Multiscale Analysis for Multiple Bifurcations of Continuous Nonconservative Mechanical Systems

A general, autonomous, continuous mechanical system, subjected to non conservative forces is considered and bifurcations from a known equilibrium path are analyzed. The system is assumed to lose stability at a bifurcation point at which several critical eigenvalues of the linear operator simultaneously assumes zero real part. Two cases can occur: (a) the critical eigenvalues are not coincident, so that the operator admits a complete set of eigenvectors (non-defective system) or (b) some of the eigenvalues are coincident, so that an incomplete set exists (defective system). In this latter case a chain of generalized eigenvector must built up to complete the base, extending to differential operators the well-known properties of nilpotent matrices. To this end, the adjoint operator and the associated boundary conditions are derived making use of the bilinear identity and the left eigenvectors evaluated. Several independent time scales depending on a perturbation parameter  are considered, and state-variables and parameters are expanded in series of . Integer power series must be used for non-defective systems while suitable fractional power series for defective systems. By solving the perturbation equations and enforcing solvability at each step, a set of p.d.e. governing the evolution of the unknown amplitudes on the various time scales are found. These can be combined to derive an o.d.e. bifurcation equation ruling the complex dynamics at the bifurcation point. To illustrate the method a planar elastic beam constrained at one end by a visco-elastic device is considered, loaded by a non conservative tangential force. The linear stability diagram is built up in a two-dimensional space of load parameters. It consists of a divergence boundary and a Hopf boundary, crossing each other at a double-zero point. There, the linear operator admits only one eigenvector, so that the system is defective (Takens-Bogdanova bifurcation). A generalized index-two eigenvector is found to complete the base. By applying the Multiple Scale Method, divergence, simple Hopf and double-zero bifurcations are studied and the whole scenario around the bifurcation is described.