Dynamic analysis for axially moving viscoelastic panels

In this study, stability and dynamic behaviour of axially moving viscoelastic panels are investigated with the help of the classical modal analysis. We use the flat panel theory combined with the Kelvin–Voigt viscoelastic constitutive model, and we include the material derivative in the viscoelastic relations. Complex eigenvalues for the moving viscoelastic panel are studied with respect to the panel velocity, and the corresponding eigenfunctions are found using central finite differences. The governing equation for the transverse displacement of the panel is of fifth order in space, and thus five boundary conditions are set for the problem. The fifth condition is derived and set at the in-flow end for clamped–clamped and clamped-simply supported panels. The numerical results suggest that the moving viscoelastic panel undergoes divergence instability for low values of viscosity. They also show that the critical panel velocity increases when viscosity is increased and that the viscoelastic panel does not experience instability with a sufficiently high viscosity coefficient. For the cases with low viscosity, the modes and velocities corresponding to divergence instability are found numerically. We also report that the value of bending rigidity (bending stiffness) affects the distance between the divergence velocity and the flutter velocity: the higher the bending rigidity, the larger the distance.     

Minimax approach to structural optimization problems

The problem of determining the shape of an elastic body which is optimal for a whole class of loads is formulated. Its general solution scheme, based on a minimax approach, is indicated. The optimization problem of an elastic beam is studied for both the simply supported case and the cantilevered case, and some specific properties of the optimal shape are determined.The author is grateful to Professor A. I. Lurie, Professor F. L. Chernousko, and Professor G. S. Shapiro for useful discussions of the results.     

Static instability analysis for travelling membranes and plates interacting with axially moving ideal fluid

The out-of-plane instability of a moving plate, travelling between two rollers with constant velocity, is studied, taking into account the mutual interaction between the buckled plate and the surrounding, axially flowing ideal fluid. Transverse displacement of the buckled plate (assumed cylindrical) is described by an integro-differential equation that includes the centrifugal force, the aerodynamic reaction of the external medium, the vertical projection of membrane tension, and the bending force. The aerodynamic reaction is found analytically as a functional of the displacement. To find the critical divergence velocity of the moving plate and its corresponding buckling mode, an eigenvalue problem and variational principle are derived. Plate divergence, both within a vacuum and when submerged in an external medium, is investigated with the application of analytical and numerical techniques.     

Mechanics of Axially Moving and Vibrating in Transverse Direction Orthotropic Thermoelastic Web

We consider deformations and stability of axially moving orthotropic thermoelastic web. The web is modelled by a thin continuous plate moving with a constant velocity with small transverse vibrations and supported by a system of rollers. It is supposed that the plate is subjected to a combined thermomechanical loading including pure mechanical in-plane tension and also centripetal forces. Thermal strains corresponding to thermal tension and bending of the moving plate are taken into account. We formulate and analytically investigate the problem of out-of-plane thermomechanical divergence of orthotropic plate.     

Calculating the flows of a viscoplastic medium in tubes using the local variation method

This article considers the problem of the motion of a visco-plastic medium in tubes and channels. The results of [1, 2] are used, which present the variational formulation and the qualitative analysis of this problem. The method of local variations suggested in [3] is used for the numerical solution of the variational problem. A more detailed presentation of the algorithm of this method in application to boundary-value and variational problems is given in [4]. Results of calculations of certain concrete problems on an electronic computer are presented.The author wishes to express his sincere gratitude to F. L. Chernous'ko for the problem formulation and helpful counsel, and G. I. Barenblatt and S. S. Grigoryan for useful discussions.     

Optimization of Axisymmetric Membrane Shells Against Brittle Fracture

The questions investigated in this paper are related to an important class of problems of optimal design of structures against brittle fracture. The primary problem of axisymmetric shell optimization under fracture mechanics constraint is formulated as the weight (volume of the shell material) minimization under stress intensity constraints. Considered problems are characterized by incomplete information concerning crack size, crack location and its orientation. Taking into account the factor of incomplete information the paper presents the formulation of optimal shell design problem based on minimax (guaranteed) approach and provides some results of analytical investigation for thin-walled shells with through cracks.     

Instability of a Longitudinal Movement along the Cylindrical Surface for a Thermoelastic Web Simulated by Stretched and Heated String

On the basis of classical methods for mathematical physics and mechanics, the stability problem of a thermoelastic web moving at a constant speed without friction along a cylindrical surface is investigated. The web is modeled by a stretched and heated string. At a sufficiently high speed and heating of a string, a loss ofmotion stability and the stringmovement in a direction normal to the cylindrical surface occur. To study the instability, a static method based on the consideration of stationary nontrivial modes of stability loss, that is, on the study of the problem for bifurcation of solutions (eigenvalue problem) for the corresponding differential equations is used. The case of the web motion along the circular cylinder is separately considered and an expression for the critical velocity leading to the instability is found.     

COMPOSITION OF INDOOR AEROSOLS AT EMPEROR QIN''''S TERRA-COTTA MUSEUM, XI''''AN, CHINA, DURING SUMMER, 2004

Particle samples were collected in August 2004 both inside and outside Emperor Qin's Terra-Cotta Museum in Xi'an, China. Mass and chemical composition of total suspended particles (JSP, particles with aerodynamic diameter less than-30μm), PM2.5(particles with aerodynamic diameter <2.5μm) were determined. The average levels of indoor PM2.5 and TSP were 108.4 and 172.4 μg·m-3, respectively, with PM2.5 constituting 62.9% of the TSP mass. Sulfate ((32.4±6.2)%), organics ((27.7±8.0)%), and geological material ((12.5±3.4)%) dominated indoor PM2.5, followed by ammonium ((8.9±2.8)%), nitrate ((7.0±2.9)%), and elemental carbon (EC, (3.9±1.5)%). Particle size distribution varied with the number of tourists in the museum. The size of sulfate, organics, EC, nitrate, and ammonium was found to vary in the range of 0.43 to 3.3 μm in fraction. Ion balance indicated that the aerosol was acidic, with insufficient ammonium ions to neutralize the sulfuric and nitric acids. High concentrations of acidic aerosols will erode the Terra-cotta warriors and horses especially in the summer season with high temperature (30℃) and relative humidity (70%) and undesirable solar radiation inside the museum. More attention should be paid to protecting these precious antiques made 2000 years ago.     

Shape optimization in contact problems of the theory of elasticity with incomplete external loading data

The contact interaction without friction of an absolutely rigid punch with an elastic half-space is considered. The external loads on the elastic medium are not fixed in advance, but a set containing all the admissible forms of applied forces is assumed to be specified. Using a guaranteed (minimax) approach, problems of optimizing the shape of the punch from the condition that its mass is a minimum are formulated. Inequality-type constraints, imposed on the total force and moments applied to the punch from the elastic-medium side, are assumed. Using Betti's reciprocal theorem and calculating the “worst” case for different types of constraints, the corresponding forces are determined and the optimum shape of the punch is obtained in analytical form.     

Localization of eigenforms and limit transformations in problems of the stability of rectangular plates

Problems of the stability (buckling) of compressed rectangular plates are considered. It is assumed that evenly distributed compressive forces are applied to the opposite hinged edges, while the remaining two edges are not fastened and are free from external actions. It is shown that an increase in the length ratio of the hinged and unfastened sides in close proximity to the free boundary results in the localization of eigenforms corresponding to the lowest eigenvalues.