Approximate methods for calculating the impact of a massive body on a plate lying on the base

Normal impact of a massive body on a uniformly stretched plate lying on the base is investigated. A hinged round or rectangular plate on an elastic base, or an infinite plate on the surface of an ideal incompressible fluid is considered. The solution to the elastoplastic impact is in good agreement with numerical calculations and experimental data. With a small parameter of elastic collapse, that is, with the developed local plastic deformations, a solution to the problem of impact with rigid-plastic local collapse can be used. Approximate formulas for calculating the main characteristics of rigid-plastic impact are set up.

Analysis and Optimization of Damping Properties of Constrained Layer Damping Structures with Multilayers

Constrained layer damping (CLD) structures, which are one of the composite structures with a softer viscoelastic material (VEM) layer sandwiched between a elastic base layer and a relatively stiffer constraining layer, are widely used in engineering applications for reducing vibration and noise radiation. To accurately predict and effectively control vibration and properly and quickly determine the design parameters, optimal designs for the CLD structures are necessary. The optimal designs depend on thoroughly understanding the damping characteristics of the CLD structures. In addition, for some cases, CLD structures with multi-constrained VEM layers are needed to suppress vibration more effectively. In this paper, an effective modeling method to accurately describe the damping properties and a quick optimization design method using COMSOL were proposed for CLD structures with multilayers in detail. The effects of nondimensional thickness ratios of the VEM and constraining layer to the base layer on the damping properties of CLD structures were analyzed. For CLD structures with different configurations, different constraints were selected to obtain the maximized damping in the optimization design. The conclusions from this research provide an insight into the effects of thicknesses of VEM and constraining layers on the damping properties of CLD structures regardless of its size. The modeling and optimal methods using COMSOL in this paper are not limited to CLD structures and can be used by other structures also.

An infinite plate loaded with a normal force moving along a complex open trajectory

Introduction. A method for solving the problem on the action of a normal force moving on an infinite plate according to an arbitrary law is considered. This method and the results obtained can be used to study the effect of a moving load on various structures.Materials and Methods. An original method for solving problems of the action of a normal force moving arbitrarily along a freeform open curve on an infinite plate resting on an elastic base, is developed. For this purpose, a fundamental solution to the differential equation of the dynamics of a plate resting on an elastic base is used. It is assumed that the movement of force begins at a sufficiently distant moment in time. Therefore, there are no initial conditions in this formulation of the problem. When determining the fundamental solution, the Fourier transform is performed in time. When the Fourier transform is inverted, the image is expanded in terms of the transformation parameter into a series in Hermite polynomials.Results. The solution to the problem on an infinite plate resting on an elastic base, along which a concentrated force moves at a variable speed, is presented. A smooth open curve, consisting of straight lines and arcs of circles, was considered as a trajectory. The behavior of the components of the displacement vector and the stress tensor at the location of the moving force is studied, as well as the process of wave energy propagation, for which the change in the Umov-Poynting energy flux density vector is considered. The effect of the speed and acceleration of the force movement on the displacements, stresses and propagation of elastic waves is investigated. The influence of the force trajectory shape on the stress-strain state of the plate and on the nature of the propagation of elastic waves is studied. The results indicate that the method is quite stable within a wide range of changes in the speed of force movement.Discussion and Conclusions. The calculations have shown that the most significant factor affecting the stress-strain states of the plate and the propagation of elastic wave energy near the concentrated force is the speed of its movement. These results will be useful under studying dynamic processes generated by a moving load.

Design of a beam of variable cross-section on the elastic base by the quasi-analytical method considering boundary conditions

Purpose. Improvement of the quasi-analytical method of nonlinear differential equation solution and its approbation with reference to beams of variable cross-section on the elastic base with two base factors. Research methods. Boundary conditions in the form of required number of correspondently transformed equations are added to the system of the linear algebraic equations which results from substitution of approximating function with constant factors (for example – power function) in the nonlinear differential equation and fixation of a set of variable values. The total number of the equations have to correspond to quantity of constant factors if the further solution will be carried out by an analytical method. Results. Deflection diagram of a trapezoid concrete beam with rectangular cross-section of variable height on the elastic base with two base factors has been calculated during approbation. Average solution error was equal to 0.06%. Distributions of the bending moments and normal stresses along the beam have been researched. Scientific novelty. The authors did not meet in literature such method of nonlinear differential equation solution. Practical value. The quasi-analytical method with realised consideration of boundary conditions that has been offered can be used for solution of differential equations of any order with various types of nonlinearity, including calculations of beams of variable cross-section on the elastic base.

Dynamic Vibration Extinguished on a Viscously Elastic Base

The aim of the work is to develop algorithms and a set of programs for studying the dynamic characteristics of viscoelastic thin plates on a deformable base on which it is installed with several dynamic dampers. The theory of thin plates is used to obtain the equation of motion for the plate. The relationship between the efforts and the stirred plate obeys in the hereditary Boltzmann Voltaire integral. With this, a system of integro-differential equations is obtained which is solved by the method of complex amplitudes. As a result, a transcendental algebraic equation was obtained to determine the resonance frequencies, which is solved numerically by the Muller method. To determine the displacement of the point of the plate with periodic oscillations of the base of the plate, a linear inhomogeneous algebraic equation was obtained, which is solved by the Gauss method. The amplitude - frequency response of the midpoint of the plate is constructed with and without regard to the viscosity of the deformed element. The dependence of the stiffness of a deformed element on the frequency of external action is obtained to ensure optimal damping of vibrational vibrations of the plate.

Temperature Measurement at Curved Surfaces Using 3D Printed Planar Resistance Temperature Detectors

In this article, novel 3D printed sensors for temperature measurement are presented. A planar structure of the resistive element is made, utilizing paths of a conductive filament embedded in an elastic base. Both electrically conductive and flexible filaments are used simultaneously during the 3D printing procedure, to form a ready–to–use measuring device. Due to the achieved flexibility, the detectors may be used on curved and irregular surfaces, with no concern for their possible damage. The geometry and properties of the proposed resistance detectors are discussed, along with a printing procedure. Numerical models of considered sensors are characterized, and the calculated current distributions as well as equivalent resistances of the different structures are compared. Then, a nonlinear influence of temperature on the resistance is experimentally determined for the exemplary planar sensors. Based on these results, using first–order and hybrid linear–exponential approximations, the analytical formulae are derived. Additionally, the device to measure an average temperature from several measuring surfaces is considered. Since geometry of the sensor can be designed utilizing presented approach and printed by applying fused deposition modeling, the functional device can be customized to individual needs.

EXPERIMENTAL-THEORETICAL CALCULATION OF THE STRESS-DEFORMED STATE OF THE BAR ON THE ELASTIC BASIS

Any strip foundation, and sometimes a slab foundation can be consideredas a be a mon an elastic base. And if with calculation of a beam – the tape base problems practically do not a rise because loading on the tape base as a rule is evenly distributed, and means and the base be haves, a sabsolutelyrigid beam. Then when considering a section of the foundation with unevenlyapplied load, some problems may arise. Today there are many publications on studies of the use of beams with an elastic base in the field of construction and the application of the features of the method of calculating the stress-strain state in the field of design. As is known, from classical soil mechanics, when a load is applied to a flexible slab, its center gives a draft of 1.24-1.57 times greater than the edges. Note that this effect can be explained by the contour work of the base and its uneven stiffness in the central and peripheral zone of the slab foundation. It should be noted that today in most cases the method of BN Zhemochkin [6] is used as the basis for the analytical solution of the problem of the interaction of the beam with the elastic base, which combines an engineering approach and strict solutions of the theory of elasticity. According to research, the method is based on the replacement of the continuous problem of the interaction of the foundation beam with the soil base, discrete, with a limited number of calculated sections within the beam and approximation of the smooth reactive pressure curve stepped with a constant value within a single section. Such simplifications allow to calculate various engineering problems on interaction of beams and plates, including with difficult geometry with the set degree of accuracy, bypassing difficult differential and integral calculations.