New modal synthesis methods are developed for calculating the exact eigenproperties of structures divided into two substructures. Unlike the conventional mode synthesis techniques, here the synthesis of modes is carried out by solving a second eigenvalue problem by nontraditional means: the eigenvalues of the combined structure are obtained by solving simple characteristic equations which,in the proposed approach, can be defined in closed-form. These equations can be easily solved by a simple Newton-Raphson iterative scheme, especially when good initial estimates of the roots are available. Herein, explicit expressions are provided to calculate these initial values; these expressions are obtained via a second order matrix perturbation analysis of the algebraic eigenvalue problems. Once the eigenvalues are known, the eigenvectors can be calculated with closed-form expressions without solving any simultaneous equations. Several variants of the methods are developed to consider various damping cases which can be encountered in practice. Furthermore, for each damping case, two parallel approaches are developed that allow one to utilize two different types of modes - free and fixed attachment modes - of one of the substructures to be synthesized with the free attachment modes of the other structure. The eigenproperties of the combined system, once calculated, can be utilized in a mode superposition approach for the calculation of the system response for any forcing function . For seismic analysis of combined primary and secondary structures, such as a main structure supporting piping or other auxiliary system, the combined structure is divided into two substructures. The eigenproperties of these two substructures are then synthesized to obtain the eigenproperties of the combined system. To obtain response for seismic design motions defined in terms of ground response spectra, response spectrum approaches are presented which utilize the eigenproperties of the combined system. Closed-form formulas are presented to obtain any displacement-related response quantity of interest for the seismic input defined in terms of ground response spectra. The seismic response calculated by the proposed procedure accounts for the dynamic interaction effects between the primary and secondary structures and the nonclassically damped characteristics of the combined system in an analytically exact way. Numerical results showing the performance of the modal synthesis methods are presented. The applicability of the response spectrum approaches is verified by numerical simulation studies. The approach is applicable to light as well as heavy secondary structures equally effectively. It can also be used for seismic analysis of tertiary systems in industrial units. The potential application of the approach for efficient and accurate vibration analysis of aerospace and aircraft structures should also be of interest. / Ph. D.