Finite element method analyses of ambient temperature effects on characteristics of piezoelectric motors

Abstract

There have been numerous experimental reports about the environmental effects on characteristics of intelligent actuators, such as the piezoelectric motors. However, the influences of temperature coefficients of material properties, which are the fundamental reasons for the changes of motors’ output characteristics in different ambient temperature condition, are specially difficult to be acquired through experiments. Thus, the optimization for piezoelectric motors driven in extreme environments is scarce till now. This article is aimed to establish one calculating method to solve this problem. First, a theoretical model is developed for investigating the effects of ambient temperature on characteristics of piezoelectric motors by the finite element method. And then, the mechanical and electrical characteristics of a piezoelectric motor are measured to demonstrate the theoretical model. After that, the changes of critical parameters caused by the ambient temperature are discussed. Based on this, the final proportion of effects of materials’ temperature coefficients on changes of characteristics of piezoelectric motors is obtained. The results obtained by the theoretical model give useful guidelines for the optimization of piezoelectric motors operating in extreme environments.

Keywords

Finite element method analyses ambient temperature characteristic piezoelectric motor

Introduction

As one typical intelligent actuator, the piezoelectric motor (Ueha and Tomikawa, 1993) has many applications in different areas (Bo et al., 2010; Uchino, 1997; Uchino et al., 2004). However, the characteristics of piezoelectric actuators are normally influenced by the environmental factors, especially the ambient temperature and vacuum conditions. Many experimental studies were undertaken to test the characteristics of piezoelectric motor in extreme environments (Morita et al., 2002, 2003; Qu et al., 2009; Six et al., 2004, 2006; Xu et al., 2003). These works give valuable experience for improving the performance of piezoelectric motors operating in extreme environments, but the corresponding theoretical analyses are not yet sufficient.

There have been a number of theoretical models for piezoelectric motors operating under normal environmental conditions. Hagood and McFarland (1995) and Flynn (1995) separately derived the equations of energy conversions in piezoelectric motors. Besides that, the finite element method (FEM) was applied to simulate the dynamic response of the stator (Jeong et al., 1997; Kagawa et al., 1996) and the contact between the stator and rotor (Maeno et al., 1992). But the above-mentioned methods cannot be directly used to analyze the characteristics of piezoelectric motors operating under extreme environmental conditions.

Our previous work deals with the temperature field analyses of piezoelectric motors (Lu et al., 2011), this article is aimed to establish one theoretical method for investigating the ambient temperature–influenced characteristics of piezoelectric motors and the proportion of effects of critical materials’ temperature coefficients on these changes. A theoretical model, which takes into account the temperature influences on material properties, is established, based on the FEM. An iterative mathematical method is introduced to consider the effect of the interface load on the resonant frequency and amplitude, which is first used for calculating output characteristics. An experimental system is set up for verifying the calculated results. Temperature-influenced critical parameters, including the static normal contact force, contact stiffness, resonant frequency, and vibration amplitude, are discussed. Finally, the proportion of effects of materials’ temperature coefficients on changes of characteristics of piezoelectric motors is obtained.

Mathematical model

The typical structure of piezoelectric motors is shown in Figure 1(a) (Ueha and Tomikawa, 1993), which is a traveling wave rotary piezoelectric motor. A ring-shaped stator is fastened on the top of the support. The piezoelectric wafer is adhered to the bottom surface of the stator in which there is a traveling wave in the circumference direction during operation. The rotor is pressed onto the top surface of the stator. The stator harnesses the traveling wave and propels the rotor through the frictional contact area between the stator and rotor. The cover is tightly fixed to the support by four bolts. The dimensions of the motor for calculation are listed in Table 1. The symbols for the dimensions in Table 1 are explained in Figure 1(b) and (c).

Table 1.

Size parameters for PIEZOELECTRIC MOTOR-60 (unit: m).

Structure	Description	Value	Description	Value
Stator	Inner radius of the middle part $r_{s 1}$	1.5 × 10⁻²	Outer radius of the middle part $r_{s 2}$	2.2 × 10⁻²
	Height of the middle part $h_{m}$	7.0 × 10⁻⁴	Outer radius of the outer ring $r_{s 3}$	3.0 × 10⁻²
	Height of the substrate $h_{s}$	2.8 × 10⁻³	Height of the tooth $h_{t}$	1.7 × 10⁻³
Piezoelectric wafer	Thickness $h_{p}$	5.0 × 10⁻⁴	Width $b_{p}$	_{$r_{s 3} - r_{s 2}$}
Bonding layer	Thickness $h_{b}$	7.0 × 10⁻⁶	Width $b_{b}$	_{$r_{s 3} - r_{s 2}$}
Friction layer	Thickness $h_{f}$	3.0 × 10⁻⁴	Width $b_{f}$	_{$r_{c 2} - r_{c 1}$}
Rotor	Inner radius of the outer ring $r_{c 1}$	26.5 × 10⁻³	Outer radius of the outer ring $r_{c 2}$	29.0 × 10⁻³
	Thickness of the thin-walled cylinder $h_{c 1}$	1.1 × 10⁻³	Width of the thin-walled cylinder $b_{c 1}$	0.3 × 10⁻³
	Thickness of the middle part	h_c ₂ = 0.9 × 10⁻³	Width of the middle part $b_{c 2}$	16.5 × 10⁻³
		h_c ₃ = 0.4 × 10⁻³
Cover	Inner radius of the bottom part $r_{v 1}$	31.0 × 10⁻³	Outer radius of the bottom part $r_{v 2}$	33.5 × 10⁻³
	Thickness of the bottom part $h_{v 1}$	14.4 × 10⁻³	Thickness of the top part $h_{v 2}$	5.0 × 10⁻³
Support	Outer radius of the bottom part $r_{u}$	34 × 10⁻³	Thickness of the bottom part $h_{u}$	4.5 × 10⁻³

A theoretical model for solving the piezoelectric motor’s characteristics is built up, in which the ambient temperature effects are taken into account. The piezoelectric motor’s characteristics referred in this article include the relationships of output mechanical torque versus rotor’s speed and electrical admittance versus driven frequency. The computing algorithm is shown in Figure 2(a). The calculation process in the model is given as follows:

First of all, the material property parameters of the piezoelectric motor are calculated based on the given ambient temperature $T$ (see section “Material property parameters”).

Then, the static contact force is analyzed in order to get the static normal force $N_{p}$ on the interface caused by the pre-pressure (the normal force load at the interface between the stator and rotor caused by the assembling) (see section “Static contact force analysis”). Meanwhile, the dynamic analysis of the free stator is also done in order to get the vibration parameters, including the resonant angular frequency $w_{n}$ , vibration amplitude $A_{s}$ , and mode shape function $Φ$ of the free stator (see section “Dynamic analysis of the free stator”). The element models for the static contact force analysis and dynamic analysis are shown in Figure 2(b) and (c), respectively.

After that, the dynamic contact analysis is done with an iterative algorithm using the above results (see Figure 2(a)). In the diagram, the three-dimensional (3D) dynamic force vector acting on each contact element at the (i− 1)th iteration $F_{j_{f}}^{(i - 1)} [= (F_{r}^{(i - 1)}, F_{θ}^{(i - 1)}, F_{z}^{(i - 1)})]$ (where $j_{f}$ is the number of contact element) is calculated under a given rotating speed of rotor $V_{rotor}$ (see section “Dynamic analysis of the stator pressed by the rotor”). Then, the dynamic contact forces are transformed into the calculation of the equivalent additional stiffness $k_{e}'$ and additional damping $c_{e}'$ (see equation (24)). Using the stator’s electromechanical coupling equation modified by $k_{e}'$ and $c_{e}'$ , the resonant frequency $w_{n}'^{(i)}$ and vibration amplitude $A_{s}'^{(i)}$ of the pressed stator are calculated. Both values are used to solve the 3D dynamic contact forces $F_{j_{f}}^{(i)} [= (F_{r}^{(i)}, F_{θ}^{(i)}, F_{z}^{(i)})]$ . Normalized contact force factor $τ$ is used to judge whether an acceptable solution has been obtained. The element model used is shown in Figure 2(d).

Since the exact contact forces and pressed stator’s vibration amplitude and resonant frequency have been obtained, the motor’s speed versus output torque and output power can be calculated using the analytical equations (see section “Mechanical characteristics”). And the motor’s admittance versus driving frequency can be calculated from the equivalent circuit parameters, which can be calculated from the simulation results of ANSYS SOFTWARE (ANSYS Inc., Canonsburg, PA) (see section “Electrical characteristics”).

Both FEM (see Figure 2(b) to (d)) and analytical calculation method (see equations (21) and (31)) are utilized, in which ANSYS SOFTWARE is used for solving the element models, and the resulting data are transferred into the MATLAB program for further solutions. The details of the above listed calculation processes are as follows.

Material property parameters

The piezoelectric motor used for our calculation is made of lead zirconate titanate (PZT)-8H (by HAIYING Enterprise Group Co., Ltd, Wuxi, China), phosphor bronze, polytetrafluoroethylene (PTFE), aluminum, and adhesive (by Shanghai Research Institute of Synthetic Resins, Shanghai, China), among which PZT-8H and PTFE are used as piezoelectric and frictional materials, respectively, and phosphor bronze and aluminum as the stator and rotor materials, respectively. For these materials, the elastic modulus, dielectric constant, piezoelectric constant, and damping ratio of stator are substantially affected by ambient temperature, and other property parameters are little affected by ambient temperature (Ando and Kagawa, 1992). Assuming that temperature dependence of the elastic modulus, dielectric constant, piezoelectric constant, and damping ratio of stator is linear (Ando and Kagawa, 1992), we have

{\begin{matrix} E = E_{0} [1 + (T - T_{0}) α] \\ ε = ε_{0} [1 + (T - T_{0}) β] \\ e = e_{0} [1 + (T - T_{0}) χ] \\ η = η_{0} [1 + (T - T_{0}) λ] \end{matrix}

(1)

where $T$ is the ambient temperature, and $T_{0} = 293 K$ is the room temperature. $E$ , $ε$ , $e$ , and $η$ are the elastic modulus, dielectric constant, piezoelectric constant, and stator’s damping ratio at ambient temperature $T$ , respectively. $E_{0}$ , $ε_{0}$ , $e_{0}$ , and $η_{0}$ are the elastic modulus, dielectric constant, piezoelectric constant, and damping ratio of free stator at the room temperature $T_{0}$ , respectively. Coefficients $α$ , $β$ , $χ$ , and $λ$ are the temperature coefficients of the elastic modulus, dielectric constant, piezoelectric constant, and stator’s damping ratio, respectively. In addition, thermal expansion coefficients are also used in this analysis, and thermal stress can be expressed as (Ando and Kagawa, 1992)

{\begin{matrix} ε_{t} = {\begin{matrix} γ (T - T_{0}) & γ (T - T_{0}) & γ (T - T_{0}) 0 0 0 \end{matrix}}^{T} \\ σ_{t} = D ε_{t} \end{matrix}

(2)

where $ε_{t}$ is the thermal strain vector, $σ_{t}$ is the thermal stress vector, $γ$ is the linear thermal expansion coefficient, and $D$ is the stress–strain matrix, which can be acquired by generalized Hooke’s law (Nowacki, 1962). The material property parameters, which need to be used in our calculation, are listed in Table 2 (Ando and Kagawa, 1992; Engeln et al., 1984).

Table 2.

Material constants (T₀ = 293 K).

Material	PZT-8H	Phosphor bronze	PTFE	Aluminum	Adhesive
Density $ρ$ (kg/m³)	7650	8800	2100	2770	1100
Elastic modulus $E_{0}$ (×10⁹ N/m²)	See below	117	0.3	71.7	2.9
Temperature coefficient $α$ of $E_{0}$ (/K)	−2.0 × 10⁻⁴	−4.0 × 10⁻⁴	−8.74 × 10⁻³	−5.1 × 10⁻⁴	−9.65 × 10⁻³
Poisson’s ratio	0.31	0.3	0.36	0.33	0.35
Thermal expansioncoefficient (m/(m K))	1.2 × 10⁻⁵	1.72 × 10⁻⁵	1.78 × 10⁻⁴	2.36 × 10⁻⁵	7.93 × 10⁻⁵
Elastic stiffness matrix(×10⁹ N/m²)	$(\begin{matrix} 12.06 & 5.35 & 5.15 & 0 & 0 & 0 \\ 12.06 & 5.15 & 0 & 0 & 0 \\ 10.45 & 0 & 0 & 0 \\ 3.13 & 0 & 0 \\ 3.13 & 0 \\ 3.46 \end{matrix})$
Dielectric constant $ε_{0}$ (×10⁻⁹ F/m)	$(\begin{matrix} 7.124 & 0 & 0 \\ 0 & 7.124 & 0 \\ 0 & 0 & 5.841 \end{matrix})$
Temperature coefficient $β$ of $ε_{0}$ (/K)	3.2 × 10⁻³
Piezoelectric constant $e_{0}$ (C/m²)	$(\begin{matrix} 0 & 0 & - 5.2 \\ 0 & 0 & - 5.2 \\ 0 & 0 & 15.1 \\ 0 & 12.7 & 0 \\ 12.7 & 0 & 0 \\ 0 & 0 & 0 \end{matrix})$
Temperature coefficient $χ$ of $e_{0}$ (/K)	1.0 × 10⁻⁴

PTFE: polytetrafluoroethylene; PZT: lead zirconate titanate.

Static contact force analysis

Static normal contact force $N_{p}$ on the interface between the stator and rotor, as shown in Figure 2(b), is a critical input parameter for solving the dynamic contact force in the next step. The calculation in this step aims at calculating $N_{p}$ versus ambient temperature from the pre-pressure of the rotor. Due to the pre-pressure, the cover’s fixed surface has a deformed downward displacement $d_{c}$ relative to its initial position without pre-pressure (see Figure 2(b)). Suitable parameter $d_{c}$ is determined by a given pre-pressure. This part consists of calculating the influence of ambient temperature on the $N_{p}$ when the cover is fixed at $d_{c}$ . The calculated result is put into dynamic contact analysis program in the following step.

In ANSYS, static force analysis function is chosen as the calculation method. Because of the symmetrical configuration, 3D model of one quarter of the whole motor is used in ANSYS, as shown in Figure 2(b). In this calculation, piezoelectric wafer is modeled with the 3D element SOLID 5. Other parts are modeled with SOLID 45 (ANSYS Inc., 2007). A cylindrical coordinate system is used, and the contact pairs are created, as shown in Figure 2(b) on the interface between the rotor and stator, in which the rotor surface is identified as the flexible surface modeled with the CONTAC174 element, and the stator surface is identified as the rigid surface modeled with the TARGE 170 element (ANSYS Inc., 2007). The friction coefficient between the rotor and stator is 0.2. Other connective surfaces between two different parts are bonded together by the glue operation. The element model shown in Figure 2(b) has totally 7056 elements, including 6624 elements of SOLID 45, 144 elements of SOLID 5, 72 elements of TARGE 170, and 216 elements of CONTAC174. In the calculation, the support’s bottom surface is fixed. The normal force in the z direction of each contact element is recorded in the element table of the software, and the total force in the normal direction on the interface $N_{p}$ can be acquired by the summation of them.

Dynamic analysis of the free stator

In this section, by solving the vibration characteristics of the free stator shown in Figure 2(c), we want to obtain the eigenvalues of the stator’s ninth flexural mode at various ambient temperatures. A combination of the analytical method and the FEM is used here. The vibration amplitude and resonant frequency of the free stator can be calculated by (Flynn, 1995; Hagood and McFarland, 1995)

{\begin{matrix} A_{s} = | \frac{θ_{e} V_{max}}{- w_{n}^{2} m_{e} + j w_{n} c_{e} + k_{e}} | \\ w_{n} = \sqrt{\frac{k_{e}}{m_{e}}} \end{matrix}

(3)

where $V_{max}$ is the maximum voltage (if not mentioned, $V_{max} = 200 V_{0 - p}$ is used for calculation and experiments); $w_{n}$ is the angular frequency, which can also be expressed by $w_{n} = 2 π f_{n}$ ; and $A_{s}$ is the vibration amplitude. Parameters $m_{e}$ is the modal mass, $c_{e}$ is the modal damping factor in forced condition, $k_{e}$ is the modal stiffness, and $θ_{e}$ is the electromechanical force factor. The modal mass $m_{e}$ , modal damping factor $c_{e}$ , modal stiffness $k_{e}$ , and electromechanical force $θ_{e}$ can be calculated with the results of ANSYS (Chen and Zhao, 2008; Ghouti, 2000) (see equations (10) and (11)). In this calculation, modal analysis function of ANSYS is used. Inner side surface of the stator is limited for all degrees of freedoms because the stator is fixed to the support. Two-phase electrical signals with phase difference of π/2 in both time and space are applied to piezoelectric elements, according to the working principles of piezoelectric motor (Ueha and Tomikawa, 1993). Based on our experimental results, the damping ratio $η_{0}$ of free stator at temperature $T_{0}$ is 0.002, and the temperature coefficient $λ$ of damping ratio is

(\begin{matrix} 8.0 \times 1 0^{- 3} & 0 & 0 \\ 0 & 8.0 \times 1 0^{- 3} & 0 \\ 0 & 0 & 8.0 \times 1 0^{- 3} \end{matrix}) / K

Table 3 lists the derived results of eigen parameters ( $m_{e}$ , $c_{e}$ , and $k_{e}$ ) and other parameters ( $θ_{e}$ and $c_{d}$ ) of dynamic calculation of free stator at T₀ (see equations (10) and (11)).

Table 3.

Critical parameters for free stator’s vibration (T ₀ = 293 K).

Description	Parameter	Value	Unit
Modal mass	_{$m_{e}$}	1.24 × 10⁻²	kg
Modal damping	_{$c_{e}$}	12.08	N s/m
Modal stiffness	_{$k_{e}$}	7.39 × 10⁸	N/m
Electromechanical force factor	_{$θ_{e}$}	0.2731	C/m
Piezoelectric capacitance	$c_{d}$	5.71	nF

The vibration mode shapes $Φ_{A}$ and $Φ_{B}$ are considered to be unchangeable with the ambient temperature, which are expressed as follows (Chen and Zhao, 2008) (see equation (13))

Φ_{A} = {\begin{matrix} Φ_{rA} (r_{c}, z_{c}, θ) \\ Φ_{θ A} (r_{c}, z_{c}, θ) \\ Φ_{zA} (r_{c}, z_{c}, θ) \end{matrix}} = {\begin{matrix} - Φ_{rf} \cdot \cos (n_{r} θ) \\ Φ_{θ f} \cdot \sin (n_{r} θ) \\ Φ_{zf} \cdot \cos (n_{r} θ) \end{matrix}}

(4a)

Φ_{B} = {\begin{matrix} Φ_{rB} (r_{c}, z_{c}, θ) \\ Φ_{θ B} (r_{c}, z_{c}, θ) \\ Φ_{zB} (r_{c}, z_{c}, θ) \end{matrix}} = {\begin{matrix} - Φ_{rf} \cdot \sin (n_{r} θ) \\ - Φ_{θ f} \cdot \cos (n_{r} θ) \\ Φ_{zf} \cdot \sin (n_{r} θ) \end{matrix}}

(4b)

The solved results about the variables in mode shape equation (4) are $Φ_{rf} = 0.2752$ , $Φ_{θ f} = 0.7566$ , and $Φ_{zf} = 0.8218$ . It can be seen that $Φ_{zf}$ is larger than other two amplitudes of shift factors, and all of them are on the same magnitude, which means primary vibration movement happens in normal and tangential directions, while the movement in radial direction should also be taken into account for the power loss consideration.

Dynamic analysis of the stator pressed by the rotor

This section is the core part in the computing program. Using the previous solutions, 3D dynamic contact forces caused by the stator’s vibration could be calculated here. The results of contact forces can be used to calculate the modified variables for vibration characteristics of pressed stator, with which the output characteristics of piezoelectric motor in different temperature conditions could be calculated more accurately.

The modal contact force is expressed in the following equation with mode shapes $Φ_{A}$ and $Φ_{B}$

F = \sum_{j_{f} = 1}^{n_{f}} [\begin{matrix} Φ_{A}^{T} \\ Φ_{B}^{T} \end{matrix}] F_{j_{f}}

(5)

where n_f = 216 is the amount of CONTAC52 element and $F_{j_{f}}$ is the vector of 3D dynamic contact forces.

Thus, the modified frequency $w'_{n}$ and amplitude $A'_{s}$ for the pressed stator can be expressed as (see equation (26)) (Ghouti, 2000; Pirrotta et al., 2005)

{\begin{matrix} {A'}_{s} = | \frac{θ_{i} V_{max}}{- w_{n}^{2} m_{e} + j w_{n} (c_{e} + {c'}_{e}) + (k_{e} + {k'}_{e})} | \\ {w'}_{n} = \sqrt{\frac{(k_{e} + {k'}_{e})}{m_{e}}} \end{matrix}

(6)

where $k'_{e}$ is the additional stiffness and $c'_{e}$ is the additional damping factor (Ghouti, 2000). Both of them are relevant to the force vector $F_{j_{f}}$ (see equation (24)).

The static force analysis method with boundary conditions at two different times $t_{1} = 0$ and $t_{2} = π / (2 w_{n})$ is used as the calculation method because of the temporal and spatial correspondences of traveling wave in piezoelectric motors, and the results of which can be used to express other dynamic contact forces at any time conditions (see equation (18)). The final element model is shown in Figure 2(d). The 3D point-to-point contact element CONTAC52 (ANSYS Inc., 2007) is applied in order to simulate the maintenance or break of the physical contact conditions (Appendices 2 and 3). The vibration mode functions $Φ$ and the normal force $N_{p}$ are used to describe boundary conditions (see equations (15) and (16)).

It is observed that the contact force depends on the vibration amplitude and frequency, both of which are also determined by the contact force (Ghouti, 2000; Pirrotta et al., 2005). This is a typical coupling problem and needs to be solved by the iterative method. The normalized modal contact force factor $τ = \sum_{j_{f} = 1}^{n_{f}} Φ_{B}^{T} {F_{s}}$ (here ${F_{s}}$ is one component part of $F_{j_{f}}$ , see equation (18)) is taken as the decoupled parameter, and the iterative algorithm shown in Figure 2(a) is programmed as follows.

In the first stage, static normal force $N_{p}$ , free stator’s vibration amplitude $A_{s}$ , and resonant frequency $w_{n}$ in equation (3) are used for setting the boundary condition (see equations (15) and (16)) for solving the first group of normalized factor $τ^{(0)}$ under the given $V_{rotor}$ . And the first modified frequency ${(w'_{n})}^{(1)}$ and amplitude ${(A'_{s})}^{(1)}$ can be computed by equation (6).

In the second stage, ${(A'_{s})}^{(1)}$ and ${(w'_{n})}^{(1)}$ are used to replace free stator’s $A_{s}$ and $w_{n}$ for solving the second group of normalized factor $τ^{(1)}$ . It is obvious that the vibration amplitude ${(A'_{s})}^{(1)}$ is smaller than $A_{s}$ , and ${(w'_{n})}^{(1)}$ is larger than $w_{n}$ because of the pressure effect. Therefore, the summation of the second normalized factor $τ^{(1)}$ is smaller compared with $τ^{(0)}$ , which can lead to larger ${(A'_{s})}^{(2)}$ and smaller ${(w'_{n})}^{(2)}$ compared with the results in the last stage. This is an infinite loop process for ${(w'_{n})}^{(i)}$ and amplitude ${(A'_{s})}^{(i)}$ (here $i$ is the number of looped calculation), both values of which will fluctuate at certain points finally.

In the third stage, if we define $| τ^{(i)} - τ^{(i - 1)} | \leq 0.01$ as the convergent condition, the loop will terminate after several computations, and the final value of ${(w'_{n})}^{(i)}$ and ${(A'_{s})}^{(i)}$ can be obtained. They will be used for calculating the output characteristics in next steps.

Mechanical characteristics

Mechanical characteristic of piezoelectric motor mainly refers to the speed versus output torque. It can be calculated by (Hagood and McFarland, 1995) (see equation (27))

T_{q} = \sum_{j_{f} = 1}^{n_{f}} F_{2 θ} (j_{f}) r_{c}

(7)

where $F_{2 θ} (j_{f})$ is the contact force in θ direction of No. $j_{f}$ contact element, $r_{c}$ is the equivalent contact radius, and $T_{q}$ is the output torque. Through this calculation, we can get $T_{q}$ versus $V_{rotor}$ in different ambient temperatures.

Electrical characteristics

The admittance of pressed stator can be calculated by (see equation (31)) (Ueha and Tomikawa, 1993)

Y_{m 1} = \frac{1}{1 / (j {w'}_{n} C_{1}) + j {w'}_{n} L_{1} + R_{1}} + j w'_{n} C_{d}

(8)

where $L_{1}$ , $C_{1}$ , and $R_{1}$ are the equivalent inductance, equivalent capacity, and equivalent resistance of pressed stator, respectively. Through this calculation, we can get $Y_{m 1}$ versus driving frequency in different ambient temperatures.

Experiments

Experimental setup

The experimental setup is shown in Figure 3. One piezoelectric motor is put into the temperature control system, which has the temperature range from 203 to 423 K and 0.5 K accuracy. The motor controller is used to control the tested piezoelectric motor working at resonant point and suitable driving voltage $V_{max} (= 200 V_{0 - p})$ . The electrical characteristic of the piezoelectric motor is measured by the Agilent 4294A precision impedance analyzer. The mechanical characteristic of the piezoelectric motor is measured by the mechanical test system, which is composed of the torque sensor, the magnetic brake, and the computer. The torque sensor measures the output torque and rotating speed of the motor’s shaft, and the magnetic brake is used to apply load on the shaft. The computer is used to give the load signal to the magnetic brake and collect the torque and speed signals from the sensor.

Results

The effects of ambient temperature on mechanical and electrical characteristics of piezoelectric motors are plotted out in Figures 4 and 5, respectively. As shown in Figure 4(a), the stalling torque decreases gradually, and the tendency is more significant in high ambient temperature. The calculated maximum stalling torque (= 0.85 N m) happens at T = 273 K. However, the influence of ambient temperature on no-load speed is complex. The no-load speed value increases as T varying from 273 to 333 K, reaches the peak point (= 210 r/min) at 343 K, and then decreases sharply in much hotter condition. The experimental data in Figure 4(b) basically agree with the theoretical results.

As shown in Figure 5(a), the resonant frequency is 41,675 Hz and peak admittance is 1.83 × 10⁻³Ω⁻¹ at room temperature (T₀ = 293 K). As temperature increases, there is a notable rising of the peak admittance, the value of which will enlarge about two times when ambient temperature varies between 273 and 373 K. Experimental results in Figure 5(b) also verify the analyzed tendency in Figure 5(a). The larger admittance value leads to larger input current under the same driven voltage when piezoelectric motor works.

The theoretical model reflects the ambient temperature effects on the mechanical and electrical characteristics of piezoelectric motors, and the calculated results are confirmed by the experiments. The following parts are show the utilization of this model for analyzing the temperature-influenced critical parameters in motor’s characteristics calculation and discussing the proportion of effects of materials’ temperature coefficients on changes of characteristics of piezoelectric motors.

Analyses and discussion

Analyses of critical parameters

Figure 6 shows the ambient temperature T versus the static normal contact force $N_{p}$ , in which $d_{c}$ is kept at 0.6 × 10⁻³ m, and the calculated ambient temperature ranges from 273 to 373 K. The theoretical curve shows that $N_{p}$ decreases with the increase of ambient temperature T, which indicates that the influences of thermal expansion and temperature on the elastic modulus mainly affect the static normal contact force between the stator and rotor. The decrease of static normal contact force negatively affects the mechanical characteristic of piezoelectric motors.

The iterative algorithm is carried out based on the solved temperature–determined $N_{p}$ and vibration parameters of free stator. Taking T = 293 K, for example, the convergent 3D contact force distribution in one wavelength is shown in Figure 7, and the contact modal forces in equation (5) are listed in Table 4. It can be seen that the vibration condition on the calculated time $t_{2} = π / (2 w_{n})$ for traveling wave movement is mainly attributed to the vibration mode shape $Φ_{B}$ , which corresponds to the wave movement excited by the phase B signal. The calculated additional stiffness and damping factors are $c_{e}' = 90.48 N s / m$ and $k_{e}' = 1.14 \times 10^{8} N / m$ . Using equation (6), the modified frequency $w_{n}'$ and amplitude $A'_{s}$ for the pressed stator can be solved. With the change of ambient temperature condition, vibration frequency and vibration amplitude of pressed stator are plotted in Figure 8.

Table 4.

Results of contact modal forces (T₀ = 293 K).

Description	Modal force_r	Modal force_θ	Modal force_z
t ₁ = 0	0.2314	2.1794	0.7188
t ₂ = π/(2w_n)	0.0143	0.1353	11.61

From Figure 8, the resonant frequency decreases as the ambient temperature increases. This may be due to the following reasons: the decrease of stator’s modal stiffness and the decrease of additional stiffness. It is also observed that the vibration amplitude increases as the ambient temperature increases. Besides the effects of variance of stiffness, this may also be caused by the decrease of additional damping factor and the increase of electromechanical force factor. The variances of resonant frequency and vibration amplitude indicate that temperature influences on the elastic modulus, dielectric constant, and piezoelectric constant mainly influence the stator’s vibration characteristics. The increase of vibration amplitude positively affects the mechanical characteristic of piezoelectric motors.

Discussion of temperature coefficient

The calculated result of stalling torque or peak admittance at T = 373 K considering all the temperature constants is defined as $X_{A}$ . Similarly, $X_{d}$ is defined for the calculated result of stalling torque or peak admittance at T = 373 K considering without the temperature constant d, which represents the temperature coefficient or thermal expansion coefficient in Table 2. Thus, the impact value of each temperature constant d on the motor’s stalling torque or peak admittance can be expressed by $| X_{d} - X_{A} |$ , and summation of them denotes the total temperature impact values. For judging the proportion of each material’s temperature coefficient effect on changes of characteristics of piezoelectric motors, an impact index is defined as

I_{m} = \frac{| X_{d} - X_{A} |}{\sum | X_{d} - X_{A} |} \times 100 %

(9)

The result of impact indexes on the stalling torque is shown in Figure 9(a). It is observed that temperature coefficient $α$ of $E_{0}$ of PTFE has the largest impact index, which means the decrease of friction material’s elastic modulus in high ambient temperature is the main reason for the decrease of output mechanical characteristics. Besides that, temperature coefficient $χ$ of $e_{0}$ of PZT-8H and temperature coefficient $α$ of $E_{0}$ of phosphor bronze and adhesive have almost the same impact indexes on the stalling torque, which are much smaller than the largest value. Other temperature constants have little impact index, which can be ignored.

The results of impact indexes on the peak admittance are shown in Figure 9(b). It is obvious that the largest impact index also belongs to the temperature coefficient $α$ of $E_{0}$ of PTFE, which means that in high ambient temperature, the increase of motor’s input current is mainly caused by the decrease of friction material’s elastic modulus. Temperature coefficient $β$ of $ε_{0}$ has the second large value of impact index on the peak admittance, and temperature coefficient $α$ of $E_{0}$ of phosphor bronze and aluminum and temperature coefficient $χ$ of $e_{0}$ of PZT-8H have the little smaller impact indexes. Additionally, temperature coefficient $α$ of $E_{0}$ of PZT-8H and adhesive also takes a few parts. Other temperature constants have little impact index, which can be ignored.

It is observed from Figure 9(a) and (b) that thermal expansion coefficient of each material in Table 2 has little impact index of both stalling torque and peak admittance, which indicates that the material’s thermal expansion coefficient can be ignored when analyzing the effects of ambient temperature on characteristics of piezoelectric motors. It is also observed that PTFE’s temperature coefficient of elastic modulus has the largest impact index of both stalling torque and peak admittance, which indicates that effects of ambient temperature on characteristics of piezoelectric motors are determined by the friction material.

Conclusion

In this article, the effects of ambient temperature on characteristics of piezoelectric motors were investigated. A theoretical model was proposed based on the FEM. The dynamic contact forces were transformed into additional stiffness and damping factors, with which an iterative mathematical method was introduced to calculate the resonant frequency and amplitude of pressed stator. The mechanical and electrical characteristics were calculated out and confirmed by the experimental results. The proportion of effects of materials’ temperature coefficients on changes of characteristics of piezoelectric motors is finally obtained. The meaningful results are summarized in the following.

As the ambient temperature increases, the stalling torque and the no-load speed decrease gradually. It implies that in high ambient temperature, the output mechanical characteristic of the piezoelectric motor is not so good as that in normal condition.

The peak admittance increases as the ambient temperature increases. The larger peak admittance value will lead to larger input current under the same driven voltage in operation.

The static normal contact force decreases as the ambient temperature increases, which negatively affects the mechanical characteristic of piezoelectric motors at high ambient temperature.

The vibration amplitude increases as the ambient temperature increases, which means high ambient temperature enhances the stator’s vibration amplitude. The increase of vibration amplitude positively affects the mechanical characteristic of piezoelectric motors.

Material’s thermal expansion coefficient can be ignored when analyzing the effects of ambient temperature on characteristics of piezoelectric motors, and the temperature coefficient of the friction material’s elastic modulus determines the effects of ambient temperature on characteristics of piezoelectric motors.

Footnotes

Appendix 1

Appendix 2

This part mainly refers to techniques in ANSYS and the solution of eigen parameters of piezoelectric motor’s stator. In ANSYS modal analysis, if the mode shapes are normalized to unity, the maximum kinetic energy named $K E_{n}$ and the maximum elastic potential energy named $S E_{n}$ of the No. n element can be obtained by commands of element table (ANSYS, 2007; Moaveni, 1999). The parameters $m_{e}$ , $c_{e}$ , and $k_{e}$ can be calculated from the equation

(10)

{\begin{matrix} \begin{matrix} \sum_{n = 1}^{N} K E_{n} = \frac{k_{e}}{2} \\ \sum_{n = 1}^{N} S E_{n} = \frac{m_{e} w_{n}^{2}}{2} \end{matrix} \\ c_{e} = 2 η w_{n} m_{e} \end{matrix}

where $N$ is the total amount of the solid elements, as shown in Figure 2(c).

The piezoelectric factor $θ_{e}$ is related to the anti-resonant frequency. In the electric model, the anti-resonant mode happens when the boundary condition of the electrical polar is open, namely, $Q = 0$ . Therefore, $θ_{e}$ can be derived from the stator’s electromechanical coupled equation (Chen and Zhao, 2008; Ghouti, 2000; Pirrotta et al., 2005) and expressed as

(11)

θ_{e} = \sqrt{(m_{e} \cdot 2 π f_{a}^{2} - k_{e}) \cdot c_{d}}

where $f_{a}$ is the anti-resonant frequency and $c_{d}$ is the piezoelectric capacity.

The anti-resonant frequency $f_{a}$ can be obtained in harmonic analysis. A finite bandwidth near the resonant frequency and the amount of sub-step are determined. After solution, the absolute electric quantity value of each element in group A is summarized together in order to get the total electric quantity at each sub-step of frequency point. We compare the total electric quantity from the first sub-step to the end. When the smallest total quantity value arrives, the relative frequency is just the anti-resonant frequency $f_{a}$ of the free stator (Ueha and Tomikawa, 1993).

Piezoelectric capacity $c_{d}$ reflects the static dielectric characteristic of the piezoelectric wafer. Considering from energy transmission, the power energy in piezoelectric ceramics is transferred from the charge work (Jaffe et al., 1971), that is, $c_{d} V^{2} / 2 = QV / 2$ . A static electrical analysis is done in ANSYS with the unit voltage load applied on Group A or B. The ± sign needs to be noted due to the reversed polarization. The electric quantity of No. ne element of group can be obtained in POST1 postprocessor, and $c_{d}$ can be calculated by equation

(12)

c_{d} = \sum_{ne = 1}^{Ne} | Q_{ne} |

where $Ne$ is the amount of the group elements.

Using the above equation, the piezoelectric factor $θ_{e}$ in equation (11) can be calculated.

The vibration mode functions are useful for applying boundary conditions in next contact analysis. In this article, vibration mode shape is considered to be unchangeable with ambient temperature. Piezoelectric elements, Groups A and B, are designed for exciting $Φ_{A}$ and $Φ_{B}$ , respectively. A series of nodes on top surface of stator are picked up with the same locations in the r directions (see Figure 2(c)). In ANSYS modal analysis, if the mode shapes are normalized to unity, the shift factors with $Φ_{rf}$ , $Φ_{θ f}$ , and $Φ_{zf}$ representing the amplitude in r, z and θ directions can be obtained. The mode shape functions $Φ_{A}$ and $Φ_{B}$ are expressed as follows (Chen and Zhao, 2008)

(13a)

Φ_{A} = {\begin{matrix} Φ_{rA} (r_{c}, z_{c}, θ) \\ Φ_{θ A} (r_{c}, z_{c}, θ) \\ Φ_{zA} (r_{c}, z_{c}, θ) \end{matrix}} = {\begin{matrix} - Φ_{rf} \cdot \cos (n_{r} θ) \\ Φ_{θ f} \cdot \sin (n_{r} θ) \\ Φ_{zf} \cdot \cos (n_{r} θ) \end{matrix}}

(13b)

Φ_{B} = {\begin{matrix} Φ_{rB} (r_{c}, z_{c}, θ) \\ Φ_{θ B} (r_{c}, z_{c}, θ) \\ Φ_{zB} (r_{c}, z_{c}, θ) \end{matrix}} = {\begin{matrix} - Φ_{rf} \cdot \sin (n_{r} θ) \\ - Φ_{θ f} \cdot \cos (n_{r} θ) \\ Φ_{zf} \sin (n_{r} θ) \end{matrix}}

where $n_{r}$ is the amount of nodal diameters of vibration in the direction perpendicular to stator’s top surface. Here, $n_{r} = 9$ is used corresponding to the stator’s ninth flexural vibration mode.

Appendix 3

This part mainly refers to techniques about dynamic contact analysis in ANSYS and derivation of pressure influenced vibration parameters and of output characteristics of piezoelectric motor.

As shown in Figure 2(d), one group of nodes on the surface of stator with the same radius r_c = 27.75 × 10⁻³ m ( $r_{c}$ is assumed to be the equivalent contact radius) is chosen out, and CONTAC52 elements are built upon them. The upper layer (UPL) nodes of CONTAC52 represent the contact surface of rotor, while the under layer (UNL) nodes represent the contact surface of stator. The value of contact stiffness is determined as a real constant for CONTAC52 and expressed as equation (14), and normal stiffness $k_{fn}$ is assumed to be equal to the tangential stiffness $k_{ft}$ . The element model is shown in Figure 2(d). There are about 216 CONTAC52 elements. The friction coefficient is also set to 0.2 as before

(14)

k_{fn} = \frac{E_{f} b_{f} (2 π r_{c})}{h_{f} n_{f}}

where $E_{f}$ is the elastic modulus of friction material, $b_{f}$ is the width of the friction layer, $h_{f}$ is the height of the friction layer, and $n_{f}$ is the amount of CONTAC52 elements in Figure 2(d).

We use the calculation method referred before to solve the vibration mode functions. These three functions are multiplied by the amplitude $A_{s}$ and then treated as the displacement boundary conditions. The displacement boundary function in z direction is applied for UNL nodes directly, while the pre-pressure is treated as the normal loads for UPL nodes. The displacement of CONTAC52 in tangential, and radial direction is determined by the velocity contrast between UNL and UPL nodes (ANSYS, 2007).

The velocity expression of any point in free stator can be referred to in Hagood and McFarland (1995) and Pirrotta et al. (2005). Both the rotational speed of rotor and the velocity of stator in r and θ directions should be transferred in order to keep the identical form with the boundary condition in z direction. The concrete boundary conditions are as follows

For UNL nodes

(15)

{\begin{cases} \begin{array}{l} U_{r} = \frac{v_{s_r} (θ, t)}{w_{n}} \\ U_{θ} = \frac{v_{s_θ} (θ, t)}{w_{n}} \end{array} \\ U_{z} = u_{s_z} (θ, t) \end{cases}

For UPL nodes

(16)

{\begin{matrix} \begin{matrix} U_{r} = \frac{v_{R_r} (θ, t)}{w_{n}} = 0 \\ U_{θ} = \frac{v_{R_θ} (θ, t)}{w_{n}} = - [V_{rotor} 2 π r_{c} / 60] / w_{n} \end{matrix} \\ F_{z} = F_{R_z} (θ, t) = - N_{p} / n_{f} \end{matrix}

where $U_{p}$ (p = r, θ, or z) is the displacement boundary condition, $F_{p}$ (p = z) is the load force boundary condition, and $V_{rotor}$ is the rotating speed of rotor.

Mode shapes $Φ_{A}$ and $Φ_{B}$ will be applied to express the modal contact force as

(17)

F = \sum_{j_{f} = 1}^{n_{f}} [\begin{matrix} Φ_{A}^{T} \\ Φ_{B}^{T} \end{matrix}] F_{j_{f}}

where n_f = 216 is the amount of CONTAC52 element.

Time is relevant to the contact analysis because of the traveling wave, two typical time points ( $t_{1} = 0; t_{2} = π / (2 w_{n})$ ) are set. When the solution is done, the basic information saved in element table contains results of $F_{nr} (j_{f})$ , $F_{n θ} (j_{f})$ , and $F_{nz} (j_{f})$ . Due to the orthogonality in time phase of two exciting signals, we define the contact force $F_{j_{f}}$ with $t$ variable as follows

(18)

{\begin{matrix} F_{j_{f}} = {F_{c}} \cos (w_{n} t) + {F_{s}} \sin (w_{n} t) \\ {F_{c}} = {[F_{2 r} (j_{f}), F_{2 θ} (j_{f}), F_{1 z} (j_{f})]}^{T} \\ {F_{s}} = {[F_{1 r} (j_{f}), F_{1 θ} (j_{f}), F_{2 z} (j_{f})]}^{T} \end{matrix}

where $F_{lr}$ , $F_{l θ}$ , and $F_{lz}$ (l = 1, 2) are the contact force in r, z and θ directions, respectively, and $j_{f}$ is the number of contact element. The contact force elements on l = 1 condition can be acquired when $t_{1} = 0$ , and those on l = 2 condition can be acquired when $t_{2} = π / (2 w_{n})$ .

Many experiments show that the resonant frequency and amplitude of stator will be changed when the pre-pressure value is not 0. In order to get the exact analysis model, the effect of pre-pressure on the dynamic characteristic of stator should be taken into account. According to Pirrotta et al. (2005), the electromechanical coupled equation of pressed stator should be written as

(19)

{\begin{matrix} M \ddot{P} + C \overset{\cdot}{P} + KP = Θ V - F \\ Θ^{T} P + C_{d} V = Q \end{matrix}

in which

M = [\begin{matrix} m_{e} & 0 \\ 0 & m_{e} \end{matrix}]; C = [\begin{matrix} c_{e} & 0 \\ 0 & c_{e} \end{matrix}]; K = [\begin{matrix} k_{e} & 0 \\ 0 & k_{e} \end{matrix}]; Θ = [\begin{matrix} θ_{e} & 0 \\ 0 & θ_{e} \end{matrix}]; C_{d} = [\begin{matrix} c_{d} & 0 \\ 0 & c_{d} \end{matrix}]

where $m_{e}$ is the modal mass, $c_{e}$ is the modal damping factor in the forced condition, $k_{e}$ is the modal stiffness, $θ_{e}$ is the electromechanical force factor, and $c_{d}$ is the piezoelectric capacitance. They are eigen parameters of stator at B_(0,9) mode and influenced by the ambient temperature. Two vectors, $V$ and $P$ , are defined as the applied voltage and modal coordinate, respectively

(20)

{\begin{array}{l} V = {[V_{A}, V_{B}]}^{T} = {[V_{\max} \cos w_{n} t, V_{\max} \sin w_{n} t]}^{T} \\ P = {[p_{A}, p_{B}]}^{T} = {[A_{s} \cos w_{n} t, A_{s} \sin w_{n} t]}^{T} \end{array}

where $V_{max}$ is the maximum voltage value; $w_{n}$ is the angular frequency, which can be expressed by $w_{n} = 2 π f_{n}$ ( $f_{n}$ is the modal frequency); $t$ is the time variable; and $A_{s}$ is the vibration amplitude.

Taking the exciting voltage $V_{B}$ , for example, the engine equation can be acquired by substituting equations (18) and (20) in equation (19) and reads as (Ghouti, 2000; Pirrotta et al., 2005)

(21)

m_{e} {\ddot{p}}_{B} + c_{e} {\overset{\cdot}{p}}_{B} + k_{e} p_{B} = θ_{e} V_{B} - \sum_{j_{f}}^{n_{f}} Φ_{B}^{T} {F_{c}} \cos (w_{n} t) - \sum_{j_{f}}^{n_{f}} Φ_{B}^{T} {F_{s}} \sin (w_{n} t)

where $\cos (w_{n} t)$ can be derived from the following equation

(22)

{\overset{\cdot}{p}}_{B} = A_{s} w_{n} \cos (w_{n} t)

Thereafter, by rearranging the terms, another type of engine equation can be written as

(23)

\begin{matrix} m_{e} {\ddot{p}}_{B} + [c_{e} + \sum_{j_{f}}^{n_{f}} \frac{Φ_{B}^{T} {F_{c}}}{(A_{s} w_{n})}] {\overset{\cdot}{p}}_{B} \\ + [k_{e} + \sum_{j_{f}}^{n_{f}} \frac{Φ_{B}^{T} {F_{s}}}{A_{s}}] p_{B} = θ_{e} V_{B} \end{matrix}

The additional stiffness and damping factors are defined as

(24)

{\begin{matrix} {c'}_{e} = \sum_{j_{f}}^{n_{f}} \frac{Φ_{B}^{T} {F_{c}}}{(A_{s} w_{n})} \\ {k'}_{e} = \sum_{j_{f}}^{n_{f}} \frac{Φ_{B}^{T} {F_{s}}}{A_{s}} \end{matrix}

Parameters $A_{s}$ and $w_{n}$ of the free stator can also be expressed as (Flynn, 1995; Hagood and McFarland, 1995)

(25)

{\begin{matrix} A_{s} = | \frac{θ_{i} V_{max}}{- w_{n}^{2} m_{e} + j w_{n} c_{e} + k_{e}} | \\ w_{n} = \sqrt{\frac{k_{e}}{m_{e}}} \end{matrix}

Thus, the modified frequency $w'_{n}$ and amplitude $A'_{s}$ for the pressed stator can be expressed as

(26)

{\begin{matrix} {A'}_{s} = | \frac{θ_{i} V_{max}}{- w_{n}^{2} m_{e} + j w_{n} (c_{e} + {c'}_{e}) + (k_{e} + {k'}_{e})} | \\ {w'}_{n} = \sqrt{\frac{(k_{e} + {k'}_{e})}{m_{e}}} \end{matrix}

Using the calculated parameter $F_{2 θ} (j_{f})$ , the output torque and output power can be expressed as (Hagood and McFarland, 1995)

(27)

{\begin{matrix} T_{q} = \sum_{j_{f} = 1}^{n_{f}} F_{2 θ} (j_{f}) r_{c} \\ P_{out} = 2 π T_{q} V_{rotor} \end{matrix}

where $T_{q}$ is the output torque and $P_{out}$ is the output power.

For free stator, the parameters of the equivalent electrical circuit are (Uchino et al., 2004; Ueha and Tomikawa, 1993)

(28)

{\begin{matrix} L_{0} = \frac{m_{e}}{θ_{e}^{2}} \\ C_{0} = \frac{θ_{e}^{2}}{k_{e}} \\ R_{0} = \frac{c_{e}}{θ_{e}^{2}} \end{matrix}

where $L_{0}$ is the equivalent inductance, $C_{0}$ is the equivalent capacity, and $R_{0}$ is the equivalent resistance.

Using the above parameters, the admittance can be expressed as (Uchino et al., 2004; Ueha and Tomikawa, 1993)

(29)

Y_{m 0} = \frac{1}{1 / (j w_{n} C_{0}) + j w_{n} L_{0} + R_{0}} + j w_{n} C_{d}

Compared with equation (28), the electrical parameters of pressed stator can be expressed as

(30)

{\begin{matrix} L_{1} = \frac{m_{e}}{θ_{e}^{2}} \\ C_{1} = \frac{θ_{e}^{2}}{(k_{e} + {k'}_{e})} \\ R_{1} = \frac{(c_{e} + {c'}_{e})}{θ_{e}^{2}} \end{matrix}

where $L_{1}$ , $C_{1}$ , and $R_{1}$ are the equivalent inductance, equivalent capacity, and equivalent resistance of pressed stator, respectively.

The admittance should also be modified as

(31)

Y_{m 1} = \frac{1}{1 / (j {w'}_{n} C_{1}) + j {w'}_{n} L_{1} + R_{1}} + j w'_{n} C_{d}

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This project was supported by the National Science Foundation of China (Nos 51205203 and 51275228) and the Foundation of NUAA (No. 56YAH12015).

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