Numerical Analysis of Vibrations of Structures under Moving Inertial Load

Computer methods are commonly used now in engineering design, manufacturing, and applications. They replace experimental methods of verification, especially if experiments are expensive, time consuming, or difficult to perform. Static analysis, plastic deformations, optimization, and free vibrations are fields sufficiently well explored, and now possess efficient numerical procedures implemented in commercial software. But the case of moving loads is not represented in such computer codes. Design engineers use simplifications and approximations known from analytical solutions. These are often adequate if the load does not change the dynamical properties of the structure, i.e. is massless. In the case of an inertial load we do not have adequate tools.

A concentrated load acting on a continuous medium is usually described by a Dirac delta function. The point force or mass whose area of influence is limited, must be described in the entire spatial domain of the structure, for example 0≤ x ≤ l. Multiplication of the force by the Dirac delta function δ(x) leads to such an effect. Then we have the load terms δ(x − x ₀)P or δ(x − x ₀)md² w/dt ² described in the domain of the problem. Unfortunately, the mathematical treatment of the term of the first type is relatively simple. It does not contain the solution variable. The treatment of a term of the second type, which describes the inertial force induced by the material particle, is much more complex. It includes the acceleration of the selected point x ₀ as the second derivative of the solution of the differential equation w.

Problems of the dynamics of moving loads can be divided into three main groups depending on the nature of the load. The first is called the Willis-Stokes [131, 140] problem, describing the motion of an inertial point load travelling along a massless Euler beam. We know its complete analytical solution. The second case is related to the load of a constant amplitude moving along an inertial beam. This task was first solved by Krylov [75]. Further works discussed the influence of the elastic foundation [1, 129] and subcritical and critical velocities of the moving force [53]. Also in the case of a moving force with periodic amplitude, the complete analytical solutions are known [30, 94, 96, 137]. An excellent summary of these works is given by Frýba in his monograph [56]. He discusses in detail the majority of types of such problems.

In this chapter we will discuss the numerical approaches to the moving load problem given in the literature. Most of them concern beam deflection. Unfortunately, comparison with exact analytical or semi-analytical results are rarely given. In most cases the authors compare their results with curves published by other researchers. The authors compute examples using different data and boundary conditions. They usually emphasize the agreement of their results with other computational methods. Unfortunately, results which coincide with an approximate method are not necessarily accurate as well.We should relate the results to analytical solutions or at least to solutions which fulfill the governing differential equations with possibly the lowest error. In this chapter we will compare the curves presented in these publications with semi-analytical results.

The development of electronics and the dissemination of computer technology has led to the development of methods for computational mechanics. First, previously published methods were implemented. Then, more effective solutions were sought. New methods were created incomparably faster. With the increasing computational power of computers, new and more complex issues were studied: the problems of geometric and material non-linearities in the dynamics of structures and problems with complex geometry.

In the previous section we discussed some classical methods for the time integration of the differential equations of motion. They have interesting properties, not appreciated by researchers and software developers. In this section we will present the space-time element method.We will give its basic concepts and how to derivate the stepwise equations for this method. We will present the displacement formulation, used in the early stages of the development of the method, and the velocity formulation, which is currently being successfully used for difficult or atypical tasks.

Non-classical problems are usually poorly treated by classical and commonly known solution methods. Time dependent problems, especially vibrations, described by partial differential equations are classically treated with the finite element method in space and the family of Newmark methods in time. Such time integration methods were described in Chapter 5. We discussed in the introduction to Chapter 6 the disadvantages of such an approach and the necessity for a more general treatment of phenomena in space and time. The space-time finite element method extends the finite element approximation of the differential equation over the time domain. The main advantage in our moving mass problems concerns its facility in treating the partial derivatives obtained from the chain rule applied to the acceleration of the inertial particle in a moving coordinate system, equations (3.119) or (3.121).

The Newmark method (see Section 5.5) is considered here as a representative example of a wide family of time integration methods. It is attractive since most of computational procedures in structural dynamics are based on this numerical scheme.

The idea of meshless methods is to eliminate the mesh generation stage, which is the main disadvantage of the finite element method (or other classical discrete methods). In a meshless method, the set of separated points is placed in the domain of the structure. Interpolation functions (shape functions) are then generated not in element subdomains, but in arbitrarily placed nodal points.

The examples of the calculations of the selected engineering problems given in this Chapter demonstrate the practice of numerical solutions. In real structures we always ask questions as to what geometry and what values of the material data are appropriate to pass from the physical model of the structure to the numerical one. Real shapes are usually complex and we try to simplify them, replacing curves with straight lines, non-uniformly distributed material parameters with homogeneous material, material damping with a numerical decay of the amplitude. Let us consider, as a first example, a track subjected to a moving vehicle. We can build a detailed three-dimensional model using cubes or tetrahedra with many degrees of freedom describing the foundation, ballast, track elements, rails, wheels, and the remaining part of the vehicle. We can include contact phenomena, friction, material nonlinearities, thermo-mechanical coupling, etc. However, such a model nowadays would be a challenge even for a static problem.Calculating the solution can last even a quarter of an hour. That is relatively long considering the computational power of multi-core processors. In a dynamic analysis, such a computation must be repeated thousands of times. The duration of the task exceeds any reasonable length of time. That is why we must still simplify our numerical models and improve the computational tools. Fortunately, a coarse discretization and a simplified mesh does not influence the frequencies significantly. The amplitudes are worse.