The Stability of Parametrically Excited Systems: Coexistence and Trigonometrification
This dissertation addresses questions regarding the stability of two degree of freedom oscillating systems. The systems being discussed fall into three classes. The first class we discuss has the property that one of the non-linear normal modes (NNM) has a harmonic solution, x(t)=A cos t. For this class, the equation governing the stability of the system will be a second order differential equation with parametric excitation. Mathieu's equation (1), or more generally Ince's equation (2), are standard examples of such systems. x'' + (d + e*cos t)x = 0 (1) (1+a*cos t)x''+(b*sin t)x'+(d+c*cos t)x=0 (2) For Ince's equation we know that the stability portraits have tongues of instability defined by two transition curves. When these two transition curves overlap, the unstable region disappears and we say that the hidden tongue is coexistent. In this thesis we obtain sufficient conditions for coexistence to occur in stability equations of the form (1+a1*cos t+a2*cos 2t+...+an*cos nt)x''+(b1*sin t+b2*sin 2t+...+bn*sin nt)x'+(d+c1*cos t+c2*cos 2t+...+cn*cos nt)x=0 (3) Ince's equation has no damping. For the second class of systems, we seek to understand how dissipation affects coexistence. Here the analysis focuses on the behavior of coexistence as damping (mu) is added. Our analysis indicates coexistence is not possible in a damped Ince equation (4). (1+a*cos t)x''+(mu+b*sin t)x'+(d+c*cos t)x=0 (4) The previous two classes address systems with a harmonic NNM. The third class of systems treated in this thesis involve two degree of freedom systems that have a periodic NNM, not in general harmonic. To accomplish this we rescale time such that the periodic solution to the NNM is transformed into the form x(tau)=A0+A1 cos(tau). We call this procedure of rescaling time trigonometrification. The power of trigonometrification is that it is exact, requiring no approximations and produces a stability equation in new time (tau) of the form (1+a1*cos tau+a2*cos 2*tau+...+an*cos n*tau)x''+(b1*sin tau+b2*sin 2*tau+...+bn*sin n*tau)x'+(d+c1*cos tau+c2*cos 2*tau+...+cn*cos n*tau)x=0 (5) Trigonometrification can be used to study any system property that is invariant under a time transformation.
Coexistence; Trigonometrification; Parametric excitation; Time transformations; stability tongue; stability
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