Problem Of Natural Vibrations Research Articles

Passive damping of vibrations of plates and cylindrical shells interacting with a flowing fluid using piezoelectric elements

The paper presents a mathematical formulation of the problem of natural and forced harmonic vibrations of electro-elastic thin-walled structures interacting with a flowing fluid. The numerical algorithm based on the finite element method is used to choose optimal parameters of a shunt electric circuit for problems of passive vibration damping. The key idea behind the proposed approach is a solution of eigenvalue problem. It allows us to obtain higher damping rate of free vibrations compared to those evaluated by the commonly used analytical expressions and leads to the smallest difference in the natural frequencies of the structure and the electric circuit.

Natural vibration of elastic metamaterial beam under arbitrary elastic boundary conditions

Elastic metamaterial (EM) is a kind of new composite material/structure that provides a new feasible way of vibration control in many scenarios. Although some analytical means for solving the vibration behavior of EM components with simple shapes and classic boundary conditions (BCs) have been presented, the analyses of those with complex shapes or BCs that are common in reality remain to be studied further. In this paper, the natural vibration problem of a typical EM beam under arbitrary elastic boundary conditions (EBC) is investigated by using the effective medium theory (EMT) and the Rayleigh-Ritz method. The inherent correlations between the EM beam and corresponding modified matrix beam together with the modified static equivalent beam are disclosed for their modal behaviors. The extraordinary natural vibration properties of the EM beam under arbitrary EBC are revealed and some common features are concluded. This work may provide a useful reference for further investigations of many other similar EM structures.

Stochastic Vibrations of a System of Plates Immersed in Fluid Using a Coupled Boundary Element, Finite Element, and Finite Difference Methods Approach.

The main objective of this work is to investigate the natural vibrations of a system of two thin (Kirchhoff-Love) plates surrounded by liquid in terms of the coupled Stochastic Boundary Element Method (SBEM), Stochastic Finite Element Method (SFEM), and Stochastic Finite Difference Method (SFDM) implemented using three different probabilistic approaches. The BEM, FEM, and FDM were used equally to describe plate deformation, and the BEM was applied to describe the dynamic interaction of water on a plate surface. The plate's inertial forces were expressed using a diagonal or consistent mass matrix. The inertial forces of water were described using the mass matrix, which was fully populated and derived using the theory of double-layer potential. The main aspect of this work is the simultaneous application of the BEM, FEM, and FDM to describe and model the problem of natural vibrations in a coupled problem in solid-liquid mechanics. The second very important novelty of this work is the application of the Bhattacharyya relative entropy apparatus to test the safety of such a system in terms of potential resonance. The presented concept is part of a solution to engineering problems in the field of structure and fluid dynamics and can also be successfully applied to the analysis of the dynamics of the control surfaces of ships or aircraft.

Concurrent optimization method of shape and topology for natural vibration problem of shell structures

In this paper, a concurrent optimization method of shape and topology for natural vibration design of shell structures is presented. The fundamental vibration eigenvalue is maximized under volume constraints for shape and topology optimization. The free-form optimization method for shells and the solid isotropic material with penalization (SIMP) method are respectively employed and combined effectively for shape and topology design, in which shape and fictitious homogenized-density variations are used as design variables and simultaneously determined. In other words, the optimal topology is determined in the variable design surface optimized by shape optimization. The optimal design problem is formulated as a distributed-parameter optimization problem, and the sensitivity functions with respect to shape and density variations are theoretically derived. Both the optimal shape and density variations are determined with the unified H1 gradient method, where the sensitivity functions are respectively applied as the Robin condition to the design surface in order to determine the optimal shape and topology. We also show an improved method for the spurious (or localized) mode problem of shell structures, which often occurs in the lower density region in topology optimization for vibration design. With the proposed method, shell structure for light weight and improved low-order vibration characteristics can be obtained without any design parameterization, free of numerical instabilities such as checkerboard pattern and zigzag-shape problems.

A NUMERICAL ANALYSIS OF FORCED STEADY-STATE VIBRATIONS OF AN ELECTRO-VISCOELASTIC SYSTEM IN CASE OF A JOINT IMPACT OF ELECTRICAL AND MECHANICAL LOADS

In the current paper an estimation of a mechanical response of a structure on joint mechanical (force and kinematic) and electrical (voltage and current) impacts taking into account different options of their combinations was carried out. A mechanical response was esteemed basing on solution to the problem of natural vibrations for piecewise-homogeneous electro-viscoelastic bodies. A cantilever plate with piezoelectric element attached to its surface was chosen as an object under study. The plate was made of viscoelastic material the mechanical properties of which are described using complex dynamic moduli. A piezoelectric ceramic material was chosen as the one for piezoelectric elements. An amplitude of forced steady-state vibrations of a component of displacement vector that is perpendicular to the surface of the plate was chosen as a quantity that characterize a mechanical response. Variational equation of motion of the object under consideration is formulated based on relations of linear elasticity, linear viscoelasticity and Maxwell’s equations for a conducting media in a quasistatic approximation. Ranges of changing of values of excitation impacts for vibration modes taken into account were chosen on the basis of values of natural vibrations frequencies obtained from the solution to natural vibration problem for electro-viscoelastic bodies. Numerical implementation of the stated problem was performed in finite element software package ANSYS. Some characteristic peculiarities of a mechanical response depending on a type and magnitude of external impact were obtained as the result of the carried out numerical experiments. There was demonstrated that in case of joint impact of two loading factors there reveals an additional capability of control a mechanical response of an electro-viscoelastic system.

AIMED CONTROL OF THE FREQUENCY SPECTRUM OF EIGENVIBRATIONS OF ELASTIC PLATES WITH A FINITE NUMBER OF DEGREES OF MASS FREEDOM BY INTRODUCING ADDITIONAL GENERALIZED KINEMATIC DEVICES

As is known, targeted regulation of the frequency spectrum of natural vibrations of elastic systems with a finite number of degrees of mass freedom can be performed by introducing additional generalized constraints and generalized kinematic devices. Each targeted generalized constraint increases, and each generalized kinematic device reduces the value of only one selected natural frequency to a predetermined value, without changing the remaining natural frequencies and all forms of natural vibrations (natural modes). To date, for some elastic systems with a finite number of degrees of freedom of masses, in which the directions of mass movement are parallel and lie in the same plane, special methods have been already developed for creating additional constraints and generalized kinematic devices that change the frequency spectrum of natural vibrations in a targeted manner. In particular, a theory and an algorithm for the creation of targeted generalized constraints and generalized kinematic devices have been developed for rods. It was previously proved that the method of forming a matrix of additional stiffness coefficients, specifying targeted generalized constraint, in the problem of natural vibrations of rods can also be applied to solving a similar problem for elastic systems with a finite number of degrees of freedom, in which the directions of mass movement are parallel, but do not lie in the same plane. In particular, such systems include plates. The distinctive paper shows that the method of forming a matrix for taking into account the action of additional inertial forces, specifying targeted kinematic devices in the problem of natural vibrations of rods can also be applied to solving a similar problem for elastic systems with a finite number of degrees of freedom, in which the directions of mass movement are parallel, but do not lie in the same plane. However, the algorithms for the creation of targeted generalized kinematic devices developed for rods based on the properties of rope polygons cannot be used without significant changes in a similar problem for plates. The method of creation of computational schemes of kinematic devices that precisely change the frequency spectrum of natural vibrations of elastic plates with a finite number of degrees of mass freedom is a separate problem and will be considered in a subsequent paper.

ВЛАСНІ ЗГИННІ КОЛИВАННЯ БАЛКИ ЗІ СПЕЦІАЛЬНИМ ЗАКОНОМ ЗМІНИ ШИРИНИ

Purpose. The aim of the work is to construct a closed analytical solution of the problem of natural vibrations of the beam, the width of which varies according to the law exp (αx2). Methodology. The approach is based on the provisions of symmetric analysis of differential equations with variable coefficients. This approach allows you to find a way to obtain an analytical solution of the corresponding differential equation and, ultimately, the boundary value problem. Findings. The main result is the construction of the algorithm and obtaining the solution of the differential equation of the IV order, which describes the transverse bending vibrations of the beam with a special law of change of width (the thickness of the beam is a constant value). Two examples of the analysis of oscillations of such beam in case of its bilateral rigid fastening and cantilever fastening are resulted. For these cases, the frequency equations are obtained, the natural frequencies and amplitude coefficients are found, which are necessary for the construction of natural forms of oscillations. Originality.The approach presented in this paper is based on the idea of symmetries of differential equations and is characterized by a sim- plified analysis of the solution of the problem of bending oscillations of the beam with a special law of width. The method proposed for solving the boundary value problem is convenient and simple, because the results are found without the use of numerical research methods. Practical value.Examples of the exact analysis of fluctuations which allow to assert about real possibility of expansion of an existing number of configurations of a beam, both on width, and on ways of fastening are resulted. Such beams can, for example, be used as prototype samples for resonant tests of materials for fatigue strength. Сonclusions. The given algorithm for constructing the solution of the problem on eigenvalues for a beam with the given special law of change of width is universal and can be extended to other constructions of beams. Since in this case the question arises only about the choice of the approximation function, which allows to use the method of symmetries and obtain an exact solution of the corresponding differential equation of the IV order, which in turn describes the transverse oscillations of the beams.

AIMED CONTROL OF THE FREQUENCY SPECTRUM OF EIGENVIBRATIONS OF ELASTIC PLATES WITH A FINITE NUMBER OF DEGREES OF FREEDOM OF MASSES BY SUPERIMPOSING ADDITIONAL CONSTRAINTS

As is known, for some elastic systems with a finite number of degrees of freedom of masses, for which thedirections of motion of the masses are parallel and lie in the same plane, methods have been developed for creatingadditional constraints that purposefully change the spectrum of natural frequencies. In particular, theory and algorithm forthe formation of aimed additional constraints have been developed for the rods, the introduction of each of which doesnot change any of the modes of natural vibrations, but only increases the value of only one frequency, without changingthe values of the remaining frequencies. The distinctive paper is devoted to the method of forming a matrix of additionalstiffness coefficients corresponding to such aimed constraint in the problem of natural vibrations of rods. This method canalso be applied to solving a similar problem for elastic systems with a finite number of degrees of freedom, in which thedirections of motion of the masses are parallel, but not lie in the same plane. In particular, such systems include plates.However, the algorithms for the formation of aimed additional constraints, developed for rods and based on the propertiesof rope polygons, cannot be used without significant changes in a similar problem for plates. The method for the formationof design constraint schemes that purposefully change the spectrum of frequencies of natural vibrations of elastic plateswith a finite number of degrees of freedom of masses, will be considered in the next work.

Development of an electrical analogue for modeling natural vibrations of a viscoelastic structure with piezoelectric element

This paper focuses on the development of an equivalent electrical model with lumped parameters capable of describing the natural vibrations of an electromechanical system comprising a viscoelastic structure with a piezoelectric element attached to its surface. The important advantage of the model is that it takes into account the energy losses associated with the viscoelastic properties of the material of the main structure. Two versions of the equivalent electric analogue of the initial electro-viscoelastic system in the form of electric circuits, the elements of which are described by the real or complex quantities, are considered. The approaches to the formulation of the problem of natural vibrations in the developed electric analogue are based on Kirchhoff’s laws for electric circuits and Ohm’s law for alternating current. Special attention is paid to the identification of model parameters. A procedure for determining the parameters for the equivalent electrical model is based on the results gained from the solution of a coupled problem of natural vibrations of the initial electromechanical system; problem formulation is also given here. The effectiveness and reliability of the developed equivalent electric models with lumped parameters for the determination of complex eigenfrequencies of the electromechanical system containing energy dissipation elements are demonstrated by analyzing the behavior of structures in the form of a rectangular plate and a semi-cylindrical shell.

Vibrations of multilayer composite viscoelastic curved pipe under internal pressure

The paper presents a mathematical statement and the methods to solve the problem of natural vibrations of thin-walled multilayer composite viscoelastic curved pipes under internal pressure. A brief overview of well-known publications devoted to this problem is given. The stress state and flexibility of the pipes are investigated for two types of boundary conditions, taking into account internal pressure and viscoelastic properties of the material and geometrical and physical parameters of the pipe. The paper provides the results of numerical analysis of the spectra of the lower complex eigenfrequencies of the pipe depending on internal pressure, geometrical, structural parameters, rheological properties and boundary conditions. Ten lower eigenfrequencies were found and the corresponding vibration modes constructed. The paper presents a comparison of solution results with known solutions and experimental results.

Dynamic stress-strain state of earth dam considering an inhomogeneous base

The design, construction and operation of important water retaining structures (high earth dams) in seismic regions, such as the territory of the Republic of Uzbekistan, require constant improvement of the methods to calculate their behavior under various loads, including seismic ones. A variational statement of the problem of natural vibrations of a plane elastic earth dam with and without an adjacent base is given. The problem is solved numerically by the finite element method. The aim of this study is to determine the effect of a piecewise-inhomogeneous section at the base of an earth dam on the stress-strain state of the dam itself. The result of solving the problem lies in the analysis of eigenfrequencies, vibration modes, stresses of a concrete earth dam.

DAMPED NATURAL VIBRATIONS OF FLUID IN A WELL INTERFACED WITH A RESERVOIR

We study natural vibrations of the fluid column in a vertical oil well that occur during sudden shutdown or start-up of the well (water hammer). In this case, the period of vibration and the intensity of vibration damping are determined not only by the length and diameter of the fluid column in the well and the rheological properties of the fluid, but also by the reservoir characteristics of the bottomhole zone (in particular, permeability, the quality of well perforation, and the properties of the hydraulic fractures formed). Solutions to the problem of damped natural vibrations of the fluid column in the well are found using a mathematical model describing the movement of the fluid column in the well and filtration in the bottomhole zone. Characteristic equations for determining complex frequencies (frequency of vibration and the damping coefficient) are obtained. The dependences of the vibration frequency, damping coefficient, and damping decrement on reservoir permeability are studied, and the amplitudes of vibrations at different points of the well are obtained.

An approach to the analysis of hydroelastic vibrations of electromechanical systems, based on the solution of the non-classical eigenvalue problem

In this paper we introduce an approach to solution of the problem of natural vibrations of piecewise homogeneous electroelastic bodies connected to external passive electric circuits and interacting with a quiescent fluid. A mathematical formulation of the problem, and a variant of its numerical implementation based on the finite element method are presented. Based on the obtained results we extend the previously developed approach to the selection of optimal parameters of shunt circuits for damping of vibrations of electroelastic structures in the case of their interaction with a fluid.

Application of Graphene Composites for Additional Vibration Damping in Smart Structures Based on Piezoelectric Elements

A smart-structure variant representing a piecewise homogeneous body composed of elastic and viscoelastic materials and piezoelectric elements the electrodes of which can be connected to shunting circuits is considered. One of the basic purposes of such structures is vibration damping. This paper deals with a variant in which the classic resistor in a shunting circuit is replaced by a graphene composite acting not only as a mechanically deformable body, but also as a resistor in a smart structure. The mathematical formulation of the problem of forced steady-state and natural vibrations in the structures composed of elastic and viscoelastic elements, piezoelectric elements, and graphene composites acting as a deformable body and a resistor is given. The results of numerical experiments demonstrate the applicability of graphene composites for implementing an additional mechanism of vibration damping in smart structures based on piezoelectric elements.

ВІЛЬНІ КОЛИВАННЯ ОРТОТРОПНИХ ПЛАСТИН

The solution of the problem of free vibrations of a rectangular orthotropic plate by the methods of boundary and finite elements under any boundary conditions. Transformation of the two-dimensional differential equation of free vibrations of an orthotropic rectangular plate to one-dimensional. Determination of the complete system of its fundamental solutions using the numerical-analytical method of boundary elements. Implementation of the algorithm on the example of a specific plate and comparison with the results of finite element analysis in ANSYS. The solution to the problem of natural vibrations of a rectangular orthotropic plate is obtained without any restrictions on the nature of the fixing of its sides. A transcendental frequency equation is obtained whose roots give the full spectrum of natural frequencies. The modeling and calculations of the orthotropic plate by the finite element method are performed. An analysis of the numerical results obtained by the author's method shows a very good convergence with the results of finite element analysis. For a plate with rigid fastening of three sides with a free fourth side, the discrepancy is slightly higher than for a plate with a hinged support along the contour. Under both variants of the boundary conditions, the frequency spectrum calculated by the boundary element method is lower than in the finite element calculations. Analytical expressions of fundamental functions are obtained that correspond to all possible solutions to the differential equation of free oscillations. For the first time, a solution to the problem of free vibrations of a rectangular orthotropic plate is presented by the numerical-analytical method of boundary elements. The results allow us to solve the problem of free vibrations of a rectangular orthotropic plate by two methods under any boundary conditions, including inhomogeneous ones.

Free-form optimization of CFRP plate/shell structures in natural vibration problem

Free-form optimization of CFRP plate/shell structures in natural vibration problem

Construction of an electric analogue for determining the dissipative characteristics of electric viscoelastic bodies with external electric circuits

The problem of damping vibrations of structural elements in various branches of technology does not lose its relevance over time but rather becomes a key issue for development of modern devices in airspace, aircraft, automotive industry. The use of piezoelectric elements and external electric circuits opens up novel possibilities for controlling the dynamic behavior of structures. Modeling the dynamic behavior of such electromechanical systems requires solving a related problem of electroviscoelasticity. Optimization of the dissipative properties of electromechanical systems under consideration is carried out through the selection of circuit parameters (tuning) that provide the best vibrations damping at a specified frequency. In this case, the most efficient tool is the problem of natural vibrations of piecewise homogeneous electroviscoelastic body with external electric circuits. However, the numerical procedure for finding the optimal parameters of electric circuits is related to multiple solution of the problem for each combination of the values of the circuit elements using complex large-sized matrices, including solving the algebraic problem of complex eigenvalues for these matrices. At the same time, the generality of mathematical equations, which describe oscillations in mechanical systems and oscillations of current and voltage in electric circuits, makes it possible to apply the method of dynamical analogies and consider equivalent electric system instead of coupled electromechanical one, replacing the equations of motion of electromechanical system with the corresponding analogues of equations for an equivalent electrical system. This allows to considerably decrease computational time and reduce the requirements for hardware resources. The developed discrete analogue is completely equivalent to the original electromechanical system with external electric circuit in terms of the spectrum of natural vibration frequencies. The results obtained with the aid of the electric analogue were verified by solving problems on natural vibrations of electroviscoelastic bodies, including those with external electric circuits, according to the mathematical statement of continuous media mechanics.

A comparative analysis of two approaches to damping of vibrations

In the current paper a comparison between two approaches to damping of vibrations is presented. A comparative analysis was carried-out on the example of a cantilever plate. In the first case, the surface of the plate was covered with a layer of viscoelastic material. Its mechanical behavior is described by equations of the linear theory of viscoelasticity based on complex dynamic moduli. In this case, the complex dynamic moduli can increase or decrease as the oscillation frequency increases, or remain constant. It is assumed that the thickness of the viscoelastic layer is equal to the thickness of the plate.In the second case piezoelectric elements shunted with resistive (R) or resonant (RL) electric circuits were applied for damping of vibrations. In this case, for each vibration mode, the optimal location of the piezoelectric element and the parameters of the external electrical circuit are determined. The comparison is carried out on the basis of analysis of the vibration damping coefficients determined by the value of the complex natural vibration frequencies obtained from the solution to the problem of natural vibrations for electro-viscoelastic bodies with external electric circuits. Numerical implementation was performed using finite element method.

Natural Vibrations Of Structurally Inhomogeneous Multi-Connected Shell Structures With Viscoelastic Elements

The methods for solving the problems of natural vibrations of structurally inhomogeneous, multi-connected, axisymmetric shell structures based on the hereditary theory of viscoelasticity are given in the paper. Using the basic relations of the hereditary theory of viscoelasticity and asymptotic methods, the problem of natural vibrations of structurally inhomogeneous, multi-connected, axisymmetric shell structures is reduced to the effectively solvable mathematical problem of complex eigenvalues, in which approximate engineering methods are proposed.

Frequency analysis of the response of a one-sheet hyperboloid to periodic longitudinal loading

The frequency analysis of the steady state forced vibrations of a thin shell of negative Gaussian curvature is performed. The one-sheet hyperboloid shell is subjected to the action of the periodic longitudinal load. The finite element model for the shell is built using flat quadrilateral shell elements with six degrees of freedom per node. The modal analysis of the shell is performed by the Lanczos method using linear formulation of the problems. Eigenfrequencies and eigenmodes of the shell are also defined using nonlinear formulation of the problems that takes into account the effect of a longitudinal under critical static force preliminary applied to the top edge of the shell. For this purpose the nonlinear statics problems and the natural vibration problems are sequentially solved by the modified Newton-Raphson method and Lanczos method respectively. A modal analysis showes that dependence of the shell eigenfrequencies on the static longitudinal loading is nonlinear. The steady state response of the shell to the periodic longitudinal loading is obtained both by means of direct and modal frequency analysis. It turned out that sensitivity of the hyperboloid of one sheet to the periodic longitudinal loading considerably increases when the static under critical longitudinal loading is taken into account. The responses of the shell are presented in the form of amplitude frequency dependences of the finite element model nodes.