On the existence of periodic solutions and normal mode vibrations of nonlinear systems

Modal Analysis of Mechanical Systems With Impact Non-Linearities: Limitations to a Modal Superposition

“…This formula is similar to the formula that obtained in the linear case, but the mass (which should be equal to 1) is replaced by a modal mass (11).…”

Normal modes, uncoupling, and stability for a class of nonlinear oscillators

“…there exists a canonical transformation (£, rj) = S(x, y) taking H(x(t), y(t)) to H(£(t), tj(*)), with £ and 17 satisfying i)-iii) in place of x and y respectively and such that Z = tj(/) Although this last condition does not necessarily eliminate all solutions which are not modes, it provides a computationally useful condition automatically satisfied by the classical modes of linear systems. We shall not investigate existence of modes in the most general setting, although it would be of considerable interest to know whether the modes shown to exist by Rosenberg [14,17,19] and Cooke and Struble [6] also satisfy our condition (iv). We shall instead apply this definition to the study of isoenergetic stability of some special modes of vibration for a class of simple conservative nonlinear oscillators.…”

Large-amplitude non-linear normal modes of piecewise linear systems

“…Since then, the existence [4,5], stability [6][7][8][9][10], and construction [11][12][13][14] of non-linear normal modes have been among the topics of investigation in this field. More recently, an alternative definition for NNMs was introduced by Shaw and Pierre [15][16][17], based on invariant manifolds.…”