Identifying External Force on Nonlinear Structures by a Lie-group Differential Algebraic Equations Method on Transformed PDE

　　For structures protection and control, it is utmost to immediately detect the external force being imposed on the structures currently in civil engineering. 　　Desiring a real-time recovery of an unknown external force in the nonlinear inverse vibration problem; in this thesis, 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 an unknown external force identification problem, of which the numerical method of lines is used to discretize the governing equation into a system of differential algebraic equations (DAEs) and with the constraints conditions. 　　A fictitious time variable transforming the Sturm-Liouville equation into a parabolic type partial differential equation (PDE) endowed with an extra fictitious time dimension. 　　Hence, we can develop an implicit GL(n, ) 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. 　　Obviously, we transform a simple ODE into a more complex PDE; however, the merits obtained in this transformation will be seen, when we examine some nonlinear inverse vibration problems with large time span and under large noise. We can alleviate the influence of noise, which is only happened at the first and the last line equations among the many 2m equations.The estimated results obtained by the novel methods are quite promising.