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The Transformation of Second-Order Linear Systems into Independent Equations


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.

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