ABOUT DYNAMIC STABILITY OF DEFORMABLE ELEMENTS OF VIBRATION SYSTEMS

The dynamics and stability of the elastic elements of vibration devices, modeled by a channel containing elastic elements, are investigated. Inside the channel ﬂows a stream of stirred liquid. The model of the device with two elastic elements is considered. The solution of the aerohydrodynamic part of the problem, based on the methods of the theory of functions of a complex variable, is given. The solution of the original problem is reduced to the study of a coupled system of partial differential equations for the deformations of elements, which makes it possible to study their dynamics. On the basis of the constructed functional for this system, the sufﬁ-cient conditions of stability are obtained. The conditions impose restrictions on the parameters of the mechanical system. Based on the Galerkin method, the numerical experiments for speciﬁc examples of mechanical systems were carried out, conﬁrming the reliability of the investigations. A special case of the model of device with one elastic element is considered. Based on this case, a comparison with the model of the vibration device considered earlier is made.


Introduction
For the design and exploitation of structures, devices, mechanisms for various applications, interacting with liquid, an important problem is to ensure the reliability of their functioning and durability. Similar problems are common to many branches of engineering. In particular, such problems arise in missilery, aircraft construction, instrumentation, and so on. A stability investigation of the deformable elements has essential value in the calculation of structures that interact with the liquid, as the impact of the liquid may lead to its loss. Examples of the loss of dynamic stability include: the flutter of an airplane wing; panel flutter of plates and shells that are streamlined by a gas or liquid, for example, flutter of an aircraft or rocket skin panel; shear flutter of turbine blades and propellers; oscillations of wires, ducts, suspension bridges, and so on.
At the same time, for the operation of some technical devices, the phenomenon of excitation of vibrations during aero-hydrodynamic effects, mentioned above as negative, is necessary. Examples of such devices related to the vibration technique and used to intensify technological processes are devices for preparing homogeneous mixtures and emulsions, in particular, devices for supplying coolant to the treatment area (see, for example, [Velmisov et al., 1996]). The main part of a wide class of such devices is a flow channel, on the walls of which (or inside it) the elastic elements are located. The operation of such devices is based on the vibration of elastic elements during the flow of liquid inside the channels.
Thus, for designing of the structures and devices interacting with the liquid, it is necessary to solve problems related to the investigation of stability required for their functioning and operational reliability.
The goal of this work is to analyze stability of elastic body correspond to the Lyapunov concept of stability of dynamical systems. The problem can be formulated as follows: for any values of the parameters characterizing the system "liquid-solid" (the main parameters are the flow velocity, strength and inertial characteristics of the body, compressive or tensile forces, friction forces) determine whether small deformation of bodies at the initial time t = 0 (i.e. a small initial deviations from the equilibrium position) correspond to small deformations at any time t > 0.
In the work, the problems of the dynamics and stability of the elastic elements of the vibration device are investigated. The study of the dynamic stability of the working elements of vibrating device is necessary for optimal control of the parameters of the mechanical system in order to increase the efficiency of its functioning. The device is a flow channel with deformable elements, simulated by elastic plates, which can be located both on the walls of the channel and inside it. The number and location of elements are arbitrary. A subsonic flow of an ideal compressible or incompressible medium flows through the channel. To study the dynamics of elastic elements, nonlinear equations are used that describe the longitudinal-transverse oscillations of elastic plates. The aerohydrodynamic load is determined from the asymptotic equations of aero-hydromechanics. At the inlet and outlet of the channel, either the laws of pressure change, or the velocity potential of the liquid, or the longitudinal components of the velocity of the liquid are given. At the inlet to the channel, the flow velocity of the liquid is assumed to be constant and directed along the channel axis ( Figure 1). As an example, let us consider a mathematical model of a hydrodynamic emitter -a vibration device designed to prepare homogeneous mixtures and emulsions. The main components of the device are two elastic elements located on the walls of the flow channel. The oscillations of elastic elements lead to mixing of the inhomogeneous medium supplied to this channel.
2 Model of Device with Two Elastic Elements

Mathematical Model
A flat flow in a straight line channel J = (x, y) ∈ R 2 : 0 < y < y 0 } is considered (Figure 2).
It is assumed that the longitudinal size of the channel considerably exceeds its transverse size, which leads to the absence of disturbances at a point sufficiently far from the elements. The velocity of the undisturbed flow will be considered equal V and directed along the axis Ox. Suppose that the elastic parts are located on the walls y = 0 and y = y 0 at x ∈ [a, b] ( Figure 2). Figure 2. A channel whose walls contain deformable elements We introduce the following notation: u i (x, t), w i (x, t) are elastic displacement of insert plates in the direction of axes Ox and Oy walls y = 0 at i = 1 and y = y 0 at i = 2; φ(x, y, t) is potential of the velocity of the disturbed flow. Functions it belongs to four times continuously differentiable functions with respect to the variable x on the interval (a, b) and twice continuously differentiable with respect to the variable t at t ≥ 0 and takes real values.
Functions u i (x, t) ∈ C 2,2 {[a, b] × R + }, i.e. it belongs to twice continuously differentiable functions with respect to the variable x on the interval (a, b) and twice continuously differentiable with respect to the variable t at t ≥ 0 and takes real values.
Function φ(x, y, t) ∈ C 2,1 {J × R + }, i.e. it belongs to twice continuously differentiable functions with respect to the variables x, y in the area J and continuously differentiable with respect to the variable t at t ≥ 0 and takes real values.
In a model of an ideal incompressible medium the potential φ of the disturbed flow satisfies the Laplace equation: (1) The linearized boundary conditions arising from the condition of impermeability have form: where i = 1, 2, y 1 = 0, y 2 = y 0 . The conditions for the absence of disturbances at an infinitely distant point: The boundary conditions corresponding to rigid fixing of the ends of the plates: Generalizing the nonlinear equations of small oscillations of elastic plates obtained in [Shmidt, 1978], we write them in the form The indices x, y, t below denote partial derivatives with respect to x, y, t; the bar and the point denote the partial derivatives with respect to x and t, respectively; ρ is density of liquid; are elasticity modulus and the linear density of the plates; ν i are Poisson coefficients; N i (t) are compressing (N i > 0) or tensile (N i < 0) forces of the plates; β 2i , β 1i are coefficients of internal and external damping; β 0i are stiffness coefficients of the bases (beds); P 0 is pressure in a uniform flow; P * is external distributed load acting on the channel walls.
Compressive (tensile) forces N i (t) elements may depend on time. For example, if a non-stationary heat exposure to the plate the N i (t) is as follows: where α T i are the temperature coefficients of linear expansion, T i (z, t) are the laws of temperature change over the thickness of the plates, N 0i are the constant components of forces generated when fixing plates. A nonlinear boundary value problem (1) -(6) was obtained for determining five unknown functions -the deformations of elastic elements u i (x, t), w i (x, t), i = 1, 2 and the velocity potential φ(x, y, t) of the liquid.

Determination of the Flow Force
In the region J, we introduce the complex potential Using the function ζ = −e −πz/y0 , we conformally map the strip J to the upper half-plane H = {ζ : Im ζ > 0} of the complex variable ζ = ξ + iη. In this case, the segments [−α, −β], [β, α] on the real axis will correspond to the elastic plates, where β = e −πb/y0 , α = e −πa/y0 . According to the boundary conditions (2), (3) we will have wherē According to (4), applying the Schwartz integral, we obtain where C(t) is arbitrary function of the variable t.
, then from (4) it follows that C(t) ≡ 0. By virtue of the same condition at ζ → 0 (x → +∞), from (8) According to the boundary conditions (5) we obtain The physical meaning (9) is that the gas flow through the boundary of region J equal to zero, which corresponds to the model of an incompressible medium.
Further, since W ζ = f z dz dζ = −f z y 0 πζ , integrating by ζ and differentiating by t, according to (4) we obtain: Using the integral representations (8), (10), we transform the right-hand side of the second equation of system (6). To this end, in (8), (10) we pass to the limits at Substituting ξ, τ , we write the expressions (11) in the form It is easy to see that the kernels are symmetric The expressions (12) are obtained for any methods of fixing elastic plates. Substituting (12) into (6), we obtain a system of differential equations with four unknown functions u 1 (x, t), w 1 (x, t), u 2 (x, t), w 2 (x, t): The system (13) is homogeneous and obtained under the assumption that P 0 = P * .

Stability Investigation
Introduce the following notation: λ 1i , η 1i are the smallest eigenvalues of the boundary value problems for the equations ψ = −λψ , ψ = ηψ, x ∈ [a, b] with boundary conditions corresponding (5) for the functions w i (x, t), i = 1, 2; where g 1 (x), g 2 (x) are the arbitrary functions integrable over a segment [a, b] chosen for reasons of attaining the smallest possible values G, K. Further an example of the choice of functions g 1 (x), g 2 (x) is given.
Proof. We introduce the functional Find the derivative of Φ by ṫ We will integrate by parts taking into account the boundary conditions (6) Similar to the last equality, it is possible to obtain equalities for all integrals with integrands containing ∂K i (τ, x) ∂x , i = 1, 2.
Remark. In a similar way, we can prove that theorem 2.1 is also true, if the functions u i (x, t), w i (x, t), i = 1, 2 satisfy any combination of the following boundary conditions: 1) rigid fastening: 2) hinged fastening:
According to conditions (16) it is necessary to find the coefficients K, G. For the calculation, the functions g 1 (x) = −1, 25 sin πx 2 + 0, 23 cos πx 2 − 0, 37, 198 were selected and the coefficients (14): K ≈ 0, 243; G ≈ 0, 956 were found using the Mathcad mathematical system. The first condition (16) is satisfied, and from the second condition we obtain If the points (V ,N i (t)) for any t ≥ 0 do not go beyond the stability region (gray area in Figure 3 below the parabola), then the solutions w i (x, t), i = 1, 2 of the system of equations (13) are stable with respect to the perturbations of the initial values w i (x, 0),u i (x, 0),
Therefore, the theorem 2.1 takes the following form.

Comparison with Finite Length Model
In the paper [Velmisov and Ankilov, 2019] we considered a model of a device of finite length x 0 (Figure 8).
We introduce the following notation: snx is the elliptic sine, K(k) is the full elliptic integral of the first kind, the module k is determined from the relation K √ 1 − k 2 y 0 = 2K(k)x 0 . Then, using methods of the theory of functions of a complex variable, we obtained a system of differential equations with two unknown functions u(x, t), w(x, t): (ẅ(τ, t) + Vẇ (τ, t)) G 1 (τ, x)dτ, x ∈ (a, b).
Similarly to the above studies, based on the study of a functional of Lyapunov type, the following theorem is proved. |G 2 (τ, x) + g 2 (x) + g 2 (τ )| dτ .
Compare the second condition (36) for the models in the figures 7 and 8. We take the following parameters of the studied mechanical systems: D = 810; ρ = 840; y 0 = 0.5; a = 0.8; b = 1.2. Under conditions (37) the smallest eigenvalue λ 1 = 25π 2 . All values are given in the SI system.
The conditions are almost identical, which confirms the adequacy of the proposed mathematical model and the obtained results .

Conclusion
On the basis of the proposed mathematical model of oscillations of elastic elements of a vibrating device in the form of a plate-strip with a single-sided flow around them by a subsonic flow of an ideal liquid, the investigation of the dynamics and stability of these elements was conducted. The obtained stability conditions impose restrictions on the mass and bending stiffness of elements, compressive (tensile) force elements, the velocity of an unperturbed flow and other parameters of the mechanical system. These conditions clearly contain the basic parameters of the mechanical system, and in this form they are most suitable for solving optimization problems, automatic control, and automated design.