We propose to study some differential systems based on previous experiments of vibrations in order to detect damage in an elastic structure. Several asymptotic expansions and numerical techniques are developped in this area. We use the vibrational response to an external actuation to try to distinguish between damaged and undamaged structures. The objective is to use very simple models in order to simulate some experimental results in [7]. The system is represented by an ordinary differentiel system that we solve with two methods of asymptotic expansion.They are compared to the numerical solution given by a step by step scheme with O.D.E package of SCILAB. The asymptotic developpement is justified with the average method . The comparison in the frequentiel plane is convenient since it provides for both approaches a qualitative and numerical interpretation