The application of the simultaneous iteration method to flexural vibration problems

Int. J. mee/t. Be/. Pergamon Press. 1974. Vol. 16, pp. 269-283. Printed in Great Britain

T H E APPLICATION OF T H E SIMULTANEOUS ITERATION METHOD TO F L E X U R A L VIBRATION PROBLEMS*t V. I~AMAMURTI

Department of Applied Mechanics, Indian Institute of Technology, Madras 36, India and O. I~AHRENHOLTZ Institute of Mechanics, Technical University, Hannover, West Germany

(Received 14 March 1973, and in revised form 29 September 1973) Summary--The simultaneous iteration method to determine the eigenfrequeneies and eigenvectors is utilized for solving the flexural vibration problems of rotationally symmetric bodies. The finite element approach is used to predict the behaviour of the system. A suitable shape function for the circumferential displacement distribution has been proposed which reduces the three-dimensional problem to a two-dimensional one. The accuracy of the method has been checked by verifying the results of known eases. It has been applied to a complicated structure and results have been compared with experimental ones. NOTATION A area of cross-section of the cylinder E Young's modulus I moment of inertia of the Cylinder l length of the cylinder cylindrical co-ordinates radial, circumferential and axial displacements vr, v ~ , Vz shear strains Cr, e ~ , Cz normal strains /z Poisson's ratio natural frequency P density of the material normal stresses shear stresses [K], [M], [D], [B] stiffness, mass, elasticity and strain-displacement matrices {8}, {e},{a} displacement, strain and stress vectors 1. I N T R O D U C T I O N THE DETERMINATION o f the n a t u r a l frequencies of flexural v i b r a t i o n s is a well-known problem. The a p p l i c a t i o n o f the transfer m a t r i x a p p r o a c h for solving flexural problems o f shafts on m a n y supports or with v a r i a t i o n s in cross-section t a k i n g shear d e f o r m a t i o n a n d r o t a r y inertia into a c c o u n t is familiar, z, ~ The governing equations are f o r m u l a t e d b y using either the flexibility m a t r i x m e t h o d or the stiffness m a t r i x m e t h o d . This a p p r o a c h poses difficulties if a t a n y cross-section one comes across a n n u l a r rings one a r o u n d * This investigation has been partly sponsored by the Ministry of "Forschung und Technologic", FRG, under the "l-ll-lT-Hochschulprogramm". Dedicated to Professor Eduard C. Pestel in honour of his sixtieth birthday (29 May 1974). 269 20

270

V. RAMAMURTI a n d O. MAHRENHOLTZ

the other. In order to solve such problems, connected with rotationally symmetric objects, this method is introduced. The finite element procedure is used to predict the behaviour of any element inside the body of revolution. The static analysis of axisymmetric solids, subjected to asymmetric loads, has been investigated by Wilson s b y expressing the loads and displacements by means of Fourier series. In the present case, the cumbersome three-dimensional problem is reduced to a two-dimensional one b y the choice of a suitable shape function. This method h a s been applied to a cylindrical shaft with various end-conditions. The results are compared with those obtained b y known methods. The method is finally employed to solve a practical problem connected with the flexural vibration of a compressor disk. The results are verified b y experiments. In all these cases the simultaneous iteration method 4 to solve the eigenvalue problem has been used. This method is well suited for problems where only a few of the dominant vectors and latent roots of a large symmetric matrix are required. The advantages of using this method over Householder's method for solving problems involving a large number of degrees of freedom has been demonstrated b y Jennings and Orr. 5 In that paper the simultaneous iteration method has been applied to: vibration problems connected with a fuselage wing combination and rectangular grillages. The effectiveness of this method for the analysis of undamped vibration problems has also been demonstrated by BrSnlund. ~ 2. ANALYSIS 2.1. M a s s and atiffneaa ~at~icea

If r, ¢ and z define the position of any point within a body of revolution, see Fig. 1, the strain vector {e} is given by ~U

~r u (

Er

~v

7+~--~ Ow Oz

£=

Ou

Vr~

~v Vzr

(t)

Ov v

~w

,

Ou Ow

N +~--;

DoI C FIo. 1. C o - o r d i n a t e s y s t e m .

Z,W

r

Simultaneous iteration method for flexural vibration problems

271

Let u s assmne u = U(r, z) cos ~, v = V(r, z) sin q~,

j

(2)

w = W ( r , z) COS¢ .

These expressions represent flexure in the A O Z plane. The points A, B, C and D, Fig. l, move vertically downwards. This does not necessarily mean that all the other points in the circle should move only vertically downwards. This will be achieved only if U is equal to - V. But it can be seen later from the results that U is very nearly equal to - V. Let us choose a triangular element (Fig. 2) with the nodal displacement vector U~ cos V~sin ¢ W~cos ¢ U# cos Vjsin ¢

{8}' =

W~cos ¢ U~ cos ¢ sin ¢ W~cos ¢ (From now on we shall use for brevity only U~, ~ , W~, etc. for displacements. This means t h a t wherever Ul a n d W~ are used to represent displacements they are accompanied b y cos • and ~ b y sin ~.) Let the expressions U, V and W within the element be represented by U -~ a l z + a 4 r = z + a T , V ~- a = z + a a r ~ z + a s ' W = aar+aerz=+ag.

• (4)

I

i

Z(W)

0

r(u)

=-

FIG. 2. Triangular element of the body.

272

V. RAM~MUR~ a n d O. ~IAltRENHOLTZ

T h e s e t e r m s r e p r e s e n t a p p r o x i m a t e l y t h e d i s p l a c e m e n t s U, V a n d W o f a c a n t i l e v e r , circular in cross-section, s u b j e c t e d t o a c o n c e n t r a t e d load a t one e n d 3 * N o w one can write Ui = ax z i + a 4 r~ z~+aT, U¢ = al z¢+a4 r~ z j + a 7, U~ = a l zk + a , r ~ z k +aT, Vi = a s z i + a 5 r ~ z ~ + a s , Vt = a s z j + a 5 r~ z¢ + as,

(5)

V~ = asz~ +a~ r~z~ +a8, W~ = a 3 r , + a a r , z~ + a ., W~ = a s r$ + a t r I z~ + a , , W~ = a s r~ + a , r~ z~ + a~.

Hence

/ al} i } / a,} a4

= [R]

a,

U¢

,

U~

a6

= [R]

as

Vj

(6)

,

V~

at

= [S]

a.

W¢

,

W~

whore

[

~11 ~12 r13 ] [R] =

rn

r22

r31

~32 r33

and

r23

IS] = [

811 812 81$ 821 8S2 823 831 832 833

0

(7)

r n , r12, etc. a r e g i v e n i n t h e A p p e n d i x . E q u a t i o n s (6) c a n b e r e w r i t t e n as {A} = [ R S ] {8} o,

(8)

where

[Rs] =

rll

0

0 0

0

rl$

0

0

r18

0

rn

0

0

0

811

0

rSl

0

0

0

rls

0

0

r12

0

0

812

0

0

813

rss

0

0

rsa

0

0

o

r21

0

0

rss

0

0

rs2

0

0

0

sst

0

0

ss2

0

0

Sss

r81

0

0

r22

0

0

ra,

0

0

0

r21

0

0

r32

0

0

r.

0

0

0

ssl

0

0

ass

0

0

sss

C9)

* T h e r e m a y b e m a n y m o r e s h a p e f u n c t i o n s w h i c h m i g h t h a v e g i v e n rise t o reliable r e s u l t s . I t s h o u l d b e p o i n t e d o u t t h a t t h e linear p o l y n o m i a l u = a 1 + a 4 z + a 7 r or t h e t w o o t h e r choices u = ax + a4 rzS+ aT rSz a n d a I + a 4 rz + aT r s z 2 d i d n o t y i e l d g o o d results.

Simultaneous iteration method for flexural vibration problems

al}

and

{A} =

a,

.

273

(10)

a9

Hence the displacement within the element can be expressed in the form V

= [N]{$}t=

W

0

z

0

0

r 2z

0

0

r

0

0

0

1

0

rz * 0

0

0

1

[R8]{$}".

(11)

I f an element triangular in cross-section subtending an a~gle d e at the centre is considered, the mass m a t r i x of such an element is given by 8

[,.].

[ N ] r dr dz d e

= p ff =

p[RS]~ff[P]

d r d z [RS] d e ,

(12)

where the elements of the symmetric m a t r i x [P] are given in the Appendix. This integral can be evaluated numerically by using the Gaussian quadrature formulae, s The strain and displacement vectors are connected by {e} -- [B] [ITS] {~}',

(13)

where the strain-displacement m a t r i x [B] reads 0

[B] --

0

0

2rz

0

0

0

0

0

z/r

0

rz

rz

0

1]r

1/r

0

0

0

0

0

0

2rz

0

0

0

--zlr

-z/r

0

-rz

rz

0

-1]r

-r/r

0

0

1

-- 1

0

rs

--z 2

0

0

1

0

1

rs

0

zz

0

0

--

(14)

lit 0

Hence the stress vector {a} is given by {a} = [D] {e}, where the elasticity m a t r i x [D] is represented by

[D] --

Px

P:

P:

0

0

0

P:

Px

P:

0

0

0

P,

Ps

Pl

0

0

0

0

0

0

P8

0

0

0

0

0

0

Ps

0

0

0

0

0

0

Ps

(15)

The p's are given in the Appendix. Now the stiffness matrix [k] ~ of the element can be written as s

= ff[B]

[D] [B] r dr dz dO

= [RS] T [Q] dr dz d¢[RS],

(16)

(17)

where the elements of the symmetric m a t r i x [Q] are given in the Appendix. Again, t h e integral can be evaluated numerically.

274

V. RAMAMURTIand O. MAHRENHOLTZ

2.2. Boundary conditions Three possible boundary conditions most commonly used in bending theory are the fixed end, free end and hinged end conditions. For fixed end, u, v, w are zero along the ~radial line, for free end all are unconstrained and for hinged end u, v and the axial stress, Oz, are zero. 2.3. Dynamic analysis The free undampe d vibrations of the problem under consideration reduce to the form

~'CM] {q} = [K] {q},

(18)

where [M] and [K] are the overall mass and stiffness matrices and {q} is the eigenvector. This c a n b e reduced to the symmetric eigenvalue problem b y introducing auxiliary variables

{u} = [L]~ {q},

(19)

where [L] is the lower triangular matrix obtained from the Choleski factorization of [K] according to [K] -- [L] ILl T. Then equation (18) reduces to

•

[A] (u} = ~ {u} = ~{u},

(20)

•where [A] --- ILl" 1 [M] I L l , T.

(21)

Equations (20) are solved b y the simuItaamons iteration method as described b y Jeunings. 4 i,It consists of the solution of the following three subsequent equations followed b y interaction analysis and orthogoualization [L] T (x} = {u}, (22) {y} - [M] {x}, [L] {v} = {y}.

(23) (24)

Here {u} are the assumed and {v} the predicted vectors. They are of the order n x m, where n is the degree of freedom a n d m is the n u m b e r of trial vectors chosen. Equations (22) and (24) are simple forward and backward substitutions. The predicted vectors {v}, obtained from equation (24), are orthogonalized to produce a new set of trial vectors (u} for subsequent iteration. To take full advantage both of symmetry and of band form of the matrix, a n unconventional form of storage as suggested by Martin and W'flkinson9 is used. T h e a l g o r l t h m for the triangular decomposition of the b a n d matrix is also given in t h a t paper. I n the case of unconstrained structures (as in the case of a frec-frec beam) equations (18) can be modified as suggested b y Cox 1° and written in t h e form

(to2 + a)[M] {q}= ([K] + aiM]) (q},

(25)

where a is a n y positive constant. 3. N U M E R I C A L

RESULTS

The flexural vibration analysis has been carried out for the following cases using the above method (i) the cylindrical bar with (a) fixed-fixed ends, (b) fixed-free ends,

(c) hinged-hinged ends, (d) hinged-fixed ends, and

. (ii) the complicated structure.

:3.1' The cylindrical bar • A b a r : o f radius one and length twelve is analysed. For the purpose of finite element analysis thirty axial locations with four radial points along each location are chosen. The

Simultaneous iteration r a tt h o d for flexural vibration problems

275

number of trial vectors chosen is eight and after eight iterations the first three natural frequencies converged up to the first four significant digits. To ascertain the accuracy of the results the natural frequencies and the eigenvectors were calculated using the bending theory with rotary inertia and shear deformation taken into account. The normal function of cosine or sine type satisfies the governing equation and the boundary conditions exactly for hinged-hinged beams. 11 F o r t h e other three cases the transfer m a t r i x approach ~ was used to obtain the eigenvalues and eigenvectors. The natural frequencies are compared in Tables 1-4 and the response along the axis in Figs. 3-6. The values of U at r = 0.02 are assumed to represent the values of U at r = 0. |.0

Bending theory -

E E. Method

3

o.

E ,K

2

-tO

0

,

0,5 L Distance along the axis

FIG. 3. Response of a fixed beam (r

|.01. =

0.02).

/ .!

I

0.5 L Distance along the axis

FIQ. 4. Response of a cantilever (r = 0.02).

I

1.0 L

276

V. RAMAMURTI and

O, I~AHRENHOLTZ

Bending theory

1.0

Q.
- 1.0

1.0L

O,5L Distance along the axis

FzG. 5. R e s p o n s e of a hinged b e a m ( r = 0.02). I.O Bending theory o

F.E. Method

E

_f/

°1.0

0

0.5L Distance along the axis

1.0 L

FIG. 6. Response of a f i x e d - h i n g e d b e a m (r = 0.02). T A B L E 1. N A T U R A L FREQUENCIES I N r R d / S e C - - F I X E D - - F I X E D BEAM = 0"1205 sec -2, p = 0.3)

(E]/ApI4

Frequency

Beam theory

F.E. method

Difference (%)

wI

7-01

7.04

÷ 0.42

oJ2 w8

17.57 31.10

17.32 30.20

-- 1 - 4 2 -- 3 . 2 2

277

S i m u l t a n e o u s i t e r a t i o n m e t h o d for f l e x u r a l v i b r a t i o n p r o b l e m s TABLE 2. NA'ru*tAL FREQUENCIES IN r a d / s e c - - H I N G E D - - H I N G E D BEAM (EI/Apl4 = 0.1205 s e c - ' , p = 0.3) Frequency

Beam theory

F.E. method

Difference ( ~o )

to, to~ toa tot

3"33 12"55 24-90 41.60

3"35 12-43 25.35 40.70

+ 0"60 -- 0"96 + 1-80 -- 2.16

TABLE 3.

NA~rRAL

FREQUENCIES rN rad/seC--FIXED--FREE BEAM 0"1205 s e c - ' , / ~ = 0.3)

(EI/ApI~ = Frequency to, to2 toa

Beam theory

F.E. method

Difference ( ~o )

1"212 7"270 18-970

1.229 7.100 17.900

+ 1-41 2"34 5-63 -

-

-

-

TABLE 4. NATURAL FREQUENCIES IN FS~I/sePr---FIXED--HINGED BEAM

(EI/ApI4 =

0"1205 sec -s, ft

=

0"3)

Frequency

Beam theory

F.E. method

Difference (%)

to, to~ tot

5"07 15.25 28.95

5-08 14.95 28.00

+0"20 -- 1-97 -- 3 . 2 8

T h e v a l u e s o f U a n d - V a t r = 0-02 w e r e t h e s a m e for all f r e q u e n c i e s a n d for all f o u r cases. T h e p e r c e n t a g e difference i n t h e v a l u e s of U a n d V, r e f e r r e d t o r = 0.02 a l o n g t h r e e r a d i a l lines for t h e h i n g e d - h i n g e d b e a m , a r c s h o w n i n T a b l e 5. T h e v a r i a t i o n o f W a l o n g t h e r a d i a l line a t a x i a l l o c a t i o n s o f o n e - s i x t h a n d o n e - t h i r d o f t h e l e n g t h for a b e a m w i t h h i n g e d - h i n g e d e n d s is p l o t t e d i n Fig. 7. TABLE 5. HINGED--HINGED BEAM--DISPLACEMENTS Difference ( % )

Radius

OJ 1

OJ2

ID3

t0 4

For u For v

For u For v

For u For v

For u For v

0.00 1-80 6.00

IL

0-28 0.60 1-00

0.17 0.34 1.02

0.00 0.34 0.85

0.30 0-90 2.70

0.33 0.90 2.70

z = ½L

0.28 0.60 1.00

0.00 0.10 0.10

0.00 0.10 0.10

0.30 1.20 2-70

0.30 1.20 2.70

z =

0.28 0.60 1.00

0.00 0.08 0.08

0.00 0.00 0.08

z =

½L

0.60 1.80 6.00

0"60 1-80 5.40

0.60 1-80 5.40

0.50 2.50 8.26

0.63 3.02 8.26

0.81 3.23 9.92

0.00 1.62 6.11

278

V. RAMAMURTI a n d O. I~AHRENHOLTZ

Z= t

Z= /

Third mode

Fourth mode

3 First mode

Second mode

FIG. 7. Displacement W along the radial line (hinged beam). The following conclusions can be drawn from these results: (i) The agreement in the amplitude of response obtained by these two approaches is excellent, see Figs. 3-6. (ii) The percentage difference in frequency values, even though negligible for the fundamental, is as high as 5.6 per cent in the case of the third frequency for the cantilever. (iii) Whereas U - a n d V-values do n o t show appreciable change along the radial line for the fundamental, they do show a noticeable difference for higher modes of vibration, see Table 5. The difference is as high as 9.9 per cent for the fourth mode. The same trend has been observed for beams With the other boundary conditions. (iv) Whereas W is essentially linear along the radial line for the fundamental mode, it becomes more and more non-linear for higher modes. The same trend has been observed for beams with other end conditions. Conclusi0ns (iii) and (iv)Clearly indicate that the improved bending theory including the effdcts of shear deformation a n d rof~ry 'inertia is not fully Valid for :higher modes. This also'explains Why t h e difference in frequencies is higher for higher modes. 3.2. The complicated structure The half-sectional elevation of the structure analysed is shown in Fig. 8. I t is assumed to be fixed at the ends. Eight trial vectors were chosen and after eight iterations the first three natural frequencies converged up to four significant digits and the fourth up to two digits. The time taken w a s about 5 m i n on a Univac 1108. To verify the results the natural frequencies were determined experimentally. The standard set of equipment from Briiel and Kjaer, consisting of the vibration exciter, power amplifier, exciter control, preamplifier and acceleration pick up, was used for this purpose. The experimental values are compared w i t h the theoretical results in Table 6. The deformed shape of the compressor disk in all the four modes is shown in Figs. 9-12. I t m a y be oberved that the TABLE 6. COMPRESSOR DISK2--COMPARISOI~

¢oI 1

2 3 4

Experimental values (counts/sec)

Computed values (eounts/sec)

Error (%)

128 842 1490 1746

149 750 1622 2070

+ 16"40 - 10"90 + 8"86 + 18"55

Simultaneous iteration method for flexural vibration problems --

105

,~

~d/N/l/Jl///

N//I/

//

tt~

~

.

J

~

f

/

J

J f f

J ~

/ I

/

JJ JJ ~z / l~ ~ l

J

/

. , 1 " q .,.,~ , , ' 1 / I , M / L , I

'250 Dimensions in mm

F I G . 8. Half sectional elevation of the compressor disk.

!

. . . .

I .

.

.

.

F ___~__ FIG. 9. Response of the compressor disk (I mode).

279

280

V. RAMAMURTZand O. !~IAHRENHOLTZ

i I I I

L FIo. 10. Response of the compressor disk (II mode).

F*o. 11. Response of the compressor disk ( I I I mode).

Simultaneous iteration method for flexural vibration problems

281

FIo. 12. Response of the compressor disk (IV mode). buckling of the cylindrical portion CD is predominant in the third and the fourth modes. The percentage difference in frequencies is relatively high. This may be due to the fact that the actual structure had to be modified to reduce the number of nodal points to meet the available storage of the computer. 4. C O N C L U S I O N S T h e s i m u l t a n e o u s iteration m e t h o d h a s b e e n applied t o t h e flexural v i b r a t i o n p r o b l e m . T h e reliability o f t h e s h a p e f u n c t i o n used h a s b e e n d e m o n s t r a t e d b y its a p p l i c a t i o n t o t h e b e n d i n g v i b r a t i o n s o f a cylindrical b e a m . T h e fairly good a g r e e m e n t b e t w e e n t h e theoretical analysis a n d e x p e r i m e n t a l results in t h e case o f t h e c o m p r e s s o r disk d e m o n s t r a t e s its a p p l i c a b i l i t y t o t h e flexural v i b r a t i o n o f c o m p l i c a t e d structures. I t is w o r t h w h i l e exploring t h e possibility o f its usage to o t h e r p r a c t i c a l p r o b l e m s . Acknowledgemenb---The authors take the opportunity of thanking Frau W. Poock for having typed the manuscript and Herr W. Pietsch for having prepared the drawings.

REFERENCES 1. W. M. JEI~KINS,Matrix and Digital Computer Methods in Structural Analysis. McGrawHill, New York (1969}. 2. E. C. PES~T. and F. A. LECgT~, Matrix Methods in Elastomechanics, pp. 132-137. McGraw-Hill, London (1963). 3. E. L. WILSON, A I A A J . 8, 2269 (1965). 4. A. J~.N~r~os, Proc. Gamb. Phil. Soc. 68, 755 (1967). 5. A. JE~cn~Gs and D. R. L. OBR, Int. J. Num. Meth. Engng 8, 13 (1971).

V. RAMAMURTIand 0. MAWRENHOLTZ

282

6. O. E. BR6~7_~UND,Syrup. Finite Elem. Techn. Institut fi~r Statik und Dynamik der Luftund Raumfahrtkonstrulction, U n i v e r s i t y of S t u t t g a r t (1969). 7. S. Tr~OSK~KO a n d J. N. GOODm~, Theory of Elasticity, pp. 37-38. McGraw-Hill, New Y o r k (1951). 8. O. C. Zm~Kmw~cz, The Finite Element Method, pp. 15-16, 171, 263-264. McGraw Hill, L o n d o n (1967). 9. R. S. M ~ T ~ a n d J. H. WV.~x~soN, Num. Math. 7, 355 (1965). 10. A. J ~ . ~ o s , A i r Engn~ 34, 81 (1962). 11. S. TI~OSHE~-~O, Vibration Problems in Engineering, pp. 334--335. V a n Nostrand, N e w York (1955).

APPENDIX Coej~cients

r

r . = (ff z ~ - r l z ~ ) / A .

r~ = (z~-z~)/A.

r.x -- z~ z~(ri-~)/A1, r~3 = (r~ z , - # z j ) / A . r~ = (z~-zj)/A. r~3 = (z,-z~)/A. Co.~ient.

r ~ = z, z~(ff - r D I A .

r~ = z~ zAr~- r$)/Ax, A1 = X z,(r~ z~- r| z~).

P E #a = 2 ( l + p ) '

~(1-p) ~'~ = ( 1 + ~ ) ( 1 - 2 ~ ) ' E~

~, = (~1+~,), ~ = (#t-p~).

/z~ = ( l + p ) ( l - - 2 p ) ' Ooe~ient.

8

Elements of matriz P P(1, 1) : p s V t s + p , V~.. P O , 2) = ~ , V . , P ( 1 , 3) = p , ~ s ,

2ptVo

P(1, ~) = w v . ÷ 2v~ v~ P(1, 7) = ~ v . , P(1, 8) P(2, 2) P(2, 3) P(2,4) P(2, 5) P(2, 6) P(2, 7) P(2, 8) P(2, 9) P(3, 3) P(3, 4) P(3, 5) P(3, 6)

= (r~.r~)/A v

~.

=

r ,.r~(z~ - zg)/A*;

s~ = r~ r~(z~- z~)/A t, Aa = X r,(ra Z~--r~ zg).

s~ = ( r i - r D / A v

P(1, 4) = p , K + p t K + P(1, 5) --- p . ~ ,

~

= p~ Vt,, = p~ V~ + p , V~, = -p~ ~, = pi~+2ptV~, = p~+p~V~, = - p a V~+ 2pt V~, = p i V~a, -- p i ~ , = - p ~ Vt~, -- 2p. V~, -- fta V~, -------/~, V~, ----2p~ V~,

P(3, P(4, P(4, P(4, P(4, P(4, P(5, P(5, P(5, P(5, P(5, P(6, P(6, P(e, P(6, P(7, P(7, P(8, P(9,

9) = Pa Vt~, 4) = 5) = O ) = (~s + ~f**) v,, 7) = 8) = ( t ~ , + 2 w ) ~ , 5) = ~4 V~ + ~s Vlo, 6) = ( - ~ , + 2 ~ ) IT,, 7) = 8) = 9) = 6) = 7) = 8) = 9) = 7) -8) = mV~,, s) ffi 9) -- t~s Vls.

Simultaneous iteration method for flexural vibration problems

Elements of matrix Q Q(1, 1) Q(1, 4) Q(1, 7) Q(4,4) Q(4, 7) Q(7, 7)

=

= = = = =

Q(2, Q(2, Q(2, Q(5, Q(5, Q(8,

2) 5) 8) 5) 8) 8)

=

-= ----

V,, v 7, V,, Vo, Vs, Vx5,

Matrices P and Q are symmetrical.

Q(3, 3) -- Va, Q(3, 6) -- V~ Q(3, o) = Vx, Q(6, 6) --- vs, Q(6, 9) = V,, Q(9, 9) = V~b. T h e e l e m e n t s n o t m e n t i o n e d a r e zero.

Values of inteerala V = j'J'(=l dr d~, Va ---- f f ( r a) dr dz,

~ I = SS(rz') @ dz,

~ = SS(~/r) ~ ,

= ff(zr a) dr dz,

v. = fS(r' ~') dr v, = SS(~' ~') dr v. = ff(r" ~') d r V, = f~(r~ z*) dr

~, a~, ~, dz,

v,, = SS(~') dr ~ ,

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