Budapest University of Technology and EconomicsFaculty of Civil Engineering - Since 1782

Nonlinear Normal Modes received increasing interest in  the vibration analysis of nonlinear structral dynamics recently. As a generalization of the linear normal modes of a multi degree of freedom system or a continuum, these vibration modes serve as a basis for many usual engineering tasks, e.g. the analysis of the resonant modes and frequencies, the analysis of the stability of the vibration. Dynamic properties of the structures play important role in structural diagnostic, thus for slender structures the effects of nonlinearities must be considered in those tasks too.

In the civil engineering praxis the nonlinearity often occurs in the form of a piecewise linear behaviour. Closing and opening gaps, slackening cables are a few examples on structural elements, which cause the separation of the structural behaviour into piecewise linear states, where the stiffness is different. Still, each separated state can be treated as a linear elastic one inside its range of interpretation, which allows the usage of modal analysis. When the stiffness of the structure changes at the boundary of the range of interpretation, the current displacement and velocity field of the structure must be transformed from one basis to the other. During a numerical calculation the transformation occurs between two finite dimensional spaces formed by the basis of the different modal shapes, thus there will be an energy loss at every state change called numerical damping. In the case of free vibration this numerical damping can be bounded between an upper and lower value based on the soultion of a generalized eigenvalue problem. In the current research the same approach must be extended to the forced vibration problems. Based on the self-similarity constraint on the vibration modes an algorithm must be formed to obtain nonlinear normal modes of periodically forced piecewise linear structures. From the further analysis of the results an equivalent coefficient of resonance must be calculated. The equivalent damping and period time of the steady-state vibration must be compared to the numerical damping and the period times of free vibration normal modes.