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This work concerns linear parametrically excited systems that involve multiple resonances. The property of such systems is that if the parameters are fixed and lie inside a resonance tongue, the motion becomes unbounded as time goes to infinity. In this work we consider what happens when the parameters are not fixed, but rather are constrained to vary slowly in time, passing into and out of the resonance tongues. One might expect that during the time in which the motion lies inside a tongue the solution grows, and that the slower the passage through the tongue the more time is spent inside the tongue, and the larger the resulting growth. We show that this is not always the case. In particular we investigate the effect of initial conditions and relative forcing amplitudes on the growth or amplification of the solution. We address the problem of how to choose these parameters so as to minimize growth (i.e., to de-amplify the solution) after passage through multiple tongues.

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In aerospace and automotive industries, structural designs are required to meet specific weight limits whilst maintaining strength and stiffness. The material of choice for such applications is fiber-reinforced composites because of their high strength-to-weight and stiffness-to-weight ratios. Composite materials used in these applications are often subjected to random dynamic loads and if the excitation sources are large enough, excessive vibration can lead to unwanted noise levels and fatigue failure. A commonly used design tool for modeling the vibration transmission and power flow in structures excited by random forces is statistical energy analysis. In order to incorporate fiber-reinforced composites within a statistical energy analysis methodology, a detailed understanding of how the modal energy levels vary when excited by a broadband random excitation source is needed. In this paper, analytical expressions are derived for the modal energy levels of a fiber-reinforced composite beam coupled in bending and torsion subjected to spatial white noise or the so-called rain-on-the-roof loading. It is shown that the modal energy levels of the fiber-reinforced composite beam are markedly nonuniform across the modal spectrum. This finding is in contrast with the well-known classical result that an isotropic beam subjected to the same type of excitation is characterized by a uniform modal energy spectrum, a result sometimes referred to as the equipartition of energy. The equipartition of energy forms the basis of statistical energy analysis and the results of this work indicate that fiber-reinforced composite beams cannot be reliably included into a conventional statistical energy analysis methodology. It is also shown that the mechanism responsible for the nonuniform modal energy distribution is related to the physical process by which power is transmitted across the boundaries. Simulations are also carried out to illustrate that the distribution characteristics of the modal energy are identical for a range of classical boundary conditions.

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