AD-Ai42 674 LONGITUDINAL'VIBRATIONS OF RODS OF FINITE LENGTH MITH i/iRADIAL DEFORNATION(U) ARMY ARMAMENT RESEARCH ANDDEVELOPMENT CENTER WATERVLIET NY L. J J WU ET AL.UNCLASSIFIED MAR 84 ARLCB-TR-848i6 F/G 28/11 NLlUlI~Ul~III

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TECHNICAL REPORT ARLCB-TR- 840 16

LONGITUDINAL VIBRATIONSOF RODS OF FINITE LENGTHWITH RADIAL DEFORMATION

r%

JULIAN J. WU

W. H. CHEN

MAY 1984

~ US ARMY ARMAMENT RESEARCH AND DEVELOPMENT CENTERLARGE CALIBER WEAPON SYSTEMS LABORATORY

BENET WEAPONS LABORATORYWATERVIET N.Y. 12189

cl- APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED

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84 0 7 002 023

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7. AUTHORS (CONT'D)

W. H. ChenDepartment of Power Mechanical EngineeringNational Tsing Hua UniversityHsinchu, Taiwan, Republic of China

20. ABSTRACT (CONT'D)

the set of two partial differential equations is recapitulated together withappropriate boundary conditions. For vibration problems, two sets ofeigenvalue problems are formulated to satisfy the simultaneous partialdifferential equations and the homogeneous boundary conditions. Suitableparameters are defined to describe the dispersion relations. These dualeigenvalue matrix equations are then solved numerically. For an infinite rod,a dispersion relation of frequency versus wave number which contains animaginary branch has been obtained. The free vibration problem of a fixed-fixed Mindlin-Herrmann rod has been solved. The numerical values of six (6)lowest frequencies, the associated wave numbers and mode shapes are tabulatedfor three different slenderness ratios.

SECURITY CLASSIFICATION OF THIS PAGE("Wen Data Entered)

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ACKNOWLEDGEqENTS

The authors wish to thank Mr. Kuen Ting for his help in obtaining the

numerical results.

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TABLE OF CONTENTS.a .

ACKNOWLEDGEMENTS ii

LIST OF SYMBOLS tit

- INTRODUCTION I

GOVERNING EQUATIONS 2

SOLUTION FORMULATIONS FOR RODS OF INFINITE LENGTH 4

NUMERICAL RESULTS FOR RODS OF INFINITE LENGTH 8

SOLUTION FORMULATIONS OF RODS OF FINITE LENGTH 9

NUMERICAL RESULTS ON VIBRATIONS OF FINITE RODS WITH AXIAL 15AND RADIAL MOTIONS

REFERENCES 16

TABLES

I. VIBRATION SOLUTIONS TO A FINITE KINDLIN-HERRMANN ROD WITH 17FIXED-FIXED END CONDITIONS (v = 0,29, Slenderness Ratio --0.05)

II. VIBRATION SOLUTIONS TO A FINITE MINDLIN-HIERRMANN ROD WITH 18FIXED-FIXED END CONDITIONS (v = 0.29, Slenderness Ratio0.10)

III. VIBRATION SOLUTIONS TO A FINITE MINDLIN-HERRMIANN ROD WITH 19FIXED-FIXED END CONDITIONS (v = 0.29, Slenderness Ratio =

0.20) "

LIST OF ILLUSTRATIONS

I. Longitudinal wave velocities of first mode of motion according 20to Mindlin-Herrmann Rod Theory (same as curves la and l'a inFigure 1, Reference 1).

2. Effect of coupling through Poisson's Ratio, on wave velocities 21of axial and radial modes (same as Figure 2, Reference 1).

3. Frequency versus real and Imaginary "wave number" in an 22infinite Mindlin-Herrmann Rod.

i4.

LIST OF SYMBOLS

a radius of the rod

w,u displacements in z and r-directions, respectively

Lame' elastic constants

p :density

k,kI correction factors

y : wave number

: frequency

L : wave length

c : phase velocity (= w /y)

cs shear wave velocity ( /-i7P)

length of the rod

W* : dimensionless frequency

Y* : dimensionless wave number

a* :slenderness ratio

13 coefficients of the associated eigenfunctions for w

B :coefficients of the associated eigenfunctions for u

iii

~ .:..~~ ---- **..~..* ..

P.. . . . . . .. .. . . . ..I

I NTRODUCTItON

This report investigates the free vibration problem of a rod of finite

length which models the axial as well as radial modes of motion. The formula-

tion is based on a rod model of infinite length developed by Mindlin and

Herrmann in 1951 (ref 1). The differential equations, boundary conditions,

and the associated energy principle have all been developed in the original

paper as well as dispersion relations (relations between wave number and

velocity) for a rod of infinite length. Subsequently, Herrmnann presented a

solution formulation for free and forced vibration of rods of finite length

(ref 2). No numerical results were given there, however. Miklowitz has

obtained formed solutions to forced vibration problems of rods of semi-

infinite and finite length by the use of Laplace transforms (ref 3). He has

also obtained numerical results for the earlier formulations (ref 4). Never-

theless, no specific numerical data were provided for free vibrations of a

finite rod. Since the free vibration information is fundamental to all linear

and nonlinear wave propagation phenomena, our first step is to study this

aspect for a Mindlin-Herrmann rod of finite length.

* 1R. D. Mindlin and G. Herrmann, "A One-Dimensional Theory of CompressionalWaves in an Elastic Rod," Proceedings of the First U.S. National Congress ofApplied Mechanics, 1950, pp. 187-191.2.Heran "Forced Motions of Elastic Rods," Journal of Applied Mechanics,September 1954, pp. 221-224.

3 J. Miklowitz, "Travelling Compressional Waves in an Elastic Rod According tothe More Exact One-Dimensional Theory," Proceedings of the Second U.S.National Congress of Applied Mechanics, June 1954, ASME, 1955, pp. 176-186.

4J. Miklowitz, "The Propagation of Compressional Waves in a Dispersive ElasticRod," June 1957, pp. 231-239.

%

First, the governing partial differential equations and pertinent

variables of the approximate theory of the rod are recapitulated. With the

original equations, an eigenvalue problem based on a wave motion is

formulated. The characteristic equation of this eigenvalue problem is shown

to be identical to the one in Reference I in terms of phase velocity and wave

number. However, the same equation can be written in terms of dimensionless

frequency and wave number which is also allowed to assume imaginary values.

Then, the dispersion relation of frequency versus wave number reveals another

* imaginary branch similar to the one corresponding to Timoshenko beam theory

(ref 5). This new dispersion relation is shown here together with other

* . results identical to the original paper by Mindlin and Herrmann. Next,

solutions for finite rods are treated. As a first example, we take a rod with

fixed-fixed ends. In order to satisfy these end conditions, another

eigenvalue problem is established. This produces a characteristic equation

for those particular frequencies so that free vibrations under the given

boundary conditions are possible. The numerical results are presented; and

among infinite number of discrete frequencies, several of the first lowest

* are recorded. Corresponding to each, the various modes (consisting of sine,

cosine, hyperbolic sine, and cosine functions) and relative amplitudes are

also presented in the last section.

1R. 0). Mindlin and G. Herrmann, "A One-Dimensional Theory of CompressionalWaves in an Elastic Rod," Proceedings of the First U.S. National Congress ofApplied Mechanics, 1950, pp. 187-191.

5S. P. Tituoshenko, "On the Correction for Shear of the Differential Equationfor Transverse Vibrations of Prismatic Bars," Philosophical Magazine,

* Series 6, Vol. 41, 1951, pp. 744-746.

GOVERNING EQUATIONS

It will be convenient here to recapitulate the full set of equations of

the Mindlin-Herrmann rod model (ref i). The displacement equations of motions

are:

a2(X+2p)w" + 2aXj' + 2aZ = pa2w• I (I)

a2k2 u" - 8k12(A+I)u - 4ak 1

2Xw' + 4aR = pa2u

where w = w(z,t) is the displacement component in z-direction; u = u(z,t) so

that ru(z,t)/a is the displacement component in r-direction; z, r denote the

axial and radial coordinates, respectively; t, the time. A prime (') denotes

partial differential with respect to z and a dot (), the same with respect to

t. X, w are Lame' elastic constants, p is the density of the rod material,

and a is the radius. R and Z denote the radial and axial components,

respectively, of the traction on the cylindrical surface of the rod. k and k1

are correction factors. They are introduced so that the dispersion relations

from this approximate theory will better match the exact solution of an

infinite rod. Their usage is discussed fully in the original paper by Mindlin

and Herrmann (ref 1).

The stress-displacement relations are

'A' 2Pr = 2P= kl2 [2a(X+pi)u + a2Xw'] )2Pz = 2aXu + a 2(X+2w)w' (2)

4Q = k 2a 2 pu'

1R. D. Mindlin and G. Herrmann, "A One-Dimensional Theory of CompressionalWaves in an Elastic Rod," Proceedings of the First U.S. National Congress of

Applied Mechanics, 1950, pp. 187-191.

.-

where Pr, Pe, PZ9 and Qare the averaged stresses which are related to the

Cauchy stress components Orr , etc., by the following relations:

.rrrdr; Pe = a °oordr

00

(3)a a °rz 2 J

ozzrdr" Q = --- r 2dr

Equation (2) will be used to obtain solutions of finite rods if stress

*boundary conditions at the ends are given.

In the following discussion, we shall assume that the cylindrical surface

of the rod is free of stress. Hence

R =Z 0 (4)

and Eq. (1) becomes:

a2(X+2p)w ' + 2aXu' = pa2w"-.. " (1' )

a2k2u** - 8kl2(X+1)u - 4akl2Xw , pa 2u

SOLUTION FORMULATIONS FOR RODS OF INFINITE LENGTHV2

To solve Eq. (l'), we divide through by p2 and obtain:

.<.2 ~X+2 2 X "".(...)w" + -- u' = w

p pa

'.

.-'.,k2 V 8kl2+) 4kl2X .

(---)u -- ---------- u- ----- W, 1p pa2 Pa

S-LetiyZe-iwt

w(zt) = Ae"""" ((6)~iyZei

t

u(z,t) = Be

4

.:... . . . ..'* . - ' - - - . . .- . . . . . . .

S - t.. . --. . ' . .,**** . 2 . .% * .* - - . . *;

47 1.1% .7 1 V

and substitute Eq. (6) into Eq. (5). One has

+2 + w2 2XIA + iY(--)B 0

P pa

4k 1

2X

1 (7)4k,2 k2a2 k 2 (

)A + - 2 + W21B 0Pa P Pa 2

Divide Eq. (7) through by Y2:

(X+2pi) w2 2X[- -2 A + i(---)B 0P Y pya

22 -- B=O }(7')4k12x k 2 8kl 2 (X+W) 2

-i(. )A+ [------------ + -21B =P4a P .Y a Y

From Eq. (7'), one obtains the following characteristic equation:

X+2p W2 k2P 8k12(X+p) W 2 8k1

2 2

. . . .2 -= 0 ( 8 )P Y P PYa 2 p2Y2a

and the amplitude ratio

B X+2 w 2 2X.... -- [- .... + -]/[ i(--- )]A p y2 pya

4k12X k211 8k1

2(X+1) W2+= + [I ) -/[--- --------- + -] (9)

pya p pY 2 a 2 y

where Eq. (8) is exactly the same as Eq. (22) in Mindlin-Herrmann's paper of

1951 (ref 1).

It would be convenient for the present analysis to introduce some

dimensionless parameters. But first, let

IR. D. Mindlin and G. Herrmann, "A One-Dimensional Theory of CompressionalWaves in an Elastic Rod," Proceedings of the First U.S. National Congress ofApplied Mechanics, 1950, pp. 187-191.

45

44°

• " " •" . -4 " "- . .. . .. " ' h ' * ". " '. " ." " -". . . .". '° ' o . * . . o. ,. ..-.-. o, ,"* .' * *- _ " ," .' '. ' ...-.vt - .' ' -•

-. " .

L,7W

= length of the rod

L = wave length

and hence the wave number can be defined as

-Y= 2t/b (10)

Now, define

a* a/Z , y* yZ= 2'n/L

(*)2= w2 a2 /(p/p) , \* = A/p

If, from Eq. (6), we define wave (phase) velocity as

c = W/Yt (12)

andC, = (lI/p)

1/2

one has the relation

C2 W2 w2a 2 W,2

2 22 (13)Cs 2 y*2a*2_P 2 P 2a2- Y - YaP P

With Eqs. (11) and (13), Eq. (8) becomes the following when divided through by

12 /p2 : cs 2 :

'" 2 8k 12 (X*+l) w*2 8k 1

2 X* 2

(* + 2 - a*)(k2 + y*2 a*2 -1 3) * 2 - 0 (14)

Equation (9) can also be written as

B *2 2X*a =-=-i(X* + 2 - - - -( ) =

A Y* a* Y*a*

4k 12 X* 8kl 2 (X*+1) #,2

-i( -- )/(k2 + y.*2 *2 r2;.- (15)

Now, we shall solve Y*a* for any given #* in Eq. (14). Let

= J(16)

6

%,...

In terms of x and y, Eq. (14) becomes

(X*+2-xy)[k 2 + 8kl 2 (X*+l)y - xy] - 8k12X*2y = 0

Or, upon expanding,

[x2 - 8k,2(X*+)x]y 2 - [(k2+X*+2)x - 8k1 2(3X*+2)]y + k'(X*+2) = 0 (17)

Or,

Ay2 - By + C = 0 (18)

with

Ak= x2 - 8k12(X*+l)x

B = 2+A*+2)x - 8k, 2(3A*+2) (19)

C = k 2 (X*+2)

Thus, one can writeB t FB2 _ 4AC

Yl,2 - 2A (20)

Before we proceed to the actual computations, some observations on the sign of

A, B, and B2 - 4AC should be helpful.

1. A = x[x - 8k12(X*+I)J, and, since x = w*2 is always positive or zero,

for w* * 0, A will be negative, zero, or positive depending on whether x is

less, equal, or greater than 8k12(X*+l):

A = 0 if x = 3k12(X*+) 16

2.

B (k2+X*+2)x 8kl 2(3X*+2)

8k12(3 X*+2)

2+*+2)[x2

14

Hence< < 8k1

2 (3X*+2)

B - 0 if x = ----------- -10> > k2 +X*+2

7

'.°

% 7 ...,z -:. -- J , - - - .--*-'-*-.................................-....-.....-...., ..... ...-.--,. -.. -..-... ,-.--.- -...- ...-..-. ...-. ... - .- ..-..- -,

- 6 -7 - -

where the symbol - indicates "in the neighborhood when k, kj and X* are

recognized to have an order of magnitude of unity."

3. Finally, let us consider B2 - 4AC:

B2 (k2+X*+2) 2x2 + 64k14(3X*+2) 2 - 16k 1

2 (3X*+2)(k 2+X*+2)x

4AC 4[x 2 - 8k12 (X*+l)x]k 2 (X*+2) = 4k2 (X*+2)x 2 - 32k 2k1

2 (X*+l)(X*+2)x

4k2 (X*+2)x 2 - 32k 12(X* 2+3X*+2)k 2

B2 - 4AC = [k2 - (X*+2)] 2x 2 + 64k 1

4 (3X*+2) 2

- 16kl 2 x[3X*+2)(X*+2) - k 2 (2X*2+3X*+2)]

a. For x being very large compared with unity, the first term

dominates, hence, B 2 - 4AC > 0.

b. For x being very small compared with unity, the second term

dominates, and hence, B2 - 4AC > 0 again.

c. For x being in the neighborhood of unity, B2 - 4AC - 4 + 1600 - 128 =

1476 > 0.

Hence, in most likelihood, B2 - 4AC > 0 for all values of x. But we shall

examine this carefully in actual computations.

NUMERICAL RESULTS FOR RODS OF INFINITE LENGTH

The solution formulations of the previous section are now carried out for

specific numerical parameters. The results are shown in Figures 1, 2, and 3.

In Figure 1, two dispersion curves (i.e., c/c vs. a/L) reported in Reference 1

are reproduced here for the purpose of numerical verifications. As in

Reference 1, we have set v = 0.29. The curve la corresponds to k I 1 and

1 R. D. Mindlin and G. Herrmann, "A One-Dimensional Theory of CompressionalWaves in an Elastic Rod," Proceedings of the First U.S. National Congress ofApplied Mechanics, 1950, pp. 187-191.

k 0.9258 which is obtained from Eq. (25) in Reference 1; the curve l'a

corresponds to the same k, but k, is ohtained by Eq. (28) of the same

reference. In Figure 2, the coupling effect of the axial and radial modes

through Poisson's ratio given in Reference I is also reproduced here. As

* shown in Figure 2, there exist two velocities for each wave number a/L in its

full range. However, it will be realized that the full range of frequency

(w*) is not covered in Figure 2. Since w* is related to the velocity c/cs by

Eq. (13), it is necessary to solve w* for given a/L = 2nY*a* in the

characteristic equation (14). Hence, another dispersion relation is obtained

and is shown in Figure 3 for w* vs. 27y*a* curves. Evidently, in this figure,

full range of frequency is covered. It is observed that for w* smaller than a

value near 4.3, one root of Y*a* is imaginary, while the other is real. The

eigenfunctions corresponding to the imaginary root are hyperbolic sine and

cosine functions. They, together with the sine and cosine functions

associated with the real root of Y*a*, will be needed for vibration solutions

for a given set of end conditions of a finite rod.

SOLUTION FORMULATIONS OF RODS OF FINITE LENGTH

From solutions of rods of infinite length, one observes that there are

four fundamental solutions in the form of sines and cosines or hyperbolic sine

and cosine functions corresponding to any given frequency w. Now, we wish to

find a particular w for a given specific length of the rod and for a given set

of boundary conditions. It is also observed that the fundamental solutions of

1R. D. Mindlin and G. Herrmann, "A One-Dimensional Theory of CompressionalWaves in an Elastic Rod," Proceedings of the First U.S. National Congress ofApplied Mechanics, 1950, pp. 187-191.

4. 9

w and u are matched as follows:

w = WI cos Y*z*e- iwt with u uI sin Y*z*e- iwt

w = w2 sin y*z*ei ~t with u = u2 cOs y*z*ei~t

-= = (21)W :3 cosh Y*z*e

- i wt with u = u3 sinh Y*z*e - iwt

w = W4 sinh y*z*ei at with u = u4 cosh Y*z*ei~t

where

u i = aiwi , 1 = 1,2,3,4 (22)

and'"2 2X*"-'"- = -t-(x,+2) +

al +

"">' 2 =-al(23)

"",2 2X*a13 = [(X*+2) + --I -- ]/( --a-)-''' * 2a*2 2] a

a4 f_ 3

From the dispersion relations one observes that (Y*a*) 2 always has two roots

for any given w*: either (I) both of these roots are real, or, (2) one is

real and the other, imaginary.

For case (I), the general solution has the form

w(zt) = (w1 cos YI*z* + w2 sin yl*Z* + w3 cosh Y*z* + w4 sinh Y*z*)e - iwt (24)

u(zt) = CuI sin Yl*zk + u2 cos Yl*Z* + u3 sinh Y*z* + u4 cosh Y*z*)e- iwt (25)

As our first example, the boundary conditions of a rod with both ends

fixed will be considered:

w(0,t) = w(X,t) = 0• .- (26)

u(O,t) , u(Xt) = 0

'..10

g'I

Using Eq. (26) for Eqs. (24) and (25), one has, respectively

w I + 0 + W3 + 0 0

w I cos Y1* + w2 sin * + w3 COS Y3* + W4 sin 13* 0

(27)0 + u2w2 + 0 + a4 w4 = 0

alW sin 'Y* + a2w2 COS Y* + a3w 3 sin Y3* + 04w4 COS Y3* 0

with

a, -u2 -[-()*+2) + .*2 ] ( a*

(28),'..' '3 -4 = -1-c( .*+2) +--------.

Y3*2a* Y3*a*

and

W1 + 0 + W3 + 0 =0

]w Cos Tl* + W2 sin Y* + w3 cosh Y3* + w4 sinh Y3* = 0

0 + Q2w2 + 0 + a4w4= 0 (29)

= =--= -

alWl sin Y* + a2w2 COS Y* + a3w3 sinh Y3* + 44W4 CoS f3* 0

with? .::.,-wlth * 2 2 A*U3 = - u 4 = [ ( X * + 2 ) + .. . / - a )( 3 0 )

while a, and Q2 are the same as above.

L4

.-.

................................................. V ,r"

. . - .*.~-*-'.* % 7. . ... ... Z *..-. .

For a nontrivial solution of Eq. (27), one has

10 0

COS Y'1* sin j1* cos Y3 sin Y3*A1 = = 0

0 a2 0 4

I sin Ml* 2 COS 71" (3 sin y3* 4 COS Y3"

-I,_ _ (31)

sin cos " Y3- Cos Y1* sin Y3*

A 1 = 02 0 (14

2 COS YI* a3 sin Y3* - (1 sin YI* a4 COS B 3

A1 = a2 sin Y3*(a3 sin Y3 * - al sin lI*)

+ a2a4 sin yl*(a 3 sin Y3 * - a4 sin 'YI*)

- a4 sin yl*(a 3 sin 3*- al sin YI*)

a24 cos Y3*(cos Y3 - COB yl*)

Or,

A1 = (03 sin Y3* - al sin YI*)(a 2 sin Y3* - (14 sin yI*)

- 12 14(cOs Y3 " - cos ll*)2 = 0 (32)

For a nontrivial solution of Eq. (29):

" 0 1 0

Cos YI* sin Y1* cosh Y3* sinh Y3*

A2 =0 (33)0 a2 0 (14

al sin il* a2 COs YI* a3 sinh "y2* a4 cosh Y3*

12

7-I

• . . U. * * * - *. . U * % U * . **-" .* * *.*-

Or,

L2 (a3 si, h y3* - L sin l*)(a 2 sinh Y3 " a4 sin Yl*)

- a2 a4 (cosh Y3* - cos Y*) 2 = 0 (34)

With Eqs. (33) and (34), one can write

A ) AI(w*) if y*2a 2 has two real roots.a = A ( * ) = (3 5)

.2(*) if T* 2a*2 has one real and one imaginary root

Hence, for a given w*, one must first find two values for Y*2a*2 and aI, a 2 ,

a3 , and a4 (or a3 and a4 ). Then with given a* = a/k, one obtains Yl* and Y3*"

Thus one can plot (w*) with respect to w* and find "p's at which A(w*) = 0.

Finally in this section, we will write down the formulas for computing

the relative amplitudes of various mode shapes. Equation (27) for case (1)

can be written as

0 1 0 w2 1

sin Y* cos Y3 * sin Y3* w3 cos -l* w (36)

a 2 0 '14 w4 0

From Eq. (36), one has= D2 =

2 = -- wI -w=D

D3 =W3 wl a3 wl (37)

D

= D4 ,

4= W4 w , ,3wl

.4

13

-4 e

4 '- ..- '- , ,' ' . .. ,' - ,- . .,,.'',- . .',' 1, . ."-. .. - -.- . '- .' '. -. . - . ,

o - - -' : .? P ° , ,, ,'¢' • " • " "o ' "o ' ," • A °° ' " .'" " -."" -.

whe re

-'Z"--0 I0

D sin YI* cos Y3* sin Y3* = 2 sin /3* - (4 sin Yl*

a(2 0 014

-1 1 0

D2 -cos YI* cos Y3* sin Y3 " = (4 (cOs YI* - cos Y3 *)

0 -I a14

0 -i 0

D3 = sin 'Y* -cos Yl* sin Y3* = a4 sin YI* - %2 sin Y3*

a2 0 0

D4 sin Yl* COS Y3* -COS = (12(cOs 13* - cos *)

-a(2 0 0

(38)'4..

For case (2), the above equation can still be used with cos Y3* and sin

Y3* replaced by cosh Y3* and sinh Y3*" Now, for ui, I = 1,2,3,4, we observe

Eqs. (22), (23), (28), and (30), and write

Ui diwi , iwi , i = 1,2,3,4 (39)

with

I-itcLji fi 1,2,3,4 (40)

14

N%

I . . . .° . . . . . ...:. . ..

where 1 = 1 when w1 is used as the basis of normalization,

Di= -- i = 2,3,4

DI

(41)1 = 1

In Eq. (40), Q3 and a4 will be used in place of "3 and u4 if the mode shape

involves hyperbolic cosine and sine as stated in the earlier discussion.

NUMERICAL RESULTS ON VIBRATIONS OF FINITE RODS WITH AXIAL AND RADIAL MOTIONS

The formulations obtained in the previous section are now used for

specific numerical examples. In conjunction with a fixed-fixed rod (i.e., w

and u are zero at both ends), several lowest vibration frequencies and the

associated eigenfunctions have been obtained. The results are tabulated in

Tables I through III for slenderness ratio a* = a/k = 0.05, 0.10, and 0.20,

respectively. In these tables, i, I = 1,2,3,4 are the coefficients for

w(x,t) and 01 for u(x,t). They are normalized with respect to the one with

largest absolute value. For any given w*, a pair of y*a* is obtained. These

are indicated by yl*a and Y3*a, respectively, as described in the previous

section. If one of them (yl*a) is imaginary, i.e., yl*a = il*a, then Yl*a is

given and it is indicated by parentheses. The coefficients of the associated.4

eigenfunctions (hyperbolic sine and cosine functions) are also indicated by

parentheses. For example, in Table I for a* = a/k = 0.5, both the lowest

frequencies have one imaginary root each. For all higher w's, both roots of

Y*a* are real.

lI51.5

,I. ,' ,2 r JL ' ¢ 4 # . ., -. . ' ' ' ' ' ' '* - " +' ' - .. . , . .; . . £ .- . ., .. . , , . . , . . . . . . , .

REFERENCES

.1. R. D. Mindlin and G. Herrmann, "A One-Dimensional Theory of Compressional

Waves in an Elastic Rod," Proceedings of the First U.S. National Congress

of Applied Mechanics, 1950, pp. 187-191.

* 2. G. Herrmann, "Forced Motions of Elastic Rods," Journal of Applied

Mechanics, September 1954, pp. 221-224.

3. J. Miklowitz, "Travelling Compressional Waves in an Elastic Rod According

to the More Exact One-Dimensional Theory," Proceedings of the Second U.S.

National Congress of Applied Mechanics, June 1954, ASME, 1955, pp. 176-

186.

4. J. Miklowitz, "The Propagation of Compressional Waves in a Dispersive

Elastic Rod," June 1957, pp. 231-239.

5. S. P. Timoshenko, "On the Correction for Shear of the Differential

Equation for Transverse Vibrations of Prismatic Bar," Philosophical

Magazine, Series 6, Vol. 41, 1951, pp. 744-746.

-,.1

16

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-..

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