PARAMETRIC VIBRATIONS OF THE UNDERGROUND OIL PIPELINE.

. This paper is devoted to obtaining a solution to the problem of parametric vibrations and dynamic stability of thin-walled underground oil pipelines, taking into account the effect of a damper. An equation has been obtained that allows estimating the dynamic stability of underground oil pipelines, which consists in constructing regions on the plane of the parameters ωmn, ω0 and the longitudinal force parameter P, as well as the internal pressure p0, where ωmn is the frequency of free vibrations, ω0 is the excitation frequency. The assessment of the dynamic stability of the oil pipeline section is made for the known values of ωmn, ω0, p0, P, » by finding a point. If the point does not fall into the outlined area, it means that dynamic stability is ensured, otherwise there is a danger of parametric resonance.


Introduction
The widespread use of main pipelines with a large diameter of more than 1200 mm interacting with various media provides the relevance of research in the field of vibrations and stability of structures.It is impossible to ensure the reliability of such structures without appropriate calculations for vibration resistance.One of the most important goals of these calculations is to avoid the dangerous phenomenon of parametric resonance.Parametric resonance is more dangerous than ordinary resonance, since parametric resonance occurs at the set of points with certain relationships between natural frequencies and disturbance frequencies.These sets cover some regions on the plane of the parameters of the <pipe -liquid= systemregions of dynamic instability.
Analysis of works on the problem of dynamic stability of large-diameter main pipelines shows that most of the works on both core and shell theory in this area are devoted to studies without taking into account the influence of the damping properties of the medium.Studies on the core theory were carried out in [1][2][3][4][5][6], on the shell theory -in [7][8][9][10][11][12].
This work is devoted to obtaining a solution to the problem of parametric vibrations and dynamic stability of thin-walled pipes (cylindrical shells) laid in a soil medium (taking into account the resistance forces of the medium) at an unsteady internal working pressure, longitudinal compressive force, the reaction of the elastic soil resistance and the flowing liquid flow.

Materials and methods
Let's consider a section of an underground oil pipeline with a length L, a cross-sectional radius R, and a thickness h.This section of the cylindrical shell is exposed to an unsteady internal working pressure P(t): ( ) where: p0 3 internal working pressure; γ 3 excitation frequency corresponding to the characteristics of the pumping station; ½ ≤ 0.5 3 excitation factor.The non-stationary internal working pressure brings the stationary longitudinal compressive force to the function: coefficient of elastic soil resistance to function: where: Flongitudinal compressive force; κ3 coefficient of elastic soil resistance.
In [13], on the basis of a geometrically nonlinear version of the semi-momentless theory of shells of mean bending, a differential equation of motion in displacements of a homogeneous shell was obtained, taking into account the effect of the internal working pressure p0, the longitudinal compressive force F, the reaction of the elastic soil resistance: where u, v, w − components of displacement of the middle surface of a homogeneous shell, referred to its radius of the middle surface, x L ø = , θ − longitudinal coordinate and polar angle, ϑ2− angle of rotation of the tangent to the median line of the shell cross-section, Е − elastic modulus of pipe material, ρ − shell material density (N•s 2 /m 4 ), •0 3 initial deformation of a straight section of a cylindrical shell for a non-deformable cross-sectional contour (m), − parameter of the relative thickness of the shell (dimensionless quantity), ¾ 3 Poisson's ratio of the shell material.Equation ( 5) contains 4 unknown functions of coordinates and time t: u, v, w and ϑ2.Adding the relations of the semi-momentless theory of cylindrical shells [14] we obtain a complete system of differential equations consisting of four equations ( 5) and (6).
Solving system ( 5) and ( 6) by the Fourier method, for the case of hinged support of the ends of the shell, let's represent (ξ,θ, ) w t in the form: from condition (6) we obtain the expressions: where , m, n 3 wave numbers in circumferential and longitudinal directions.
where ω 2 mn is the square of the frequency of free vibrations of the underground section of the oil pipeline without taking into account the side coefficients In ( 12), the following dimensionless parameters were introduced: − parameter of internal working pressure (dimensionless); − parameter of elastic soil resistance (dimensionless); Excitation factor •mn is defined by the expression from ( Damping factor 2¸0 from (10): By solving the Mathieu differential equation (11), it is reduced to constructing regions of dynamic instability, i.e.Haynes-Strett diagrams and converging under conditions conducive to the occurrence of parametric vibrations.Assessment of the dynamic stability of underground oil pipelines consists, firstly, in the construction of regions of dynamic instability on the plane of the parameters ωmn, γ and internal pressure p0, the parameter of the longitudinal force P, the velocity of the fluid flow V0, the reaction of the elastic soil resistance, where ωmn− the frequency of free vibrations for a given mode of vibration (m = 1,2,3… n = 1,2,3….),γ -excitation frequency determined by the operation of pumping stations.Second, the dynamic stability of the underground oil pipeline section is assessed at known values of ωmn, γ, p0, P, V0, » by finding a point.If a point does not fall into the stability region, this means that dynamic stability is ensured, otherwise stability is not ensured, i.e. there is a danger of parametric resonance.
The solution of the Mathieu equation (11) for instability regions obtained in [1,16] is an inequality for finding the boundaries of the first, second, third, and so on instability regions: -the main first region of instability ( ) ( ) -second region of instability The analysis of the obtained values, as noted [16], shows that in the presence of damping, in order for the resonance ωmn ≈ γ to be noticeable, a greater modulation depth •mn•ω 2 mn is required than in the case of the resonance ωmn ≈ γ/2, and it is even harder to realize the resonance at ωmn ≈3 γ/2.

Results and their discussion
The construction of regions of dynamic instability for a large-diameter oil pipeline consists in finding the lower and upper boundaries of these regions by substituting values in (15), ( 16), (17).
The definition of dynamic instability consists in determining the position of the point (ω0, or η).If the point is in a free, unshaded region, then the stability of the pipeline in question is ensured.If the point falls into the shaded regin, then the parameters of the oil pipeline should be changed to exclude the possibility of the appearance of parametric resonance.
Based on this technique, below are the regions of dynamic instability for large-diameter oil pipelines laid in a soil medium with a coefficient of dynamic viscosity η =0.    1 and Figure 1 shows that for thick-walled pipelines (h/R=1/30), the upper and lower boundaries of the region converge at the value of the internal working pressure p0=4.2 mPa, and for the parameter of wall thinness h/R=1/50, at p0=2.20 mPa, therefore, with a decrease in the parameter of wall thinness, the region of dynamic instability increases.This indicates that pipelines with a larger wall thickness are less susceptible to parametric resonance.
The values in Tables 1 and in Figure 1 show that for the wall thinness parameter h/R=1/30 for the first instability region at the value of the longitudinal force parameter P=0.1, the lower and upper boundaries converge when the internal working pressure p0=4.2 mPa.Without taking into account the influence of the damper, this boundary is at a value of p0=2.0 mPa, which means that the region of dynamic instability, for calculations without taking into account the influence of the damper, as can be seen in Figure 1, expands.Therefore, taking into account the forces of resistance of the medium gives a narrower region of instability, which means that for certain values of the excitation frequencies, it excludes the possibility of the occurrence of parametric resonance from a part of the region without taking into account the damper (the third region of instability is not reflected in the table, since the values are relevant only at a pressure p0>30mPa).Next, we will construct regions of dynamic instability depending on the parameter of the longitudinal force, taking into account the influence of the damping properties of the soil medium.The values obtained are shown in Table 2.  Analysis of the values in Table 2 and Figure 2 shows that with an increase in the longitudinal force parameter, the dynamic instability regions expand for all thinness parameters, which can lead to parametric resonance and loss of dynamic stability of the pipeline.
Studies of the influence of the dynamic viscosity coefficient on the excitation frequency, the results of which are listed in Table 3 and reflected in Figure 3, show that with an increase in this coefficient, the dynamic instability region narrows and at some values, the boundaries of the upper and lower zones converge at one point, at which the region ends its existence.Therefore, with a subsequent increase in the values of the viscosity coefficient of the soil medium, the dynamic instability of the gas pipeline will be ensured.For pipelines with a lower thinness parameter, this boundary is shifted towards higher values of dynamic viscosity.For example, at h/R=1/30 and a pressure of 6.0 mPa for the 1st instability region, convergence was found for the value η=1.2•104Pa•s, and at the parameter h/R=1/50, for η=3.1•104Pa•s.This suggests that pipelines with a larger wall thickness are less susceptible to parametric resonance.With an increase in the internal working pressure, the instability regions for all thinness parameters increase.This is explained by an increase in the natural vibration frequencies, which were studied in [17,18].

Conclusions
1.The problem in the field of dynamic instability of underground thin-walled oil pipelines of various diameters and wall thicknesses, under the unsteady effect of the internal working pressure, the parameter of the longitudinal compressive force and the flow of the flowing liquid is solved.A separable system of Mathieu differential equations is obtained, with the help of which regions of dynamic instability are constructed to assess the possibility of the occurrence of parametric resonance.
2. The dynamic stability of underground oil pipelines laid in a soil medium with different coefficients of dynamic viscosity at various parameters of the pipeline thinness, internal working pressure, and longitudinal force parameter has been studied.Research shows that: -with an increase in the viscosity of the soil medium, the region of dynamic instability narrows and at some values ceases to exist, i.e. for pipelines laid in the soil with a dynamic viscosity coefficient of more than η>8•10 5 Pa•s, the occurrence of parametric resonance is excluded; for unstable soil media 0.1•10 4 <η <8•10 5 Pa•s, in which the pipeline is laid, the loss of dynamic stability is possible, and the probability of its occurrence is the higher, the lower the dynamic viscosity of the soil medium.Pipelines with a thicker wall h/R=1/30 are more advantageous from the standpoint of dynamic stability, since the region of dynamic instability is 1.5-2.0times less than with h/R=1/40 or h/R=1/50.
coefficient of elastic soil resistance for a pipeline subjected to the action of internal working pressure (N/m 3 ), ¾0 3 soil Poisson's ratio, Е0 − modulus of soil elasticity (N/m 2 ), γ 3 soil density (N/m 3 ), H 3 pipeline laying depth (m), ρ0 − density of the flowing liquid (N•s 2 /m 4 ), V0 3 flow rate (m/s), ρ0Φm 3 added fluid mass, Φm = I(¼0) / ¼0 I'(¼0) 3 parameter of dependence on wavenumbers in circumferential and longitudinal directions (m, ω 3 determined by the ratio of the Bessel function to its derivative depending on ¼0=nπR/L, η 3 damping factor, we obtain a system of separable differential equations of the shell motion with respect to the time functions φ(t): 5•104 Pa•s (for clarity: ηwater=1.7•10-3Pa•s, ηmercury=1.55•10-3Pa•s, ηoil=0.128Pa•s), with different parameters of wall thinness.For comparison, the calculation was made taking into account the influence of the damper and without it.See table 1 for results.

Table 1 .
The dependence of the disturbance frequencies ω0 (Hz) for the regions of dynamic instability on the internal working pressure at the dynamic viscosity coefficient η = 0.5•104 Pa•s with and without taking into account the damper for the underground oil pipeline.Taking into account the damper η =0.

Fig. 1 .
Fig. 1.Regions of dynamic instability depending on the internal pressure for an underground oil pipeline laid in soil with a viscosity coefficient η=0.5•104Pa•s.

Fig. 2 .
Fig. 2. Regions of dynamic instability for an underground oil pipeline depending on the value of the longitudinal force parameter.

Fig. 3 .
Fig. 3. Regions of dynamic instability for an underground oil pipeline laid in soil with a different coefficient of dynamic viscosity, for different parameters of thinness.

Table 2 .
Dependence of the disturbance frequencies ω0 (Hz) for the dynamic instability regions on the longitudinal force parameter for various parameters of wall thinness.

Table 3 .
Dependence of disturbance frequencies ω0 (Hz) for regions of dynamic instability on the coefficient of dynamic viscosity for an underground oil pipeline