On the mechanical impedances for torsional wave propagation in a solid elastic cylinder

An analysis of torsional wave propagation in a solid elastic cylinder from a new standpoint is undertaken. It reveals a set of mechanical impedanceversus wave-length curves for each member of the set signifying a separate mode of torsional wave propagation. A few expressions for the input impedances for terminated cylinders have been obtained and some numerical results for first few lower order modes have been presented. The method has been successfully utilized for evaluating impedances for a combination of two cylinders perfectly bonded together by an end-to-end joint.     

A matrix algorithm for computing the coefficients of the transformation of waves on a reinforcing rib

In the work a matrix algorithm is proposed for computing the transformation coefficients (the coefficients of reflection, transmission, and transformation into waves of other types) for any vector fields and any angles of incidence. The method is illustrated by computing the transformation coefficients of vibrational waves on a rectilinear reinforcing rib reinforcing a plate and an annular rib reinforcing a thin, cylindrical shell.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 134–151, 1979.     

A simple analytic method for computing the natural frequencies and mode shapes of tall buildings

A simple, yet accurate analytical approach based on energy principles is developed for quick computation of natural frequencies and mode shapes of multistory buildings constructed using framed tube, shear core and double belt trusses systems. The approach here is based on development of a continuum model that would be equivalent, in major motions, to the actual multistory building. Models studied here are cantilever beams with concentrated moments placed at belt truss locations. Governing equation and boundary conditions of the equivalent beam and moment system were derived using the energy method and Hamilton’s principle. Separation of variables technique is then applied to model’s partial differential equation to obtain the required eigensystem. Robustness and correctness of the proposed method are demonstrated through several numerical examples. Here, 40, 55 and 70-storey tall buildings with combined system of framed tube, shear core and double belt trusses, in which the results obtained from the proposed method, are compared with those obtained from three-dimensional analyses using SAP2000 software. Comparative analyses reveal that the proposed method is simple and efficient; and it provides reasonably accurate results quickly, a feature that is vital during the early stages of building design.     

Algorithm for computing the covering constant of a linear operator on a cone

An algorithm for computing the covering constant for the restriction of a linear operator to a cone defined by a finite set of inequalities is proposed. After a finite number of steps, the algorithm reduces the original problem to one of finding the eigenvalues of linear operators.     

Dispersion of the mechanical characteristics of plastics at various temperatures

The results of an investigation of the dispersion (standard deviation s and coefficient of variation V) of the static and dynamic strengths of plastics are described. The effect of temperature, state of stress, and type of plastic on the dispersion characteristics is considered.Moscow. Translated from Mekhanika Polimerov, No. 1, pp. 29–34, January–February, 1970.     

Comparative analysis of the structural order and defectiveness of reinforcing fibers on the basis of normalized values of mechanical properties

Experimental data on mechanical properties referred to their limiting values for a flawless polymer crystal with long molecular chains is used in a comparative analysis of the degree of ordering of the structure (crystallinity, orientation) and the defectiveness of the reinforcing fibers. The actual elastic moduli and limiting (theoretical) elastic modulus are used to obtain coefficients that characterize the overall order of the structure of the fibers but are independent of their defectiveness. Values of true strength in tension and the limitingly attainable or theoretical strength are used to calculate conditional coefficients that depend both on the overall order and the defectiveness of the fiber structure. The difference in the coefficients makes it possible to detect dangerous local defects that lead to fiber failure. Results are presented from calculations for more than 20 types of reinforcing fibers. Despite the approximate nature of these representations, the data that is obtained permits comparisons of different types of fibers, characterization of their quality, and evaluation of the degree of perfection of the fiber production technology.Paper presented at the IX International Conference on the Mechanics of Composite Materials, Riga, October, 1995.St. Petersburg State University for Technology and design, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 4, pp. 444–453, July–August, 1996.     

Homotopies for resolutions of skew-hook shapes

We present characteristic-free resolutions and splitting homotopies for the Weyl modules associated to skew-hook shapes.     

Reversal of the reinforcing effect of fillers and the relation between the flexibility of the macromolecules and the mechanical properties of filled polymers

The author describes various effects which he has detected involving the reversal of the reinforcing action of fillers in polymers in different subregions of the glassy state. It is suggested that these effects can be explained in terms of, among other things, the flexibility of the macromolecules. The concept of an optimal flexibility at which the maximum value of a given mechanical property can be realized is introduced. It is shown that it is possible, in principle, to reinforce a polymer in any subregion of the glassy state and even below T_b, as well as to obtain equal-strength, equal-modulus, equally reinforced, etc., composites containing different (large and small) amounts of fillers and plasticizers.     

Bounds for computing the tame kernel

The tame kernel of the of a number field  is the kernel of some explicit map , where the product runs over all finite primes  of  and is the residue class field at . When is a set of primes of , containing the infinite ones, we can consider the -unit group  of . Then has a natural image in . The tame kernel is contained in this image if  contains all finite primes of  up to some bound. This is a theorem due to Bass and Tate. An explicit bound for imaginary quadratic fields was given by Browkin. In this article we give a bound, valid for any number field, that is smaller than Browkin's bound in the imaginary quadratic case and has better asymptotics. A simplified version of this bound says that we only have to include in  all primes with norm up to  , where  is the discriminant of . Using this bound, one can find explicit generators for the tame kernel, and a ``long enough' search would also yield all relations. Unfortunately, we have no explicit formula to describe what ``long enough' means. However, using theorems from Keune, we can show that the tame kernel is computable.

     

Nonvanishing of Kronecker coefficients for rectangular shapes

We prove that for any partition (λ₁,…,λd²) of size ?d there exists k?1 such that the tensor square of the irreducible representation of the symmetric group Sk?d with respect to the rectangular partition (k?,…,k?) contains the irreducible representation corresponding to the stretched partition (kλ₁,…,kλd²). We also prove a related approximate version of this statement in which the stretching factor k is effectively bounded in terms of d. We further discuss the consequences for geometric complexity theory which provided the motivation for this work.     

Effect of sign-variable thermocycling on the mechanical properties of carbon-fiber plastics with various layups

Conclusion An economic method for the ST of carbon-fiber-reinforced plastics is proposed in this study. It is established as a result of investigations of the mechanical properties of carbon-fiber-reinforced plastics with different layups, which are subjected to ST in the range to 100 cycles, that in the majority of cases, the number and duration of thermal cycles has no effect on the stiffness and strength in tension, compression, bending, and shear. The effect of number of thermal cycles was manifested only on the stiffness in tension and bending and also on the tensile, compressive, and bending strengths of the material with the obliquely reinforced structure loaded in the direction of the diagonal. It is permissible to use the dynamic method to assess variations in the stiffness of a material subjected to ST.Translated from Mekhanika Kompozitnykh Materialov, Vol. 29, No. 1, pp. 66–76, January–February, 1993.