Nonlinear Vibration of an Elastic Soft String: Large Amplitude and Large Curvature

De-Min Zhao,Yun Zhang,Jian-Lin Liu,Shan-Peng Li

doi:10.1155/2018/7909876

De-Min Zhao, Yun Zhang + Show 2 more

Open Access

https://doi.org/10.1155/2018/7909876

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Abstract

Mechanical nonlinear vibration of slender structures, such as beams, strings, rods, plates, and even shells occurs extensively in a variety of areas, spanning from aerospace, automobile, cranes, ships, offshore platforms, and bridges to MEMS/NEMS. In the present study, the nonlinear vibration of an elastic string with large amplitude and large curvature has been systematically investigated. Firstly, the mechanics model of the string undergoing strong geometric deformation is built based on the Hamilton principle. The nonlinear mode shape function was used to discretize the partial differential equation into ordinary differential equation. The modified complex normal form method (CNFM) and the finite difference scheme are used to calculate the critical parameters of the string vibration, including the time history diagram, configuration, total length, and fundamental frequency. It is shown that the calculation results from these two methods are close, which are different with those from the linear equation model. The numerical results are also validated by our experiment, and they take excellent agreement. These analyses may be helpful to engineer some soft materials and can also provide insight into the design of elementary structures in sensors, actuators and resonators, etc.

Highlights

Mechanical vibration of slender structures, such as beams, strings, rods, plates, and even shells, occurs widely in a variety of areas, spanning from aerospace, automobile, cranes, ships, offshore platforms, and bridges to MEMS/NEMS [1,2,3,4,5]
At the nanoscale, Wang et al used a situ transmission electron microscopy (TEM) to measure the dynamic deflection of a cantilever made of multiwalled carbon nanotube, which was excited to resonance in TEM [7, 8]
The nonlinear vibration of a soft elastic string with large amplitude and large curvature has been systematically investigated in this study

Summary

Introduction

Mechanical vibration of slender structures, such as beams, strings, rods, plates, and even shells, occurs widely in a variety of areas, spanning from aerospace, automobile, cranes, ships, offshore platforms, and bridges to MEMS/NEMS (micro/nanoelectromechanical system) [1,2,3,4,5]. Vlajic and Fitzgerald et al [30] studied the prestressed beam with large variable curvature, and they provided the analytical formulation for static configurations, natural frequencies, and mode shapes, which were validated by the experiment and finite element method. We do not concentrate on the vibration of beams or plates but mainly on an elastic string or rod made of soft materials such as rubber materials, where large deformation and curvature can often happen when it vibrates. In this situation, the string will experience a very large displacement and especially a large curvature. The analysis is aiming to investigate the string vibration, the route of line can be extended to explore the vibration of some other elastic structures, such as rods, beams, plates, and shells

Model Formulation

Mode Discretization with Exact Mode Shape

Semianalytical Solution Using CNFM Method

Results and Discussion

Conclusion