An iterative GL(n, R) method for solving non-linear inverse vibration problems

For the inverse vibration problem, a differential-algebraic equation (DAE) method is proposed to simultaneously estimate the time-dependent damping and stiffness coefficients by using two sets of displacement and velocity as input data. We combine the equations of motion and the supplemental data into a set of DAEs. We develop an implicit scheme and a Newton iterative algorithm to stably solve the DAEs to find the unknown structural coefficients. The unknown force is also recovered by the present method. A linear oscillator and a non-linear Duffing oscillator are used as testing examples. The estimated results are rather accurate and robust against random noise; hence, the new method can be used in the solutions of non-linear inverse vibration problems.