Static and dynamic responses of a microcantilever with a T-shaped tip mass to an electrostatic actuation

Abstract

In this study, nonlinear static and dynamic responses of a microcantilever with a T-shaped tip mass excited by electrostatic actuations are investigated. The electrostatic force is generated by applying an electric voltage between the horizontal part of T-shaped tip mass and an opposite electrode plate. The cantilever microbeam is modeled as an Euler–Bernoulli beam. The T-shaped tip mass is assumed to be a rigid body and the nonlinear effect of electrostatic force is considered. An equation of motion and its associated boundary conditions are derived by the aid of combining the Hamilton principle and Newton’s method. An exact solution is obtained for static deflection and mode shape of vibration around the static position. The differential equation of nonlinear vibration around the static position is discretized using the Galerkin method. The system mode shapes are used as its related comparison functions. The discretized equations are solved by the perturbation theory in the neighborhood of primary and subharmonic resonances. In addition, effects of mass inertia, mass moment of inertia as well as rotation of the T-shaped mass, which were ignored in previous works, are considered in the analysis. It is shown that by increasing the length of the horizontal part of the T-shaped mass, the amount of static deflection increases, natural frequency decreases and nonlinear shift of the resonance frequency increases. It is concluded that attaching an electrode plate with a T-shaped configuration to the end of the cantilever microbeam results in a configuration with larger pull-in voltage and smaller nonlinear shift of the resonance frequency compared to the configuration in which the electrode plate is directly attached to it.

Appendices

Appendix 1

Here the equation of motion and associated boundary conditions for a cantilever beam, shown in the figure below, is derived using the Hamilton principle. It is assumed that the length of the microbeam is $l+\varepsilon _0$, and a concentrated force and moment, $F_\mathrm{s} ,M$ at distance l from the clamped side are applied to it. According to the Hamilton principle [52]:

$$\begin{aligned} \int _{t_1 }^{t_2} {\delta (T-V+W_\mathrm{f} } +W_\mathrm{v}) \hbox {d}t=0, \end{aligned}$$

(54)

where T, V, $W_\mathrm{v}$, and $W_\mathrm{f}$ are the kinetic energy, potential strain energy, external work due to equivalent viscous damping and external work due to concentrated force and moment, respectively. The variation of these terms are as below (Fig. 20)

$$\begin{aligned} T= & {} \int _0^{l+\varepsilon _0 } {\frac{1}{2}} \rho A\left( \frac{\partial w_1 }{\partial t}\right) ^{2}\hbox {d}x, \nonumber \\ V= & {} \int _0^{l+\varepsilon _0 } {\frac{1}{2}} EI\left( \frac{\partial ^{2}w_1 }{\partial x^{2}}\right) ^{2}\hbox {d}x, \end{aligned}$$

(55)

Fig. 20

The microbeam under concentrated force and moment

$$\begin{aligned} \int _{t_1}^{t_2 } {\delta (T})= & {} \delta \int _{t_1 }^{t_2 } \int _0^{l+\varepsilon _0 } {\frac{1}{2}} \rho A\left( \frac{\partial w_1 }{\partial t}\right) ^{2}\hbox {d}x\hbox {d}t \nonumber \\= & {} \int _0^{l+\varepsilon _0} \int _{t_1 }^{t_2 } {\rho A\left( \frac{\partial w_1 }{\partial t}\right) \delta \left( \frac{\partial w_1 }{\partial t}\right) \hbox {d}t\hbox {d}x} \nonumber \\= & {} \underbrace{\int _0^{l+\varepsilon _0 } \rho A\left( \frac{\partial w_1}{\partial t}\right) \delta w_1 \left| {_{t_1 }^{t_2 } } \right. \hbox {d}x}_{=0} \nonumber \\&-\int _0^{l+\varepsilon _0} \int _{t_1 }^{t_2 } {\rho A\left( \frac{\partial ^{2}w_1 }{\partial t^{2}}\right) \delta w_1 \hbox {d}t\hbox {d}x}, \end{aligned}$$

(56)

$$\begin{aligned} \int _{t_1}^{t_2 } {\delta (V} )= & {} \delta \int _{t_1 }^{t_2 } \int _0^{l+\varepsilon _0 } {\frac{1}{2}} EI(\frac{\partial ^{2}w_1 }{\partial x^{2}})^{2}\hbox {d}x\hbox {d}t \nonumber \\= & {} \int _{t_1 }^{t_2 } \int _0^{l+\varepsilon _0 } EI\left( \frac{\partial ^{2}w_1 }{\partial x^{2}}\right) \delta \left( \frac{\partial ^{2}w_1 }{\partial x^{2}}\right) \hbox {d}x\hbox {d}t \nonumber \\= & {} \int _{t_1}^{t_2 } EI\left( \frac{\partial ^{2}w_1 }{\partial x^{2}}\right) \delta \left( \frac{\partial w_1 }{\partial x}\right) \left| {_0^{l+\varepsilon _0 } } \right. \hbox {d}t \nonumber \\&-\int _{t_1 }^{t_2 } \int _0^{l+\varepsilon _0 } EI\left( \frac{\partial ^{3}w_1 }{\partial x^{3}}\right) \delta \left( \frac{\partial w_1 }{\partial x}\right) \hbox {d}x\hbox {d}t \nonumber \\= & {} \int _{t_1 }^{t_2 } EI\left( \frac{\partial ^{2}w_1 }{\partial x^{2}}\right) \delta \left( \frac{\partial w_1 }{\partial x}\right) \left| {_0^{l+\varepsilon _0}} \right. \nonumber \\&- \int _{t_1 }^{t_2 } EI\left( \frac{\partial ^{3}w_1 }{\partial x^{3}}\right) \delta w_1 \left| {_0^{l+\varepsilon _0 } } \right. \nonumber \\&+\int _{t_1}^{t_2 } \int _0^{l+\varepsilon _0 } EI\left( \frac{\partial ^{4}w_1}{\partial x^{4}}\right) \delta (w_1 )\hbox {d}x\hbox {d}t, \end{aligned}$$

(57)

$$\begin{aligned} \int _{t_1 }^{t_2 } {\delta W_\mathrm{f} }= & {} \int _{t_1 }^{t_2 } \left( F_\mathrm{s} \delta w(l,t)-M\delta \frac{\partial w_1 (l,t)}{\partial x}\right) \hbox {d}t, \end{aligned}$$

(58)

$$\begin{aligned} \int _{t_1 }^{t_2 } {\delta W_\mathrm{f} \hbox {d}t}= & {} \int _{t_1 }^{t_2 } \left( F_\mathrm{s} \delta w(l,t)-M\delta \frac{\partial w_1 (l,t)}{\partial x}\right) \hbox {d}t \nonumber \\= & {} \int _{t_1 }^{t_2 } {\int _0^{l+\varepsilon _0 } } F_\mathrm{s} \delta w\hbox { Dirac}(x-l)\hbox {d}x\hbox {d}t \nonumber \\&-\int _{t_1 }^{t_2 } {\int _0^{l+\varepsilon _0 } } M\delta \left( \frac{\partial w}{\partial x}\right) \hbox {Dirac}(x-l)\hbox {d}x\hbox {d}t \nonumber \\= & {} \int _{t_1 }^{t_2 } {\int _0^{l+\varepsilon _0 } } F \delta w\hbox { Dirac}(x-l)\hbox {d}x\hbox {d}t \nonumber \\&-\underbrace{\int _{t_1 }^{t_2 } M\delta w_1 \hbox {Dirac}(x-l)\left| {_0^{l+\varepsilon _0}} \right. }_{=0} \nonumber \\&+\int _{t_1}^{t_2 } {\int _0^{l+\varepsilon _0}} M\frac{\partial \hbox {Dirac}(x-l)}{\partial x} \delta w_1 \hbox {d}x\hbox {d}t, \end{aligned}$$

(59)

$$\begin{aligned} \int _{t_1 }^{t_2 } {\delta (W_\mathrm{v} } )= & {} -\int _{t_1 }^{t_2 } \int _0^{l+\varepsilon _0 } {c\frac{\partial w_1 }{\partial t}\delta w_1 \hbox {d}x} . \end{aligned}$$

(60)

Now, if one substitutes Eqs. (56), (57), (59), and (60) into Eq. (54), the equation of motion for microbeam and associated boundary conditions will be:

$$\begin{aligned} \begin{aligned}&EI\frac{\partial ^{4}w_1 }{\partial x^{4}}+\rho A\frac{\partial ^{2}w_1 }{\partial t^{2}}+c_1 \frac{\partial w_1 }{\partial t}, \\&\quad =F_\mathrm{s}\,\hbox {Dirac}(x-l)+M\frac{\partial \hbox {Dirac}(x-l)}{\partial x}, \\&w_1 \Big | {_{x=0}} =0, \quad \frac{\partial w_1 }{\partial x} \Big | {_{x=0} } =0, \\&\frac{\partial ^{2}w_1 }{\partial x^{2}} \Big | {_{x=l+\varepsilon _0 } } =0,\quad \frac{\partial ^{3}w_1 }{\partial x^{3}} \Big | {_{x=l+\varepsilon _0 } } =0. \end{aligned} \end{aligned}$$

(61)

It is clear that when $\varepsilon _0 \rightarrow 0$ i.e., one assumes that the external force and moment are at the free boundary, then Eq. (61) results to Eq. (1). It can be considered that if one sets $\varepsilon _0 =0$, and then substitutes Eq. (58) instead of Eq. (59) into Eq. (54) then the equation of motion for microbeam will be:

$$\begin{aligned} \begin{aligned}&EI\frac{\partial ^{4}w_1 }{\partial x^{4}}+\rho A\frac{\partial ^{2}w_1 }{\partial t^{2}}+c_1 \frac{\partial w_1 }{\partial t}=0 ,\quad 0\le x\le l_1 , \\&w_1 \Big | {_{x=0} } =0, \quad \frac{\partial w_1 }{\partial x}\Big | {_{x=0} } =0 ,\\&-EI\frac{\partial ^{2}w_1 }{\partial x^{2}} \Big | {_{x=l} } =M, \quad -EI\frac{\partial ^{3}w_1 }{\partial x^{3}} \Big | {_{x=l} } =F_\mathrm{s}. \end{aligned} \end{aligned}$$

(62)

Appendix 2

$$\begin{aligned} S_1= & {} \int _0^{\hat{{l}}} \left\{ \frac{\partial ^{4}\varphi _1 }{\partial \hat{{x}}^{4}}-2\alpha _2 V_\mathrm{p} ^{2}\left[ {\int _{\hat{{l}}_3 }^{\hat{{l}}_4 } {\frac{\varphi _2 (\hat{{x}},t)}{(1-w_{\mathrm{s}2} (\hat{{x}}))^{3}}} \hbox {d}\hat{{x}}} \right] \right. \\&\times \left. \hbox {Dirac}(\hat{{x}}-\hat{{l}})\right\} \varphi _1 \hbox {d}\hat{{x}}+2\alpha _2 V_\mathrm{p} ^{2} \\&\times \,\int _0^{\hat{{l}}} { \left[ {\int _{l_3 }^{l_4 } {\frac{(\hat{{x}}-l_c )\varphi _2 (\hat{{x}},\hat{{t}})}{(1-w_\mathrm{S2} (\hat{{x}}))^{3}}} \hbox {d}\hat{{x}}} \right] \frac{\partial \hbox {Dirac}(\hat{{x}}-\hat{{l}})}{\partial \hat{{x}}}\varphi _1 }\hbox {d} \hat{{x}}, \\ S_2= & {} -3\alpha _2 V_\mathrm{p} ^{2}\int _0^{\hat{{l}}} \left[ {\int _{\hat{{l}}_3 }^{\hat{{l}}_4 } {\frac{\varphi _2 ^{2}}{(1-w_{\mathrm{s2}} (\hat{{x}}))^{4}}} \hbox {d}\hat{{x}}} \right] \\&\times \,\hbox {Dirac}(\hat{{x}}-\hat{{l}})\varphi _1 \hbox {d}\hat{{x}}+ 3\alpha _2 V_\mathrm{p} ^{2} \\&\times \,\int _0^{\hat{{l}}} \left[ {\int _{l_3 }^{l_4 } {\frac{\left( \hat{{x}}-\hat{{l}}_c\right) \varphi _2 ^{2}}{(1-w_{s2} (\hat{{x}}))^{4}}} \hbox {d}\hat{{x}}} \right] \frac{\partial \hbox {Dirac}(\hat{{x}}-\hat{{l}})}{\partial \hat{{x}}}\varphi _1 \hbox {d}\hat{{x}}, \\ S_5= & {} \int _0^{\hat{{l}}} \varphi _1^2 \hbox {d}\hat{{x}}+\alpha _3 \int _0^{\hat{{l}}} {\varphi _1^2} \hbox {Dirac}\left( \hat{{x}}-\hat{{l}}\right) \hbox {d}x-\alpha _4 \\&\times \,\int _0^{\hat{{l}}} \left( {\frac{\partial \varphi _1 }{\partial \hat{{x}}}\Big | {_{\hat{{x}}=l} } } \right) \frac{\partial \hbox {Dirac}(\hat{{x}}-\hat{{l}})}{\partial \hat{{x}}} \varphi _1 \hbox {d}\hat{{x}}, \\ S_4= & {} \int _0^{\hat{{l}}} {\hat{{c}}_1 } \varphi _{_1 }^2 \hbox {d}\hat{{x}}+\int _0^{\hat{{l}}} {\left( {\int _{_{\hat{{l}}_3 } }^{\hat{{l}}_4 } {\hat{{c}}_2 \varphi _2 \;\hbox {d}\hat{{x}}} } \right) } \\&\times \,\hbox {Dirac}\left( \hat{{x}}-\hat{{l}}\right) \varphi _1 \hbox {d}\hat{{x}} \\&-\,\int _0^{\hat{{l}}} {\left[ {\int _{_{\hat{{l}}_3 } }^{\hat{{l}}_4 } {\hat{{c}}_2 \left( {\hat{{x}}-\hat{{l}}_c } \right) \varphi _2 \;\;\hbox {d}\hat{{x}}} } \right] } \frac{\partial \hbox {Dirac}(\hat{{x}}-\hat{{l}})}{\partial \hat{{x}}}\varphi _1 \hbox {d}\hat{{x}}, \\ S_5= & {} \int _0^{\hat{{l}}} \varphi _1^2 \hbox {d}\hat{{x}}+\alpha _3 \int _0^{\hat{{l}}} {\varphi _1^2 } \hbox {Dirac}(\hat{{x}}-\hat{{l}})\hbox {d}x-\alpha _4 \\&\times \,\int _0^{\hat{{l}}} \left( {\frac{\partial \varphi _1 }{\partial \hat{{x}}}\left| {_{\hat{{x}}=l} } \right. } \right) \frac{\partial \hbox {Dirac}\left( \hat{{x}}-\hat{{l}}\right) }{\partial \hat{{x}}} \varphi _1 \hbox {d}\hat{{x}}, \end{aligned}$$

$$\begin{aligned} S_6= & {} -2\alpha _2 V_\mathrm{p} \int _0^{\hat{{l}}} \left[ {\int _{\hat{{l}}_3 }^{\hat{{l}}_4 } {\frac{1}{(1-w_{\mathrm{s2}} (\hat{{x}}))^{3}}} \hbox {d}\hat{{x}}} \right] \\&\times \hbox {Dirac}(\hat{{x}}-\hat{{l}})\varphi _1 \hbox {d}\hat{{x}}+2\alpha _2 V_\mathrm{p} \\&\times \int _0^{\hat{{l}}} \left[ {\int _{\hat{{l}}_3 }^{\hat{{l}}_4 } {\frac{(\hat{{x}}-l_c )}{(1-w_{\mathrm{s2}} (\hat{{x}}))^{3}}} \hbox {d}\hat{{x}}} \right] \frac{\partial \hbox {Dirac}(\hat{{x}}-\hat{{l}})}{\partial \hat{{x}}}\varphi _1 \hbox {d}\hat{{x}}, \\ \quad S_7= & {} -\alpha _2 \int _0^{\hat{{l}}} \left[ {\int _{\hat{{l}}_3 }^{\hat{{l}}_4 } {\frac{1}{(1-w_{\mathrm{s2}} (\hat{{x}}))^{3}}} \hbox {d}\hat{{x}}} \right] \\&\times \hbox {Dirac}(\hat{{x}}-\hat{{l}})\varphi _1 \hbox {d}\hat{{x}}+\alpha _2 \int _0^{\hat{{l}}} \left[ {\int _{\hat{{l}}_3 }^{\hat{{l}}_4 } {\frac{(\hat{{x}}-l_c )}{(1-w_{\mathrm{s2}} (\hat{{x}}))^{3}}} \hbox {d}\hat{{x}}} \right] \\&\times \frac{\partial \hbox {Dirac}(\hat{{x}}-\hat{{l}})}{\partial \hat{{x}}}\varphi _1 \hbox {d}\hat{{x}}, \\ \quad S_8= & {} -2\alpha _2 V_\mathrm{p} \int _0^{\hat{{l}}} \left[ {\int _{\hat{{l}}_3 }^{\hat{{l}}_4 } {\frac{\varphi _2 }{(1-w_{\mathrm{s2}} (\hat{{x}}))^{3}}} \hbox {d}\hat{{x}}} \right] \\&\times \hbox {Dirac}(\hat{{x}}-\hat{{l}})\varphi _1 \hbox {d}\hat{{x}}+2\alpha _2 V_\mathrm{p} \int _0^{\hat{{l}}} \left[ {\int _{l_3 }^{l_4 } {\frac{(\hat{{x}}-l_c )\varphi _2 }{(1-w_{\mathrm{s2}} (\hat{{x}}))^{3}}} \hbox {d}\hat{{x}}} \right] \\&\times \frac{\partial \hbox {Dirac}(\hat{{x}}-\hat{{l}})}{\partial \hat{{x}}}\varphi _1 \hbox {d}\hat{{x}}. \end{aligned}$$