Moving Mass Problem Research Articles

Dynamic interaction and instability of two moving proximate masses on a beam on a Pasternak viscoelastic foundation

The study analyzes the dynamic interaction of proximate masses using a new form of semi-analytical results for the moving mass problem developed by the author. This paper presents the results for a particular case of two moving masses of equal value acted upon by constant forces of equal values. The interaction level was demonstrated to be substantially high, making it impossible to superimpose the results obtained for one mass unless the masses were located at a significantly large distance apart. In the undamped case, the onset of instability occurred at the critical velocity, as with one moving mass; however, the onset of instability could be shifted significantly into the subcritical velocity range in the damped case. The critical distance between the moving masses was determined as the value at which the exponential increase in the amplitudes was severe. The implementation of dimensionless parameters revealed that this distance was slightly dependent on the damping. Moreover, the limiting moving mass ratio for such unexpected instability was derived.The results were validated by eigenmode expansion analysis on long finite beams, for which computational time savings were proposed by the rearrangement of the terms involved. Excellent agreement between the results was obtained, validating the new formulae. Furthermore, extension to more general cases and additional moving masses can easily be achieved.

Semi-analytical analysis of vibrations induced by a mass traversing a beam supported by a finite depth foundation with simplified shear resistance

In this paper the new semi-analytical solution for the moving mass problem on massless foundation, published by the author of this paper, is extended to account for inertial foundation modelled by a continuous homogeneous finite depth foundation with simplified shear resistance. Derivations are presented for infinite as well as finite homogeneous beams. Mode expansion method is used to solve the problem on finite beams, thus vibration modes, the corresponding orthogonality condition, reengagement of coupled equations to ensure significant calculation time savings are derived. Methods of integral transforms and contour integration are exploited to obtain the solution on infinite beams. Resulting vibrations are derived as a sum of the steady and unsteady harmonic vibrations and a transient contribution. The unsteady harmonic vibration is proven to be a useful indicator of unstable behaviour through the mass induced frequencies. Besides frequency lines also discontinuity lines are determined and their influence on the proximity of harmonic and full solutions is discussed. Even if the differences between these two versions are larger than for the massless foundation, it is shown that the harmonic solution provides a very good estimate of the full solution (in several cases perfect match is achieved) with the advantage to be obtainable by a simple evaluation of the derived closed-form results. Like for the massless foundation, also here, vibrations on infinite beams can be obtained on long finite beams with eliminated effect of its supports. All mentioned approaches are also validated by the finite element method.

Parametric Vibration of a Flexible Structure Excited by Periodic Passage of Moving Oscillators

Abstract Flexible structures carrying moving subsystems are found in various engineering applications. Periodic passage of subsystems over a supporting structure can induce parametric resonance, causing vibration with ever-increasing amplitude in the structure. Instead of its engineering implications, parametric excitation of a structure with sequentially passing oscillators has not been well addressed. The dynamic stability in such a moving-oscillator problem, due to viscoelastic coupling between the supporting structure and moving oscillators, is different from that in a moving-mass problem. In this paper, parametric resonance of coupled structure-moving oscillator systems is thoroughly examined, and a new stability analysis method is proposed. In the development, a set of sequential state equations is first derived, leading to a model for structures carrying a sequence of moving oscillators. Through the introduction of a mapping matrix, a set of stability criteria on parametric resonance is then established. Being of analytical form, these criteria can accurately and efficiently predict the dynamic stability of a coupled structure-moving oscillator system. In addition, by the spectral radius of the mapping matrix, the global stability of a coupled system can be conveniently investigated in a parameter space. The system model and stability criteria are illustrated and validated in numerical examples.

Vibration under variable magnitude moving distributed masses of non-uniform Bernoulli-Euler beam resting on Pasternak elastic foundation

The dynamic response to variable magnitude moving distributed masses of simply supported non-uniform Bernoulli-Euler beam resting on Pasternak elastic foundation is investigated in this paper. The problem is governed by fourth-order partial differential equation with variable and singular coefficients. The main objective of this work is to obtain closed form solution to this class of dynamical problem. In order to obtain the solution, a technique based on the method of Galerkin with the series representation of Heaviside function is first used to reduce the equation to second order ordinary differential equations with variable coefficients. Thereafter the transformed equations are simplified using (i) The Laplace transformation technique in conjunction with convolution theory to obtain the solution for moving force problem and (ii) finite element analysis in conjunction with Newmark method to solve the analytically unsolvable moving mass problem because of the harmonic nature of the moving load. The finite element method is first used to solve the moving force problem and the solution is compared with the analytical solution of the moving force problem in order to validate the accuracy of the finite element method in solving the analytically unsolvable moving mass problem. The numerical solution using the finite element method is shown to compare favorably with the analytical solution of the moving force problem. The displacement response for moving distributed force and moving distributed mass models for the dynamical problem are calculated for various time t and presented in plotted curves.

Complete semi-analytical solution for a uniformly moving mass on a beam on a two-parameter visco-elastic foundation with non-homogeneous initial conditions

In this paper the new semi-analytical solution for the moving mass problem, published by the author of this paper, is extended to account for the non-homogeneous initial conditions. Derivations are presented for infinite homogeneous beams placed on a two-parameter visco-elastic foundation. Methods of integral transforms and contour integration are exploited to obtain the final closed-form solution, which is presented in form of a sum of the truly steady-state part, mass induced harmonic part, initial conditions induced harmonic part and transient vibration. Except for the transient part that is obtained by numerical integration, full evolution of the transversal vibrations can be quickly and accurately obtained by simple evaluation of the presented closed-form results.Newly derived formulas for infinite beams are validated by analysis of long finite beams, where the problem is solved by the eigenmode expansion method. Excellent agreement between the results is obtained validating the new formulas.

Moving mass problem: Complete solution with the effect of initial conditions

In this contribution, recently published new semi-analytical solution for the moving mass problem [1] is extended to account for the transient terms that adapt the initial part of the complete solution in a way to match the initial conditions. It is assumed that a mass and a vertical force with harmonic component move by constant velocity along a horizontal infinite beam posted on a two-parameter visco-elastic foundation. The new semi-analytical solution is presented as a sum of truly steady-state terms, harmonic terms induced by the moving mass and transient terms adapting the initial conditions. Closed-form formula is given for the first two types of vibrations. It is concluded that transient terms have in most cases almost negligible effect on the full solution and that the initial conditions can significantly affect the amplitudes of the induced harmonic vibrations, but the induced frequencies are kept without any changes.

Dynamic Response to Moving Distributed Masses of Pre-Stressed Uniform Rayleigh Beam Resting on Variable Elastic Pasternak Foundation

The dynamic response to moving distributed masses of pre-stressed uniform Rayleigh beam resting on variable elastic Pasternak foundation is examined. The equation governing this problem is a fourth order partial differential equation with variable and singular co-efficients. To solve this cumbersome equation, the method of Galerkin approach is adopted to reduce the governing differential equation to a sequence of coupled second order ordinary differential equation which is then simplified further with modified asymptotic method of Struble. The more simplified equation is solved using the Laplace transformation technique. The closed form solutions obtained are analyzed in order to show the conditions of resonance, and to show that resonance is attained earlier in moving mass system than in the moving force system. The results in plotted graphs show that as the axial force, the rotatory inertia, foundation modulus and shear modulus increase, the deflection of the elastically supported non-uniform Rayleigh beam decreases in each case. The transverse deflections of the beam on variable Pasternak elastic foundation are higher under the action of moving masses than those when only the force effects of the moving load are considered. This implies that resonance is reached faster in moving mass problem than in moving force problem.

New Semi-analytical Solution for a Moving Mass Problem: the Effect of Initial Conditions and Abrupt Change in Foundation Stiffness

In this contribution, the new semi-analytical solution for the moving mass problem from [1] is extended to account for the influence of initial conditions. The essential part of the new closed-form formula derived in [1] is the term governed by the mass-induced frequencies, which cannot be obtained when a common solution method for this kind of problems like double Fourier transform is used. Apart the term given by the closed-form formula, there is a transient part of natural vibrations originated by a line discontinuity in the Laplace image of the displacement under the load. Analytical expression is derived for such a defined line cut. Then the transient part is determined by numerical integration and added to the response given by the closed-form formula. Very good coincidence between results on finite and infinite beams is obtained, which is further exploited for the analysis of the influence of the abrupt change in the foundation stiffness.

New semi-analytical solution for a uniformly moving mass on a beam on a two-parameter visco-elastic foundation

In this paper a new semi-analytical solution for the moving mass problem is presented. Firstly, the problem of a mass traversing a finite beam on an elastic foundation is reviewed and some new aspects are added. Then, the new semi-analytical solution is deduced for an infinite beam. The semi-analytical solution of the displacement under the mass is derived with the help of integral transforms and the full deflection shape is obtained by linking together two semi-infinite beams. An iterative procedure is suggested for the determination of the frequency of the oscillation induced by the moving mass. Results deduced for infinite beams are confirmed by analysis of long finite beams, with the help of derivations given in the first part of this paper. Convergence analysis on finite beams is also presented, and, in addition, the effects of the normal force, of the harmonic component of the vertical force and of the foundation damping are discussed.

Damage Detection Using Measurement Response Data of Beam Structure Subject to a Moving Mass

THE MOVING LOAD PROBLEM IS DIVIDED INTO TWO TYPICAL TYPES: MOVING FORCE AND MOVING MASS. THE MOVING MASS PROBLEM INCLUDES THE TIME-VARYING MASS EFFECT. THIS WORK CONSIDERS DAMAGE DETECTION OF THE BEAM STRUCTURE SUBJECT TO A MOVING LOAD INCLUDING THE INERTIA EFFECT BASED ON THE ONLY MEASUREMENT DATA FROM STRAIN GAGES AND ACCELEROMETERS WITHOUT ANY BASELINE DATA. THIS EXPERIMENT COMPARES THE FEASIBILITY OF DAMAGE DETECTION METHODS DEPENDING ON THE MEASUREMENT SENSORS OF STRAIN GAGES AND ACCELEROMETERS. THE MEASUREMENT DATA ARE TRANSFORMED TO THE PROPER ORTHOGONAL MODES (POMS) IN THE TIME DOMAIN AND THE FREQUENCY DOMAIN, RESPECTIVELY. THE MAGNITUDE OF THE MOVING MASS AND ITS VELOCITY ARE ALSO EVALUATED AS TEST VARIABLES IN THIS EXPERIMENT. IT IS SHOWN IN THE BEAM TESTS THAT THE MEASURED STRAIN DATA CAN BE MORE EXPLICITLY UTILIZED IN DETECTING DAMAGE THAN THE ACCELERATION DATA, AND THE MASS MAGNITUDE AND ITS VELOCITY AFFECT THE FEASIBILITY OF DAMAGE DETECTION.

Instability and vibration of a vehicle moving on curved beams with different boundary conditions

ABSTRACTThe instability and vibration of a concentrated mass moving along the curved beam is investigated. In most literature, the moving mass model is approximated by the moving load model. The semi-analytical method for this moving mass problem is presented here. Comparison between the two models is made. The mechanism of instability of the vehicle separating from a curved bridge is studied. Moreover, the effects of several parameters and boundary conditions on the vibration and instability are investigated.

Response to Concentrated Moving Masses of Elastically Supported Rectangular Plates Resting on Winkler Elastic Foundation

Abstract The dynamic response to moving concentrated masses of elastically supported rectangular plates resting on Winkler elastic foundation is investigated in this work. This problem, involving non-classical boundary conditions, is solved and illustrated with two common examples often encountered in engineering practice. Analysis of the closed form solutions shows that, for the same natural frequency (i) the response amplitude for the moving mass problem is greater than that one of the moving force problem for fixed Rotatory inertia correction factor R0 and foundation modulus F0, (ii) The critical speed for the moving mass problem is smaller than that for the moving force problem and so resonance is reached earlier in the former. The numerical results in plotted curves show that, for the elastically supported plate, as the value of R0 increases, the response amplitudes of the plate decrease and that, for fixed value of R0, the displacements of the plate decrease as F0 increases. The results also show that for fixed R0 and F0, the transverse deflections of the plates under the actions of moving masses are higher than those when only the force effects of the moving load are considered. Hence, the moving force solution is not a save approximation to the moving mass problem. Also, as the mass ratio Γ approaches zero, the response amplitude of the moving mass problem approaches that one of the moving force problem of the elastically supported rectangular plate resting on constant Winkler elastic foundation.

Dynamic Analysis under Uniformly Distributed Moving Masses of Rectangular Plate with General Boundary Conditions

The problem of the flexural vibrations of a rectangular plate having arbitrary supports at both ends is investigated. The solution technique which is suitable for all variants of classical boundary conditions involves using the generalized two-dimensional integral transform to reduce the fourth order partial differential equation governing the vibration of the plate to a second order ordinary differential equation which is then treated with the modified asymptotic method of Struble. The closed form solutions are obtained and numerical analyses in plotted curves are presented. It is also deduced that for the same natural frequency, the critical speed for the system traversed by uniformly distributed moving forces at constant speed is greater than that of the uniformly distributed moving mass problem for both clamped-clamped and simple-clamped end conditions. Hence resonance is reached earlier in the uniformly distributed moving mass system. Furthermore, for both structural parameters considered, the response amplitude of the moving distributed mass system is higher than that of the moving distributed force system. Thus, it is established that the moving distributed force solution is not an upper bound for an accurate solution of the moving distributed mass problem.

Flexural motions under moving concentrated masses of elastically supported rectangular plates resting on variable winkler elastic foundation

The flexural motions of elastically supported rectangular plates carrying moving masses and resting on variable Winkler elastic foundations is investigated in this work In order to solve the fourth order partial differential equation governing the problem, a technique based on separation of variables is used to reduce the governing fourth order partial differential equations with variable and singular coefficients to a sequence of second order ordinary differential equations. These equations are then solved using a modification of the Struble's technique and method of integral transformations. Numerical results are then presented in plotted curves. The results show that response amplitudes of the plate decrease as the value of the rotatory inertia correction factor Ro increases and for fixed value of Ro, the displacements of the elastically supported rectangular plates resting on variable elastic foundations decrease as the foundation modulus Fo increases. Also, for fixed Ro and Fo, the transverse deflections of the rectangular plates under the actions of moving masses are higher than those when only the force effects of the moving load are considered. Therefore, the moving force solution is not a safe approximation to the moving mass problem. Hence, safety is not guaranteed for a design based on the moving force solution. Furthermore, the results show that the critical speed for the moving mass problem is reached prior to that of the moving force for the elastically supported rectangular plates on Winkler elastic foundation with stiffness variation.

Dynamic response to moving masses of rectangular plates with general boundary conditions and resting on variable winkler foundation

The dynamic response to moving masses of rectangular plates with general classical boundary conditions and resting on variable Winkler elastic foundation is investigated in this work. The governing fourth order partial differential equation is solved using a technique based on separation of variables, the modified method of Struble and the integral transformations. Numerical results in plotted curves are then presented. The results show that as the value of the rotatory inertia correction factor Ro increases, the response amplitudes of the plate decrease and that, for fixed value of Ro, the displacements of the plate decrease as the foundation modulus Fo increases for the variants of the classical boundary conditions considered. The results also show that for fixed Ro and Fo, the transverse deflections of the rectangular plates under the actions of moving masses are higher than those when only the force effects of the moving load are considered. For the rectangular plate, for the same natural frequency, the critical speed for moving mass problem is smaller than that of the moving force problem for all variants of classical boundary conditions, that is, resonance is reached earlier in moving mass problem than in moving force problem. When Fo and Ro increase, the critical speed increases, hence, risk is reduced.

Active suppression of a beam under a moving mass using a pointwise fiber bragg grating displacement sensing system

This paper investigates active vibration control of a beam under a moving mass using a pointwise fiber Bragg grating (FBG) displacement sensing system. Dynamic responses of the proposed FBG displacement sensor are demodulated with an FBG filter and verified with measurement results obtained from a noncontact fiber-optic displacement sensor. System identification of the beam is first performed with a piezoceramic actuator and positive position feedback (PPF) controllers are designed based on the identified results. Then, transient responses of the beam under a moving mass with different moving conditions are measured using the FBG displacement sensor. A high-speed camera is used to detect the speed of the moving mass for further discussions about its influence on the transient response. Finally, active vibration control of the beam under the moving mass is performed and fast Fourier transform (FFT) as well as short-time Fourier transform (STFT) are employed to demonstrate control performances. For the case in which a rolling steel ball is directed from a slide to the beam to generate the moving mass, reductions of the vibration up to 50% and 60% are achieved in the frequency domain for the first and second modes of the beam, respectively. Based on the control experiments on the smallscale beam, results in this work show that the proposed FBG displacement sensing system can be used in research on the moving mass problem.

The Response of Viscously Damped Euler-Bernoulli Beam to Uniform Partially Distributed Moving Loads

The paper investigates the response of non-initially stressed Euler-Bernoulli beam to uniform partially distributed moving loads. The governing partial differential equations were analyzed for both moving force and moving mass problem in order to determine the behaviour of the system under consideration. The analytical method in terms of series solution and numerical method were used for the governing equation. The effect of various beam observed that the response amplitude due to the moving force is greater than that due to moving mass. It was also found that the response amplitude of the moving force problem with non-initial stress increase as mass of the mass of the load M increases.

An approach for modeling long flexible bodies with application to railroad dynamics

Considering track flexibility in railroad vehicle simulations can lead to improved results. In modeling a railroad vehicle as a multibody system, track flexibility can be incorporated by using the floating frame of reference formulation (FFRF), which describes rail deformations in terms of shape functions defined in the track body frame of reference. However, the FFRF method is subject to two serious shortcomings, namely: it uses unreal track boundary conditions to calculate shape functions and requires a large number of functions to describe deformation. These shortcomings can be circumvented by defining shape functions in the trajectory frame of reference. Based on this notion, a new form of FFRF that can be used to describe the dynamics of long bodies subjected to moving loads (cable cars, zip-lines, elevator guides, pantograph catenary mechanism, etc.) was developed here. The shape functions selected in this work are based on the steady deformation exhibited by a beam on a Winkler foundation under the action of a moving load. However, other sets of shape functions more appropriate for transient dynamics are suggested. The definition of the deformation shape functions in a frame that moves with respect to the flexible body produces new terms in the generalized inertia forces of the flexible track. The proposed approach was applied to an unsuspended wheelset traveling on a tangent track supported on an elastic foundation. The results thus obtained under variable foundation stiffness conditions are discussed and comparisons made with the case of a rigid track. The new approach is also compared with the moving mass problem as solved with the mode superposition method.

Dynamic behaviour under moving concentrated masses of simply supported rectangular plates resting on variable Winkler elastic foundation

The response of simply supported rectangular plates carrying moving masses and resting on variable Winkler elastic foundations is investigated in this work. The governing equation of the problem is a fourth order partial differential equation. In order to solve this problem, a technique based on separation of variables is used to reduce the governing fourth order partial differential equations with variable and singular coefficients to a sequence of second order ordinary differential equations. For the solutions of these equations, a modification of the Struble's technique and method of integral transformations are employed. Numerical results in plotted curves are then presented. The results show that response amplitudes of the plate decrease as the value of the rotatory inertia correction factor R0 increases. Furthermore, for fixed value of R0, the displacements of the simply supported rectangular plates resting on variable elastic foundations decrease as the foundation modulus F0 increases. The results further show that, for fixed R0 and F0, the transverse deflections of the rectangular plates under the actions of moving masses are higher than those when only the force effects of the moving load are considered. Therefore, the moving force solution is not a safe approximation to the moving mass problem. Hence, safety is not guaranteed for a design based on the moving force solution. Also, the analyses show that the response amplitudes of both moving force and moving mass problems decrease both with increasing Foundation modulus and with increasing rotatory inertia correction factor. The results again show that the critical speed for the moving mass problem is reached prior to that of the moving force for the simply supported rectangular plates on variable Winkler elastic foundation.

Dynamic behaviour of non-uniform Bernoulli-Euler beams subjected to concentrated loads travelling at varying velocities.

This paper investigates the dynamics behaviour of non-uniform Bernoulli-Euler beams subjected to concentrated loads ravelling at variable velocities. The solution technique is based on the Generalized Galerkin Method and the use of the generating function of the Bessel function type. The results show that, for all the illustrative examples considered, for the same natural frequency, the critical speed for the system consisting of a non-uniform beam traversed by a force moving at a non-uniform velocity is greater than that of the corresponding moving mass problem. It was also found that, for fixed axial force, an increase in foundation moduli reduces the response amplitudes of the dynamical system. Furthermore, it was shown that the transverse-displacement amplitude of a clamped-clamped non-uniform Bernoulli-Euler beam traversed by a load moving at variable velocities is lower than that of the cantilever. The response amplitude of the same dynamical systems which is simply supported is higher than those which consist of clamped-clamped or clamped-free (Cantilever) end conditions. Finally, an increase in the values of foundation moduli and axial force reduces the critical speed for all variants of the boundary conditions Journal of the Nigerian Association of Mathematical Physics Vol. 9 2005: pp. 79-102