Synge's concept [J.L. Synge, On the geometry of dynamics, Philos. Trans. R. Soc. London, Ser. A 226 (1926) 33–106] of stability is introduced and shown to be equivalent to the orbital stability in holonomic conservative systems of two-degrees-of-freedom. This furnishes an analytical tool to study the orbital stability in strongly non-linear systems. This concept is shown to be applicable to the stability analysis of non-linear normal modes, for which Liapunov's first method generally fails. Integrally related numbers are found such that, if the ratio of linear natural frequencies is close to one of the numbers, then a normal mode may lose stability at a small amplitude. These numbers depend on the symmetry or asymmetry of system with respect to the origin of the configuration space. Some examples are given to demonstrate the stability analysis of the normal modes and to verify the integrally related numbers.

Synge's concept of stability applied to non-linear normal modes

An analytical procedure is presented to find approximately the nonlinear normal modes in conservative two-degree-of-freedom system by using the principle of least action and by assuming that the modal curve is straight. The results are compared with those of numerical experiments by utilizing the 4th order Runge-Kutta method, and it is found that there are good agreements between them. By utilizing this procedure, it is demnostrated to compute the normal modes which are analytically extended from the linearized modes and to find the generically or non-generically bifurcated modes which do not have any counterpart in the linear theory.

An Approximate Method to Compute the Nonlinear Normal Modes and Bifurcation by the Principle of Least Action

In a nonlinear holonomic conservative system having two-degree-of-freedom, the modal curves of normal mode vibrations are investigated investimgated by the harmonic balance method. The general procedure to compute the modal curve is suggested. Even if the linearized frequencies of the system are satisfied with the commensurability condition under which the approaches using the perturbation method have the problem of small divisor, the modal curve can be obtained by this method, provided that the fundamental harmonics are dominant when the normal modes are expanded in Fourier series in time domain. As an example, in a system with cubic nonlinearity the modal curves are computed analytically and numerically to compare both results.

Sun Jae Park

Papers

On the Modal Curve of Nonlinear Normal Modes