Bartłomiej Dyniewicz

Numerical Analysis of Vibrations of Structures under Moving Inertial Load

Introduction.- Analytical solutions.- Semi-analytical methods.- Review of numerical methods of solution.- Classical numerical methods of time integration.- Space-time finite element method.- Space-time finite elements and a moving load.- The Newmark method and a moving inertial load.- Meshfree methods in moving load problems.- Examples of applications.

Vibration Control of Double-Beam System with Multiple Smart Damping Members

A control method to stabilize vibration of a double cantilever system with a set of smart damping blocks is designed and numerically evaluated. The externally controlled magnetorheological sheared elastomer damping block is considered, but other smart materials can be used as well. The robust bang-bang control law for stabilization the bilinear system is elaborated. The key feature of the closed loop controller is the efficiency for different types of initial excitement. By employing the finite element model, the performance of the controller is validated for strong wind blow load and concentrated impact excitement of the particular point of one of the beams. For each of the excitations, the closed loop control outperforms the optimal passive damping case by over 27% for the considered energy metric.

Mass splitting of train wheels in the numerical analysis of high speed train–track interactions

We demonstrate that the dynamic simulation of a vehicle moving on a track requires the correct mass distribution in the wheel–rail system. A wheel travelling on a rail should be modelled as a pair of masses coupled as a double mass oscillator. One of the masses is attached to the rail and carries the moving inertial load, while the second one is treated classically, being connected to the rail only through an elastic spring. This model is called the ‘mass splitting model’. The classical approach overestimates the accelerations by a factor of 10. The presented method produces displacements and velocities which agree well with the results of a precise finite element method and with measurements. Some real-life problems of a vehicle moving on a track at high speed are solved numerically by own computer program and the results are compared with measurements and with the solutions obtained using other codes.

Space-Time Finite Element Method

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.

A Gao beam subjected to a moving inertial point load

A model for the dynamics of a Gao elastic or viscoelastic nonlinear beam that is subject to a horizontally moving vertical point-force is modeled and computationally studied. In particular, the behavior and vibrations of the beam as the mass is moving on it is investigated. Such problems arise naturally in transportation systems with rails. A time-marching finite element numerical algorithm for the problem is developed and implemented. Results of representative simulations are depicted and compared to the behavior of a linear Euler beam with a moving mass.

Meshfree Methods in Moving Load Problems

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.

Examples of Applications

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.

Classical Numerical Methods of Time Integration

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.

The Newmark Method and a Moving Inertial Load

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.

Semi-analytical Methods

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.