Dynamic modelling and analysis of V- and Z-shaped electrothermal microactuators

Abstract

This paper presents a dynamic model and design analysis for V- and Z-shaped electrothermal microactuators operating in vacuum and in air conditions. The model is established with a coupled-field analysis combining the electrothermal and thermomechanical analyses for both heating and cooling processes. The electrothermal behaviors that dominated the overall dynamics are described by hybrid partial differential equations for three serially connected segments. The equations are solved subjected to the boundary, continuity, and initial conditions, and a unique method based on Fourier series is utilized to solve the temperature increase in each arms. The thermomechanical responses, i.e., the displacement and force, of the actuator are then calculated under the assumptions of quasi-static inertia. The analytical evaluations of the temperature and displacement are compared with the ones from finite element analysis via ANSYS software. A good agreement is found between analytical and simulation results. By virtual of the finite-element simulation, local high-frequency low-amplitude vibrations are demonstrated along the overall dynamic response for both V- and Z-shaped actuators with specific dimensions. Moreover, distinct dynamic behaviors between U- and V- and Z-shaped beams are observed and discussed using a proposed comparison benchmark. Finally, based on the dynamic model, the influences of structural as well as material parameters on the dynamic behaviors are analyzed to pave the way for improving the design and optimizing the dimensions of V- and Z-shaped electrothermal microactuators.

Appendix

In this appendix, dynamic modeling for the line-shaped microbeam is investigated. The governing equation is in (2). The temperature evolution $u(x,t)$. is decomposed into two parts: the steady-state and transient response $v(x,t)$ as in (6). The ODE of $w(x)$ and PDE of $v(x,t)$ are in (7) and (8) respectively. The transform of (9) is used to allow using of the method of separation of variables, and (10) is derived for describing the transient response of $v_{0} \left( {x,t} \right)$.

The main difference for the line-shaped beam is e non-existence of the middle shuttle and therefore the continuity conditions need not to be taken into account. The boundary an i conditions for the line-shaped beam are listed in Table 6 as follows

Table 6 Boundary and initial conditions for line-shaped beams

The steady-state temperature response of cooling process for both in air and in vacuum cases is $T_{0}$. The steady-state for the heating process in cases of the vacuum and air conditions are in (50) and (51) as follows

$$ \hat{w}\left( x \right) = ax^{2} + bx + c, $$

(50)

$$ \hat{w}\left( x \right) = ae^{rx} + be^{ - rx} + c, $$

(51)

where $a$, $b$, and $c$ are calculated using the boundary conditions. Using the method of separation of variables, $v_{0} \left( {x,t} \right)$ is decomposed into two functions with separated variables as in (13), and therefore the general solution of $v_{0} \left( {x,t} \right)$ has the form as in (16). Introducing the boundary conditions to (16), we have $A_{n} = 0$ and

$$ sin\lambda_{n} L = sin\sqrt {\frac{\lambda }{{\hat{k}}}} L = 0. $$

(52)

Clearly, we have $\lambda_{n} L = n\pi$, and thus

$$ \lambda_{n} = \frac{n\pi }{L},\quad n = 1,2,3, \ldots , $$

(53)

(16) becomes

$$ v_{0} \left( {x,t} \right) = \mathop \sum \limits_{n = 1}^{\infty } e^{{ - \hat{k}\lambda_{n}^{2} t}} B_{n} sin(\lambda_{n} x), $$

(54)

in which, $\lambda_{n} = n\pi /L$, $n$ = 1, 2, 3, … For heating process, applying the initial conditions to (54), we have

$$ T_{0} - \hat{w}(x) = \mathop \sum \limits_{n = 1}^{\infty } \hat{B}_{n} \sin \left( {\lambda_{n} x} \right), $$

(55)

where $\hat{B}_{n}$ is solved with the Fourier series method as follows

$$ \begin{aligned} \hat{B}_{n} &= \frac{2}{L}\mathop \int\limits_{0}^{L} \left[ {T_{0} - w\left( x \right)} \right]\sin \left( {\lambda_{n} x} \right)dx \hfill \\ & = \frac{2}{{\varepsilon_{0} \hat{\rho }L^{3} }} \cdot \left[ {\cos \left( {\lambda_{n} L} \right) - 1} \right]\frac{{r^{2} }}{{\lambda_{n} \left( {\lambda_{n}^{2} + r^{2} } \right)}} \cdot U^{2} . \hfill \\ \end{aligned} $$

(56)

Therefore,

$$ \hat{v}\left( {x,t} \right) = \frac{{U^{2} }}{{k\hat{\rho }}} \cdot \left[ {\mathop \sum \limits_{n = 1}^{\infty } \frac{{e^{{ - \hat{k}\left( {\lambda_{n}^{2} + r^{2} } \right)t}} }}{{\left( {r^{2} + \lambda_{n}^{2} } \right)\lambda_{n} }} \cdot D^{{\prime }} \cdot sin\left( {\lambda_{n} x} \right)} \right] , $$

(57)

where $D^{{\prime }} = 2\left[ {\cos \left( {\lambda_{n} L} \right) - 1} \right]/L^{3}$. For the cooling process, ${\check{B}}_{n} = - \hat{B}_{n}$, and ${\check{v}} \left( {x,t} \right) = - \hat{v}\left( {x,t} \right)$.