This chapter and the next two use linear matrix algebra in the analysis. Appendix D gives a brief introduction to matrix algebra for those readers who may have lost and need to regain familiarity with the subject. This chapter develops from first principles all the required finite element stiffness and inertia matrices required for calculating natural frequencies and mode shapes, sinusoidal forced response, and transient response to any defined sets of applied torque histories (input of torques as a function of time). The finite elements are defined in terms of geometric and material property information.

Inertia matrices are produced using both linear and quadratic shape functions. In the former case each element has 2 degrees of freedom corresponding to rotational motion at each end of the element, and in the latter case each element has 3 degrees of freedom by including an additional one at the element midspan. In each case the inertia matrices are banded and more powerful than the traditional diagonal inertia matrices. Equivalent stiffness matrix derivations are provided.

The creation of the global inertia and stiffness matrices from the individual element matrices is demonstrated using conventional finite element assembly methodology.