Vibration analysis of a non-linear beam subjected to moving loads by using the galerkin method

This paper presents the analysis of dynamic deflections of a beam, including the effects of geometric non-linearity, subjected to moving vehicle loads. The beam is assumed to be elastic and simply supported with immovable ends and the vehicles are assumed to be single degree of freedom spring-mass-damper. With the vehicles moving on the beam from one end to the other, the dynamic reflections of the beam and vehicles are computed by using the Galerkin method. Thus the dynamic deflections are assumed to be a set of time functions multiplied by approximate functions, respectively, and the time functions are numerically computed by solving the non-linear differential equations by the Newmark-β method.     

Codimension-two bifurcation of axial loaded beam bridge subjected to an infinite series of moving loads

A novel model is proposed which comprises of a beam bridge subjected to an axial load and an infinite series of moving loads. The moving loads, whose distance between the neighbouring ones is the length of the beam bridge, coupled with the axial force can lead the vibration of the beam bridge to codimension-two bifurcation. Of particular concern is a parameter regime where non-persistence set regions undergo a transition to persistence regions. The boundary of each stripe represents a bifurcation which can drive the system off a kind of dynamics and jump to another one, causing damage due to the resulting amplitude jumps. The Galerkin method, averaging method, invertible linear transformation, and near identity nonlinear transformations are used to obtain the universal unfolding for the codimension-two bifurcation of the mid-span deflection. The efficiency of the theoretical analysis obtained in this paper is verified via numerical simulations.     

Axisymmetric dynamic behaviour of thick plate subjected to impulsive loads

Analysis is carried out for the case where a thick circular plate is subjected to impulsive load. The horizontal displacement u is assumed to be in the form of u = zu₁ + zⁿu_n, where n is an odd integer. The fundamental equations governing the displacement are introduced, with use of equilibrium conditions. The relationships between frequencies and R/h (radius/thickness) are obtained and they are compared with those given by Mindlin theory. For the case where the fundamental equation is introduced by the variational calculus, it is necessary to notice whether restrained conditions are satisfied or not.     

Vibration of curved plate assemblies subjected to membrane stresses

In a previous paper [1] the finite strip method was applied to the prediction of the natural frequencies of vibration of longitudinally invariant, rigidly connected assemblies of circularly curved and flat strips having diaphragm end supports. This work is extended here to include the presence of an initial membrane stress field. An individual curved strip may be subjected to a biaxial direct stress field comprising a uniform stress acting in the circumferential direction and a non-uniform stress acting in the longitudinal direction. The presence of the membrane stress field is accommodated in the analysis by the inclusion of an initial stress or geometric stiffness matrix. A further extension included here is a facility to delete in-surface inertia terms. Results are presented for the application of the strip method in predicting the frequencies of vibration of a circular cylinder subjected to a complicated membrane stress system.     

Suspended bridges subjected to moving loads and support motions due to earthquake

The paper deals with the vibration of suspended bridges subjected to the simultaneous action of moving loads and vertical support motions due to earthquake. The basic partial integro-differential equation is applied to the vertical vibration of a suspended beam. The dynamic actions of traffic loads are modelled as a row of equidistant moving forces, while the earthquake is considered by vertical motions of supports. The governing equation is solved first analytically to receive an ordinary differential equation and next numerically. Moreover, the designed world's largest suspended bridge—Messina Bridge—is investigated (central span of length 3.3 km). The paper studies the effect of various lags of the earthquake arrival because the earthquake may appear at any time when the train moves along a large-span bridge. The modified Kobe earthquake records have been applied to calculations. The results indicate that the interaction of both the moving and seismic forces may substantially amplify the response of long-span suspended bridges in the vicinity of the supports and increase with the rising speed of trains.     

Dynamic stability of thermoelastic coupling moving plate subjected to follower force

The dynamic characteristics and stability of the moving thermoelastic coupling rectangular plate subjected to uniformly distributed tangential follower force are investigated. Based on the heat conduction equation containing the thermoelastic coupling term and the thin plate theory, the thermoelastic coupling differential equation of motion of the rectangular plate under the action of uniformly distributed tangential follower force is established. Dimensionless complex frequencies of the moving thermoelastic coupling rectangular plate with four edges simply supported, two opposite edges simply supported and other two edges clamped are calculated by the differential quadrature method. The effects of the dimensionless thermoelastic coupling factor and dimensionless moving speed on the stability and critical load of the moving plate are analyzed. The results show that the divergence loads of the first order mode increase with the increase of the dimensionless thermoelastic coupling factor, and decrease with increasing the dimensionless moving speed.     

Vibration and stability of a free circular plate subjected to non-conservative loading

This paper is concerned with the stability and vibration of a completely free circular plate subjected to a non-conservative edge loading. The eigencurves and mode shapes of the plate are obtained for various values of the non-conservativeness parameter. Numerical results are presented for the asymmetrical mode shapes of the plate. Interesting conclusions are drawn from these results some of which are verified analytically.     

Vibration of Timoshenko beam on hysteretically damped elastic foundation subjected to moving load

The vibration of beams on foundations under moving loads has many applications in several fields, such as pavements in highways or rails in railways. However, most of the current studies only consider the energy dissipation mechanism of the foundation through viscous behavior; this assumption is unrealistic for soils. The shear rigidity and radius of gyration of the beam are also usually excluded. Therefore, this study investigates the vibration of an infinite Timoshenko beam resting on a hysteretically damped elastic foundation under a moving load with constant or harmonic amplitude. The governing differential equations of motion are formulated on the basis of the Hamilton principle and Timoshenko beam theory, and are then transformed into two algebraic equations through a double Fourier transform with respect to moving space and time. Beam deflection is obtained by inverse fast Fourier transform. The solution is verified through comparison with the closed-form solution of an Euler-Bernoulli beam on a Winkler foundation. Numerical examples are used to investigate:(a) the effect of the spatial distribution of the load, and(b) the effects of the beam properties on the deflected shape, maximum displacement, critical frequency, and critical velocity. These findings can serve as references for the performance and safety assessment of railway and highway structures.     

Vibration serviceability of stadia structures subjected to dynamic crowd loads: A literature review

This paper is a critical review of information pertinent to the behaviour of stadia structures subjected to dynamic crowd loading. It is intended to introduce and explain key concepts in the field as well as summarise the development of current guidance and methods for modelling and assessing crowd occupancy.The review is structured by presenting a field overview and then rationalising the materials found into a standard vibration serviceability problem defined by the vibration source, path and receiver. After briefly introducing the problems associated with dynamic crowd loading, the loads generated by various actions of occupants are discussed. This leads into methods for modelling these activities. The derivation of relevant dynamic structural properties, through both FE modelling and field dynamic testing, is briefly investigated. The concept of human-structure interaction, the merging of dynamic properties of occupants and structure, is considered in depth.Finally, methods for assessing vibration levels are discussed and currently available codes and guidance are appraised in the context of the issues outlined.Further work is required at all stages of the excitation/source (crowd load models which account for flexible structures and large, distributed crowds), system/path (methods for accounting for the effects of occupancy upon system dynamic properties) and response/receiver (occupant tolerance levels) serviceability assessment approach in order to be able to understand and model the outlined phenomena accurately.     

Vibration and stability of a variable thickness annular plate subjected to a torque

The free vibration and stability of a variable thickness annular plate subjected to a torque are analyzed by the Ritz method. For this purpose, the transverse deflection of an annular plate is written in a series of the deflection functions of a uniform thickness annular plate without the action of a torque. The kinetic and strain energies of the plate are evaluated analytically and the frequency equation of the plate is derived by the conditions for a stationary value of the Lagrange functional. The present method is applied to annular plates with two types of radial thickness variation, power law and exponential, and the natural frequencies (the frequency parameters) and the divergence torques are calculated numerically, from which the effects of the varying thickness, inner/outer radii ratio and edge conditions are studied.     

Vibration analysis of horizontal self-weighted beams and cables with bending stiffness subjected to thermal loads

This paper aims at proposing an analytical model for the vibration analysis of horizontal beams that are self-weighted and thermally stressed. Geometrical nonlinearities are taken into account on the basis of large displacement and small rotation. Natural frequencies are obtained from a linearization of equilibrium equations. Thermal force and thermal bending moment are both included in the analysis. Torsional and axial springs are considered at beam ends, allowing various boundary conditions. A dimensionless analysis is performed leading to only four parameters, respectively, related to the self-weight, thermal force, thermal bending moment and torsional spring stiffness. It is shown that the proposed model can be efficiently used for cable problems with small sag-to-span ratios (typically , as in Irvine's theory). For beam problems, the model is validated thanks to finite element solutions and a parametric study is conducted in order to highlight the combined effects of thermal loads and self-weight on natural frequencies. For cable problems, solutions are first compared with existing results in the literature obtained without thermal effects or bending stiffness. Good agreement is found. A parametric study combining the effects of sag-extensibility, thermal change and bending stiffness is finally given.     

Vibration and parametric excitation in asymmetric circular plates under moving loads

The natural frequencies and modes of transverse vibration of circular plates containing small imperfections are determined through a perturbation method. Incision of equally spaced, equal-size radial slots at the rim of the plate creates asymmetry in some, but not all, of the vibration modes, and it causes the repeated natural frequencies of these modes in the symmetric plate to split into two distinct values. These vibration modes are called the split modes, and those associated with the repeated natural frequencies are called the repeated modes. A relationship identifying the split and repeated modes for any configuration of slots is presented. The vibration of a plate containing any number of thin slots cut into it at the rim and with any number of rotating linear springs is analyzed. Parametric instability can be excited in the split modes of the plate by the springs rotating below critical speed, but it cannot be excited in the repeated modes. The response of the plate in forms such as traveling or standing waves at parametric resonance is discussed. The theoretical predictions of split and repeated vibration modes and of the excitation of parametric instability are confirmed by experiments.     

Flow past a continuously moving flat plate in transverse magnetic field

The analysis of flow past a continuously moving semi-infinite flat plate in the presence of transverse magnetic field has been presented. Model solutions have been derived and the resulting equations are integrated numerically. Velocity profiles are displayed graphically and the numerical values of the transverse velocity skin friction are given in the form of tables. It has been observed that an increase inM (magnetic field parameter) leads to a decrease in velocity.The author wishes to express his thanks to Dr. H. N. Siddiquie, Director, and Dr. J. S. Sastry, Head, Physical Oceanography Division, for their keen interest and encouragement throughout this work.     

Dynamic analysis of a rectangular plate subjected to multiple forces moving along a circular path

A multi-degree-of-freedom (m.d.o.f.) system excited by a rough moving surface has been developed to study friction-induced oscillations. The normal degrees of freedom allow for oscillatory normal forces, while the normal-tangential coupling of friction produces parametric excitation in the slipping equations of motion. After a modal change of variables, first order averaging has been used to produce a set of autonomous equations of motion. Eigenvalue analysis of the averaged equations has produced stability predictions for the steady sliding position. Numerical integration of the original system of equations has verified the existence of locally unstable oscillations for a system excited by a rough surface input. The combination of velocity-dependent friction and a harmonically varying normal force have been shown to produce large-amplitude oscillations, in some cases leading to stick-slip responses.     

The electrostatic capacitance of an inclined plate capacitor

Considering the dimension and relative position of the two plates, the capacitance of an inclined plate capacitor is precisely calculated. Conformal mappings are employed and elliptic functions are used for achieving the general result. Taking the fringing effect into account, the inner capacitance and outer capacitance of the inclined angle are added to obtain a reliable result. The expression for computing the fringing capacitance of a parallel-plate capacitor and the capacitance of a parallel-strip transmission line in a plane are presented as special cases.     

Nonlocal plate model for nonlinear bending of single-layer graphene sheets subjected to transverse loads in thermal environments

This paper investigates the nonlinear bending behavior of a single-layer rectangular graphene sheet subjected to a transverse uniform load in thermal environments. The single-layer graphene sheet (SLGS) is modeled as a nonlocal orthotropic plate which contains small scale effect. Geometric nonlinearity in the von Kármán sense is adopted. The thermal effects are included and the material properties are assumed to be size dependent and temperature dependent, and are obtained from molecular dynamics (MD) simulations. The small scale parameter e ₀ a is estimated by matching the deflections of graphene sheets observed from the MD simulation results with the numerical results obtained from the nonlocal plate model. The numerical results show that the temperature change as well as the aspect ratio has a significant effect on the nonlinear bending behavior of SLGSs. The results reveal that the small scale parameter reduces the static large deflections of SLGSs, and the small scale effect also plays an important role in the nonlinear bending of SLGSs.     

Response of an elastically supported plate strip to a moving load

A theoretical analysis is presented of the steady state response of a plate strip constrained elastically along its edges against rotation and translation under the action of a moving transverse line load. Within the classical plate theory the solutions are obtained by using Fourier and Laplace transformation methods with respect to space variables. Numerical results are given for a plate strip with both edges identically constrained and a normal line load of constant intensity travelling along the plate strip with a constant speed. The first five speeds of the applied load for which a resonance effect occurs in the system are plotted as functions of the edge constraint parameters. The profiles of the displacement and the moment of the plate are also shown graphically for several values of the load speed and the edge constraint parameters.