Structures may subject to both static and dynamic loading. Unlike static analysis, in which only static structural displacement is considered, acceleration and velocity are introduced as well in dynamic analysis. For a system that has only one degree of freedom (DOF), as shown in Figure 17.1, the forces resisting the applied loading are considered as the following:

1. A force proportional to displacement (the stiffness), which can be expressed as ky

2. A force proportional to velocity (the damping), which can be considered as cy

3. A force proportional to acceleration (the inertia), which can be expressed as my¨

So, as shown in Figure 17.1, the fundamental dynamic equilibrium equation is

m t cy t ky t f tÿ( ) ( ) ( ) ( )+ + = (17.1a)

where y, y, and ÿ are displacement, velocity, and acceleration, respectively. For a system that has multiple DOFs, the equation corresponding to

17.1a can be rewritten as

M t t t tä Ca Ka f( ) ( ) ( ) ( )+ + = (17.1b)

where: M is the global mass matrix C is the global damping matrix K is the global stiffness matrix a(t) is the displacement vector f(t) is the external load vector

In comparison with the static equation 3.3, forces due to acceleration and damping are introduced in dynamic analysis.