Our solution method of choice is to discretize all problems and solve the resulting simultaneous equations numerically. Sometimes deeper insights into a problem can be obtained by solving the continuous problem directly. This is not always feasible, does not generalize very well, and typically is restricted to linearized systems, but when it can be accomplished, the results can be very rewarding. This chapter therefore develops this aspect of dynamic analysis.

The first task is to derive adequate dynamic models to describe continuous systems; we use Hamilton's principle in conjunction with the Ritz method to derive these models in a consistent rational way. Section 3.1 developed the energies for a number of structural components. Hamilton's principal is used to convert these energy representations into a set of governing differential equations plus the associated boundary conditions. This is called the strong formulation of problems.

The derived models are in the form of a system of partial differential equations. Partial differential equations are notoriously difficult to solve in general; we introduce spectral analysis as a powerful tool for simplifying and solving problems arising in the analysis of continuous systems. In essence, dynamic problems are reduced to a series of pseudostatic problems, and thus they are amenable to the solution procedures that are standard for static problems.

Strong Formulation of Problems

The strong formulation of a problem comprises the set of governing equations plus the appropriate geometric and natural boundary conditions.