Since systems involving multi-degrees-of-freedom (MDOF) require two or more coordinates to describe their motion, Chapter 4 begins with the general development of the equations of motion using an elasticity approach. The equations of motion are presented in the matrix form for simplifying solution procedures. The use of generalized coordinates to solve MDOF systems is discussed in terms of Lagrange's equations for generalized forces as well as their application to potential energy. Free and forced (damped and undamped) vibrations are discussed as well as the frequency response function with applications to a unit impulse and an arbitrary forcing function. The general theory of multiple-degree-of-freedom systems with emphasis on matrix notation is then discussed. The use of matrix notation is used in the presentation of the orthogonality of natural modes in which all parts of the vibrating system move sinusoidally with the same frequency and fixed phase relation. Subsequently undamped and damped (with proportional and viscous damping) forced vibrations are discussed.