A Lie-Group Shooting Method Estimating Nonlinear Restoring Forces in Mechanical Systems

For an inverse vibration problem of nonlinear mechanical system to estimate displacement- and velocity-dependent restoring force, we transform the equation of motion into a parabolic type partial differential equation (PDE). Then by a semi-discretization of the PDE, the inverse vibration problem is formulated as a multi-dimensional two-point boundary value problem with unknown sources, allowing a closed-form estimation through a Lie-group shooting method to construct the restoring force surface over phase plane. Only one set of displacements measured at sampling time points is used in the estimation. The new method does not require to assume a priori the functional form of unknown restoring force, and more importantly, it is free of iteration. The estimated results are very accurate for stably identifying the restoring force under noise, which can be well used in the engineering of vibrational mechanics.