Torsional vibrations of a rotating viskoelastic rod

. A homogeneous and isotropic round viscoelastic rod rotating around its axis of symmetry with a constant angular velocity is considered in a cylindrical coordinate system. It is believed that the behavior of the rod is described within the framework of the linear theory of viscoelasticity, where the relationship between stresses and deformations is given in the form of relations Boltzmann-Volterra. In this case, the condition of its reversibility is imposed on the kernel of the integral operator. The equations of motion of the rod concerning non-zero stress components are written, taking into account the centrifugal force caused by the rotation. It is assumed that torsional vibrations are caused by given stress on its surface. A general equation of torsional vibrations of such a rod is obtained, which is an integral-differential equation of infinitely high order for the main part of the torsional displacement. Limiting the general equations to the zero and first approximations, the equations of the second and fourth orders are obtained, which, in the case of the absence of rotation, exactly coincide with the known equations of other authors. The resulting refined equation of the fourth order in its structure considers the angular velocity of rotation, the deformation of the transverse shear, and the inertia of rotation. Based on the derived oscillation equations, a particular problem is solved to study the influence of rotation and viscoelastic properties of the material on the stress-strain state of the rod, according to the results of which graphs of the dependencies of elastic and viscoelastic changes on time at points of three different sections of the rod are constructed. A comparative analysis of the results obtained with the results of other authors is performed.


Introduction
Many works have been devoted to studying nonstationary vibrations of plates, shells, and rods [1][2][3].At the same time, a few papers have been devoted to studying problems with vibrations of rotating rods and shells, taking into account their viscoelastic properties [4,5].
In work [4], the dynamic behavior of a beam rotating at a constant angular velocity around its longitudinal axis was investigated for free, restrained, freely supported and other types of end fixing and their combinations.The problem of vibrations of a rod rotating around its axis, considering the geometric nonlinearity caused by the hinge supports fixed in the longitudinal direction, was solved in [6].Free and forced vibrations of a rotating layered cylindrical shell were investigated in [7][8][9].
A numerical study of the influence of rheological parameters on the nature of oscillations of hereditarily deformable systems is of great importance [10,11].In this regard, work [12] shows that the differential dependence between the forces and deformations arising in the solution of dynamic problems of viscoelasticity leads to a certain error, especially at the initial moment.Studies of vibrations of structural elements taking into account nonlinearity, play an important role in applied applications [13,14].Studies of nonlinear stability and vibrations of a viscoelastic beam displacement in the axial direction were carried out based on the Kelvin and Maxwell models in work [15].Numerical solutions of complete nonlinear and linearized oscillation equations are compared.The work [16] is devoted to solving the spectral problem of describing small transverse vibrations of a homogeneous viscoelastic rod.In [17], a method is proposed for calculating the effective viscoelastic characteristics of composite materials under steady cyclical vibrations, based on the asymptotic averaging of periodic structures and the finite element solution of local viscoelasticity problems on the periodicity cell of composites.
The given brief review of the literature shows the relevance of studies of torsional vibrations of rotating elements of engineering structures, such as cylindrical shells and rods, taking into account various anisotropic, inhomogeneous, viscoelastic, and other properties of their material.Studies conducted on the basis of refined vibrations theories that consider certain physical, mechanical or geometric factors are also relevant [18,19].Based on these considerations, the proposed article aims to develop a theory of unsteady torsional vibrations of a circular viscoelastic rod rotating at a constant angular velocity around its own axis of symmetry and based on the results obtained.To achieve the goal, you should solve the following tasks: the derivation of a refined integro-differential equation of torsional vibrations of a viscoelastic round rod, which in its structure takes into account the angular velocity of rotation, transverse shear deformation, and inertia of rotation; development of an algorithm for calculating the stress-strain state of the rod points by spatial coordinates and time; solution of an applied problem on torsional vibrations of a round viscoelastic rod excited by an end kinematic effect; conducting a comparative and numerical analysis of the results obtained.

Method
The method used in the research is based on the use of a general exact solution in transformations of the problem of torsional vibrations of a round viscoelastic rod, posed within the framework of the linear theory of viscoelasticity.The essence of the method is the approximate satisfaction of the dynamic conditions set on the surface of the rod.

Statement of the problem
In a cylindrical coordinate system z r , ,T , we consider a homogeneous and isotropic circular viscoelastic rod of radius 0 r , rotating around the axis of symmetry with a constant angular velocity : .It is believed that the behavior of the rod is described in the framework of the linear theory of viscoelasticity, where the Boltzmann-Volterra relations give the relationship between stresses and deformations.The torsion vibrations of a circular cylindrical rod may be considered separate from the problem of its longitudinal-radial vibrations [19].Due to the ax symmetricity of the problem, the components of the stress and displacement fields do not depend on the angular coordinate T .In this case, only the where \ is the component of the vector potential of transverse waves; -viscoelastic operator; P is shear modulus; t R is the core of the viscoelastic operator.It is assumed that the operator M is invertible.
Taking into account the centrifugal force caused by the rotation of the rod, the equation of its motion concerning non-zero stress components has the form [20] where U is density of the rod material.
It is assumed that torsional vibrations are caused by stress t z f r , T on its surface, i.e., the boundary condition task at 0 r r the initial conditions are considered to be zero.

The vibration equations
Substituting the expressions of displacement and stresses -(1) into the equation of motion (2) gives the following integro-differential equation The sought function in ( 4) is represented in the form [5] ³ ³ where 0 I is modified Bessel function of zero order; B is the constant of integration; Representing the displacement T U as well as (5), expressing transformed in this way displacement 0 T U through the general solution ( 6) and expanding the Bessel function in it in a series in degrees of the radial coordinate will have Where B U Then presenting stress T also in the form of (5) from the boundary condition (3), we get where 1   M is operator, inverse operator M .
If consider that parameter J has the form (7), it is not difficult to see that the operators n O in variables t , z have the form [18]   ,...
Taking into account the form of the introduced integro-differential operators ), the resulting equation ( 10) is an integro-differential equation infinitely high order relative to the main part of the twisting displacement.Therefore, it is called a general equation of torsional vibrations of a viscoelastic circular rod rotating at a constant angular velocity about the axis of symmetry.Note that this equation contains derivatives of any order of the coordinate z and on time t.In addition, the right-hand side considers the external force applied to the surface of the bar and reflects the dependence on the viscoelastic operator if known for a particular medium.

Formulas for displacement and stress
Similarly, for non-zero stress component will have Formulas ( 12) and ( 13) allow, with a given accuracy to the radial coordinate and time, to determine the displacement and stress at points of an arbitrary section of the rod through the solution of equation (10).

Approximate equations of vibrations
As already noted, equation (10) has a very high order, and naturally, in this form, it can not be applied in engineering calculations.Hence it follows that it is necessary to limit the number of its members, i.e., limit them to zero ( 0 n , and, i.e., approximations. Assuming that for the case of a rotating rod, the conditions are also fulfilled concerning the range of applicability of the thus "truncated" equations, similar to those obtained in [19], one can obtain various approximate equations from (10).Thus, limiting ourselves to the zero approximation in it, we obtain the second-order equation which, in the absence of rotation, coincides exactly with the equation derived by Professor I.G.Filippov [18].A first approximation of equation (10) gives the fourth order equation which is refined concerning ( 14) and takes into account its structure, in addition to the influence of the angular velocity of rotation, the deformation of the transverse shear, and the inertia of rotation.By limiting zero, first or other approximations vibration equation (10) should also be limited to the corresponding approximation in the formulas for translation (12) and stress (13): If found solving equation ( 14) or (15), then it is easy to determine the stress-deformed condition of the rod by the formulas ( 16) or (17).

Formulation of the boundary value problem
We investigate the influence of rotation and viscoelastic properties of the material on the stress-deformed condition of the rod based on derived equations vibration.Let us assume that the surface of the rod is free from external loads.Then, putting 0 T r f the right-hand side of equation ( 14 For further calculations, for the viscoelastic operator, we take the weakly singular kernel of A. R. Rzhanitsyn [

Algorithm for numerical solution of the problem
To solve the boundary value problem, we express the integro-differential equation (18) in the implicit Crank-Nicholson scheme [22], wherein the integral is calculated using the quadrature formula of rectangles at each time step where The boundary conditions (20) are expressed as at 0 z and the initial conditions (21) at 0 t The resulting system of algebraic equations ( 22) -(24) was solved using the sweep method.The following values of the dimensionless parameters were adopted for calculations 1 0 r ; 10 l The results are shown in Fig. 1-6.

Results and Discussions
Thus, we can note the following results obtained in work and comparisons with the results of other researchers: -a general equation of torsional vibrations of a viscoelastic round rod rotating with a constant angular velocity around the symmetry axis is obtained, an integral-differential equation of infinitely high order relative to the main part of the torsional displacement.This equation coincides with a similar equation obtained in [19]; -limiting the general equations to the zero approximation, a second-order equation is obtained, which, in the absence of rotation ( 0 : ), exactly coincides with the equation obtained by Professor I. G. Filippov [18]; -limiting the general equations to the first approximation, a refined equation of the fourth order is obtained, which in its structure takes into account the angular velocity of rotation, the deformation of the transverse shear, and the inertia of rotation.In the special case of the absence of rotation, this equation exactly coincides with the equation obtained by Professor K. Khudoynazarov [19]; -formulas are proposed that allow determining the displacement and stresses at points of an arbitrary cross-section of the rod with a given accuracy in the radial coordinate and time [1,3]; -on the basis of the derived oscillation equations, a particular problem was solved to study the effect of the material's rotation and viscoelastic properties on the rod's stressstrain state, according to the results of which the graphs of dependencies presented in Fig. 1-6 were obtained.
Depending move U Viscoelastic rod against time (Figure 1) for different values of the parameter E exponential core ( 1 D ), show a decrease in the amplitude of displacement when E tends to zero.In the section z=8, the maximum displacement value is reduced to 33%.The displacement curves change to time at various values of the parameters E and D in a viscoelastic rod with a weakly singular core (Fig. 2, Fig. 3) also show that, in this case, a decrease in the parameter values leads to a decrease in the displacement amplitude.
At close to zero values of the singularity exponent D , there is  ).Hence it follows that the influence of the parameter D on changes in the amplitude of displacement is more significant than the influence of the parameter E (fig.3).Which confirms the result of work [1] on the preference for using weakly singular viscoelastic core.With removal distance from the end face, it is observed viscoelastic damping time stress z T V (fig. 4)in fixed sections (z=2, 5, 8) ( example, the maximum stress value per section z=2 is achieved at t=4,8 а in sections z=5 and z=8 at t=8 and t=1.12, respectively.In this case, the maximum values of viscoelastic displacement in the sections z=2 and z=8 differ more than 52%. In the case of an elastic rod rotating around the axis of symmetry (Fig. 5), from the dependence of the change in stress over time at different angular velocity values : , it follows that an increase in the angular velocity increases displacement amplitude.In section z=8, at the angular velocity value 3 .0 : , the displacement increases by 1.5 times, and the vibration shape is disturbed.An increase in the amplitude of displacement is observed with an increase in the angular velocity, and in the case of taking into account, the viscoelastic properties (Fig. 6).At 3 .0 : the amplitude of displacement in a viscoelastic rod with a weakly singular core decreases by 1.12 times, versus 1.5 times in the elastic case.

Conclusions
The paper presents a new mathematical model of the torsion vibrations of a viscoelastic circular rod rotating at a constant angular velocity about the axis of symmetry.Explore the effect on vibrations viscoelastic characteristics, and rotation.Established that taking viscoelasticity into account leads to a decrease in the vibration amplitude and delay of the wave, while rotation leads to an increase in the vibration amplitude.An algorithm has been developed that allows it to vary viscoelasticity parameters, angular velocity, and external forces acting on the rod during numerical experiments.
), taking into account the views of the viscoelastic operator and introducing dimensionless variables according to the formulas propagation of shear waves in the rod material,