 4 kHz, which is called the resonant frequency of the motor and it is higher than the resonant frequency of the stator (about 39.5 kHz). In the case of 250 V, the maximum speed of the motor occurs at the driving frequency about 40. 2 kHz, which is lower than the resonant fre-

Fig. 10. The motion trajectories of the contact tip under different excitation voltages at a fixed excitation frequency of 39.5 kHz.

Please cite this article as: X. Li, C. Kan, Y. Cheng et al., Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j. ymssp.2019.106475

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Fig. 11. The motion trajectories of the contact tip under different excitation frequencies at a fixed excitation voltage of 400 V.

quency for the case of 100 V. In particular, an obvious resonant frequency drift towards to the left side is observed when the excitation voltage increases to 400 V, which decreases to about 39. 6 kHz. In other words, the driving frequency for the maximum motor speed changes with the excitation voltage, which is caused by the resonant frequency drift of the motor under high excitation voltage. However, the inherent reason for the resonant frequency drift of the motor is due to the softening behavior of the amplitude response for the stator caused by nonlinear electroelasticity under high excitation voltage, as described in Fig. 9. Fig. 15 illustrates how the resonant frequency of the motor drifts in a nonlinear manner under excitation voltage, and the simulation and experimental results indicate that the drift tendency of the frequency is more obvious for a higher excitation voltage and the nonlinear frequency drift phenomenon can be accurately predicted and explained by the nonlinear model proposed in this study. This result cannot be predicted by a linear model based on the linear theory. It is noted that the resonant frequency drift of the motor could be very useful for the choice of the optimal working frequency of the motor under high-power input conditions from mechanical characteristics improvement aspect. The simulation and experimental results for the comparisons of the mechanical characteristics of the motor under high-power input conditions with the driving frequencies considering and without considering resonant frequency drift are presented in Fig. 16, where the excitation voltage peak-peak value is 300 V and 400 V in the plots (a) and (b), respectively. For comparison, the mechanical characteristics of the motor are simulated and measured with the driving frequency of 40.4 kHz which is very close to the resonant frequency of the motor under a relative lower excitation voltage (see Fig. 14). According to Fig. 16, one can see that the motor has better mechanical characteristics under high-power input conditions when the driving frequency is chosen to be 40.06 kHz and 39.6 kHz (both lower than 40.4 kHz) considering the effect of the resonant frequency drift of the motor as described in Fig. 15, and the improvement tendency is more remarkable for the case of 400 V (see Fig. 16(b)) due to a relative larger resonant frequency drift tendency towards to the left side as shown in Fig. 15. As a result, the motor can operate under a lower working frequency when the excitation voltage is relative higher comparing to the case with a relative lower excitation voltage, in order to improve the mechanical characteristics of the motor. It should be pointed out that, however, the experiments in this paper are implemented within a relative short time for avoiding obvious temperature rise of the motor under high-power input conditions and serious wear behavior. Actually, the motor temperature will rise remarkably under a long-term running and a high-power input condition, which could further aggravate the resonant frequency drift tendency of the motor and degrade the motor performance due to the changes of the material parameters caused by temperature rising effect [12]. The performance changes of the motor induced by temperature rise cannot be predicted with the present model, and the interaction effect between nonlinear vibration and temperature increase will be studied in the future research. On the other hand, the contact parameters such as roughness

Table 3 Parameters used in simulation. Parameter

Value

Applied preload F p Mass of the mover m Damping coefficient c Equivalent spring stiffness ke

24.8 N 0:2 kg 1.3 Ns/m

Kinetic friction coefficient lc Friction coefficient of the linear guide Sliding constant factor a

ls

2:58  107 N=m 0:35 0:04 6

Please cite this article as: X. Li, C. Kan, Y. Cheng et al., Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j. ymssp.2019.106475

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Signal Generator Data Acquisition

Laser Sensor

Mover Stator

Power Amplifier

Loading Device

SWUM Fig. 12. Experimental setup for measuring motor performance.

Fig. 13. No-load speed of the motor with different excitation voltages at a fixed frequency of 40 kHz.

Fig. 14. No-load speed of the motor versus driving frequency under different excitation voltages.

parameters and friction coefficient will dramatically change under continuous operation if the improper friction pairs are adopted due to serious wear behavior [32], then the contact parameters should be modified for avoiding resulting in erroneous prediction by the proposed model. Fortunately, the linear ultrasonic motor is usually applied in precision positioning field [1], which means that it often operates under a relative short time. As a result, the temperature rise and wear problems could be ignored upon most occasions, and the proposed model in this paper is feasible. Please cite this article as: X. Li, C. Kan, Y. Cheng et al., Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j. ymssp.2019.106475

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Fig. 15. Driving frequency drift for maximum no-load speed of the motor under high excitation voltages.

Fig. 16. Comparison of mechanical characteristics of the motor under high-power input conditions with the driving frequencies considering and without considering resonant frequency drift. (a) Excitation voltage V p - p ¼ 300 V. (b) Excitation voltage V p - p ¼ 400 V.

6. Conclusions In this work, a nonlinear whole-machine dynamic model is developed to evaluate the performance of a typical bimodal linear SWUM. Differing from the previous dynamical models of SWUMs, the inherent nonlinear electroelasticity of piezoelectric vibrator under high excitation voltage is taken into account by employing nonlinear constitutive equations of piezoelectric element in modeling. Furthermore, the measured vibration responses for the stator along with analytical approximations for the nonlinear vibration responses by harmonic balance method are used to identify several critical nonlinear parameters that determine the nonlinear saturation and softening behaviors of the stator vibration responses under high excitation voltage. Finally, simulation and experiment are conducted to validate the proposed nonlinear dynamical model, and the results indicated that the simulations agree well with the experimental results. It is shown that, in particular, the speed saturation phenomenon and the resonant frequency drift of the motor can be accurately predicted and analyzed with the developed model when the motor is driven at a higher voltage, which cannot be predicted by a linear model based on the linear theory. It is believed that the proposed nonlinear whole-machine model considering nonlinear electroelasticity of the stator could be very useful for the performance evaluation and working range choice of such motors under high-power input conditions. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Please cite this article as: X. Li, C. Kan, Y. Cheng et al., Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j. ymssp.2019.106475

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Acknowledgements This research was supported by the Natural Science Foundation of Anhui Province (No. JZ2019AKZR0219), and the Fundamental Research Funds for the Central Universities. The authors would like to thank Dr. Z. Liu with his assistance of 3D Doppler vibrometer system. Appendix A The modal mass, the stiffness coefficients, the electrical field excitation terms, interface forces factors and the parameters for nonlinear terms in (12) and (13) are expressed as follow:

M 1 ¼ ðqs As þ 4qp Ap Þ  M2 ¼ ðqs As þ 4qp Ap Þ 

RL 0

ð/L ðxÞÞ2 dx;

0

ð/B ðxÞÞ2 dx;

RL

RL 2 K 1 ¼ 0 ðEs As þ 4cE11 Ap Þð/0L ðxÞÞ dx; RL 2 2 K 2 ¼ 0 ðEs Is þ 13 cE11 hp Ap Þð/00L ðxÞÞ dx; R a0 ðtÞ ¼  S 8

Xi

R b0 ðtÞ ¼  12 S 8

ð34Þ

e31 E3 ðtÞ/0L ðxÞdV p ;

i¼1

Xi

ð33Þ

e31 E3 ðtÞ/00B ðxÞydV p ;

ð35Þ

i¼1

gN ¼ /L ð0Þ  /L ðLÞ; gT ¼ /B ðLÞ þ L1 /0B ðLÞ; a1 ¼ 6 b1 ¼ 8 1 b2 ¼ 10

RL 0

RL

ð36Þ

Ap ð/0L ðxÞÞ dx; 3

4

A ð/0L ðxÞÞ dx; 0 p RL 4 4 h A ð/0L ðxÞÞ dx; 0 p p

R a1 ðtÞ ¼ 2 S 8

Xi

ð37Þ

E23 ðtÞ/0L ðxÞdV p ;

i¼1

R a2 ðtÞ ¼ 4 S 8

2

Xi

E3 ðtÞð/0L ðxÞÞ dV p ;

i¼1

R a3 ðtÞ ¼ 4 S 8 R

2

Xi

E23 ðtÞð/0L ðxÞÞ dV p ;

ð38Þ

i¼1

a4 ðtÞ ¼ 6 S 8

3

Xi

E3 ðtÞð/0L ðxÞÞ dV p ;

i¼1

R b1 ðtÞ ¼  S 8

Xi

E23 ðtÞ/00B ðxÞydV p ;

i¼1

R b2 ðtÞ ¼  S 8

2

Xi

E3 ðtÞð/00B ðxÞÞ y2 dV p ;

i¼1

R b3 ðtÞ ¼  S 8

2

Xi

E23 ðtÞð/00B ðxÞÞ y2 dV p ;

ð39Þ

i¼1

R

b4 ðtÞ ¼  34 S 8

3

Xi

E3 ðtÞð/00B ðxÞÞ y3 dV p ;

i¼1

Appendix B The coefficients in Eqs. (31) and (32) are expressed as follows:

1 1 2 a2 ¼ ½ðK 1  M 1 x2 Þ þ ðC 1 xÞ2 ðe31 V 0 Þ2  e3111 ðe31 V 0 Þ3 þ e311 ðe31 V 0 Þ2 ; 2 4

ð40Þ

Please cite this article as: X. Li, C. Kan, Y. Cheng et al., Performance evaluation of a bimodal standing-wave ultrasonic motor considering nonlinear electroelasticity: Modeling and experimental validation, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j. ymssp.2019.106475

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a4 ¼

h

3 ðK 1 2

  M 1 x2 Þ C E1111 b1 þ

3 E C 8 111



i

a1  118 e23111 ðe31 V 0 Þ2

2

 12 ½ðK 1  M 1 x2 Þ þ 3ðC 1 xÞ2 ð2e311 þ e3111 Þe31 V 0 ; a5 ¼

15

3 E C a1 ðK 1  M 1 x2 Þe3111 ðe31 V 0 Þ2 ; 8 111

ð42Þ

a6 ¼

 2 5 9 2 ðK 1  M1 x2 ÞðC E1111 b1 þ 9C E111 a1 Þ þ e3111 ðe31 V 0 Þ2 16 16

  3 3 1 2 E 2 2 e311 þ C 1111 e3111 e31 V 0 þ e31 V 0 e33111 ;  ðK 1  M 1 x Þ þ ðC 1 xÞ 4 8 16

ð41Þ

ð43Þ

a7 ¼

2 5 E ðC a1 Þ ðK 1  M 1 x2 Þe3111 ðe31 V 0 Þ2 ; 16 111

ð44Þ

a8 ¼

2 3 9 3 E 9 4 ðK 1  M 1 x2 ÞC E1111 b1 e23111  ð C 111 a1 þ C E1111 b1 Þ e3111 e31 V 0  e ; 32 32 2 256 3111

ð45Þ

a10 ¼

2 9 ðC E b Þ e2 256 1111 1 3111

1 3 2 e311 ðe31 V 0 Þ2 ; b2 ¼ ½ðK 2  M2 x2 Þ þ ðC 2 xÞ2 ðe31 V 0 Þ2 þ e3111 ðe31 V 0 Þ3  2 16

ð46Þ ð47Þ

b3 ¼

3 xðK 2  M2 x2 Þe3111 ðe31 V 0 Þ2 ; 64

ð48Þ

b4 ¼


2  i
1 3 E 9 2 1 1h 3 2 C 1111 b2 ðK 2  M2 x2 Þ  e3111 e31 V 0  ðK 2  M2 x2 Þ þ 3ðC 2 xÞ2 e311 þ e3111 e31 V 0 ; 4 16 4 2 4 32

ð49Þ

b6 ¼


2 5 E 9 2 1 C 1111 b2 ðK 2  M 2 x2 Þ þ e3111 e31 V 0 64 128 16

  3 1 5 2  ðK 2  M 2 x2 Þ þ ðC 2 xÞ2 e311 þ C E1111 e3111 e31 V 0 þ e31 V 0 e33111 ; 64 64 32

b8 ¼  b10 ¼

2 3 5 E 1 4 ðK 2  M 2 x2 ÞC E1111 b2 e23111 þ ðC e b Þ e3111 e31 V 0  ; 128 64 1111 2 256 3111

2 3 ðC E b Þ e2 : 512 1111 2 3111

ð50Þ

ð51Þ ð52Þ

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