ABSTRACT
The equation of motion of a vibrating system specifies the requirement of equilibrium between the applied force, the force of inertia, the damping force, and the spring force. For a single-degree-of-freedom system, the equation of dynamic equilibrium is given by
fI + fD + fS = p(t) (20.1)
where fI is the force of inertia, fD is the damping force, and fS is spring force. When the damping is viscous in nature and the displacements are small, Equation 20.1 can be expressed as
mu¨+ cu˙+ ku = p(t) (20.2)
wherem is the mass of the system, c is the damping constant, and k is the stiffness. Since the state variables, namely acceleration u¨, the velocity u˙, and the displacement u appear in Equation 20.2 raised only to the power 1, that equation is linear as long as the mass, damping constant, and stiffness do not change with time. The methods for solving the equations of motion developed in the previous chapters for single-as well as multidegree-of-freedom systems are applicable when the equations are linear. In particular, methods that involve the superposition of responses are valid only when linearity is ensured. Superposition of responses is involved whenever we express the total solution as the addition of a free vibration component, or complementary solution, and a forced vibration component, or particular solution. The Duhamel integral method and the mode superposition method also rely on the superposition of responses. When the system equation is nonlinear, methods that involve superposition are no longer valid.
