Subharmonic responses in harmonically excited rectangular plates with one-to-one internal resonance

Non-linear flexural vibrations of a rectangular plate with uniform stretching are studied for the case when it is subharmonically excited at nearly three times one of the linear natural frequencies. The interest here is in the subharmonic response of the plate when two distinct linear modes are near one-to-one internal resonance. Using the method of averaging, it is shown that, depending on the spatial distribution of the external forces, the plate can undergo harmonic motions, subharmonic motions in the directly excited spatial mode, or subharmonic motions in which both the internally resonant modes participate. The coupled-mode subharmonic oscillations can also undergo Hopf bifurcation to complicated amplitude-modulated subharmonic solutions which exhibit period-doubling route to chaos. Numerical results are presented specifically for one-to-one resonance in the (1,2) and (3,1) plate modes. The two-mode discretization of the von Karman plate equations are also directly simulated to verify some of the predictions of the averaged equations.