| The influence of modal interactions on the response of harmonically excited flexible L-shaped metallic and composite structures has been investigated analytically and experimentally. Each metallic structure possesses a two-to-one internal resonance, while each composite structure possesses a three-to-one internal resonance and either a two-to-one or a one-to-one internal resonance. For the metallic structures, a weakly nonlinear analysis is used to derive the autonomous system of equations which describe the evolution of the amplitudes and phases of the internally resonant modes. These equations are obtained for primary- and secondary-resonant excitations. The excitation frequency or amplitude is used as a control parameter and the resulting bifurcations (saddle-node, pitchfork, and Hopf bifurcations) are studied. Theoretical analyses for internally resonant systems are used to predict and explain the responses of the composite structures.;In the experiments, during primary-resonant excitations, low excitation (mili g) levels are used, while during secondary-resonant excitations, high excitation (one to two g) levels are used. During primary-resonant excitations of the metallic structures, the saturation phenomenon and its breakdown, two-period quasiperiodic motions, and chaotically modulated motions are observed. The experimentally observed locations of jumps and transitions to quasiperiodic motions are in good agreement, respectively, with the analytically predicted pitchfork (saddle-node) and Hopf-bifurcation points. During subharmonic excitations of the higher mode, the metallic structure exhibits modal saturation and its breakdown, leading from periodic to modulated motions. The metallic structure also exhibits nonlinear periodic, quasiperiodic, and chaotically modulated responses to combination-resonant excitations. The experimental observations made during secondary-resonant excitations are in good agreement with the analytical predictions. All the observed responses of the metallic structures are planar. During primary-resonant excitations of a composite structure, nonlinear planar and nonplanar motions, nonplanar periodically and chaotically modulated motions, and modal saturated are observed. Nonlinear responses of a composite structure could not be excited during secondary-resonant excitations even at high excitation levels. In all cases, the transition from periodic to chaotically modulated motions occurred via quasiperiodic motions.;Analytical approximations for representative third-order and fourth-order autonomous systems are derived to study motions near the Hopf-bifurcation points of each of these systems. |
