Moving Load Problem Research Articles

Physics-informed machine learning for moving load problems

Abstract This paper presents a new approach to simulate forward and inverse problems of moving loads using physics-informed machine learning (PIML). Physics-informed neural networks (PINNs) utilize the underlying physics of moving load problems and aim to predict the deflection of beams and the magnitude of the loads. The mathematical representation of the moving load considered involves a Dirac delta function, to capture the effect of the load moving across the structure. Approximating the Dirac delta function with PINNs is challenging because of its instantaneous change of output at a single point, causing difficulty in the convergence of the loss function. We propose to approximate the Dirac delta function with a Gaussian function. The incorporated Gaussian function physical equations are used in the physics-informed neural architecture to simulate beam deflections and to predict the magnitude of the load. Numerical results show that PIML is an effective method for simulating the forward and inverse problems for the considered model of a moving load.

Nonlinear Dynamic Analysis of a Curved Sandwich Beam with a Time-Dependent Viscoelastic Core Using the Generalized Differential Quadrature Method (GDQM)

This paper focuses on the geometrically nonlinear dynamic analyses of a three-layered curved sandwich beam with isotropic face layers and a time-dependent viscoelastic core. The boundary conditions and equations of motion governing the forced vibration are derived by using Hamilton’s principle. The first-order shear deformation theory is used to obtain kinematic relations. The spatial discretization of the equations is achieved using the generalized differential quadrature method (GDQM), and the Newmark-Beta algorithm is used to solve the time variation of the equations. The Newton–Raphson method is used to transform nonlinear equations into linear equations. The validation of the proposed model and the GDQM solution’s reliability are provided via comparison with the results that already exist in the literature and finite element method (FEM) analyses using ANSYS. Then, a series of parametric studies are carried out for a curved sandwich beam with aluminum face layers and a time-dependent viscoelastic core. The resonance and cancellation phenomena for the nonlinear moving-load problem of curved sandwich beams with a time-dependent viscoelastic core are performed using the GDQM for the first time, to the best of the authors’ knowledge.

An exploration of the super-critical speed dynamic behaviours of the pantograph-catenary system

The train operating speed is traditionally limited to lower than 0.7 times the critical speed of the catenary. The material’s strength limits the tensile forces of the tension wires, preventing the further improvement of the train speed. A novel super-critical speed operating strategy of the pantograph catenary system (PCS) is proposed, using the relatively low tensile values of the catenary to make the pantograph run at a speed ratio larger than 1. The analytical results of a string under a moving load show the theoretical possibility of the super-critical speed operating strategy in moving load problems. The finite element method (FEM) is adopted for the model of the catenary. The differences between the simple catenary and the stitched catenary running at the super-critical speed regime are compared. Then, the tensile forces designation of the stitched catenary with different speed ratio values beyond 400 km/h is studied. Finally, the dynamic behaviours of the PCS reaching the super-critical speed regime are discussed, consisting of the continuous acceleration process and the transition process from the high-tension section to the low-tension section. The proposed super-critical speed operating strategy may be promising for the next-generation high-speed PCS.

The Moving Element Method for the Dynamic Responses of Floating Double-Plate Structures on the Shallow Water

In this paper, the dynamic behavior of floating double-plate structures under a moving load is investigated in great detail with different factors utilizing the moving element method (MEM). The structures in this study are very large floating structures (VLFSs). The Kirchhoff–Love theory is used to model two floating plates connected by a Winkler-type elastic core, whilst the water surrounding the structure is simulated by the shallow linear water wave theory. In order to improve the computational efficiency of moving load problems, the MEM is adopted. Since the MEM uses fewer discrete elements than those of the traditional finite element method (FEM), the computational cost is reduced. The proposed two-plate model helps to reduce the influence of the surrounding water on the upper plate. Several numerical results are presented to investigate the influence of the ratio of thickness between layers as well as core stiffness on the displacements with different water depths, and load speeds.

Analysis of Critical Speed for High-Speed Railway Pantograph-Catenary System

The pantograph-catenary system is used in modern electrified railways to power electric trains. Common sense is that the increase of train speed leads to a deterioration of the interaction performance. But recent studies indicate that a higher speed may result in a better performance in some local speed regions. This paper is the first attempt to provide a theoretical explanation for this phenomenon. A validated pantograph-catenary model is built to reproduce this phenomenon. Referring the definition of critical speed in moving load problems, the catenary's critical speed is defined. The wavelength components can be extracted through the Fourier transformation to the contact wire's vertical mode shape. The speed that satisfies the resonance condition can be regarded as the potential critical speed. The corresponding contact wire amplitude is further evaluated to determine the critical speeds at a given mode. The analysis indicates that a good interaction performance happens when the speed falls in the transition procedure between the critical two speeds. Based on this idea, a simple indicator that describes the overall contribution of all modes subjected to a moving pantograph at a given speed is proposed to facilitate the determination of the local optimal speed. The variation of the proposed indicator with respect to the speed shows good consistency with the trend of the simulation results. The main finding in this paper can be used in the design phase to determine the optimal speed of the pantograph-catenary system in a local speed range.

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.

An investigation into the moving load problem for the lifting boom of a ship unloader

While the moving load problem has received a lot of attention in the context of bridges traversed by road or rail vehicles, the same cannot be said about ship unloaders where research is limited and typically focused on displacements. In ship unloaders, the moving system is composed of a trolley, attached grab, and payload. The interaction between the moving system and the lifting boom on which the trolley travels can lead to high stresses. Given that the stress ranges and associated number of cycles experienced by a section of the lifting boom determine its fatigue life, these stresses must be assessed as accurately as possible. For this purpose, dynamic transient analysis of the effects of a payload-carrying trolley moving along the boom is carried out using an equivalent planar finite element model. Results show how bending stresses are affected by the inertial forces of the structure and the moving load, the ratio of the mass of the load to the mass of the boom, the flexibility of the moving system, and the payload mass and its swing angle.

Dynamic analysis of axially functionally graded porous beams under a moving load

In presented paper, moving load problem of functionally graded beams is investigated with porosity effects based on the first shear beam theory. The material properties of beam vary along the axial direction. The porosity is depicted by two different distributions along axial direction. The governing equations of problem are derived by using the Lagrange procedure. In the solution of the problem the Ritz method is used and algebraic polynomials are used with the trivial functions for the Ritz method. In the solution of the moving load problem, the Newmark average acceleration method is used in the time history. In the numerical examples, the effects of material graduation, porosity distribution, porosity coefficients and velocity of moving load on the dynamic responses of axially functionally graded beam are presented and discussed. The dynamic responses are obtained for different boundary conditions.

Vibration analysis of an experimental double bridge crane system with artificial neural networks

In crane systems, lifting, carrying and lowering the load from one place have different dynamic effects on the system. One of these dynamic effects is the moving load problem caused by the movement of the load on the crane system. With the increasing technology in recent years, production speeds have increased. For this reason, it has made the requirements for fast-running cranes mandatory for the transportation and loading of products. Therefore, it is important to know the dynamic effects of the moving load in fast working conditions. In this experimental study, the dynamic effects occurring on the crane beams with different loads and different working speeds during the transportation of the load on the crane are analysed. Here, there are multiple cars on the crane, and these cars are designed in different numbers on the crane and can be operated at different speeds. Under these conditions, the dynamic effects that have arisen have been tested. Also, vibration measurements were carried out at different points on the bridges. And then, these parameters obtained were used in two different proposed neural network types to predict the vibrations that occur on the crane system. Simulation results show that two approaches suggested that a radial basis neural network type can be used as an adaptive predictor for such systems in the experimental applications.

DSC regularized Dirac-delta method for dynamic analysis of FG graphene platelet-reinforced porous beams on elastic foundation under a moving load

This work presents a novel computational approach, the DSC regularized Dirac-delta method, for the vibration analysis of functionally graded graphene-platelet reinforced (FG-GPLR) porous beams resting on a Winkler–Pasternak elastic foundation under a moving load. Based on the Timoshenko beam theory, the energy functional of the beam model is represented by a newly constructed basis function and is minimized under the variational principle. To account for the properties of composite materials, the Halpin–Tsai model is used to predict the elastic modulus of graphene-reinforced composites. A coupling of the DSC regularized Dirac-delta method and the Newmark–β integration scheme is then adopted for solving the dynamic problem. The DSC-based approach exhibits controllable accuracy for approximations and shows excellent flexibility in handling time-dependent moving load problems, because the equally spaced grid system used in the DSC numerical approach can achieve a preferable representation of moving load sources. An intensive parametric study is provided with a particular focus on the influence of moving loads, foundation supports and material properties (e.g., weight fraction, porosity distribution, dispersion pattern and geometry size of graphene reinforcements). First-known solutions reported in tabular and graphical forms should be useful for researchers and engineers in designing such beam problems.

Hilbert transform-based semi-analytic meta-model for maximum response envelopes in dynamics of railway bridges

This paper presents an innovative meta-model for fast assessment of dynamic design envelopes of railway bridges. The proposed approach allows the computation of instantaneous amplitudes of dynamic response time series through the integral Hilbert transform (HT). Based upon the semi-analytic solution of the moving load problem of linear bridge structures, which is analytical in the time domain, the HT is obtained in the Cauchy principal value sense in closed-form. Therefore, the proposed HT-based semi-analytic (HTSA) solution provides a direct estimation of the instantaneous response envelopes of railway bridges under high-speed trains. Since the HT demodulates the signals by suppressing the fast-varying oscillating components, the sampling rates needed for tracing the instantaneous amplitudes are considerably lower than those required for sampling the original signals. Thus, the reduction in computational time achieved by the HTSA approach stems from the sub-sampled approximation of instantaneous envelopes. In order to appraise the effectiveness of the proposed HTSA meta-model, four validation case studies are presented, including mono-dimensional and realistic three-dimensional bridge structures. The presented results report substantial reduction in computational cost with limited accuracy loss, and demonstrate the usefulness of the HTSA meta-model for fast comparison of bridge design alternatives during early design stages.

A true PML approach for steady-state vibration analysis of an elastically supported beam under moving load by a DLSFEM formulation

This paper concerns a computational implementation for solving a Moving Load (ML) problem on an infinite Euler–Bernoulli elastic beam on a Pasternak visco-elastic support. A steady-state dynamic response in convected coordinate is sought, by a numerical approach with discretization over a finite domain, implying spurious boundary reflections of non-evanescent waves. This is effectively solved by: (a) analytically formulating a new, true Perfectly Matched Layer (PML) approach, toward handling the underlying fourth-order differential problem and the corresponding far-field conditions, without adopting special boundary conditions; (b) outlining a local Discontinuous Least-Squares Finite Element Method (DLSFEM) formulation, apt to provide a robust approach for the present non self-adjoint problem and to conveniently handle the jump condition in the shear force at the concentrated ML position. Consistent numerical results are illustrated and compared to an available analytical solution, showing a perfect match, with a complete removal of spurious boundary effects and a proof of theoretical a priori error estimates. Further results are produced for a case with multiple MLs. The paper shows that the present innovative DLSFEM-PML formulation is effectively suitable to numerically solve a steady-state ML problem on an infinite beam, setting up a new computational tool in such a challenging mechanical context.

Further insights into moving load problem on inclined beam based on semi-analytical solution

This paper presents the semi-analytical solutions for a vertical load moving on an inclined simply supported beam. By adopting the energy approach along with the Fourier sine series for the beam deflection shape, the governing equation of motion is obtained as an easy-to-solve ordinary differential equation. The frequencies of the inclined beam excited by the moving load can be expressed in analytical form which elucidates the intrinsic and fundamental features of the solution. The moving load problems solved herein show that the dynamic responses (deflections) at any point on the inclined beam with respect to the moving load position are smaller with increasing angle of inclination. For more insights into the axial load effect on the beam stiffness, the exciting frequencies are found to decrease as the load moves up the inclined beam. Interestingly, while the moving load is traversing over a certain portion of the beam, the frequencies stay constant rather than decrease monotonically; in other word, the stiffness of the inclined beam remains virtually unaffected when the moving load passes in this portion of the inclined beam. The stability issues of the inclined beam with a moving load are discussed by examining the determinant of the global stiffness matrix of the beam. Moreover, the maximum magnitude of moving load with allowable deflection is obtained to provide design recommendations.

An accurate differential quadrature procedure for the numerical solution of the moving load problem

In moving load-type problems, the moving point load is modeled mathematically by a time-dependent Dirac-delta function. A key and difficult step in solving this type of problems using a point discrete method such as the differential quadrature method is the discretization of the Dirac-delta function in a simple and accurate manner. This paper is conducted to facilitate this step and to present a new way to do this task. In this way, the Dirac-delta function is approximated by orthogonal polynomials such as the Legendre and Chebyshev polynomials. Unlike the original Dirac-delta function, which is a generalized singularity function, the resulting approximation function is a non-singular function which can be discretized simply and efficiently. The proposed procedure is applied herein to solve the moving load problem in beams and rectangular plates. Comparisons with available analytical and numerical solutions prove that the proposed approach is highly accurate and efficient.

A DSC Regularized Dirac-Delta Method for Flexural Vibration of Elastically Supported FG Beams Subjected to a Moving Load

This research presents a numerical approach to address the moving load problem of functionally graded (FG) beams with rotational elastic edge constraints, in which the regularized Dirac-delta function is used to describe a time-dependent moving load source. The governing partial differential equations of the system, derived in accordance with the classical Euler–Bernoulli beam theory, are approximated by the discrete singular convolution (DSC) method. The resulting set of algebraic equations can then be solved by the Newmark-β integration scheme. Such a singular Dirac-delta formulation is also employed as the kernel function of the DSC method. In this work, the material properties of FG beams are assumed to be changed in the thickness direction. A convergence study is performed to validate the accuracy and reliability of the numerical results. In addition, the effects of moving load velocity and material distribution on the dynamic behavior of elastically restrained FG beams are also studied to serve as new benchmark solutions. By comparing with the available results in the existing literature, the present results show good agreement. More importantly, the major finding of this work indicates that the DSC regularized Dirac-delta approach is a good candidate for moving load problems, since the equally spaced grid system adopted in the DSC scheme can achieve a preferable representation of moving load sources.

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.

Optimization of multilayer rail substrate under moving load, using metamaterials.

Abstract Paper presents the optimization of linear elastic multilayer foundation of the Timoshenko beam, which can be considered as a rail model. The moving load traveling along a beam with particular velocity, which distribution is characterized by Gauss function, is implemented in the model. The displacement caused by the load is dependent on multilayer foundation parameters such as layer thickness, a type of material and load parameters such as moving velocity as well its distribution on a beam. The optimization, i.e. minimization of the total displacement sum, as the result of objective function determined regarding chosen parameters of a foundation and load, as a design variable was carried out. The Comsol Multiphysics [1], is applied in order to solve the beam and moving load problem by using the finite element method [2,3,4,5,6]. Using genetic algorithm [7,8,9] searches the minimum of objective function.

DLSFEM–PML formulation for the steady-state response of a taut string on visco-elastic support under moving load

The numerical solution of the steady-state response of a uniform taut string on visco-elastic support under a concentrated transverse moving load is addressed. By recasting the governing second-order differential equation as a first-order system in convected coordinate, a local Discontinuous Least-Squares Finite Element Method (DLSFEM) formulation is developed within a complex-valued function space, to overcome numerical instabilities linked to high-velocity loads and handle far-field conditions through an effective Perfectly Matched Layer (PML) implementation. As an original advancement of the present DLSFEM–PML formulation, a coercivity theorem is proven for any first-order ordinary differential system and uniform error estimates are established for the finite element approximation for both$$L^2$$- and$$H^1$$-norms. Thus, the formulation newly joins a DLSFEM approach and a PML implementation, for solving the above-mentioned moving load problem. Numerical examples illustrate feasibility and accuracy of the method in reproducing the expected trends of solution and a priori error estimates.

Structural damage detection in moving load problem using JRNNs based method

Damage detection in structure using vibration signature is quiet smart method for condition monitoring of structure. In this problem, Recurrent Neural Networks (RNNs) based method has been implemented for damage detection in moving load problem as inverse method. A multi-cracked simply supported beam under a traversing load has been considered for the present problem. The localization and severities of the supervised cracks on the structure are determined using the adapted Jordan’s Recurrent Neural Networks (JRNNs) approach. The mechanism of Levenberg-Marquardt’s back propagation algorithm has been implemented to train the networks. To check the adoptability of the proposed JRNNs method, numerical analyses along with laboratory test verifications have been conducted and found to be well emerged.

Optimal vibration control of moving-mass beam systems with uncertainty

A linear optimal regulator for uncertain system is designed through the application of the probability density evolution method to linear quadratic regulator controller. One important background of this work is bridge-vehicle/gun-projectile system. This type of optimal problem is currently transformed into a moving load problem. The developed optimal regulator can provide the law of probability densities of outputs varying with time. In order to make the advocated method reach an optimal performance, the beneficial weighting matrix pair (Q, R) is selected using a trade-off sense. The designed regulator is then applied to a coupled simply supported beam-moving mass system, choosing the mid-span deflection as an output response and considering stochastic system parameters. The numerical example shows that the robustness of the proposed optimal regulator cannot be overestimated in comparison with a deterministic linear quadratic regulator controller. Further, the proposed method can produce an efficient solution channel for modern optimal control theory, especially, when compared with different uncertain optimal control techniques.