In Chapters 3 and 4, a theory was developed to explain the static response of microstructures, and linear and cubic stiffnesses, which relate to the static response of commonly used microstructures, were expressed in closed form. In this chapter we describe displacement, velocity, and acceleration and discuss the governing equations for the dynamic behavior of microstructures. Combining these governing equations with the linear and cubic stiffnesses developed in Chapter 4, we derive closed-form solutions for the dynamic response of microstructures.
5.1 CUBIC EQUATION
In Chapters 3 and 4, the static response of nonlinear structures was expressed in terms of the linear stiffness k1 and the cubic stiffness k3. The dynamic responses of the structures are expressed as a function of the linear and cubic stiffnesses. The static and dynamic behavior of the structures is then expressed in cubic equations. To calculate the static and dynamic response, it is necessary to obtain the roots of a cubic equation.
In general, a cubic equation is expressed as
where x, L, A1, B1, and C1 denote an unknown and its coefficients. Dividing equation (5.1) by the leading coefficient L, we have a simple form of the cubic equation as follows:
where a, b, and c represent new ...