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-□ □Розроблено математичну модель поздовжтх коливань триступеневог колони насосних штанг iз застосуванням функцш перемiщення та наван-таження окремих ступеней. Приведено формулу для визначення коефщенту дисипаци коливань колони та дослджено його змшу в залежностi вид жорсткостей гг ступеней. Встановлено, що безрезонансна робота колони при певнш дисипаци коливань ступеней забезпечуеться шляхом вда-лого тдбору гх жорсткостей
Ключовi слова: демпфування, дисипащя, змт-т навантаження, колона насосних штанг, мехатчна система, резонанс
Разработана математическая модель продольных колебаний трехступечатой колонны насосных штанг с применением функций перемещения и нагружения отдельных ступеней. Приведена формула для определения коэффициента диссипации колебаний колонны и исследовано его изменение в зависимости от жесткостей ее ступеней. Установлено, что безрезонансная работа колонны при определенной диссипации колебаний ступеней обеспечивается путем удачного подбора их жесткостей
механическая система, резонанс -□ □UDC 622.24.058
|DOI: 10.15587/1729-4061.2017.96528]
DETERMINING THE PARAMETERS OF OSCILLATION DISSIPATION IN A COLUMN OF SUCKER RODS
Ja. Grydzhuk
PhD, Associate Professor* E-mail: jaroslav.gridzhuk@gmail.com M. Lyskanych Doctor of Technical Sciences, Professor*
B. Kopey
Doctor of Technical Sciences, Professor** Yu Chuahjuy
Postgraduate Student** *Department of Applied Mechanics*** **Department of oil and gas equipment*** ***Ivano-Frankivsk National Technical University of Oil and Gas Karpatska str., 15, Ivano-Frankivsk, Ukraine, 76019
In the process of oil extraction by downhole rod pump installations (DRPI), a column of sucker rods (SR) in a string of pumping-and-compression pipes (PCP) is exposed to both static and dynamic load. At present, reliable information about the loading of a SR column can be obtained by the application of dynamometry at different DRPI operational modes [1]. This approach has a significant shortcoming in that it is impossible to employ the results of dynamometry at the stage of completing a SR column when the cost and time of making changes in its design is minimal. A development of dynamic processes over time known in advance makes it possible to evaluate the capability of mechanical system "SR column - fluid - PCP string" to resist external stresses. In addition, it contains information about the structure of the system, degree of its nonlinearity, dissipation mechanism of oscillation energy, allows evaluation of its stability and the capability for the self-excitation of oscillations. An alternative method of estimating the dynamic loading of a SR column is mathematical modeling of oscillating processes, which basically comes down to constructing and solving a system of differential equations of motion.
Study of parameters of dynamic instability of a SR column and its corresponding vibrational characteristics based on a comprehensive mathematical model was conducted in [2].
Determining the nature of influence of dynamic instability on the law of motion of a SR column in the directed well and numerical examining of parameters of its work was expanded in articles [3] and [4]. Paper [5] explored dynamic behavior of rod systems taking into account external force factors that are described by multivalued (subdifferential) relations. The statement of a boundary problem with nonlinearities for friction in the form of variation and quasi-stationary inequalities was presented. The algorithm of numerical calculation of rod columns of DRPI was proposed. Dynamics of the motion of a SR column in the twisted column of PCP is described in [6]. The mathematical model is presented in the form of a system of differential equations of motion with partial derivatives and geometric equations of a spatial curve. A problem on the distribution of elastic waves of impact character in a rod, one end of which moves with acceleration by the assigned law, while the other one is loaded with mass and rests on a spring, was tackled in [7]. The propagation of elastic waves and the development of deformations in a rod are determined using the normal functions. Assessment of the magnitude of dynamic loads for single- and two-stage columns of SR was conducted in [8]. By the results of theoretical studies, the authors proposed ways to reduce the dynamic loads and defined particular conditions for preventing the parametric resonance. Article [9] determines the damping coefficient of oscillations in a column of sucker rods based on the dynamograms obtained during experimental research. The impact of change in the amplitude and period of load oscillations over time on the damping coefficient along the entire length of a SR column was established. By three natural frequencies of longitudinal oscillations
they determine the location and dimensions of a cross cut in the vertical rod on an elastic suspension exposed to the action of its own weight [10]. According to research results, the authors substantiated the possibility to determine location and to diagnose a damage in a vertical rod.
The above enumerated articles did not investigate one of the important characteristics of dynamic behavior of a SR column - its capability to dissipate in the irreversible form some part of oscillation energy. Damping properties of oscillations in a SR column are predetermined by certain dissipation parameters whose quantitative assessment still requires a number of studies. That is why determining the dissipation characteristics of multistage SR columns of large length that influence the intensity of their oscillations, as well as defining the parameters of dissipation of a SR column, is an important practical task. Its solution will help increase the accuracy of assessment of the strength and durability of its elements.
The aim of present work is to determine the coefficient of dissipation of oscillation energy of a SR column based on examining it as a mechanical system with a finite number of degrees of freedom.
To achieve the set aim, the following tasks are to be solved:
- to substantiate basic principles in determining the coefficient of dissipation using the equations of motion of a three-stage SR column;
- to evaluate the coefficient of dissipation of oscillation energy for different configurations of SR columns, formed of fiberglass and steel rods.
In the dynamic calculations of a multistage SR column, it is very important to estimate the intensity of its oscillations during transition modes of DRPI operation. In most cases, it is under these modes, over a relatively short period of time, the resonance and near-resonance oscillations may occur [11]. The levels of such oscillations, as a rule, can exceed the oscillations of the systems under the modes of established operation. Quite often high levels of oscillations of a SR column during transitional modes cause the occurrence of damage in the elements of the column and their subsequent destruction under the action of alternating load.
Studying of the dynamics of a SR column during transition modes is, strictly speaking, a partial case of the calculation of mechanical system under the action of random loads [12]. A problem on the oscillations of a SR column during transition modes under the action of alternating load can be reduced with sufficient accuracy to the calculation of the systems with a finite number of degrees of freedom. Given this, further research is conducted for a conditionally vertical three-stage SR column. It is modeled in the form of a mechanical system with three degrees of freedom (Fig. 1). For this purpose, the following designations are accepted for the parameters of a SR system:
- m1, m2, m3 are the masses of the first, second and third stages, respectively;
- y1, y2, y3 are the displacement of masses in the system, respectively;
- m1, m2, m3 are the damping coefficients of the stages, respectively;
- k1, k2, k3 are the rigidities of stages that are brought to the point of suspension of the column, and of the joints between stages, respectively;
- F1(t), F2(t), F3(t) are the external loads, applied to the stages.
Fig. 1. Three-stage column of sucker rods: a — set-up; b — dynamic model
With regard to the basic principles of analytical mechanics [13], differential equations of motion for the constructed estimated scheme take the form:
F (t) = m1ddy1+(l+] ki (y, - y2 ); F2(t) = m2ddy2 + ^ + )k2(y2 -ya)--(1+) ki (y,- y 2 );
Fa (t )=m
+ ^ + kaya -(( 1 + )k2(y2 -ya). (1)
Given the nature of oscillating process of a SR column as a long complex rod, the functions of displacement and load are represented as:
y (t) = Yai (œ) eœt,
F (t) = F
where Yai, Fai are the amplitude values of displacements and forces of the i-th stage of the column; w is the frequency of forced oscillations.
Substituting the force function (3) and second derivative of displacement function by time (2) into system (1), we, after transformations, received:
Fai = -m4ra2 Yal +(1 + k (Yal - Ya2 );
Fa2 =-m2^2 Ya2 -(1 + 2^)kl (Yal-Ya2) + (l + 2^®i)k2 (Ya2- Ya3);
Fa3 =-m3œ Ya3- (l + 2^) k2 ( Ya2 - Ya3 ) + (l + 2^) kgYa3 .
+n3fm3P3 (pi2+p2 )]=B;
The determinant of the system of equations (4) takes the form:
D (ra) = [-m4ra2 + (1 + 2^4rai) k4 ]x
x [-m2ra2 + (1 + 2^1rai) k1+(1 + 2|i2rai) k2 ] x
x[-m3ra2 + (1 + 2|i2rai) k2 +(1 + 2|i3rai) k3 ]-[-m1ra2 + (1+ 2^1rai) k1 ](1+ 2|i2rai)2 k2 -[-m3ra2 + (1 + 2|i2rai) k2+(1 + 2|i3rai) k3 ](1 + 2|i1rai)2 k42. (5)
For convenience, the last expression is recorded in the form:
D(ra) « mlm2m3 (ra2 - 2|ilfp2mi - p2 ) x x (ra2 - 2^2fP2rai - P2 ) (ra2 - 2l3f P^i - P2 ),
Then, taking into account (9) and the accepted designations, the system of equations (7) can be represented as:
If we accept for coefficients n1f, n2f, naf the mean value of nrf, which matches the r-th form of natural oscillations, then we can obtain from (10) an approximated formula for the estimation of attenuation coefficient:
AmlESPlPM + Cp2 -B . P2 & .
mlm2m3 (P2 - P2 )(p3 - P2 ) (P2 - P2 )
where mrf is the damping coefficient that matches the r-th form of natural oscillations; grf is the dissipation coefficient; pj is the frequency of natural oscillations.
Revealing the brackets in expression (5), we shall equal coefficients at wi. Excluding the magnitudes of second order of smallness, after transformations, we write the following system of equations:
Ilf + l2f + l3f = ll + I2 + 1
mlm2m3 [ilf Pi2 (p2+p3 )+i2f p2 (p2+p2 )+i3f p3 (p2+p2 )]=
= (|l +|2 ) klk2 (ml + m2 + m3 ) +
+ (il +I3 ) klk3 (ml + m2 ) + (l +l3 ) k2k3ml&-mlm2m3 [llfP2 +l2fP2 +l3fP2 ] =
= mlm2 (|2k2 +|3k3 ) + mlm3 (|lkl + |2k2 ) + |lklm2m3. (7)
For convenience, the right sides of equations of the system will be denoted as follows:
A = Ii + I2 +I3;
B = (|l + |2 ) klk2 (ml + m2 +m3 ) +
+(il+I3 ) klk3 (ml+m2 )+(i+I3 ) k2k3ml ;
C = mlm2 (|2k2 +|3k3 ) +
+mlm3 (|lkl + |2k2 ) + |lklm2m3. (8)
We also note that the attenuation coefficient of oscillations of stages of the column is associated with the damping coefficient by dependence:
where Rem(r,3) is the remainder after dividing the number of natural form r by 3.
The basic oscillations of a SR column are longitudinal. The dissipation of energy of these oscillations occurs as a result of friction between rods and PCP in viscous medium and internal friction in the material of rods. Damping the oscillations of a column through the dissipation of energy leads to a decrease in their amplitude and frequency. That is why intensive dampening of longitudinal oscillations of a SR column is observed at their constant dissipation. This feature is expressed by a direct dependence of damping coefficient mrf on the coefficient of dissipation grf:
irf = Y rf/ (2rao )(l2)
where w0 is the main operational frequency.
After comparing dependences (9), (11) and (12), the coefficient of dissipation is determined as:
Y rf = 4mrao x
+Cp2-B . P2 r .
Pl+Rem(r+l,3) -Pl+Rem(r,3)
mlm2m3 (p2 -P2)(P2 -P2)(P2 -p2)
As can be seen from (13), the coefficient of dissipation depends on the masses, frequencies of natural oscillations of a SR column and the rigidities of stages of a SR column. The frequency of natural oscillations of a SR column depends only on the geometrical dimensions of its stages, while the mass and rigidity of each degree depends on its geometrical dimensions and material. Therefore, the study of change in the coefficient of dissipation for the configurations of SR columns with different rigidity is essential to ensure their resonance-free operation.
For further research we selected a three-stage SR column. Length and diameters of the column&s stages, equipped with steel rods according to [1], are given below:
- the first stage - 3329 feet (1015 m) and 1 inch (25 mm);
- the second stage - 4325 feet (1318 m) and 0.875 inch (22 mm);
- the third stage - 1525 feet (465 m) and 1 inch (25 mm).
A SR column enables a descent of pump with conditional diameter of 2.25 inches (56 mm) for the depth of 9300 feet (2835 m). Given current trends regarding the use of SR made of composite materials, analytical study was conducted for four variants of the set-up of a three-stage column (Table 1).
Variants of the set-up of a three-stage SR column
Column stage Materials of stages for configuration
No. 1 No. 2 No. 3 No. 4
When calculating the parameters B and C (8), rigidities of stages of a SR column were determined by formulas:
where Aj is the cross-sectional area of the stage; Ej is the modulus of elasticity of material of the stage; 1 is the length of the stage.
The masses of stages in a SR column were determined taking into account their length and the resultant mass of one meter of rods.
Parameters of a three-stage SR column, equipped with fiberglass and steel rods
Stage Length, m Diameter, mm Material Mass mi, kg Rigidity of the stages kj, N/m
Fiberglass 1339.8 2.418-104
Fiberglass 1291.6 1.442-104
According to [1], the main circular frequency of forced oscillations of a SR column is w0=0.398 rad/s. Circular frequencies that match the first, second and third form of natural oscillations of a SR column are equal to, respectively:
- for set-up No. 1 pi=4.405 rad/s; p2=10.8016 rad/s; p3= =72.747 rad/s;
- for set-up No. 2 pi=3.178 rad/s; p2=7.078 rad/s; p3= = 12.088 rad/s;
- for set-up No. 3 p1=2.739 rad/s; p2=6.761 rad/s; p3= = 15.956 rad/s;
- for set-up No. 4 p1=2.877 rad/s; p2=7.806 rad/s; p3= = 12.461 rad/s.
Guided by the theoretical provisions, given in [8], in order to determine and analyze the dissipation parameter (13), we shall use coefficients in the dimensionless form. The first coefficient is determined by relations between rigidities of the adjacent stages a^k^/k^ and a2=k3/k2, and the second
one - by the relations between dissipation coefficients of the adjacent stages ^=72/^ and c2=g}/g2.
By the results of research [9] for a column, equipped with fiberglass rods, in contrast to the steel column, there is a noticeable damping of the amplitude of oscillations due to internal friction. That is why of practical interest here are determining and examining the dissipatio coefficients of oscillations of a SR column with fiberglass rods. For set-up No. 2, as graph in Fig. 2 shows, with a decrease in relation a1=k2/k1 (that is, with increasing the rigidity of first stage k^, coefficient of dissipation gf decreases and approaches 0.12. At the same time, gf and g^ increase, accordingly, to values 0.42 and 0.35. With an increase in relation a1=k2/k1 (that is, with decreasing the rigidity of k1 of the first stage), gf and g^ decreases to values 0.14 and 0.1, respectively; while gf increases to value 0.38. It follows from graph in Fig. 3 that with a decrease in the ratio a2=k3/k2 (that is, with increasing the rigidity of the second stage k2), coefficients of dissipation of the second and third stage increase, accordingly, to values g2f=0.41 and g3f=0.51; while g^ for the first stage is reduced to value 0.12. With an increase in the relation a2=k3/k2 (that is, when reducing the rigidity of the second stage k2), g3f approaches gj. As illustrated by the graph in Fig. 4 (set-up No. 3) and as an analysis of formula (13) confirms, grf depends on g linearly and at q=
= g2/g1=1, g2f»g2; g1f»g3f»(g2+g3)/2 = 0.4. It follows from the
graph in Fig. 5 and formula (13) for set-up No. 4 that gH depends on g also linearly, and at c2=gj/g2=1; g1f»g3f»g3=0.3.
Y rfifli)
Fig. 2. Graph of dependences of dissipation coefficients gn on the dimensionless parameter ai (at ai=2.5 g=0.14; g2=0.28; g3=0.24): 1 - curve g^); 2 - curve g2^); 3 — curve g3f(ai)
Fig. 3. Graph of dependences of dissipation coefficients gf on the dimensionless parameter a2 (at a2=2.5 g=0.14; g2=0.28; g3=0.24): 1 — curve g^fo); 2 — curve g2^); 3 — curve g3f(a2)
Fig. 4. Graph of dependence of dissipation coefficients grf on the dimensionless parameter c1: 1 — curve g1f(c,); 2 — ^(q); 3 — curve g3f(c1)
lrf<c2)
Fig. 5. Graph of dependence of dissipation coefficients grf on the dimensionless parameter c2: 1 — curve g1f(c2); 2 — g2f(c2); 3 — curve g3f(c2)
Based on the above results of calculation and constructed graphic dependences, we can argue that by a proper selection of rigidities of a column&s stages, it is possible to provide the required energy dissipation of its oscillations and, in addition, to prevent the occurrence of resonance. A
change in rigidities of separate stages of a SR column may be carried out either by changing their geometrical dimensions or changing the material that they are made of. As research results reveal, the inclusion in set-up No. 2 of the first fiberglass stage instead of the steel one (at a1=2.5) causes an increase in the attenuation coefficient in the fiberglass stage to g1f =0.38 and its simultaneous reduction in the steel ones to g3f=0.1 The use of the first fiberglass stage instead of the steel one reduces its rigidity by 4.2 times and leads to an increase in the coefficient of dissipation of oscillations by 3.8 times. An increase in the rigidity of the weighted steel bottom (at a2=3.7) in a narrow range reduces coefficient of dissipation of the upper fiberglass stage by 3.6 times. However, the linear increase in dissipation coefficients in set-ups No. 2, No. 3 and No. 4 with fiberglass stages predetermines a corresponding decrease in the amplitude and frequency of their oscillations. Such approach makes it possible to prevent the phenomenon of resonance in the operation of conditionally vertical SR column during transition modes of its work under the action of alternating load. On the other hand, it will help to minimize the probability of occurring fatigue damage in the elements of a SR column and their subsequent destruction.
References
Наведено основш дефекти метале-вих гофрованих водопропускних труб, як виникають внаслидок експлуатацй, та висвтлено проблеми забезпечення ïx дов-говiчностi та мщности Проаналгзовано проблеми адаптацй закордонних норма-тивних документiв щодо проектування металевих гофрованих конструкцш на залiзничниx та автомобшьних дорогах Украши. Наведено результати експери-ментальних та теоретичних розрахун^в несучоï здатностi металевих гофрованих конструкций
Ключовi слова: залишковi деформацп, проектування, пластичний шартр, рухо-мий склад залiзниць, щЫьтсть грунто-воï засипки
Приведены основные дефекты металлических гофрированных водопропускных труб, возникающие вследствие эксплуатации, и освещены проблемы обеспечения их долговечности. Проанализированы проблемы адаптации иностранных нормативных документов по проектированию металлических гофрированных конструкций на железных и автомобильных дорогах Украины. Приведены результаты экспериментальных и теоретических расчетов несущей способности металлических гофрированных конструкций
UDC 624.014.27-422.12.13
[DOI: 10.15587/1729-4061.2017.96549|
THE STUDY OF STRENGTH OF CORRUGATED METAL STRUCTURES OF RAILROAD TRACKS
V. Kovalchuk
E-mail: kovalchuk.diit@mail.ru R. Markul
Department "Track and track facilities" Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan Lazaryana str., 2, Dnipro, Ukraine, 49010 E-mail: guaranga_mr@mail.ru O. Bal PhD, Associate professor* E-mail: olenabal79@gmail.com A. М i ly a n y c h PhD*
E-mail: milyan_74@ukr.net
A. Pe n tsak
PhD, Associate Professor** E-mail: apentsak1963@gmail.com
B. Parneta PhD, Associate professor** E-mail: f_termit@yahoo.com
O. Gajda PhD, Associate professor** E-mail: gajda@ukr.net *Department "The rolling stock and track" Lviv branch of Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan I. Blazhkevych str., 12a, Lviv, Ukraine, 79052 **Department of Construction industry National University «Lviv Polytechnic» S. Bandery str., 12, Lviv, Ukraine, 79013
Metal corrugated structures (MCS) have been known since the end of the XIX century [1, 2]. In Russia, the first
mention of the MCS constructions were found as early as in 1875, when about 1300 linear meters of pipes were laid on the Transcaspian railway. From 1887 to 1914, another 64000 linear meters were laid, which comprised five thou©
