Bogolyubov averaging and normalization procedures in nonlinear mechanics. IV

In this paper, we apply the theory developed in parts I-III [Ukr. Math. Zh.,46, No. 9, 1171–1188; No. 11, 1509–1526; No. 12, 1627–1646 (1994)] to some classes of problems. We consider linear systems in zero approximation and investigate the problem of invariance of integral manifolds under perturbations. Unlike nonlinear systems, linear ones have centralized systems, which are always decomposable. Moreover, restrictions connected with the impossibility of diagonalization of the coefficient matrix in zero approximation are removed. In conclusion, we apply the method of local asymptotic decomposition to some mechanical problems.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1044–1068, August, 1995.This research was partially supported by the International Science Foundation, grant No. UB2000.     

Comparative micromechanical analysis of nonelastic behavior of unidirectional plastics with tetragonal and hexagonal structures

A mathematical model for determining the effective elastic properties and describing the processes of inelastic deformation and damage accumulation of unidirectional fiber-reinforced composites with tetragonal and hexagonal structures is developed. A comparative analysis of the effective elastic moduli of glass, boron, organic, and carbon unidirectional plastics shows that, if the fiber volume fraction does not exceed 0.5, the effective elastic properties calculated by the models presented give closely related results. The calculation results for nonlinear fields of deformation and failure are presented and the limiting strength surfaces of fibrous glass plastics with hexagonal and tetragonal structures are obtained for different transverse loading paths. It is found that the structure of a composite affects significantly its strength properties.Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000).Perm' State Technical University, Perm', Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 455–464, July–August, 2000.     

Nonlinear constitutive behavior of orthotropic materials

The problem of optimal material orientation is studied in the case of nonlinear elastic materials. Optimal orientations corresponding to extreme (maximal or minimal) energy density are obtained for orthotropic materials. The material behavior (the stress-strain relation) is simulated in a general form, which includes, as particular cases, different versions of the power law stress-strain relations. The optimality conditions are derived for the general cases. Local and global extremums are determined for particular cases.Presented at the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000).Tartu University, Estonia. Published in Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 445–454, March–April, 2000.     

Positive difference operators of any preassigned order of approximation for the second derivative

Difference operators are stuied, constructed by the method of block approximation for the differential operator –d ²/dt ² with homogeneous first boundary conditions. For separators of any order of approximation the positivity property is established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1590–1597, November, 1992.     

Bending features of a sandwich panel with asymmetric structure under local loads

The bending characteristics of a composite panel with asymmetric layered structure under local surface loads are obtained. A refined version of the applied theory is developed using the analytical solution of the bending problem of a sandwich plate with arbitrary asymmetric structure under a point load. Local effects are investigated within the limits of a discrete model allowing for the specific character of elastic properties of a soft filler. The advantages of the solution are expressions of bending characteristics — layer curvatures, displacements, and stresses — in a closed form. It is shown that these characteristics can vary several times depending on the asymmetry parameters of the structure. Degeneration peculiarities of the solution, stemming from the slipping of layers or, otherwise, their rigid linking by the Kirchoff—Love hypothesis, as well as from account of the transverse shear and compression of the normal, are examined in line with the degeneration of geometric and physical parameters of the discrete model adopted. The results obtained are illustrated by curves and surfaces for the characteristics studied.Submitted for the 11th International Conference on the Mechanics of Composite Materials (Riga, June 11–15, 2000).Institute of Polymer Mechanics, Latvian University, Riga, LV-1006 Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 717–742, November–December, 1999.     

Transverse compression of PPTA fibers

Results of single transverse compression testing of PPTA and PIPD fibers, using a novel test device, are presented and discussed. In the tests, short lengths of single fibers are compressed between two parallel, stiff platens. The fiber elastic deformation is analyzed as a Hertzian contact problem. The inelastic deformation is analyzed by elastic-plastic FE simulation and by laser-scanning confocal microscopy of the compressed fibers ex post facto. The results obtained are compared to those in the literature and to the theoretical predictions of PPTA fiber transverse elasticity based on PPTA crystal elasticity.Presented at the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000).3TEX, Inc. 109, MacKenan Drive, Cary, North Carolina 27511, USA. Published in Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 533–544, March–April, 2000.     

Stability in first approximation of stochastic systems with after effect : PMM vol. 40, n 6, 1976, pp. 1116–1121

The theorem on existence of the Liapunov functionals and the theorem on stability in first approximation for a stochastic differential equation with aftereffect are proved.The suggestion of the replacement of Liapunov functions by functionals [1] in the investigation of the stability of ordinary differential equations with lag, has been widely utilized in dealing with determinate systems, as well as in the case of linear and nonlinear stochastic systems (see e. g. [2 – 11]). Results concerning the stability in the first approximation were obtained for stochastic systems in [12 – 18] and others. Use of Liapunov functionals for the differential equations with aftereffect was first encountered in [1, 19, 20] where the inversion theorems were proved and conditions for the stability in first approximation were obtained.Below a stochastic differential equation with aftereffect is investigated where the random perturbations represent an arbitrary process with independent increments.     

On Distribution of Error of Linear Wavelet Estimator of Probability Density

We establish the asymptotic normality of the squared L ²-norm of the approximation error of a linear wavelet estimator of the density of a distribution. The calculations are based on the smallness of correlations between the coefficients of the high-frequency part of the multiresolution expansion of the estimator.Supported by the FCT Foundation (Portugal) in the framework of the project Probability and Statistics (2000–2002), Centro de Matematica, Universidade da Beira Interior.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 2, pp. 184–207, April–June, 2005.     

A 3/4-Approximation Algorithm for Multiple Subset Sum

The Multiple Subset Sum Problem (MSSP) is the variant of bin packing in which the number of bins is given and one would like to maximize the overall weight of the items packed in the bins. The problem is also a special case of the multiple knapsack problem in which all knapsacks have the same capacity and the item profits and weights coincide. Recently, polynomial time approximation schemes have been proposed for MSSP and its generalizations, see A. Caprara, H. Kellerer, and U. Pferschy (SIAM J. on Optimization, Vol. 11, pp. 308–319, 2000; Information Processing Letters, Vol. 73, pp. 111–118, 2000), C. Chekuri and S. Khanna (Proceedings of SODA 00, 2000, pp. 213–222), and H. Kellerer (Proceedings of APPROX, 1999, pp. 51–62). However, these schemes are only of theoretical interest, since they require either the solution of huge integer linear programs, or the enumeration of a huge number of possible solutions, for any reasonable value of required accuracy. In this paper, we present a polynomial-time 3/4-approximation algorithm which runs fast also in practice. Its running time is linear in the number of items and quadratic in the number of bins. The core of the algorithm is a procedure to pack triples of large items into the bins. As a byproduct of our analysis, we get the approximation guarantee for a natural greedy heuristic for the 3-Partitioning Problem.     

Free oscillations of plastic structures taking into account the elastic and damping properties of the material

The problem of the free oscillations of a plastic bar with a weight at one end is examined. A method is proposed for calculating the free oscillations of plastic structures, taking into account the special elastic properties and hysteresis losses of plastics. The original equation for the motion of the system, simplified by certain transformations and assumptions, is solved to a first approximation by the Krylov-Bogolyubov asymptotic method (following G. S. Pisarenko). The calculated data are in good agreement with the experimental findings.Mekhanika polimerov, Vol. 1, No. 1, pp. 136–145, 1965     

Worst case complexity of multivariate Feynman–Kac path integration

We study the multivariate Feynman–Kac path integration problem. This problem was studied in Plaskota et al. (J. Comp. Phys. 164 (2000) 335) for the univariate case. We describe an algorithm based on uniform approximation, instead of the L₂-approximation used in Plaskota et al. (2000). Similarly to Plaskota et al. (2000), our algorithm requires extensive precomputing. We also present bounds on the complexity of our problem. The lower bound is provided by the complexity of a certain integration problem, and the upper bound by the complexity of the uniform approximation problem. The algorithm presented in this paper is almost optimal for the classes of functions for which uniform approximation and integration have roughly the same complexities.     

Solution of two-dimensional boundary-value problems of the theory of thin composite shells

Discrete analogues of the boundary-value problems of a two-dimensional refined theory of anisotropic shells taking into account the transverse shear deformation are presented. The systems of resolving equations in the general form are obtained for arbitrary nonshallow shells of variable curvature whose coordinate lines of the reduction surface may not coincide with the lines of principal curvatures. The algebraic problems of determining the stress-strain state in shells made of composite materials with stress concentrators under various kinds of loads are obtained as particular cases of the schemes presented. The results of calculating the stress concentration near a nonsmall circular hole in a transversely isotropic nonshallow spherical shell under internal pressure are presented. The dependences of stress concentration factors on the hole dimension and on a change in the shear stiffness of the shells are studied. A comparison between the calculation results obtained within the framework of the theories of shallow and nonshallow shells is given.Presented at the 11th International Conference on the Mechanics of Composite Materials (Riga, June 11–15, 2000).Timoshenko Institute of Mechanics, Ukranian National Academy of Sciences, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 465–472, July–August, 2000.     

Dynamic stability of a hinged viscoelastic bar

The averaging method is used to show that the region of dynamic stability of a viscoelastic bar contains the region of dynamic stability of the elastic bar.M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 2, pp. 293–300, March–April, 1971.     

Global Error Estimators for Order 7, 8 Runge–Kutta Pairs

Dormand, Prince and their colleagues [3–5] showed in a sequence of papers that the approximation of an initial value differential system propagated by a Runge–Kutta pair, together with a continuous approximation obtained using additional derivative values could be utilized to obtain estimates of the global error. They illustrated the results using pairs of orders p–1 and p for several values of p. The current authors [13] have developed a more direct representation of the order conditions, characterized families of global error estimators for Runge–Kutta pairs of arbitrary values of p, and showed that efficient global error estimating Runge–Kutta methods are based on the nodes of a Lobatto quadrature formula. Here, formulas for a good 7, 8 pair, interpolants of each of orders 7 and 8, and global error estimators of orders 10 and 12 illustrate how to obtain global error estimates of orders 9, 10, or 11, for arbitrary initial value systems. One set of graphs indicates that the stated order of the global error estimators is achieved numerically, and a second set illustrates the relative efficiency for several global error estimators when the approximation is propagated with a variable stepsize.     

On the rate of rational approximation of functions on tangent continua

Upper and lower bounds are established for the rate of rational approximation of functions piecewise analytic on tangent continua. In some special cases, these bounds are coordinated depending on the mutual location of the continua.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1522–1533, November, 1993.     

Ultrasonic method of determining the elastic characteristics of anisotropic materials

An ultrasonic method of determining the elastic characteristics, developed for isotropic materials and based on the measurement of the angles of total internal reflection of ultrasound at a liquid-solid interface, is applied to an anisotropic Compreg of the DSPA type. The results obtained for the shear modulus using the ultrasonic method at 2.5 MHz and an independent method based on measuring the frequency of the damped torsional vibrations of a prismatic bar at 10–30 Hz are compared. It is shown that the shear modulus varies only slightly (10–12%) on the frequency interval investigated.Leningrad Kirov Forest Engineering Academy. Translated from Mekhanika Polimerov, No. 4, pp. 728–731, July–August, 1969.     

Vibrational frequency shifts in the stretched polypropylene molecule

The GF matrix method is used to compute the changes in the skeletal vibration frequencies for a simplified model of isotactic polypropylene stretched by a loadf. The series of frequencies obtained atf=0 are identified with those experimentally observed. A linear approximation is used for the deformation potential energy atf=0–2·10^–4 dyneperchain. The results of the calculations are presented in graph form.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, Vol. 4, No. 4, pp. 586–594, July–August, 1968.     

A theorem concerning the joint approximation of real numbers by rationals

In this note we prove a theorem concerning the joint approximation of irrational numbers by rationals.Translated from Matematicheskie Zametki, Vol. 11, No. 3, pp. 313–317, March, 1972.     

Approximation of the eigenvalues of a singular Sturm-Liouville problem nonlinear in the parameter

Two auxiliary problems are solved on a truncated interval in the domain of definition with boundary conditions of first and second kind. The eigenvalues of the auxiliary problems provide a two-sided approximation of the eigenvalues of the original problem.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 11–16, 1990.     

Some problems in approximation theory

This dissertation consists of three parts. The first part is devoted to the approximation of a fixed element by a fixed approximating set. In the second part we study approximation by finite sets (Chapter I, Secs. 3–6) and by finite-dimensional subspaces (Chapter II, Secs. 7–15). Selected problems with geometric (Secs. 1–3) and variational (Secs. 4–6) content are considered in the third chapter.The author's summary of dissertation for the degree of Doctor of Physicomathematical Sciences. The dissertation was defended on December 18, 1970, at a session of the Scientific Council of the Mechanics-Mathematics Faculty of Moscow State University. The official opposition was made up of Academician A. N. Kolmogorov, K. I. Babenko, and S. B. Stechkin.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 593–607, May, 1971.