The problem of acquiring elastic properties of a composite material from the data of the vibrational testing is considered. The specimen is considered to abide by the linear elasticity laws and subject to viscous-like damping. The BVP for transverse movement of such a specimen under harmonic load is formulated and solved with finite-element method. The problem of acquiring the elastic parameters is then formulated as a nonlinear leastsquare optimization problem. The usage of the automatic differentiation technique for stable and efficient computation of the gradient and hessian allows to use well-studied first and second order optimization methods, namely Newton's and BFGS. The results of the numerical experiments on simulated data are analyzed in order to provide insights for the experiment planning.