Linear second-order ordinary differential equations arise from Newton's second law combined with Hooke's law and are ubiquitous in mechanical and civil engineering. Perhaps the most prominent example is a mathematical model for small oscillations of particles around their equilibrium positions. However, second-order systems also find applications in such diverse areas as chemical engineering, structural dynamics, linear systems theory or even economics. Very large second-order systems appear, for example, in mathematical modeling of complex structures by finite-element methods.

In general, any system of second-order equations is coupled. Each equation is linked to at least one of its neighbors and the solution of one of the equations requires the solution of all equations. The "classical decoupling problem" is concerned with the elimination of coordinate coupling in linear dynamical systems. The decoupling transforms the system of equations into a collection of mutually independent equations so that each equation can be solved without solving any other equation. In "The Theory of Sound" in 1894, Lord Rayleigh already expounded on the significance of system decoupling. Since then, the problem has attracted the attention of many researchers.

Mathematically, the system of differential equations is defined by three coefficient matrices. The equations are coupled unless all three matrices are diagonal. The "classical decoupling problem" is thus equivalent to the problem of simultaneous con- version of the coefficient matrices into diagonal forms. Current theory emphasizes simultaneous diagonalization of the coefficient matrices by equivalence or similarity transformations. However, it has been shown that no time-invariant linear trans- formations will decouple every second-order system. Even partial decoupling, i.e. simultaneous conversion of the coefficient matrices into upper triangular forms, is not ensured with time-invariant linear transformations.

The purpose of this work is to present a general method and algorithm to de- couple any second-order linear system (possessing symmetric and non-symmetric co- efficients). The theory exploits the parameter "time," characteristic of a dynamical

system. The decoupling is achieved by a real, invertible, but generally nonlinear map- ping. This mapping simplifies to a real, linear time-invariant transformation when the coefficient matrices can be simultaneously diagonalized by a similarity transformation. A state-space reformulation of the mapping is also derived. In homogeneous systems the configuration-space decoupling transformation is real, linear and time-invariant when cast in state space. In non-homogeneous systems, both the configuration and associated state transformations are nonlinear and depend continuously on the excitation. The theory is illustrated by several numerical examples. Two applications in earthquake engineering demonstrate the utility of the decoupling approach.

Decoupling a second-order linear dynamical system requires that one develop a transformation that simultaneously diagonalizes the coefficient matrices that define the system in terms of its distribution of inertia and viscoelasticity. A traditional approach to decoupling a viscously damped system uses the eigenvectors of the corresponding undamped system to diagonalize the mass, damping, and stiffness matrices through a real congruence transformation in the configuration space, a process known as classical modal analysis. However, it is well known that classical modal analysis fails to decouple a linear dynamical system if its damping matrix does not satisfy a commutativity relationship involving the system matrices. Such a system is said to be non-classically damped. We demonstrate that it is possible to decouple any non-classically damped system in the configuration and state spaces through generally time-dependent transformations constructed using spectral data obtained from the solution of a quadratic eigenvalue problem.

When a non-classically damped system has complex but non-defective eigenvalues, the effect of non-classical damping is that it introduces constant phase shifts among the components of the system's free response. Decoupling of free vibration in the configuration space is achieved through a real, linear, time-shifting transformation that eliminates these phase differences, yielding classical modes of vibration. This decoupling transformation, referred to as phase synchronization, preserves both the eigenvalues and their multiplicities. When cast in a state space form, the transformation between the coupled and decoupled systems is real, linear, but time-invariant. Through the concept of real quadratic conjugation, we illustrate that there is no fundamental difference in the representation of the free response of a system with complex eigenvalues and one with real eigenvalues, and thus systems with non-defective real eigenvalues can also be decoupled by phase synchronization. When phase synchronization is extended to forced systems, the decoupling transformation in both the configuration and state spaces is nonlinear and depends continuously on the applied excitation.

If a non-classically damped system is defective, it may only be partially decoupled if one insists on preserving the geometric multiplicities of the defective eigenvalues. We present the first systematic effort to decouple defective systems in free or forced vibration by not demanding invariance of the geometric multiplicities. In the course of this development, the notion of critical damping in multi-degree-of-freedom systems is clarified and expanded. It is shown that the decoupling of defective systems is a rather delicate procedure that depends on the multiplicities of the system eigenvalues. A generalized state space-based decoupling transformation is developed that relates the response of any non-classically damped system to that of its decoupled form. In principle, one could extract from the state space a decoupling transformation in the configuration space, but it generally does not have an explicit form. The decoupling transformation in both the configuration and state spaces is real and time-dependent. Several numerical examples are provided to illustrate the theoretical developments.

In this thesis, a novel investigation of deterministic nonlinear systems is presented. When autonomous dynamical systems are randomized by white-noise excitation, their behaviors are governed by diffusion equations. The core idea behind the method of randomizationinvolves replacing a nonlinear ordinary differential equation of motion by a linear partial differential equation of diffusion.

Exact analytical solutions are feasible for only a limited number of nonlinear systems. However, some nonlinear systems, which are difficult to analyze using deterministic methods, possess stationary diffusion equations with exact solutions. These diffusion responses provide deeper insights into the qualitative behaviors of nonlinear systems, such as stability at multiple equilibria, limit cycles, and bifurcations.

To demonstrate the potential and feasibility of randomization, numerous nonlinear oscillators are considered. When both the deterministic systems and the associated randomized systems can be analyzed, there is complete agreement in their qualitative properties. Furthermore, some example systems inaccessible through deterministic approaches can be readily examined using randomization, suggesting that randomization could be an alternative tool for investigating nonlinear oscillations.