Modal interaction of resonantly forced oscillations of two-degree-of-freedom structure.
The author studies the nonlinear response of internally resonant two-degree-of-freedom systems with quadratic and cubic nonlinearities to harmonic external excitations. More precisely, if \(X_{1}\) and \(X_{2}\) are two generalized coordinates describing the motion of the system, it is assumed that the kinetic energy can be expressed as a polynomial of the form \(T = \frac{1}{2}M_{1}(1+m_{1}X_{1}+m_{2}X_{2})\dot{X}_{1}^{2} + \frac{1}{2}M_{2}(1+m_{3}X_{1}+m_{4}X_{2})\dot{X}_{2}^{2}+ M_{3}X_{1}\dot{X}_{1}\dot{X}_{2} + M_{4}X_{2}\dot{X}_{1}\dot{X}_{2},\) and that the potential energy has the form \(V = d_{1}X_{1}^{2} + d_{2}X_{2}^{2} + \frac{1}{4} d_{3}X_{1}^{4} + d_{4} X_{1}^{3}X_{2} + \frac{3}{2}d_{5} X_{1}^{2}X_{2}^{2} + d_{6} X_{1}X_{2}^{3} + \frac{1}{3} d_{7} X_{2}^{4}.\) Moreover, the author considers the effect of weak damping described by the dissipation function \(D = \frac{1}{2} c_{1}\dot{X}_{1}^{2} + \frac{1}{2} c_{2}\dot{X}_{2}^{2}.\) In the proof, the method of multiple scales is used to determine four first-order ordinary differential equations which describe the modulation of amplitudes and phases caused by resonances. Also, numerical results are presented.
scientific article; zbMATH DE number 1861296
scientific article