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* TECHNICAL REPORT ARBRL-TR-02391

ASCENDING AND ASYMPTOTIC SERIES FOR

SQUARES, PRODUCTS AND

CROSS-PRODUCTS OF

MODIFIED BESSEL TECHNICAL FUNCTIONS

A. S. Elder K. L Zimmerman £. M. Wineholt

LIBRARY

February 1982

US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMAND BALLISTIC RESEARCH LABORATORY ABERDEEN PROVING GROUND, MARYLAND

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The findings in this report are not to be construed as an official Department of the Army position, unless so designated by other authorized documents.

The use jf tmde names or mmufaaturere ' namec in thin 'vport wea not jonetitute Cndoi'sement of any aormeraial product.

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REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM

1. REPORT NUMBER

Technical Report ARBRL-TR-02591

2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

«. TITLE (and Sublltle)

ASCENDING AND ASYMPTOTIC SERIES FOR SQUARES, PRODUCTS AND CROSS-PRODUCTS OF MODIFIED BESSEL FUNCTIONS

5. TYPE OF REPORT & PERIOD COVERED

Technical Report 6. PERFORMING ORG. REPORT NUMBER

7. AUTHORfa;

A. S. ELDER K. L. ZIMMERMAN E. M. WINEHOLT

8. CONTRACT OR GRANT NUMBERf*)

9. PERFORMING ORGANIZATION NAME AND ADDRESS

U. S. Army Ballistic Research Laboratory ATTN: DRDAR-BLI Aberdeen Proving Ground, MD 21005

10. PROGRAM ELEMENT, PROJECT, TASK AREA ft WORK UNIT NUMBERS

1L162618AH80 11. CONTROLLING OFFICE NAME AND ADDRESS

U. S. Army Armament Research & Development Command U. S. Army Ballistic Research Laboratory (DRDAR-BL) Aberdeen Proving Ground, MD 21005

12. REPORT DATE

February 1982 13. NUMBER OF PAGES

93 14. MONITORING AGENCY NAME ft ADDRESS^//d///er«n> Irom ContntUng OIUcej 15. SECURITY CLASS, (of thlm report)

UNCLASSIFIED 1S«. DECLASSIFI CATION/DOWN GRADING

SCHEDULE

16. DISTRIBUTION STATEMENT (of Oil* Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (ot thm abttract ttlermd In Block 20, II different horn Report)

18. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on rereree aide If neceaaary and Identify by block number)

Bessel Functions Residue Theory Asymptotic Expansions Elasticity Cross Products Fourier Integrals Thin-Walled Cylinders

/JlM ^^ <7~

20. ABSTRACT fCantfaua eta revaraa atda It nmceeeary antt fdeatlfy by block number) ■ i

Asymptotic series for squares and products of modified Bessel functions were programmed as well as the ascending and asymptotic series for the cross products These programs reduced round off error and led to precise calculation of stresses m thm-walled cylinder according to the equations of elasticity

DD , FORM JAN 73 1473 EDITIOM OF » NOV 65 IS OBSOLETE UNCLASSIFIED

SECURITY CLASSIFICATIOK OF THIS PAGE (When Data Entered)

TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS 5

LIST OF TABLES 7

I. INTRODUCTION 9

II. SOURCES OF NUMERICAL DIFFICULTIES 9

III. ASCENDING SERIES FOR CROSS PRODUCTS OF MODIFIED BESSEL FUNCTIONS 10

IV. ASYMPTOTIC EXPANSIONS OF CROSS PRODUCTS FOR MODIFIED BESSEL FUNCTIONS 16

V. ASYMPTOTIC SERIES FOR SQUARES AND PRODUCTS OF MODIFIED BESSEL FUNCTIONS 39

VI, SUMMARY AND CONCLUSIONS 86

REFERENCES 87

DISTRIBUTION LIST 89

Figure

1. Graph of R- n(s

2. Graph of R (s

3. Graph of R0 0(s

4. Graph of R, -(s

5. Graph of Rn n(s

6. Graph of R (s

7. Graph of R- n(s

8. Graph of R (s i , u

9. Graph of R0 0(s

10. Graph of R (s

11. Graph of Rn n(s

12. Graph of ^ .(s

13. Graph of L- n(s

14. Graph of L, n(s

15. Graph of L (s

16. Graph of L1 n(s

17. Graph of L- n(s

18. Graph of 1^ (s

19. Graph of L (s

20. Graph of I. 0(s

21. Graph of Ln n(s

22. Graph of L. ^fs

23. Graph of L0 0(s

24. Graph of L (s

LIST OF

and R- , (s

and R-

and R

and R

and R

and R

and R

and R

and R

and R,

and R

and R,

and L

and L,

and L

and L,

and L

and L,

and L

and L,

and L

and L,

and L

and Ln

1

L(S

L^3

L^5

l(S

l^5

l(S

l^5

l(S

(s

(s

(s

(s

(s

(s

(s

(s

(s

(s

(s

(s

ILLUSTRATIONS

Page

for Wall Ratio = 1.1 20



for Wall Ratio ■ 1.2 23


for Wall Ratio = 1,25 25



for Wall Ratio =1.4 28





for Wall Ratio = 1.1, .

for Wall Ratio =1.2. .


for Wall Ratio = 1,25 .

for Wall Ratio = 1,25 .





for Wall Ratio =1.5, ,

for Wall Ratio =1,5, .

58

59

60

61

62

63

64

65

66

67

68

LIST OF TABLES

Table , Page

1. COEFFICIENTS FOR THE ASCENDING SERIES FOR THE CROSS PRODUCTS OF MODIFIED BESSEL FUNCTIONS FOR WALL RATIOS B = I.1 AND B = 1-2 17

2. COEFFICIENTS FOR THE ASCENDING SERIES FOR THE CROSS PRODUCTS OF MODIFIED BESSEL FUNCTIONS FOR WALL RATIOS B = 1 25 AND B = i'3 ' 18

3. COEFFICIENTS FOR THE ASCENDING SERIES FOR THE CROSS PRODUCTS OF MODIFIED BESSEL FUNCTIONS FOR WALL RATIOS B = 1 4 AND B= ^ ' 19

4. COEFFICIENTS FOR ASYMPTOTIC EXPANSIONS OF CROSS PRODUCTS OF MODIFIED BESSEL FUNCTIONS FOR WALL RATIO B = 1.10 40






10. COEFFICIENTS FOR THE ASYMPTOTIC SERIES FOR THE SQUARES AND PRODUCTS OF MODIFIED BESSEL FUNCTIONS 74

11. CHARACTERISTIC ROOTS FOR THIN-WALLED CYLINDERS, WALL RATIO = 1-010 76

a. REAL VALUES 76

b. IMAGINARY VALUES 77

12. CHARACTERISTIC ROOTS FOR THIN-WALLED CYLINDERS, WALL RATIO = 1-020 78

a. REAL VALUES 78


7

LIST OF TABLES (continued)

13. CHARACTERISTIC ROOTS FOR THIN-WALLED CYLINDERS, WALL RATIO = 1.100 80

a. REAL VALUES 80



a. REAL VALUES 82



a. REAL VALUES 84


I. INTRODUCTION

In the Fourier integrals used to calculate stresses in a hollow cylinder, the variable of integration ranges from zero to infinity along the real axis; consequently the Bessel functions in the integrand must be computed for very large and very small values of the argument. Calculation of stresses in a thin-walled cylinder by residue theory requires Bessel functions of complex argument and very large modulus. Use of the Bessel function previously described1 to calculate the integrand for these extreme ranges leads to exponential overrun and cancellation of significant digits. Axial stresses due to discontinuous internal shear loading are calculated from a Fourier integral with an apparent singularity at the lower limit of integration. Evaluation of the integral by quadratures leads to the numerical difference of nearly equal numbers for points near the discontinuity of loading.

Exponential overrun is especially critical in calculating stresses for thin-walled cylinders according to the theory of elasticity. Asymptotic series have been used to calculate stresses for cylinders with wall ratios in the range 1.01 £ B < 1.25. The lower part of this range is not accessible when conventional tables or subroutines are used to calculate the integrand of the Fourier integral.

This report describes three algorithms which have been used to over- come the numerical difficulties described above. The corresponding computer programs are listed in the thick-walled cylinder code2 and will not be repeated.

II. SOURCES OF NUMERICAL DIFFICULTIES

The characteristic function for a hollow cylinder with stress-free cylindrical surfaces illustrates the numerical difficulties in evaluating determinants arising in thick-walled cylinder theory.

I0(P) K0(P) Vi^p) o^Cp)

A(s] = VP) I0(q) K0(q)

PI0CP] -PK0(P)

S^Cq) IjCq) "^(q) qi0(q) -qK0(q)

where

(1)

K.L. Zimmerman, A.S. Elder, A.K. Depue, "User's Manual for the BRL Subroutine to Calculate Bessel Functions of Integral Order and Complex Argument", Ballistic Research Laboratory Report No. ARBRLTR-02068 May 2978. CAD A056369)

2 A.S. Elder and K.L. Zimmerman, "Stresses in a Gun Tube Produced by Internal Pressure and Shear", BRL Memorandum Report No. 2495, June 1975. AD M012765.

p = sa , q = sb , a = p + (2-2v)/p , B = q + (2-2v)/q ,

and

v = Poisson's ratio , a = inner radius of cylinder , b = outer radius of cylinder.

If we evaluate A(s) by approximating the Bessel functions with Hankel asymptotic series, we find products of the type

sa -sb , -sa sb e e and e e

occurring, leading to exponential overrun when |sa| and |sb| are very large. We also find, after considerable algebra, that the first and second terms of the Hankel asymptotic series are lost by subtraction, leading to a loss of significant figures.

The characteristic function A(s) has a pole at the origin, but is analytic elsewhere in the finite part of the complex s plane. However, the Bessel functions of the second kind in Eq. (1) have logarithmic singularities at the origin. The logarithmic singularities will cause a loss of significant figures when |s| is small. The algorithm for calculating cross products of the modified Bessel functions for small |s| eliminates this difficulty.

III. ASCENDING SERIES FOR CROSS PRODUCTS OF MODIFIED BESSEL FUNCTIONS

Cross products for the modified Bessel functions are defined by the following equations^

L0j0(s) = I0(sa)K0(sb)-K0(sa)I0(sb)

L0)1(s) = I0Csa)K1(sb)+K0(sa)I1(sb)

Llj0(s) = I1(sa)K0(sb)+K1(sa)I0(sb)

Lj jCs) = I1(sa)K1(sb)-K1(sa)I1(sb) ,

3 S. Timoshenko and J.N. Goodier, Theory of Elasticity, Second Edition, McGraw-Hill Book Company, New York, 1951.

10

where s is the variable of integration in the Fourier integrals. We now show that the L1 n(s) functions are analytic in s except for a simple

pole at the origin. We have

T , ,. . (sa) (sa) fsa) 1 (sa) = 1 + -^—^y + -^—i— + -£*—iy + . . .

2 (1!) 2 (2:)z 2D(3!)

i0(sb) - i + -^+-A^l\^)l+ ... 2 (10 2 (2!) 2 (3!)

(3)

I.Csa) = ^ + i^i-+ll^!_+ ... 2 1121 2 2!3!

^(sb) = ^ + (sb) (sb)3 (sb)5

23i:2! 252!3! +

for modified Bessel functions of the first kind. In the series for modified Bessel functions of the second kind, we separate the logarithmic terms to facilitate the algebraic manipulation. For the outside radius we have^

K0(sb) = G0(sb) - M0(sb) - S0(b,s) , (4)

where

oo

V* _1 ^sb\2r All

r=l Lr■-,

M0(sb) = I0(sb) loge (j) , (6)

S (b,s) = In(sb) [y + log s] , (7)

4 V.K. Prokopov, "Equilibrium of an Elastic Axisyrmetriaally Loaded Thiak- Walled Cylinder", Priladnaya matematika i mehanika. Vol. XIII, 1949., pages 135-144. Institute of Mechanics of the Academy of Sciences, USSR. FTIO Translation No. J-2589, Aberdeen Proving Ground, Maryland, Translation dated 22 August 1967.

British Association for the Advancement of Science, Bessel Functions, Part I, Function of Orders Zero and Units, Mathematical Tables, Volume VI, University Press, Cambridge, 1937.

11

where y is Euler's constant.

Similarily we have

K^sb) = - G^sb) + M^sb) + SjCb.s] , (8)

where

1 v-Zx^r^Y'1 (^*h-*k-k).

M^sb) = I^sb) loge (y) + ^- , (10)

S^b^s) = I^sb) [y + loge s] . (11)

Similar formulas are obtained for the inside radius.

K0(sa) = G0(sa) - M()(sa) - S0(a,s) , (12)

(13)

M0(sa) = I0(sa) loge |jj , (14)

S0(a,s) = I0(sa) [Y + loge s] , (15)

and

K^sa) = - G^sa) + M^sa) + S^a^) , (16) a

where

12

SjCa.s) = I^sa) [y + loge sj . (19)

We now eliminate Bessel functions from the formulas for the cross products. We obtain

Lo,o(s) = Io(sa)[Go(:sb)-Mo(sb)]-Io(sb;)IGo(sa)"Mo(sa:i] t20)

L0,l(s) = Io(:saH-G1Csb3+M1Csb3.]+I1Csb)lGQ(sa)-M0Csa)] (21)

Ll,0(s) = Ii(saHG0(sb)-M0(sb)] + I0(sb)[-G1(sb)+M1(sb)] (22)

Lj^Cs) = I1(sa)[-G1(sb)+M1(sb)]-I1(sb)[-G1(sa)+M1(sa)] . (23)

The logarithmic singularity in s is eliminated. The only remaining singularity is a simple pole at the origin due to the functions NL (sa) and M,(sb). 1

The first few terms of the ascending series for the Li 4(s) functions can be obtained by multiplication of the appropriate series! We find

Loys) = - loge(b/a) + [- (a2+b2)loge(h/a) + (b

2-a2)Js2/4 + ... (24)

L0jl(s) = l/(sb) + [2b loge (b/a) - (b2-a2)/b]s/4 + ... (25)

Llj0(s) = l/(sa) + [-2a loge (b/a) + (b2-a2)/a]s/4 + ... (26)

13

Ll,l(s) =-(b2-a2)/(2ab)+[ab loge(b/a]-(b4-a4l/(4ab)]s2/4 + ... (27)

Calculation of literal coefficients for higher powers of s requires excessive algebraic manipulation. Numerical values are readily calculated from recursion formulas, using the first terms in the above expansions as starting values.

The recursion formulas for the coefficients are derived from a set of first-order, linear differential equations satisfied by the cross products.

L oys) = aLlj0(s) - bL0jl(s) (28)

l'0>1is) = aL^OO - bL0>0(s) - L0>1(s)/s (29)

L lj0(s) = aL0j0(s) - bL^ts) - Lj 0(s)/s (30)

L'^JCS) = aL0jl(s) - bLlj0(s) - 2L1^1(s)/s (31)

We assume

00

L0,0(s) =2J 'n ^ (32) n=0

00

Lo,i(s) =X/U" ^ C33)

a=-l

00

L1,0(S) =2^ vn ^ (34) n=-l

00

1,1(S) =Z^wn n=0

n s (35)

14

We substitute these series into the preceding differential equations, and the coefficients of each power of s are set equal to zero. The following recursion formulas are obtained:

(n+2)wn - a u^j + b v^ = 0 (36)

(n+2)vn+1 - a tn + b wn = 0 (37)

(n+2)un+1 - a wn + b tn = 0 (38)

n ^ " a Vl + b un-l = 0 (39)

The indices are shifted for convenience in computation.

- a un + b vn + fn+3lwn+l = 0 WQ3

- a tn + (n+2)vn+1 + b wn =0 (41)

b tn + (n+2)un+1 - a wn =0 (42)

(n+l)tn+1 + b un - a un =0 (43)

We find that L0j0(s) and 1^ ^s) are even functions of s, whereas Ln (s)

and Lj 0(s) are odd functions. The initial values for the recursion' formulas are obtained from Eqs. (24 - 27)

t0 = " loge(b/a) (44) u_l = Vb (45)

v.l ■ i/3 (46)

w0 = - (b -a )/(2ab) (47)

15

h ■ 0

uo - 0

vo = 0

wl = 0

and

(48)

(49)

(50)

(51)

The coefficients are listed in Tables 1, 2, and 3 for several values of b and a=l. The series converge exponentially for all values of s. The convergence is very rapid when the wall ratio b/a is close to unity, facilitating calculations for thin-walled cylinders. Eqs. 24 thru 27 can be expressed as

L0j0(s) = - loge(b/a) + R0)0(s)

L0jl(s) = l/(sb) + R0)1(s)

Llf0(S) = l/(sa) + R1)0(s)

Ll,l(s) = ' (b2-a2)/2ab + Rj jCs)

The graphs of the R function for b = 1.1, 1.2, 1.25, 1.3, 1.4, and 1.5 are shown in Figures 1 thru 12.

IV. ASYMPTOTIC EXPANSIONS OF CROSS PRODUCTS FOR MODIFIED BESSEL FUNCTIONS

We obtain asymptotic expansions of the cross products by expressing the Bessel functions of Eq. (2) in terms of Hankel asymptotic expansions. We assume

Im (s) > 0 (52)

to be consistent with formulas for calculating stresses by the theory of residues. Then6

G.N. Watson, A Treatise on the Theory of Bessel Functions3 The MaaMillan Co. New York, 1948. See Eq. (2) Page 203, and the discussion of Stokes phenomena which follows.

16

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(53) I0(sa) = [esaP0(sa) + ie"saqoCsa)] //liii"

K0(sa) = T7e_saQ0(sa)/ / 27rsa , (54)

I^sa) = [e^P^sa) - i e"3^ (sa) ] //l^il , (55)

K^sa) = Tre'Sa Q^sa) / / Zusa , (56)

where

I2 I2 32 P0(sa) = 1 + l!(8sa) + 2" + •'• . (57) L J 2:(8sa)

2 2 2 Qo(sa) = 1 " ITTsiiT + ,7^2- '•• ' (58)

Similar formulas for P0(sb), Q0(sb), P^sb), and Q1 (sb) are obtained by substituting b for a in the preceding equations.

The following formulas for the L. .(s) functions are obtained by successive substitutions. 1'-'

Lo,o(s) = [e-sCb-a)P0Csa)Q0Csb)-es^-aipQCsbiqQCsa]l / ^4ahs2 (61)

L0jl(s) = [e-s(b-a)P0(sa)Q1(sb) + es (b-a)P1 (sb)Q0(sa)'[ / ^4abs2 (62)

Ll,0(s3 = [e"S(b"a)piCsa)Q0(sb) + esCb-a)P0(sb)Q1(sa)]/^labs2 (63)

Ll,l(s) = [e'^'^PiCsalQ^sh) - esCb-a)P1(sb)qi(sa)l/^b7 (64)

32

Next, we eliminate the exponential functions from the subsequent analysis. Assume

then U._.(s) and V.^s) will also satisfy Eqs. (2). Finally we write

Ui;j(s) =e-^ E^.Cs) , (66)

and

V. .(s) ^e5^) F (s) (67)

On substituting these expressions for "^.(s) and V. . (s) we obtain the

following sets of differential equations'for E. .(s/and F. . (s). 1»J 1

)j

^0,0 = aEl,0 - bE0,l + (b-a) Eo,0 ' (68)

'Vi = a Ei,i " bEo,o + ka)-F]Eo,i .

E,i,o =aEo,o-bEi,i + [t^-j] h,

(69)

0 ' (70)

''l,! = a E0,l - ^1,0 + bb-^- |] h,l ' (71)

and

F 0,0 = a Fl,0 - bF0,l - (b-a) Fo,0 ' (72)

F o,l =aFi,i-bFo,o- [(b-a) -7]Fo,i • (73)

33

P'I.O =aFo,o-bFi,i - I»-a> -|J Fl,0 ' C?41

F,l,l =aF0,l -bFl,0- t(b-a) -|j Fljl . (75)

We assume*

Eo,o n=l

oo

'.»■"

Eo,i = E u s

n n=l

1,0 = 2 Vn S"n ' E i,U

n=l

n=l

We can prove that

n=l

F0 E, ^n+1 -n (-i) v n=l

n=l

(76)

(77)

(78)

EI,I = X) Wn s"n • (79)

Fo,o = Z C-^^'V"" ' (80)

(81)

\ , ^n+l -n ^i,o = 2-J f-^ v ' (82i

''The ooeffiaients t.3 u.3 v.} w. are new and cere not the same as in Eqs. (32)-(35). ^ ^ ^ ^

34

Fl,l = 12 t-Vn+\s'n , (83) n=l

so a separate derivation for the F (s) functions is not required. The initial terms are :L'J

tj = i/Aib , (84]

ui = 1/,^F , (85)

v1 = 1/^b , (86)

and

wl = V^ • (87)

The series expansions for E^, E^, E^^ and Ej 1 are

substituted in Eqs. (68) - (71) and the'coefficients of'each power of

s set equal to zero. We examine the coefficients of 1/s, 1/s2 and 1/s3 in detail. We find from terms in 1/s / . / , u i/b

av1 - b^ + (b-a)t1 = 0 , C88)

aw1 - bt1 + (b-a)u1 = 0 , (89)

atj - bWj + (b-a)v1 = 0 , ^

aul - bvl + (b-a)wi = 0 • (91)

35

We see these equations are consistent with the initial conditions given in Eqs. (84) - (87).

2 The coefficients of 1/s lead to the following equations:

av0 - bu2 + (b-a)t2 ■ - t, , (92)

aw2 - bt + (b-a)u2 = 0 , (93)

at9 - bw + (b-a)v9 = 0 , (94)

au2 - bv2 + (b-a)w2 = w. . (95)

We find these equations are not linearly independent, and must be supplemented by equations obtained from the coefficients of 1/s .

(96)

(97)

(98)

(99)

(b-a)t3 -bu3 +av3 = -2t2

-bt3 + (b-a)u3 +aw3 = -U2

at3 +(b-a)v3 -bw3 = -V2

au3 -bv3 + (b-a)w = 0

Consequently,

2t2 + u- + v2 = = 0 • (100)

We now solve Eqs. (92), (93), (94), and (100), obtaining

t2 = t1(b-a)/8ab , (101)

36

u2 = -^(Sa+tO/Sab ) (102)

v2 = - tjCa+Sbj/Sab , (103)

w2 = tj^CSa-S^/Sab . (104)

We proceed with the general case in a similar manner, drawing on coefficients from 1/s and 1/s for a set of linearly independent equations.

'"Cn"13tn-1 (105)

+ awn =-(11-2)^^ (106)

b wn =-(n-2)vn_1 (107)

+ (b-a)wn=-(n-3)wn_1(108)

On adding, we find

Cn"1)tn-1 + (n"2)un-l + (n-2)vn-i + (n-3;iwn-l = 0 • C109)

Now replace n by (n+1). We find

n tn + (n-l)un + (n-l)vn + (n-2)wn = 0 . (no)

To simplify, add Eq. (105) to Eq. (106], (107) and (108) in turn. We find

at - au + av + aw = x , fnn n n n n n-1 » U-i-iJ

57

(b-a)tn • -bun + av n

- b t + n (b-a)un

atn + (b-a)vn

aun - b v n

bt - bu + bv - bw = y , (112') n n n n 'n-l » u-**j

(b-a)tn - (b-a)un - (b-a)vn + (b-a)wn = zn_1 , (113)

where

Vl = " ^Vl " (n-2)Vl • (114)

yn-l = " fr-^n-l ' ^-2K.l ' (115)

Vl = - ^^n-l - ^-3K-1 ■ (116)

We now solve Eqs. (110) - (113) algebraically by Cramer's rule

Atn = [(4n-6)a2- (12n-16)ab + (4n-6)b2][n-l]tn

+ [(4n-6)(-ab+b2)][n-2]un_;l

+ [(4n-6)(a2-ab)][n-2]vn_1 + [(-4n-4)(ab)]In-3]wn_1 ,

Aun = [-(4n-6)(ab-b2)][n-ljtn_1

+ [(4n-2)a2 - (12n-12)ab + (4n-6)b2][n-2]u

+ [-(4n-4)ab][n-2]vn_1

+ [(4n-2)(a2-ab)][n-3]wn_1 ,

(117)

(118)

38

Avn = [(4n-6)(a2-ab)][n-l]tn_1 + [-(4n-4)ab][n-2]un

2 2 (119)

+ [(4n-6)a -(12n-12)ab+(4n-2)b ][n-2]v , n-1

+ [-(4n-2)(ab-b2)][n-3]wn_1 ,

Awn = [-(4n-4)ab][n-l]tn_1 + [(4n-2)(a2-ab)][n-2]un

(120)

+ [-(4n-2)(a/-ab)][n-2]v , J n-1

+[(4n-2)a2-(12n-8)ab+(4n-2)b2J[n-3]w , J J n-1 >

where

A = 16(n-l)ab (b-a), n>l, (121)

These formulas were used in an algorithm for the l^ . (s) functions

which has been incorporated in the thick-walled cylinder problem. The variable s was taken sufficiently large, so the asymptotic series could be truncated when a given term was sufficiently small, well before the smallest term of the asymptotic series. Although the formulas are cumbersome, computations were extremely rapid.

The coefficients are listed for various wall ratios in Tables 4 thru 9 and graphs of the ^ .(s) given in Eqs. 61 thru 64 for positive real

s are shown in Figures 13 thru 24.

V. ASYMPTOTIC SERIES FOR SQUARES AND PRODUCTS OF MODIFIED BESSEL FUNCTIONS

The asymptotic series for the ^.(s) functions eliminated exponential

overrun for all wall ratios of interest, but cancellation of leading terms still occurred, and led to serious computational difficulties for thm-walled cylinders when the arguments for the Bessel functions were large. New dependent variables were selected in which the differences of nearly equal numbers were obtained by analysis, thus bypassing the numerical problem. These new variables were suggested by the Laplace

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expansion of the determinant in Eq. (1); we note for instance

I0(P) OjIjCp)

^(p) p I0(p)

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2Cp)]-(2-2v)I12Cp)/p

Cancellation occurs in the bracketed term when |p| is large.

Let

Fj = TT [z I02(Z) - 2 I^Cz)] .

F2 = [z I0(z) K0(z) * zl^z) KjCz)] ,

F3 " [zKo2(z) " ^^COl/ir .

F4 = [zl0(z) KjCz) ♦ zl^z) K0(z)] .

The last equation and the Wronskian relation

(122)

(123)

(124)

(125)

(126)

I^KjC*) + I1(z)K0(z) = 1/z (127)

give

F4 = 1-

Derivatives of F^, F2, and F3 are given by

PJ = " [I02(z) - I^C?)] ,

Fj = ^ [4 I0(z) IjCz) - 2I12(z)/z] ,

(128)

(129)

(130)

Thomas Muir, "A Treatise on the Theory of Determinanta". Dover Fublioa- tvons NY, 1960. See p. 94 for expansion of a 4th order determinant.

69

F2 = [I0(Z) K0(z) - I^z) KjCz)] , (131)

F2 = 2 [I1(z)K0(z)-K1(z)I0(z) + I1(z)K1(z)/Z] , (132)

F3 = [K0Cz) + ^{W* > (133)

F3 = - [4 K0(z)K1(z)+2K^z)/z]/T7 . (134)

The squares and products of the individual Bessel functions are readily obtained from these formulas. We note that cancellation will occur in Eqs. (123), (125), (131), and (132) if |z| is large.

Let

F=A1F1+A2F2+A3F3+A4F4 ' C135D

then F satisfies the third order differential equation

3 tii 7 11 5 1

z F + 2z F - (4z%z)F + F = A. . (136)

The functions F^ F2, and F, satisfy the homogeneous equation obtained

by setting A4 = 0; F4 is a particular solution of the non-homogeneous equation.

Next, set sa = z in Eqs. (53) - (60), and define the functions

GjCz) = P^z) - p2(z) , (137)

G2(z) = P0(z) Q0(z) + P^z) Q^z) , (138)

G3CZ) = Qo(z) - Qj(z) , (139)

V2) - P0(z)Q1(z) + P1(z)Q0(z) . (140)

70

It is evident that

G4(Z) = 1 (141)

on account of Eq. (127).

We also define the functions

H^) - e2^) . (142)

H2(z) = G2(z) . (143)

H3(z) = e-2zG3(z) . (144)

Then

Fl(z) = J tHi(z) * 2i H2(z) - H3(z)] ,

F2(Z) = | [H2(z) ♦ iH (z)] ,

(145)

(146)

F3^ -2H3^ ' (147)

equation6" "^ ^ fUnCti0nS Hl' H2' a^ H3 satisfy the differential

zV" + 2Z2H" - (4z2+z) H' + H = 0 , (148)

since they are linear combinations of F, , F and F

Assume

H= eaz G (149)

71

z-'G + (3az0+2z^)G + I(3aZ-4) z^ + 4 a z -z] G

+ [(a5-4a)z3 + (2a2z2) - az + 1] G = 0

(150)

We let a = 2, 0, and -2 in turn, obtaining the following differential equations:

Z Gj + (6 z +2z )Q1 + (8z3 + 8z2-z)G^ + (8z2-2z+l)G1 = 0 , (151)

z^" * 2z2G2 + (4z3+z)G2 + G2 = 0 , (152)

z G^" + (-6z3+12z2)G3+(-8z

3-8z2-z)G3+(8z2+2z+l)G3 = 0 . (153)

We assume

G1(Z) = al,0 + al,l Z'1 + al,2Z'2 + ••• • (154)

G2(z) = a2j0 + a2jl z"1 + *2.2Z'2+'-' ' C1S5)

G3(Z) = a3,0 + a3,2 "^ + a3,2Z'2+--- • C1563

The coefficients are obtained by substituting each series in the appropriate equation, carrying out the required algebraic manipulation, and then setting the coefficients of each power of z equal to zero. The initial terms are obtained from the asymptotic expansions for Pn(z), Qr.(z), P (z), and Q1(z). We find U 0 '1

(8k-8)a1>k-(6k2-14k+6)a1)k_1+(k

3-5k2+7k-3)a1 k.2 = 0 , (157)

al,0 = 0' al,l = 2 ' al,2 "" 8 • f158)

4ka2jk-(k3-5k2+7k-3)a2>k_2 = 0 , (159)

72

a2,0 = 1' a2,l = 0' a2.2 = 'I ' (160)

The coefficients of z'1, z~5, z'S etc. in Eq. (155) are zero.

We note that

G3CZ) = G^-z) , (161)

so that

a3,k = f-13 al,k ' (162)

The derivatives of G^z), G2(z), and G3(z) are obtained by

differentiating the asymptotic series in Eq. (152), (153) and (154) term by term. Derivatives of H^z), H^z), and H3(z) are then obtained by

differentiating Eqs. (141) and (143). The detailed derivatives and resulting formulas will not be given here.

Coefficients for these asymptotic series are listed in Table 10 The early coefficients, which are the most important, do not increase rapidly; consequently these series offer an effective method of computation. Characteristic roots for a thin-walled cylinder are listed in Tables 11 thru 15 for several wall ratios. These characteristic roots could not be obtained either with the original Bessel function subroutine or with the asymptotic series for the L. .(s) functions.

^ > J

When |z| is large, it is unnecessary to calculate FjCz), F (z), and

F3(z), as the characteristic function A(s) can be expressed directly in

terms of the P and Q functions with exponential factors. Combinations involving differences of nearly equal numbers can then be expressed in terms of the G functions. All similar determinants occurring in the stress analysis of thick-walled cylinders by Fourier methods can be treated in a similar manner. First, replace the modified Bessel functions in Eq. (1) with the appropriate Hankel asymptotic series. These are given m Eqs. (55) - (60) for the argument(sa); similar formulas hold for the argument (sb). Next, eliminate recessive terms like ie_saQ (sa)//2^ii:

by manipulating columns of the determinant. We finally obtain

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85

A(s) =

esaP0(sa) e-saQ0(sa)

e^P^sa) -e'^Q^sa)

esbP0(sb) e"sbQ0(sb)

e^P^sb) -e'sbQ1(sb)

a1eSaP1(sa)

sae P0(sa)

g^^P^sb)

sbesbP0(sb)

a1e QjCsa)

-sae Q^Csa)

-sbe Qn(sb)

(163)

Reduction of this determinant by Laplace's method leads to the required result**.

VI. SUMMARY AND CONCLUSIONS

In essence, these algorithms permit precise calculations in an extended region of the complex s plane, so the Fourier integrals can be evaluated either by quadratures or by residue theory. The ascending series for the L. .(s) functions express A(s) and related determinants

in a form which is free of logarithmic singularities. Consequently these formulas are valid in the entire finite part of the complex s plane; the cut along the negative real axis, needed for the logarithmic function, is not needed. The asymptotic expansions of the L. .(s) functions eliminated

exponential overrun and also the Stokes phenomena." Finally the F^(z) functions eliminated cancellation of leading terms which occurred for large |s|.

It is clear that accurate values of the Bessel functions do not ensure accuracy in evaluating the Fourier integrals arising in thick- walled cylinder theory when |s| is very large or very small; special attention for various combinations of Bessel functions is also required. The obvious alternative, the use of multiple precision calculations, is not standard practice at BRL and consequently was not considered.

A.S. Elderj J.N. Walbert, K. Zimmerman^ "Stresses near a Disaontinuity of Loading in Thick and Thin Walled Cylinders", BRL Report in preparation.

86

REFERENCES

1. K.L. Zimmerman, A.S. Elder, A.K. Depue, "User's Manual for the BRL Subroutine to Calculate Bessel Functions of Integral Order and Complex Argument", Ballistic Research Laboratory Report No. ARBRLTR-02068, May 1978. (AD A056369)

2. A.S. Elder and K.L. Zimmerman, "Stresses in a Gun Tube Produced by Internal Pressure and Shear", BRL Memorandum Report No. 2495 June 1975. AD #A012765.

3. S. Timoshenko and J.N. Goodier, Theory of Elasticity. Second Edition McGraw-Hill Book Company, New York, 1951.

4. V.K. Prokopov, "Equilibrium of an Elastic Axisymmetrically Loaded Thick-Walled Cylinder", Priladnaya matematika i mehanika, Vol. XIII, 1949, pages 135-144. Institute of Mechanics of the Academy of Sciences, USSR. FTIO Translation No. J-2589, Aberdeen Proving Ground, Maryland, Translation dated 22 August 1967.

5. British Association for the Advancement of Science, Bessel Functions Part I, Function of Orders Zero and Units. Mathematical Tables ' Volume VI, University Press, Cambridge, 1937.

6- ?/N-.]i*tSOn' A Treatise on the Theory of Bessel Functions. The MacMillan Co. New York, 1948. See Eq. (2) Page 203, and the discussion of Stokes phenomena which follows.

7. Thomas Muir, A Treatise on the Theory of Determinant. Dover Publications NY, 1960. See p. 94 for expansion of a 4th order determinant.

8. AS Elder, J.N Walbert, K. Zimmerman, "Stresses near a Discontinuity of Loading in Thick and Thin Walled Cylinders", BRL Report in preparation. r

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