Figure Ja shows a linear structure of mass m, lateral stiffness k, and viscous damping c subjected to an externally applied dynamic force p(t). This notation indicates that the force p varies with time t. Under the influence of such a force, the roof of the structure displaces in the lateral direction by an amount u(t), which is also the deformation in the structure (displace• ment of roof relative to base). Because the force p varies with time, so does the displacement u.
The various forces acting on the mass at some instant of time
are shown in a free-body diagram of the mass (Fig. Jb). These include the external force p(t), the elastic resisting force fs, the damping force f0, and the inertia force f1. The elastic and damping forces act to the left because they resist the deforma• tion and velocity, respectively, which are taken as positive to the right. The inertia force also acts to the left, opposite to the direc•
tion of positive acceleration. At each instant of time, the mass is in equilibrium under the action of these forces at that time. From the free body diagram, this condition of dynamic equi• librium is