A systematic approach to obtain the analytical solution for coupled linear second-order ordinary differential equations: Part II

Coupled systems of second-order ODEs are found in many areas, such as vibrations and electric circuits. Thus, accurate and efficient solutions are of great interest. This manuscript extends the concept of generalized integrating factor, recently introduced by the authors for one-dimensional problems, to coupled n-dimensional systems of second-order ODEs. The general formulation for time-varying coefficients is found and particularized to constant coefficients. Closed-form integrating factors are obtained for the constant coefficient case under mild assumptions about the damping coefficient matrix. Analytical solutions for different types of continuous and discontinuous excitations are presented for problems with constant coefficients and proportional damping, disregarding the damping level. Examples are provided for each type of excitation studied in this manuscript. Results obtained with the proposed approach are compared to analytical solutions obtained using the order reduction by state variables approach and to numerical methods. Results show that the proposed procedure is accurate, not suffering with interpolation errors found in traditional numerical methods. The results also show that the proposed approach can be very efficient when compared to other analytical or numerical methods.