Lyapunov-Based Boundary Control of Strain Gradient Microplates

Abstract

Purpose

Due to the significance of the effect size on microstructures and also due to the fact that classical continuum theory cannot predict their behavior, higher order continuum theories should be used for modeling and obtaining the governing partial differential equations of microplates. For considering the size effect, the strain gradient theory is used. Vibration control of a microplate is studied using a linear boundary control law, and by utilizing a Lyapunov functional, the boundary stabilization of the system will be proved. The subject of vibration control of a strain gradient microplate utilizing the PDE control theory. The boundary action, in which the actuator is placed on the free sides of the plate, is used to stabilize mechanical vibrations.

Method

The governing equation and boundary condition of strain gradient microplate using Hamilton principle are derived. The boundary control problem of a CFFC (Clamped-Free-Free-Clamped) non-classical microplate using Lyapunov stability theorem and the design of a proper controller for the system are stablisged. A Lyapunov-based functional is used to demonstrate closed-loop system boundary stabilization. The verification of the theoretical results is accomplished using the finite-element method. New non-classical microplate element stiffness and mass matrices are developed based on the strain gradient theory. To derive strain gradient plate element stiffness and mass matrices, a four-node rectangular element is utilized. The verification of the theoretical results is accomplished using the finite-element method. Numerical method (Finite Element Method) is used for microplate simulation.

Results and Conclusion

Boundary stabilization of a vibrating strain gradient plate has been achieved using the Lyapunov stability theorem and boundary control. The boundary action is used to stabilize mechanical vibrations. The boundary PDE control technique used in this study provides various advantages. This simulations shows good agreement with the theoretical results. The deflection, slope, velocity and angular velocity are comprised in open loop and closed loop. Appling control force {F} at the free sides of the plate, causes to stabilize the vibrating system.

Appendix: The Components of Stiffness Matrix

The stiffness matrix is obtained in Eq. (45)

$$K={K}_{\varepsilon }+{K}_{\eta }+{K}_{\gamma }+{K}_{\chi }.$$

(51)

That

$${K}_{\varepsilon }= \underset{V}{\overset{ }{\int }}({{B}^{\varepsilon })}^{T}{D}^{c}{B}^{\varepsilon }\mathrm{d}V , {K}_{\eta }= \underset{V}{\overset{ }{\int }}({{B}^{\eta })}^{T}{D}^{\eta }{\Upsilon }^{\eta }{B}^{\eta }\mathrm{d}V {K}_{\gamma }= \underset{V}{\overset{ }{\int }}({{B}^{\gamma })}^{T}{D}^{\gamma }{B}^{\gamma }\mathrm{d}V , {K}_{\chi }= \underset{V}{\overset{ }{\int }}{{(B}^{\chi })}^{T}{D}^{\chi }{\Upsilon }^{\chi }{B}^{\chi }\mathrm{d}V.$$

(52)

The components of ${K}_{\varepsilon }$, ${K}_{\eta }$,${K}_{\gamma }$ and ${K}_{\chi }$ are as follows:

$${B}^{\varepsilon }=-z\left[\begin{array}{c}{N}_{,xx}\\ {N}_{,yy}\\ {N}_{,xy}\end{array}\right], {D}^{c}=\left[\begin{array}{c}\begin{array}{ccc}1& \nu & 0\end{array}\\ \begin{array}{ccc}\nu & 1& 0\end{array}\\ \begin{array}{ccc}0& 0& \frac{1-\nu }{2}\end{array}\end{array}\right],$$

(53)

$${B}^{\gamma }=\left[\begin{array}{c}0\\ 0\\ {-N}_{,xx}+{N}_{,yy}\end{array}\right]-\mathrm{z}\left[\begin{array}{c}{N}_{,xxx}+{N}_{,xyy}\\ {N}_{,yyy}+{N}_{,xxy}\\ 0\end{array}\right], {D}^{\gamma }=2\mu {l}_{0}^{2}{z}^{2}.$$

(54)

The ${\gamma }^{\chi }$ and ${\gamma }^{\eta }$ are diagonal matrices.

$${B}^{\chi }=\left[\begin{array}{c}{N}_{,xy}\\ {-N}_{,xy}\\ \frac{1}{2}\left({N}_{,yy}-{N}_{,xx}\right)\end{array}\right], {D}^{\chi }=2\mu {l}_{2}^{2}{z}^{2}, {\mathrm{diag}(\gamma }^{\chi })=\left[1 1 2\right]$$

(55)

$${B}^{\eta }=\left[\begin{array}{c}0\\ 0\\ \begin{array}{c}0\\ 0\\ \begin{array}{c}0\\ 0\\ \begin{array}{c}\frac{1}{15}\left({N}_{,yy}-4{N}_{,xx}\right)\\ \frac{-1}{3}\left({N}_{,xy}\right)\\ \begin{array}{c}\frac{1}{15}\left({N}_{,xx}-4{N}_{,yy}\right)\\ \frac{1}{5}\left({N}_{,xx}+{N}_{,yy}\right)\end{array}\end{array}\end{array}\end{array}\end{array}\right]-\frac{\mathrm{z}}{5}\left[\begin{array}{c}3{N}_{,xyy}-2{N}_{,xxx}\\ 3{N}_{,yyy}-4{N}_{,xxy}\\ \begin{array}{c}{N}_{,xxx}-4{N}_{,xyy}\\ 3{N}_{,xxy}-2{N}_{,yyy}\\ \begin{array}{c}{N}_{,xxx}+{N}_{,xyy}\\ {N}_{,yyy}+{N}_{,xxy}\\ \begin{array}{c}0\\ 0\\ \begin{array}{c}0\\ 0\end{array}\end{array}\end{array}\end{array}\end{array}\right] , {D}^{\eta }=2\mu {l}_{1}^{2}{z}^{2} \mathrm{diag}\left({\gamma }^{\eta }\right)=\left[1 3 3 1 3 3 3 6 3 1\right].$$

(56)