Friction-induced self-sustained oscillations, also known as stick-slip vibrations, occur in mechanical systems as well as in everyday life. On the basis of a one-dimensional map, the bifurcation behaviour including unstable branches is investigated for a friction oscillator with simultaneous self- and external excitation. The chosen way of mapping also allows a simple determination of Lyapunov exponents.
Steady thermal stresses in a plate made of a functionally gradient material (FGM) are analyzed theoretically and calculated numerically. An FGM plate composed of PSZ and Ti-6Al-4V is examined, and the temperature dependence of the material properties is considered. A local safety factor is used for evaluation of the FGM's strength. It is assumed that top and bottom surfaces of the plate are heated and kept at constant thermal boundary conditions. The pairs of the surface temperatures, for which the minimum local safety factor can be of more than one, are obtained as available temperature regions. The temperature dependence of the material properties diminishes, available temperature region as compared with that for an FGM plate without it. The available temperature region of the FGM plate is wider than that of the two-layered plate, especially for the surface temperatures which are high at the ceramic surface and low at the metal side. The influence of different mechanical boundary conditions is examined, and available temperature regions are found to be different, depending on the mechanical boundary conditions. The influence of the intermediate composition on the thermal stress reduction is also investigated in detail for the surface temperatures which are kept at 1300 K at the ceramic surface and 300 K at the metal side. Appropriate intermediate composition of the FGM plate can yield the local safety factor of one or:more for the four mechanical boundary conditions at once. For the two-layered plate there does not exist, however, any appropriate pair of metal and ceramic thicknesses which would yield the local safety factor of one or more for the four mechanical boundary conditions at once. The influence of the intermediate composition on the maximization of the minimum stress ratio depends on the mechanical boundary conditions. Finally, the optimal FGM plates are determined.
Nonlinear excitations cause angular vibrations in torsional strings. In long strings, the vibrations are characterized by different dynamic behavior over the length. For a general case of a long torsional string, a simplified mathematical model is introduced and numerically simulated. In order to gain insight into the complex spatio-temporal dynamics, the method of proper orthogonal decomposition is applied. A short description of this powerful technique fur continuous as well as discrete systems follows. By proper orthogonal decomposition, the dynamic response is projected on a subset of the state space in which the most dominant dynamic effects take place. The time-invariant eigenfunctions represent the most persistent structures in the system. By this method the eigenfunctions of long torsional strings are investigated. The reduction of the system's dimension as well as the approximation of the system state is presented.
The paper explores the theory of reactive porous media for the modelling of creep and plasticity due to chemo-mechanical couplings at the macro-level of material description. The formulation is based upon thermodynamics of open porous media composed of a skeleton and several fluid phases saturating the porous space. This theoretical framework allows to introduce the kinetics of a chemical reaction directly at the macro-level of material description. In turn, it is used to model creep due to chemo-mechanical couplings within a closed reactive porous continuum, as well as ageing creep due to two chemical reactions, one associated with the apparent creep phenomenon, the other with the apparent ageing phenomenon. Furthermore, it is shown how the modelling can be extended to account for plastic (i.e. permanent) phenomena, including hardening/softening and damage phenomena, coupled with a chemical reaction (chemical hardening).
This work presents an exact piezothermoelastic solution of infinitely long, simply supported, cylindrically orthotropic, piezoelectric, radially polarised, circular cylindrical shell panel in cylindrical bending under thermal and electrostatic excitation. The general solution of the governing differential equations is obtained by separation of variables. The displacements, electric potential and temperature are expanded in appropriate Fourier series in the circumferential coordinate to satisfy the boundary conditions at the simply-supported longitudinal edges. The governing equations reduce to Euler-Cauchy type of ordinary differential equations. Their general solution involves six constants for each Fourier component. These are solved from the algebraic equations obtained by satisfying the boundary conditions at the lateral surfaces. The solution of the inverse problem of inferring the applied temperature field from the given measured distribution of electrical potential difference between the lateral surfaces of the shell has also been presented. Numerical results are presented for typical thermal and electrostatic loadings for various values of radius to thickness ratio.
The complete equations describing simple relations between forces and stresses at the weld spot of overlap joints are presented, and applied to specimens used in industry as an example. The equations are also modified in respect to a measuring method for determining the weld spot forces. The essential components of the theoretical description are: an orthogonal reference system at the weld spot positioned according to the principal loading direction, the decomposition of the weld spot forces into joint face forces and >>eigenforces>eigenforces<<. The assumptions and possible complications of the method under engineering aspects and the details of the rigid core model are described in separate publications.
The fundamental solutions of the displacement discontinuity for three-dimensional problems in Laplace space are deduced in this paper. The displacement discontinuity method and the equivalent stress method were combined and used to determine dynamic stress intensity factors for three-dimensional time-dependent crack problems. The stress intensity factors were calculated for dynamically loaded cracks with rectangular, circular, and elliptical crack fronts. The influence of elasticity waves (in particular surface waves) on the magnitude of the stress intensity factor and on the displacement of the crack surfaces was analysed.
Although the phenomenon of effective stresses was known for a long time, the theoretical foundation has remained unsatisfactory until now. Due to new experimental and theoretical findings in the porous media theory, the concept of effective stresses will be reexamined. This is necessary for porous media such as concrete and rock which show at high pressure a significant deviation of the real effective stresses from those calculated with von Terzaghi's concept due to the compressibility of the true material. A second feature of the present paper is the investigation of the effective stress ''principle'' in unsaturated porous media.
The deformation of a short helix in contact with a rigid cylinder is investigated. Deformations occur due to bending, torsion and longitudinal elasticity of the helix. Shear deformation is neglected. Some of the equations describing the problem have been given already in Love's Treatise on the Mathematical Theory of Elasticity, in terms of curvature changes. Nevertheless, the equations for small deformations have to be reformulated in terms of displacements and rotations, because contact constraints cannot be expressed in terms of curvature. Friction is neglected, thus the problem is symmetric, and it is sufficient to determine its solution for one half of the helix. Without friction between the cylinder and the helix, the contact problem arises only for a helix longer than one length of twist. For a shorter helix, the global equilibrium conditions cannot be satisfied for nonvanishing contact forces. For the minimum length, there are two noncontact zones, and the helix is in contact with the cylinder only at three points: at the ends and in the middle. For a slightly longer helix, four contact points and three noncontact regions are found. The dependence of the noncontact zones and the contact forces, which are of practical interest, can be calculated as a function of the length of the helix and its geometrical parameters. The case of a very long helix with more than four contact points remains unsolved.
The changes of mechanical properties of filled polymers and their dependence on deformation history are the subject of this paper. For most of filled polymers in practical use, the theory of linear viscoelasticity cannot be applied, even at small deformations. In this work, samples of glass bead filled polybutadiene rubber with different filler levels were investigated at small strains (epsilon < 10%). The evolution of the relaxation modulus and Poisson's ratio was observed in cyclic experiments, which were also applied in inducing a defined deformation history for the succeeding relaxation experiments. In these experiments, the relaxation modulus and Poisson's ratio were measured as functions of time, with strain, strain rate, filler level and the preceeding deformation history as parameters. The results indicate dewetting as the main reason for the changes of the mechanical properties of the filled materials.
The aim of this paper is to show that multibody systems with a large number of degrees of freedom can be efficiently modelled, taking conjointly advantage of a recursive formulation of the equations of motion and of the symbolic generation capabilities. Recursive schemes are widely used in the field of multibody dynamics since they avoid the ''explosion'' of the number of arithmetical operations in case of large multibody models. Within the context of our field of applications (railway dynamics simulation), explicit integration schemes are still prefered and thus oblige us to compute the generalized accelerations at each time step. To achieve this, we propose a new formulation of the well-known Newton/Euler recursive method, whose efficiency will be compared with a so-called ''O(N)'' formulation. A regards the symbolic generation, often decried due to the size of the equations in case of large systems, we have recently implemented recursive multibody formalisms in the symbolic programme ROBOTRAN . As we shall explain, the recursive nature of these formalisms is particularly well-suited to symbolic manipulation. All these developments have been successfully applied in the field of railway dynamics, and in particular allowed us to analyse the dynamic behaviour of several railway vehicles. Some typical results related to a completely non-conventional bogie will be presented before concluding.
By means of a combined method it is demonstrated for regular perturbation problems how the higher order terms of an asymptotic expansion may be determined from numerical solutions of the non-expanded basic equations. The method is applied to heat transfer effects in a laminar boundary layer and to the analysis of its stability. All first- and second-order coefficients of the problem are determined from numerical solutions of the basic set of equations.
The critical load of the divergence instability, eigenfrequency curves and the Rayleigh's quotient for a column with one clamped end, subject to a generalized load, have been established in the paper. In particular, it has been found that the way the external force is being applied results in the linear dependence of both the shearing force and the bending moment upon both the deflection and the deflection angle of the loaded end. Such a case of loading exists e.g. for a column which supports a transversally stabilized structure when the contact surfaces are cylindrical. Furthermore, it has been theoretically and experimentally proven, that the slope of some eigencurves can change from positive to negative with the increasing load value.
Nonlinear dynamic buckling of nonlinearly elastic dissipative/nondissipative multi-mass systems, mainly under step load of infinite duration, is studied in detail. These systems, under the same loading applied statically, experience a limit point instability. The analysis can be readily extended to the case of dynamic buckling under impact loading. Energy, topological and geometrical aspects for the total potential energy V, which is constrained to lie in a region of phase-space where V less than or equal to 0, allow conclusions to be drawn directly regarding dynamic buckling. Criteria leading to very good, approximate and lower/upper bound dynamic buckling estimates are readily established without solving the highly nonlinear set of equations of motion. The theory is illustrated with several analyses of a two-degree-of-freedom model.
The aim of this work is to investigate the thermal stress intensity factor of a functionally gradient half space with an edge crack under a steady heat flux. All material properties of the functionally gradient half space, except for the coefficient of linear thermal expansion, are exponentially dependent on the distance from the boundary of the plate. The coefficient of linear thermal expansion is assumed to be two-dimensionally dependent. The problem is reduced to a singular integral equation by using the Fourier transform. The thermal stress intensity factor versus the nonhomogeneous material parameters is calculated and represented in figures. The numerical results show that thermal stress intensity factor is dramatically decreased when the material nonhomogeneous parameters are appropriately selected.
In this paper, the global motion of rigid bodies subjected to small perturbation torques, either conservative or dissipative, is investigated by means of Melnikov's method. Deprit's variables are introduced to transform the equations of motion into a standard form which is rendered suitable for the application of Melnikov's method. The Melnikov method is used to predict the transversal intersections of stable and unstable manifolds for the pertubed rigid-body motion. The chosen examples are a self-excited rigid body subject to a small periodic torque in a viscous medium, and the heavy rigid body. It is shown in both cases that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.
A computational scheme for determining the dynamic stiffness coefficients of a linear, inclined, translating and viscously/hysteretically damped cable element is outlined. Also taken into account is the coupling between inplane transverse and longitudinal forms of cable vibration. The scheme is based on conversion of the governing set of quasistatic boundary value problems into a larger equivalent set of initial value problems, which are subsequently numerically integrated in a spatial domain using marching algorithms. Numerical results which bring out the nature of the dynamic stiffness coefficients are presented. A specific example of random vibration analysis of a long span cable subjected to earthquake support motions modeled as vector gaussian random processes is also discussed. The approach presented is versatile and capable of handling many complicating effects in cable dynamics in a unified manner.
The aim of the contribution is to formulate an engineering theory describing the dynamic behaviour of periodically waved shell-like elements, called wavy plates. On the basis of the proposed theory, the effect of coupling between free macro- and micro-vibrations of a wavy plate is investigated. It is also shown that the homogenized model of wavy plates (obtained by scaling down the wavelength parameters) cannot be applied in the analysis of dynamic problems.