The nonlinear equations of motions of a parallel robot with six degrees-of-freedom (DOF) are presented for the use in real-time computation of the inverse dynamics. The formulation is based on the Newton–Euler equations that provide the presentation of the forces exerted by each body. Therefore, with regard to real-time needs, small influences can be detected and assessed. The presented equations of motion result in a model containing as the DOF six independent tool center point (TCP) coordinates. The requirement that the mechanical parts should be free from backlash involves the existence of increased friction forces due to pre-stressed joints. The frictional behavior is modelled, and the parameters describing the friction model are identified and optimized. Some experiments are presented, and the comparison between the measured and the online-simulated actuation forces shows good accordance.

The Bubnov-Galerkin method is applied to reduce partial differential equations governing the dynamics of flexible plates and shells to a discrete system with finite degrees of freedom. Chaotic behaviour of systems with various degrees of freedom is analysed. It is shown that the attractor dimension of a system has no relationship with the attractor dimension of any of its subsystems.

In this paper, the steady-state dynamic response of an embedded railway track to a moving train is investigated theoretically. The model for the track consists of a flexible plate performing vertical vibrations, two beams that are connected to the plate by continuous visco-elastic elements and an elastic foundation that supports the plate. Two harmonic loads that move uniformly along the beams describe the train load. The plate, the beams and the elastic foundation are employed to model a concrete slab of an embedded track, the rails and the ground reaction, respectively. The problem is studied by employing the Fourier integral transforms in the following way. Firstly, the dispersion analysis of waves that may propagate along the system is accomplished in the frequency-wavenumber domain. On the basis of this analysis, critical velocities of the loads are found both for the in-phase and anti-phase vibrations of the loads. Secondly, the vertical displacement of the rails and the slab, along with the stresses in the slab, are investigated as functions of the velocity and frequency of the loads. Finally, the response of the two-dimensional model is compared to that of a simplified one-dimensional model.

The interaction of railway vehicles and track in the mid-frequency range is not fully understood yet. However, this range is important for wear problems, therefore more knowledge is required. Since the conventional modeling of the vehicle-track system is not sufficient, this paper presents some methods of modeling, which consider the elasticity of wheelsets and track. The results obtained with this model show the influence of the elasticity on the behaviour of the system.

Vibrations may be undesirable in dynamical systems for several reasons. They may affect the security, such as vibrations in primary cycle parts of (nuclear) power plants. They may decrease the quality and functionality of products, such as those manufactured by machine tools. And they may lower the comfort, such as vibrations in car wheel suspension systems or in power trains of cars. One possibility to attenuate these vibrations is by employing active suspension elements. Mounted at appropriate places inside the systems or with respect to their environment, they are able to interchange or dissipate kinetic and potential energy in an effective way with moderate control effort. Their effectiveness depends greatly on the control scheme applied to change damping and stiffness characteristics of the suspension elements. The control schemes, however, very often need information on the state variables involved in the mathematical modeling. On the other hand, it is mostly the acceleration or speed of certain parts that can be sensed reasonably and measured with sufficient accuracy. We propose here a control scheme which is solely based on the derivative of the state variables, provided that active suspension elements or actuators with the above-mentioned properties may be employed within the system. Furthermore, we only use control actions within a discrete set of possible values, which aids the real-time implementation of the designed control algorithms. And, last but not least, the number of control inputs (actuators) may be arbitrary, that is, the system may be mismatched. The scheme is based on the Lyapunov stability theory, which involves discontinuities of the Lyapunov function candidates along trajectories of the state derivative. The effectiveness and behavior of the control scheme is demonstrated on a two-DOF model of an active car seat suspension in order to enhance the driving comfort.

The concept of base forces is introduced in order to replace various stress tensors for the description of the stress state at a point. Basic equations are written in terms of the base forces. The elastic law is given that can directly express the base forces by strain energy. The Cauchy and Piola stresses are expressed through base forces by dyadic vectors. Further, a purely physical expression of stress tensors is obtained without using a coordinate system. Finally, as examples, the potential energy principle of elasticity for the case of large strain is re-stated in a simple manner and the explicit expressions of stiffness and compliance matrices of the finite element method for linear elasticity are obtained.

Sliding friction between railway wheels and rails results in considerable contact temperatures and gives rise to severe thermal stresses at the surfaces of the wheels and rails. An approximate analytical solution is presented for a line contact model. The increased bulk temperature of the wheel after a long period of constant operating conditions is also taken into account. The thermal stresses have to be superimposed on the mechanical contact stresses. They reduce the elastic limit of the wheel and rail, and yielding begins at lower mechanical loads. When residual stresses build up during the initial cycles of plastic deformation, the structure can carry higher loads with a purely elastic response in subsequent load cycles. This phenomenon is referred to as shakedown. Due to the distribution of temperature, the rail surface is generally subjected to higher stresses than the wheel surface. This can cause structural changes in the rail material and hence rail damage.

Considered in a straightforward manner multibody systems with many multiple unilateral contacts involve a combinatorial problem of huge dimensions, which can be solved reasonably only by the introduction of the complementarity idea. It states that for unilateral contacts either relative kinematics is zero and the corresponding constraint forces are not zero, or vice versa. This leads to a complementarity problem which is related to linear programming problems. This paper discusses the theoretical and practical background of complementarity.

Problems of solid mechanics are most generally formulated within 3D continuum mechanics. However, engineering models favor reduced dimensions, in order to portray mechanical properties by surface or curvilinear approximations. Such attempts for dimensional reduction constitute interactions between theoretical formulations and numerical techniques. A classical reduced model for thin bodies is represented by shell theory, an approximation in terms of resultants and first-order moments. If the shell theory, with its inherent errors, is considered as qualitatively insufficient for a particular problem, a further improvement is given by solid shell models, which are gained by direct linear interpolation of the 3D kinematic relations. They improve considerably the analytic capabilities for shells, especially when their congenital locking effects are handled by variational `convergence tricks'. The next step towards 3D quality are layered shells or solid shell elements. The present paper compares these three approximation stages from the point of view of multi-director (integral) transformations of classical continuum mechanics. It offers physical convergence requirements for each of the treated models.

By introduction of a special dependent variable and separation of variables technique, the electroelastic dynamic problem of a nonhomogeneous, spherically isotropic hollow sphere is transformed to a Volterra integral equation of the second kind about a function of time. The equation can be solved by means of the interpolation method, and the solutions for displacements, stresses, electric displacements and electric potential are obtained. The present method is suitable for a piezoelectric hollow sphere with an arbitrary thickness subjected to arbitrary mechanical and electrical loads. Numerical results are presented at the end.

The paper presents a model of the UIC link suspension for freight wagons with emphasis on its longitudinal and lateral characteristics, which influence the lateral dynamics of the vehicle. The functioning of the suspension in the horizontal plane is realised by a number of technical (pivoted) pendulums composing linkages. The main feature of the joints of linkages is internal rolling/sliding in the presence of dry friction. The dissipation of energy by dry friction in the joints is the only source of damping, which influences the lateral dynamics of the vehicle. After detailed modelling of the technical pendulum, phenomenological models of the suspension are built, which reproduce the characteristics of the suspension using simple elements. A three-parameter model with one dry-friction slider and two linear springs reproduces the lateral characteristic of the suspension. A nine-parameter model with four dry-friction sliders and five springs reproduces the longitudinal characteristic. The models, using a method of non-smooth mechanics, may be directly implemented to vehicle/track dynamic simulations.

A theory of plasticity is proposed for cellular metals to describe their elastic-plastic transition zone at small strain. Under certain conditions, only a plane strain test is necessary to determine the yield surface. The method to derive the elastic–plastic behaviour [14, 15] was originally proposed for classical metals. A simple cubic model of a cellular metal is used to demonstrate the method by the finite element method. Recommendations for the numerical simulation are given. The influence of the relative density and the hardening behaviour of the cell wall material is investigated.

A boundary element formulation is presented for the solution of the equations of fully coupled thermoelasticity for materials of arbitrary degree of anisotropy. By employing the fundamental solutions of anisotropic elastostatics and stationary heat conduction, a system of equations with time-independent matrices is obtained. Since the fundamental solutions are uncoupled and time-independent, a domain integral remains in the representation formula which contains the time-dependence as well as the thermoelastic coupling. This domain integral is transformed to the boundary by means of the dual reciprocity method. By taking this approach, the use of dynamic fundamental solutions is avoided, which enables an efficient calculation of system matrices. In addition, the solution of transient processes as well as, free and forced vibration analysis becomes straightforward and can be carried out with standard time-stepping schemes and eigensystem solvers. Another important advantage of the present formulation is its versatility, since it includes a number of simplified thermoelastic theories, viz. the theory of thermal stresses, coupled and uncoupled quasi-static thermoelasticity, and stationary thermoelasticity. The accuracy of the new thermoelastic boundary element method is demonstrated by a number of example problems.

We are interested in the stability properties of the equilibrium states of a simple mechanical system with unilateral contact and Coulomb friction. First, we give a complete description of the set of solutions that the equilibrium problem may have, depending on the stiffness coefficients, on the external forces and on the friction coefficient. In the second part, we present the stability properties of all these equilibrium states, obtained by appropriate numerical experiments.

Sliding friction between railway wheels and rails results in considerable contact temperatures and gives rise to severe thermal stresses at the surfaces of the wheels and rails. An approximate analytical solution is presented for a line contact model. The increased bulk temperature of the wheel after a long period of constant operating conditions is also taken into account. The thermal stresses have to be superimposed on the mechanical contact stresses. They reduce the elastic limit of the wheel and rail, and yielding begins at lower mechanical loads. When residual stresses build up during the initial cycles of plastic deformation, the structure can carry higher loads with a purely elastic response in subsequent load cycles. This phenomenon is referred to as shakedown. Due to the distribution of temperature, the rail surface is generally subjected to higher stresses than the wheel surface. This can cause structural changes in the rail material and hence rail damage.

An efficient one-dimensional model is developed for the statics of piezoelectric sandwich beams. Third-order zigzag approximation is used for axial displacement, and the potential is approximated as piecewise linear. The displacement field is expressed in terms of three primary displacement variables and the electric potential variables by satisfying the conditions of zero transverse shear stress at the top and bottom and its continuity at layer interfaces. The deflection field accounts for the piezoelectric transverse normal strain. The governing equations are derived using a variational principle. The present results agree very well with the exact solution for thin and thick highly inhomogeneous simply supported hybrid sandwich beams. The developed theory can accurately model open and closed circuit boundary conditions.

Impact response encompasses a variety of complicated dynamic effects including wave propagation, structural vibrations and rigid-body motion. For efficient simulation of impact response with sufficient accuracy, the methods of wave propagation and multibody systems should be combined. This paper deals with an adaptive simulation of impact response during the transition from wave propagation to rigid-body motion. For modeling structural vibrations, the approach of flexible multibody systems with floating frame of reference formulation is used and the impact-induced elastic deformations are assumed to be small. In the simulation of transient impact response, contributions of the elastic coordinates are monitored with regard to their response bounds. When response bounds are reached, the corresponding elastic coordinates are deleted. As a consequence, the number of degrees of freedom of the flexible system is reduced and the efficiency of the simulation improved. Due to material damping, the impact-induced structural vibrations decay and only the rigid-body motion remains. This adaptive simulation approach is experimentally validated for the longitudinal impact of a rigid body against an elastic rod.

Two self-consistent schemes (effective medium method and effective field method) are applied to the problem of monochromatic elastic shear wave propagation through matrix composite materials containing cylindrical unidirected fibers. Dispersion equations of the mean wave field in such composites are derived by both methods. In the long-wave and short-wave ranges, analytical solutions of these equations are obtained and compared with each other, while numerical solutions are constructed for a wide range of frequencies. In particular, velocities and attenuation factors of the mean wave fields obtained by the two methods are compared for various volume concentrations, elastic properties and densities of inclusions in a wide range of frequencies of the incident field. The main discrepancies in the predictions made by the two methods are indicated, analyzed and discussed.

In this paper, the dynamic anti-plane crack problem of two dissimilar homogeneous piezoelectric materials bonded through a functionally graded interfacial region is considered. Integral transforms are employed to reduce the problem to Cauchy singular integral equations. Numerical results illustrate the effect of the loading combination parameter λ, material property distribution and crack configuration on the dynamic stress and electric displacement intensity factors. It is found that the presence of the dynamic electric field could impede of enhance the crack propagation depending on the time elapsed and the direction of applied electric impact.

In a previous contribution, higher-order strain-gradient models for linear elasticity have been studied in statics and dynamics [9]. In this paper, the extension towards damage mechanics is made. A damage model is derived from a discrete microstructure. In the homogenisation process, higher-order strain gradients appear both in the linear and in the nonlinear parts of the constitutive equation. Similar to the elastic models, stabilising and destabilising gradients can be distinguished. The stabilising or destabilising effect of each gradient term is determined. Opposite (competing) effects on the stability are found for the gradients of the elastic and the gradients in the damage response. Various truncations of the two strain-gradient series are studied, with the aim to arrive at a continuum model that fulfills the following requirements (i) it is derivable from a discrete microstructure, (ii) it is able to describe wave dispersion in elastic and damaging media properly, and (iii) it can be used to model strain-softening phenomena, i.e. it is a regularised model. The response of the various models is studied analytically and numerically. For the analytical investigation, dispersive waves are studied and critical wave lengths are derived. Numerical simulations are carried out with the element-free Galerkin method. This combined analytical/numerical approach allows to establish the role of the critical wave length both for mechanically stable and mechanically unstable models. For stabilised models, the critical wave length sets the width of the damaging zone. On the other hand, for destabilised models, the critical wave length sets a periodicity in the response that leads to divergence of the numerical scheme. The influence of the individual gradient terms on the stability and the structural ductility is verified in static and dynamic analyses.