Czesław I. Bajer

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.

Smart Suspension System for Linear Guideways

This paper presents a new method for the semi-active control of the span system of linear guideways subjected to a travelling load. Two elastic beams are coupled by a set of controlled dampers. The relative velocity of the spans provides an opportunity for efficient control via semi-active suspension. The magnitude of the moving force is assumed to be constant by neglecting inertial forces. The response of the system is solved in modal space. The full analytical solution is based on the power series method and can be given over an arbitrary time interval. The control strategy is formulated by using bilinear optimal control theory. As a result, bang-bang controls are taken into account. The final solution is obtained as a numerical mean value. Several examples demonstrate the efficiency of the proposed method. The controlled system outperforms passive solutions over a wide range. Due to the simplicity of its design, the presented solution should be interesting to engineers.

The soft way method and the velocity formulation

Abstract In this paper the new approach to dynamic contact problems is described. The velocity formulation was assumed and a new time integration scheme was elaborated. The space-time finite element method used in derivation enables control of the accuracy (order of the error) and stability. Methods for the solution of contact problems were discussed. A discretized approach, prepared for large displacements and large rotations, enabled real engineering problems to be solved in a relatively short time.

Adaptive space-time elements in the dynamic elastic-viscoplastic problem

Abstract An adaptive technique for the solution of the dynamic elastic-viscoplastic problem has been developed. The mesh modification is performed by the use of the space-time element method according to error estimation. The number of joints is preserved and the mesh is refined in regions of high stress gradients. This enables the size of the problem to be reduced and increases the speed of computations. The incremental procedure in the case of a small time step allows the nonlinear path iteration to be associated with the time marching scheme. The remesh and remap problems related to stresses are described. Numerical examples of a plane strain rolling contact problem and collision of the plane strain object prove the efficiency of the approach.

Adaptive mesh in dynamic problems by the space-time approach

Abstract Stress fields varying in time are typical for dynamic wave problems. Nonclassic problems involve changing of structure properties, especially wave reflection zones or dissipative zones. Stress field propagation requires a variable mesh that allows one to approach the phenomenon with the smallest error in each time step. The space-time approximation of the differential equation of motion enables the modification of the spatial partition into finite elements in a continuous way. Error estimation was the reason to refine and coarsen the spatial partition, moving the nodes towards the zone of higher error. Applying the simplex-shaped space-time elements one can gain the triangular form of coefficient matrix directly in the element matrix assembly process. Consistent characteristic matrices are used. The approach presented was successfully applied for bar, beam and plane strain analysis. The method is more powerful for materially nonlinear cases for which element matrices should be calculated in each time step. Good accuracy of the movable mesh approach was proved in several testing examples.

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.