A description of the play, that may be applied to many other problems, is the basis of the model of a beam system presented in this paper. Equations describing the motion of the system of spring elements under dynamic load have been derived taking into consideration the play occurring between the elements. The equations have been derived according to the Hamilton's variational prineiple. The play has been treated as a force interacting between the elements whose value depends non-linearly on the mutual distance of the contact places. The function that defines the elements' interacting force may be easily enriched with elements responsible for the energy dissipation, eg. friction. The value of the method presented was proved by the carried out comparative analysis. The equation obtained has been used for an example which the finite elements method (modal technique) has been applied too for comparison. In order to make the comparison more complete, the calculations have been performed not only for the beam model but for a full spatial model as well (basing on the shell model).

Ch. Yi, P. Dietz and X. Hu. Vibrational power flow in beam-plate structures. Part I: Basic theory. CAMES 1998 (5)

This paper presents the expressions of Vibrational Power Flow (VPF) in Beam-Plate Structures (BPS). These expressions are derived based on structural dynamics. BPS is composed of a constant cross-section beam and a thin flat rectangular plate. In expressions the plate is four-edge simply suppored and four-edge fixedly supported, respectively. The numerical calculation of these expressions is implemented. IțIeanwhile, the different parameters of beam and plate are also considered in the calculation. Finally, the numerical solutions of VPF in BPS are compared with the corresponding the measured VPF. The numerical VPFs have a good agreement with the measured VPF: The results of VPF provide a tool for analyzing vibra- tional energy transmission between the support roller and the armored plate of tracked vehicles.
*Keywords:* vibrational power flow (VPF), mechanical mobilities, beam-plate structures (BPS).

Ch. Yi, P. Dietz and X. Hu. Vibrational power flow in beam-plate structures. Part II: Beam-plate structure with isolating components. CAMES 1998 (5)

The expressions of vibration power flow (VPF) in beam-plate structures (BPS) have been derived in the Part I of this paper. Based on those results got in Part I, Part II presents the expressions of VPF in BPS under the following three cases as shown Figs. 1, 2 and 3 below. (1) the isolating component 1 (ICl) being added at the free end of the beam; (2) isolating component 2 (IC2) being embedded between the beam and the plate; and (3) IC1 and IC2 being in BPS simultaneously (IC1 and IC2). According to these expressions, the corresponding numerical calculations are completed. The influence of parameter of the isolating components on VPF is also considered in this part. And then numerical calculations of VPF are verified by the measurements of VPF. Some valuable conclusions have been applied to vibration control of tracked vehicles mentioned in Part I.
*Keywords:* vibrational power flow (VPF), beam-plate structures (BPS), isolation.

Z. Sienkiewicz and B. Wilczynski. Minimum weight design of vertically vibrating 3-D machine foundation coupled to layered half-space. CAMES 1998 (5)

Dynamic optimization problem for a machine rigid block foundation on an inhomogeneous soil is considered. The soil deposit under the base of block corresponds to a layer with linearly varying properties overlying a uniform half space. Furthermore, the block may be surrounded by a backfill. The optimal designs of a vertically excited rectangular block foundation are found by iterative application of a sequential linear programming for a number of rationally inhomogeneous supporting media as well as for a uniform half space. It illustrates the problem of adequate modelling of the nature of the soil profile and provides an insight into the action of the soil-foundation-machine system from the point of view of the long-term satisfactory performance and safety.

O.D. Makinde and P. Sibanda. On suction driven flow and heat transfer in a pipe filled with porous media. CAMES 1998 (5)

The axisymmetric flow of a viscous fluid and heat transfer in a pipe filled with porous media driven by suction at the pipe wall is examined. For low suction Reynolds number flow, asymptotic solutions are developed. Using MAPLE, the solution series is extended and a bifurcation study is performed. Our results show that a decrease in the permeability of porous media may reduce the magnitude of heat transfer across the wall. The absence of real solutions of the given type between two turning points is also noticed and this gap of no solution disappears as the permeability of the porous media decreases.

M. Kowalczyk. New predictor-corrector method used as a solver for decohesion problem. CAMES 1998 (5)

In certain problems of loading of elastic-perfectly plastic thin sheets a continuous displacement solution may not exist. The evolution of plastic zone is then connected with the evolution of discontinuity lines in both velocity and displacement fields. In the present paper it is assumed that in the presence of discontinuity lines the localized plastic zones start to proceed. A numerical study of decohesion within thin elastic-plastic sheets is conducted to total collapse. It is shown, that the localized plastic flow may develop simultaneously with the diffuse plastic zones. The structural softening caused by decohesive cracks is coupled with a complex elasto-plastic deformation process, where the previously developed diffuse plastic zones are subjected to unloading. The post-critical analysis is performed using a new reliable algorithm of a continuation method. The algorithm is based on a rank analysis of the rectangular matrix of the homogeneous set of incremental equations.

S. Parman and H. Koguchi. Rest-to-rest attitude maneuvers of a satellite with flexible solar panels by using input shapers. CAMES 1998 (5)

A hybrid system of coordinates and a relatively general *Lagrangian formulation* for studying the dynamics and control of spacecraft with flexible members is developed. Versatility of the formulation is illustrated through a dynamical study of the satellite with two symmetrical flexible solar panels, where the *finite
element method* is used to describe elastic deformations of solar panels modelled as flat plate structures in bending. The performance of the satellite undergoing roll maneuver is simulated. Results indicate that, under an unshaped input, the maneuvers induce undesirable roll motion of the satellite as well as vibration of the solar panels. A *zero vibration input shaper* is then applied to reduce the largest magnitude of residual oscillation of roll motion. Once the shaped roll torque input is applied to the satellite, the performance improves significantly. When the longest distance of impulsing time sequences in the input shaper is close enough to the period of large amplitude vibration of flexible members, its maximum deflection during attitude maneuver will also be close enough to the amplitude of vibration with this period under the bang-bang input.

Z. Kulpa, A. Pownuk and I. Skalna. Analysis of linear mechanical structures with uncertainties by means of interval methods. CAMES 1998 (5)

One of the simplest ways of representation of uncertain or inexact data, as well as inexact computations with them, is based on *interval arithmetic*. In this approach, an uncertain (real) number is represented by an *interval* (a continuous bounded subset) of real numbers which presumably contains the unknown exact value of the number in question. Despite its simplicity, it conforms very well to many practical situations, like tolerance handling or managing rounding errors in numerical computations. Also, the so-called *alpha-cut* method of handling fuzzy sets membership functions is based on replacing a fuzzy set problem with a set of interval problems.
The purpose of this paper is to investigate possibilities of and problems with application of interval methods in (qualitative) analysis of linear mechanical systems with parameter uncertainties, in particular truss structures and frames. The paper starts with an introduction to interval arithmetic and systems of linear interval equations, including an overview of basic methods for finding interval estimates for the set of solutions of such systems. The methods are further illustrated by several examples of practical problems, solved by our hybrid system of analysis of mechanical structures. Finally, several general problems with using interval methods for analysis of such linear systems are identified, with promising avenues for further research indicated as a result. The problems discussed include estimation inaccuracy of the algorithms (especially the fundamental problem of matrix coefficient dependence), their computational complexity, as well as inadequate development of methods for analysis of interval systems with singular matrices.

P. Breitkopf, G Touzot and P. Villon. Consistency approach and diffuse derivation in element free methods based on moving least squares approximation. CAMES 1998 (5)

The paper concerns shape functions formulations in the scope of the recent methods generalizing finite elements and whose common feature is the absence of a mesh. These methods may also be interpreted as a generalization of the finite difference approach for irregular grids. The shape functions obtained by the Moving Least Squares and by the GFDM (Generalized Finite Difference Method) approach exhibit a number of interesting properties, the most interesting being a local character of the approximation, high degree of continuity and the satisfaction of consistencu contraints neccesary for exact reproduction of polynomials. In the present work we formulate the shape functions directly as solutions of the minimization of a weighted quadratic form subjected to the consistency contraints explicitly introduced by Lagrange multipliers. This approach gives similar results as the standard moving least squares algorithm applied to the Taylor series expansion where the consistency is automatically satisfied but is more general in the sense, that an explicit specification of wished properties permits an introduction of additional arbitrary constraints other than consistency. It also leads to faster and more robust algorithms by avoiding matrix inversion. On the other hand, the consistency based formulations naturally lead to diffuse (or incomplete) derivatives of the shape functions. They are obtained at a significantly lower cost than full derivatives and their convergence to extact derivatives is demonstrated.