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The Transformation of Linear Second-Order ODEs into Independent Second-Order Equations: Scientific Computing and Matrix Computations Seminar
Seminar: Departmental | January 23 | 12:10-1 p.m. | 380 Soda Hall
Matthias Morzfeld, LBL
The class of linear second-order dynamical systems is considered. These systems of coupled ordinary differential equations (ODE) are characterized by three symmetric positive definite (SPD) coefficient matrices: one arises from inertial terms, the second from Hooke's law, and the third from velocity-proportional energy dissipation. The goal is to transform the coupled set of n second-order ODEs into n independent (scalar) second-order equations. In the absence of damping, this goal can be achieved by congruence transformation because two SPD matrices can be simultaneously diagonalized. However, three SPD matrices can not be simultaneously diagonalized by congruence transformations unless certain restrictive conditions apply, so in this case we must consider more general transformations. I will present a method that utilizes a real, invertible but nonlinear mapping to transform any set of coupled second-order linear ODEs into independent equations. Two examples from earthquake engineering are provided to indicate the utility of this approach.
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