Experimental Random Excitation of Nonlinear Systems with Multiple Internal Resonances

Abstract

Small nonlinear modal interactions may have considerable effect on the pattern of forced vibration response of certain types of dynamic systems possessing low damping characteristics. The design of these systems may create two or more linear algebraic relationships between the system’s normal mode frequencies. These relationships are referred to as internal (or autoparametric) resonances, and usually take the form £ $\sum {\rm K}_{\rm j} {\rm \omega}_{\rm j} = 0$. The constant ${\rm K} = \sum {\Bbb K}_{\rm j} {\rm l}$ is referred to as the order of internal resonance. where k_j are integers. and ω_j are the system normal mode frequencies. Third-order internal resonance results from quadratic nonlinear coupling of normal modes. while fourth-order is due to cubic nonlinearities. Cubic nonlinearities can also cause second-order internal resonance (i.e., one-to-one). Simultaneous internal resonances are classified as independent and interacting. Independent internal resonances usually take the form

$$\sum\limits_{\rm j}^{\rm m} {{\rm K}_{\rm j} {\rm \omega }_{\rm j} } = 0,\,\,\,\,\,\,\,\sum\limits_{\ell}^{\rm n} {{\rm K}_{\ell} {\rm \omega }_{\ell} } = 0$$

(1)

such that j=1,2,.., m. and l=m+1. m+2,..., n