Simulation of the transverse vibrations of a cantilever beam with an eccentric tip mass in the axial direction using integral transforms

The main purpose of the current work is to employ an integral transform approach based on eigenfunction expansion and on an implicit filter scheme in order to solve the governing equations for the transverse vibrations of a cantilever beam clamped at one end and with an eccentric tip mass in the axial direction at the other end. Numerical results are obtained for both the undamped and damped natural frequencies of the system, as well as for its transverse displacement due to arbitrarily time-varying load and imposed displacement at the clamped end. The numerical results reported in the current work are highly accurate and new in the literature. New exact results are also provided for the transient displacement and its higher-order spatial derivatives to allow computation of bending stresses and strains. The relative merits of the proposed approach are finally pointed out.     

Existence of solutions for a cantilever beam problem

We are concerned with the fourth-order nonuniform cantilever beam problem

(I(x)WΔ∇(x)⁾Δ∇=f(x,W(x)),     

具有强非线性源非牛顿多方渗流方程解的极限性质

研 究如下第一边值 问题ｕ ｔ ＝ ｄｉｖ（｜ Ｄｕ ｍ ｜ｐ － ２ Ｄｕｍ ） ＋ ｆ （ｘ ,ｕ）ｕ （ｘ ,ｔ） ＝ ０ｕ （ｘ ,０） ＝ ｕ０  ０ 　 　　　（ｘ ,ｔ） ∈ Ｑ Ｔ ＝ Ω× （０, Ｔ）（ｘ ,ｔ） ∈ Ω× （０, Ｔ）ｘ ∈ Ω解的极限性质 （ｔ → ∞）,推 广了文献［１ ～ ７］ 的结果     

Energy decay of solutions for an extensible beam equation with a weak nonlinear dissipation

In this paper, we study the decay property of the solutions to an extensible beam equation with a weak nonlinear dissipation. We establish an explicit and general decay result, depending on nonlinear function g and positive function σ, using some properties of convex functions. Copyright © 2012 John Wiley & Sons, Ltd.     

Blow-up of solutions for a nonlinear beam equation with fractional feedback

A nonlinear beam equation describing the transversal vibrations of a beam with boundary feedback is considered. The boundary feedback involves a fractional derivative. We discuss the asymptotic behavior of solutions. In fact, we prove that solutions blow up in finite time under certain assumptions on the nonlinearity.     

An analytical technique for solving a class of strongly nonlinear conservative systems

Recently, an analytical technique has been developed to determine approximate solutions of strongly nonlinear differential equations containing higher order harmonic terms. Usually, a set of nonlinear algebraic equations is solved in this method. However, analytical solutions of these algebraic equations are not always possible, especially in the case of a large oscillation. Previously such algebraic equations for the Duffing equation were solved in powers of a small parameter; but the solutions measure desired results when the amplitude is an order of 1. In this article different parameters of the same nonlinear problems are found, for which the power series produces desired results even for the large oscillation. Moreover, two or three terms of this power series solution measure a good result when the amplitude is an order of 1. Besides these, a suitable truncation formula is found in which the solution measures better results than existing solutions. The method is mainly illustrated by the Duffing oscillator but it is also useful for many other nonlinear problems.     

Positive solutions for a beam equation on a nonlinear elastic foundation

Positive solutions for a fourth-order differential equation with nonlinear boundary conditions modeling beams on elastic foundations are considered. The results are shown by using variational methods and a maximum principle for fourth-order equations.     

一类非线性奇摄动方程解的渐近表示

应用匹配渐近方法讨论一类非线性奇异摄动方程的边值问题解的渐近表示,得到了边界层或冲击层解的刻画,阐述了边界参数对边界层或冲击层位置的影响.     

Predicting nonlinear dynamic response of internal cantilever beam system on a steadily rotating ring

This paper is focused on nonlinear dynamic response of internal cantilever beam system on a steadily rotating ring via a nonlinear dynamic model. The analytical approximate solutions to the oscillation motion are obtained by combining Newton linearization with Galerkin's method. Numerical solutions could be obtained by using the shooting method on the exact governing equation. Compared with numerical solutions, the approximate analytical solutions here show excellent accuracy and rapid convergence. Two different kinds of oscillating internal cantilever beam system on a steadily rotating ring are investigated by using the analytical approximate solutions. These include symmetric vibration through three equilibrium points, and asymmetric vibration through the only trivial equilibrium point. The effects of geometric and physical parameters on dynamic response are useful and can be easily applied to design practical engineering structures. In particular, the ring angular velocity plays a significant role on the period and periodic solution of the beam oscillation. In conclusion, the analytical approximate solutions presented here are sufficiently precise for a wide range of oscillation amplitudes.     

Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary differential equations

Summary Conditions are given for the nonlinear differential equation (1)L _n y+f(t, y, ..., ...,y ^(n–1)=0to have solutions which exist on a given interval [t₀, )and behave in some sense like specified solutions of the linear equation (2)L _n z=0as t.The global nature of these results is unusual as compared to most theorems of this kind, which guarantee the existence of solutions of (1)only for sufficiently large t. The main theorem requires no assumptions regarding oscillation or nonoscillation of solutions of (2).A second theorem is specifically applicable to the situation where (2)is disconjugate on [t ₀, ),and a corollary of the latter applies to the case where Lz=z ⁿ.     

The investigation of free vibrations for an asymmetric beam equation using category theory

We investigate free vibrations for an asymmetric beam equation. We find them by applying linking inequalities and the limit relative category theory.     

Multi-harmonic measurements and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions

This study presents a direct comparison of measured and predicted nonlinear vibrations of a clamped–clamped steel beam with non-ideal boundary conditions. A multi-harmonic comparison of simulations with measurements is performed in the vicinity of the primary resonance. First of all, a nonlinear analytical model of the beam is developed taking into account non-ideal boundary conditions. Three simulation methods are implemented to investigate the nonlinear behavior of the clamped–clamped beam. The method of multiple scales is used to compute an analytical expression of the frequency response which enables an easy updating of the model. Then, two numerical methods, the Harmonic Balance Method and a time-integration method with shooting algorithm, are employed and compared one with each other. The Harmonic Balance Method enables to simulate the vibrational stationary response of a nonlinear system projected on several harmonics. This study then proposes a method to compare numerical simulations with measurements of all these harmonics. A signal analysis tool is developed to extract the system harmonics’ frequency responses from the temporal signal of a swept sine experiment. An evolutionary updating algorithm (Covariance Matrix Adaptation Evolution Strategy), coupled with highly selective filters is used to identify both fundamental frequency and harmonic amplitudes in the temporal signal, at every moment. This tool enables to extract the harmonic amplitudes of the output signal as well as the input signal. The input of the Harmonic Balance Method can then be either an ideal mono-harmonic signal or a multi-harmonic experimental signal. Finally, the present work focuses on the comparison of experimental and simulated results. From experimental output harmonics and numerical simulations, it is shown that it is possible to distinguish the nonlinearities of the clamped–clamped beam and the effect of the non-ideal input signal.     

A result on quasi-periodic solutions of a nonlinear beam equation with a quasi-periodic forcing term

In this paper, a quasi-periodically forced nonlinear beam equation \({u_{tt}+u_{xxxx}+\mu u+\varepsilon\phi(t)h(u)=0}\) with hinged boundary conditions is considered, where μ > 0, \({\varepsilon}\) is a small positive parameter, \({\phi}\) is a real analytic quasi-periodic function in t with a frequency vector ω = (ω ₁,ω ₂ . . . , ω _m ), and the nonlinearity h is a real analytic odd function of the form \({h(u)=\eta_1u+\eta_{2\bar{r}+1}u^{2\bar{r}+1}+\sum_{k\geq \bar{r}+1}\eta_{2k+1}u^{2k+1},\eta_1,\eta_{2\bar{r}+1} \neq0, \bar{r} \in {\mathbb {N}}.}\) The above equation admits a quasi-periodic solution.     

Convergence rates to asymptotic profile for solutions of quasilinear hyperbolic equations with nonlinear damping

The quasilinear hyperbolic equation with nonlinear damping is considered in this paper, a new asymptotic profile for the solution to the equation is obtained by suitably choosing the initial data of the corresponding parabolic equation, the convergence rates of the new profile are better than that obtained by Nishihara (1997, J. Differential Equations 137, 384–395) and H.-J. Zhao (2000, J. Differential Equations 167, 467–494).     

Quasi-periodic solutions to nonlinear beam equations on compact Lie groups with a multiplicative potential

The goal of this work is to study the existence of quasi-periodic solutions to nonlinear beam equations with a multiplicative potential. The nonlinearity is required to only finitely differentiable and the frequency is along a pre-assigned direction. The result holds on any compact Lie group or homogeneous manifold with respect to a compact Lie group, which includes standard torus ${\mathbb{T}}^d$, special orthogonal group $SO(d)$, special unitary group $SU(d)$, spheres ${\mathbb{S}}^d$ and the real and complex Grassmannians. The proof is based on a differentiable Nash–Moser iteration scheme.