Transient dynamics, damping, and mode coupling of nonlinear systems with internal resonances

Abstract

The study of both linear and nonlinear structural vibrations routinely circles the concise yet complex problem of choosing a set of coordinates which yield simple equations of motion. In both experimental and mathematical methods, that choice is a difficult one because of measurement, computational, and interpretation difficulties. Often times, researchers choose to solve their problems in terms of linear, undamped mode shapes because they are easy to obtain; however, this is known to give rise to complicated phenomena such as mode coupling and internal resonance. This work considers the nature of mode coupling and internal resonance in systems containing non-proportional damping, linear detuning, and cubic nonlinearities through the method of multiple scales as well as instantaneous measures of effective damping. The energy decay observed in the structural modes is well approximated by the slow-flow equations in terms of the modal amplitudes, and it is shown how mode coupling enhances the damping observed in the system. Moreover, in the presence of a 3:1 internal resonance between two modes, the nonlinearities not only enhance the dissipation, but can allow for the exchange and transfer of energy between the resonant modes. However, this exchange depends on the resonant phase between the modes and is proportional to the energy in the lowest mode. The results of the analysis tie together interpretations used by both experimentalists and theoreticians to study such systems and provide a more concrete way to interpret these phenomena.

Appendix

The coefficients $I_{ijkl}^{\bullet ,\bullet }$ identified in Eq. (24) are defined as the Fourier components of terms in the slow-flow equations arising from the cubic nonlinearities. With $\psi _{q} \equiv \omega _{q} \, \eta _{0} + \phi _{q}^{(0)}$, these reduce to

$$\begin{aligned} I_{ijkl}^{s,\cos }&\equiv \frac{\omega _{i}}{\pi } \, \int _{0}^{2 \, \pi /\omega _{i}} \left\{ \cos \psi _{i} \, \sin \psi _{j} \, \sin \psi _{k} \, \sin \psi _{l} \right\} \, d\eta _{0},\end{aligned}$$

(42a)

$$\begin{aligned} I_{ijkl}^{c,\cos }&\equiv \frac{\omega _{i}}{\pi } \, \int _{0}^{2 \, \pi /\omega _{i}} \left\{ \cos \psi _{i} \, \cos \psi _{j} \, \cos \psi _{k} \, \sin \psi _{l} \right\} \, d\eta _{0},\end{aligned}$$

(42b)

$$\begin{aligned} I_{ijkl}^{v,\cos }&\equiv \frac{\omega _{i}}{\pi } \, \int _{0}^{2 \, \pi /\omega _{i}} \left\{ \cos \psi _{i} \, \sin \psi _{j} \, \sin \psi _{k} \, \cos \psi _{l} \right\} \, d\eta _{0},\end{aligned}$$

(42c)

$$\begin{aligned} I_{ijkl}^{d,\cos }&\equiv \frac{\omega _{i}}{\pi } \, \int _{0}^{2 \, \pi /\omega _{i}} \left\{ \cos \psi _{i} \, \cos \psi _{j} \, \cos \psi _{k} \, \cos \psi _{l} \right\} \, d\eta _{0},\end{aligned}$$

(42d)

$$\begin{aligned} I_{ijkl}^{s,\sin }&\equiv \frac{\omega _{i}}{\pi } \, \int _{0}^{2 \, \pi /\omega _{i}} \left\{ \sin \psi _{i} \, \sin \psi _{j} \, \sin \psi _{k} \, \sin \psi _{l} \right\} \, d\eta _{0},\end{aligned}$$

(42e)

$$\begin{aligned} I_{ijkl}^{c,\sin }&\equiv \frac{\omega _{i}}{\pi } \, \int _{0}^{2 \, \pi /\omega _{i}} \left\{ \sin \psi _{i} \, \cos \psi _{j} \, \cos \psi _{k} \, \sin \psi _{l} \right\} \, d\eta _{0},\end{aligned}$$

(42f)

$$\begin{aligned} I_{ijkl}^{v,\sin }&\equiv \frac{\omega _{i}}{\pi } \, \int _{0}^{2 \, \pi /\omega _{i}} \left\{ \sin \psi _{i} \, \sin \psi _{j} \, \sin \psi _{k} \, \cos \psi _{l} \right\} \, d\eta _{0},\end{aligned}$$

(42g)

$$\begin{aligned} I_{ijkl}^{d,\sin }&\equiv \frac{\omega _{i}}{\pi } \, \int _{0}^{2 \, \pi /\omega _{i}} \left\{ \sin \psi _{i} \, \cos \psi _{j} \, \cos \psi _{k} \, \cos \psi _{l} \right\} \, d\eta _{0}, \end{aligned}$$

(42h)

The evaluation of these integrals depends on the relationship between the frequencies $(\omega _{i}, \omega _{j}, \omega _{k}, \omega _{l})$. Specifically, there exist terms that appear for any set of frequencies. For example,

$$\begin{aligned} I_{iiii}^{d,\cos } = \frac{3}{8}, \quad I_{iijj}^{d,\cos } = I_{ijij}^{d,\cos } = I_{ijji}^{d,\cos } = \frac{1}{4}, \end{aligned}$$

(43)

while $I_{ijkl}^{d,\cos } = 0$ otherwise. Likewise one can show that $I_{ijkl}^{d,\sin } \equiv 0$. However, additional terms arise when there exists a 3 : 1 resonance between two modal frequencies. For example, if $\alpha$ and $\beta$ denote mode numbers such that $3 \, \omega _{\alpha } = \omega _{\beta }$, then there are additional components of $I^{d,\cos }$ and $I^{d,\sin }$ such that

$$\begin{aligned} \begin{aligned} I_{\beta \alpha \alpha \alpha }^{d,\cos } = I_{\alpha \beta \alpha \alpha }^{d,\cos } = I_{\alpha \alpha \beta \alpha }^{d,\cos } {=} I_{\alpha \alpha \alpha \beta }^{d,\cos }&{=} \frac{1}{8} \, \cos \left( \psi _{\alpha \beta }^{(0)} \right) ,\\ -I_{\beta \alpha \alpha \alpha }^{d,\sin } {=} I_{\alpha \beta \alpha \alpha }^{d,\cos } = I_{\alpha \alpha \beta \alpha }^{d,\cos } {=} I_{\alpha \alpha \alpha \beta }^{d,\cos }&{=} \frac{1}{8} \, \sin \left( \psi _{\alpha \beta }^{(0)} \right) , \end{aligned}\nonumber \\ \end{aligned}$$

(44)

where $\psi _{\alpha \beta }^{(0)} \equiv 3 \, \phi _{\alpha }^{(0)} - \phi _{\beta }^{(0)}$. These are in addition to the terms that exist for any set of frequencies. Similar conclusions hold for the remaining parameters.