Eventos Anais de eventos
COBEM 2023
27th International Congress of Mechanical Engineering
ON THE USE OF THE GENERALIZED INTEGRATING FACTOR FOR SOLVING COUPLED SYSTEMS OF ORDINARY SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS
Submission Author:
Matheus Janczkowski Fogaça , SC , Brazil
Co-Authors:
Matheus Janczkowski Fogaça, Eduardo Lenz Cardoso
Presenter: Matheus Janczkowski Fogaça
doi://10.26678/ABCM.COBEM2023.COB2023-0777
Abstract
Systems of coupled linear second order Ordinary Differential Equations, ODEs, appear in fields like vibration, electric circuitry, applied mathematics and physics. Analytically solving these differential equations, especially for nontrivial excitation functions, is paramount for problems that depend on solution accuracy. Nonetheless, there is no general analytic solution procedure for non-homogeneous linear ordinary second order differential equations. This work proposes an extension of the Leibniz integrating factor to account for ODEs of higher orders. The approach consists in splitting the coefficients of intermediate derivatives to allow for systematic use of the original concept of integrating factors. Analytical solutions are obtained through a double convolution, avoiding the need of proposing a candidate solution. The method is then particularized for second order ODEs with constant coefficients. Closed-form particular solutions due to Heaviside multiplied by polynomials and Dirac’s delta excitations are obtained by using the proposed approach. Unlike existing methods as uncoupled eigen decomposition and Fourier series, the proposed approach can be used to derive analytical solutions by means of exponential maps due to the use of the integrating factor. It is shown that proportional damping renders many efficient particularization to the analytical solutions. Two test cases are addressed and compared to numerical approximations. It is shown that the proposed methodology is accurate, provides the homogeneous and the particular solutions in separate, and is not sensitive to time discretization like numerical methods.
Keywords
Integrating factor, constant coefficients second order differential equations, vibration differential equations, electric circuitry differential equations
