Vibration of Flexible Structures Under Nonlinear Boundary Conditions

Abstract

The nonlinear response of a flexible structure, subjected to generally supported conditions with nonlinearities, is investigated for the first time. An analytical procedure is proposed first. Moreover, a simulation technique usually employed in static analysis is developed for confirmation. Generally, ordinary perturbation methods could analyze dynamics of flexible structures with linear boundary conditions. As nonlinear boundaries are taken into account, they are out of operation for the modal shape that is hardly to be obtained, which is the key to the analysis. In order to overcome this, nonlinear boundary conditions are rescaled and the technique of modal revision is employed. Consequently, each governing equation with different time-scales could be analyzed exactly according to corresponding rescaled boundary conditions. The total response of any point at the flexible structure will be composed by harmonic responses yielded by the analytical method. Furthermore, the differential quadrature element method (DQEM), a numerical simulation technique could satisfy boundary conditions strictly, is introduced to certify analytical results. The comparison shows a reasonable agreement between these two methods. In fact, the accuracy of the analytical method for nonlinear boundaries could be explained in theory. Based on the certification, boundary nonlinearities are discussed in detail analytically and found to play an important role in responses. Because of the important role played by the nonlinear factors in the vibration and control of the flexible structure, this paper will open the vibration analysis and numerical study of the flexible structure with nonlinear constraints.

In-Plane Free Vibration of Functionally Graded Rectangular Plates with Elastic Restraint

“…In this section, the DQM is used to solve the equations. In DQM, we can express the derivative of a function at a given point as a weighted linear summation of the values of the function at all the points in the domain [29]. Therefore, the differential equation can be transformed into a set of algebraic equations expressed by the value of the function at the node.…”

Study on Vibration Response of a Non-Uniform Beam With Nonlinear Boundary Condition

“…Song uses the theorem of infinitely many critical points to study the existence of infinitely many solutions to the boundary value problem of elastic beam deflection on a fourth-order nonlinear elastic foundation that depends on two real parameters [29]. Mao et al studied the nonlinear response of flexible structures with nonlinear general support conditions by using the modal revision method [30]. Rahman et al studied the forced nonlinear vibration of Euler Bernoulli beam on nonlinear elastic foundation by using the improved multilevel residual harmonic balance method [31].…”

Post-Buckling Spring Vibration Isolator Using Silicone Gel Column: A Theoretical and Experimental Study

“…The generation of harmful vibration is inevitable, which reduces the performance of machines used in engineering [1][2][3]. Harmful vibration during engineering not only affects the use and working efficiency of precision instruments but also leads to precision instruments being damaged [4].…”

“…Both simulation and experiments show that the aforementioned nonlinear design methods can achieve better vibration isolation effects and have a high bearing capacity in the corresponding frequency band. Even so, the nonlinearity of this kind of special design needs higher application requirements and is hard to control, evaluate, and realize [2,[13][14][15]. Based on recent research investigating the contradiction between low stiffness and low static load in vibration isolation theory, we aim to analyze and design the isolator in a manner that reduces stiffness and increases its bearing capacity, thus improving the vibration isolation performance of the isolator.…”