A real-time Lie-group differential algebraic equations method to solve the inverse nonlinear vibration problems. Issue 9 (21st November 2016)

• Record Type: Journal Article
• Authors: Liu, Chein-Shan; Chang, Chih-Wen
• Is Part Of: Inverse problems in science and engineering. Volume 24:Issue 9(2016)
• Page Start: 1569
• Page End: 1586
• Publication Date: 2016-11-21
• Subjects: Nonlinear inverse vibration problem -- real-time recovery of external force -- Duffing nonlinear oscillator -- Duffing-van der Pol nonlinear oscillator -- Lie-group differential algebraic equations (LGDAE) method; 65M; Engineering mathematics -- Periodicals; Inverse problems (Differential equations) -- Periodicals; 620.001515357
• DOI: 10.1080/17415977.2015.1130043
• Languages: English
• ISSNs: 1741-5977

Abstract

Abstract : Desiring a real-time recovery of an unknown external force in the nonlinear inverse vibration problem, we transform the nonlinear ordinary differential equation (ODE) of motion into a nonlinear parabolic type partial differential equation (PDE), which can raise the robustness against large noise. Then, we come to a heat source identification problem, of which the numerical method of lines is used to discretize the PDE into a system of differential algebraic equations (DAEs). Consequently, we can develop an implicit G L ( n, R ) Lie-group scheme and a Newton algorithm to stably solve the DAEs for finding the unknown force, damping function, or stiffness function, which is well recovered even under a large noise. Superficially, we seem to transform a simple ODE to a more complex PDE; however, the advantages gained in this transformation will be seen when we test some nonlinear inverse vibration problems with large time span and under large noise.