Showing papers in "Meccanica in 1992"

Determination of source parameter in parabolic equations

Abstract: The authors consider the problem of finding u=u(x, t) and p=p(t) which satisfy u = Lu + p(t) + F(x, t, u, x, p(t)) in Q T=Ω×(0, T], u(x, 0)=o(x), x∈Ω, u(x, t)=g(x, t) on ∂Ω×(0, T] and either ∫G(t) Φ(x,t)u(x,t)dx = E(t), 0 ⩽ t ⩽ T or u(x0, t)=E(t), 0≤t≤T, where Ω∋R n is a bounded domain with smooth boundary ∂Ω, x 0∈Ω, L is a linear elliptic operator, G(t)∋Ω, and F, o, g, and E are known functions. For each of the two problems stated above, we demonstrate the existence, unicity and continuous dependence upon the data. Some considerations on the numerical solution for these two inverse problems are presented with examples.

A Computational Model for the Limit Analysis of Three-Dimensional Masonry Structures

Abstract: This paper extends previous work on the limit analysis of ductile frames and plane masonry arches to the limit analysis of three-dimensional masonry structures. A lower-bound approach is developed which can handle three-dimensional collapse mechanisms involving any combination of sliding, twisting and hingeingt at the block interfaces. A computer program for determining the collapse load of such structures is used to study (a) the equilibrium limits of a block with four contact points resting on an inclined plane and (b) the collapse of a semicircular arch of four blocks. The paper also describes experimental and computational work on a radially symmetric model dome of 380 blocks subject to foundation settlement.

Modelling the dynamics of large block structures

Abstract: This paper summarizes the main critical points that arise when the problem of modelling the dynamics of block structures is tackled. In the first sections, a rigorous formulation of dynamics and impact problem is presented for a single rigid block freely supported on rigid ground, in order to illustrate the basic difficulties concerning the modelling of more complicated structures. Then, a critical review is presented on the numerous researches performed on this subject and the results achieved, and the problems still open, are put in evidence.

New trends in the analysis of masonry structures

Abstract: The modern theory of masonry structures has been set up on the hypothesis of no-tension behaviour, with the aim of offering a reference model, independent of materials and building techniques employed This hypothesis gives rise to inequalities which have to be satisfied by the stress tensor components and, as a dual aspect, to the kinematic behaviour characteristics of media which can be classified as lying between solids and fluids: the structure of the masonry material consists of particles reacting elastically only when in contact An examination of the plane-stress problem leads us to define, within the prescribed domain under admissible loads, three different subdomains with null, ‘regular’, or ‘non-regular’ principal stress tensors, respectively As the boundaries of such subdomains are not known α priori, the problem can be classified as a free boundary value problem The analysis concerns mainly the subdomains where the stress tensor is ‘non-regular’; and a ‘non-regularity’ condition det σ = 0 is added to the equilibrium equations This condition makes the stress problem ‘isostatic’ and leads to a violation of Saint-Venant’s compliance conditions on strains Hence there is a need to introduce a strain tensor, not related to the stress tensor, which can be decomposed into an extensional component and a shearing component; we prove that such strains, of the class γσ c , are similar to those of the theory of plastic flow From the point of view of computational analysis the anelastic strains are considered as given distortions; they are computed by means of the Haar-Karman principle, modified for computational purposes by an idea of Prager and Hodge

Equilibrium and Collapse Analysis of Masonry Bodies

Abstract: This paper gives a general formulation of the statics of the masonry continuum within the conceptual framework set up by J. Heyman in his fundamental and pioneering studies of masonry arches and vaults. Here the masonry body will be represented by an assemblage of rigid particles of stones, held together only by compressive forces, and liable to crack as soon as tensile stresses begin to develop. The very small size of the stones, compared to the overall dimensions of the body, permits a treatment in terms of a continuum.

Basic features of the time finite element approach for dynamics

Abstract: Very general weak forms may be developed for dynamic systems, the most general being analogous to a Hu-Washizu three-field formulation, thus paralleling well-established weak methods of solid mechanics. In this work two different formulations are developed: a pure displacement formulation and a two-field mixed formulation. With the objective of developing a thorough understanding of the peculiar features of finite elements in time, the relevant methodologies associated with this approach for dynamics are extensively discussed. After having laid the theoretical bases, the finite element approximation and the linearization of the resulting forms are developed, together with a method for the treatment of holonomic and nonholonomic constraints, thus widening the horizons of applicability over the vast world of multibody system dynamics. With the purpose of enlightening on the peculiar numerical behavior of the different approaches, simple but meaningful examples are illustrated. To this aim, significant parallels with elastostatics are emphasized.

The (3+1)-dimensional Monge-Ampère equation in discontinuity wave theory: Application of a reciprocal transformation

Abstract: It is shown that the complete exceptionality condition for discontinuity waves associated with a second-order non-linear hyperbolic equation of the form $$u_u + f(x_i ,{\text{ }}t,{\text{ }}u,{\text{ }}u_i {\text{ }}u_u ,{\text{ }}u_{jk} ) = 0,{\text{ }}i = 1,{\text{ 2, 3; }}j \leqslant k$$ leads to a Monge-Ampere-type equation in 3+1 dimensions. Application of a novel reciprocal transformation shows that an important subclass may be reduced to linear canonical form. Specialization to 1+1 dimensions yields linearization of a Boillat-type equation satisfying the complete exceptionality criterion. In this last case the transformation allowing the linearization coincide with the one introduced by Hoskins and Bretherton in the theory of atmospheric frontogenesis and so-called geostrophic transformation. Finally, always in 1+1 dimensions, we show that the Monge-Ampere equation is also strictly exceptional, i.e. the only possible shocks are characteristic.

Strengthening Buildings of Stone Masonry to Resist Earthquakes

Abstract: Stone masonry buildings are common in many areas in the Alpine-Himalayan earthquake zone, and their failure in recent earthquakes has been the cause of many deaths. Poverty and lack of alternatives prevent the replacement of stone masonry with more ductile materials, but the brittleness of unreinforced stone masonry can be considerably reduced by the incorporation of horizontal lacings of timber or reinforced mortar.

An analytical approach to the dynamics of rotating shafts

Abstract: An analytical procedure, based on the dynamic stiffness method, is proposed for the study of rotor dynamics problems. On the grounds of the governing differential equations of a continuous beam, the dynamic stiffness matrix of the rotating Timoshenko beam is derived, including the effects of translational and rotational inertia, gyroscopic moments and shear deformation of the shaft. Concentrated disks and isotropic, elastic bearings are taken into account in the element formulation. The results obtained by the proposed method are compared with both classical closed form solutions and finite element analyses taken from the literature.

A new viewpoint on mixed elements

Abstract: Mixed elements are special finite elements for vector fields, whose shape functions are not necessarily continuous in all their components. They are used in connection with ‘two-field’ formulations, for instance when one tries simultaneously to compute displacement and stress in elasticity. We present here a family of finite elements which originate from a chapter of differential geometry apparently unconnected with numerical analysis (Whitney forms) and insist on the structural properties of this family considered as a whole. This sheds a new light on mixed elements, under which they acquire a more natural character than in previous presentations.

Fractals and fractal approximation in structural mechanics

Abstract: The present paper presents several new applications of the theory of fractals in structural mechanics. Until now most of the existing applications of the theory of fractals concern the calculation of fractal dimensions in physical phenomena, especially with respect to dynamical systems. The present paper deals with several other aspects of the theory of fractals which, from the standpoint of mechanics, seem to be of greater importance. Indeed the methods of fractal analysis permit the formulation and solution of difficult or yet unsolved mechanical problems or their treatment from an entirely new point of view. This paper is a first attempt towards this direction.

A new kinematic model for the closure equations of the generalized Stewart platform mechanism

Abstract: The paper deals with the direct position analysis of the six degrees of freedom parallel manipulator known as generalized Stewart Platform Mechanism. When a set of actuator displacements is given the mechanism becomes a statically determined structure and the analysis solves for the closure of the structure. The governing equations are non-linear and many solutions are possible. Kinematic models reported in the literature relate to systems of six equations in six unknowns, which are solved numerically because of their complexity. Based on a novel approach, a new kinematic model of the structure is presented in this paper. It leads to a system of three equations in three unknowns that greatly reduces the computational burden. Finally, a case study has been reported.

Stick-slip instability analysis

Abstract: A theoretical investigation of friction-induced self-excited oscillations for systems with one degree of freedom is proposed. The friction force is assumed as an odd function of the relative sliding velocity with a jump discontinuity at a value of zero for the relative sliding velocity. The friction characteristic is approximated with a piecewise linear function, i.e. straight line segments with a suitable slope. For the generic system belonging to the class in question, the stick-slip instability region is located on a suitable dimensionless map.

Difference equations approach to the analysis of layered systems

Abstract: The analysis of chain-like structures is tackled on the basis of difference equations. The advantages of such an approach are outlined. Difference equations for layered systems are derived both in terms of tractions and displacements. Interconnections with other methods are stated. Detailed analysis is given for the case when solution for a single layer is presented by Fourier series (integrals). Practical conclusions are driven at.

Static and dynamic behaviour of a rigid rotor on journal bearings

Abstract: This experimental investigation concerns the ‘static’ and dynamic behaviour of a rigid balanced rotor on journal bearings. Running conditions were chosen in order to obtain relatively high eccentricity ratios near the instability threshold speeds.

Internal variable formulation of a backward difference corrector algorithm for piecewise linear yield surfaces

Abstract: The formulation of a backward difference algorithm based on an internal variable description for piecewise linear yield surfaces is presented. Attention is restricted to an associated flow rule and isotropic material behaviour. The Tresca and Mohr-Coulomb yield surfaces with perfectly plastic and linear hardening rules are considered in detail. The algorithm has the advantages of being fully linked to the governing principles and avoiding the inherent problems associated with corners on the yield surface. It is used to identify return paths in stress space for the Tresca and Mohr-Coulomb yield surfaces with perfectly plastic and linear hardening rules. These return paths provide a basis against which heuristically developed algorithms can be compared.

Mass and rated characteristics of planetary gear reduction units

Abstract: A sample of planetary gear reduction units for industrial appliances is analysed from a statistical point of view. A correlation between the variables ‘mass/rated torque’ M/T and ‘rated torque’ T of the above machines is pointed out, which is independent of the number of stages z. Parallel correlations for single values of z are moreover pointed out between the variables P t/T 2/3 and T, where P t is the thermal capacity, with natural cooling, of the above machines. The results are compared with those found for a sample of ordinary gear units, chosen and analysed with the same criteria. The well-known advantages of planetary over ordinary gear units, as far as mass is concerned, are confirmed but a corresponding disadvantage is pointed out as far as thermal capacity is concerned.

A fractional derivative model for single-link mechanism vibration

Abstract: A model for a flexible pinned-free link is defined, which is based on a set of linear uncoupled equations and which is even valid for large rotations. A stress-strain relationship based on fractional derivatives is used to define the material properties. Experimental findings and numerical results are compared.

Repair of Masonry Structures

Abstract: Besides the traditional repair techniques of craftsmen for masonry structures, engineering methods and procedures such as grouting and reinforcing of old masonry are available. These technical measures can help to save the monumental value of historically important buildings more effectively than the procedure of dismantling and rebuilding; and, as a rule, they are distinctly less costly. Nevertheless, too much technical aid can destroy what is meant to be preserved. For that reason the investigations described in this paper on both improvement and development of engineer-like repair techniques have been focused on the goal of minimizing interventions and modern additions as far as possible.

On the numerical solution of structures with fractal geometry: The FE approach

Abstract: The scope of the present paper is to present the numerical aspects of the theory developed in [1]. The fractal geometry of structure(s) is approximated either through the IFS (iterated function system) method or through the FI (fractal interpolation) method. These approximations of the fractal through classical curves and surfaces are combined with the FEM in order to get numerical results for important technical problems, which cannot be satisfactorily treated by other methods.

Gear tooth stress analysis by the complex potentials method

Abstract: The complex potentials method of plane elasticity theory is applied to spur gear stress analysis. The concept is that the stress field in a gear tooth loaded by a point force can be obtained using the Boussinesq solution and a transformation function mapping a bell-shaped curve approximating the real tooth shape in the z-plane into the semi-infinite ζ-plane. The main features of an original computer code implementing this formulation are presented along with applications.

Weathering of rock, corrosion of stone and rusting of iron

Abstract: The central purpose of this paper is to present a survey of the extrinsic and intrinsic factors which influence the durability of masonry. In approaching this subject other themes are developed; in particular a study is made of the damage due to the volume changes which accompany all biological, physical and chemical changes. Light can be thrown on the corrosion of stone from a knowledge of the weathering of rocks throughout geological time, and this aspect is explored in the opening section of the paper. The final part of the paper consists of a study of the stresses and cracking of stone which can result from the expansive rusting of iron or steel reinforcements. Although mechanical damage dominates the discussion, some comments are made on the staining and dissolution of stone and examples are illustrated.

Modelling and simulation of nonpremixed turbulent flames

Abstract: A computational model for nonpremixed turbulent flames is presented. It is based on the conserved scalar approach and on a convenient specification of the probability density function, which allows the mean density to be recovered in closed (algebraic) form. The k-e1 model is adopted for turbulence, and the resulting equations for parabolic flows are solved via a block implicit algorithm. The computed results are compared with experimental data and other authors' predictions.

Bias in modal parameters due to aliasing in the time domain

Abstract: This paper deals with some aliasing effects in the time domain that can lead to unacceptable misestimations of modal parameters. When a frequency response function of a vibrating system is sampled and inverse Fourier transformed, the resulting impulse response is given by an infinite geometric series, the single term of which is the impulse response itself shifted in time. For this reason, some modal parameters, if estimated in the time domain, are biased; in particular, while the damping factor and the natural frequency are not influenced by the aliasing phenomenon, the magnitude and phase of the residue can be highly biased. Corrective terms are theoretically evaluated and their efficiency is shown in numerical simulations.

A note on uniqueness of the normal form for quasi-integrable systems

Abstract: We provide a simple argument that at non-resonant actions the normal form for a quasi-integrable Hamiltonian system, as defined by von Zeipel-Poicare and Lie perturbation algorithms, is unique.

Uniqueness theorems for the entropy in any differential body of complexity one

Abstract: By using in an essential way a certain condition of mutual physical equivalence between admissible response functions for the heat flux, in a previous paper uniqueness theorems were proved for the response functions of the internal energy and of the equilibrium stress, in connection with differential bodies of complexity 1. It was then pointed out that the equality expressing the vanishing of the static internal dissipation uniquely determines the rate of entropy variation in terms of the rate of the internal energy variation and of the equilibrium stress. This paper shows, in a threefold manner, that the last result also holds if one does not impose the condition of physical equivalence. The first proof uses the assumption that the response functions are Euclidean invariant. The second proof uses (i) the weaker assumption of Galilean invariance and (ii) a greater degree of smoothness of the response function for the internal energy. Both of these proofs use an axiom postulating the possibility of putting the body in contact with a vacuum. The third proof of the uniqueness property for the entropy is independent of the isolation axiom and uses the assumptions of the second proof. Whereas any of the first two proofs is a consequence of the uniqueness theorem for the internal energy-proved here by using the afore-mentioned axiom-the third proof does not depend on this theorem. Rather, disregarding the above isolation axiom, it implies that uniqueness of the entropy is compatible with non-uniqueness of both the stress and internal energy.

A mathematical model for an alloy solidification problem in the presence of an electric field

Abstract: A one-dimensional mathematical model for a process of solidification of a binary alloy in the presence of an electric field is studied. A situation in which the thermal properties of each phase are different and the latent heat is non-zero is considered. A quasi-static approximation for the thermal and electric fields is used. Local existence and uniqueness of a classical solution to the resulting free boundary problem are proved for two kinds of boundary conditions. Moreover, under particular hypotheses, the monotonicity of the free boundary and the global existence of the solution is proved.

Simple discrete-continuous model of machine support subject to transversal kinematic excitation

Abstract: In this paper a simple discrete-continuous layered model is proposed in dynamic investigations of a machine support subjected to transversal kinematic excitation. The model consists of an elastic finite-layered system connected to two rigid bodies. The layers undergo only shear deformation and may differ in their mechanical properties. In the discussion, the classical wave equation and the wave solution are applied after the estimation of the support dimensions by a comparison of the frequencies of free vibration for the Timoshenko beam and the classical wave equation. Numerical calculations are carried out for a four-layered model.