In the previous two chapters we focused on issues of modeling, equilibrium, and stability largely associated with SDOF systems. As such, the dynamic and stability behavior of a mass was largely characterized, for example, by a single frequency. However, there are many examples of systems with more than a SDOF, in which dynamics and stability issues are more involved. Certain characteristics of a physical system can be lumped at discrete locations, and this leads to sets of coupled ordinary differential equations. Linear algebra plays a key role in their analysis. Of course, most real structures are continuous and have an infinite number of DOFs. Governing equations of motion are typically partial differential equations (depending on both space and time), with boundary, as well as initial, conditions needing to be satisfied for a complete solution. Unlike typical (static) bending problems, which lead to inhomogeneous differential equations, the systems of primary focus in this book concern nontrivial homogeneous differential equations and often, in the analysis process, we will need to formulate and solve an eigenvalue problem: algebraic, for finite DOFs, and differential, for infinite DOFs. This will also typically involve the use of various approximation techniques, discretizations, and computational methods. This chapter serves the purpose of expanding the theoretical basis of dynamics and stability to this wider class of problems.
Returning to the Lagrangian description (Section 2.5), we again focus attention on conservative systems and develop Lagrange's equation in matrix form.