Dynamic Analysis of Micro -Scale Parallelogram Flexures Using Beam Constraint Model and Modified Strain Gradient Theory

In this paper, the dynamic behavior of a small-scale parallelogram (P) flexure is studied. First, using the beam constrain model and the modified strain gradient theory, the nonlinear strain energy of a small-scale beam is obtained in terms of its tip displacements. This energy expression is utilized to derive the strain energy of a P-flexure. Then the governing dynamic equations of motion are derived using Lagrange equations and are linearized around the operating equilibrium point. This linear model is employed to determine the allowable forces which do not lead to instability of the system. Moreover, the natural frequencies of the system are also extracted and the size effect as well as the static components of the applied loads on them are studied in detail. It is observed that by reducing the dimensions, the normalized transverse natural frequency of the system is increased. However, since there is no strain gradient in an axial mode, the axial normalized frequency is remained constant reducing the dimensions of the system. Moreover, it was observed that the tensile static forces lead to an increase, and transverse forces lead to a decrease in normalized natural frequency of the system. The procedure utilized for dynamic modeling of parallelogram flexures in this paper can be further extended for modeling more complex flexure systems.