This dissertation evaluates higher-order modal methods for predicting thermal and structural response. More accurate methods or ones which can significantly reduce the size of complex, transient thermal and structural problems are desirable for analysis and are required for synthesis of real structures subjected to thermal and mechanical loading. A unified method is presented for deriving successively higher-order modal solutions related to previously developed, lower-order methods such as the mode-displacement and mode-acceleration methods. A new method, called the force derivative method, is used to obtain higher-order modal solutions for both uncoupled (proportionally-damped) structural problems as well as thermal problems and coupled (non-proportionally damped) structural problems. The new method is called the force-derivative method because, analogous to the mode-acceleration method, it produces a term that depends on the forcing function and additional terms that depend on the time derivatives of the forcing function. The accuracy and convergence history of various modal methods are compared for several example problems, both structural and thermal. The example problems include the case of proportional damping for: a cantilevered beam subjected to a quintic time varying tip load and a unit step tip load and a muItispan beam subjected to both uniform and discrete quintic time-varying loads. Examples of non-proportional damping include a simple two-degree-of-freedom spring-mass system with discrete viscous dampers subjected to a sinusoidally varying load and a multispan beam with discrete viscous dampers subjected to a uniform, quintic time varying load. The last example studied is a transient thermal problem of a rod subjected to a linearly-varying, tip heat load. / Ph. D.