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Graduate Studies Legacy Theses

1994

Dynamic stability analysis of a sucker rod string in

the deviated well

Cao, Wilfred

Cao, W. (1994). Dynamic stability analysis of a sucker rod string in the deviated well

(Unpublished master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/12852

http://hdl.handle.net/1880/30437

master thesis

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THE UNIVERSITY OF CALGARY

DYNAMIC STABILITY ANALYSIS OF A SUCKER ROD

STRING IN THE DEVIATED WELL

BY

WILFRED CAO

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS

FOR THE DEGREE OF MASTER OF SCIENCE IN ENGINEERING

DEPARTMENT OF MECHANICAL ENGINEERING

CALGARY, ALBERTA

MARCH 1994

© WILFRED CAO 1994

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The University of Calgary

Faculty of Graduate Studies

The undersigned certified that they have read, and recommend to the Faculty of

Graduate Studies for acceptance, a thesis entitled, "Dynamic Stability . Analysis of a

Sucker Rod String in the Deviated Well", submitted by Wilfred Cao in partial fulfilment

of the requirements for the degree of Master of Science in Engineering.

Dr. SA. Lukasiewicz(SupersTi) Department of Mechanical Engineering

/44 f42

Dr. M.C. Singh Department of Mechanical Engineering

C Dr. JD.M. Beigrave Department of Chemical and Petroleum Engineering

AO Date

11

Abstract

This thesis presents a model for predicting and analyzing the dynamic stability

behaviour of sucker rod pumping installations in inclined wells. This model incorporates

the dynamics of a curved sucker rod string and the post buckling behaviour of the rod

inside the tubing.

The governing differential equations are derived from dynamic virtual work

principle, and analyzed using the analytical (Galerkin) method and numerical (forth-order

Runge-Kutta) method.

The model thus developed predicts the behaviour of the rod string. It incorporates

the effects of the rod's inertia, its weight, fluid inertia and viscosity. The model is able

to simulate a wide variety of pump conditions, geometry and material of the rod string.he

results predicted by the model can be used in the design and operation of sucker rod

pumping installations to prevent the string from buckling. A computer program was

developed to calculate the distances between two supports along the rod string and the

lateral deflections of the rod. For analyzing and optimizing the performance of an existing

well, the polished rod dynamometer data must be known before this program is used. If

the program is used in the design of a new rod string, the maximum and minimum forces

acting on the rod must be known.

111

Acknowledgements

I would like to express my sincere gratitude and appreciation to my supervisor,

Dr. S. Lukasiewicz, for his conscientious guidance, support, understanding and

encouragement throughout the course of this work.

Thanks are due to the fellow students of the Department of Mechanical

Engineering for their advice and cooperation. I would also like to pay tribute to my wife,

Suejean, who typed the manuscript. The financial assistance provided by the Department

of Mechanical Engineering of The University of Calgary, as well as by Dr. S.

Lukasiewicz, are gratefully appreciated and acknowledged.

iv

Table of Contents

Approval Page

Abstract

Acknowledgements iv

Table of Contents v

List of Tables vii

List of Figures viii

Nomenclature ix

Chapter 1 Introduction

1.1 General background 1

1.2 Literature review 4

1.3 Objective 8

Chapter 2 Variational Formulation of the Governing Equations 14

2.1 Fundamental assumptions 14

2.2 Development of equations 16

2.2.1 Expression of the rod axial force 16

2.2.2 Mathematic model for the initial shape of the rod 21

V

2.2.3 Calculation of force on rod due to the fluid pressure 25

2.2.4 Development of equations 26

2.3 Static analysis 41

2.4 Dynamic analysis 43

2.5 Buckling behaviour of sucker rod in the tubing 45

Chapter 3 Numerical Solutions and Techniques 57

3.1 Numerical techniques 57

3.2 Comparison 60

Chapter 4 Application 63

Chapter 5 Concluding Remarks and Summary 70

5.1 Remarks and limitations 70

5.2 Summary 72

References 73

Appendices 77

vi

List of Tables

Table 1 Input data 66

Table 2 The distance between two supports along the rod 67

Table 3 The distribution of supports 69

vi'

List of Figures

Figure 1 Components of a sucker rod pumping system 2

Figure 2 Pumping cycle 11

Figure 3 Tapered rod string 12

Figure 4 Polished rod load 19

Figure 5 Ideal axial force of the rod 20

Figure 6 Transverse deflection of the rod 22

Figure 7 Model of the rod string 23

Figure 8 One segment of the rod 27

Figure 9 Comparison of the deflections due to the different assumptions 46

Figure 10 Comparison of three components of the deflections due to different

deflection function assumptions 47

Figure 11 Deflections against the axial forces in the rod 48

Figure 12 Deflections against the forced frequency parameters 49

Figure 13 Deflections against the fluid pressure parameters 50

Figure 14 Geometry of the rod after contacting the tubing 52

Figure 15 Reaction loads from the tubing 56

Figure 16 Comparison of the results from Galerkin and R-K 62

Figure 17 Geometry of the rod in case 1 and case 2 65

vi"

Nomenclature

Distance between two supports (for the wells in which both wheeled

couplings and guides/scrapes are installed, the supports are wheeled

couplings, the transverse displacement of the rod must be smaller than the

gap between the guides/scrapes and the tubing in the neutral position; for

the wells in which no wheeled coupling is installed, the supports are the

guides/scrates, the transverse displacement of the rod must be smaller than

the gap between the rod and the tubing in the neutral position)

R Radius of the curved rod string

P Axial force along the string, at the top of the strings, called polished rod

force

CO Angular frequency of the string's up and down movement

Pmax Maximum axial force along the string

Pmin Minimum axial force along the string

W 0 Initial deformation of the string due to the inclined well, ie, the distance

from the straight line between two centres of supporters

X The distance measured along a straight line between two supports

r Radius of the rod string's cross-section

p Fluid static pressure on the surface of the string

Fp Fluid pressure force per unit length of the string

ix

Nomenclature (cone d)

M The bending moment at an arbitrary section of the rod

W(x,t) The deformation function of the rod string

T1(t), T2(t) The deformation function of the rod string versus time

The derivative with respect to x

The derivative with respect to t

V Strain energy

W F The work of the axial force

W f The work of the transverse damping force

W, The work of the fluid pressure

T The kinetic energy of rod and fluid

pf Mass density of fluid

Pr Mass density of the rod string

A Cross-section area of the rod string

E Young's modulus

a, b Coefficients of deformation function

Dimensionless factor of average axial force

Dimensionless factor of axial force difference

Dimensionless factor of fluid pressure

Dimensionless factor of the transverse damping force

Nomenclature (cont' d)

Dimensionless factor of the radius of the curved rod

Dimensionless factor of kinetic energy

Shear force of cross-section of the rod string

Modulus of cross-section area of the rod

The gap between the rod (or guides/scrapes if available) and the tubing in

the neutral position

Supporting force from supporter after the rod contacts the tubing

Axial force along the rod after the rod contacts the tubing

Damping factor

I

Chapter 1 Introduction

1.1 General Background

When oil wells cease flowing, some means of artificial lift are required to produce

from the well. According to a recent survey, there are, in the North America,

approximately 85 percent of producing wells which are on artificial lift. Of these wells,

a vast majority (80 to 85 percent) are being produced by sucker rod pumping (shown in

Fig. 1). Proper design and maintenance of sucker rod pumping installation is, therefore,

as of major importance in the field of oil production.

As shown in Fig. 1, the prime mover makes the cranks rotate. The crank is

connected to the pitman side members and causes the walking beam to pivot about the

saddle bearing, thereby causing the polished rod to move up and down through its

connection to the wireline and horsehead. In this way, the surface equipment transfers

energy for pumping the well from the prime mover to the sucker rod string.

The sucker rod is composed by numbers of bars connected together by couplings.

Each bar is 25' or 30' in length. The sucker rod is the connecting link between the

surface pumping unit and the subsurface pump which is located at or near the bottom of

the oil well. The vertical motion of the surface pumping unit is transferred to the

2

fl.Ow LINE

TUBING

- MORSE HEAD

- POLISHED POD CLAMPN_

CAR PIEP BAR

POLISHED ROO-

StUFFING BOX

TEE

CASING HEAD

CASING

SUCKER ROD

SADDLE SEARING-tf I \.J 97 -tAIL BEARING

SAMPSON POST

CRANKSHAFT

Figure 1 Components of a Sucker Rod Pumping System

GEAR REDUCER

V-BELT

R.

PIWiC MOVER

3

subsurface by the sucker rod.

As shown in Fig. 1, the prime mover makes the cranks rotate. The crank is

connected to the pitman side members and causes the walking beam to pivot about the

saddle bearing, thereby causing the polished rod to move up and down through its

connection to the wireline and horsehead. In this way, the surface equipment transfers

energy for pumping the well from the prime mover to the sucker rod string.

The subsurface pump (see Fig. 2) consists of the usual simple combination of a

working barrel, liner and plunger with a suitable intake valve (standing valve) and

discharge valve ,(traveling valve) for displacing the well fluid into the tubing and to the

surface.

The sucker rod used to be in the vertical well, but now more and more oil

companies interest the ones in the inclined and deviated well, which can decrease the

spending on land lease.

The rod string in the inclined and deviated well is under different static and

dynamic forces than that in the vertical well. It is supported by the scrapers/guides and

(or) wheeled couplings (if available) which are distributed along the rod string in a certain

distance, but it is also transversely loaded by its own weight. Therefore, the dynamic

behaviour of the sucker rod in the deviated well is much more complicated and different

4

from the dynamic behaviour in a vertical well.

These differences are caused by the friction between the rod string and the tubing,

and the rod curvature. If the tubing is curved, the dynamically-loaded rod string deforms

and comes into contact with the wall of the tubing. The curvature causes lateral

displacement of the rod inside the tubing. It couples the longitudinal vibrations with the

transverse vibrations. The axial compressive forces at the bottom of the well can buckle

the rods and wear the rods and tubings. To prevent rod and tubing wearing problems, the

distance between two close supporters(couplings/scrapers/guides), as well as rod size,

must be determined.

1.2 Literature Review

The walking beam sucker rod pumping system is one of oldest mechanical system

known to man, having been used by the Chinese at least 3,000 years ago. The standard

analysis and design procedure for these systems used by the industry for the past 20 to

25 years was completed by the Midwest Research Institute in 1962. The Institute, under

contract to Sucker Rod Pumping Research Inc., developed a method for computing forces

and displacements using an analog computer simulation. The method is called the API

RPIIL method (Ref. [6]).

5

These standards and recommendations are based on dynamometer cards derived

from electronic analogue computer analyses of many combinations of well parameters,

including depths from 2,000 ft (610 m) to 12,000 ft (3658 m) and production rates from

100 bbl to 1,500 bbl per day. From the result of these analyses, nondimensional

parameters were computed and plotted to show the relationship between the various

parameters, and a procedure was developed for evaluating stress in a given rod string.

The API method has been a powerful engineering tool but has limited application.

It cannot be used to design deeper wells or the rod strings made of two different

materials. It can be used only for vertical wells.

These limitations include the simplified pumping unit geometry and polished rod

motions, low slip prime movers and full pump fillage (Ref. [6]).

Pioneering work by Gibbs in the 1960's resulted in the development of two

mathematical models which form the basis for most of today's technology. These two

models, which include solutions to the damped wave equations, are a diagnostic analytical

model and a predictive finite difference model.

The diagnostic analytical model is based on a classical Fourier Series solution of

the damped wave equation. Stated simply, the input to this model is the surface

dynamometer card, and the output is the subsurface dynamometer card. The primary use

6

of this model is to diagnose downhole pump problems. The predictive finite difference

model is a solution to the same equation but with a different statement of the boundary

conditions. The input to this model is the surface polished rod motion and load at the

pump. The output is the load at the surface and motion at the pump. The primary use of

this model, when used with the kinematic description of the surface unit and

characteristics of the prime mover, is to predict loads at any point in the system such that

the suitability of a particular design may be verified.

For deep-well pumping applications, it may be found that a single sucker rod size

is impractical because a polished rod stress of sufficient magnitude may be exerted by the

weight of the rods themselves such that only a very small diameter plunger can be

tolerated. A method of overcoming this difficulty is to taper the rod string, placing large-

sized rods at the top of the hole but reducing the size (and so the weight) by stages down

the hole.

The design of such tapered strings to take into account the various dynamic

stresses becomes very complex, and for that reason tapered strings are frequently

calculated on static loads alone.

As computer use grew, there was a substantial effort made to develop sucker rod

pumping models that could be solved with the aid of a digital computer. These efforts

have been involved in not only the dynamics of sucker rod strings in vertical wells but

7

also the sucker rod string in the deviated and inclined wells as well.

An improved model for the sucker rod in vertical wells was presented by Dale

Russel Doty and Zelimir Schmidt (Ref. [11]). Their study overcame one of the

shortcomings of previous models. Whereas previous models ignored the dynamics of the

fluid and tubing columns as well as the fluid physical properties, they included the

dynamics of the fluid as well as the rods. They analyzed the effects of fluid physical

properties on a sucker rod pumping installation.

The mathematical model was developed by performing a standard momentum

balance on the string which yields a first-order partial differential equation and by

applying the experiment results of dimensionless factors performed in 1976 by Valeev and

Repin. They developed a modified method of characterizations to solve the system of

partial differential equations and their associated boundary conditions, and generated an

accurate and efficient computer program by using a numerical solution technique. The

technique relies on the fact that the rod string and fluid columns are only weakly coupled

along their length through the various frictional forces, to solve the rod equations and

fluid equations separately.

A much more careful mathematical model of the dynamic behaviour of the rod in

the deviated and inclined well was presented by S.A. Lukasiewicz(Ref. [9]) in 1990. This

model deals with two partial differential equations: one concerns the motion in the

8

direction tangential to the rod, the other vertical to the rod. It is only by using this model

that analysis of lateral deflections can be performed not only in theory but also in

practice. The model includes the situations where the rod contacts the tubing.

A computer program, SRPUMP, based on the above-mentioned model, was

developed by Lukasiewicz (1990) and used to simulate the behaviour of the sucker rod

string in deviated wells. The developed model was applied to the design of new wells and

to the evaluation of the behaviour of existing wells.

1.3 Objectives

Summarizing the above, there are two methods for analyzing a sucker rod system:

the API method and the analytical method.

The API method (Ref. [6]) is based on correlations from data ( from steel sucker

rods ) which are presented as nondimensional parameters plotted in curves. This method

was originally designed for hand calculations and involves a trial and error approach to

the design of sucker rod pumping systems. A particular design for a sucker rod string and

pumping system is chosen and then evaluated using the correlations of the API method

to determine if the loads and stresses on the chosen system are acceptable.

9

The analytical method actually designs the appropriate system using complex

stress and vibration analysis techniques that apply the basic principles of material,

properties and mechanics. Since much more complex assumptions are used in the

analysis, the analytical method is much more rigorous than that of the API method and

is not restricted to steel sucker rod strings. The analytical method, as applied in SRPUMP,

is able to design the optimal rod string.

As the plunger ( see Fig. 2) travels downwards, the travelling valve is open and

the principal force acting in the bottom of the rod string is a compressive load caused by

hydrostatic pressure. The compressive load can cause buckling or, for some materials such

as fibreglass, fracturing of the rod string. To overcome this problem, a sinker bar, which

is a thick, stiff section, may be installed at the bottom of the rod string ( see Fig. 3 ) to

provide the additional weight and stiffness necessary to prevent buckling.

Buckling can cause problems, especially in the case of directional or deviated

wells, where the sucker rod does not hang vertically, but is in an inclined position

according to the curvature of the tubing. Due to its own weight, the curvature of the

tubing and the compressive forces, the sucker rod comes into contact with the wall of

tubing; therefore, the friction forces between the tubing and the rod affect the loads in the

rods, torques and power consumption.

The purpose of this thesis is to present a mathematic model for analysis of the

10

buckling of the sucker rod, and to study the deflections and the dynamic stability of

sucker rod strings in deviated and inclined wells, with the effects of the fluid pressure on

the rod, weights of the fluid and the rod itself.

Discussion of the influences of a series of dimensionless factors is also presented

in this thesis as an analysis of transverse displacements is given.

An attempt has been made to develop a computer program, which takes into

account both the transverse displacements of the rods and the buckling compressive

forces, and calculates the maximum distance between the wheeled couplings/guides which

prevent the contact of the rod with the tubing along the rod string. Installation of the

guides or wheeled couplings in the areas predicted by the program is recommended. Wear

of the rods and the tubing as well as power consumption can be significanly reduced.

The maximum distance between the two wheeled couplings/guides has been

determined by assuming that the transverse displacements of the rod must be smaller than

the transverse distance between the rod and the tubing in the neutral position.

The author hopes that the model and the program, which need further

developments to connect it to other pumping programs providing additional

information, can be used to replace the current commonly used rules for determining

the number of wheeled couplings/guides/scrapers along the string or segments of the rod

11

4,

VZO

(a)

Tubing

Sucker rods

Working barrel and liner

Traveling valve

Plunger

Standing valve

it"

S•_ -

(b) (C) (d)

Figure 2 Pumping Cycle

12

Normal Rod Rod string with

string Sinker Bar

-r

L 4 44

L5

-F Plunger

Length i

D3

05

D6 Plunger Diameter

Figure 3 Tapered Rod Strings

Sinker Bar

13

string.

This report also investigates and illustrates the effects of the dimensions of the

sinker bar on the distribution of supporters (couplings/guides/scrapers) along the rod string

under the circumstances that the string does not contact the tubing.

14

Chapter 2 Variational Formulation of the

Governing Equations

2.1 Fundamental Assumptions

An accurate prediction of the performance of the sucker rod string requires careful

modelling of the dynamic behaviour of the rod. The rod must be considered as an elastic

vibrating beam which has to satisfy the dynamic conditions at the polished rod and at the

pump. The rod in the deviated tubing undergoes not only longitudinal vibrations but it

also deforms in the direction perpendicular to the directions of the rod string. During the

pumping period, the rod vibrates simultaneously both longitudinally and transversely. Two

different situations are considered: in the first case, the deformed rod is still in contact

with tubing along a certain length; in the second case, the rod itself does not contact the

tubing but is supported by the wheeled couplings or guides along its length in certain

points.

In the first case, the lateral vibrations are prevented. The rod vibrates

longitudinally, and the friction forces between the rod and tubing, or between the

coupling/guide and the tubing, affect the rod's motion. In the second case, the rod vibrates

both transversely and longitudinally.

15

The mathematical solution for the first case is much easier to obtain because the

equations are quasi-linear. Only Coulomb friction forces which change the sign with the

velocity, produce the nonlinearities. In the second case, two coupled nonlinear differential

equations of lateral and longitudinal vibrations must be solved. The problem is much

more difficult to solve because of nonlinearities and unknown contact zones of the rod

and the tubing. However, the solution can be simplified by the introduction of certain

assumptions. Since we are mostly interested in a steady-state solution, we can assume that

the frequency of lateral vibrations of the rod string is equal to the pumping frequency (the

forced frequency in the string's axis direction). That assumption is justified by the fact

that the natural frequency of the longitudinal vibrations is usually much higher than the

frequency of the highly damped lateral vibrations. Thus, we are able to separate both

fundamental equations of motions and approximately solve the nonlinear problem of

lateral vibrations, using, for example, the Galerkin method. In this way, we are able to

determine the dynamic behaviour of the curved rod and its buckling load and analyze the

stability of the rod. Also, we are able to determine where along the rods should be

supported by couplings to prevent large bending deformations and contact with the tubing.

The major features of the model dynamic behaviour are summarized as follows:

1. Elastic slender beam theory is used to relate bending moment to the

curvature.

2. The string is assumed to deform in a plane.

16

3. Only the transverse damping force is included in the calculation of

transverse vibrations. The viscous and dry friction forces acting along the

string length are taken into account in the calculations of the equilibrium

of the axial forces in the string.

4. The beam is simply supported at each end with some restriction due to the

installations of the wheeled coupings/guides/scrapers. See the definition of

distance 1 in page VII for the definition of the supports at each end.

5. The tubing is assumed inextensible. If there is any strain in the tubing,

that amount of strain is taken into account in the calculations of the

effective plunger stroke.

6. Temperature is constant along the rod string.

7. The period of lateral vibrations of the rod string is equal to the pumping

period.

2.2 Development of equations

2.2.1 Expression of the rod axial force

When a well test is performed, a dynamometer installed at the polished rod records

the force at the top of the rod string and its position versus time. A diagram of the

polished rod forces for different rod positions due to the up-down movement can be

17

produced, this diagram, which is called "dynamometer card". It is introduced to describe

the rod string movement. The card is a continuous plot of polished rod load vs. polished

rod displacement, or it may be a continuous plot of polished rod vs. time (see Fig. 4). The

load can be presented by means of a Fourier series as a function of time as follows:

P((at) = P0 + E1 Pcos(n()t) + Tsin(nwt)

Where P0----static component of the load

P,T ---- coefficients of dynamic component of the load

co ----angular frequency of the string's up-down movement

n ----number of terms

(2.2.1.1)

The load (axial force) changes from the top to the bottom along the rod because

of contributions from other forces acting on the rod in this downward direction. The

relation for the load (axial force) versus time along the rod length can be presented in a

similar form. However, the values of the coefficients (PQ,P,T) depend on the position or

the coordinate of the sucker rod in the well. There are some programs and methods to

calculate these coefficients at anywhere along the rod, but they are not discussed here.

If the force is measured in a field, the load-displacement diagram is more

complicated because a variety of known and unknown forces act on the rod (see Fig. 4).

18

To simplify the present analysis, the diagram of the force versus time can be assumed to

be a constant for the half the period. The minimum force is assumed to be a constant for

the other half of the period (see Fig. 5). This ideal diagram of the axial load ignores fluid

acceleration and assumes instantaneous valve action, completely rigid rods, no time lag,

no dynamic vibration effects and 100% efficiency.

The mathematic expression for the force P(o t) can be expressed by

/

P IM (2.2.1.2)

Where for P and Pm positive sign means tension force, negative compression force. The

P.in at the bottom of the well may cause rod string buckling.

Presenting the force by means of the Fourier series gives:

(2.2.1.3)

19

4000

r- 3000

2000

1000

:I4 0

-1000

POLISHED ROE

24000

18000

12000 L

6000 -

C)

I I I I I I I I

4 2 0 -2 -4

PUMP DISPL10EME, ft

(a) Dynamometer Card

. . -IV, 1•j

2 4 6 8

TIME, S

(b) Polished Rod Load and Time Histories

Figure 4 Polished Rod Load

P(wt)

PMAX

P MIN

0

PMAX

P MIN

0 displacement

(a) Axial Force vs. Displacement (b) Axial Force vs. Time

Figure 5 Ideal Axial Force of the Rod

21

2.2.2 Mathematic model for the initial shape of the rod

As the string is mounted in the deviated well, each segment of the rod is curved

and its radius of the curvature depends on the position along the rod. Assuming that the

rod string is placed in the tubing, its initial curvature is the same as the curvature of the

tubing.

Fig. 6 defines the geometry of the rod; it yields

R2 = ( R2_()2 + W0)2 + (f_x)2 (2.2.2.1)

where 1 the distance between two supports

In which the initial deflection is obtained

wo = R2_(.)2_x2+lx -

at x=112, the maximum deformation is:

W, = R - R2 ()2 IX.

(2.2.2.2)

(2.2.2.3)

22

0 X

I I

wheeled coupling

Tubing

wheeled coupling

Rod

W(x,t)

normal deflection

Figure 6 Transverse Deflection of the Rod

23

y

P(wt

M*

P(wt)

P (wt) Ill-v

o L

1

initial deformation

(a) Rod Deflection in the Form of Sin(icx/1)

w

1

initial deformation

(b) Rod Deflectoion in the Form of Sin2(irx/l)

Figure 7 Model of the Rod String

x

P(wt

x

24

In order to introduce the initial deformation into this analysis, the following

expression is used to define the initial shape of the rod

W=(R_R2_(f)2 ) Sm 1x -

The derivation of Eq. (2.2.2.1) with respect to x yields

WI0 -

1 - -x 2

W0 +

(2.2.2.4)

(2.2.2.5)

In practical cases, the initial deformation W0, due to the curvature of the well, is

much smaller than the radius of well curvature(W0<<R). Therefore, Eq.(2.2.2.5) can be

simplified as

WI0 = 1 1

R2- ( l)2

The derivative of Eq. (2.2.2.5) with respect to x yields

(2.2.2.6)

W I' = -

R

2.2.3 Calculation of force on rod due to the fluid pressure

25

(2.2.2.7)

When the sucker rod vibrates transversely as well as longitudinally, there is also

a force acting on the rod which affects the rod's behaviour. This force is caused by the

pressure of fluid. We observe from Fig. 8 that the length of the arc, AB, on the surface

of the curved sucker rod is longer than the length of the arc CD. That difference cause

some transverse force (Fr) to affect the rod.

The force from the pressure acting on a finite area of the rod is shown in Fig. 8.

The resultant force normal to the rod can be calculated

FRdA = f ' p(R + r cos,&)d,& r cost d$

The Fp expression from above equation is

(2.2.3.1)

26

F = pr27c

R (2.2.3.2)

where Fp ---- fluid pressure force per unit length of rod

p ---- fluid pressure per square unit

r ---- radius of rod cross-section

R ---- radius of rod curve, i.e., R=-1/W' (where W" is second derivative of

deflection of rod)

2.2.4 Development of equations

To develop the governing equations, let us assume the deformation function in

the form W(x,t)=X(x) T(t). The function X(x) must satisfy the boundary conditions

which are determined by the type of the supports mounted, wheeled couplings, guides

and(or) scrapes. Where there are supports, sucker rod can almost move freely along the

length of the tubing, but suffers some rotational restriction in the x-y plane, which results

from the certain length of the support. As shown in Fig. 7, it is possible for the sucker

rod to deform in the two shapes of a sinusoid (see (A) in Fig. 7) and a sinusoid square

(see (B) in Fig. 7). Thus the deflection expression has the form

A Direction

Y

Figure 8 One Segment of the Rod

prdct(R+rcos(d ))d

28

1 W(x,t) = WO + T1(t) sin 7X — + T2(t)Sin2 WX 1 (2.2.5.1)

where: W0 is initial deformation

T1(t) and T2(t) are the amplitude of vibration which are function of time

1 is the distance between two supports

Derivative of Eq. (2.2.5.1) with respect to x gives:

t 1tX 'r W'(x,t) = W'0 + T1(t)--cos-1- + sin 'Lux

Derivative of Eq. (2.2.5.2) with respect to x gives

(2.2.5.2)

W"(x,t) = WI' - T(t)( 2 --) Sm 7X— + 2(.1.)2cos 2itx (2.2.5.3)

Derivative of Eq. (2.2.5.1) with respect to time t gives

7tX 21tX W(x,t) t1(t)sin— + £2(t)sin -

(2.2.5.4)

29

Derivative of Eq. (2.2.5.4) with respect to time t gives

7tX - = ti(t)sin_ + (t)sm (2.2.5.5)

The strain energy and the works done by all of the forces can be calculated as follows:

The strain energy

V = f(W"-W,,)dx 2°

Substituting expressions (2.2.2.7) and (2.2.5.3) into (2.2.5.6) yields

The work of the axial force

V = EIl(!t)4[Ti2(t) + 81T2 2(t)]

= P((O,t) f 11 WF (W ,2_W l 0 2 )dX 2 O

(2.2.5.6)

(2.2.5.7)

(2.2.5.8)

30

Substituting expressions (2.2.2.6), (2.2.1.3) and (2.2.5.2) into (2.2.5.8) yields:

WF P in

2

+ 9(!.)2T22(t) +

The work of the damping force

+ , 2(P - P) sin(nt)] fl7t

2T(tl + 2T2 tl

R2 - (1)2 jR2 - ( 1)2

Wf = fIdxf'ji#7 Wdt

Substituting expression (2.2.5.4) into (2.2.5.10) yields

W = Llft[3i2(t) + t1(t)i(t) + / 24

The work of the fluid pressure

w,=f'w- W0)dx

(2.2.5.9)

(2.2.5.10)

(2.2.5.11)

(2.2.5.12)

31

Substituting expressions (2.2.5.1), (2.2.4.8) and (2.2.3.2) into (2.2.5.12) gives

= - pr21i'v {( 752[T2(t) + 97 2(t)] + j[Ti(t) + 2

(2.2.5.13)

The kinetic energy of rod and fluid:

T - P,Af t#Ad i + 2 0 2 fO'#Ad

where p ---- mass density of fluid

Pr mass density of rod

A cross section area of rod

Substituting expression (2.2.5.4) into (2.2.5.14) yield

T - pAl 3j,12 +

(2.2.5.14)

(2.2.5.15)

32

where PPf+Pr

The dynamic virtual work principle requires

ft (V- 7) dt = ftowdt o to

where 8W = 8WF + 6Wf + SWP

= P(0) ,t)öu + FUf +

where P (o) ,t)----the generated axial force

Ff ----viscous damping force

FP ----fluid pressure force

6u, & i, Su ---- virtual displacement of the three forces respectively

Removing the right side of equation (2.2.5.16a) to the left side yields

V - T) - ÔW]dt = 0 to

(2.2.5.16a)

(2.2.5.16b)

With the application of usual integration and variational procedures to the dynamic

virtual work equation (2.2.5.16b), the following equation is obtained:

33

ft{EJ(7)4l[T(t) + !_ T2(t)] 3ir

1 + + 2(P - P) 11 sine(nwt)] 2 2 fig

.[(7t)2lT(t + )T2(t) + 41

ir4 2 R2 - (1)2

+ pr 2 t[ 21- + (—)2T1(t)l + 7(t)] Rit

- ill 16 t [2i'1(t) + 16 t + 2[2 + 3 3t g

+ ft{(iE.)41F!Ti(t) + 4T2(t)]

lr 1 lflaXfllfl 2(P - mm' . max

+ sine(nvit)] V 2 fig

[(5T1(t) + 1 - - + (i)2lT(t)]

\I R2_(L)2 2

+ pr2ig[_!_ + j(1)T1(t) + (2t_)21T2] 2R 3 1

- j [.i(t) + .!. i'1(t)] + -P[.I(t) + -!.4 1(t)]}8T2dt 0 37r

(2.2.5.16C)

In order to make equation (2.2.5. 16c) is tenable at any time, the both integrated

functions in the equation must be naught. Thus two differential equations for the dynamic

behaviour of sucker rods can be obtained as follows:

34

Li:

+ ._!_T2(t)]

1 max + 1min + 2(P - sinncot] 21: 2 nit

L2:

[(lC)21T(t) + +

2 it[- + 21 iv + pr (-j-)2T1(t)l + 1JT2(t)] Kit

- ..E[2j'1(t) + i•'2(t)] + 2t(t) + - 2()} = 0 3,t

+ 4T2(t)] 2 1 3iv

iFPmax +Pmin 2(P — Pmm' .i 2 2 + - sütJ nit

[8()T1(t) 1 it )21T • - - + + (-12(t)]

k.j R2_(f)2

+ pr2Jt[_L + !(.!.)T1(t) • (it)21T(t)] 2R 31

- 1 I—t(t) + 11 3 -.i'1(t)] + 2A[. i2(t) + J.l1(t)} = 0 3 3ic3-n

(2.2.5.17)

(2.2.5.18)

where Li, L2 is the left side expression of the two differential equations respectively

To solve these two equations, five terms of Fourier Series for the force are used

35

in Eq.(2.2.5.17) and (2.2.5.18), i.e.:

P(t) - smcAn + + 2(P - - 2(P - P)

2 It In

.sin3W + 2(P - P) + 2(P - Pmin) sinAat Sit 7it

(2.2.5.19)

The dimensionless parameters are used to describe the behaviour of the rod with

two dimensionless deflection functions, which are defined as follows

t1(t) = T(t) - A0 + A1siruM + A2cosot + A3sin&ot

+ A4cos3t + A5sjn5c,t + A6cos5c1t + A7sjn7c,t + A8cos7øt

(2.2.5.20)

t2(t) = T(t) - B0 + B1sinct + B2cosot + B3sin3cot

+ B4cos3t + B5sin5cot + B6cos5t + B7sin7ct + B8cos7øt

(2.2.5.21)

36

From the energy balance method, the following equations yield:

2it

10

2jt

JO 'L1 dt =

fL2 dt = 0

0 L1 sinnt dt = 0 n=1,3,5,7

0 L2 sinnt dt = 0 n=1,3,5,7

fo 'Ll cosnt dt = 0 n=1,3,5,7

0 L2 cosnt dt = 0 n=1,3,5,7

(2.2.5.22a)

(2.2.5.22b)

(2.2.5.22c)

(2.2.5.22d)

(2.2.5.22e)

(2.2.5.22t)

37

After the Eq. (2.2.5.20) and (2.2.5.21) are substituted into the equations form (2.2.5.22a)

to (2.2.5.220, a set of eighteen linear algebraic equations is obtained as follows:

(1 + a ± 20) A0 + 3A1 + -PA3 + --A5 + -A + -(1 + a + 20)B0 3 5 77 37r

+ + 8p B5 + JLB = - 4a 40 37v 9t 15%2lit 7C 3V82 - 0.25 78

(2.2.5.23a)

2A0 + (1 + a + 20 - 2?.)A1 + 2yA2 + --PB0 3it

+ --(1 + a + 20 -2?)B + J.2LB - 8f3 1 2

3ir 3 /o2 - 0.25

(2.2.5.23b)

-2?A1 + (1 + a + 20 - 2X)A2 - + ---(1 + a + 20 - 2X)B2 = 0 3ic

RPAO + (1 + a + 20 - 18).)A3 + 6yA + 4 9 0

+ a + 20 - 18? 8 )B3 + 16y — B = -

4 J 7 TC 37r 3V82 -0.25

(2.2.5.23c)

(2.2.5.23d)

38

-6yA3 + (1 + a + 20 - 18).)A4 - 1B + + a + 20 - 18).)B4 = 0 it 3ir

.?A0 + (1 + a + 20 50).)A + 10yA6 + 16DB l5ir °

kiL + a + 20 - 50A)B + !PIB - 8 3ir 3n 5,x 3V62 -0.25

(2.2.5.23e)

(2.2.5.23±)

+ a + 20 - 50) --B + i.(1 + a +20-5O?)B6 =0 -1OyA5 + (1 t.)A6 3it 3ir

?PAØ + (1 + a 20 - 98))A7 + 14yA + 16 P 21it °

+ a + 20 - 98X)B7 + 102 B8 = -

3it 77c 3V82 - 0.25

-14yA7 + (1 + a + 20 - 98?.)A8 - 102B + 3ic 3ir

(2.2.5.23g)

(2.2.5.23h)

1 + a + 20 - 98)B3 = 0

(2.2.5.23i)

39

---(1 + a + 20)A + !JA + A + 8 PA 3n 37c 9a 157E

+(4 +: + 20)B0 + PB I + B3 + P B 5 +

2lic a

7t 2V82 - 0.25

(2.2.5.23j)

.P-A0 .--- a + (1 + + 20 - 2X)A1 + 2 3it 3i 3it

+ B0 + (4 • a + 20 - 1.5?)B1 + 1.5yB2 = -

TC 2V82 - 0.25

(2.2.5.23k)

_1A1 + .i.(1 + a + 20 - 2).)A2 - 1.5yB1 + (4 + a + 20 - 1.5X)B2 = 0 3it 3it

!JQnr 13AO + ---(1 i- a + 20 - 18X)A3 +

JB0 + 4 + a + 20 - 13.5)B3 + 4.5yB4 = -

(2.2.5.23m)

213

3,t Io2 - 0.25

(2.2.5.23n)

40

_1A3 • + a + 20 - 18?)A4 - 4.5yB3 + (4 + a + 20 - 13.5?.)B4 0 7t 3t

(2.2.5.23o)

+ --(1 + a + 20 - 50A)A +

l5it ° 3ic 3it 6

+ !Bo + (4 + a + 20 - 37.5?)B5 + 7.5yB6 = - 2 5 5n 2V,52 - 0.25

(2.2.S.23p)

- ..P1A5 + -!(I + a + 20 - 50X)A6 - 7.5yB5 + (4 + a + 20 - 37.5X)B6 = 0 3,t '37t

(2.2.5.23q)

+ -!_(1 + a + 20 - 98?)A7 21ir 0 3t

+ 3ic B

+ + (4 + a + 20 - 73.5A)B7 + 10.5yB8 213

7 2 /8 2 - 0.25

(2.2.5.23r)

- 102 A7 + ---(1 + a + 20 - 98?.)A8 - 10.5yB7 + (4 + a + 20 - 73.5X)B8 0 In 3t

(2.2.5.23s)

41

where 0=(Pmax+Pmin)/2E1(lt/1)2

(Pmax+Pmin)/1tEI(7t1O2

e=pr¼fEI@/l)2

6=R/l

te(l/it)2IEI(ir/l)2

A=a 2 A(l/ir)2/2EI(it/l)

When certain values are given to the six dimensionless factors, the eighteen

cofefficients of two deflection functions can be solved. The results are shown in Section

2.4.

2.3 Static Analysis

When the term maxmjfl in force Fourier series is zero, the behaviour of the sucker

rod is static. The dimensionless deflection functions t1(t) and t2(t) become constants. In

this case, from the equations (2.2.5.17) and (2.2.5.18), we have

+ 8—t2) - 2 2

41 2 21 7t2 + J + pr 7c[— + C—) t11 + t2l = 0

R7t 1 (1)2

(2.3.1)

42

+ 4t2] - 1 Pm + 1'wjn 8 7c t, 1 3ir 2 2 [31 t1 •

+pr it[— -4- 2 1 8 7 --t + 2R 3 1 ' (7r)21t] = 0

these two equations yield

1

3,t a -

= 8(1 + a + 20) it 2 ,J 82 - (1)2 8712

2

0.0082( a 8

82 (1)2 2

- (4 + a + 20)

t -

2 3.28 + 0.28a + 0.560

3.28 + 0.28a + 0.560

0.0082( a

\j 82_(.)2 6

o - pr 2 7c

EI(1t)2

}

]

(2.3.2)

43

where

2.4 Dynamic Analysis

6

a - 'n,ax + nhIn

R

1

2E1( 1t)2

Fig. 9 shows the mid-deflection (the deflection at the mid of the span) results

obtained by the different assumptions for deflection functions. The rod curve changes

from ten to one hundred times in the span of the rod. The results obtained from two

term's deflection function show a good agreement with those using three terms. In most

cases, except for those close to resonance range, the differences are within 5 percent. In

practice, the pumping frequency is much smaller than the sucker rod natural frequency,

so the two terms' deflection function makes the result accurate enough for use in field

engineering. The curves shown in Fig. 9 lead to the following conclusion. The deflection

is in inverse proportion to the radius of the rod curvature.

In Fig. 10, the results are compared for each of three components in the two

assumed deflection functions, both with three terms. It shows that the deflection

differences between each pair of three components are not significant, and both the third

44

terms (t3sin2(3icx/l) and t3sin(37tx/l)) contribute much less than the first and second to the

deflections.

The deflection amplitude of the steady-state motion is generally the most important

result. The axial force-deflection relationship is plotted in Fig. 11, where both signs of

the force difference parameter ( 3) and the average force parameter (a) are negative, ie,

the axial force is a compressive force. If the deflection is given, the sum of a and

remains almost constant, ie, the level of contribution to the deflection from the static part

of the force is the same as that from the dynamic part of the force.

Fig. 12 shows that the forced frequencies do not affect the deflection. There are

four peaks. The peak on the right is the point at which the first resonance happens (forced

frequency is equal to the rod's natural frequency).

The influence of the fluid static pressure (0) on the rod mid-deflection is plotted

in Fig. 13. It is observed that the pressure in the low range has little effect on the

deflection as the damping force increases. When the damping factor is over a certain

value, the deflection changes very smoothly as the damping factor increases. In practice,

the deeper the well, the greater the pressure; the lower the pumping, the more pressure.

Thus. for pressure, the deeper and lower pumping well will make the sucker rod more

stable.

45

2.5 Buckling Behaviour of Sucker Rod String in the tubing

When the sucker rod string deforms enough to contact with the tubing, the

deflection is constrained by the inisde of the tubing, It makes the rod's lateral movement

of the string slow down. Thus, the deflection expression of the sucker rod string after

contacting the tubing, is considered and simulated as a static form, ie, a function of only

coordinate x as follows:

W(x) = a + bx + c cos(ax) + d sin(ax)

where a, b, c, d, a are the constant coefficients to be determined

(2.5.1)

Fig. 14 illustrates the plane buckling problem: a length of rod string, 1, is

constrained by the supporters and tubing at each end. Because of applied load, F, the

string has buckled in the x-y plane, resulting in reaction loads FR! at the couplings and

2FR1 where the string contacts the tubing interior wail.

In the rectangular coordinate system illustrated in Fig. 14, the force and moment

equation associated with a bent can be expressed by

-4-M = N dx

(2.5.2)

0.0080

0.0060 02

ITJ

0.0040

0

C) 4) -4 C44

4)

0.0020

0.0000

t sin(Tvx/L) +t2*s1n2(ivx/L) t1* sin( ivx/L) +t2*sin2(irx/L) +t3*sin2( 3rvx/L)

t1 sin(irx/L) +t2*sin2(ivx/L) +t3*sin ( 3ivx/L)

0.0 20.0 40.0 60.0 Curve radius over column span ( 6)

a=O.O fl=O.5 v=O.0005 i=6.8 X=O.014

Figure 9 Comparison of the Deflections due to the Different Assumptions

80.0 100.0

Mid—deflection over the rod span

0.008

€ sin( ivx/L) +t2*5in2 ( rvx/L) +tsin2 ( 3irx/L) t1 sin(icx/L) +t2*sin2 (icx/L) +t3*sin(37cX/L)

0.006

0.004

0,002

0.000 0.0 20.0 40.0 60.0 80.0

Curve radius over column span (o) oc=0.0 =0.5 v=0.0005 =6.8 ?=0.O14

Figure 10 Comparison of Three Components of the Deflections due to Different Deflection Function Assumptions

100.0

the rod span

0.2 0.4 Dimensionless force different

v=O.0005 =O.O X=O.014

Figure 11 Deflections against the Axial

0.6 parameter () o=100

Forces in the Rod

0.8 1.0

0.020

Cd 0.015

0.010

0 +3

4-4

C.) Q)

0.005

0.000 —

0.00

1.69

3.38

- 5.07

6.76

8.45

0.05 0.10

Dimensionless damping parameter(?)

cx=O 3=—O.4 ö=100 =O

Figure 12 Deflections against the Forced Frequency Parameters

0.0008 .1,

Mid—deflection over the rod span

0.0006

0.0004

0.0002

0.0000

-...- -.-.-----

-

0.0

0.4

0.8

1.2

1.6

---------------------------------------------

-------------------------------------------

0.0 0.5 1.0 1.5 Dimensionless damping parameter(v)

a=0.0 =-0.5 6=100'X=0.2

Figure 13 Deflections against the Fluid Pressure Parameters

2.0

51

where

M----moment of internal forces,

N----shear force

x----coordinate along the centre line of the two supporters

The motion equations of the buckled rod obtained from Eq. (2.5.2) are

d3W dW fpr271_dX - Fd2wElRdx 3 +F,--dx2 0 2

d3W dw

El 3 + F- - fpr27 ''d x + FR = 0

(2.5.3a)

(2.5.3b)

where the first term on the left side is the shear force in the selected cross section

the second term on the left side is the projection of axial force in the direction

of the selected cross section

the third term on the left side is the projection of fluid pressure in the

direction of the selected cross section

the forth term is the reaction force from the support

Y

A rod string

F2

o2>

FR1

2F1

\initial deformation

tubing

Figure 14 Geometry of the Rod after Contacting the Tubing

FR1

53

Substituting expression (2.5.1) into E. (2.5.3a) yields

Fb - FR + pr27rxd = 0

and

EIa2 - F + pr2lc = 0

(2.5.4)

(2.5.5)

Because of the symmetry of the problem, the solution to Eq. (2.5.3b) is the same

as that of Eq. (2.5.3a). We consider only the solution of Eq. (2.5.3a) for the remainder

of this section because solutions to Eq. (2.5.3b) can be easilly determined by the same

methods. For a cantilever tubing, Eq. (2.5.3b) must satisfy the following boundary

conditions.

and

which require

W(0) = 0

dW(0) = 0 dx

a + c =0

(2.5.6)

(2.5.7)

54

b + ad = 0

Further Eq. (2.5.1) must satisfy

(2.5.9)

(2.5.10)

(2.5.11)

Where rb is the gap between the rod (or guide/scrape if available) and the tubing in the

neutral position

When the Equations (2.5.8) through (2.5.11) are combined, they yield following

two equations

F = EIa2 + pr21t

2 rba FR = (F - Pr "I

sinu (u - sinu) - (1 - cosu)2

(2.5.12)

(2.5.13)

55

where u=a x

Fig. 15 shows the relation between the axial force (FR2) and reaction force (FR!)

with a fluid pressure effect. As the reaction force must be positive, the curves under FR!

less than zero are invalid. The relation between the axial force and the reaction force is

nearly linear. The larger the axial force, the larger the reaction force. In this Fig., the axial

force is compressive, the fluid pressure decreased the reaction force.

FRI*L/(EI(3.14/L)**2)/R_r

10.0

5.0

0.0

—5.0

0.0

•.•

I I I

e

0

0.157

0.314

1.0 2.0 3.0

FE2/EI(3.14/L)**2

Figure 15 Reaction Loads from the Tubing

4.0' 5.0

57

Chapter 3 Numerical Solutions and Techniques

3.1 Numerical Techniques

As we have seen in Chapter 2, the results of the displacement were obtained from

five terms in Fourier series for the axial force in the rod. Therefore, the obtained results

are only approximate. The axial forces measured in pratice or inferred from design theory

are often complicated and at least dozens of terms are needed to present the forces

exactly, by means of the Fourier series.

To be more accurate for the expressions of sucker rod axial force and deflection,

the numerical technique is used. The time-deflection function relationship of the rod can

be found by solving Eq.(2.2.5.19) and (2.2.5.20) with the Runge-Kutta scheme. The

fourth order integration scheme is employed. To apply this scheme, the equations are

modified into a set of first order simultaneous differential equations.

= T, (t)

= T2 (t)

Y3 = "I (t)

Derivative of Eq. (3.1.lc) and (3.1.ld) with respect to time t yields

13 = p1 (t)

= I(t)

Substituting expressions (3.1.la), (3.1.lb), (3.1.1e), and (3.1.lf) into Eq. (2.2.5.19) and

(2.2.5.20) yields

8 7 + 1 -Y2)- l'[(1)2ly +

3it

4! +

7Vk,J R2_(-)2

+ pr2ir[-! it J1(2Y3 + Ric + ( 7) Ll + .11y2 2 3ir

+--[2t3 + 14j=O 3ir

Elic 4 8 it 2 1 3ir ' 41'2)- P [8Y + 1 + (2!)2 —Y + lY2]

\JR2 _(f )

+ • IEy• )2,y ±13 Mi[.Y4 + J.y3] 2R 31

pAl3. 16 + —[—Y • = 0 4 2 3ic

(3.1.2)

(3.1.3)

59

Eq. (3.1.2) and (3.1.3) yield

• - 50.9 { EI()2 1 (0.0195Y1 - 1.84Y2)

pAl

+ [0.039( it )2lY1 - 0.283Y +

2

0.141

R2 ( 1)2

I

+ pr27r [O.O71 + 0.039()2lY1 - 0.283Y] + 0.03951aY3 }

Y4 - { - 1.93EI(L)4 1 Y pAl

+ .![_0.329(.)2lY2 + 0.095! 2-

R2 - (1)2

2 0.0476! + pr R - 0.33(.)2lY2] + 0.0345ilY4 }

(3.1.4)

(3.1.5)

The two equations must satisfy the set of the initial conditions as follows

Y1(t=t0) = Y 10 (3.1.6a)

Y2(t=t0) = (3.1.6b)

Y(t=t0) =

Y4(t=t0) =

(t=t0) =

14(t=t0) =

60

(3.l.6c)

(3.l.6d)

(3.1.6e)

(3.1.60

Using the initial conditions presented above and integrating from the initial time

point in a small integration step (0.2 second), numerical values of time histories of

deflection and velocity of the rod are obtained over a time span. A Fortran program is

developed to execute this integration.

Both free and forced vibrations of the string were analyzed. In free vibrations

(where the axial force vanishes), the vibration frequency varies with its oscillation

amplitude. Thus, it is also valid to state that the free vibration frequency is dependent

on both its initial conditions and its geometric configuration. The resonance occurs when

the forcing frequency is close to that of free oscillation with the same initial condition.

3.2 Comparison

Fig. 16 shows the comparison of deflections from 4th order of the Runge-Kutta

61

method and the Galerkin method. They give similar, although not identical, results. The

difference between the two results from both methods is about 5 percent. This is due to

different mathematical algorithms and some assumptions about initial conditions.

0.0040 Cd rI

ITJ 0

0.0030

zi 0.0020 C.) a)

0.2 0.4 0.6 0.8 Dimensionless force different parameter(j)

cx=O v=O.0005 ô=100 i5'=O X=O.015

Figure 16 Comparison of the Solutions from Galerkin and R-K

63

Chapter 4 Application

In Chapter 2, a slender beam simply supported at each end was introduced to

represent any segment of the sucker rod. A two term function of time and coordinate x

was used to describe the rod's deflection. If a series of slender beams are connected to

each other, they can be used for the entire sucker rod, even though different sucker rod

materials and different sizes are used.

The friction force acting on the sucker rod has three important components, they

are: ( 1) the viscous force of the fluid acting on the surface along the rod's axis; (2) the

viscous force of the fluid due to the rod couplings/guides/scrapes; (3) the friction between

the tubing and the rod. A complete sucker rod model cannot ignore these effects.

However, these forces are highly dependent on the deviation of a well as well as many

other factors that are difficult to determine. The formulas used for calculating the above

three forces are presented in other papers (Ref. [14]). In the two cases that follow for

which the calculations were performed, the variation of the rod axial force due to the

contribution of these three friction forces was assumed to decrease by three Newtons per

meter (3 N/rn) from top to bottom along the rod.

The data shown in Table I were collected from a actual pumping system (Ref.

[10]). These necessary data were used in the calculations of all the dimensionless.

64

parameters in the governing equations. Fig. 17 presents the geometry of a simulated rod

string which was used for the two cases that are discussed below. The case 1 is without

a sinker bar, the case 2 includes a sinker bar. The computer program (3) presented in

Appendix was used to compute and determine the distances between each two supports

along the rod. The program is based on the Galerkin method. The distances were

calculated based on the assumption that the maximum mid-deflection of the rod between

each two supports was equal to the allowed deflection (the gap between the rod [or

guides/scrapers if availabe] and the tubing in the neutral position). The effect of the rod

weight on the deflection was included.

Table 2 shows the calculated distances between the supports for the Case 1 and

Case 2. Comparing the distances in Case 1 and Case 2, the effect of sinker bar is visual.

Table 3 shows that the well with the sinker bar, can save 42 guides, assuming the wax

problem is not serious enough to need be taken into account. At the same time, because

of the sinker bar, the system needs more power to move the string up and down, ie, more

energy consumption. The use of sinker bar will cause higher cost to pump the oil from

subsurface to surface.

65

x

45 0

I

/

Y I

Figure 17 Geometry of the Rod in Case 1 and Case 2

66

Table 1 Input data

Case 1 Case 2

Peak polish rod load 33500 (N) 34500 (N)

Minimum polish rod load 12300 (N) 13300 (N)

Rod cross-section diameter 0.022 (m) 0.022 (m)

Mass density of rod 8490 (kg/rn3) 8490 (kg/rn3)

Mass density of fluid 814 (kg/n3) 814 (kg/rn3)

Damping factor 0.1 NS/m3 0.1 NS/m3

Strokes per minuter 4.6 4.6

Young's modulus of elasticity 2.OxlO" (Pa) 2.Oxlo" (Pa)

Tubing inside diameter 0.061 (m) 0.061 (m)

67

Table 2 The distance between two supports

along the rod (unit: m)

The distance along the rod from top to bottom

Case 1 Case 2

0.0 16.50 16.50

50.0 16.50 16.48

100.0 16.48 16.42

150.0 16.44 16.36

200.0 16.36 16.24

250.0 16.22 14.24

300.0 13.60 10.46

350.0 9.38 6.68

400.0 6.16 4.27

450.0 4.02 3.10

500.0 3.08 2.58

550.0 2.54 2.26

600.0 2.24 2.02

650.0 2.02 1.86

700.0 1.86 1.72

750.0 1.74 1.64

68

The distance along the rod from top to bottom

Case 1 Case 2

800.0 1.64 1.54

850.0 1.56 1.48

900.0 1.48 1.42

950.0 1.44 1.38

1000.0 1.38 1.34

1050.0 1.34 1.30

1100.0 1.30 1.26

1150.0 1.28 1.24

1200.0 1.26 1.22

1250.0 1.24 1.20

1300.0 1.22 1.18

1350.0 1.20 1.16

1400.0 1.20 1.16

1450.0 1.18 1.16

1500.0 1.18 1.14

1550.0 1.18 1.14

1600.0 1.18 1.14

69

Table 3 The distribution of supports

Number of supports per rod string (25 " )*

Case Case 2**

0 1-40 1-47

1 41-58 48-60

2 59-66 61-73

3 67-79 74-85

4 80-105 86-112

5 106— 138 113 - 151

6 139 - 206 152 - 206

total number of coupling/guides

750 708

*The numbers of supports per rod string are calculated based on the assumption that there

are supports at every couplings which connect two strings

**The numbers in columns of Case 1 and Case2 are the numbers of rod strings (25" in

length for each) in order from top to bottom

70

Chapter 5 Concluding Remarks and Summary

5.1 Remarks and Limitations

This thesis presents a non-linear analysis of dynamic behaviour and stability

behaviour of the sucker rod in the deviated well. It also discusses post buckling behaviour

of the rod taking into account interaction of forces from the tubing. The focus was

concentrated on the effects of the curvature of the rod, the fluid pressure, damping force,

dynamic inertia and rod weight. By computing the lateral deflection, the distribution of

the supporters was obtained, assuming that the rod string just contacts the tubing. The

effects of the dry frictions between rod and coupling, between the coupling and tubing

on the buckling behaviour were neglected. Also the effect of viscous friction in the

tangential direction on the deflections of the rod string was not considered. If necessary,

any of the above effects could be taken into account in the calculations of rod deflections.

No analytical form of expression can be obtained to define the relations between

the parameters that describe the deformed configuration and the deflection of the rod in

the dynamic analysis. A simple expression was established only to describe the

relationship between the static deflection and the parameters for the static behaviour of

the rod string.

71

It was found that the effect of curvature of the rod on the deflection is significant,

whenever the rod was in static and dynamic states. The ratio of the maximum deflection

to the rod curvature was almost constant if the other parameters stayed the same.

The two governing differential equations (Eq. 2.2.5.17 and Eq. 2.2.5.18) are

relatively simple in appearance. The assumed deflection function, contained only two

terms, and therefore it can not completely represent the actual deflection. When using

analytical method, only seven terms of Fourier series were used to present the axial load

along the string in this thesis. In most cases, many more terms are needed to describe the

force versus time relation. That leads to many equations and solutions with accompanying

convergence problems. This task can be carried out using numerical techniques. The exact

initial conditions are not known, however they can be achieved by taking the data from

a real pumping system. The different initial conditions lead to different solutions. It is

possible to specify these conditions using the coefficents of the deflection function

directly from Galerkin approach and transfering them into the initial conditions for the

differential equations in numerical approach. The two deflection solutions from two

approaches match very well in this means.

It is proved that while using Galerkin method, the two terms of the series for the

deflection function make the solution accurate enough for design and evaluation purposes.

Although the results obtained from the numerical approach are more accurate that from

the Galerkin approach, the numerical approach takes much more time for calculations.

72

In order to analyze the dynamic behaviour and stability of the sucker rod, two

programs written in Fortran 77 were developed, as shown in the Appendix. The first one

gives the solution using Fourier Series approach. The second one uses Runge-Kutta

method. The program for calculating the distances between two supporters from the top

to the bottom of the string is also shown in the Appendix: It is developed only for the

geometry of tubing that is an arc in shape, but it is not difficult to make further

developments to apply to any curve shape.

5.2 Summary

The work presented in this thesis can be considered as an initial part of a study on

the transverse vibration behaviour and stability behaviour of sucker rod in inclined well.

It shows how to determinate the distance between two supports along the rod and the

effect of the sinker bar.

It is the author's hope that this work will pave ground for further study of sucker

rod behavior where the effects of stretch of the rod and the tubing as well as the wax

problem are considered. It is anticipated that a more accurate Fourier series for the force

can present more realistic representation of the axial load along the string.

73

References

1. Arthur Lubinski, "Developments in Petroleum Engineering", Gulf Publishing

Company, 1987

2. Frick, T.C., "Petroleum Production Handbook", McGraw-Hill Book Company,

1962

3. Dym, C.L., "Stability Theory and its Applications to Structural Mechanics",

Noordholf International Publishing, 1974

4. Ziegler, Hans, "Principles of Structural Stability", Blaisdell Publishing Company,

1966

5. Meirovitch, L., "Analytical Method in Vibrations", MacMillan Company, 1967

6. "Recommended Practice for Design Calculations for Sucker Rod Pumping Systems

(Conventional Units)", API RPIIL, American Petroleum Institute, 1988

7. Press, W.H., "Numerical Recipes", Cambridge University Press, 1988

8. Gibbs, S.G., "A Review of Methods for Design and Analysis of Rod Pumping

74

Installations", Society of Petroteum Engineers, pp. 2931-2940 trans., December

1982

9. Lukasiewicz, S.A., "Dynamic Behaviour of the Sucker Rod String in the Inclined

Well"(5PE21665), presented at the Production Operations Symposium held in

Oklahoma City, Oklahoma, April 7-9; 1991

10. Lukasiewicz, S.A., "Computer Model Evaluation Oil Pumping Units in Inclined

Wells", Journal of Canadian Petroleum Technology, pp. 76-79, November-

December 1990

11. Doty, D.R., "An Improved Method for Sucker Rod Pumping", Society of

Petroleum Engineers, pp. 33-41, February 1983

12. Tripp, H.A., "Mechanical Performance of Fibreglass Sucker-Rod String", Society

of Petroleum Engineers, pp. 346-350 trans., August 1988

13. Pickford, K.H., "Hydraulic Rod-Pumpling Units in Offshore Artificial-Lift

Applications", Society of Petroleum Engineers, pp. 131-134 trans., May 1989

14. Mien, L.F.,"Rod Pumping Optimization Prorgam Reduces Equipment Failures and

Operating Costs" (5PE13247), presented at the 59th Annual Technical Conference

75

and Exhibition held in Houston, Texas, September 16-19, 1984

15. Gibbs, S.G., "Predicting the Behavior of Sucker-rod Pumping System", Journal of

Petroleum Technology, pp. 769-778 trans., July 1963

16. Schafer, D.J., "An Investigation of Analytical and Numerical Sucker Rod Pumping

Mathematical Models", presented at the 62nd Annual Technical Conference and

Exhibition of the Society of Petroleum Engineers held in Dallas, TX, September

27-30, 1987

17. Wilkins, D.S., "Welisite Determination of Bottomhole Condition for Rod Pumping

Wells" (SPE 14502), Presented at the Society of Petroleum Engineers 1985 Eastern

Regional Meeting held in Morgantown, West Virginia, November 6-8, 1985

18. Gibbs, S.G., "Computer Diagnosis of Down-Hole Conditions in Sucker Rod

Pumping Wells", Journal of Petroleum Technology, pp. 93-96, January 1966

19. Bellow, D.G., "Bending Stresses in Otherwise Straight Sucker Rods", Journal of

Canadian Petroleum Technology, pp. 53-57, September-October 1988

20. Vierck, R.K., "Vibration Analysis", Harper & Row Publishers, 1979

76

21. Meirovitch, L., "Analytical Methods in Vibrations", MacMillan Company, 1967

22. Bishop, R.E.D., and Johnson, D.C., "Mechanics of Vibration", Cambridge

University Press, 1960

23. Harris, C.M. and Crede C.E., "Shock and Vibration Handbook", McGraw-Hill

Book Company, 1976

24. Myldestad, N.O., "Fundamentals of Vibration Analysis", McGraw-Hill Book

Company, 1956

25. Chen, Waifan, "Theory of Beam-Columns", McGraw-Hill Book Company, 1977

77

Appendix

Computer Program

1. Calculation the deflections of the rod with Galerkin approach

2. Calculating the deflections of the rod with Runge-Kutta approach

3. Determination of the distances between two supporters along the rod with Galerkin

approach

78 C THIS IS THE PROGRAM FOR COMPUTING THE DEFLECTIONS BY GALERKIN APP.

PARAMETER ( N=].8, NP=18,MP=1,M=1P=3.1415) DIMENSION A(NP,NP),C(NP,NP),B(NP,MP),E(NP,Mp) REAL Dl, D2, D3, D4, D5, D6, WT, Y, Z, Q,EF REAL D1S,D2S,D3S,D4S,D5S,D6S REAL D1E,D2E,D3E,D4E,D5E,D6E INTEGER I, J, F

5 PRINT * ,'ENTER INITIAL,STEP AND END VALUES OF ALPHA' READ *, D1,D1S,D1E

PRINT * ,'ENTER INTIAL,STEP AND END VALUES OF BETA' READ * D2D2sD2E PRINT * ,'ENTER INITIAL,STEP AND END VALUES OF GRANMA' READ *, D3,D3s,D3E

PRINT *, ENTER INITIAL,STEP AND END VALUES OF DELTA' READ *, D4,D4S,D4E

PRINT *, ENTER INITIAL,STEP AND END VALUES OF THETA'

RE[AAD *, D5,D55,DSE PRINT *, ER INITIAL,STEP AND END VALUES OF LAMBDA' READ *, D6,D6S,DGE

PRINT *, 'ENTER THE ALLOWED MAXIMUM DEFLECTION BY L' READ , XMAX

ND1= ( D1E-Di) /D1S+l ND2= (D2E-D2) /D2S+l ND3=(D3E-D3) /D3S+1B ND4= ( D4E-D4) /D4S+1

ND5= (D5E-p) 1D55+l ND6= ( D6E-D6) /D6S+1 DO 106 Nl=1,ND1 DO 105 N2=1,ND2 DC 104 N3=1,ND3 DC 103 N4=1,ND4 DO 102 N5=1,ND5 DO 101 N6=1,NDG

7 PRINT * 'ALPHA= ', Dl,'BETA= ', D2,'GRAMMA= ', D3 PRINT ', 'DELTA= ', D4,'TEETA= ', D5,'LANBTA= ', D6 DO 96 F=1,10 DO 10 I=1,N DO 10 J=1,N

A(I,J)=0.0 C(I,J)=0.0

B(J,NP)=O.0 E(J,MP)=0.O

10 CONTINUE C THE ASSUMPTION IS ( SIN(PAI*X/L))**2 & ( SIN(3*PAI*X/L))**2

A(1,1)=1+D1+2*D5 A(1,2)2*D2/P A(l,4)=2*D2/3/P

A(1,6)=2*D2/5/P A(1,8)=2*D2/7/P A(2,.1)=4*D2/P

A(2, ) =1+D1+2*DS_1. 5*DG A(2,3)=1.5*D3

A(2,11)=-D6

A(2,12)=D3 A(3 , 2) =-1 5*D3 A(3 , 3)=1+Dl+2*D5_1 . 5*D6

A(3, 11)=-D3 A(3,12)=-D6 A(4,1)=4*D2/3/P

A(4, 4)=1iD1+2*D5_13 . 5*DG A(4,5)=4.5*D3 A(4,13)=_9*D6 A(4,14)=3*D3 A(5,4)=_4.5*D3 A(5, 5)=1iDl+2*D5_13 5*D6 A(5,13)=_3*D3

79 A(5,14)=_9*D6 A(6.1)=4*D2/5/p A(6, 6)=1+D1+2*D537 5*D6 A(6,7)=7.5*D3 A(6, 15)=_25*D6 A(6, 16)=5*D3 A(7,6)=_7.5*D3 A(7, 7)=1+D1+2*D5_37 5*D6 A(7, 15)=.5*D3 A(7, 16) =_25*D6 A(8,1)=4*D2/7/p A(88)=1+D1+2*D5_73.5*D6 A(89)=1O.5*D3 A(817)=_49*DG A(818)=7*D3 A(9,8)=_1O.5*D3 A(9, 9)=1+D1+2*D5.73 5*DG

A(917)=_7*D3 A(918)=_49*D6

A(1O, iO)=81+9*D1+18*D5 A(1O,11)=18*D2/p A(1O,13)=6*D2/p A(1O,15)=18*D2/5/p A(lO, 17)=18*D2/7/p

A( 13. 2) =-D6 A(11,3)=D3 A(11,3.0)=36*D2/p A(11, 11)=81i9*D1_1.5*D6+i8*D5 AC ii, 12) =3.. 5 *D3 A( 12 2) =-D3 A(12,3)=-D6 A(3.2..11)=_i.5*D3 A(12, 12)=81+9*Di+18*D5_1 . 5*D6 AC 13 4) *D6 A(13, 5)3*D3 A(13,1O)=12*D2/p A(13, 13)=81+9*Di_13 A(13,14)=4.5*D3 Acid , 4)..3*D3 A(14,5)=_9*D6 A(1413)=_4.5*D3 A(id, 14)=81+9*D1+18*D5_13 . 5*D6 AC 15 6) =_ 25 *DG A(15,7)=5*D3 A(15,1O)=+36*D2/5/p A(15, 15)=81+9*D1..37.5*D6+18*D5 AC 15 , 16) =7 . 5*D3 A(16, 6)_5*D3 A(16,7)=_25*D6 A(16,15)=_7.5*D3 A(16,16)=8i+9*D1+18*D_37.5*D6 AC 17 , 8) = 49 *D6 A(17, 9)=7*D3

A(171O)=36*D2/7/p A(17, 17)=81+9*D1_73 5*D6+18*D5 A(17,18)=1O.5*D3 AC 18 , 8) =.. 7 * D3 A(189)=_49*D6

A(18,17)=_1O.5*D3 A(18, 18)=81+9*D1+18*D5.73 . 5*D6 B(l,MP)=_Di/((p**2)*sQRp(fl4*D4_O.25))_D5/(p**2)/D4 B(1O,MP)=B(1,Mp)

B(2,MP)=_4*D2/((p**3)*sQRr(D4*D4_o.25)) B(11,MP)=B(2,Mp) B(4,MP)=B(2,Mp)/3 B(13,MP)=B(4,Mp)

80

B(6,MP)=8(2,MP)/5 B(15.MP)=B(6,MP) B(8,MP)=B(2,MP)/7 B(].7,MP)=B(8,MP)

C THE ASSUMPTION IS SIN(PAI*X/L) & (SIN(3*PAI*X/L))**2 C(1, ].)= 1+4*D1+8*D5 C(1,2)=8*D2/P C(1.4)=8*D2/3/P C(1.6)=8*D2/5/P C(18)=8*D2/7/P C(1,1O)=(8/3/P)+(32*D1/3/P)+(8*8*D5/3/p) C(1,11)=64*D2/3/(P**2)

C(1,13)=64*D2/9/(P**2) C(1, 15)=64*D2/15/(P**2) C(1.17)=64*D2/21/(P**2) C(2,1)=16*D2/P C(2 , 2)=].+4*D1+8*D5.8*D6 C(2,3)=8*D3 C(2.1O)=16*8*D2/3/(P**2)

C(2,11)=(8/3/P)+(4*8*D1/3/P)+(8*8*D513/p) + _( 4*16*D6/3/P)

C(2,12)=4*16*D3/3/P C(3,2)=_8*D3

C(3 3)=1+4*D1+8*D5_8*D6 C(3 111)=_4*16*D3/3/P

C(3,12)=(8/.3/P)+(4*8*D1/3/p)+(8*8*D5/3/p) + _( 4*16*D6/3/p)

C(4,1)=16*D2/3/P C(4, 4)=1+4*D1+8*D5_72*D6 C(4,5)=24*D3 C(4,1O)=16*8*D2/3/3/(P**2) C(4,13)=(8/3/P)+(4*8*D1/3/P)+(8*8*D5/3/P)

+ _( 4*16*9*D6/3/P)

C(4,14)=4*16*3*D3/3/P C(54)=_8*3*D3

C(5, 5)=1+d*D1+8*DS_72*D6 C(5, 13)=_4*16*3*D3/3/P C(5, 14)=(8/3/P)+(4*8*D1/3/P)+(8*8*D5/3/P)

+ _( 4*16*9*DS/3/p) C(6,1)=16*1D2/5/P

C(6, 6)=1+4*D1+8*D5_8*25*D6 C(6,7)=8*5*D3

C(6, 1O)=16*8*D2/5/3/(P**2) C(6, 1S)=(8/3/P)+(4*8*D1/3/P)+(8*8*05/3/p)

+ _( 4*16*25*DS/3/p)

C(6, 16)=4*16*5*D3/3/P C(7,6)=_8*5*D3

C(7, 7)=1+4*D1+8*D5_8*25*D6 C(7, 15)=_4*16*5*D3/3/P C(7,16)=(8/3/P)+(4*8*Di./3/P)+(8*8*D5/3/p)

+ _( 4*].6*25*D6/3/P) C(8,1)=16*D2/7/P C(8, 8)=1+4*D1+8*D5_8*49*D6 C(8,9)=8*7*D3 C(8, 1O)=16*8*D2/7/3/(P**2) C(8,17)=(8/3/P)+(4*8*D1/3/P)+(8*8*D5/3/p)

+ _( 4*16*49*D6/3/p)

C(8, 18)=4*16*7*D3/3/P C ( 9 8) =- 8 * 7 * D3 C(9, 9)=1+4*D1+8*D5_8*49*DG C(9, 17)=_4*16*7*D3/3/p C(9.,18)=(8/3/P)+(4*8*D1/3/P)+(8*8*D5/3/p)

+ ...( 4*16*49*DG/3/p)

C(1O, 1)=(8/3/P)+(4*8*D1/31P)+(8*8*D5/3/p) C(1O,2)=8*8*D2/3/(P**2) C(1O.4)=8*8*D2/3/3/(P**2)

81 C(1O, 6)=8*8*D2/5/3/(P**2) C(1O8)=8*8*D2/7/3/(P**2) C(1O, 1O)=2**2+4*D1+8*D5 C(1O11)=8*D2/P C(1O, 13)=8*D2/3/P C(1O,15)=8*D2/5/P C(1O, 17)=8*D2/7/P C(11, 1)=8*8*D2/3/(P**2) C(112)=(8/3/P)+(4*8*D1/3/P)+(8*8*D5/3/P)

+ _( 4*16*D6/3/p)

C(11,3)=4*16*D3/3/P C(11,1O)=8*D2/P C(11, 11)=2**2+4*D1+8*D5..6*D6

C ( 11, 12) = 6 * D3 C(12, 2)=_4*16*D3/3/P C(12, 3)=(8/3/P)i(4*8*D1/3/P)+(8*8*D5/3/P)

+ ...( 4*16*D6/3/p) C(12, 11)=.6*D3 C(12, 12)=2**2+4*D1+8*D5_6*D6

C(13, 1)=8*8*D2/3/3/(P**2) C(i.3, 4)=(8/3/P)+(4*8*D1/3/P)+(8*8*D5/3/P)

+ _( 4*5,6*9*DG/31p) C(13,5)=4*16*3*D3/3/P

C(13, 1O)=8*D2/3/P C(13, 13)=2**2+4*D1+8*D5_6*9*D6 C(1314)=6*3*D3 C(14.4)=_4*16*3*D3/3/P

C(14,5)=(8/3/P)+(4*8*D1/3/P)+(8*8*D5/3/P) + _( 4*16*9*D6/3/p)

C(14,13)=_6*3*D3

C(14, 14)=2**2+4*D1+8*D5_6*9*D6 C(15. 1)=8*8*D2/5/3/(P**2) C(15, 6)=(8/3/P)+(4*8*D1/3/P)+(8*8*D5/3/P)

+ _( 4*16*25*D6/3/p) C(15,7)=d*16*5*D3/3/P

C(15, 1O)=8*D2/5/P C(15 15)=2**2+4*D1+8*D5_6*25*D6 C ( 15, 16) =6 C(16,6)=_4*16*5*D3/3/P C(16,7)=(8/3/P)+(4*8*D1/3/P)+(8*8*D5/3/P)

+ _( 4*16*25*D6/3/p) C(16,15)=_6*5*D3

C(i.6, 16)=2**2+4*D1+8*DS_6*25*D6 C(17, 1)=8*8*D2/7/3/(P**2)

C(17, 8)=(8/3/P)+(4*8D1/3/P)+(8*8*D5/3/P) + _( 4*16*49*D6/3/p)

C(17, 9)4*16*7*D3/3/p C(17,1O)=8*D2/7/P C(17,17)=2**2+4*D1+8*D5_6*49*D6 C(17,18)=6*7*D3 C(18, 8)=_4*16*7*D3/3/P

C(18, 9) =( 8/3/P) + (4*8*D1/3/P) +( 8*8*D5/3/P) + _( 4*16*49*DG/3/p)

C(18.17)=_6*7*D3 C(18, 18)=2**2+4*D1+8*D5_6*49*D6 E(1,NP)=_16*D5/D4/(P**3)_16*D1/((P**3)*SQRT(D4**2_O.25)) E(2.NP)=_64*D2/((P**4)*SQRT(D4**2_O.25))

E(4,MP)=E(2,MP)/3 E(6,NP)=E(2,MP)/5 E(8,NP)=E(2,MP)/7 E(1O,MP)=(_4*D1/SQRT(D4**2_O.25)/(P**2))_4*D5/D4/(P**2) E(11,MP)=_16*D2/SQRT(D4**2_O.25)/(P**3)

E(13,MP)=E(11,MP)/3 E(15,MP)=E(11,NP)/5 E(17,MP)=E(11,MP)/7 CALL GAUSSJ(A,N,NP,B,M,MP)

82

CALL GAUSSJ(C,N,NP,E,M,MP)

CALL AVER(LZ,N,MP) CALL AVER(E,Q,N,MP)

C IF (ABS(Q) . GT. XMAX) GOTO 97 40 DO 95 I=1,N

PRINT , I, B(I,MP),' ', E(I,MP) 95 CONTINUE

PRINT *,' (SIN(PAI*X/L))**2 SIN(PAI*X/L)

PRINT * 'DYNAMIC DIFECTION BY L=' PRINT , ' ', Z, ' PRINT

C PAUSE D1E=D1 D2E=D2 D3E=D3 D4E=D4 D5E=D5 D6E=D6 EF=1+(16*0.0000000119*D1**2/(0.5*E(1,MP)**2

+ +0.849*E(1,MP)*E(10,MP)+2.*E(10,MP)**2)) PRINT * EF=EF D1=D1*EF

D2=D2 *EF D3=D3*EF

D4=D4 *EF D5=D5*EF D6=D6*EF

96 CONTINUE D1=D1E D2=D2E D3=D3E D4=D4E D5=DSE D6=D6E

97 D6=D6+D6S 101 CONTINUE

D6=D6_D6S*ND6 D5=D5+D5S

102 CONTINUE D5=D5_D5S*ND5

D4=D4+D4S 103 CONTINUE

D4=D4_D4S*ND4


D3=D3_D3S*ND3


D2=D2_D2S*ND2

D1=D1+D1S 106 CONTINUE 20 PRINT *, NEXT COMPUTING?(1:YES 2:NO)'

READ , F

IF ( F . EQ. 1) GOTO 5 STOP END

SUBROUTINE GAUSSJ(A,N,NP,B,M,NP) PARANETER(NNAX=50) DIMENSION A(NP,NP) , B(NP,MP)

DIMENSION INDXC(NMAX), INDXR(NMAX), IPIV(NMAX) DO 1011 J=1.N IPIV(J)=0

1011 CONTINUE

DO 1022 I=1,N BIG=0 . 0

83

DO 1013 J=1,N IF(IPIV(J) .NE. 1) THEN

DO 1012 K=1,N IF(IPIV(K) . EQ. 0) THEN IF(ABS(A(J.K)) . GE. BIG) THEN

BIGABS(A(J,K)) IROWJ ICOLK ENDIF ELSE IF(IPIV(K) . GT. 1) THEN

PAUSE SINGULAR MATRIX' ENDIF

1012 CONTINUE ENDIF

1013 CONTINUE IPIV(ICOL) =IPIV(ICOL)+1 IF(IROW . NE. ICOL) THEN DO 1014 L=1.N DUNA(IROW.L) A(IROW,L)A(ICOL,L.) A(ICOL,L)DUM

1014 CONTINUE DO 1015 L=1,N DUMB(IROW,L) B(IROW,L)B(ICOL,L) B(ICOL,L)1?M

1015 CONTINUE ENDIF INDXR(I)IROW INDXC(I)ICOLi IF(A(ICOL,ICOL) . EQ. 0 ) PAUSE ' SINGULAR MATRIX.' PIVINV=1. /A(ICOL, XCOI,)

A(ICOL,ICOL)1. DC 1016 L1,N A(ICOL,L)A(ICOL,L) *pIVIJ

1016 CONTINUE DO 1017 I.,=1,M B(ICOL,L)B(ICOL,L)*PIVINV

1017 CONTINUE DO 1021 L.L=1N IF(LL .NE. ICOL) THEN

DUN=A(LL, ICOL)

A(LL, ICOL)0. DO 1018 L1,N A(LL,L)A(LL,L)_A(IC0L sL)*D

1018 CONTINUE DO 1019 L=1,M B(LL,L)B(LL,L)B(ICOL,14*D

1019 CONTINUE ENDIF

1021 CONTINUE 1022 CONTINUE

DO 1024 L=N,1,-1 IF(INDXR(L) . NE. INDXC(L)) THEN

DO 1023 K=1,N DUN=A(K,INDXR(L')) A(K,INDXR(L))A(K,INDXC(L))

A(K,INDXC(L))DUN 1023 CONTINUE

ENDIF

1024 CONTINUE RETURN END

SUBROUTINE AVER(W, Z,N,MP)

84

INTEGER I, J, N REAL Y, WT, Z DIMENSION W(N,MP) WR=0.48 YIi=0. Y12=0. Y13=0. Y14=0. YI5=0. Y16=0. DO 91 1=1,4

YI1=YI1+W(2*I_1,MP) Y12=Y12+W(2*I+8 , MP) Y13 =Y13 + (2*I_1)*WR*W(2*I,MP) Y14=Y14+(2*I_i) * Jj*(2*I+9Mp)

Y15=YI5-( ( 2*1_i) * fl)**2*W(2*I+1Mp) Y16=Y16-( ( 2*1_i) *WR) ** 2*W(2*I+1OMP)

91 CONTINUE YI1=YIi+W(9,MP) Y12=Y12+W(18,MF) PRINT * ,'INITIAL CONDITIONS FOR RK APPROICH' PRINT *, yI1,yI2,YI3,YI4 PRINT *, yI5,yI6

C PAUSE Z=0.0 DO 90 J=i,181 '1=0.0 WT=3.14*(J_1.0)1180 DO 100 1=1,4 Y=Y+(W(2*I,MP)+W(2*X+9,N?) ) *SXN((2*I_1)*WT) Y=Y+ (W(2*I+1,MP)+W(2*I+iO,MP))*COS((2*I_i)*WT)

100 CONTINUE C PRINT *, J_1,y+W(1,MP)+W(10,MP)

Z=Z+Y+W(1,MP)+W(10,MP) 90 CONTINUE

Z=Z/181 RETURN END

85

C THIS IS THE PROGRAM FOR COMPUTING DEFLECTIONS BY R-K APP. INTEGER NMAX,NEQ,F PARAMETER (NMAX=10 , NEQ=4) REAL D1,D2,D3,D4,D5,D6,W

REAL D1S,D2S,D3S,D4S,D5S,D6S,WS REAL D1E,D2E,D3E,D4E,DSE,D6E,WE DIMENSION Y(NEQ) , DYDX(NEQ) , YOUT(NEQ) EXTERNAL DERIVS

5 PRINT * , 'ENTER INITIAL, STEP READ *, D1,D1S,D1E

PRINT *, ER INTIAL,STEP READ *, D2,D25,D2E

PRINT *,' ENTER INTIAL,STEP READ *, D3,D3S,D3E

PRINT *, ENTER INTIAL,STEP READ *, D4,D45,04E PRINT *, ER INTIAL,STEP READ *, D5,DSS,DSE PRINT *,' ENTER INTIAL,STEP READ *, D6,D6S,D6E

PRINT *,' ENTER INTIAL,STEP READ *, W,WS,WE

ND1=(D1E-D1) /D1S+1 ND2= ( D2E-D2) /D2S+1

ND3=(D3E-D3) /D3S+1 ND4= ( D4E-D4)/p4S+1 ND5= ( D5E-D5) /D5S+1

ND6=(D6E-D6) /D6S+1 NW= (WE-W) /WS+1 DO 106 N1=1,ND1

DO 105 N2=1,ND2 DO 104 N3=1.ND3 DO 103 N4=1,ND4 DO 102 N5=1,ND5 DO 101 N6=1,ND6

DO 100 N7=1,NW PRINT ',Dl,'BETA= PRINT *, DELTA ', D4,'THETA=

PRINT *, ALMEGA ', W

H=0.2 PRINT *, 'ENTER EIGHT INTIAL DATAS' READ , Y(1),Y(2),Y(3),Y(4) READ ', DYDX(1) , DYDX(2) , DYDX(3) , DYDX(4) PRINT , ' T YOUT(1) DO 10 ISTEP=1,1300

T=H* ISTEP

CALL RK4(Y,DYDX,NEQ,T,H,YOUT,DERIVS,D1,D2,D3 , D4,D5,DG,W) PRINT ', T,YOUT(1) , YOUT(2) , YOU(1)+YOUT(2)

10 CONTINUE w=w+ws

100 CONTINUE W=W-WS *NW


D6=D6_D6S*ND6 D5=D5+DSS

102 CONTINUE D5=D5_D5S*NDS


D4=D4_D4S*ND4


D3=D3_D3S*ND3

D2=D2+D2S

105 CONTINUE

AND

AND

AND

AND

AND

AND

AND

VALUES OF

END

END

END

END

END

END

VALUES

VALUES

VALUES

VALUES

VALUES

VALUES

ALPHA'

OF BETA'

OF GRANMA'

OF DELTA'

OF THETA'

OF LAMBDA'

OF ALMEGA'

',D2,'GANMA= ', D3 ',DS,'LANBDA= ', D6

YOUT ( 2)'

86

D2=D2_D2S*ND2


PRINT * ,'NEXT COMPUTIING?(1:YES 2:NO)' READ * F IF ( F . EQ. 1) GOTO 5 STOP END

SUBROUTINE RK4(Y,DYDX,N,X,H,YOUT,DERIVS, Dl, D2,D3,D4,D5,D6,W) PARANETER(NNAX=10) DIMENSION Y(N) , DYDX(N) , YOUT(N),YT(NMAX) , DYT(NMAX)DYM(NMAX) HH=R*0.5

H6=H/6. XH=X+HH DO 11 I=1,N

YT(I)=Y(I)+HH*DYDX(I)

11 CONTINUE CALL DERIVS(XH,YT,DYT,D1,D2,D3,D4,D5,D6,W) DO 12 I=1,N

YT(I)=Y(I)+HH*DYT(I)

12 CONTINUE CALL DERIVS(XH,YT,DYN,D].,D2,D3,D4,D5,DG,W) DO 13 I=1,N

YT(I)=Y(I)+H*DYM(I)

DYM(I)=D?i(I)+DYM(I) 2.3 CONTINUE

CALL DERIVS(X+I!,YT,DYT,D1,D2.D3,D4,D5,D6,W) DO 14 I=1,N

YOUT(I)=Y(I)+H6*(DYDX(I)+DYT(I)+2.*DYM(I)) 14 CONTINUE

DO 20 I=1,N DYDX(I)=DYT(I) Y(X)=YOUT(I)


SUBROUTINE DERIVS(X,Y,DYDX,D1,D2,D3,D4,DS,D6,W) PARAMETER (NEQ=4) DIMENSION Y(NEQ) , DYDX(NEQ) A=4*D1

DO 15 N=1,4 A=A+(16*D2/3.1416/(2*N_1))*SIN((2*N_1)*W*X)

15 CONTINUE DYDX ( 1) =Y ( 3) DYDX(2)Y(4) DYDX(3) = (W**2/O.278/D6)*(0.278*D3*Y(3)/W+(_0.035_0.035*A +_0.28*D5) *y(1)_O .0507*D5/D4_0 . 0127*A/SQRT(D4**2_0.25) ++ (3 . 25+0 . 25 *A+2 *D5 ) (2) DYDX(4) = (W**2/0.236/D6)*(0.236*D3*Y(4)/W+(_3.28_0.279*A +_2.23*D5)*Y(2)+0.033*D5/D4+0.0082*A/SQRT(D4**2_0.25))

RETURN END

87

C THIS IS THE PROGRAM FOR DETERMINE DISTANCE BETWEEN GUILDS COMMON/BLOK/LAFA,BETA(4) , PETA,GAMA,LADA, DETA

COMNON/DIM/RES,BL(18,1),UMAX REAL LADA,K PARANETER(PI=3 . 1415 , NN=32500)

5 WRITE(6,*) ' PMAXI= PNINI='

READ(6,*) PMAXI,PMINI WRITE(6,*) ' CURVE RADIUS= TUBE DIA='

READ(6,*) R,DTUBE

WRITE(6,*) ' ROD DIA=, DISTANCE BETWEEN COUPLINGS' READ(6,*) D,L

WRITE(6,*) 'MASS DENSITY OF ROD AND FLUID=' READ(6,*) GMR,SGF WRITE(6,*) ' DAMPING FACTOR=, SRTIKES PER MINITER=' READ(6,*) VIS,OMEGA WRITE(6,*) ' YOUNG MODELUS=' READ(6,*) E

OMEGA=0. 1047 *OMEGA

RODLO . 02 RODXO . 001 DX=0 . 0001 W=GMR*PX*D**2*9 . 8/4

P=GMR*PI*D**2*R*9.8/4

10 DO 15 I=1,NN RODX=RODX+DX SPCRE*PI.3 *D* * 4/64 /RODX**2

ALFA ( PMAXI+PMINI) /2/SPCR BETA(1)=(PMAXI-PMINI) /PI/SPCR BETA(2)=BETA(1)/3 BETA(3)=BETA(1)/5 BETA(4)BETA(1) /7

DETA=R/RODX LADAOMEGA**2* (G+SGF) *D**2*RODX**2/pI/SPCR/8

GANA=D*VIS*OMEGA* (RODX/PI) **2/SPCR TETA= (1E7+R*SIN(RODL/R)*9.8*SGF)*(D/2.)**2*PI/SPCR

PW=P*SIN(RODL/R) K=SQRT(PW/E/ ( PI*D**4/64)) UMAX=W*COS(RODL/R) /PW/K**2* ( 1/COS(K*RODX/2) -1)

& _W*COS(RODL/R)*RODX**2/8/PW+K**2*(R_SQRT(R**2_(RODX/2)**2))

& /(( PI/RODX)**2_K**2)

UMAXUMAX/RODX FX=RES CALL DEFLE write(6,*) ' res(res),fx',res(dtube-d)/2/rodx,fx

IF(RES.GT. (DTUBE-D)/2/RODX) THEN

XB=RODX-DX GOTO 20 ELSE GOTO 15 ENDIF

15 CONTINUE 20 AB=(DTUBE-D)/2/XB

WRITE(6,*) 'ALFA= BETA=',ALFA,BETA(1),BETA(2) ,BETA (3) ,BETA (4) WRITE(6,*) ' PLADA,GAMA=',LADA,GAMA

WRITE(6,*) 'THETA,DETA=' , TETA,DETA WRITE(6,*) 'THE MAX DISTANCE OF WHEEL=',XB WRITE(6,*) ' THE POSION ALONG THE ROD= ', RODL WRITE(6,*) 'THE DEFLEXION AT THIS DISTANCE=',FX WRITE(6,*) ' THE ALLOWED MAX DEFLEXION AT THIS DISTANCE=',AB WRITE(6,*) '

RODL=RODL+XB IF(RODL.GT.R*PI/2) THEN

GOTO 30 ELSE RODX=0 . 001 GOTO 10

88

ENDIF 30 PRINT *, NEXT COMPUTING?(1:YES 2:NO)'

READ(6,*) F

IF(F.EQ.1) GOTO 5 STOP END

SUBROUTINE GAUSSJ(AA,N,NP,BL,M,MP) PAR1METER ( NMAX=50) DIMENSION INDXC(NMAX) , INDXR(NMAX) , IPIV(NMAX) DIMENSION AA(NP,NP) , BL(NP,MP) DO 1011 J=1,N IPIV(J)=0

1011 CONTINUE DO 1022 I=1,N BIG=0 . 0 DO 1013 J=i.,N IF(IPIV(J) .NE. 1) THEN DO 1012 K=1,N IF(IPIV(K) . EQ. 0) THEN IF(ABS(AA(tT,K)) . GE. BIG) THEN BIG=ABS(AA(J,K)) IROW=J ICOL.=K ENDIF

ELSE IF(IPIV(K) . GT. 1) THEN PAUSE ' SINGULAR MATRIX' ENDIF

1012 CONTINUE ENDIF

1013 CONTINUE IPIV(ICOL) =XPIV(ICOL)+1 IF(IROW . NE. ICOL) THEN DO 1014 L=1,N DUM=AA(XROW, L) AA(IROW,L)AA(ICOL,L) AA(ICOL,L)DUM

1014 CONTINUE DO 1015 L=1,M DUMBL(IROW, L) BL(IROW,L)=BL(ICOL,L) BL(ICOL,,L)DUN

1015 CONTINUE ENDIF INDXR(I)=IROW INDXC ( 1) =XCOL IF(AA(ICOL,ICOL) . EQ. 0.0) PAUSE ' SINGULAR MATRIX' IC=ICOL PIVINV=1.0/AA(IC,IC) AA(ICOL,ICOL)=1. DO 1016 L=1,N AA(ICOL,L)=AA(ICOL tL)*PIVINV

1016 CONTINUE DO 1017 L=1,M BL(ICOL,L)=BL(ICOL,L)*PIVINV

1017 CONTINUE DO 1021 LL=1,N IF(LL . NE. ICOL) THEN DUM=AA(LL,ICOL) AA(LL, ICOL)=0. DO 1018 L=1,N AA(LL,L)=AA(LL,L) -AA(ICOL,L) *DJN

1018 CONTINUE DO 1019 L=1,M

89

BL(LL,L)=BL(LL,L)_BL(ICOL,,L)*DUM

1019 CONTINUE ENDIF

1021 CONTINUE 1022 CONTINUE

DO 1024 L=N,1.-1 IF(INDXR(L) . NE. INDXC(L)) THEN DO 1023 K=1,N DUM=AA(K,INDXR(L)) AA(K,INDXR(L))=AA(K,INDXC(L)) AA(K,INDXC(L))=DUN

1023 CONTINUE ENDIF


SUBROUTINE AVER(BL,Z,UMAX) PARPNETER (MP=1) INTEGER I, J REAL Y, WT,Z

DIMENSION BL(18,3.) Z1=0.0 Z2=0.O DO 90 .3=1,180

Y=o.0 WT=3.1415*(J_1.0)/180 DO 100 1=2.4 Y=Y+ (BLa (2*1, 1) +flT ( 2*1+9, 1) ) *SIN( ( 2*I_3.) *WT) Y=Y+(BL(2*I+1,2.)+BL,(2*I+10,1))*COS((2 *I_1) *WT)

100 CONTINUE Z1=Z1+Y+BL(1,1)+BL(1O, 1) Z2=Z2—Y+BL(1,1)+BL(10, 1)

90 CONTINUE Z1=Z1/180+UMAX Z2=Z2/180+UMAX IF (ABS(Z1) . GT. ABS(Z2)) GOTO 92

Z=ABS(Z2) GOTO 94

92 Z=ABS(Z1) 94 RETURN

END

SUBROUTINE DEFLE C THIS SUBRUTINE IS FOR COMPUTING THE DEFLECTION OF ROD AT MID

COMMON/BLOK/ALFA, BETA(4) , TETA,GANA, LADA, DETA COMNON/DIM/RES,BL(18, 1) , UMAX PARAMETER (N=18, NP=18, MP=1, M=1, P1=3.1415) DIMENSION AA(NP,NP)

7 DO 10 I=1.N DO 10 J=1,N AA(I,J)=0.0 BL(J,MP)=0.0

10 CONTINUE C THE ASSUMPTION IS SIN(PAI*X/L) & ( SIN(3*PAI*X/L))**2

AA(1, 1)=1+ALFA+2*TETA

( 1, 2) =BETA ( 1) AA ( 1, 4) =BEPA ( 2 AA ( 1, 6) =BETA ( 3 AA ( 1, 8) =BETA ( 4) AA(1,10)=(8/3/PI)+(8*ALFA/3/PI)+(8*2*TETA/31P1)

90

AA(1,11)=8*SETA(].)/3/pl

AA(1,13)=8*BETA(2)/3/pl AA(1,15)=8*BETA(3)/3/pl AA(1,17)=8*BETA(5)/3/pl AA ( 2 , 1) =2 *BETA ( 1 AA(2,2)=I+ALFA+2*TETA-2*PLADA AA ( 2 , 3) =2 *GAMJ AA(2, 1O)=16*BETA(1)/3/pl

AA(2, ll)=(8/3/PI)+(8*ALFA/3/pl)+(8*2*TETA13/pl) * _( 4*4*pLA/3/pI)

AA(2,12)=4*4*GjJ/3/pI

AA ( 3,3) =1+ALFA+2 *TETA_2 * pLJWA AA(3 ,11)=_4*4*GAI4A/3,pl A1(3, 12)=(8/3/pI)+(8*ALFA/3/pl)+(8*2*TETA/3/pl)

* _( 4*4*pLJA/3/pI) (4 , 1) =2 *BETA ( 2)

AA(4, 4)=l+ALFA+2*TETA_18*pLA AA(4,5)=6*GAMj AA(4,10)=16*EpA(2)/3/pI

* (16*9*PLADA/3/pI)

AA(4, 14)=16*G/pI AA(5, 4)=-6*GAMA AA(5, 5) =1+ALFA+2 *TETA_18*P]JDA

(5 13)*3*GxJ/3/pI

AA(5, 14 )=( 8/3/PI)+(8 *ALFA/3/pl)+(8*2*TETA/31p1) * _( 16*9*PLADA/3/px)

AA(6,1)=2*BETA(3) AA ( 6,6) =1+ALFA+2 *TETA_2 *25 AA(6,7)=2*5*G?A

AA(6,10)=16*ETA(3)/3/pI A(6, lS)=(8/3/PI)+(8*ALFA/3/pI)+(8*2*TEpA,3,pl)

*

AA(6, 1G)=16*5*GANA/3/px AA(7, 6)=_2*5*GAM1

AA(7, AA(7,15)=_16*5*GNA/3/pI

A7(7, lG)(8/3/PI)+(8*M,1A/3/pI)+(8*2*TEpA/3/pI) *

1) =2 *BETA ( 4) A(8, 8) =1+ALFA+2 *TETA_2 * 49 *pLJA AA(8,9)=2*7*GAMA AZ(8, 1O)=16*BEPA(4) /3/PI AA(8, l7)=(8/3/PI)+(8ALpA/3/pI)+(8*2*TETA/3/pI)

* ..( 16*49*PLADA/3/pI) A1(8, 18)=16*7*GANh/3/pl

A7(9, 9)=1+ALFA+2*TETA_2*49*pLA AA(9, 17)=_16*7*GNA/3/pI

* _( 16*49*PLADA/3/pI)

AA(1O,2)=8*BETA(1)/3/pl AA(1O,4)=8*BETA(2)/3/pl AA(10,6)=8*BETA(3)/3/pl A(10,8)=8*BErA(4)/3/pI

AA(10, 1O)=2**2+ALFA+2*TETA ( 10 , 11) =BETA ( 1)

AA ( 10 , 13) =BETA ( 2) AA ( 10 , 15) =BETA ( 3 AA ( 10 , 17) =BETA ( 4) A(11, 1)=8*BETA(1)/3/pl

* _( 16*PL,ADA/3/pl) AA(11, 3)=16*GAMA/3/pI

91

AA ( 11, 10) BETA ( 1) AA(11, 11)=2**2+ALFA+2*TETA_1. 5*pJDA AA(11,12)1.5*GANA AA(12,2)_16*GANA/3/PI AM12,3)(8/3/PI)+(8*ALFA/3/PI)+(8*2*TETA/3/P1)

* _( 16*PLADA/3/PI) AA(12.11)=_1.5*GANA AA(12, 12)=2**2+ALFA+2*TETA1.5*PL1.DA

A(13,1)=8*BETA(2)/3/PI AA(13,4) = (8/3/PI)+(8*ALFA/3/P1)+(8*2*TETA/3/P1)

* _( 16*9*pL7A/3/pI)

AA(13, 5)=16*3*GAW/3/PI

AA ( 13 10) =BETA ( 2) AA(13 , 13) =2**2+ALFA+2*TETA_1.5*9*PLADA AA(13,14)=1.5*3*GANA AA(14,4)_16*3*GPNA/3/PI AM14.5)=(8/3/PI)+(8*ALFA/3/P1)+(8*2*TETA/3/P1)

* _( 16*9*PLADA/3/PI) AA(14, 13)=_1.5*3*GAMA AA(14 , 14)=2**2+ALFA+2*TETA_1. 5*9*PLA AA(151)=8*BETA(3)/3/PI AA(15, 6)=(8/3/PI)+(8*ALFA/3/PI)+(8*2*TETA/3/PI)

* _( 16*25*PLADA/3/PI) AA(15,7)16*5*GAM/3/PI

AA ( 15 10) 8ETA ( 3) AA(15, 15)=2**2+ALFA+2*TETA_1. 5*25*PLA AA(15,16)=1.5*5*GNA AA(16,6)=_16*5*GPN1/3/PI AA(16,7) = (8/3/PX)+(8*1LFA/3/PI)+(8*2*TETA/3/P1)

* _( 16*25*PLADA/3/PI) AA(16,15)=_1.5*5*GJaNA AA(16,16)2**2+ALFA+2*TETA_1.5*25*PLA

AA(17,1)=8*BETA(4)/3/PI AA(17, 8)=(8/3/PI)+(8*LFA/3/PI)+(8*2*TETA/3/P1)

* _( 16*49*PLADA/3/PI) AA(17.9)=16*7*GAMA/3/PI AA ( 17 10) =BETA ( 4) AA(17, 17)=2**2+ALFA+2*TETA_1. 5*49*PA AA.(17.18)1.5*7*GJMA AA(18,8)_16*7*GAM/3/PI AA(18, 9)=(8/3/PI)+(8*ALFA/3/PI)+(8*2*TETA/3/P1)

* _( 16*49*PLADA/3/PI) AA ( 18 17) =- 1. 5*7 *GAMA AA(18,18)4.+M.FA+2.TETA_1.5*49.*PLADA BL(1,NP)_4*TETA/DETA/(P1**3)_4*1FA/((p1**3)*SQRT(

* DETA**2_O.25)) BL(2,HP)_8*BETA(1)/((PX**3)*SQRT(DETA**2_0.25)) BL(4,MP)_8*BETA(2)/((PI**3)*SQRT(DETA**2_0.25)) BL(6.MP)_8*BETA(3) / (( PI**3)*SQRT(DETA**2_0.25)) BL,(8,NP)_8*BETA(4) / (( PI**3)*SQRT(DETA**2_0.25)) BL(10,MP) =_(ALFA/SQRT(DETA**20.25)/(P1**2))_TETAIDETAI(PI**2) BL(11,MP) =_2*BETA(1)/SQRT(DETA**2_0.25)/(P1**2) BL.(13,M?)_2*BETA(2)/SQRT(DETA**2_0.25)/(P1**2) BL(15,MP)_4*BETA(3)/SQRT(DETA**2_0.25)/(P1**2) BL,(17.MP)_4*BETA(4)/SQRT(DETA**2_0.25)/(P1**2)

CALL GAUSSJ(AA,N,NP,BL,LMP) CALL AVER(BL, RES,UMAX) RETURN END

Date post:	30-Nov-2023
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