Published: 07 October 2016

Mathematical model of elastic ribbed shell dynamics interaction with viscous liquid under vibration

Dmitry V. Kondratov1
Anna V. Kalinina2
Lev I. Mogilevich3
Anna A. Popova4
Yulia N. Kondratova5
1, 2Russian Presidential Academy of National Economy and Public Administration, Saratov, Russian Federation
3, 4Yuri Gagarin State Technical University of Saratov, Saratov, Russian Federation
5Saratov State University, Saratov, Russian Federation
Corresponding Author:
Dmitry V. Kondratov
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Abstract

The mechanical model of the system, formed by two surfaces of the coaxial cylindrical shells interacting with viscous incompressible liquid between them under vibration, is considered. The outer shell is geometrically irregular, and inner one is an absolutely rigid cylinder. The mathematical model of this system, consisting of differential equations in partial derivatives for describing dynamics of viscous incompressible liquid and an elastic ribbed shell together with boundary conditions is constructed. The expressions for amplitude frequency characteristics of outer geometrically irregular shell are discovered.

1. Introduction

The pipelines are one of the basic elements in modern devices and various engineering constructions applied in aviation. Such devices are represented by pipelines of the hydraulic drive, cooling, greasing, fuel supply, dosaging system, power cylinder and ram-type pairs [1]. The studies of the mechanical systems, consisting of coaxial geometrically regular cylindrical shells with pressure pulsations or under vibrations were conducted earlier [2-8].

The present paper considers the pipeline system of ring like profile with elastic geometrically irregular outer shell of final length and absolutely rigid inner cylinder, interacting with viscous incompressible liquid between them under vibration for defining and regulating oscillations of “shell-fluid” system.

It should be noted that this problem is relevant because real mechanical systems are constantly exposed to different vibration impacts from both external and internal sources. Vibration leads to emerging substantial oscillations of pipeline systems. At resonance frequencies the walls velocities of the pipeline elastic shells will be maximum, and it can lead to ruptures in a fluid and emergence of vibrational cavitation. Gas bubbles bursting on walls, lead to pipeline walls damage and therefore to cavitation wear [9].

2. Statement of the problem

Let us consider the mechanical model presented on Fig. 1.

The outer shell is an elastic cylindrical geometrically irregular shell. The inner shell can be considered as absolutely rigid cylinder. Viscous incompressible liquid is placed between the indicated cylindrical shells. The radius of outer ribbed shell medium surface equals R, and its thickness at the places, without rigid ribs equals h0. The length of cylindrical shells l are equal, and elastic shifts of an outer ribbed shell are much less than the width δ of the cylindrical slot. The width δ=R1-R2R2 of the ring crossed section cylindrical slot, formed by two shells, is less than outer radius R2 of the inner shell and inner radius R1 of an outer shell. The medium surface radius R is much larger than the width of an outer shell h0 =2R-R1. The temperature effects in the system discussed above are not considered.

In the present mechanical model we will determine amplitude-frequency characteristic of outer geometrically irregular shell sagging.

Fig. 1The mechanical model

The mechanical model

3. The theory

The movements of viscous incompressible liquid in coordinate system r, θ, y rigidly connected with the center of coordinates, are presented by Navier-Stokes equations and by continuity equation for an axes symmetric case in a scalar state [10, 11]:

1
Wr=-1ρpr+ν2Vrr2+1rVrr+1r22Vrθ2+2Vry2-2r2Vθθ-Vrr2,
Wθ=-1ρrpθ+ν2Vθr2+1rVθr+1r22Vθθ2+2Vθy2+2r2Vrθ-Vθr2,
Wy=-1ρpy+ν2Vyr2+1rVyr+1r22Vyθ2+2Vyy2,
Vrr+Vrr+1rVθθ+Vyy=0,

where:

Wr=W1z1cosθ+W1x1sinθ+Vrt+VrVrr+VθVrrθ+VyVry-Vθ2r,
Wθ=-W1z1sinθ+W1x1cosθ+Vθt+VrVθr+VθrVθθ+VyVθy+VrVθr,
Wy=Vyt+VrVyr+VθrVyθ+VyVyy,

where p – pressure of viscous incompressible liquid; Vy, Vr, Vθ – are the components of liquid velocity vector n-r, j- in which the beginning is located in the center of the inner shell O; t – time; ν – kinematic coefficient of viscosity; r – distance from an axis Oy; ρ – liquid density, W1x1=Exω2sinωt, W1z1=Ezω2sinωt – vibroacceleration of a base which is attached to a mechanical system into inertial space.

The boundary conditions for cylindrical axes on impenetrable surface of outer and inner shells in a cylindrical clearance for Eq. (1) are:

2
Vr=u3 t, Vθ=u2 t, Vy=-u1 t at r=R2+δ+u3`,
Vr=0, Vθ=0, Vy=0 at r=R2,

where u1 =u1 y, θ, t – elastic longitudinal shell shift which is positive in the direction n-s, opposite in the direction j-; u2 =u2 y, θ, t – circumferential elastic movement in the direction n-θ, u3 =u3 y, θ, t –shell sagging which is positive in the direction n-, coinciding with n-r and opposite in a direction to a curvature center; u-=u1n-s+u2n-θ+u3n- – vector displacement of the elastic shell.

The outer surface of a pipe outer shell represents geometrically irregular shell, having n of rigidity ribs which altitude changes stepwise. The ribs present outer frames. The strengthening of geometrically irregular shell at butt ends has free attaching [11].

The dynamic equations of a ribbed shell based on Kirchhoff-Love hypotheses, proceeding from the received Hamilton integral variational principle, in cylindrical axes are:

3
yEh01-μ02-u1y+μ01Ru2θ+u3k1(y)
+μ01R2u2θ-2u3θ2-2u3y2h0k2(y+1RθEh021+μ0u2y-1Ru1θk1y
+2Ru2y-2u3yθh0k2(y)=-h0ρ02u1t2k1y-qs,
yEh02(1+μ0)u2y-1Ru1θk1(y)+2Ru2y-2u3yθh0k2(y)
+1RθEh01-μ021Ru2θ+u3-μ0u1yk1(y
+1R2u2θ-2u3θ2-μ02u3y2h0k2(y)
+1R2θEh03121+μ021R2u2θ-2u3θ2-μ02u3y2k3(y
+Eh01-μ02-μ0u1y+1Ru2θ+u3h0k2(y)
+2RyEh03121+μ01Ru2y-2u3yθk3(y)+Eh02(1+μ0)u2y-1Ru1θh0k2(y)
=h0ρ0W1x1sinθ-W1z1cosθ2u2t2k1y-qθ,
2y2Eh0312(1-μ02)μ01R2u2θ-2u3θ2-2u3y2k3(y)
+Eh01-μ02u1y+μ01Ru1θ+u3h0k2(y)
+1R22θ2Eh0312(1-μ02)1R2u2θ-2u3θ2-μ02u3y2k3(y)
+Eh01-μ02-μ0u1y+1Ru2θ+u3h0k2(y)
+2RθyEh0312(1+μ0)1Ru2y-2u3yθk3(y)
+Eh02(1+μ0)u2y-1Ru1θh0k2(y)-1REh01-μ021Ru2θ+u3-μ0u1yk1(y)
+1R2u2θ-2u3θ2-μ02u3y2h0k2(y)
=-h0ρ0W1x1sinθ+W1z1cosθ+2u3t2k1y-qn,

where:

k1j=1-h0hpjhpjh0, k2j=1-h0hpjhpj22h02, k3j=1-h0hpj4-2h0hpj+h02hpj2hpj3h03.

ΔΓyj=Γy-yj-Γy-yj-ε0j*, Γy – is a unit function Heaviside on the longitudinal coordinate y; μ0 – Poisson’s ratio of shell material; yj – is a point, where the rib appears on the longitudinal coordinate; ρ0 – shell material density; E – shell material Young modulus:

4
qs =-Pry cos n-,n-r+Pθy cos n-,n-θ+Pyy cos n-,j- r=r1,
qθ =Prθ cos n-,n-r+Pθθ cos n-,n-θ+Pθy cosn-,j- r=r1,
qn =Prr cos n-,n-r+Prθ cos n-,n-θ+Pry cosin-,j- r=r1,
r1=R2+δ+u3 , Prr=-p+2ρνVrr, Prθ=ρνVθr-Vθr+1rVrθ,
Pθθ=-p+2ρν1rVθθ+Vrr, Pry=ρνVyr+Vry, Pyy=-p+2ρνVyy,
Pθy=ρνVθy+1rVyθ, cos n-,n-r=rN-, cosn-,n-θ=-1N-u3 θ,
cosn-,j-=-rN-u3 y, N-=r21+u3 y2+u3 θ212, a0 2=h0 212R2.

The boundary conditions of outer geometrically irregular shell free attaching look like:

5
u1 y=0, u2 =0, u3 =0, 2u3 y2=0 at y=±l2.

The bound hydroelasticity problem Eq. (1-5) is solved in the undimensional variables by perturbation method [12-15] with the use of two small parameters describing the relative width of liquid layer and relative outer elastic geometrically irregular shell sagging by presupposing the harmonic law of system vibration. Such approach allows linearizing the problem of fluid dynamics. Liquid movements velocity and pressure in liquid layer, received in the course of hydrodynamics equations solution, allow to find the stresses acting on the shell. They are substituted in elastic geometrically irregular cylindrical shell dynamic equations. As a result, the system of integrodifferential equations is received.

Bubnov-Galerkin method in the longitudinal coordinate is applied to the solution of integrodifferential equations system [11, 15], where elastic shells movements are presented as trigonometric series in ζ and harmonic function in θ:

6
u1 =um U1 =um k=1sin2k-12πζa10C cosθ+a10S sinθsinτ+φu1 +a10O ,
u2 =vm U2 =vm k=1cos2k-12πζa20S cosθ+a20C sinθ+a20O sinτ+φu2 ,
u3 =wm U3 =wmk=1cos2k-12πζa30C cosθ+a30S sinθsinτ+φu3 +a30O .

Thus, as a result of the solution of hydroelasticity problem, we shall get the expressions for outer elastic ribbed shell elastic shifts.

It was shown earlier that for the study of the resonance amplitude-frequency characteristic in solving the equations of the dynamics of the shell it is sufficient to take only the first term of the series Eq. (6) [16].

By solving the considered system of simple equations, we get new coefficients in the expressions of outer geometrically irregular shell sagging, which in the dimensional state will become:

7
u30 =cosπζ2a30O +AnDn×Ezω2cosθsinωt+φz0+ηω+Exω2sinθsinωt+φx0+ηω.

We get the corresponding matching amplitude-frequency characteristic of outer shell sagging:

8
Aω=AnDn,

where An, Dn– expression system-dependent parameters are not discussed here due to where cumbersomeness; ηω – phase frequency characteristic.

4. Conclusions

Thus, the mathematical model of the mechanical system consisting of two coaxial cylindrical shells of final length, with the free attaching at butt ends is constructed. The outer shell is geometrically irregular, and the inner one is an absolutely rigid cylinder. The mechanical system is exposed to vibration. Such model of the system considers the liquid movement inertias and elasticity of final length outer shell, having rigid ribs. The constructed model allowed to define the expressions for outer geometrically irregular shell amplitude frequency characteristics. The proposed model allows to choose optimal parameters, proceeding from the known frequency range of vibration and necessary parameters of wear hardness, at the stage of mechanisms projects.

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About this article

Received
08 June 2016
Accepted
24 August 2016
Published
07 October 2016
SUBJECTS
Flow induced structural vibrations
Keywords
mathematical modeling
hydroelasticity
viscous liquid
coaxial cylindrical shells
geometrically irregular shell
vibration
Acknowledgements

The research is made under the financial support of RFFI Grants No. 15-01-01604-а, No. 16-01-00175-а and President of Russian Federation Grant No. MD-6012.2016.8.