## No 2 (2018)

**Year:**2018**Articles:**13**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/13**DOI:**https://doi.org/10.15593/perm.mech/2018.2

Numerical damping of oscillations of beams by using multiple point actuators

#### Abstract

Methods of damping the oscillations of complex mechanical systems’ elements, such as strings, membranes, beams, plates, have intensively been developing since the 1970s. In particular, the beam oscillations are modeled by the fourth-order partial differential equation, which is Petrovsky-hyperbolic. The minimized functional is the energy integral of an oscillating beam. Control is implemented via certain function appearing in the right side of the equation. Previously, it was shown that the solution of the problem exists for any given damping time, but as soon as this time decreases, the optimal control becomes more complicated. To obtain approximate numerical solutions, the so-called point actuators were considered. Control was considered via single point actuator placed at some point of the beam, but in this case it turned out that it is not always possible to dampen it. Therefore, control was also considered via point actuator moving along a small section of the beam. However, the implementation of such an actuator is very difficult. In this work, the numerical damping of beam oscillations is implemented via several fixed point actuators. Computational algorithms have been developed on the basis of the matrix sweep method and the second order Marquardt minimization method of finding the minimum of functions of many variables. To find a good initial approximation, when minimizing the energy integral, empirical functions with a small number of variables are used. This made it possible to significantly reduce the calculation time of the task. The examples of damping the oscillations via a different number of actuators are given. It is shown that the amplitude of the oscillations of any control functions increases with the reduction of the given damping time. The examples of damping the oscillations in the presence of constraints on control functions are given; in this case the minimum damping time exists. The damping of oscillations is considered also in the case when different combinations of actuators are switched on at different time intervals of oscillation damping.

**PNRPU Mechanics Bulletin**. 2018;(2):5-15

Modelling of the amplitude of transverse vibrations of a homogeneous rod upon impact against a rigid obstacle taking into account its own weight

#### Abstract

The accurate solution of dynamic problems in a nonlinear formulation is associated with certain mathematical difficulties. The available research works devoted to the problem solution on the dynamic deflection and stability of the rods under shock wave loading have considered perfect rods with a straight axis and have not considered any external effects. The account of additional external factors influencing the state of a shock system makes this task more complicated. By using the method of the initial parameters and the wave model of the longitudinal impact, this work attempts to develop the calculation method for dynamic deflection of a homogeneous rod producing lateral vibrations under the longitudinal impact against a hard obstacle, also taking lateral loads into account. Depending on the pre-impact state of the rod, the corresponding initial parameters are set, such as initial deflection, rotation, bending moment and shear force. By using the wave model of the longitudinal impact and method of characteristics, it becomes possible to take the short-term exposure of the shock compressing load into account, after which the rod is considered as a vibratory system having the initial velocity and the deviation from the equilibrium position. In this state, the rod makes transverse shock vibrations. The present article describes the calculation method of the amplitude of transverse vibrations of the rod with a constant longitudinal stiffness under the impact against an absolutely rigid barrier. It is of a particular interest to use this method in order to calculate the system of uniform and stepped rods making lateral vibrations under the longitudinal impact. Such systems of rods are often found in present impact mechanisms, they are used as structural elements, e.g. core elements of trusses, frames, pillars, columns, piles, etc.

**PNRPU Mechanics Bulletin**. 2018;(2):16-23

The properties of the structure of disperse-filled polymer composites

#### Abstract

The effect of the structure on the properties of the metal-polymer composite consisting of a polymer matrix in the form of epoxy resin (ED-20) with butadiene-styrene rubber (BSK), dispersed-filled particles of nanomedium is investigated. Within the framework of the fractal analysis, the actual diameter of the aggregates of the initial filler particles is calculated for different degrees of concentration and size of the filling particles, as well as for different compositions of a polymer matrix. At the same time, the concept of the structure of a polymer composite is used as a set of two fractals (multi-fractals), which allows one to determine the nature of the change in the plasticity of a polymer matrix and simultaneously to reveal the main factors affecting the degree of perturbation of its structure. Using the fractal analysis methods, we investigated the influence of the factors on the fractal dimension of the surface of aggregates of the initial filler particles and on the nature of its dependence on both the degree of aggregation and the fractal, the dimensionality of the aggregate frame of the particle aggregate. The proposed approach allows to predict the final parameters of aggregates of nanoparticles as a function of the size of the initial particles, their concentration and chemical properties of the surface of the polymer matrix. Using the scaling nature of the size distribution of coils of polymeric macromolecules, it is shown that when polymeric composites are filled with dispersed micro- and nanoscale particles, the aggregates of these particles form a fractal framework, that is analogous to a fractal lattice. The degree of local order of the structure determined in the early cluster model controls the most important properties of the polymer matrix and the composite, in general. As the size of the polymer and metal particles decreases, substantially all physical and chemical properties of both the original components and the resulting composite material change substantially. The adjustment of the ratio between the polymer matrix and the fillers, with the adaptation of the synthesis condition to a given characteristic value, promotes the wide use of metal-polymer composites and enables the creation of new technologies for designing materials with the required properties, including a decrease in the independence of the products obtained from them. The use of fractal dimensions to characterize the structural equations of polymers makes it possible to obtain a number of quantitative analytical relationships between them.

**PNRPU Mechanics Bulletin**. 2018;(2):24-31

Stability of orthotropic doubly curved shallow shells with a movable hinged fixing of the border

#### Abstract

Shell structures are often used in different fields and their studies are important for many applications. To eliminate the stress concentration near the contour, especially at the corner points of the shell, the border of the structure has a fixed movable hinge support. This paper considers double-curved shallow shells, square in a plan, made from orthotropic materials with their border having a fixed movable hinge support. The mathematical model is based on the hypotheses of the theory of Tymoshenko - Reisner shells, which takes into account the transverse shifts and represents the mixed-form equations. In addition, the model takes into account the geometric nonlinearity. To solve the system of differential equations we used the method of Bubnov - Galerkin, that makes it possible to reduce the problem to the solution of a system of nonlinear algebraic equations. The convergence of the method is also shown for the increasing number of terms of approximation. The resulting system is nonlinear and solved by the Newton method. The developed algorithm is implemented in Maple 2017. The proposed algorithm is verified by comparing the calculation results of the test problem with the result obtained by other authors. The combination of the load-deflection curve showed a good consistency of the data. The stability analysis of three variants of shallow shell structures with a double curvature is carried out; each of them is made of four orthotropic materials. The outer uniformly distributed transverse load acts on the shell, the border fixing is hinged-movable. For all the structures studied, the critical buckling load, the maximum value of the deflection, corresponding to this load, and load-deflection curves are given. Conclusions are drawn about the stress-strain state of the shells under consideration.

**PNRPU Mechanics Bulletin**. 2018;(2):32-43

Mechanics of collisions of solids: influence of friction and adhesion. I. Review of experimental and theoretical works

#### Abstract

Collisions of solid bodies are of significant interest for a great variety of physical and engineering applications. This review is devoted to non-elastic collisions of solid bodies when the energy dissipation is caused by the inner or interface friction, plasticity, adhesion, or other damping mechanisms. We consider only two-particle collisions. This impact problem can be reduced to the collision of a solid particle with a half-space. We discuss the collision without slip (infinite coefficient of friction) and in the presence of a finite coefficient of friction, as well as in the presence of adhesion between contacting bodies. A review is given of the previous theoretical and experimental work on both elastic and non-elastic impacts. However, the focus of the present work is on collisions of elastic particles. We consider the general oblique impact with non-zero normal and tangential velocity components of impacting particles. Analytical expressions for the restitution coefficient obtained by many authors are presented. Generally, the restitution coefficient depends on adhesive and plastic properties of contacting bodies. High velocity impact with destruction of particles does not belong to the scope of the paper. This paper is the first part of a two-parts-review. In the second part the results of analytical and numerical simulations will be presented, which have been obtained by the authors using the method of dimensionality reduction, allowing for reduction of three-dimensional contact problems to a much simpler equivalent problem in a one-dimensional space.

**PNRPU Mechanics Bulletin**. 2018;(2):44-61

The aerodynamic component of the damping of cantilevered test specimens oscillating near a rigid shield

#### Abstract

A numerical technique has been developed to process the experimental vibrogram of damped flexural vibrations of test specimens to determine the experimental lower frequency and the amplitude dependence of the logarithmic decrement of oscillations, which determines the damping properties of the test specimen. To determine the logarithmic decrement, the experimental envelope of damped flexural oscillations of the specimen’s free end has been used. The experimental envelope was approximated by the sum of two exponents with four independent parameters, which was determined by the direct search for the minimum of the objective function that depends on these parameters. Numerical experiments were performed to show the reliability and sufficient accuracy of the developed procedure. It is shown that to determine the experimental aerodynamic component of the damping of a test specimen reliably, it is necessary that a test material has stable and low damping properties. Such requirements are fully met by duralumin. The experimental amplitude dependences of the logarithmic decrement for a series of duralumin test samples located at different distances from an absolutely rigid shield have been determined. On their basis, a theoretical-experimental method for determining the aerodynamic component of damping has been proposed by modifying the structural formula obtained earlier for determining the aerodynamic component of the damping of a thin rectangular planar elongated plate (test specimen) in the absence of a shield. Three additional parameters determined from the condition of a minimum objective function representing a quadratic discrepancy between the calculated and experimental values of the aerodynamic component of the damping of the test sample for several values of the length of its working part, and the distance to the rigid shield has been introduced into the formula. To find the minimum of the objective function, the Hook-Jeeves method has been used. This method does not require calculating its gradient at the current point in the space of the required parameters. Polynomial dependences of the found parameters on the dimensionless lower vibration frequency of the test specimen and the relative distance to the rigid shield are constructed. Numerical experiments have been carried out to confirm the validity of the developed method.

**PNRPU Mechanics Bulletin**. 2018;(2):62-71

Resonant diagnostics of temperature distribution by the piezo-electro-luminescent fiber-optical sensor according to the solution of the Fredholm integral equation

#### Abstract

The mathematical model of the resonant diagnostics of the non-uniform temperature field using the piezo-electro-luminescent fiber-optical sensor is developed. The sensor is an optical fiber with electroluminescent and piezoelectric layers. The sensor’s first thin cylindrical photo passing electrode is located between the optical fiber and the electroluminescent layer, and the second electrode is located on an external cylindrical surface of the piezoelectric layer. The mechanoluminescent effect is a result of the interaction between the electroluminescent and piezoelectric layers in case of axisymmetric forced vibrations of the sensor. In case of vibrations, the electroluminescent glow penetrates through an inner electrode into the optical fiber and reaches the receiver-analyzer of light intensity on the output of the optical fiber. The model is based on the amplitude-frequency characteristic for the stationary electroelastic axisymmetric forced vibrations of a local section of the sensor. The forced oscillations are caused by the harmonious part of the control voltage on the electrodes of the sensor; the constant part of the control voltage is necessary for the setup of the sensor for an operating mode in the considered range of temperatures. When heating a local section of the sensor, the diagram of its amplitude-frequency characteristic displaces along the frequency axis (at a value of resonance frequency’s change), proportionally to the temperature change of this section. As a result, the problem of finding the required function of temperature distribution density along the sensor is reduced to the solution of the Fredholm integral equation of the 1-st kind, based on the results of the measured values of a derivative of amplitude of the luminescence intensity at the output of the optical fiber. Fredholm's kernel is calculated using the known amplitude-frequency characteristic of the sensor and the dependence of the resonant frequency on temperature. The numerical modeling results are presented and the influence on amplitude of the luminescence intensity at output of the sensor’s optical fiber are studied for various models and real laws of distribution of the diagnosed temperatures on a sensor ‘s length.

**PNRPU Mechanics Bulletin**. 2018;(2):72-82

Analysis of stress-strain state of dental implants by the boundary integral equations method

#### Abstract

The boundary integral equations method is applied for the computational analysis of the stresses and strain of dental implants. Isoparametric quadratic boundary elements are used for the numerical solution of the boundary integral equations. The numerical algorithm has been implemented in the form of computer software for solving problems of elasticity and thermoelasticity with mixed boundary conditions and conditions of non-ideal interface between sub-regions of a structure. The implants with a crown fixed by cement using various materials for the junction were considered. Computations were performed for the plane strain state of the structure and consisted of the two stages. The first stage was the analysis of the whole structure with a smoothed screw in the join between the implant and bone; the second one was the analysis of stress concentration in the screwed join at the contact zone between the implant and bone. The model of the first stage contained 7 sub-domains, which are conforming to various parts of the implant. The analysis of the stress concentration of the screw and bone joint is performed at the second stage of this research. It was assumed that those hollows in the spongy bone, which were formed in the bone after the implant’s penetration, are conformed to the screw thread on the implant. Also it was assumed that there is the formation of the full materials joint on the boundary line of the implant and bone. At the first stage of calculations of the implant structure with components made from various materials made it possible to determine, that the greatest stresses occur in implants with a predominant component of titanium. An estimate of the stress concentration in the screw turns of the thread and in the bone tissue was obtained at the second stage of calculating the screw joint of the implant and bone tissue. It was also established that the greatest stresses occur in the zone of the first turn of the thread of the implant screw.

**PNRPU Mechanics Bulletin**. 2018;(2):83-95

Experimental and FEM short-term tensile strength assessment of U- and sharp V-notched specimens made of ductile material

#### Abstract

In this work, the experimental and calculation studies of the short-time strength of tensile rods made of a plastic material with U- and V-shape sharp notches were made. Thermoplastic-acryl-butadiene-styrene (ABS) was chosen as the model material. Samples with a diameter of about 5 mm were obtained on a single-screw extruder by melting granules. In the first part of the paper, the dependence of the mechanical properties of ABS in a wide range of quasi-static strain rates (0.02 ... 10 min-1) was studied. With these conditions, under single-step loading, this material can be considered as an elastoplastic one with an elastic modulus of 2200 MPa, with the yield strength of about 41 MPa and independent of the given strain rates with an error of less than 5 %. For the tested smooth samples, the residual longitudinal strain at rupture was 15 ... 25 %, the residual decreasing of a cross-section (necking) was 30 ... 50 %. The next group of samples had U-notches with a radius of 1.6 mm. The angle of sharp V-notches was 60°. The depth of the single-sided notches was varied in the range of 0 ... 3.5 mm. It is obtained that the ultimate load held by the samples with V-notches exceeds the corresponding load of the samples with U-notches of the same depth due to a greater constraint of the plastic strains in a notch zone. Notches up to 1 mm depth practically do not reduce the ultimate load of the samples. In the second part of the work, with the help of the FEM, the elastic-plastic deformation and fracture kinetics were calculated using a non-local approach, an explicit integration scheme in the ANSYS Workbench package. The calculations showed that the ultimate load is determined only by the yield strength of the material and the configuration of the notch. Rupture of samples (in ANSYS this is the technology of removing critically deformed finite elements) occurred at lower loads, depending on the plasticity resource of the material and the configuration of the notch. The calculated values of the ultimate and rupture loads are in a good agreement with the experimental data. The technique can be recommended for evaluating the strength of complex shape samples of plastic materials with arbitrary stress concentrators.

**PNRPU Mechanics Bulletin**. 2018;(2):96-106

A method for converting an experimental torsion diagram for a cylindrical specimen to the stress-strain diagram of the material

#### Abstract

The main problem with diagnostics and testing is that the overwhelming number of physical quantities cannot be measured directly. Only a limited number of physical quantities can be measured directly, the values of these physical quantities being indirectly influenced by the other (unmeasurable) parameters. Hence the problem arises to determine physical quantities by the results of their manifestations. The same problem concerns the determination of material properties in all the stages of deformation, including the softening stage. This problem is rather laborious. The complexity of the problem is that the material is physically instable at the stage of strain softening. Thus, special devises are needed to obtain material characteristics. Often, they cannot be obtained even by means of non-trivial technical tools. One of the real ways to solve the problem is the testing of special structural components followed by the conversion of obtained data into material properties. The article deals with a well-known methodology of solving inverse ill-posed problems, which was developed by A.N. Tikhonov and V.K. Ivanov. The method is based on the trial-and-error method and the concept of quasi-solution. The problem of determining the stress-strain diagram with a negative slope in the “principal shear stress - shear strain” coordinates by the diagram of torsion of a cylindrical specimen is discussed as an example. It is shown that the problem requires the solution of the first-kind Volterra integral equation. Therefore, the problem is ill-posed. The problem is reduced to a system of linear algebraic equations with an inaccurate right-hand side. After solving the system by the trapezium method, we obtain a particular sawtooth solution. The solution is regularized by a special interpretation of the trial-and-error method. Experimental data obtained from torsion of cylindrical specimens made of steel St3sp is shown in the article. The stated method is used to convert the diagram of torsion of a cylindrical specimen to the material stress-strain diagram with a negative slope.

**PNRPU Mechanics Bulletin**. 2018;(2):107-113

Models of molecular dynamics: a review of EAM-potentials. Part 2. Potentials for multi-component systems

#### Abstract

This article is the second part of the review of modern approaches and works devoted to the construction of interatomic interaction potentials using the embedded atom methodology (the so-called EAM potentials). This part of the review is devoted to one of the most relevant problems in molecular dynamics, which is the problem of constructing potentials that would be suitable for describing the structure, physical and mechanical properties of multi-component (binary and ternary) materials. We have outlined the emerging papers, in which the approaches to the construction of cross-interaction functions for Ni and Cu alloys were proposed, both with the use of the EAM methodology, and a potential of the Finiss-Sinclair type, that differed in the construction procedure. The works, in which different approaches to the construction of potentials are compared, as well as the procedure for identifying parameters using the example of the same multicomponent systems (such as Al-Ni or Cu-Au). In addition, some ternary systems, for example, Fe-Ni-Cr, W-H-He or U-Mo-Xe are of a particular interest as key materials for nuclear energy; and recently they have been actively studied as materials that could be used in thermonuclear rectors. This is to present the examples of works, which offer and investigate the potentials for the description of multicomponent systems, suitable for the aerospace industry, which are made, first of all, on the basis of Ni. The results of the investigations of various intermetallic compounds are considered, and studies have been performed, in which it was possible to accurately describe the phase diagrams of compounds and calculate the characteristics of phase transitions.

**PNRPU Mechanics Bulletin**. 2018;(2):114-132

Modeling of dynamic behavior of reinforced cylindrical shells under elastic-plastic deformation of materials of composition components

#### Abstract

The initial boundary value problem of elastic-plastic deformation of flexible fibrous cylindrical circular shells is formulated. The crossed reinforcement is located on equidistant surfaces. The mechanical behavior of the materials of the composition components is described by the equations of the theory of flow with isotropic hardening. The geometric nonlinearity is taken into account in the Karman approximation. The weakened resistance of the fibrous shells by the transverse shear is taken into account. The system of governing equations and the corresponding boundary and initial conditions, which allow determining the stress-strain state in the components of the composition of flexible cylindrical shells with varying degrees of accuracy, are obtained. The relations of the traditional non-classical Freddy theory follow from the obtained equations, boundary and initial conditions in the first approximation. The solution of the formulated initial-boundary problem is based on an explicit numerical "cross" scheme. The features of inelastic dynamic and quasi-static deformation are studied for very short, short and long fibrous cylindrical shells of different relative thicknesses for different reinforcements. It is found that under dynamic loading of such structures by internal pressure, Reddy theory can lead to inadmissible results. The difference in the calculations of the Reddy theory and the refined theories increases with the increasing time interval. It is shown that for the dynamic calculations of very thin cylindrical reinforced shells, it is necessary to take into account the change of their metrics on the thickness of the structure. It is shown that due to the geometric and physical nonlinearity of the formulated problem, the maximum deflections in thin shells can occur after several tens of oscillations of the fibrous structure, and not only the neighborhood of the initial moment of time when the cylindrical shell is under short-term intensive dynamic loading.

**PNRPU Mechanics Bulletin**. 2018;(2):133-146

Mathematical models of nonlinear viscoelasticity with operators of fractional integro-differentiation

#### Abstract

Based on the method of structural modelling and the Boltzmann-Volterra hypothesis of the hereditary elasticity deformable solid, the article considers linear and nonlinear fractional analogues of classical rheological models, such as Newton (the so-called model of Scott Blair), Voigt, Maxwell, Kelvin and Zener using the tool of fractional integro-differentiation of Riemann Liouville. The classes of nonlinear mathematical models are distinguished, for which the solution of the creep problem can be obtained explicitly in terms of known special functions. A technique for identifying the parameters of the proposed mathematical models is developed on the basis of known experimental data on uniaxial stretching of samples at different and constant load levels. In the presence of explicit solutions to the creep problem, the parameters of mathematical models are determined from the solution of the problem of approximating the experimental values of deformation using the method of least squares with subsequent refinement by the coordinate-wise descent method at all time points for all stress values in a series of experiments. For nonlinear mathematical models of viscoelastic deformation, which do not allow to find the solution of the creep problem in an explicit form, a method for determining the model parameters based on the coordinate-wise descent method with inversion at each step to the numerical solution of the defining integral equation has been developed. The method of identifying model parameters with operators of fractional integro-differentiation is realized on the example of creep of polyvinylchloride plastic compound. The values of the parameters for all the models studied are given, their adequacy to the experimental data is checked, and errors in the deviation of the calculated data from the experimental values are analyzed. As an example, a comparative analysis of the relative error in approximating the experimental creep curves by theoretical deformation values is made in the framework of a linear, nonlinear integrable and nonlinear nonintegrable fractional analog of the Kelvin model. The article oulines the appropriateness of using the viscoelastic deformation models with the operators of fractional integro-differentiation, based on the comparison of calculations of the considered models with the calculations of the viscoelasticity models having integral operators of integro-differentiation.

**PNRPU Mechanics Bulletin**. 2018;(2):147-161