The response of two-degree-of-freedom systems with quadratic non-linearities to a parametric excitation

The method of multiple scales is used to analyze the response of two-degree-of-freedom systems with quadratic non-linearities to a parametric harmonic excitation having the frequency Ω. Four ordinary differential equations are derived to describe the modulation of the amplitudes and the phases when ω₂ ≈ 2ω₁ and either $\text{Ω ≈ 2ω}{}_1$ or $\text{Ω ≈ 2ω}{}_2$, where ω₁ and ω₂ are the linear undamped natural frequencies of the system. Two critical values ζ₁ and ζ₂ of the amplitude F of the excitation are identified in the analysis. When F >ζ₂, the amplitude of the directly excited mode grows exponentially with time according to the linear analysis, whereas the amplitudes of both modes achieve steady state constant values, irrespective of the initial amplitudes, according to the non-linear analysis. When F < ζ₁, the motion decays to zero according to both the linear and non-linear analyses. When ζ₁ ? F ? ζ₂, the motion decays to zero according to the linear analysis, whereas it achieves a periodic steady state or decays to zero depending on the initial amplitudes according to the non-linear analysis. This is an example of subcritical instability. When $\text{Ω ≈ 2ω}{}_2$, the steady state value of the higher mode, which is directly excited, is a constant that is independent of the excitation of amplitude F, whereas the amplitude of the lower mode, which is indirectly excited through internal resonance, grows with the excitation amplitude F. This is another example of saturation.