Large free vibrations of a beam carrying a moving mass

The focus of this work is to develop a technique to obtain numerical solution over a long range of time for non-linear multi-body dynamic systems undergoing large amplitude motion. The system considered is an idealization of an important class of problems characterized by non-linear interaction between continuously distributed mass and stiffness and lumped mass and stiffness. This characteristic results in some distinctive features in the system response and also poses significant challenges in obtaining a solution.

In this paper, equations of motion are developed for large amplitude motion of a beam carrying a moving spring–mass. The equations of motion are solved using a new approach that uses average acceleration method to reduce non-linear ordinary differential equations to non-linear algebraic equations. The resulting non-linear algebraic equations are solved using an iterative method developed in this paper. Dynamics of the system is investigated using a time-frequency analysis technique.     

Dynamic analysis of a tethered satellite system with a moving mass

This paper presents a dynamic analysis of a tethered satellite system with a moving mass. A dynamic model with four degrees of freedom, i.e., a two-piece dumbbell model, is established for tethered satellites conveying a mass between them along the tether length. This model includes two satellites and a moving mass, treated as particles in a single orbital plane, which are connected by massless, straight tethers. The equations of motion are derived by using Lagrange’s equations. From the equations of motion, the dynamic response of the system when the moving mass travels along the tether connecting the two satellites is computed and analyzed. We investigate the global tendencies of the libration angle difference (between the two sections of tether) with respect to the changes in the system parameters, such as the initial libration angle, size (i.e. mass) of the moving mass, velocity of the moving mass, and tether length. We also present an elliptic orbit case and show that the libration angles and their difference increase as orbital eccentricity increases. Finally, our results show that a one-piece dumbbell model is qualitatively valid for studying the system under certain conditions, such as when the initial libration angles, moving mass velocity, and moving mass size are small, the tether length is large, and the mass ratio of the two satellites is large.     

On the dynamic behaviour of a flexible beam carrying a moving mass

The behaviour of a system containing a mass traveling on a cantilever beam is considered. The mass is induced to move by an applied force as opposed to the case which has been considered in most literature where the position of the moving mass is assumed to be known and independent of the motion of the beam. Furthermore, the system to be discussed has the unique characteristic that the motions of the mass and the beam are coupled. The mathematical model of the system includes two coupled nonlinear integral/partial differential equations which are impossible to solve analytically and are difficult to solve numerically in their original form. As a remedy, the solution is discretized into space and time functions and the equations of motion are reduced to a set of ordinary differential equations. The shape function is chosen so that it satisfies the boundary conditions of the beam as well as the transient conditions imposed by the traveling mass. This choice of the shape function, which considers the mass-beam interaction, provides an improvement over the conventional method of using a simple cantilever beam mode shapes.The ordinary differential equations of motion using the improved shaped functions, are solved numerically to obtain the dynamic behaviour of the system. The results illustrate the validity of the model, and demonstrate the advantages of the improved model to the un-improved equations.     

On the rotational motion of a gyrostat about a fixed point with mass distribution

The perturbed rotational motion of a gyrostat about a fixed point with mass distribution near to Lagrange’s case is investigated. The gyrostat is subjected under the influence of a variable restoring moment vector, a perturbing moment vector, and a third component of a gyrostatic moment vector. It is assumed that the angular velocity of the gyrostat is sufficiently large, its direction is close to the axis of dynamic symmetry, and the perturbing moments are small as compared to the restoring ones. These assumptions permit us to introduce a small parameter. Averaged systems of the equations of motion in the first and second approximations are obtained. Also, the evolution of the precession angle up to the second approximation is determined. The graphical representations of the nutation and precession angles are presented to describe the motion at any time.     

Dynamics of viscoelastoplastic soil under a moving wheel

A rate-dependent inelastic behavior or viscoelastoplastic behavior of soil subjected to dynamic loading of a moving rigid wheel is studied herein. A rational approach to this seemingly complex problem is treated in the context of continuum mechanics. Together with the concept of critical state soil mechanics for inelastic behavior we incorporate the so-called internal state variables for viscous effects. It is assumed that the free energy functional containing inelastic behavior may not be smooth for its entire domain of time histories. Rather, we assume a form of discretized free energy as a function of elastic strains, plastic strains, and internal or hidden variables of incremental quantity considered to be valid only for a small time interval or a fraction of loading increments. The finite element method is used to derive governing equations of motion. Exhaustive numerical examples are presented to demonstrate effectiveness of the present method.     

关于点的合成运动速度合成定理两种推导的辨析

针对И.М.伏龙科夫著的理论力学教材中点的合成运动速度合成定理推导,指出推导过程在概念上和数学上出现的错误. 并给出正确的推导过程,以引起力学教学工作者注意.     

日地系统L_2点Halo轨道绳系卫星编队动力学

研究平动点附近周期轨道上旋转多体绳系卫星编队系统的非线性耦合动力学问题。编队系统为各卫星质量接近的轮辐状结构,位于日地系统第二平动点附近,整个系统的旋转保持系绳处于张紧状态,建立Hill限制性三体问题的绳系编队系统动力学模型。针对处于留位阶段的典型对称三星编队,在位于较大Halo轨道上无控制力作用的情况下,进行母卫星轨道运动与系绳摆动耦合运动的动力学模拟,分析轨道方向、卫星质量比、系绳长度以及初始旋转速度对编队系统整体稳定性的影响。     

Crack detection of a double-beam carrying a concentrated mass

This paper presents the influence of a concentrated mass location on the natural frequencies of a cracked double-beam. The double-beam consists of two different beams connected by an elastic medium. The concentrated mass is located on the main beam. The relationship between the natural frequency and the location of concentrated mass is established and called “Frequency–Mass Location” (FML). The numerical simulations show that when there is a crack, the frequency of the double-beam changes irregularly when the concentrated mass is attacked at the crack position. This irregular change can be amplified by the wavelet transform and this is useful for crack detection: the crack location can be detected by the location of peaks in the wavelet transform of the FML. Finite element model for the cracked double-beam carrying a concentrated mass is presented and numerical simulations are also provided.     

Dynamic analysis of suspended cables carrying moving oscillators

An efficient procedure for analyzing in-plane vibrations of flat-sag suspended cables carrying an array of moving oscillators with arbitrarily varying velocities is presented. The cable is modelled as a mono-dimensional elastic continuum, fully accounting for geometrical nonlinearities. By eliminating the horizontal displacement component through a standard condensation procedure, the nonlinear integro-differential equation governing vertical cable vibrations is derived. Due to the dynamic interaction at the contact points with the moving oscillators, such equation is coupled to the set of ordinary differential equations ruling the response of the travelling sub-systems. An improved series representation of vertical cable displacement is proposed, which allows to overcome the inability of the traditional Galerkin method to reproduce the kinks and abrupt changes of cable configuration at the interface with the moving sub-systems. Following the philosophy of the well-known “mode-acceleration” method, the convergence of the series expansion of cable response in terms of appropriate basis functions is improved through the introduction of the so-called “quasi-static” solution. Numerical results demonstrate that, despite the basis functions are continuous, the improved series enables to capture with very few terms the abrupt changes of cable profile at the contact points between the cable and the moving oscillators.     

On the motion of a material point on a moving surface

Summary Sufficient conditions are given for the stability and instability of the equilibrium position x=y=z=0 in the mechanical system consisting of a material point constrained to move on the moving surface z=−λ(t)(x²+y²) (λ(t)>0) in a constant field of gravity (the axis 0z is directed vertically upward) under the action of viscous friction of total dissipation.

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Sommario Si danno condizioni sufficienti per la stabilità e la instabilità della posizione di equilibrio x=y=z=0 nel sistema meccanico che consiste di un punto materiale vincolato a muoversi sulla superficie mobile z=−λ(t)(x²+y²) (λ(t)>0) in un campo di gravità costante (l'asse 0z è diretto verticalmente e orientato verso l'alto) sotto l'azione di attriti viscosi con dissipazione completa.

     

关于对动点的动量矩定理------理论力学札记之八

介绍质点系对动点的动量矩定理并讨论它的一些应用.     

On a theory concerning the dynamical behavior of structures carrying moving masses

Summary Two new methods are presented in order to determine the behavior of a structure carrying moving masses. The first method is analytic in nature and represents a modified asymptotic method in the theory of nonlinear phenomena. The second method is an exact numerical technique general enough to be used for solving exactly a set of differential equations with singular coefficients.The results show that the analytical method is in excellent agreement with the exact one obtained by means of a numerical technique. Furthermore, it is shown that the effect of the response of the structure to the moving mass has to be properly considered.

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Übersicht Es werden zwei neue Methoden zum Bestimmen des Verhaltens von Strukturen mit bewegten Massen behandelt. Die erste Methode trägt analytischen Charakter und kann als eine Modifikation der asymptotischen Methode der Theorie nichtlinearer Schwingungen betrachtet werden. Die zweite Methode basiert auf einer exakten numerischen Integration. Sie ist allgemein genug, um das beschreibende System gewöhnlicher Differentialgleichungen exakt zu lösen. Ein Vergleich der Ergebnisse zeigt, daß ausgezeichnete Übereinstimmung erreicht wird. Die Auswertungen zeigen, daß die Einwirkung des Struktur-Verhaltens auf die bewegte Massc berücksichtigt werden muß.

     

Dynamics of a prestressed incompressible layered half-space under moving load

The paper addresses a plane problem: a concentrated force acts on a plate resting on an elastic half-space with homogeneous prestrain. The equations of motion of the plate incorporate shear and rotary inertia. The half-space is assumed to be incompressible and isotropic in the natural state. The elastic potential is given in general form and is only specified for numerical purposes. The dependence of the critical velocity of the load and the stress-strain state on the prestresses is analyzed for different ratios between the stiffnesses of the layer and half-space and different contact conditions. The calculations are carried out for a half-space with Bartenev-Khazanovich potential __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 36–54, March 2008.