Journal of Sound and Vibration (1995) 185(1), 93–106

AN IMPROVED TRANSFER MATRIXTECHNIQUE AS APPLIED TO BHA LATERAL

VIBRATION ANALYSIS

S. L. C M. G´

Aerospace Laboratory, University of Liege, 21, Rue E. Solvay, 4000 Liege, Belgium

(Received 7 September 1993, and in final form 7 July 1994)

In previous work on BHA (bottomhole assembly) lateral vibration, the effects of axialloads have often been neglected. In this paper an improved transfer matrix technique isapplied to study the dynamics of a BHA system which is always subject to a large axialload. The transfer matrix developed, which is derived directly from the consistent equationsof motion of rotating Timoshenko shafts subject to axial loads, includes the effects ofrotary inertia, gyroscopic moments, transverse shear and axial load. The mud–drillstringinteraction is represented by an added mass coefficient. The influences of rotation speed,weight on bit and added mass coefficient on the lateral vibration are investigated. Thebuckling loads of horizontal BHA’s are calculated by observing the disappearance of thecorresponding eigenfrequencies.

7 1995 Academic Press Limited

1. INTRODUCTION

The BHA (bottomhole assembly) used for the drilling of oil and gas wells is often madeup of different sizes of drill collars, stabilizers and downhole tools. It is usually subject tolarge axial compressive force or weight on bit (WOB) and torque (TOB).

The lateral vibration of the BHA has become a subject of investigation later than axialand torsional vibration because it does not appear uphole [1, 2]. Lateral vibration can onlybe observed downhole because of the low travelling speed of lateral waves; it takes manywavelengths for the vibration to reach the top of the drillstring [3, 4]. The lateral motionis thus completely damped out by wall contact losses, and by structural and fluid dampingalong the drillstring. However, downhole measurements have given clear evidence thatsevere lateral vibrations occur in the BHA [5–8].

The lateral vibration analysis of such structures has usually been carried out by a finiteelement method (FEM) [9, 10] or by a finite difference technique [2]. There is no doubtthat the FEM is very attractive in most cases. The main drawback associated with the FEMis that it requires large computer memory for a long BHA system. The memory neededincreases rapidly with the number of elements. Therefore, this method may not beconvenient when used in the drilling field.

The transfer matrix method (TMM), which may be considered as the simplest methodin the analysis of shaft and beam systems, was first proposed by Myklestad [11]. Thismethod has been modified by many researchers during the past few decades. Twomodifications have been made recently by Lee et al. [12]. The first modification involvesrepresenting the transfer matrix of a shaft by considering a continuous system instead ofa lumped element system. The second modification is made in an effort to fit a synchronouselliptical orbit and a non-synchronous multi-lobed whirling orbit. However, the effects of

93

0022–460X/95/310093+13 $12.00/0 7 1995 Academic Press Limited

. . . ´94

Figure 1. A typical BHA system.

axial loads were not considered in their work. Such a treatment may lead to errors whenTMM is used in the analysis of BHA systems which are subject to very large axial forces.

This paper is concerned with the use of the TMM to study the lateral vibration of aBHA. Based on the consistent equations of motion of a rotating Timoshenko shaft subjectto axial loads [13] and following the similar procedure used in reference [12], the TMMis further modified by including the effects of axial loads.

The analysis to be presented is based on the following assumptions.

(1) The bit–hole wall contact, the stabilizer–hole wall contact and the BHA–hole wallcontact are not considered. Such contact usually leads to strong non-linearity. Althougha way to treat the effect of such a contact in a linear fashion has been proposed [2], themore effective way may be the finite element method.

(2) The torque is not considered. This is because the torque has little influence on thelateral vibration [2].

(3) The lateral–torsional coupling and the lateral–axial coupling are not considered.Subject to these assumptions, a typical BHA system is analyzed, with consideration of

the lateral vibration behaviors and the buckling loads.

2. MECHANICAL MODEL OF BHA

A sketch of a typical BHA system, which consists of drill collars, stabilizers and a drillbit, is shown in Figure 1.

The main purpose of drill collars for vertical wells is to apply force to the drill bit forpenetration. Compression developed within drill collars causes them to buckle inside thewell bore and this creates side loads on the bit. The use of stabilizers is to control thedirection of the borehole and to increase the buckling stability. Depending on the locationof the stabilizers, BHA’s can be categorized as building, holding or dropping assemblies [3].

3. TRANSFER MATRICES OF ELEMENTS

3.1.

3.1.1. Equation of motion

Consider first a uniform shaft subject to constant compressive axial load, as shown inFigure 2. For such a shaft, the consistent equations of motion have been derived recently

Figure 2. A rotating Timoshenko shaft subject to axial load.

95

by using finite strain beam theory [13]. In the X–Z plane one has

14X1Z4 −0 r

KG+

r

E1 14X1Z2 1t2 +

r2

KEG14X1t4 +

rAEI

12X1t2 −

2rv

E 0 13Y1Z2 1t

−r

KG13Y1t3 1

+PEI 01+

PKAG1 12X

1Z2 +rP

EIKG12X1t2 −

P2

E2AI12X1Z2

+0−2PEA

−P2

EAKAG+

P2

(EA)21 rAEI

12X1t2 =0. (1a)

In the Y–Z plane,

14Y1Z4 −0 r

KG+

r

E1 14Y1Z2 1t2 +

r2

KEG14Y1t4 +

rAEI

12Y1t2 +

2rv

E 0 13X1Z2 1t

−r

KG13X1t3 1

+PEI 01+

PKAG1 12Y

1Z2 +rP

EIKG12Y1t2 −

P2

E2AI12Y1Z2

+0−2PEA

−P2

EAKAG+

P2

(EA)21 rAEI

12Y1t2 =0. (1b)

Here E is Young’s modulus, G is the shear modulus, K is shear factor, r is the mass density,A is the cross-sectional area and P is the axial compressive load. Note that not onlythe gyroscopic moments but also the axial load terms are captured consistently inequations (1a, b).

3.1.2. Transfer matrix of a BHA sectionThe steady state solutions of equations (1) may be represented in the form

X(Z, t)=Xc (Z) cos Vt+Xs (Z) sin Vt, Y(Z, t)=Yc (Z) cos Vt+Ys (Z) sin Vt, (2)

where V is the whirling angular frequency. Substituting equations (2) into equations (1),then separating the terms associated with cos Vt and sin Vt results in four homogeneousequations, as follows. In the X–Z plane,

d4Xc

dZ4 +0rV2

E+

rV2

KG1 d2Xc

dZ2 +0r2V4

KGE−

rAV2

EI 1Xc +2rvV

Ed2Ys

dZ2 +2r2vV3

KGEYs

+PEI 01+

PKAG1 d2Xc

dZ2 −rPV2

EIKGXc −

P2

E2AId2Xc

dZ2

−rAV2

EI 0−2PEA

−P2

EAKAG+

P2

(EA)21Xc =0, (3a)

d4Xs

dZ4 +0rV2

E+

rV2

KG1 d2Xs

dZ2 +0r2V4

KGE−

rAV2

EI 1Xs −2rvV

Ed2Yc

dZ2 −2r2vV3

KGEYc

+PEI 01+

PKAG1 d2Xs

dZ2 −rPV2

EIKGXs −

P2

E2AId2Xs

dZ2

−rAV2

EI 0−2PEA

−P2

EAKAG+

P2

(EA)21Xs =0. (3b)

. . . ´96

In the Y–Z plane,

d4Yc

dZ4 +0rV2

E+

rV2

KG1 d2Yc

dZ2 +0r2V4

KGE−

rAV2

EI 1Yc −2rvV

Ed2Xs

dZ2 −2r2vV3

KGEXs

+PEI 01+

PKAG1 d2Yc

dZ2 −rPV2

EIKGYc +

P2

E2AId2Yc

dZ2

−rAV2

EI 0−2PEA

−P2

EAKAG+

P2

(EA)21Yc =0, (3c)

d4Ys

dZ4 +0rV2

E+

rV2

KG1 d2Ys

dZ2 +0r2V4

KGE−

rAV2

EI 1Ys +2rvV

Ed2Xc

dZ2 +2r2vV3

KGEXc

+PEI 01+

PKAG1 d2Ys

dZ2 −rPV2

EIKGYs +

P2

E2AId2Ys

dZ2

−rAV2

EI 0−2PEA

−P2

EAKAG+

P2

(EA)21Ys =0. (3d)

The solutions of equations (3) take the form

Xc =Uc elZ, Xs =Us elZ, Yc =Vc elZ, Ys =Vs elZ, (4)

where Uc , Us , Vc and Vs are arbitrary real constants and l is the root of the determinantalequation

G GG G

l4 + fl2 + g00

−(hl2 + k)

0l4 + fl2 + g

hl2 + k0

0hl2 + k

l4 + fl2 + g0

−(hl2 + k)00

l4 + fl2 + g

=0. (5)G GG GG G

Here

f=(rV2/E)+ (rV2/KG)+ (P/EI)+ (P2/KAGEI)− (P2/E2AI),

g=(r2V4/KGE)− (rAV2/EI)− (rPV2/EIKG)+ (rA/EI)(2PV2/EA)

+ (rP2V2/E2IAKG)− (P2rV2/E3AI),

h=2rvV/E, k=2r2vV3/KGE.

If one compares equation (5) above with equation (5) in reference [12], it is found thatthe only difference lies in the expressions for f and g due to the axial force P. Hence, theprocedure used in reference [12] may be directly employed here.

Like the displacements the slopes, moments and shear forces may be defined as follows.For the slopes,

a(Z, t)= ac (Z) cos Vt+ as (Z) sin Vt, b(Z, t)= bc (Z) cos Vt+ bs (Z) sin Vt. (6a)

For the moments,

Mx (Z, t)=Mxc (Z) cos Vt+Mxs (Z) sin Vt,

My (Z, t)=Myc (Z) cos Vt+Mys (Z) sin Vt. (6b)

97

For the shear forces,

Qx (Z, t)=Qxc (Z) cos Vt+Qxs (Z) sin Vt,

Qy (Z, t)=Qyc (Z) cos Vt+Qys (Z) sin Vt. (6c)

The state vector is constructed as follows:

S=[Xc , Xs , Yc , Ys , ac , as , bc , bs , Mxc , Mxs , Myc , Mys , Qxc , Qxs , Qyc , Qys ]T.

The state vector on the right side of the shaft is related to the same quantities on the leftside by the matrix equation

Si =[Tbha]Si−1, (7)

where [Tbha] is called the transfer matrix of a BHA section with dimension 16×16, andis given by

[Tbha]= [A]−1[N][A]. (8)

The matrices [A] and [N] are given in the Appendix. The transfer matrix may be expandedto the size of 17×17 to calculate the response:

0Si

11=0Ts

00110Si−1

1 1. (9)

3.2.

3.2.1. Transfer matrices of a stabilizer

The stabilizer may be modelled as a spring support, as shown in Figure 3. The transfermatrix for such a support has been obtained [12] as

0Si

11=[Ts]17×170Si−1

1 1, (10)

Figure 3. A model of a stabilizer.

. . . ´98

where [Ts] is the transfer matrix of a stabilizer. The non-zero elements of the matrix[Ts] are

Tsi,i =1, i=1, 2, . . . , 17, Ts

13,3 =Ts14,4 =−Kxy , Ts

14,1 =−Ts13,2 =CxxV,

Ts13,4 =−Ts

14,3 =−CxyV, Ts16,1 =−Ts

15,2 =Cyx , Ts16,2 =Ts

15,1 =Kyx ,

Ts16,3 =−Ts

15,4 =Cyy , Ts13,1 =Ts

14,2 =mbV2 −Kxx , Ts

15,3 =Ts16,4 =mbV

2 −Kyy .

All other elements are equal to zero.

3.2.2. Transfer matrices of a bit

Compared to the flexibility of the BHA, the bit may be considered as a lumped mass.The bit transfer matrix [Tb] relates the state vector at both ends as follows:

0Si

11=[Tb]17×170Si−1

1 1. (11)

The non-zero elements of the transfer matrix [Tb] of a bit are

Tbi,i =1, i=1, 2, 3, . . . , 17, Tb

13,1 =Tb14,2 =Tb

15,3 =Tb16,4 =MbV

2,

Tb14,17 =−Tb

13,17 =Mbv2ex , Tb

15,17 =Tb16,17 =−Mbv

2ey .

Here Mb is the mass of the bit. The unbalance forces Mbv2ex and Mbv

2ey are functionsof the rotational speed v.

4. MUD–DRILLSTRING INTERACTION

The drilling fluid confined between the drillstring and the hole wall undergoes a highlyintricate motion. When lateral vibration occurs, the confined fluid is squeezed so that acomplicated flow takes place and interacts with the structure.

It is obvious that the mud–drillstring interaction is a very complicated phenomenon.However, under some conditions, the interaction effect may be represented by an addedmass and a fluid damping coefficient [9, 14, 15]. According to Allen [9], the effective densityof the string is

reff = r+ rmud (CmAo +Ai )/(Ao −Ai ), (12)

where rmud is the mud density, Cm is the added mass coefficient (usually equal to 1), Ao andAi are areas defined by the outer and inner diameters of the drillstring, respectively. It canbe seen that the added mass increases when clearance decreases because the fluid isrelatively more squeezed. If the mud–drillstring interaction is taken into account, the massdensity r in equation (1) should be replaced by the effective density reff given in equation(12). It is very complicated to estimate the added fluid damping. It is usually taken as anequivalent damping value.

5. BHA LATERAL VIBRATION ANALYSIS

5.1.

The BHA system shown in Figure 1 is used for simulation. The left end of the BHAis assumed to be fixed. Other boundary conditions can be simulated without difficulty. Thegeometry of the BHA is listed in Table 1. The stiffness of the stabilizer is taken as 106 N/m.

99

T 1

Geometry of the BHA (see Figure 1)

Section numberZXXXXXXXXXXXXXXCXXXXXXXXXXXXXXV

1 2 3 4 5

Outer radius (m) 0·085725 0·085725 0·085725 0·1016 0·1016Inner radius (m) 0·071438 0·071438 0·071438 0·071438 0·071438Length (m) 3·81 19·812 10·67 10·67 6·238

5.1.1. Influence of rotation speed

The first four natural frequencies predicted by the model are depicted in Figure 4 versusthe rotation speed. For comparison, the FEM results calculated by Samcef software [16]are also given. It is seen that (1) the results predicted by the model agree well with thosepredicted by the FEM model, (2) as long as the rotation speed V is small enough (usually50–400 r.p.m. in practice), the gyroscopic effects may be negligible. Under this condition,the rotating drillstring may be studied as a static beam.

5.1.2. Influence of WOB (weight on bit)

The influence of the WOB on the natural frequencies is shown in Figure 5. With theincrease of the WOB, the natural frequencies decrease. It is seen that the WOB decreasesthe first, second and third natural frequencies by about 41·7%, 35·6% and 8% respectively.Since the first three natural frequencies are usually within the range of the rotation speed,the effects of the WOB must be taken into account.

5.1.3. Influence of surrounding mud

A direct comparison of the cases of with and without the surrounding mud is given inTable 2. It is obvious that the surrounding mud decreases the eigenfrequencies, especiallythe first one.

Figure 4. Natural frequencies versus rotation speed of the BHA (WOB=0). (a) First mode; (b) second mode;(c) third mode; (d) fourth mode. ——, Present method; w, FEM (Samcef).

. . . ´100

Figure 5. Natural frequencies versus WOB (v=0). Key as Figure 4.

5.2.

A very common phenomenon in drilling is that the bit moves forwards or backwards.The initiation mechanisms of whirling vibration of drillstrings have been analyzed [15].Several methods of identifying the bit whirl parameters have been developed recently [17].In order to simulate the effects of the bit induced motion on the BHA dynamics, thewhirling motion of the bit is considered as an excitation source in this study. It is of interestto observe the whirl trajectory of different points on the BHA.

It is assumed that the bit has an eccentricity with Mber =0·1 kg and the bit center hasa known whirl orbit represented by Xb = r cos Vt and Yb = r sin Vt, where r=0·001 m.From the superposition of the unbalance response and the bit whirl response, themulti-lobed whirling orbits for various bit whirling motions at the four stabilizers (seeFigure 1) have been obtained by the transfer matrix method and are shown in Figures 6and 7. In Figure 6, the rotation speed v and the forward whirl speed V are takenrespectively as 2 Hz and 6 Hz. In Figure 7, the rotation speed v and the backward whirlspeed V are taken respectively as 2 Hz and −8 Hz. It is observed that, under the sameexcitation, the whirl orbits of different points on the BHA are different. The trajectoriesmay supply useful information for the stabilizer design.

T 2

Influence of surrounding mud on lateral eigenfrequencies (WOB=0)

ModeZXXXXXXXXXXXXCXXXXXXXXXXXXVV1 V2 V3 V4

Without mud 1·836 2·169 3·993 4·797With mud 1·404 1·855 3·390 4·065

101

Figure 6. Non-synchronous forward whirl orbits of the four stabilizers. Stabilizer number: (a), 1; (b), 2; (c),3; (d), 4. v=2 Hz, V=6 Hz, ——, WOB=0; –·–·, WOB=16 000 kg.

6. BUCKLING LOADS OF BHA

As stated before, the BHA is usually subject to large axial compression force or weighton bit (WOB). The magnitude of the WOB has a significant influence on the rate ofpenetration (ROP) and on the drillstring failures. Therefore, it is very important toestimate the critical loads before the planning of drilling and during drilling.

6.1.

The buckling loads are obtained by observing the disappearance of the naturalfrequencies for a given load. For example, if the first eigenfrequency disappears for a givenload P1, then P1 is the first critical load. If the load is continuously increased, one may

Figure 7. Backward whirl orbits of the four stabilizers. Key and (a)–(d) as Figure 6. v=2 Hz, V=−8 Hz.

. . . ´102

find the second and higher critical loads. Obviously, the procedure for this method isexactly the same as in the case of free lateral vibration analysis, but the effects of rotationspeed can be disregarded. This procedure is validated through the study of a simplysupported beam with two internal elastic supports (see Figure 8). The buckling loadsobtained respectively by the FEM [17] and by the present method are listed in Table 3.

6.2.

Two horizontal BHA’s have been studied. The first BHA (see Figure 9(a)) is simplysupported at both ends. The second BHA (see Figure 9(b)) is supported with two internalstabilizers. The purpose here is to show the influence of the stabilizer on the buckling loads.

Because the gravitational force of the BHA may be neglected in the horizontal case, auniform BHA section may be taken as one element in the transfer matrix. Therefore onlyfour elements for the first BHA and six elements for the second BHA are needed.

The first four buckling loads for the two BHA’s are listed in Table 4. It is seen that thestabilizer has a significant influence on the buckling loads.

Because of the low dimension matrix and hence less computing effort, the method maybe used in the drilling field to calculate the buckling loads. However, the procedure maybe tedious if the number of buckling loads wanted is large. A simplified procedure in whichan improved dynamic stiffness method is used has been proposed recently [18].

Figure 8. A simply supported beam with two internal springs.

T 3

Buckling loads of the beam shown in Figure 8

Load (N)ZXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXV

P1 P2 P3 P4 P5 P6

FEM 33 011 64 919 97 201 120 621 192 592 252 735Present 33 010 64 900 97 150 120 510 192 450 252 255

Figure 9. Two horizontal BHA’s.

103

T 4

Buckling loads of horizontal BHA’s (v=0)

PcZXXXXXXXXXXXXXCXXXXXXXXXXXXXV

P1 P2 P3 P4

BHA 1 31 525 143 920 307 400 561 080BHA 2 284 000 495 370 630 020 778 200

7. CONCLUSIONS

A BHA used in oil well drilling is always subject to a large axial load. To account forthe effects of axial force on the BHA lateral vibrations, an improved transfer matrixtechnique has been presented in this paper. The transfer matrix of a shaft subject to axialloads has been derived directly from the consistent equations of motion. The whirling orbitof any point on the BHA induced by the bit motion can be obtained by using this method.The buckling loads can also be calculated by observing the disappearance of the associatedeigenfrequency for a given load. It is shown that (1) both the WOB and the surroundingmud decrease the eigenfrequencies significantly, (2) the gyroscopic effects may be neglectedbecause the rotation speed of the BHA is usually low, and (3) the placement of thestabilizer has a considerable influence on the buckling loads of the BHA. The numericalresults show that the improved transfer matrix method is quite appropriate for linear BHAsystems.

ACKNOWLEDGMENTS

The first author is financially supported by Security DBS (Belgium). The authors thankDBS (Belgium) Company for giving them permission to publish this paper. Thanks arealso given to Mr E. Lamine (Security DBS) and to Mr D. Rixen (LTAS) for their usefulsuggestions.

REFERENCES

1. W. C. C 1988 (May) Petroleum Engineer International. Why drill strings fail at the neutralpoint.

2. R. J. S, 1989 Ph.D. Thesis, MIT. Bending vibration of rotating drill string.3. D. W. D 1984 Journal of Petroleum Technology 8, 637–644. Drill collar length is a major

factor in vibration control.4. S. F. W, M. Z and A. A 1985 SPE 14330. Field measurements of downhole

drillstring vibrations.5. D. A. C and S. C. O 1988 SPE/IADC 17273. Measurement of BHA vibration using

MWD.6. J. D. J 1993 Ph.D. Thesis, Delft University, Holland. Nonlinear dynamics of oilwell

drillstrings.7. J. K. V, J. W. N and R. J. S 1988 SPE 18652. Case studies of the bending

vibration and whirling motion of drill collars.8. J. F. B and T. M. W 1989 SPE 19571. Bit whirl: a new theory of PDC bit failure.9. R. F. M and M. B. A 1985 World Oil 2, 101–106. Lateral vibration: the key to BHA

failure analysis.10. M. B. A 1987 (March) SPE/IADC 16110. BHA lateral vibration: case studies and

evaluation of important parameters.11. N. M 1944 (April) Journal of Aeronautical Science, 153–162. A new method of

calculating natural modes of uncoupled bending vibration of airplane wings and other types ofbeams.

. . . ´104

12. A. C. L, Y. K and S. L. L 1991 Transactions of the American Society of MechanicalEngineers, Journal of Applied Mechanics 58, 776–783. A modified transfer matrix method forlinear rotor-bearing systems.

13. S. H. C, C. P and A. G. U 1992 Transactions of the American Society of MechanicalEngineers, Journal of Vibration and Acoustics 114, 249–259. Consistent modeling of rotatingTimoshenko shafts subject to axial loads.

14. S. S. C and M. W. W 1976 Transactions of the American Society of MechanicalEngineers, Journal of Applied Mechanics 42, 326–329. Added mass and damping of vibrating rodin confined viscous fluids.

15. J. D. J 1991 Journal of Sound and Vibration 147, 115–135. Nonlinear rotor dynamics asapplied to oilwell drillstring vibrations.

16. Samcef version 5.0 1994. User’s guide.17. S. L. C, M. G´ and A. L 1993 Proceedings of the 64th Shock and Vibration

Symposium, 261–269. Identification of downhole bit whirl while drilling.18. S. L. C, M. G´ and A. L 1994 Machine Vibration 3, 1–9. An improved dynamic

stiffness method with application to vibration analysis of linear drillstrings.

APPENDIX

The non-zero elements of matrix [A] are

A1,1 =A2,5 =A5,2 =A6,6 =A9,3 =A10,7 =A13,4 =A14,8 =1,

A2,13 =A6,14 =A10,15 =A14,16 =1/KGA, A3,1 =A7,2 =A11,3 =A15,4 = rV2/KG,

A3,9 =A7,10 =A11,11 =A15,12 =1/EI, A4,8 =−A8,7 =A12,6 =A16,5 =−2rvV/E,

A4,15 =A8,16 =A12,13 =A16,14 =2rvV/KGAE,

A4,13 =A8,14 =A12,15 =A16,16 = (1/EI)− (r/KGA){(1/E)− (1/KG)}V2,

A4,5 =A8,6 =A12,7 =A16,8 = (rV2/KG)− (rV2/E).

The other elements are equal to zero.The matrix [N] is given as follows:

F JG G

[N]=

N11

00

N21

0N'11

N'21

0

0N'12

N'22

0

N12

00

N22

, 0N11

N21

N12

N221=[Ha ][Ma ]−1,G GG Gf j

0N'11

N'21

N'12

N'221=[Hb ][Mb ]−1.

The matrices [Ha ] and [Hb ] are

[Ha ]=0 H1

−H1

H2

H21, [Hb ]=0H1

H1

H2

−H21,with the submatrices [H1] and [H2]:

F J F J

[H1]=G G G GG G G G

C1

laC2

l2aC1

l3aC2

C2

laC1

l2aC2

l3aC1

C3

−lbC4

−l2bC3

l3bC4

C4

lbC3

−l2bC4

−l3bC3

, [H2]=

G G G Gf j f j

C5

lcC6

l2cC5

l3cC6

C6

lcC5

l2cC6

l3cC5

C7

−ldC8

−l2dC7

l3dC8

C8

ldC7

−l2dC8

−l3dC7

.

105

The matrices [Ma ] and [Mb ] are similarly given by

[Ma ]=0 M1

−M1

M2

M21, [Mb ]=0M1

M1

M2

−M21,with the submatrices [M1] and [M2]:

F J F JG G G G

[M1]=

10l2

a

0

0la

0l3

a

10

−l2b

0

0lb

0−l3

b

, [M2]=

10l2

c

0

0lc

0l3

c

10

−l2d

0

0ld

0−l3

d

.G G G GG G G Gf j f j

Here Ci , i=1, 8, are constants:

C1 = cosh laL, C2 = sinh laL, C3 = cos lbL, C4 = sin lbL,

C5 = cosh lcL, C6 = sinh lcL, C7 = cos ldL, C8 = sin ldL.

la , lb , lc and ld are the roots of equation (5).