Structure identification with respect to linear mechanical systems in general is reduced to identification of the type of damping and the order of the model, taking into account the mechanical principles for the equations of motion. It is more difficult to model the nonlinear behaviour of a system than a linear one. Often the structure of a model is chosen and its adequacy is then tested by physical and statistical significance.Starting with a brief review of structure identification and its properties, the nonlinearity under consideration is assumed to be modelled within the class of polynomials. Nonlinearities then deal with the restoring and damping forces. Instead of the usual parameter estimation of the polynomials with a priori assumed powers, the powers of the polynomials are to be determined along with their coefficients. One approach is based on approximation theory, the other defines a recursive procedure using a ratio test. The proposed techniques are illustrated through application to examples by simulation with different noise levels and it is shown that, even with noisy data, the given methods yield satisfactory results.