Free vibration analysis of a fully clamped rectangular plate-type nano/micro-resonator in presence of the Casimir force

Abstract

The Casimir force is resulting from the propagation of retarded electromagnetic waves and acts in sub-micron separations. This force is inversely proportional to the fourth power of the nano/micro-plate’s deflection. Hence, the oscillatory behavior of nano/micro-plates is strongly influenced by this force at sub-micron separations. Due to the high stiffness and simplicity of the production procedure of fully clamped rectangular nano/micro-plates, they represent major structural components of nano/micro-resonators. The objective of present paper is to investigate the free vibrations of fully clamped rectangular nano/micro-plates under the effect of the Casimir force. To this end, the Kirchhoff thin plate theory is employed to derive the governing partial differential equation (PDE) of motion of the system. To find the free vibration characteristics of the nano/micro-plate, the linear terms in the equation of motion are retained and the extended Kantorovich method is applied. This method, which is based on the separation of variables concept, obtains ordinary differential equations (ODEs) from the governing PDE of motion on the basis of the variational form of the problem. The solution of the main problem can also be extracted using the solutions of these resulting ODEs, iteratively. It is found that the present analytical approximate procedure rapidly converges and produces high accurate results. The present findings are compared with available results in the literature and excellent agreements between them are observed. A parametric study is also conducted to show the significant effects of the Casimir force on natural frequencies of fully clamped nano/micro-plates.