For linear structural dynamics problems, an application of the minimizing properties of calculus to Rayleigh’s quotient leads to the model equations of the system. The modal equations, which are previously undefined in the literature, are a set of simultaneous cubic equations with constant coefficients. They can be solved by means of a converging iteration scheme to obtain all of the mode shapes. Each mode shape can be successively substituted back into Rayleigh’s quotient to obtain the corresponding natural frequency. It is therefore possible to determine all of the mode shapes and natural frequencies of a system without developing or solving the characteristic polynomial; there are some significant advantages to this approach.