In this chapter, we consider time-dependent problems of discrete systems with N DOF. We will show how the finite element formulation is used to construct the element mass matrices, which are assembled into the global mass matrix. We will consider free vibration, to determine natural frequencies and natural modes of a finite element model through eigenvalue analysis, and numerical methods for integrating the equation of motion in time which can be used to determine dynamic response under applied loads and given initial conditions. The Lagrange equation will be shown to demonstrate how it can be applied to construct equations of motion. Once again we will consider slender bodies undergoing uniaxial vibration, torsional vibration, and bending vibration. A formal derivation of the Lagrange equation will be considered in a later chapter.
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