Lateral vibrations of an extending rod

April

1976

LATERAL VIBRATIONS OF .AN EXTENDmG ROD

by

Miodrag S. Jankovic

v.

16

.

_•

LATERAL VIBRATIONS OF AN EXTENDING ROD

by

Miodrag S. Jankovic

Submitted: . April,

1976

June,

1976

•

UTIAS Technical Note No. 202 eN ISSN 0082-5263

Acknowledgement

I express my sincere appreciation to Prof. P. C. Hughes for his help and guidance throughout the investigation and for the considerable time and effort which he expended on my behalf. Without his support and encourage-ment this work could not have been completed. I would also express my gradi tude to P .K. Nguyen for many helpful disçussions and comments.

This work was supported by the National Research Council of Canada under Operating Grant

8-7309.

TABLE OF CONTENTS

PAGE

List of Symbols iv

l·

INTRODUCTION 1

..

II.

EXTEND TIJG PENDULUM 1

..

2.1 Motion Equations 1

2.2 AA Approximation ₃

•

2.3 The WKBJ Approximation 4

2.4 The Bessel Funçtion Solution when

t

₌

0 ₇

2.5 Error Analysis(t

f

0) 0 8

2.6 Conc1usion 0 ₁₀

lIL EXTENDING CANTILEVERED ROD 10

3.1 Motion Equations 10

3.2 Dimensional Analysis 15

3·3 Contribution of the Second Mode 35

3.4 The WKBJ Approximation ₃₅

3.5 Conc1usion 37

rY. References ₃₈

•

..

•

B D E M N R e m n .. l.J t T p 9

LIST OF SYMBOLS

Flexural stiffness Matrix defined by (3.1.12) Error defined by (2.5.1) or (3.5.1) Matrix defined by (3.1.18) Matrix defined by (3.1.19) Defined by (3.2.4) Extension rat~ Acce1eration of gravity Leng th

Mass of pendulum bob E1ements of M

E1ements of N Time

Nondimensiona1 time Mass per unit 1ength Lateral tip displacement Deflection ang1e

I. INTRODUCTION

The dynarnics of an extending rod has been investigated in Refs. 1 & 2. The equation of lateral motion of a rod whos e length changes wi th time has been derived there and was shown to be a fourth-order linear partial diffe-rential equation, under t~e assumption ofaxial rigidity and small deflections. The exact solution has not been found yet althcu gh a similarity solution has been found for specific axial speed history. ~uasi-modal* analysis was usedto transform the partial differential equation into an infinity of linear diffe-rential equations wi th variable coefficients. One solution was obtained in the form of Bessel function for the first mode only.

One purpose of this work is to assess the contribution of the second mode to the deflection of a flexible rod during extension, ani to find an approximate solution using tre first and second mode which will be more

appropriate then the Bessel function approximation. As a prelude, the analysis of an extending pendulum is done in section Ir because of the similari ty with the 'case of an extending rod and because of its simplici ty, namely only en e degree of freedom. This is equivalent in some sense to the one-mode represen-tation in the rod case because i ts length changes in a similar manner to that of an extending rod. This allows certain techniques and concepts to be evalua-ted prior to the more challenging case which has an infini te number of degrees of freedom. There is no exact solution available for the pendulum wi th arbi-trary extension rate, so certain approximations are exarnined.

The resulting error is defined in nondimensional form, and

plotted against nondimensional parameters to ensure generality of the analysis. The approximate analysis is tren applied in section 111 to the object of real interest - the extending rod.

II. EXTENDING PENDULUM

2.1 Motion Equations

Consider a simple pendulum of mass m and varying length

t(t),

as shown in Fig. 2.1~ Tt is constrained to oscillate in a vertical x-y plane aboutthe point O. Since the angular rriomentum about 0 is IDt2_{8, and the moment} due to gravity on m is -mgtsin8, the motion equation is

d

dt -mgtsin8

Under the additional conditions that of (i) small angles, and (ii) the extention rate

t

is denoted bye, the governing equation becames

*

The te rm I quasi-modal' was not us ed in the references; however i t is in troduced here as a useful nomenclature and eill be described presently.

y

FIG. 2.1

'

.

..

8 +

2

(e)' (g)

J

8.+

J

8

=

0

Note tl1e pseudo - damping term. 2.2 An Approximation

The equation (2.1.1) does not a;ppear ta have a c10sed-form so:;Lution so we will attempt to find an approximate one in the form

8 = f(t)

cp(t)

(2.2.1)

Substïtute (2.2.1) into (2.1.1) to get

.~

+ A(

t)

~ +

B(

t)

cp

=

0 (2.2.2)

The coefficient A(t) and B(t) are defined as

(2.2.3)

The equation (2.2.2) has the same form as (2.1.1) but a subs·tantial silll]:llification can be made by choosing f( t) in s'Ucl1 a way: that

A=O

(2.2.4)·

[Setting B

=

0 leads nowhere because one gets the same equatien for f as (2.1.1)].

From (2.2.4), f(t) is faund to be

f

f(t) ::;

T

where f

=

const is re1ated to initial condition. o

The substitution of (2,2 •

.5)

into (2.2.2) gives

·~

+

n2( t)

cp

=

0 where

.

n2(t)

=

g

i

~

3

(2.2.5)

(2 •

.

2.6)

Although

(22.6)

does not in general, have a alosed-form selection, the a.bsence of the cp term [c.f.

(2 .. 1..1)]

is a considerable simplification.

In particular , for constant extension rate e

n

2

_(t)

₌₌

'

.

]-g

+ et o

Even more particularly, if e

=

0 (i:"e., constant length) the distinction between

e

add cp is not necessary, and the solution for cp is elementaryt

cp

=

cpl..Cosnt

+

cp sinnt

2-where the consta..nts

cf>1

and cp depend on ini tial conditions • This does, however, suggest the idea tha;t

(2.2.6~

might have an 'approximate solution of the form

(2.2.7),

even when n = net) .One might suppose that such all approxi mation would be especially valid for small e. Denoting approximate solutions by a carat,

("), and noting (2.2.1), (2.2,,5) and (2 .. 2.7), the approximate solution

/\.

e

=

chcosn(b) t

+ C₂sinn(t) t

:eet)

(2.2.8)

will be examined. The constants C~ and C₂ depend on ini tial condi tiom; in '

fac't,

j; e + e e

C~ == 1, _o e

C

₂ 0 0

°

0 n

0

..r

gij;

"

e(t )

"

eet )

,where n ₀

=

_{' 0}

,

e

=

and e

=

0 0

,

0 0

The solution (2.2.8) is now shown to be qui te diffQrent in phase and atnplitude from the numerical solution which will be referred to as the 'exact' one. The difference grows larger as time t increases. Also the extension rate effects thequality ofthe approximation by making i t better for smaller values of e. 'Ihe numerical integration is performed using

tre

-R~ e-Kutta me th od (RKGIL) with s,tep-size :.H = 0.003. The calculation with H/2 gives the same result to

f i ve digi ts ~d for 2H the accuracy is to three digit s D '

" Tt is obvious from Figso (2.1) and

(2.2)

that the approximat:ion

, (2.2.8)

isnot ,a particularly good one even for 'e/(n 1, ) .. 060352~ One can

", concludethat the approximation shoilld not be consid~r~d sati.sfactory for

el

(n 1, )

>

0.0352, and that a search for a better approximation shruld be

under-ak

°

o - ,

t e n . , ' ,

2.3 The WKBJApproximation

The WKBJ approximation mentioned by cunningham

3 ,

which gives an appr0ximate solution'for Eq.

(2.2.6)

is now examined. It is assumed that

((

....

FlG.U S 8 Ol I -DACT t-""-"O ... TlOIf,m. (2.2." nl I " \', ti! \ l ,-VUICT 1-N'f'IIII)ICI"""" m (!..UI ,8 i . c C.nt+C 'inOt S,.O.I roet &.-Q.I rad/a 1-0.2 ft/s j _ I.Oft !: ",,>0, " .0.005 .... FIG. Z2 6. c..codlt ... e.Û'Q, J L,OIrad 8.·-01 rodl'l • • Ot fth J_Lott 1-0.0'51: " .. 0.005 -Oi § 8 AG. 2.~ 8 8

..

-a' FIG. 2.4 DAICT ..., nc ..., AIIfRI)X''''''''''' 9,8 i.CCOlht.~·t s.,QJ rad 8.--0.1 rad/, . . . 02 ft" 1.. LO ft &,."""" H" 0.003 1DCACT""'~""'~TIOtt

s.c .

.:.;

+C,aln; O"l g.-Q1rod 8.--OJ rad,. • -Ql ft/, t.· .. o ft &,.".030. " -0.00' -Ol

l.

-!

'" -e» F1G. 2.6

EXACT AfC) TH[ . . . , 1I'P'IIIOXr1rM'1"IOH

8,ê

, ..

c.co.~:..;tlnt /100 .. "'" é.,,-0.1 roeI/, . . . 0.2 ft/I t..lO" '!: 0.0562 H .0.00'

, I

net)

>

°

a1ways • . The WKBJ approximtte solution is

.. ,

where Cl. and

C~

are arbi trary c ons

t~s

and

t

7fJ(t)

=

Jo

neT) d'r

(2.3.1)

This approximation presupposes thatn(t) is slow1y time varying to the extent that

For constant extension rare e

f

0,

t=~ + e t

o

7fJ

=

,g

.r

g (,J] -

Jl )

e 0

ConseCluent1y, from (2.2.1), (2'.2.5) and (2.31) one gets

A

8

=

The first derivative of (2.3.5) may be expressed as

8

n

=

- [2sil1'ljJ

+

(e/nt) cos7fJ]CJ.

+

[2cos7fJ - (e/n.e) sin7fJ] C2

2t

'

n

1/ 2

where it has been observed that

n

= -

=

_-21

en 7fJ=n

.

Using the ini tial condi tions

ê(o)

=

e

o ~

.

8(0) ~

e

o (2.3·.3) (2.3.4) (2.3.5) (2.3:6) ,

..

..

the constants Cl, and C2 are easily derived from

(2.3.5)

and

(2.3

'

.6):

.

C

2

=

ion~/2

[

~: +~

(n:to) 8

0 ]

The WKBJ approximation is now shown to be an excellent one giving virtually the same results as the 'exact' ones obtained by numerical integration. The results are identical to five digits for the inertial condi tions specified

in Figs.

(2.4)

and

(2.5).

The accuracy lessens for higher values of the

exten-sion rate e, as expec ted. In view of tre condition

(2.3.3)

one would expect

the WKBJ approximation to be valid wherever

e

n/;

«

2.3 .•.•... (2.3.8)

The Fig.

(2.6)

shows

8

as a function of

8/n

which is useful,

being the functional relation of two nondimensional variab~es. For example,

at point M for which

.

8 8 ;::: 8M '

n-o :;::

(~

)

,

o M ~ _o

f

0

any combinati?n of (

8

_M,nO)

which gives

(s/nO)M

is possible. If

ë~

corresponds

to ~

0

then

2 8

M corresponds to 4~

0

and so on. The relation

e

=

8(

8/~0) is also

used on the error analysis which will be carried out in Section

(2.5).

The error

will.be defined as an integral at.a distanc~ between two points, M on the curve

(8, 8/n )

and M' _on the curve (ê ;~/n

),

taken over time to h l l infini ty,

multipl~ed by

Jn .

0

o

2.4

_{~---~~~~~~~~~~---~---~~o} The Bessel Function Solution when ~ _{- - - -}:;:: 0

. When ~ :;:: 0 the WKBJ approximation is of no use. [From (2.3.7),

the constants Cl. and

0

C2 are both zero, and from

(2.3.5),8

is zero also.]

The solution exists in the form of Bessel functions. Thus, for ~ :;:: et, (2.

ê

:

:6)

beco:rres

+ b cp:;:: 0 t

where. b :;:: gje and in Ref. ~ the solution is found as

7

where JJ. and YJ. are first-order Bessel fundions of the first and second kinds.

2.5

Error Analysis

{.eo

r

0)

There is not a unique defini tion of the error con:ani t ted by the

process of approximately the exact solution. WE: shall us e the folloVll ng

defi-nition, which roughly corresponds to an integrated error in a 'nondimensionalized

phase space' whose coordi nates are (8,

êjn) :

2 1 \ 2

+

(8-8) _{] dt}

t

-; .

Note .that the definition has been arranged so that E is already

nondimensional. One knows that E can only depend on the parameters of the

problem - namely.,

e,.e

,n. ,8 ,8 - and dimension analysis indicates that

o 0 0 0

E

=

E

.

(

rr

o 0

, 8 ,

o

~:

)

(2.5.2)

Tt is possible therefore to plot E as a function of each of these variables :

1) E

=

E

(rr) ,

with ( 8₀,

~

0 ) 0 0 0 fixed;

.

2) E = E

(8 ) ,

o with (

~o

o

rr)

fixed; o 0

3)

E=E

(~),

o

rr )

fixed.

no

0 with ( 8 0 ,

These are plotted in Figs.

(2.7), (2.8)

and

(2.9).

They show approximate ly

linear character. With respect to Fig.

(2.8),

the same plot was obtained by

independently varying e and.e as to be expected fram dimension analysis;

this seemed as a partial checf{ on the results .

The step-size used for numerical integration was H

=

0.01

because it proved to be optimal among the tested ones. The trancation error

was shown to be negligible, results being the same to four digits for H = 0.01

and

H/2.

All programs were run in double precision .. Using

(2.38)

and Fig.

(2.8)

i t was obtained that the error E was ra ughly

0.54

for e/n

J

==

Jïtl"3;

E is bigger when e increases or.e decreases. 0

o

Figure

(2.7)

shows that E is proportional to 8 and it is equal

to zero for 8

=

O. From Fig.

(2.9)

one ge'ts bigger values flor E when 8

increases. I~ is worth mentioning that the same plot is obtained by var~ng

8 and keeping

n

constant as by varying

n

and keeping 8 constant. Thus, the

0.1 E 0.. 0.. 0.< 0.0 E 0.1 0.' 0."

0.'-..

CU QJ 0. ... 0.. 0.0 0.0.0 E 0.10 ,uo. 0..10 '.00 0..40 0..., 0.10 0..00 0.01 '0. 0.0

t .

-0.0141

-t:r:

o.on. ... Cl COl +(t) + c..In+Ct} 9. tI"" H -o.Ot J...o.lrad ~.QO"'5 § • Clco.+(~~ ~nt(t) H. QOI

...

8.-QI roef ~.o.l4S §. C1Cott(tÖ:'int(t} H _0.01 0.. FICl Z.9 AG. 2.7 lO AG. 2.8 LO

...

-I. Te 0 •• • lt;L t.O

expected results are proved quantitatively and qualitatively. 2.6 Conclusians

The approximation (2.2.8) is not acceptable even for e/nt

=

0.0352.

Nevertheless, it has a very simple form and its accuracy increases for smaller

values of e/nt until i t becomes equal to the exact solution for e/nt = O. The

WKBJ approximation is an excellent one for e/nt

<

.Jï6/3 and i t is much superior

to (2.28) for all values of e/nt

r

o.

However, i t requires the integral (2.3.2)

to be found in closed form (which is possible in our case). Otherwise we have

still to 'U3 e some numerical methods which may be as laborions as solving the

basic equation (2.1.1) numerically. The Bessel function solution should be

used for

t

= O.

o

III. EXTENDING CANTILEVERED ROD

The object of this study is to treat an extending rod. The treatment of a simple pendulum made in the last section il1ustrated certain concepts ascertained the accuracy of candidate approximations.

3.1 Motion Equations

It is shown in Ref. 2 that the motion equation for a uniform

rod of mass/length p and flexural stiffness B, extending at uniform rate e is

By"" + p(y' + 2e

y'

+ e2 y")

=

0 (3.1.1)

Primes denote differentiation wi th respect to the axial coordinate x; overdots are time differentiations, and y is the small lateral deflection (Fig. 3.1). For a cantilevered rod, the appropriate poundary condtions are:

y(

0,

t)

o

y'(O,t)

By"(t,t) = 0 By"' (0,

t)

(3.1.2)

One approach to the solution of (3.1.1) will be called the

'quasi-modal' approach. In this approach, it is recalled that when e

=

0, the solution

of (3.1.1) takes the form of a series of 'modes' in which each mode shape ~.(x)

oscillates sinusoidally in time for a nonextending rod 1.

00

y(x,

t)

=

I

i=l

~.(x,t) f.(t)

1. 1. (3.1.3)

The dependence on the parameter

t

has been explicitly indicated in anticipation

of the next step, which is to apply (3.1.3) even when th.e length

t

is

time-varying - the quasi-modal approach.

The functions ~.(x,t) in (3.1.3) are defined by r

y

CP.

(x,};)

=

11ïi [CCOSZ.X - coshZ.x) - d. (sinZ.x-sinhZ.x)

n

1. "JIJ 1. 1. 1. 1. 1.

J

(3.1.4)

where

Z1.' = E. /}; and E. are the roots of the transcendental equation

1. 1.

COShE COSE + 1

=

0 (3.1.5)

Thus El.

=

1.875 •.• , E2

=

4.694, ..• and so on. by

The

a.

in (3.1.4) are defined 1.

a.

1.

=

COSE. + coshE. 1. =I-sinE. + sinh€. 1. 1.

The orthonorma1ity properties of

cp.

(x,};) are noted: 1.

These functions satisfy the ordinary differentia1 equations

cp

~'"

1.

z'?'

1.

cp.

1.

=

0 with the boundary condi tions

" CP.(O,~)

=

cp~(e,~) = cp~'(};,};) = cp~,,(};,};) = 0 1. 1. 1. 1. (3.1.6 ) (3.1.7) (3.1.8) (3.1.9)

The quasi-modal approach to (3.1.1) is to substitute (3.1.3) even t}'lQugh

}; =

};(t). It is not c1ear that tnis procedure can be rigörous1y justified, a1-though one would expect useful resu1ts for same range of values of e

>

O. When one does so, one obtains (Ref. 2):

when the motion f contains the functions f.: 1.

,I

(3.1.10)

and the matrices _D, Mand N have e1ements. (d;J'}' (m .. } and (n .. } defined as .Jo ~J ~J

t

m ..

=J

[-cp./2

+

(t-x)cp~]

[-cp./2

+

(t-x)cp'.]

dx ~J 0 ~ ~ J J =

fot

n .. ~J

cp.

~

[-CP

J

./2

+

(t-x)cp '. ]

J dx -.

Moreover , D is diagona1, ~ is symmetrie, and N is skew-synnnetric;

d ..

=

0 ~J m .. = m .. ~J J~ n .. = -n .. ~J J~

The numerical va1ues of these matrices are

M = N = 0.504995 0.527068 -0.559038 0.000000 0.654848 0,228501 0.527067... -0.559042 •.... 4.087497... 3.441512 ...•• 3.441515... 13.93843 .••••• -0.654852 ..••. -0.228555 .•.•• -0.000002 ....• -1.636973 .•... 1.636984 •...• 0.000028 .•... (3.1.12) (3.1.13 ) (3.1.15) (3.1.16) (3.1.17) (3.1.18) (3.1.19)

In view of (3.1.15) - (3.1,:"17) it is evident that the numerical va1ues of Mand

~ in (3.1.18) and (3.1.19) are accurate to at least four significant digits.

Thus it remains to solve (3.1.10), a set of 1inear ordindary

differentia1 equations with variab1e coefficients (sinee

t

=

t(t) ). They are in

practice truncated to a finite number of modes. I t remains to get the initia1 conditions f. and f. at t

=

°

from the :lnitia1 conditions y(x,O) and y(x,O). To this end/- it isJ.noted from (3.1.3) ·that

.00

y(x,o)

= \ '

_L

cp.(x,ft )

f.(o)

J. 0 J.

i=i

; 1

Multip1ying through by

cp.

(x,ft o) and integrating yie1ds

J . ft f.(O)

=

J

y(x,O) CP.(x,ft ) dx J. J 0 o ' Simi1ar1y, we ca1cula:te 00

Y

(x,t)

=

L

[cp. (x,ft _{J. OJ.} )f. + cp. (x,t) _J. f. ] _J. i=l where (x,t)

It may be shown from the form of (3.1.4) that

(3.1.20)

(3.1.21)

(3.1.22)

(3.1.23)

Next, multip1y (3.1.22) through by cp.(x, ft) and integrate over x from

°

t/o ft • J

Using the orthonormality conditions, one obtains

a:t

t = 0,

.

.

J

to

f

.(0)

=

y(x,y)

cp

.(x,

ft ) dx

J 0 J 0

(3.1.24 )

wh~re the f.(O) are a1ready avai1ab1e from (3.1.21). · This comp1etes the

'

.

The series soluticn, (3.103), must be truncated to a finite' number

of terms for practical reasons • . The severest possib1e truncation is to çn e

I me de I , . for which the differentia1 ~quation for fl. is a particular case of

(3.1.10), name1y

•

_(3.1.25)

The function y

=

Y = CPJ.fJ. is p10tted* fOT specified initial conditions and

parameters in Fig ~

(3.1).

3.2 Dimensional Analysis

Equation (3.1.10),can be stuOied mere efficient1y with the aid

of nondimensional,ized variables .Lengths wf11 be nondimensiona1i~ed by

t

(assumed

f

0), anel times by the t;i.me factor. 0

(3.2.1)

Thus a nondimens;i.o:pa1 time, T, and a nondimensional extension rate, e*, are

defined by T

=

tlt* e* = et*/t o (3.2.2) (3.2.3)

The ratio of the instanta.neous 1ength to the initia1 1ength

t

is defined as R:

o

R

=

j-

=

1 + e* (T-T

o) o

With these definitions, Eq. (3.1.10) becomes

where (.) now denotes (iifferentiai;;ion wi th respe ct to T, and

f*

=

!Jt

_o

3/

2

(3.2.4)

(3.2.6)

* Tt appears that the authors of Refs. 1 & 2 have inadvertent1y made their graph

scale ten times the true va1ue for t ~ 0.5.

YI

0.1 Ist. MODF.

p

= 108 f~ , 10 = O. ft to

=

0.5 sec

ti

(tol = I. ft 3/2 fdfol = O. ft 3/2 .

J

= 1.56x 108

JS

H.'" 10-51.8C Hp= 50H

•

r

.~-...

.c::::::+---

~.-. O • '4lII\i!\I'M/4~. "'\NVV '''\I\ {\,(\ ~ ~C7 A . C).'~ 7 " '>. ...~ ' ==--=-=-f~

,

_._.

,

.

:::::::===-'""

--=r==

,

o

0.1 (>:3- 0.4 0.5 0.6-'- 0.7 _.-0.8 Q9 LO

t

.

·-0.1 FIG, 3.20

I~

r

_q N ~ ~

.!::;

~ ~

-r

t~

LIJ C) ~

...

_x LIJ Z LIJ :J:

...

Z 0 Z

~

Z ;:: +- Z 0 U ~ ,..;

ci

LA:

CD N ~ N ~ C\! IC) N +-u N rt) <.!) ~

..

As to the nondimensional form of

cp.

,

define

1. cp~ = .,{}; _{o CPi} 1. ~ = xl}; 0 (3.2.8) As mentioned earlier,

~.

-

-

(

e* ) _{(cp~ + 2 ~}

_cp

_{~ * )}

2R

. 1. 1. 1. (3.2.10)

where (') denotes differ·entiation with respect to ~.

The one-mode approximation to the solution, given dimensionally by (3.2.25) is

(3.2.11)

while thetwo-mode approximation written ou~ is scalar form is

(3.2.12)

(3.2.13)

Recall the elements m .. and n .. were given in Eqs. (3.1.18) and (3.1.19).

l.J l.J

A numerical study of (3.2.11) - (3.2.13) is now undertaken to assess the contribution of the second mode to the solutian. As a measure of the size of the solution, we use the tip deflection, y($, t). As a step in

this direction, one can ask; if either of the first two modes is excited what is the suppliëd exci tation at. the other via the coupling terms in

(3.2.12) and (3.2.13)?

To answer that question, the initial conditions of each of them are confirmed to zero, in turn. If the second mode is not excited the following

two cases exis"t. * (0.5) * (0.5) fl. :::: 1. fl.

=

O.

* _(0.5) * _(0.5) f 2

=

O.

_{f 2}

₌

O.

'* _{fl. (0.5)}

₌

_o.

(3.2.14) _{fi (0.5)} .*-,

₌

_1- (3.2.15) .* .*, f 2 (0.5)

=

o.

f

₂

(0.5)

=

o.

*

.

where fl. (0.5)

=

1 in (3.2.14) and f*(0.5)

=

1. in (3.2.15) are set up for convenience. The ini"tial value for.T

=

0.5 is chosen because smaller values require much smaller s"tep-size and coRsequentlY "the compu"tation cost increases. If T

=

0 instability in integration occurs in some cases. Since'the

differen-tial 0 equations are linear, any case in which the fir st mode is exci ted may be writrter). as a superposition at (3.2.14) and (3.2.15). Similarly, the infl.'uence of the second mode on the first one was assessed by using "tb:! following initial condi tions • * * _(0.5) fl. (0.5)

=

O.

fl.

=

O.

* * _(0,5) f 2: (0.5)

₌

1- _{f 2}

=

O.

.* (3.2.16) .* (3.2:16) fl (0.5) ;::::

O.

_{fl. (0.5)}

=

o.

'* (0.5) ;* (0.5) f 2

=

O.

_{f 2}

=

1.

.

*

.

Tip deflection due to first mode,

yt,

and due to second mode, Y2 are plotted for the interval 0.5

<

T

<

20. The upper bound in,this ,ihterval is proved to be

large enough for the purposes of "the anlaysis'.

extension rate:

The Eqs. (3.2.12) and (3.2.13) are now examined for the following

1) e* = 10-4, which is the practically interesting case;'

2) e*

=

1, which gives a very'highvalue of exteneion rate for ordinary values of ~/p iof the order of 10 ft/s for ~ 0 = 1 and B/ P. .... 10 ff;t~

I

rP

~)

.

,

3) e*

=

100" which is of academie in"terest only.

For the

ers

satelli te, the \vfue of

BI

p is

~bout

106 f.t/ s',' S,o, that

a nondimensional extension rate of e*

=

10-, one gets from (3.2.3) e

=

0.1/~ .

ff,/sec. In order to obtain smoo"ther graphs for 0.5 ::: T

<

20 the step-~ize 0

H

= 0.003 was chosen. For the initial condi"tions (3.2.l4) and e*

= 10-

4 the

"tip deflections shown in Fig. (3.3) - ~3.61 are ohtained. The deflection l1as oscillatory character with frequency Ol. (Ol. will be defined precisely later) and almost constant amplitude as expected. The influence of the fir~t mode on the second one, which is observed through y~, is of the order of 10-0 and has

oscillatory character also. If there were no coupling terms, y~ wruld be zero for all';"time.

•

y; { ., ::::~.

!

f~Ct·.) -I. ;:(1'.) -0. 1 - I~ 10 .. 1' I.

!

,; 11'.1 .0. 2 model ti (T.'. 0. f; (T.o,-a.

O:f

t

\,

r

\,

1

~

\

) ,\ I \

~

\, ( \ ' ,,\

~

,

)\

{ \ l

\

k

'

,

~.

J --0.& AG. l!.3 f~ (Te) • I I;(T'/';

I

r. (1<.) • o. ? model ij ('re) • o.

'I:

{.1. . .. 10- 4 \ Te .o.~ I mode f~(T.). I. I,"(T.). O.

(

t

11 " '

li

l

11 11 11 11 • ..;..-l-+++ I I I I I I I I I I I I I I I I I I ~d~ I ~~ I I I I d~ I I I I 1 y; FIG. 3.4 Y. . .0-. {. ,

.*.

KT4 T.t-0.5 fl~(T.). I. f.a(T.). o. 11* (Te)-Q i.' «.).0

..

all, \

,I \ 1

f

f \,

r \\

,I

lr!

\

1 \

~

\\' I, \

'

I, \

~

'T ' , tD·-FlG. 3.5 FIG. 3.F {.

,

e*.10-4 1'.-0.5

'''(1'.)

-

I. '1"(1'.'-0. I,·(T.)'O ;t(T.).O .

'

~

.

Y.

* . * The phase diagram

(ii,

y~),

Fig.

(3.4)

is en ellipse but

(Y2, y~) has a rather strange shape altho~h it is bounded, Fig.

(3.6).

From

Figs.

(3.3)

and

(3.5)

it is obvious that Y2 is negligible in comparison to yf

which proves very small coupling. If those coupling terms are neglected the

following equations are obtained:

"* *2 *

fJ. + n~ fJ.

=

0

(3.2.17)

when the rotation

(:1 y

(~*

) 2 = . _mJ.J.

(3.2.18)

(:2y

_-c;Y

m22 is used.

*

*

The graphs nJ. 2 and n22 are plotted for different values at e*

in Fig.

(3.7)

and (~.8). It is obvious from these figures that there are some

values of nf2 and n22 which are negative so it shoul~ be clear that the notation

(3.2.18)

does ~ot imply positi ve values of n12 and n22 for ' all values' of T. ; t •

Both nf2 end n22 tend to zero when T --700 which shows asymptotically

non-oscillatory behaviour. The frequency n; is considerably higher than ni, as expec~

'ted.

The equations

(3.2.12)

and

(3.2.13)

are rewritten to help clarify

Figs.

(3.6)

ani

(3.10):

e* .* W*2 * - 2n1.2 R f 2 ·- ,2 f 2

(3.2.19)

e* .* w~2 * - 2n2J. R f~

,.

fJ. where

(~*)

2 ( Illl.2 + nJ.2) =

Figure

(3.6)

may then be explained as the phase diagram of

+

o

n" I IQ -IQ

..

n. '-·0 - - - -__ .~..!'ïO=4'-"

---_._----FIG. 3.7 e -=10-2.

.

.

T FIG. 3.8

1 CI ... do

~~~~1~~~~

." d.Jod _~~ ii~î .:.~':r~~!:-!:

I

0'" d d

::~~~~~~

g " i

.*

Y2

o

y, ( 0' "'10-' T.: o.~

f:

('r.). O. 2 rnodH fi(,..l-o. i~(To) -.. i;(To) -0. f~ (Tol oal , _ ;~ (t"J .. I. J

f\

A

i\ (\

0.1 _{1 L}

I

'

I 'I

"

_{1 r.} I, _,

f \,

I

'

J:

_10.,

.I

_I

i

_I

/1,

_1',

r \,

_{I '}

1

,\

_I

,I,

_20.

, ,.

-0.1 I . I 1 , , I . \ 0.1 T FIG. 314 f~ (T,!o 0 I Ir tr.) • ,:, I modi

y;

l'1G. 3.'5 \

V

\

\;

I T." 0.5

~.:

'10-'

I

1;(To)-o. 2 _ 1;(,..)-0. ;,'(To)_" +;tTo)'O. y;

y;

'0" ( 0' '-"10-4 Te" 0.5 fl·{t'.)·Q f,*(T.)-o. t,* (T.).'. f.*(T.,) .0. T

-

I

i

~~~~,n, '~f~yl ~~~~

I

I

'

P

W

'1'

I

11

~

I·

I ,

'

" "

.'

-16&' FIG. 3.16

y;

l~

'"

( 0' '*.0- 4 1'." 0.5 ft (To)-0. h*(T.). O.

',0

(To)O .. I: (To)-o· ocr· y.

y; (., .... 10- 4 T.= 0.5 ,,*(1ö.): 0. fa*(r.},",O-t.-(T.),",o. f.·CT.). I. FIG. 3.18 y; ( .

,

e-= 10-4 T." 0.5 f~ (T.)" O. 1"(,-,)=0. i,·(T.)' O. ft*{r.)"'I. T y , {

.,

Y; .~::~~. ~I· (T. ).0f,* (T.l·o. .

!

2 modes

',*

(T.).o. ; ... (T.)-I. ,; (~l '"

0

'

1'

mode f; (T.) .. I. Ol T "IJjIil.DI\I\f1i1j' ~~~\!IMf\AAM~WiIiI\MI\IIAAMI\I\Iit;:-+----+-~ _o~[1m!1n t _T , ; (TJ • 0.

I,

mode fi (Tel -I. FIG. 3.20 y;

~:'IO-'

!

T.,.. 0.5 ,; (1ö.) .. D. 2 modes ft*(T.l-O. fl-(T.,) .. o. i"(T.l. I. Y. FIG. 3.2'

.

" ;:!

H

... "" • ..;!.: /? , , "-., . "':; i a ol ....

_,

ol 11 ol ~

.

_..

.

... d cl Cf

~~!.~~l~

•

"

... cl cl ó

~i~~~~~

.

.

,

''':;

/

I

...

_.J(I.e.t:. .:.,: ï'

.:.:..-\

d~dd

~~

·

~~~~~

.

4

',:: '':: " ..;

''':; '':: .. ,::

I

;;~ll w· ..... ..:!.= '",

..

..

d ó·~ ~

I

~ I • i

:,;. '';' \ \

\

'", Cf d d ... ;-:~ill:; ..., .. .,\:-vs.:~ ~~ _"~;H ':',;'.::~ Cf IS Cf ..: ~,~~~~~ '~ '':: a • y 8,

..

""doo _s~~i]i .:Ä.i.-=~~-'o!' '':

..

_hH

.:,.;;!.: ... 9 ',;: • 1

•

i

•

I

I

r

l

;

t

9

.

..

..

; ~

..

..

'':: , / ol

\

\

\ J r" ! • !

.

"

~~~

.

~É

",;i; .. ~ ... d '1 \ \ • 1

/

/ /

y~ ( ., '-.10·" T.ao", TI' tt.,· U. t,* (~). 0.

f.-

(T.). Q 0It f.-''tol· 0. '.D (T.)-I. f,-,1'<o) -I.

',*

tr.,: I. i.-lt.) _0.

·

AAA

Ar.r'il

·

AAr

,

~

I

Qt , ,. 0 0 : .'

·

V

V

V

vV

J

\I

V

V

V

V

....

·M AG. 3.~ j: FG, 3. 57 { ., ".10,4 'f.ao.5 ,,·(r.).o. ;~:~:~:~: ft-tT.)· I. y; "'-T Y·.'f.+Y; :

..

.

(. , '-·10T._a.5 -" ''-('1".)-0. 't9 (T.).a. 1:( • .).,. i.-h.).'. j; Q' -0.& AG. 3.~' ( ., ' •• 0-4 .... O~ f,4(T.). O. ,,·(,..).0. fl·lT.)-'. ,,*(T.)·I. y:

I

~1\!\~~'1rlili

:

.

I

\

I ( '\

· ·

\ I

\

[I

1

\'

·

;

0., -Ol

I

/

r\·

Jp,~~,~

FIG. 3.51 y; ( ., ft. 10"4 "·0.5 ""{1".).o. ''-('1'.'-0. I,-tr.)-.. OR I,Dh'>_" ft (t;J -0. f,-(1OJ .0. ""h"I.,o. irDlr.) a •• flG. 5. 56 { ., ".10"4 T._O", '''(T.)Oo. ';(To)·o. ''-(T.)O', t.*tr.) a I. y' T

•

.

*

*

w:Lth the frequency

n

2 , to which is added the existing frequency w1 , (Fig. 3.11).

A similar explanation applies to Fig. (3.10).

For e*

=

1, the coupling terms in (3.2.12) and (3.2.13) have much

more influence so that the approximation (3.2.17) is not accurate for all T.

Figures (3.21) and (3.22) show a similari ty at the beginning of the inte rval as

db Figs. (3.28) and (3.29). Referring to Figs. (3.7) ~nd (3.8) one can see that

n;2

and

n~2

tend to zero much faster than for e*

=

10-4. So, asymptotically,

yf and y~ became straight lines much earlier as i·t is shown on Figs. (3.21)

-(3.39), and the oscillatory character of yf and y~ disappears. An important

conclusion is that the nonexcited mode is not negligible anymore being of the

same magnitude as the exëited one. Compare Figs. (2.21) with (2.23), (2.26)

with (2.28), (2.31) with (2.34) and (2.36) with (2.38).

The results for e*

=

100 are not of practical interest. For B/P ... l06

~~jI:!S2: the extension rate is of the order of 10

5

/t

ft/S which is very large

and far from extension rates encountered in practice. 0 The approximation

(3.2.17) are not valid anymore because of the considerable coupling influence. [Cornpare Figs. (2.41) with (2.42), (2.44) with (2.45), (2.45) with (2.46),

(2.47) with (2.48) and (2.49) with (2.50)].

3.3 Contribution of the Second Mode

The contribution of the second mode is shown to be negligible

for e* =:

~06~

.:which gi yes the real extension rate of the order 0.1/

t

0 ft/sec

for B/p ~O. f~/s.The coupling effect is very small so that the same graph

is obtained for ini tial conditions (3.2.15) and (3.3 •. 1)

* (0.5) fl.

o.

* _(0.5) f2 = O. * (3.3.1) fl. (0.5) =: _1. * _(0.5) f 2 =: _1.

*

Also,the contribution of the second mode Y2 , Fig. (3.55), is shown to be the

same for initial conditions (3.2.15) étnd (3.3.1). The amplitude at Yl. is about

five-times biggerthan the amplitude at y~. The contributim of the second iHode

.increases wi th time. However , for e* =: 1 and e* =: 100, the contribution of

the second mode cannot be neglected.

3.4 The WKBJ Approximation

The WKBJ approximation can be used for Eqs. (3.2 .• 17) and

conse-querrtly the first and second mode approximation are gi ven in explicit form:

*

y.

1. (3.4.1)

where i = 1 for the first mode, and i = 2 for the second mode. ~i is given by

=

JË.

1 e*

n.

was def.ined by (3.2.17) and

1

*

Q. dt

=

1

:it

(:~ )~c

cos (

1 1

B.

1 E. 1 arc cos(

B.

2.R

E. 1 B. 1 E. 1

The constants Cl.i and C2i are found from the initial conditions

Y*(T ) _o

=

Y

0 = (3.4.2) (3.4.2) (3.4.3)

y. was calculated over the interval 0.5

~

T

~

20, for e* =,10-

4

and gives an

1

exc~llen t accuracy having the same four significant digi ts as the I exac t'

solutions. From Figs. (3.7) and (3 .. 8) i t was concluded that the WKBJ approximation should be a good one for all·.T

>

1 when e*

>

10-2 and for 0

<

T

<.co

when

e*

<

10-2 i.e., for all T for which

nf

is slowly varying. ~

-The error ana1ysis in paragraph (.2.5) can be applie d to the a.pproximate solution (3.4.1). The error is defined as

n .

--- - - - --- - - --- - - - --- - -- -- --- ---- --- - - - . . ,

-and similar lin~ar functions of (3.5.1) can be obtained for different variables

ejn ~ , ~ and ~jn

.

0 0 0 0 0 3.5 Conclusion

It ts evident that for extension rates characteristic of actual

rod-like (e*

<

10- ) the coupling between the modal coordinates can be neglected

as an excellent approximation. Therefore, if the ini tial conditions on the rod

indicate that only the first mode is initially present [consult Eqs. (3.1."21)

and (3.1.24)], then a one-mode analysis is adequate. Similarly, even if both

modes are present the coupling between them can be neglected for e*

<

10-

4,

although the numerical integration of the coupled equations, (3.1.10), is not difficult.

Whenever, a single-mode equation is used, the WKBJ approximation to its seleçtion is an excellent one if an approximation is desired.

• 1. 2. Leeeh, C. M. Tabarrok, B. Leech, C. M. Kim, Y. I . REFERENCES

"Dynamics of Beams Under the Inf1uence of Convecting Inertia1 Forces", Ph.D. thesis, University of Toronto, Department of Mechanica1 Engineering, 1970.

"On the Dynamics of an Axia11y Moving Beam", Pergamon Press, 1974.

3·

Curmingham, W.J. "Introduction to Non1inear Ana1ysis", McGraw-Hi11, New

York, 1958.

4.

Abramavitz, M.

Stegun, A .

/'

UTIAS Teehn1eal Nate No. 202

Institute far Aeraspaee Studies, Uni versi ty of Toranto

LAXERAL VIBRATIONS OF AN EXTENDING ROD

~

42 pages 70 f1gures

Ja.nkov1c, M. S.

1. Vibratlons 2. Canti lever beams

I. Jankovie, M. S. H. UTIAS Technieal Nate No. 202

The dynomies of an extending rad has been investigated in Refs. 1 & 2. The equation of lateral motion of a rod whose length changes wlth time has been deri ved there and was shown tO' be a fourth-order linear

partlat d.1fferential equo.tion, under the assunption ofaxial. rlg1d1ty and small defiections. The exact

selutton has not been found yet a.lthough a similarity Belutton has been found ror speciflc axial speed

history. Quasi-moda.l analysls was used to transfarm the partial. dif'ferentlal equation lnto an lnflnl ty of l.inear dlfferentlal equations with varlable coeff'iclents. One selutton was obtained in the farm of'

Bessel function for the first mode only.

One purpase of this wark is ta "ssess the contribution of the seeand rode ta the defleetian of a flexible rad during extension, and ta find an spproximate solutlon uslng the first and seeond rode whieh will be more appropri"te than the Bessel function approximation. As a prelude, tbe analysis of an extending

pendulum is done in Sectian H beeause of the sim111ll'ity with the case of an extending rad and because

of its simplicity, nallEly only one degree of freedom. This is equivalent in some sense to the ene-mode

representation in the rod case because 1 ts length changes in a s1m11ar menner to that of all extending rod. This allows certain techniques end concepts te be evBl.uated prior to the more challenging case which has an lnfinite number of degrees of freedom. There ls na exact Bolution available for the JendulUlll with arbltrary &tenslon rate, sO certa1n approximatlons are examined. The resultlng error ls

defined in nondimensional ferm, and plotted against nondimenslonal parameters to ensure generali ty of

the analysis. The spproximate analys is is then applied in Seetion III ta the object of real interest -the extending rad.

Available copies O~ th is report are limited. Return this card to UTIAS, if you require a copy.

UTIAS Techn1cal Nate No. 202

Institute for Aerospace Studies, University of Toronto

LAXERAL VIBRATIONS OF AN EXTENDING ROD

~

42 pages 70 figures

Jankovic, M. 3.

1. Vibrations 2. Cantilever beams

I. Jankovic, M. S. II. Ul'IAS Technical Nate No. 202

The dynamica of an extendlng rod bas been investigated in Refs. 1 & 2. Tbe equation of lateral motien of

a rod ",hase length changes with time has been derived there and was shawn te be a fourth-order linear

part1al differential equation, under the &B8U11ptian ofaxial rigidity and small defleetians. The exact

solution has not been faund yet althaugh a .1m11arity solutian has been found for speeific axial speed

history. QUasi-modal analysis was used to transform the partial differential equation inta an infinity of linear differential equatians with variabie eoefficients. One solution was obtained in the form of

Bessel function for the first made only.

One purpose of this wark ls to asseSB the contribution of the second mde to the deflection of a flexible

rod dw'ing extension, ond to flnd an approximate solutlon using the first and sccond oode whieh will be more appropriate than the Bessel .function approximation. As a prelude, the analysis of an extending pendulum 1s dO!1e in Sectlon II because of the s1.m11arlty with the case of an extending rod and because

of its simpllcity, namaly orü.y one degree of freedom. Thls is equivalent in some sense to the one-mode

representatlon in the rod case because i ts length changes in a similar manner to that of an extending rod. This allows certain techniques and concepts to be evaluated prior te the more challenging case

which has sn infinite munber of degrees of freedom. There is no exact solution available for the

pmdUlum \dtb arbitrary extension rate, so certain approximations are examined. The resUlting error is

defined in nondimensional form, and plotted against nandimensional parameters to ensure generality of

the ar.alysis. The spproxime.te analysis is then appl1ed in Section III to the object of real interest

-the extending rad.

Available copies of this report are limited~ Return this card to UTIAS, if you require a copy.

UTIAS Teehn1cal Note No. 202

In8ti tute for Aeraspaee studie8, Uni ver8i ty of Taronto

~

LAXERAL VIBRATIONS OF AN EXTENDmG ROD

42 pages 70 f'igures

Jankovic, M. S.

1. Vibrations 2. Cantllever beoms

I. Jankovie, M. S. H. UTIAS Technieal Note No. 202

The dynamics of an extending rad has been investigated in Refs. 1 & 2. The equation of lateral motion of

a rod whose length changes with time has been derived there and was shown to be a fourth-order linear partial d.1fferential equation, Wlder the assunptlon ofaxial rigld.1ty and small defiections. The exact

solution has not been found yet al.though a sim11ari ty solution bas been found :for specific axial. speed

history. QUasi-modal analysis was ufied to transform the partial. d.1fferential equation into an in1"lnl ty

of linear d.1fferential equations w1 th varlable coefflcients. One solution was obtained in the form of

Bessel function for the first mode only.

One purpose of thls work is to assess the contribution of the seeond rode to thc deflection of e. flexible

rod during extension, and to find an spproximate solution using the first and seeond rode whieh will be more approprie.te than the Bessel functian approximation. As a prelude, the analysis of an extending

pendulum is dane in Section H because of the s1m11arity with the case of an extending rad and because

of its simplicity, naJlEly' only one degree of freedom. This is equivalent in some sense to the one-mode

representation in the rod case because its leng th changes in a sim1lar menner to that of an extending roeL This allows certain techn1ques and concepts to be evaluated prior te the more challenging case

which has all infinite number of degrees of freedom. There is no exact solution available for the JendUlum with arbitrary extension rate, so certain approx1mations are exam1ned. The resulting error is

defined in nandimensianal form, and platted against nandimensional parameters· to ensure generali ty of

the analysis. The spproxime.te analysi8 is then applied in Seetian III to the object of real interest

-the extending rad. f h' h' A f

f\vallable coples 0 t IS report are limited. Return t IS card to UTI S, i you require a copy.

UTIAS Techn1cal Note No. 202

Insti tute for Aerospace Studies, Uni versi ty of' Toronto

~

LAXERAL VIBRATIONS OF AN EXTENDING ROD

42 pages 70 figures

Jankovic, M. S.

1. Vibration. 2. Canti lever beams

I. Jankovic, M. S. II. Ul'IAS Technical Note No. 202

The dynamics o:f all extending rod has been investigated in Ref's. 1 & 2. The equation of lateral motion 0: a rod whose length changes with time has been derived there and was shown to be a fourth-order linear

part1al differential equation, under the assu.;>tion afaxial rigidity end small deflections. The exact

solution has not been found yet although a similarity solution ha. been found for specific axial speed

hl.tory.· QUasi-modal analysis was used to transform the partial differential equation inta an infinity

of linear differential equatians with variabie eoeffieients. One solution was abtained in the form of

Bessol function for the first made anly.

One purpose of this work ls to assess the contribution of the second rode to the deflection of a flexible

rod during extension, and to find all approximate solution uslng the first and second mode which will be more appropriate than the Bessel function approximation. As a prelude, the analysis of all extending pendul:wrI is done in Section II because of the similarity with the case of en extending rod and because

of lts simpl.icity, namely only one degree of' f'reedom. This is equivalent in same sense to the one-mode

representation in the rod case because its length changes in a similar marmer to that of all extending

rod. This aJ.lows certain techniques and concepts to be evaluated prior te the more challenging case

which has all lnfinite number of degrees of freedom. There is na exact solution available for the

pendulum with arbitrary extension rate, 50 certain approximations are examined. The resUltlng error is

defined in nandimensianal form, and plotted against nondimensional parameters ta ensure generality of

the analysis. The approximate analysis is then sppl1ed in Sectian III ta the object of real interest

-the extending rad.