The response of multidegree-of-freedom systems with quadratic non-linearities to a harmonic parametric resonance

An analysis is presented of the response of multidegree-of-freedom systems with quadratic non-linearities to a harmonic parametric excitation in the presence of an internal resonance of the combination type ω₃ ≈ ω₂ + ω₁, where the ω_n are the linear natural frequencies of the systems. In the case of a fundamental resonance of the third mode (i.e., $\text{Ω ≈ω}{}_3$, where Ω is the frequency of the excitation), one can identify two critical values _{ζ1 and ζ2, where ζ2 ? ζ1}, of the amplitude F of the excitation. The value F = ζ₂ corresponds to the transition from stable to unstable solutions. When F < ζ₁, the motion decays to zero according to both linear and non-linear theories. When F >ζ₂, the motion grows exponentially with time according to the linear theory but the non-linearity limits the motion to a finite amplitude steady state. The amplitude of the third mode, which is directly excited, is independent of F, whereas the amplitudes of the first and second modes, which are indirectly excited through the internal resonance, are functions of F. When ζ₁ ? F ? ζ₂, the motion decays or achieves a finite amplitude steady state depending on the initial conditions according to the non-linear theory, whereas it decays to zero according to the linear theory. This is an example of subcritical instability. In the case of a fundamental resonance of either the first or second mode, the trivial response is the only possible steady state. When F ? ζ₂, the motion decays to zero according to both linear and non-linear theories. When F >ζ₂, the motion grows exponentially with time according to the linear theory but it is aperiodic according to the non-linear theory. Experiments are being planned to check these theoretical results.