Speed Of Moving Load Research Articles

Effect of moving loads on dynamic response of multi-span sandwich beams with porous functionally graded material

This paper analysis dynamic response of porous functionally graded material (p-FGM) multi-span sandwich beams subject to two moving loads. The core of sandwich beam is fully ceramic and skins are composed of a porous functionally graded material. The vibration differential equations of beams are established based on the finite element method (FEM). Using the Newmark direct integration method, the dynamic response of the beam is calculated. A detailed study is performed to investigate the influences of material, moving load speed, porous parameter and distance between two moving loads on the dynamic response of p-FGM multi-span sandwich beams.

The dropper stress characteristics under different moving load speeds for a high-speed railway

Dropper failure seriously threatens the operation safety of a high-speed railway. In this work, for a simple chain suspension catenary, one span with five droppers is performed to establish a model and thus the effects of the moving load speed on dropper stress are investigated. First, the partial differential vibration equation of dropper is obtained through the mechanical analysis and converted into the finite difference equation. Then, we consider contact line as a beam element to obtain its motion equation. Furthermore, the boundary and initial conditions of five droppers are determined. Finally, the stresses of five droppers are numerically calculated and the effects of the moving load speed on dropper stress are investigated by writing a MATLAB code. The results suggest that the dropper location significantly affects its stress. Compared with other droppers, droppers II and IV have much more severe vibration amplitudes. Different moving load speeds could cause different stress change of each dropper. With the increasing speed, dropper experiences longer bending compression stage and the bending amplitude increases. The impact of the moving load speed on dropper stress is significant.

A complex solution on the dynamic response of sandwich graphene‐reinforced aluminum‐based composite beams with copper face sheets under two moving constant loads on an elastic foundation

AbstractAn analytical solution was used to investigate the elastic response of a sandwich beam with a graphene‐reinforced aluminum‐based composite (GRAC) on an elastic foundation using copper as the face layer of the functionally graded composite beam and a simply supported boundary condition. Mantari's higher‐order shear deformation theory was utilized to derive the equations, which were solved in Laplace space and then converted into space–time using Laplace inversion. The exact response of the GRAC sandwich beam was obtained by considering the displacement at the mid‐span of the sandwich beam. Two moving loads with different speed ratios were applied at a single point, and the effect of various parameters, including the spring constant, the speed ratio, the percentage of graphene, the moving load speed, and the distribution pattern, was investigated. This study aimed to eliminate any overlap and improve the accuracy of the results. The exact solving method presented has not been reported in other articles so far. Additionally, due to the difficulty of solving mathematical equations, this method is only applicable to simple boundary conditions.

Dynamic Response of Anisotropic Multilayered Road Structures Induced by Moving Loads Based on a Novel Spectral Element Method

This contribution is devoted to the analysis of three-dimensional elastodynamic problems in multilayered road systems caused by moving loads with static and dynamic components. For this purpose, a novel spectral element method (SEM) is derived to model the layers of a road and the half-space. To reduce the order of wave equations, a state vector is introduced, and then a state transition matrix is derived for the upper and lower boundaries of a layer. Based on the use of a precise integration algorithm (PIA), very accurate numerical solutions can be obtained. The proposed method was derived within 3D Cartesian coordinates, which facilitate the analysis of complex loads or road structures. The stiffness matrix does not depend on tedious and lengthy analytical solutions, which is conducive to the analysis of anisotropic materials. Numerical examples are presented and compared with those of previous studies to show the accuracy and effectiveness of the proposed SEM. Additionally, this model is further refined by examining the effects of moving load speed and the properties of the road on the displacement and stress field of a stratified flexible pavement.

Thermo-hydro-mechanical response of a multi-layered pavement with imperfect interface based on dual variable and position method

The pavements are subjected to the moving vehicle, pore pressure and temperature loadings, the analytical solutions of thermo-hydro-mechanical response for the pavements are paid more attentions by the road engineers while the pavements are maintained. The analytical solutions of the multi-layered pavement model are derived by the novel DVP (duel variable and position) method in this paper. Starting with the basic equations of the pavement system, obeying Biot's dynamic poroelastic theory and generalized thermoelastic theory, the original problem is divided into two sub-systems and solved twice to avoid tangling two different loading frequencies. With the aid of moving coordinate and vector functions, the decomposition expressions of stress, displacement, temperature and pore pressure variations are deduced and propagated by using DVP method, and the transformed solutions are numerically inverted. In the numerical examples, the obtained solutions are firstly verified. Then the difference among the results of several pavement models are compared and analyzed. Finally, the effects of the surface temperature, moving load speed and interface conditions on the displacement, stress, and strain distributions are discussed by the solutions of thermo-hydro-mechanical pavement models in the numerical results.

In Plane Radial Vibration of Uncracked and Cracked Circular Curved Beams Subjected to Moving Loads

The dynamic response of structures subjected to moving load is a subject of great importance from a practical point of view. In this work, the in-plane dynamic response of a cracked isotropic circular curved beam subjected to moving loads is investigated using the finite element method. The curved beam is modeled using curved beam elements, which is developed based on the Timoshenko beam theory. Furthermore, a cracked curved beam element is developed to incorporate the presence of cracks in the structure. The effect of moving load speed, depth, and the location of the crack on the dynamic response of the beam is investigated. The outcome of the work can be useful in the study of real-life moving load problems like bridges and railways and also in the field of condition monitoring using moving loads.

Filtering property of periodic pile barriers under moving loads

The attenuation property of periodic pile barriers for elastic-wave propagation in certain direction induced by stationary loading sources has been extensively investigated. However, the corresponding studies under moving-load excitations are rather limited due to the complexity caused by the time-varying feature of the induced wave-propagation directions. To address this issue, this paper investigates the vibration isolation performance of periodic pile barriers under moving loads. Using the periodic structure theory, a novel method for deriving the attenuation zones of periodic pile barriers under harmonic moving loads is proposed, which allows calculation of the directional attenuation zones (DAZs) for waves propagating in different directions. This is realized by combining the moving-load speed line with the dispersion properties of periodic pile barriers. A simplified numerical model is also developed to analyze the dynamic responses of the periodic pile barriers in frequency domain. The responses of periodic pile barriers under different moving loads are also conducted in time domain to validate the theoretical predictions. The two-dimensional (2D) Fourier transform in both time and space domain is used to study the filtering property of periodic pile barriers under harmonic moving loads and to reveal their vibration isolation mechanism. It is found that different from the stationary loading situation, for which the induced elastic waves mostly propagate in one certain direction, the filtering property of periodic pile barriers under moving loads is related to its dispersion properties along each possible wave-propagation direction. Besides, the attenuation zones for the moving loads are dependent not only on the dispersion properties of the periodic pile barriers, but also on the loading itself, namely the velocity and excitation frequency of the moving load.

Analyzing dynamic response of nonlocal strain gradient porous beams under moving load and thermal environment

This research presents dynamic response analysis of a porous functionally graded (FG) nanobeam under a moving point load. The nanobeam formulation has been established with the use of a higher-order refined beam model and nonlocal strain gradient theory (NSGT) including two scale factors named nonlocal and strain gradient factors. The porous FG material has been modeled via modified power-law functions which contain porosity volume according to even or uneven porosity dispersions. Moreover, graded nonlocality has been considered in order to provide a better modeling of size effects for FG nano-size structures. The governing equations of the nanobeam have been discretized with the use of differential quadrature method (DQM) and inverse Laplace transform approach has been utilized to calculate the dynamic deflections. The main findings of the present research indicate the influences of nonlocal strain gradient factors, moving load speed, pore amount, porosity distribution and elastic medium on dynamic deflection of FG nanobeams.

VIBRATION OF 2D-FGSW TIMOSHENKO BEAMS UNDER A MOVING HARMONIC LOAD

Vibration of two-directional functionally graded sandwich (2D-FGSW) Timoshenko beams under a moving harmonic load is investigated. The beams consist of three layers, a homogeneous core and two functionally graded skin layers with the material properties continuously varying in both the thickness and length directions by power functions. A finite element formulation is derived and employed to compute the vibration characteristics of the beams. The obtained numerical result reveals that the material inhomogeneity and the layer thickness ratio play an important role on the natural frequencies and dynamic response of the beams. A parametric study is carried out to highlight the effects of the power-law indexes, the moving load speed and excitation frequency on the vibration characteristics of the beams. The influence of the beam aspect ratio on the vibration of the beams is also examined and discussed.

Investigation of axle-span ratio and moving load speed affecting bridge extreme response using a moving load amplitude spectrum method

In this paper, firstly the moving load amplitude spectrum is established by the Fourier transform for the two-axle moving loads, an equivalent model of the moving vehicle, acting on a simply supported beam bridge. The established spectrum is very useful for the rapid and effective analysis of the bridge free vibration response due to the excitation of moving vehicle. Then, in order to analyze the extreme response of the bridge free vibration, an efficient method is presented based on the moving load amplitude spectrum. For the moving load amplitude spectrum, the axle-span ratio (the ratio of the load wheelbase to the bridge span) and dimensionless speed are two important factors. Corresponding to the maximum value of the moving load amplitude spectrum, the relationship between the dimensionless speed and the axle-span ratio is found by the combination of theoretical derivation and numerical analysis. Further, according to the relationship, a speed formula corresponding to the maximum displacement response of bridge free vibration under moving loads is proposed, and obviously different from the resonance speeds. Finally, taking two simply supported girder bridges with different spans as examples, the results of the frequency-domain analysis are verified by a large number of the time-domain calculations.

Vibration characteristics of Timoshenko stepped beam under moving load considering inertial effect

The vibration of gun barrels would result in the change of impact point, which would further reduce the firing accuracy of weapons. In the past, the calculation model based on the Euler beam theory could not satisfy the accuracy requirements. Based on the Timoshenko beam theory, the vibration equation of the stepped beam is established by invoking continuum transfer matrix method. The forced vibration of the stepped beam under the inertial moving load is solved. The model has better precision than the Euler beam model. The endpoint of the cantilever beam is analyzed. It is shown that the endpoint response increases with the increasing mass and acceleration of moving load, so does the inertial coefficient. With the increase of moving load speed, the endpoint response decreases, and the inertia coefficient increases. Among the three parameters, the mass of moving load is the main factor affecting the inertia coefficient. Furthermore, both free and forced vibrations of other stepped beam shaped structures with arbitrary segments and boundary conditions can be explored by using the proposed method.

Optimization and sensitivity of TMD parameters for mitigating bridge maximum vibration response under moving forces

In order to effectively reduce the vertical vibration response of the bridge, this paper proposes a frequency-domain analysis method of bridge vibration that can consider the influence of the tuned mass damper (TMD) device. In this method, the time-domain motion equation of bridge-TMD under moving concentrated loads’ excitation is first established. Then, the equation of motion is Fourier transformed to obtain the displacement response spectrum of the bridge with the TMD device installed. Further, the 2-norm of the displacement response spectrum of the bridge is optimized by the augmented Lagrangian optimization method to obtain the optimal TMD parameters. Finally, numerical simulations verify the superiority of this method over the traditional optimization method (Den Hartog’s method) in terms of maximum displacement control. And, the effects of bridge damping ratio, moving load number, and axle-span ratio and moving load speed on the optimal parameters of TMD to control the maximum displacement response of the bridge were studied. The results show that using the augmented Lagrangian optimization method to optimize the 2-norm of the displacement response spectrum of the bridge can not only effectively analyze the influence of each bridge and load parameters on the selection of TMD optimal parameters, but also it can reasonably configure TMD parameters for all kinds of simply supported beam bridges, which can provide references for practical engineering applications.

Dynamic response analysis of girder-orbit systems considering the constraining effect of orbit extension under moving load

In order to examine the dynamic response of a girder-orbit-moving load coupling system considering the constraint effect of the orbit extension, an analytical calculation method for the dynamic response of the girder-orbit system with arbitrary boundary conditions under a moving load is proposed based on the finite Fourier transform. Using the analytical calculation method, the dynamic response of the girder-orbit system with orbit extension at different moving load speeds is calculated, and the calculated results are compared with ANSYS finite element numerical calculation to examine the effect of moving load speed, orbit and girder flexural stiffness, interlayer spring stiffness, and orbit extension length on the dynamic response of the system. Results show that the analytical calculation results are in good agreement with the ANSYS calculation results, verifying the correctness of the analytical calculation method. The amplitude-moving load velocity relationship curve of the girder-orbit system dynamic response is a waveform curve. The interlayer spring stiffness, orbit flexural stiffness, and orbit extension length have significant effects on the dynamic response of the orbital layer, but the effect on the dynamic response of the girder layer is not significant.

Dynamic Response of an Euler-Bernoulli Beam Coupled with a Tuned Mass Damper under Moving Load Excitation

In this study, dynamic analysis of Euler-Bernoulli beam and Tuned Mass Damper (TMD) interaction problem under the effect of moving load was carried out by the mode superposition method. After the differential equations of TMD are derived by Lagrange method, beams and TMD motion equations are integrated and matrices belonging to the motion equation of the whole system are obtained. The motion equation of the system is solved in the time domain using the Newmark-β algorithm. The effect of TMD on damping vibrations has been examined in terms of parameters such as frequency, damping rate, mass ratio and moving load speed. In addition, the effect of TMD on Dynamic Amplification Factor (DAF) was examined. As a result, with the TMD application carried out in this study, approximately 14% to 24% improvement was achieved in beam deformations and accelerations.

Dynamic Behavior of a Bidirectional Functionally Graded Sandwich Beam under Nonuniform Motion of a Moving Load

A bidirectional functionally graded Sandwich (BFGSW) beam model made from three distinct materials is proposed and its dynamic behavior due to nonuniform motion of a moving point load is investigated for the first time. The beam consists of three layers, a homogeneous core, and two functionally graded face sheets with material properties varying in both the thickness and longitudinal directions by power gradation laws. Based on the first-order shear deformation beam theory, a finite beam element is derived and employed in computing dynamic response of the beam. The element which used the shear correction factor is simple with the stiffness and mass matrices evaluated analytically. The numerical result reveals that the material distribution plays an important role in the dynamic response of the beam, and the beam can be designed to meet the desired dynamic magnification factor by appropriately choosing the material grading indexes. A parametric study is carried out to highlight the effects of the material distribution, the beam layer thickness and aspect ratios, and the moving load speed on the dynamic characteristics. The influence of acceleration and deceleration of the moving load on the dynamic behavior of the beam is also examined and highlighted.

VIBRATION OF FGSW BEAMS UNDER NONUNIFORM MOTION OF MOVING LOAD USING AN EFFICIENT THIRD-ORDER SHEAR DEFORMATION FINITE ELEMENT FORMULATION

Vibration of functionally graded sandwich (FGSW) beams under nonuniform motion of a moving load is studied using a third-order shear deformation finite element formulation. The beams consists three layers, a homogeneous ceramic core and two functionally graded faces. Instead of the rotation, the finite element formulation is derived by using the transverse shear rotation as a unknown. Newmark method is used to compute the dynamic response of the beams. Numerical result reveals that the derived formulation is efficient, and it is capable to give accurate vibration characteristics by a small number of the elements. A parametric study is carried out to illustrate the effects of the material distribution, layer thickness ratio and moving load speed on the dynamic behavior of the beams. The influence of acceleration and deceleration of the moving load on the vibration of the beams is also examined and discussed.

VIBRATION OF FGSW BEAMS UNDER NONUNIFORM MOTION OF MOVING LOAD USING AN EFFICIENT THIRD-ORDER SHEAR DEFORMATION FINITE ELEMENT FORMULATION

Vibration of functionally graded sandwich (FGSW) beams under nonuniform motion of a moving load is studied using a third-order shear deformation finite element formulation. The beams consists three layers, a homogeneous ceramic core and two functionally graded faces. Instead of the rotation, the finite element formulation is derived by using the transverse shear rotation as a unknown. Newmark method is used to compute the dynamic response of the beams. Numerical result reveals that the derived formulation is efficient, and it is capable to give accurate vibration characteristics by a small number of the elements. A parametric study is carried out to illustrate the effects of the material distribution, layer thickness ratio and moving load speed on the dynamic behavior of the beams. The influence of acceleration and deceleration of the moving load on the vibration of the beams is also examined and discussed.

A New Three-Dimensional Moving Timoshenko Beam Element for Moving Load Problem Analysis

Abstract A new three-dimensional moving Timoshenko beam element is developed for dynamic analysis of a moving load problem with a very long beam structure. The beam has small deformations and rotations, and bending, shear, and torsional deformations of the beam are considered. Since the dynamic responses of the beam are concentrated on a small region around the moving load and most of the long beam is at rest, owing to the damping effect, the beam is truncated with a finite length. A control volume that is attached to the moving load is introduced, which encloses the truncated beam, and a reference coordinate system is established on the left end of the truncated beam. The arbitrary Lagrangian–Euler method is used to describe the relationship of the position of a particle on the beam between the reference coordinate system and the global coordinate system. The truncated beam is spatially discretized using the current beam elements. Governing equations of a moving element are derived using Lagrange’s equations. While the whole beam needs to be discretized in the finite element method or modeled in the modal superposition method (MSM), only the truncated beam is discretized in the current formulation, which greatly reduces degrees-of-freedom and increases the efficiency. Furthermore, the efficiency of the present beam element is independent of the moving load speed, and the critical or supercritical speed range of the moving load can be analyzed through the present method. After the validation of the current formulation, a dynamic analysis of three-dimensional train–track interaction with a non-ballasted track is conducted. Results are in excellent agreement with those from the commercial software simpack where the MSM is used, and the calculation time of the current formulation is one-third of that of simpack. The current beam element is accurate and more efficient than the MSM for moving load problems of long three-dimensional beams. The derivation of the current beam element is straightforward, and the beam element can be easily extended for various other moving load problems.

Moving Load Spectrum for Analyzing the Extreme Response of Bridge Free Vibration

In order to more effectively establish the relationship between moving load speed and the extreme response of bridge free vibration, a novel analysis method is presented based on the moving load spectrum, which is deduced from Fourier transform in this paper. By analyzing the moving load spectrum in detail, the moving load velocities which lead to the extreme responses of bridge free vibration under single moving constant force or harmonic force are obtained, and the corresponding formula for calculating the moving load velocity which leads to the maximum response of bridge free vibration is put forward. Finally, the moving load spectrum for analyzing the extreme response of bridge free vibration is validated by a large number of calculations in the time domain in this paper. The results show that the moving velocities corresponding to the extreme points in the moving load spectrum are consistent with the velocities corresponding to the extreme points of the bridge vibration response obtained in the time domain, and the forced and free vibration displacement responses of bridge are not the largest when the single moving constant force or harmonic force passes through the bridge at the resonant velocity compared to other speeds.

On the dynamics of periodically restrained flexural structures under moving loads

The dynamic response of periodically restrained flexural structures, modeled as Euler–Bernoulli beams supported by flexible and dissipative supports at equally spaced points, to moving loads is investigated. Bloch–Floquet theorem and modal analysis are used to examine the dispersion relation and band structure of the infinitely long periodic beam and the frequency spectrum of the finite beam, respectively. It is shown that the natural frequencies of the finite beam fall within the propagation bands of the corresponding infinite beam and get closer to each other as the number of spans increases ultimately covering the propagation bands. The dynamic response to moving load is obtained via a novel method based on Floquet transformation in the frequency domain for infinite periodic beams and through modal summation for finite periodic beams. Through numerical examples, the effect of boundary conditions, e.g. support stiffness and damping, and the impact of moving load speed on the flexural response as well as the energy stored and dissipated within the supports are discussed.