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This article explores the nonlinear vibration of beams with different types of nonlinearities. The beam vibration was modeled using Hamilton’s principle, and the equation of motion was solved using method of multiple time scales. Three models were developed assuming (a) geometric nonlinearity, (b) material nonlinearity and (c) combined geometric and material nonlinearity. The material nonlinearity also included both third and fourth nonlinear elasticity terms. The frequency response equation of these models were further evaluated quantitatively and qualitatively. The models capture the hardening effect, i.e., increase in resonant frequency as a function of forcing amplitude for geometric nonlinearity, and the softening effect, i.e., decrease in resonant frequency for material nonlinearity. The model is applied on the first three bending modes of the cantilever beam. The effect of the fourth-order material nonlinearity was smaller compared to the third-order term in the first mode, whereas it is significantly larger in second and third mode. The combined nonlinearity models shows a discontinuous frequency shift, which was resolved by utilizing a set of transition assumptions. This results in a smooth transition between the material and geometric zones in amplitude. These parametric models allow us to fine tune the nonlinear response of the system by changing the physical properties such as geometry, linear and nonlinear elastic properties.
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Energy Exchange between Two Sub-systems Coupled with a Nonlinear Elastic Path
The chief objective of this paper is to explore energy transfer mechanism between the sub-systems that are coupled by a nonlinear elastic path. In the proposed model (via a minimal order, two degree of freedom system), both sub-systems are defined as damped harmonic oscillators with linear springs and dampers. The first sub-system is attached to the ground on one side but connected to the second sub-system on the other side. In addition, linear elastic and dissipative characteristics of both oscillators are assumed to be identical, and a harmonic force excitation is applied only on the mass element of second oscillator. The nonlinear spring (placed in between the two sub-systems) is assumed to exhibit cubic, hardening type nonlinearity. First, the governing equations of the two degree of freedom system with a nonlinear elastic path are obtained. Second, the nonlinear differential equations are solved with a semi-analytical (multi-term harmonic balance) method, and nonlinear frequency responses of the system are calculated for different path coupling cases. As such, the nonlinear path stiffness is gradually increased so that the stiffness ratio of nonlinear element to the linear element is 0.01, 0.05, 0.1, 0.5 and 1.0 while the absolute value of linear spring stiffness is kept intact. In all solutions, it is observed that the frequency response curves at the vicinity of resonant frequencies bend towards higher frequencies as expected due to the hardening effect. However, at moderate or higher levels of path coupling (say 0.1, 0.5 and 1.0), additional branches emerge in the frequency response curves but only at the first resonant frequency. This is due to higher displacement amplitudes at the first resonant frequency as compared to the second one. Even though the oscillators move in-phase around the first natural frequency, high amplitudes increase the contribution of the stored potential energy in the nonlinear spring to the total mechanical energy. The out-of-phase motion around the second natural frequency cannot significantly contribute due to very low motion amplitudes. Finally, the governing equations are numerically solved for the same levels of nonlinearity, and the motion responses of both sub-systems are calculated. Both in-phase and out-of-phase motion responses are successfully shown in numerical solutions, and phase portraits of the system are generated in order to illustrate its nonlinear dynamics. In conclusion, a better understanding of the effect of nonlinear elastic path on two damped harmonic oscillators is gained.
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- Award ID(s):
- 1738723
- PAR ID:
- 10060449
- Date Published:
- Journal Name:
- NOVEM 2018 Conference
- Volume:
- 171785 - 2
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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