CSDL-R-1629 

NASA-CR_ 175566 

19850013383 

PLANAR DYNAMICS OF A UNIFORM BEAM 

WITH RIGID BODIES AFFIXED TO THE ENDS 

by 

Joel Storch 

Stephen Gates 

May 1983 

• ' \" .. '" I C'; 1 \11." I r ~ ,Jl 

The Charles Stark Draper Laboratory, Inc. 

Cambndge, Massachusetts 02139 

Approved for public release; 

distributIOn unlimited. 

NASA(PA), No 83-071,6/3/83 IIUI\I\I \IU 11\1 U\I\ UI\I IUI\ 11\\\ 1\\1 lUI 

NF00729 
TECHNICAL REPORT STANDARD TITLE PAGE 

1. Report No. 2. Government Accession No. 3. Recipient's Catalog No. 

4. Title and Subtitle 5. Report Date 

Planar Dynam~cs of a Un~form Beam w~th R~g~d Bod~es May 1983 

Aff~xed to the Ends 6. Performing Organization Code 

7 Author{s) 8. Performing Organization Report No. 

Joel Storch and Stephen Gates CSDL-R-1629 

9. Performing Organization Name and Address 10. Work Unit No. 

The Charles Stark Draper Laboratory, Inc. 

555 Technology Square 11. Contract or Grant No. 

Cambr~dge, MA 02139 NAS9-l6023 

13. Type of Report and Period Covered 

12. Sponsoring Agency Name and Address Techn~cal Report 

Nat~onal Aeronaut~cs and Space Adm~n~strat~on 

Johnson Space Center 

Houston, TX 77058 14. Sponsoring Agency Code 

15. Supplementary Notes 

16 Abstract 

Analyses of the planar dynamics of a uniform elastic beam subJect to a 

variety of geometric and natural boundary cond~t~ons and external exc~tat~ons 

are presented. The beams are ~nextens~ble and capable of small transverse 

bend~ng deformat~ons only. Classical beam vibration e~genvalue problems for a 

cant~lever W1th t~p mass, a cant~lever ~~th t~p body and an unconstra~ned beam 

w~th r~g~d bod~es at each end are exam~ed. The character~st~c equat~ons, 

e~genfunctions and orthogonal~ty relations for each are der~ved. The forced 

v~brat~on· of a cant~lever with t~p body subject to base accelerat~on is analyzed. The exact solut~on of the govern~ng nonhomogeneous part~al d~fferential 

equation with t~e dependent boundary cond~t~ons ~s presented and compared 

with a Rayle~gh-R~tz approximate solution. The arbitrary planar motion of an 

elast~c beam with r~gid bod~es at the ends is addressed. Equations of mot~on 

are der~ved for two modal expans~ons of the beam deflect~on. The mot~on equat~ons are cast ~n a f~rst order form suitable for numer~cal integrat~on. Selected FORTRAN programs are prov~ded. 

17. Key Words Suggested by Author 18. Distribution Statement 

Spacecraft Dynam~cs 

Flex~le Body Mechanics Approved for publ~c release; 

Beam E~genvalue Problems distr~but~on unl~mited. 

Shuttle Payloads 

Forced V~rat~on 

19. Security Classlf. (of thiS report) 20. Security Clossif. (of this page) 21. No of Pages 22. Price 

Unclass~f~ed Unclassif~ed 
CSDL-R-1629 

PLANAR DYNAMICS OF A UNIFORM BEAM 

WITH RIGID BODIES AFFIXED TO THE ENDS 

by 

Joel Storch 

Stephen Gates 

May 1983 

The Charles Stark Draper Laboratory, Inc. 

Cambr~dge, Massachusetts 02139 
Chapter 

1 

2 

TABLE OF CONTENTS 

INTRO DUCT ION •....•........•.....••.••.•.•....••...••......• 1 

EIGENVALUE PROBLEMS •.•••..•...••••••.......•.•............. 4 

2.1 Natural Frequenc~es and Mode Shapes of a 

Cant~levered Beam w~th Tip Mass................... 4 

2.2 Natural Frequenc~es and Mode Shapes of a 

Cant~levered Beam with T~p Body................... 8 

2.3 Natural Frequenc~es and Mode Shapes of an 

Unconstra~ned Beam w~th Two Tip Bod~es ..•.•...•... 15 

3 FORCED VIBRATION OF A CANTILEVER BEAM WITH 

TIP BODY SUBJECT TO BASE ACCELERATION .•....•.••.•.......••• 24 

3.1 Exact Solut~on.................................... 24 

3.2 Modal Parameter Ident~t~es •••••....•••....•••.•... 37 

3.3 Approximate Solut~on.............................. 44 

4 A MODEL OF THE SHUTTLE ORBITER WITH DEPLOYED 

PAYLOAD. . . . . . . . . . • . . • . . . . . • . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 

4.1 Mot~on Equat~ons in Terms of ClampedT~p Body E~genfunctions ........•.....••....••..... 50 

4.2 Mot~on Equat~ons in Terms of ClampedFree E~genfunct~ons............................... 65 

4.3 D~fferent~al Equat~ons of Mot~on -

F~rst Order Form................... . • . • . . . . . . . . . . . 68 

5 FORTRAN PROGRAMS........................................... 71 
TABLE OF CONTENTS (Cant.) 

Append~x 

A EIGENFUNCTION NORMALIZATION OF CLAMPED BEAM 

WITH TIP BODy.............................................. 73 

B "EIGENVALUE"-MODAL PARAMETER FORTRAN PROGRAM ............•.. 76 

C MOTION EQUATION FORTRAN PROGRAM .........•.................• 81 
LIST OF FIGURES 

F~gure 

1.1 Representat~ve cant~1evered beam-l~ke structures 

w~th t~p bodies................................. . • . • . . • • . • 2 

2.1 Clamped beam w~th t~p mass.. . . . . . . • . . . . . . . . . • . • . . . . • . . . . . . 4 

2.2 Clamped beam w~th t~p body - undeformed................... 8 

2.3 Clamped beam w~th tip body - deformed..................... 8 

2.4 Free body d~agram of t~p body ..•.......•.•..••...•........ 11 

2.5 Unconstra~ed beam w~th r~g~d bod~es attached 

to each end - undeformed.................................. 16 

2.6 Beam and t~p body 1 - deformed............................ 18 

3.1 Cantilevered beam w~th t~p body subJect to base 

accelerat~on and external force and torque ......••.••.•.•. 24 

4.1 Planar dynam~cs model of Shuttle Orb~ter with 

deployed pay load. . . . . . • . . . . . . • . . • . . . . . • . • . . • . . . . . . . . . • . . • • 51 

4.2 Free body d~agrams........................................ 53 

4.3 Deformed beam + t~p body - vector geometry ........••..•... 54 

v 
CHAPTER 1 

INTRODUCTION 

An ~portant class of Shuttle deployed payloads conS1sts of cant1-

levered beam-11ke structures w1th maSS1ve t1P bod1es. Th1s report presents analyt1cal dynam1cs models for the planar mot1on of such Orb1terpayload systems (F1gure 1.1). The models are spec1fically lntended for 

use 1n stab1l1ty stud1es of the Orb1ter f11ght control system. Well 

estab11shed eng1neer1ng approX1mat10ns are 1nvoked 1n the 1nterests of 

s1mp11c1ty and tractab1l1ty. The format 1S a stepw1se progress1on of 

mechan1cs problems each prov1ding useful results and 1ns1ght and forms a 

bas1s from wh1ch to address more complicated s1tuat10ns. 

The payload beam-like structures are taken throughout to be long 

slender uniform beams capable of transverse bend1ng only. A sequence of 

class1cal beam v1bration e1genvalue problems are exam1ned, namely: a 

cant1lever W1th t1P mass, a cant1lever w1th t1P body and an unconstra1ned 

beam w1th r1g1d bod1es at each end. In each case the frequency equations, e1genfunct10ns and orthogonality relat10ns are derived. The analytical treatment perm1ts the free vibrat10n character1zat1on 1n terms of 

a min1mum number of dimens10nless parameters. As a precursor of the 

ult1mate problem, the forced v1brat1on of a cant1levered beam with t1P 

body subJect to base acceleration 1S stud1ed. The exact solut10n to the 

nonhomogeneous part1al d1fferent1al equat10n w1th t1me dependent boundary 

cond1t10ns 1S presented. Natural "modal parameters" are def1ned and 

~portant 1dent1t1es in terms of these quant1t1es der1ved. An apprOX1-

mate solut1on uS1ng the assumed modes method proves revea11ng and serves 

1 
TETHERED SATELLITE 

,.t, 

SPACE STRUCTURE 

CONSTRUCTION ARTICLES 

RMS OPERATIONS 

ANTENNA CONCEPTS 

F1gure 1.1. Representat1ve cant11evered bearn-l1ke structures 

w1th t1P bod1es. 

2 
to check the exact solut~on. F~nallYJ the planar dyna~cs of an elast~c 

beam w~th r~g~d bod~es aff~xed to the ends ~s addressed. Th~s model of 

the Orbiter-payload system ~s capable of arb~trary mot~on ~n the plane 

accompanied by small elast~c deformat~ons. External forces and torques 

act~ng on the r~gid bod~es are accommodated. The veh~cle mot~on equat~ons are der~ved for two d~sparate modal expansions of the beam deflect~ons. The equat~ons of mot~on are cast ~n a f~rst order form s~table 

for numer~cal ~ntegrat~on. FORTRAN computer programs ~mplement~ng the 

mot~on equat~ons are g~ven. 

3 
CHAPTER 2 

EIGENVALUE PROBLEMS 

2.1 Natural Frequenc1es and Mode Shapes of a Cant1levered Beam w1th 

T1P Mass 

The part1al d1fferent1al equat10n govern1ng the transverse v1brat10n of a beam 1S g1ven by 

Th1S equat10n assumes a un1form d1str1but10n of st1ffness EI and mass per 

un1t length p. The beam is clamped at x = 0 

u(O,t) au 

ax (O,t) o for all t > 0 

at the other end of the beam: x t, there 1S a p01nt mass m

t (F1gure 2.1). 

y 

~----===---------__ -L __________________ ~x 

F1gure 2.1. Clamped beam w1th t1P mass. 

4 
and 

3 

EI ~ (i,t) 

ax3 

~wt Assume a solut~on of the form u = ~(x)e . We then obtain the follow~ng 

boundary value problem for the mode shapes ¢ 

o ~(O) :c ~'(O) o 

(A 

2 

pw lEI) ~ I I (i) = 0, ¢ I I I (i) = 

(2-1) 

The orthogona1~ty cond~t~on ~n th~s case can be arr~ved at by a cons~derat~on of the k~net~c energy express~on for the system 

T = 1 i. 2 1· 2 2 p ~ u(x,t) dx + 2 mtu(i,t) 

Expand~ng u(x,t) ~n a series of modes 

5 
the klnetlc energy can be wrltten as a quadratlc form In {qk} 

T 

00 

1 l: 

2 n=l 

l: p J <p (x) <p (x) 00 { Q, 

m=l 0 n m 

dx + mt<p (Q,)<p (Q,)}q q n m n m 

We suspect that the natural modes wlll dlagonallze thls form, 1.e. 

Q, 

p J <p (x)4> (x) dx + m <p (Q,)<p (Q,) o n m tn m 

o (2-2) 

where 4> (x) and 4> (x) are characterlstlc functl0ns correspondlng to dlsn m 

tlnct characterlstlc numbers. Indeed thlS orthogonallty condltl0n can be 

verlfled dlrectly by appeallng to the dlfferentlal equatlon and boundary 

* conditl0ns (2.1). 

If we set A 4 = a the general solution of the dlfferentlal equatlon 

lS 

<p (x) = 

(A = 0 lS not an elgenvalue) 

The boundary condltl0ns at x = 0 reqUlre 

and 

ApplYlng the boundary condltlons at x 

equatl0n 

* 

o 

o 

Q, we are lead to the frequency 

ThlS alternate method of arrlvlng at the orthogonallty condltlon was 

flrst suggested by P.C. Hughes of UTIAS. 

6 
mt cos at + cosh at + p- a(s~nh at - s~n at) mt s~ at - s~n at + p- a (cosh at - cos al) 

o s~n ai + s~nh at cos at + cosh at 

Expans10n and s~p11f1cat1on Y1elds 

m

t PI S(sin S cosh S - cos S s1nh S) = 1 + cos S cosh S (2-3) 

where we have 1ntroduced the d1mens1onless parameter S = a~. Note that 

the roots of th1s equat10n only depend upon the rat10 of the t1P mass to 

the mass of the beam. 

The natural frequencies wk are g1ven by 

(2-4) 

and the correspond1ng mode shapes ~k(x) are 

= 

(2-5) 

If we set m

t 

= 0, the orthogona11ty cond1tion (2-2) and the frequency 

equat1on, Eq. (2-3), reduce to those for an ord1nary clamped-free beam. 

In the l1m1t as m

t ~ 00 the boundary cond1t10ns at x = ~ approach those of 

a p1nned connect10n: zero d1splacement and zero moment. And 1ndeed the 

frequency equat10n, Eq. (2-3), takes the form appropr1ate to a cant1-

levered-p1nned beam (tan S = tanh S). 

7 
2.2 Natural Frequenc~es and Mode Shapes of a Cant~levered Beam w~th 

T~p Body 

F~gure 2.2 dep~cts the s~tuat~on ~n the undeformed pos~t~on. A 

un~form beam of mass dens~ty p, bend~ng st~ffness EI, and length t co~nc~des w~th the x ax~s. At the t~p (P) a r~gid body ~s attached of mass m 

and ~nert~a J (about P). The d~stance between P and the r~g~d body mass 

center ~s c and th~s d~rected l~ne segment makes an angle y w~th the 

pos~t~ve x axis. 

y 

x 

F~gure 2.2. Clamped beam w~th t~p body - undeformed. 

F~gure 2.3 shows the system ~n a deformed pos~t~on. Note that the 

t~p body is 'r~g~dly' attached at P. Denote the ~nert~al veloc~ty of P 

by v and the angular veloc~ty of a body frame (f~xed ~n the t~p body) by p 

w 

p 

~~~~~----~~----r---------~x 12,O 

F~gure 2.3. Clamped beam w~th t~p body - deformed. 

8 
v 

p 

. 

u(t,t)], w 

p = 

where u(x,t) ~s the elastic deflection of the beam and ~, ], k are the 

un~t vectors along axes x, y, z. Denot~ng the veloc~ty of the mass center of the t~p body by v~ and the vector from p to the mass center by c 

we have 

v + w x C 

p P 

observing that ~ = c cos' (y + 6 )1:" + c s~n (y + 6 )], where 6 ~s the 

p p p 

angle between the pos~t~ve x-a~s and the beam t~p tangent. Not~ng that 

16 I « 1 so that s~n 6 - 6 , cos 6 ~ 1 we arr~ve at 

p p p p 

[~(t,t) + 

a

2u v c (t,t) (cos y - 6 s~n y)J"J ~ ax at p 

a

2

u (t , t) (s~n y + 6 y)~ - c ax at cos 

p 

2 

The nonl~near term 6p a! ~t (t,t) 

= 

. 6 6 w~ll be dropped p p 

(2-6) 

The l~ear and angular accelerat~ons of the t~p body follow d~rectly by 

d~fferent~at~on of the above express~ons. It w~ll be observed that ~n 

general the mass center of the t~p body will have a component of accelerat~on along the x ax~s 

9 
Th1S ~p11es a force act1ng on the beam along the x direct1on. 

A problem ar1ses at th1s po1nt 1f we w1sh to accommodate the affect ofax1al load1ng on the transverse bend1ng of the beam S1nce non11near equat10ns would result 1n the context of the present 1nvest1gat10n. For s1mp11c1ty we therefore assume that y = 0 throughout the rema1nder of th1s sect1on. The mass center offset of the t1P body 1S 

therefore restr1cted to be d1rected axially from p and th1s results 1n no 

aX1al force be1ng app11ed to the beam. 

[

_a 2_~_ (.9, It) + c _a_

3

.....;;u"'"7'2 

at ax at 

w = 

p 

a~ 1S the accelerat10n of the mass center 

angular accelerat10n of the t1P body. 

of the t1P body and w 

p 

(2-7) 

1S the 

In order to wr1te the boundary cond1t1ons for u(x,t) at the endp01nt x = t we cons1der a free body d1agram of the t1P body. As 1nd1-

cated in F1gure 2.4 the beam exerts a force S d1rected along the y aX1S 

at p and a moment M d1rected along z. The equat10n of mot10n of the t1P 

body along the y aX1S 1S 

S [

a2 

m ~ (t,t) + C 

at 

10 

2 ax at 
F1gure 2.4. Free body d1agram of t1P body. 

The shear1ng force 1n the beam at x = t 1S g1ven by S 

Th1S g1ves one of the requ1red boundary cond1t10ns 

(t,t) - m [a2~ (t,t) + C _a_

3

...,:u'-2=-

at ax at 

o (2-8) 

The second boundary cond1t10n 1S obta1ned by wr1t1ng rotat10nal mot10n 

equat10ns for the t1P body. Let [I] denote the 1nert1a matr1x of the t1P 

body about 1tS mass center and h 1tS angular momentum both referred to 

body axes at the mass center. 

d 

dt l! . [I]w + w x [I]w -p -p -p 

Dropp1ng the non11near term 1n w 1t follows that 

p 

A dk • dt h = 

where J 1S the moment of 1nert1a of the t1P body about an aX1S parallel 

to z and pass1ng through P. 

From F1gure 2.4, the net moment about the mass center 1S 

M - c x S 

11 
M can be calculated from beam theory ~ terms of st~ffness and curvature. 

Spec~f~cally 

M 

From the work above we read~ly obta~n 

c x S 

[

13 2 

cm -1 (t,t) + c 

at 

Tak~ng the z component of the rotat~onal mot~on equat~on 

d - -h dt M - c x S 

results ~n the 2nd boundary cond~t~on at x t 

o (2-9) 

S~nce the beam ~s clamped at x 

cond~t~ons 

o we have the two geometr~c boundary 

u(O,t) = 0 and au 

ax (O,t) = 0 (2-10) 

The part~al d~fferent~al equat~on for free v~brat~on ~s of course 

o (2-11) 

12 
We now proceed to solve the part~al d~fferent~al equat~on (2-11) subJect 

to the geometr~c boundary cond~t~ons (2-10) and the natural boundary 

~wt 

cond~t~ons (2-8) and (2-9). Seek~ng solut~ons of the form e ~(x) we 

are led to the e~genvalue problem 

o 

A cp' II (R,) + m - [CP(R,) + cCP' (R,)] 

p 

A 

cp I I (i) - [me ~ (i) + J ~ I (i) ] 

p 

CP(O) ~ I (0) o 

(2-12) 

o (2-13) 

o (2-14) 

(2-15) 

Proceed~ng as ~n prev~ous sect~ons, the orthogona1~ty cond~t~on can be 

arr~ved at by cons~der~ng the k~net~c energy, T, of the system. 

1 i • 2 1 [. a

2

u T = -f u(x,t) p dx + 2 m u(i,t) (i,t)]2 + c ax at 20 

1 2 [ a

2

+- u ] 2 (J - me ) ax at (i,t) 2 

Expand~ng u(x,t) ECPk(x)qk(t) the orthogonal~ty cond~t~on ~s 

k 

i 

p f cP (x) cP (x) dx + mcp (i) cP (i) + Jcp I (R,) cP I (R,) 

o ~ J ~ J ~ J 

+ mc[~ (i)~' (i) + CP' (i)~ (i)] 0 ~ J ~ J 

13 

(2-16) 
where ¢ (x), ¢ (x) are e1genfunct1ons correspond1ng to d1st1nct e1gen- 1 ] 

values. 

Returning to the e1genvalue problem, 1t can be read1ly shown that 

A = 0 1S not an e1genvalue. The general solut1on of (2-12) can be wr1tten as 

¢ (x) = c

1 sin ax + c 2 cos ax + c 3 s1nh ax + c 4 cosh ax 

(A 

The boundary cond1t1ons (2-15) require c = -c and c = -c1 . ApplY1ng 4 2 3 

the boundary condit10ns (2-13), (2-14) and e11m1nat1ng c 3 and c 4 we arrive at the follow1ng s1multaneous equat10ns 1n c1 and c 2 

[~ m 

ca (cos t (cos a + cosh a)] c1 (s1n a - s1nh a) + Pi a - cosh a) --

a 

+ [% (cos a - cosh a) - m ca(sin t (s1n 13 - s1nh 13)] c pt 2 13 + s1nh 13) + i3 

r 

+ [-

(2-17) 

mc (s1n 13 - s1nh 13) J t 2 

p pt (s1n 13 + smh Bl]C1 S(cos 13 - cosh 13) - ~ 

a 

mc (cos a - cosh 13) J a(S1n a + s1nh 13) - t

2 

+- (cos 13 + cosh Bl]C, = p pt 

a

2 

(2-18) 

where we have 1ntroduced the symbol S = at. Sett1ng the determ1nant of 

the system (2-17), (2-18) to zero g1ves the transcendental frequency 

equat10n 

14 

0 

0 
2 4 m*(J* - m*c* )S (1 - cos S cosh S) + m*S(cos S s1nh S - S1n S cosh S) 

- 2m*c*S2 S1n S s1nh S - J*S3(s1n S cosh S + s1nh S cos S) 

+ 1 + cos S cosh S = 0 

(2-19) 

where the d1mens10n1ess t1P body parameters are def1ned by 

m* m 

pR, c* 

The natural frequenc1es are g1ven by 

and the correspond1ng e1genfunct1ons by 

+ S1n Sk - s1nh Sk] • (s1n 

+ [m*Sk(s1nh Sk - S1n Sk) + 

+ cos Sk + cosh Sk] • (cos 

J* 

(2-20) 

x x) Sk I - smh Sk I 

2 m*c*Sk(cosh Sk - cos Sk) 

x 

Sk I - cosh Sk I) 

(2-21) 

Note: the e1genfunct1ons g1ven 1n (2-21) are not norma11zed. 

2.3. Natural Frequenc1es and Mode Shapes of an Unconstra1ned Beam w1th 

Two T1P Bod1es 

F1gure 2.5 dep1cts the s1tuat10n 1n the undeformed state. A un1-

form beam of mass per un1t length P, bend1ng st1ffness El, and length R, 

15 
------------------------

F~gure 2.5. Unconstra~ned beam w~th r~g~d bod~es 

attached to each end - undeformed. 

l~es on the x ax~s (0 ~ x ~ ~). A r~g~d body of mass ml and ~nert~a J I 

about its mass center, ~s attached at x = O. The mass center offset ~s 

c I d~rected ax~ally. A second r~g~d body ~s attached at x = ~ w~th assoc~ated parameters m2 , J 2 and c 2 def~ned s~m~larly. The mot~on ~s planar 

and the beam ~s capable of bend~ng only. 

The part~al d~fferent~al equat~on for free v~brat~on ~s 

where u(x,t) ~s the transverse d~splacement of the beam from the neutral 

ax~s. The boundary cond~t~ons at x = ~ can be obta~ned d~rectly from the 

analys~s ~n sect~on 2.2. Due to subtle s~gn changes the k~nemat~cs of 

body I w~ll be der~ved here so as to obta~n the correct boundary cond~­

t~ons at x = O. Denote the ~nert~al veloc~ty of the attachment point at 

x = 0 by VI and the angular veloc~ty of a body frame f~xed ~n body 1 by WI 

. ~ 

VI = u(O,t)] 

16 
au 

ax (O,t) 

Referr1ng to Figure 2.6, the angle of 1nc11nat1on of the beam w1th respect to the x ax1S 1S 81 (t) and the vector from the attachment po1nt to 

-1 the mass center of body 1 1S c1 . If ve 1S the veloc1ty of the mass center of body 1 we can wr1te 

= 

where 

c1 -c1 (cos 8

1

1 + S1n 81 J) 

for 

18

1 1 « 1; c l - -C1 (1 + 81 J) 

Hence 

-1 [au a

2

u 

ve = at (O,t) - C (O,t)]; 1 ax at 

. where we have dropped the non11near term 8181 . By stra1ghtforward d1fferent1at10n the accelerat10n of the mass center of body 1 (~~) and the 

angular accelerat10n of body 1 (WI) are given by 

-1 [,2u a

3

u 

a (O,tl} e 

--2 (O,t) - c l 2 

at ax at 

(2-22) 

wI 

a

3

u 

= (O,t)k 

ax at

2 (2-23) 

17 
F1gure 2.6. Beam and t1P body 1 - deformed. 

We now proceed to wr1te dynam1c equ1l1br1um equat10ns for body 1. For 

th1s purpose let S be the force exerted by the beam on the t1P body and M 

the moment exerted by the beam on the t1P body. 

S (2-24) 

By the S1gn convent10ns of shear1ng forces m beams a

3

u 

S -EI 

ax

3 so we 

obta1n the following boundary cond1t1on at x = 0 

a

3 [a 2

u a

3

u EI~ (O/t) + m (O,tl] l --2 (O/t) - c l 0 (2-25) 

ax

3 at ax at2 

The z component of the t1me rate of change of the angular momentum of 

body 1 about 1tS mass center 1S 

Net moment about mass center M -

18 
From beam theory M = 

2 

EI ~ (O,t)k 

ax

2 

= 

and from (2-24) 

Equat1ng the t1IDe rate of change of the angular momentum to the net moment we arr1ve at the second boundary cond1t1on at x = 0 

(2-26) 

The boundary cond1t1ons at x = t can be obta1ned from Eqs. (2-8) and 

(2-9) (recall that 1n these equat10ns J was the moment of inert1a of the 

t1P body about the attachment p01nt) 

a

3

u [a 2

u a

3

EI (t,t) - m

u (t,t)] 0 (2-27) 2 ---2 (t,t) + c 2 = 

ax

3 at ax at2 

2 

a

2

u (t,t) + (J2 + m2c~) a

3

EI~ u (t,t) 0 

ax

2 (t It) + m2

c 2 ---2 

ax at2 

at 

(2-28) 

We now proceed to the e1genvalue problem assoc1ated w1th the beam bend1ng 

equat10n and boundary cond1t1ons (2-25) - (2-28). Assum1ng a solut10n 

1Wt ' u(x,t) = e ~(x) we arr1ve at the eigenvalue problem 

= o 

2 

pw lEI) 

19 
Boundary cond1t10ns at x 0 

4 

m1Cl 

</> I I I (0) + -p - [-</> (0) + c

1 

</> I (0) ] 0 

Boundary cond1t10ns at x ~ 

Orthogona11ty Relat10n 

The k1net1c energy, T, of the system 1S g1ven by 

T 

(2-29) 

(2-30) 

(2-31) 

(2-32) 

Expand1ng u(x,t) 1n a ser1es of e1genfunct10ns and assuming the quadrat1c . . form 1n q (t)q (t) 1S d1agonal we arr1ve at 

1 J 

20 
9., 

P f <p. (x) <p. (x) dx + m1 [<P~ (0) - C1 <P! (0)] [<P (0) - c 1 <P

J

! (0)] o ~ J .L ~ J 

+ J1<P' (0) <P' (0) + m2 [<P (9.,) + c2<P' (9.,)] [<P (9.,) + c2<P' (9.,)] ~ J ~ ~ J J 

+J2<P'(9.,)<P'(9.,) = 0 ~ J 

(2-33) 

where <p (x) and <P (x) are e~genfunct~ons correspond~ng to d~st~nct e~genl. J 

values. 

It can be read~ly shown that w = 0 ~s an e~genvalue of the problem 

correspond~ng to two l~nearly ~dependent e~genfunct~ons: r~gid body 

translat~on, and r~g~d body rotat~on. 

Nonzero Bend~ng Modes (a # 0) 

The eigenfunct~on has the form 

<P (x) 

Apply~ng the boundary condit~ons at x = 0 (2-29), (2-30) y~elds 

o 

where we have ~ntroduced the d~mens~onless t~p body parameters 

m* 1 c* 1 J* 1 

21 

and 

(2-34) 

(2-35) 

(2-36) 
Apply~ng the boundary cond~t~ons at x 1 (2-31), (2-32) y~elds 

[miS s~n S - (1 - mic2S2) cos SJA1 + [miS cos S + (1 - miciS2) s~n SJA2 

+ [miS s~nh S + (1 + m*c*e

2

) 2 2 cosh S JA3 

+ [miS cosh S + (1 + m*c*S2) s~nh SJA4 = 2 2 

(2-37) 

-[(1 + mic2S2) s~n S + (J* + m*c*2)S3 222 cos SJAl 

+ [( J 2 + mici2)S3 sin S - (1 + m*c*S2) 2 2 cos SJA2 

+ [(1 - m*c*S2) 2 2 s~nh S - (J2 + m*c* 2) S 3 

2 2 cosh SJA3 

(2-38) 

The d~ens~on1ess t1P body parameters m2, c2' J 2 are def1ned exactly as 

those for body 1. 

Equat10ns (2-35) - (2-38) const1tute a system of 4 homogeneous 

l~near equat~ons 1n Al , A2 , A3 and A4 . A nontr1v1al solut10n eX1sts 1f 

and only 1f the coeff1c1ent matr1x 1S s1ngular. 

Wr1t1ng the equat10ns 1n matr1x-vector form w1th 

A ) T 

4 

[M]A 0 

Note that 

22 

0 
The perm1ss1ble values of S are determ1ned from det[M] = 0 and are funct10ns of only the S1X d1mens10nless t1P body parameters even though there 

are n1ne system parameters. The natural frequenc1es w

k are obta1ned from 

(k 1, 2, ... ) 

-(k) - Correspond1ng to Sk' there w1ll be a vector A # 0 wh1ch 1n con]Unct1on 

w1th Eq. (2-34) g1ves the correspond1ng eigenfunct10n. 

23 
CHAPTER 3 

FORCED VIBRATION OF A CANTILEVERED BEAM WITH TIP BODY 

SUBJECT TO BASE ACCELERATION 

3.1 Exact So1ut~on 

In Sect~on 2.2 the natural frequenc~es and mode shapes for a cant~levered beam w~th t~p body were ~vest~gated. We now w~sh to exam~ne 

the more general s~tuat~on ~n wh~ch the root of the beam (x = 0) ~s not 

~nert~ally f~xed and external forces are present. As noted ~n Sect~on 2.2 we must be careful that no ax~al loads (~mpressed or ~nert~al) 

are act~ng on the beam ~f we want to use the s~mple bend~ng theory. 

Referring to F~gure 3.1 let the x-y frame be a body f~xed frame attached 

to the left end of the beam (x = 0) and denote by u(x,t) the elast~c 

deflect~on of the beam along the y ax~s. The beam has a translat~onal 

accelerat~on aO(t) at x = 0 d~rected along the y ax~s and the body frame 

F~gure 3.1. Cant~levered beam w~th t~p body subJect to base accelerat~on 

and external force and torque. 

24 
. has an 1nert1al angular accelerat10n wO(t) perpend1cular to the plane of 

mot1on (along z aX1s). Let f (t) denote the external force act1ng upon p 

the t1P body through 1tS center of mass d1rected along the y aX1s, and 

g (t) the external moment on the t1P body d1rected along the z aX1S. 

p 

The part1al d1fferent1al equat10n for the elast1c d1splacement 

u(x,t), 1S essent1ally (2-11) mod1f1ed by D'Alembert's pr1nc1ple . 

. -paO(t) - pxwO(t) (3-1) 

By def1n1t1on of the body frame we have two geometr1c boundary cond1t10ns 

at x = 0 

u(O,t) au 

ax (O,t) o for all t > 0 (3-2) 

The natural boundary cond1t1ons at x = tare obta1ned exactly as . 1n Sect10n 2.2 tak1ng 1nto account the effects of aO(t) and wO(t) 

= 

. - (mct + J)wO(t) 

(3-3) 

. = -fp(t) + mao(t) + m(t + c)wO(t) 

(3-4) 

25 
The boundary value problem (3-1) - (3-4) cons~sts of a nonhomogeneous part~al d~fferent~al equat~on and t~me dependent boundary cond~­

tions. A key ~n solv~ng th~s problem cons~sts of f~nding solut~ons of 

the assoc~ated homogeneous problem for wh~ch the base accelerat~on and . external exc~tat~on ~s zero, ~.e., ao(t) = wo(t) = gp(t) = fp(t) = o. 

Th~s problem has been solved ~n Sect~on 2.2 where the e~genfunct~ons 

¢k(x) and e~genvalues Sk were der~ved (see Eqs. (2-21) and (2-19». It 

w~ll prove more conven~ent to work w~th the d~mens~onless e~genfunct~ons 

Sk(n) where n = x/to These are essent~ally g~ven by (2-21) w~th the 

x replacement £ ~ n. The e~genvalue problem for Sk(n) can be obta~ned from 

(2-12) - (2-15) 

o 

Sk(O) OJ 

d 

dn Sk(O) = 0 

d 3 

m<Akrk (1) 

d 

n~l] - - S (n) + c* dn Sk(n) dn3 k n=l 

d 2 A [..e<s (1) + J' d 

-2 

dn 

Sk(n) 

n=l k k dn Sk (n) n~l] (3-5) 

The e~genvalue problem for Sk(n) ~s ~n terms of the dimens~onless t~p 

body parameters m*, J* and c*. The natural frequency of v~brat~on ~ ~s 

related to the e~genvalue Ak by 

26 
D3 

The orthogonal~ty cond~t~on (2-16) g~ves the natural ~nner product for 

the e~genfunct~ons S (n). We assume henceforth that the e~genfunct~ons ~ 

are orthonormal w~th respect to th~s 1nner product. * 

1 

f S (n)S (n) dn + m*S (l)S (1) o 1 J 1 J 

+ J*S' (l)S' (1) + m*c*[S (1)S' (1) + S' (l)S (1)] = 0 

1 J 1 J ~ J 1J 

(3-6) 

The boundary value problem (3-1) - (3-4) can be wr1tten 1n operator notat1on as 

. D[u(x,t)] -pao(t) - pxwo(t) 

o 

= 

where the part1al d1fferent1al operators D, D1 , D2 , D3 , D4 are 

D 

a

2 a

2 

- EI --+ mc --+ 

ax2 

at

2 

and 

* 

1; 

a

3 

J 2 ax at 

a 

ax 

a

2 

m -- - mc ---2":"" 

at

2 ax at 

The deta~ls of th1s norma11zat1on are g1ven 1n Append1x A. 

27 
. gp(t) + cfp(t) - mcaO(t) - (mct + J)wO(t) 

. -fp(t) + maO(t) + m(t + c)wo(t) 

We wr1te u(x,t) as the sum 

The approach is to choose the funct10ns h 1 , h2 such that the boundary 

cond1t10ns on v(x,t) are rendered homogeneous. In order that 

D1[v(x,t)]X=0 = 0 we must have h1(0)f1(t) + h 2 (0)f2 (t) = 0 for all t > O. 

Th1s w1ll be sat1sf1ed 1f h1(0) = 0 and h 2 (0) = O. S1m1larly 

D2 [v(x,t)]X=0 o 1f hi (0) 

.. + [mch2 (t) + Jh2(t)]f2 (t) 

The above cond1t1on w1ll be sat1sf1ed 1f 

EIh' '(t) - 1 1 

0; 

0; 

28 

o 

o 

h' , (t) 2 

o 

(for all t > 0) 

0; 

o 
the 

o 1f the follow1ng cond1t10ns are met 

h"'(2,) = 0; 1 

0; 

EIh' , '(2,) - 1 2 

In summary, we must f1nd funct10ns hI (x) 

respect1ve equat10ns 

h' , , (2,) 1 0 h2 (0) 

hI(O) 0 h2 (0) 

hi (0) 0 h"(2,) 2 

Elh' , (2,) = 1 h2 (2,) 1 

hi (2,) = 0 EIh'" (2,) 2 

hi (2,) 0 h2 (2,) 

0; 

o 

and h2 (x) wh1ch sat1sfy 

0 

0 

0 

0 

= 1 

0 

Clearly, these cond1t10ns do not determ1ne hi and h2 un1quely. For conven1ence we use f1fth degree polynomLals. ApplY1ng the above cond1t1ons 

to a general polynom1al of f1fth degree we obta1n 

1 (3 2 4 3 7 4 _ ~ x5 ) EI 2 x -Ix + 

22,2 

x 

2,3 (3-7) 

1 (- 2, 2 1 3 1 4 

-x +- x - - x + --1 x 5) EI 6 2 2Q, 62,2 

(3-8) 

29 
We must st1ll construct the funct10n v(x,t) wh1ch 1n add1t1on to 

sat1sfY1ng a part1cular d1fferent1al equat10n (cons1dered below) has to 

sat1sfy the homogeneous boundary cond1t10ns 

If we formally wr1te 

v (x,t) .Q, L: qk (t) Sk (n) 

k=l 

OJ 

the f1rst two boundary cond1t1ons are sat1sf1ed for arb1trary qk(t). 

However, S1nce the d1fferent1al operators D3 and D4 conta1n t1IDe der1vat1ves as well as spat1al der1vat1ves the last two boundary cond1t10ns can 

not be met for arb1trary qk' 1ndeed constra1nts amongst these coord1nates 

are obta1ned by demand1ng that these two boundary cond1t1ons are sat1sf1ed. Insert1ng th1s expans10n for v(x,t) 1nto the last two boundary 

cond1t10ns (assum1ng term by term d1fferent1at1on 1S va11d) and uS1ng the 

results (3-5) we obtain the follow1ng 

o (3-9) 

o (3-10) 

Recall that 

v (x,t) u (x,t) - hl (x) fl (t) - h2 (x) f2 (t) 

30 
and 

. D[u(x,t)] = -pao(t) - pxwo(t) 

Hence the part~al d~fferent~al equat~on for v(x,t) ~s 

R(x,t) ° < x < R. (3-11) 

where 

R(x,t) 

(3-12) 

(The superscr~pt "~v" ~nd~cates a fourth der~vat~ve w~th respect to the 

argument of the funct~on). 

Insert~ng the ser~es expans~on for v(x,t) and us~ng the d~fferent~al equat~on for the e~genfunct~on Sk(n); (3-11) becomes 

a We observe that R(O,t) ~ 0 and ax R(O,t) ~ ° ~ general. Therefore the 

above expans~on for R(x,t) cannot hold po~ntw~se on ° ~ x < R.. We w~ll 

~nterpret the above expans~on ~n the sense of convergence ~n the mean. 

Th~s w~ll suff~ce for our purposes s~nce we w~ll ~rnrned~ately form an 

~ntegral of both s~des w~th S (n). ~ 

1 

f R(x,t)S (n) dn 

° ~ 

(3-l3) 

31 
The orthogona11ty cond1t1on (3-6) can be rewr1tten w1th the a1d of (3-5) 

as 

1 

f 5 (n) 5

k (n) dn 

o 1 

5 I I (1) k + 5 (1) 1 

5 I I I (1) k 

Th1S result 1n cOn]unct1on w1th (3-13) and observance of the 1dent1t1es 

(3-9), (3-10) uncouples the ffioda1 coord1nate equat10ns for v(x,t). 

1 

fR(x,t)5 (n) dn 

o 1 

Th1s modal coord1nate equat10n can be expanded as 

where 

-F (t) 

1 

= 

1 

EI f 1V ( ) d f ( ) + Pi" 0 h1 (x) 51 n n 1 t 

EI 1 1V 

+ - f h2 (x) S, (n) dn f2 (t) 

pR, 0 ... 

32 

( 3-14) 

1, 2, ... ) 
It ~s poss~b1e to make a transformat~on to a new set of coord~nates Pk(t) 

such that the d~fferent~a1 equat~ons on Pk(t) do not conta~n f 1 (t) and 

f 2 (t). Indeed ~f we set 

q (t) 

~ 

(3-15) 

The modal coord~nate equat~on (3-14) transforms to 

(3-16) 

In terms of the new modal coord~nates Pk(t) the express~on for v(x,t) ~s 

v (x,t) 

The beam def1ect~on u(x,t) assumes the form 

w~th the subst~tut~on x nt Eqs. (3-7), (3-8) become 

t 

2 

(3 2 3 7 4 5) EI 2 n - 4n + 2 n - n 

1 3 

+ - n 2 

33 

1 4 - 2 n 1 n5) +6" 

(3-17) 
For future reference we also have the express~ons 

1 

Eu,2 

(84 - 120n) 

(3-18) 

1 (-12 + 20n) EU, 

Cons~der the expans~on of h1 (x) (gl(n» ~n terms of the eigenfunct~ons 

Sk (n) • 

tak~ng ~nner products w~th Sk(n) y~elds 

= 

s~nce 

= = o 

hence 

00 

= 

S~m~larly ~t can be shown that 

00 

34 
Hence the beam deflect10n 1S express1ble as a ser1es 1n the mode shapes 

u(x,t) 

00 

R, L Pk (t) 8

k (11) 

k=l 

Return1ng to the modal coordinate Eq. (3-16) we have 

1 1 1. 

- f 8 (11) dl1 • a O (t) - f 118 (n) dn Wo (t) R,O 1 0 1 

Now 

4 1 = - f (21 - 30n) 8 (n) dn 

pR, 2 0 1 

35 

(3-19) 

(3-20) 

(3-21) 
1 

f 

0 

1 

f 

0 

1 

! 

0 

and 

1 1 

-- f (-12 + 20n)S (n) dn 

p£ 0 1 

A 1 ( 1 2 1 3 1 4 1 5) -p~: -6" n + 2" n - 2" n + 6" n Sl (n) dn 

(3-22) 

We must evaluate four we1ghted 1ntegrals of the e1genfunct10n. Th1S can 

be read1ly accompl1shed through 1ntegrat1on by parts and use of the geometr1c boundary cond1t10ns. 

2 (n) dn l:...-S"'(l) 2 n S - -- S' , (1) + ~ S'(l) 1 A 1 A 1 A 1 1 1 1 

3 1 n 8 1 (n) dn = [8 ~ , '(1) - 38! ' (1) + 6S' (1) - 6S1 (1)] A 1 1 1 

(3-23) 

4 l [5'" (1) 1 

dn] n 8 (n) dn - 4S" (1) + 128' (1) - 24S.(1) + 24 f 8 (n) 1 A 1 1 1 1 

0 1 1 

1 5 l [5'" (1) - 1 

dn] f n S (n) dn 5S" (1) + 20S' (1) - 60S (1) + 120 f nS1 (n) o 1 A 1 1 1 1 1 0 

US1ng these results 1n (3-21) and (3-22) we obta1n 

1 - --- 8' (1) 

p£2 1 

~ S (1) p"" 1 

36 
Insertlng the expresslons for fl (t) and f

2 (t) ~nd uSlng the above results, the Eq. (3-20) can be wrltten In ltS flnal form 

where the dLmenS10nless modal parameters are deflned by 

1 

f Sk(n) dn + m*Sk(l) + m*c*sk(l) 

o 

1 

f nsk(n) dn + m*(l + c*)Sk(l) + (m*c* + J*)Sk(l) 

o 

3.2 Modal Parameter Identltles 

(3-24) 

(3-25) 

The modal parameter ldentltles derived In thls sectlon are obtal ned by many formal operatlons and all serles expanSlons are to be 

lnterpreted In the sense of convergence In the mean. Expandlng 

1 ~ C1s1(n) + c 2s 2 (n) + ••. and formlng the lnner product of both sldes 

wlth Sk(n) we flnd 

1 

f Sk(n) dn + m*sk(l) + m*c*sk(l) 

o 

37 
Hence 

00 

1 

Proceed~ng ~n the same fas~on we have 

00 

n 

Keep~ng the orthogona1~ty cond~t~on (3-6) ~n m~nd we can wr~te 

1 = 

also 

m* E E u 3 u 3 m*S (l)S (1) ~ J ~ J ~ J 

D~fferent~at~ng the ser~es expans~on of 1: o 

o = E E u3~u3JJ*S~(1)S~(1) 

~ J 

Mu1tiply~ng the ser~es expans~on of 1 w~th the above for 0 

1 • 0 E E u 3 u3 S (l)S'(l) ~ J ~ J ~ J 

38 
therefore 

o = E E u 3 u 3 m*c*(5 (1)5' (1) + 5' (1)5 (1») ~ J ~ J ~ J ~ J 

Addlng the above results we obta~n 

1 + m* = 

00 

2 E u 3k k=l 

To get an ~dentity on u 4k we proceed as above. F~rst1y 

2 E E u 4 U4 T] = 5 (n) 5 (T]) 

~ J ~ J ~ J 

therefore 

1 1 

E E u 4 u 4 f 5 ~(n)5J(T]) dn 3 ~ J ~ J O 

also 

m* = E E u

4 ,u4 m*5 (1)5 (1) 

~ J J ~ ~ J 

D~fferent~at~ng the expans~on for n 

00 

hence 

J* LEu u J*5! (1) 5' (1) 4~ 4J ~ J ~ J 

39 

(3-26) 
We also have 

Mult1plY1ng by the expansion above 

1 = E E u 4 ,u4 S (l)S'(l) 1 J 1 J 

1 J 

Not1ng the symmetry of th1s expreSS10n in 1, J 

2m*c* E E u 4 u 4 m*c*(s (1) s' (1) + S (1) s' (1») 1 J 1 J J 1 

1 J 

It follows by add1t1on of the above results that 

00 

2 E u4k k=l 

1 3 + m* + J* + 2m*c* (3-27) 

We now proceed to get an 1dent1ty on u

3k 0 u4k o From the f1rst expanS10ns 

Integrat1llg over (0,1) 

1 

2 

n 

m* 

1 J 

E E U 3 u 4 S (n)S (n) 1 J 1 J 

1 

E E u 3 u4 f s (n) S (n) dn 1 J 0 1 J 

1 J 

o 

E E u u 4 m*S (l)S (1) 

31 J 1 J 

1 j 

= E u

3 

s' (1) 

.i 1 1 

40 
and 

Mu1tip1Y1ng these ser1es 

o = 

1 = 

and 

o 

Hence 

1 = L U

4 

5' (1) J J J 

L L U 3 u 4 5'(1)5'(1) 

1 J 1 J 1 J 

1 J 

L L U

3 

u 4 S (1) 5 ' (1) 1 J 1 J 

1 J 

L L u 3 u4 5' (1) 5 ( 1) 

1 J 1 J 

m*c* = L L u 3 u 4 m*c*(5 (1)5'(1) + 5'(1)5 (1») 1 J 1 J 1 J 1 J 

Add1ng the above results we obta1n 

1 "2 + m* + m*c* (3-28) 

We can obta1n a s1IDp1e 1dent1ty amongst the parameters {u1k} by cons1der1ng the expans10n of n2 

over (OJ1) 1n terms of the e1genfunct1ons, 

wr1t1ng 

41 
we have 

~ n

2S

k (n) dn + m*sk(l) + 2J*Sk(1) + m*c*(sk(l) + 2Sk (1») 

o 

Evaluat1ng the 1ntegral by the f1rst of relat10ns (3-23) and US1ng the 

boundary cond1t10ns (3-5) we obta1n 

Different1at1ng th1s result and evaluat1ng at n 1 we obta1n 

1 (3-29) 

Evaluat1ng the 1ntegral by the second relat10n 1n (3-23) and 1nvok1ng 

(3-5) we obta1n 

co 

n3 L 1..6 [Sk(l) - Sk(l)]Sk(n) 

k=l k 

42 
or equl.valently 

00 

n

3 

6

k

:

1 

:k [(1 + c*)u1k - U2k]Sk(n) 

Dl.fferentl.atl.llg this result and evaluatl.ng at n = 1 we obtal.n 

1 

2 = (1 + 

The fl.rst serl.es l.S gl.ven by (3-29). Hence 

L 

k=l 

1 "2 + c* (3-30) 

3 Evaluatl.llg the expansl.on for n and its derl.vatl.ve at n=l we can wrl.te 

1 1 "6 +"2 c* 

1 1 

- + - c* = 6 2 

USl.ng (3-30) we obtal.n 

L 

k=l 

1 2 

- + c* + c* 

3 

43 

(3-31) 
The model parameter 1dent1t1es der1ved above w111 prove extremely useful 

for nurner1cal val1dat1on of d1g1tal S1IDulat10ns. 

3.3 Approx1IDate Solut10n 

In the prev10us sect10n the dynam1cs of the exc1ted beam w1th t1P 

body was solved 1n an exact fash1on. The partial d1fferent1al equat10n 

of mot10n was der1ved along w1th the t1IDe dependent boundary cond1t1ons. 

A great deal of labor was expended 1n construct1ng a solut1on wh1ch sat1sf1ed the govern1ng equat10n and all boundary cond1t1ons. Indeed, 1t 

w111 be recalled, that sat1sfact10n of the two natural boundary cond1-

t10ns at x = i proved most diff1cult. The end result, (3-19), was that 

the structural deformat1on could be expressed as a ser1es 1n the mode 

shapes Sk(n) - mode shapes for a clamped beam w1th t1P body. Although we 

were able to solve the boundary value problem (3-5), (3-6) for Sk(n); 

these funct10ns depend upon the parameters m*, c* and J* 1n a compl1cated 

fash10n. The quest10n ar1ses whether we can expand the deforrnat1on 1n a 

ser1es of s1mpler funct1ons. Spec1f1cally, the mode shapes for a clamped 

beam w1thout t1P body suggest themselves, S1nce they w111 serve the purpose for all beams w1th any t1P body. S1nce these later e1genfunct1ons 

sat~sfy s~mple homogeneous boundary cond~t~ons at x = 1, the boundary 

cond1tlons (3-3), (3-4) will not be satisfied when the structural deformat10n 1S expanded In terms of these modes. Th1s w111 lead to poor convergence of the ser1es solut1on compared w1th the serles solut1on based 

upon the former elgenfunct1ons. 

We use the symbol ()o to lndlcate a functlon or parameter based 

upon the s1mple clamped-free elgenfunctlons as opposed to the clarnped-t1p 

body elgenfunct1ons. Hence 

S~ (0) SO I (0) k 

44 

o 
o at n 1 

These funct~ons enJoy the s~mple orthonormal~ty cond~t~on 

1 

f Sa (n) Sa (n) dn O~J o ~ J 

Analogous to (3-25) we have the t~p bodyless modal parameters 

u a So, (1) 1k k 

a u2k S~(l) 

1 

a u f s~(n) dn 3k 

0 

1 

a u = f ns~(n) dn 4k 

0 

Expand~ng the structural deforrnat~on ~n terms of these modes 

u(x,t) = i L s~(n)Pk(t) 

k 

(3-32) 

(3-33) 

(3-34) 

It should be understood that th~s ~s only an approx~at~on to 

u(x,t) and that we sum over a f~~te number of terms. The general~zed 

coord~nates Pk(t) w~ll be determ~ed v~a Lagrange's Equat~ons for wh~ch 

express~ons are requ~red for the k~et~c and potent~al energ~es ~n terms 

45 
The potent~al energy ~n the beam ~s g~ven by 

v ! EI f .L£ dx ~ ( 2 )2 

2 0 dX2 

Us~ng the expans~on (3-34), ~ntegrat~g by parts, and ~nvok~ng (3-32) we 

arr~ve at 

V(t) (3-35) 

If we neglect terms of the order structural deformat~on x angular rate 

then the k~net~c energy of the beam T can be wr~tten r 

where VO(t) ~s the ~nert~al veloc~ty at x = 0, d~rected along the y-ax~s. 

Insert~ng the expans~on (3-34) we arr~ve at 

T (t) 

r 

(3-36) 

S~m~larly the k~net~c energy of the t~p body, T , can be expressed as 

p 

T 

P 

46 
or 

T 

P 

(3-37) 

The last quadrat~c form ~n (3-37) can be rewr~tten ~n symmetr~c form as 

Add~g the express~ons (3-36) and (3-37) we obta~n the total k~net~c 

energy. The v~rtual work ow performed by f and g ~s g~ven by 

p P 

ow = f ou(l,t) + (g + cf )o(u (l,t» p p p x 

Us~ng (3-34) we can wr~te 

where 

P ~ 

ow = L P op 

~ ~ 

~ 

Lagranges equat~ons are g~ven by 

av - -- + P

apk k 

47 

(k 1, 2, 3, ... ) 

(3-38) 
Insert1ng the appropr1ate express10ns for T, V and Pk we arr1ve at the 

system of d1fferent1al equat10ns 

.. EI 

[M]E + -4 [K]E 

pQ. 

(3-39) 

where the symmetr1c pos1t1ve def1n1te matr1ces [M] and [K] are g1ven by 

(1, J 1, 2, ... ) 

(3-40) 

[K] = 

and the column vectors ~1' ~2' ~3 and ~4 are def1ned as 

V1k ° u

1k 

V2k = 0 c*uo u 2k + lk 

V3k ° m*c*u o u + * ° 3k + lk m u 2k 

V4k = 0 u 4k + m* (1 + c u *) ° 2k + (m*c* + J*)u~k 

(3-41) 

AssoC1ated w1th the system (3-39) we have the e1genvalue problem 

[K]e = \l[M]e (3-42) 

48 
Here the e1genvalues ~l' ~2' ... are all pos1t1ve. Denote by [E] the 

matr1x whose columns are the e1genvectors e l , e 2 , ... and normal1ze the 

eigenvectors accord1ng to [ET] [M] [E] = [I]. Hence 

d1ag(~l' ~2' ... ) 

The equat10ns (3-39) can now be uncoupled by applY1ng the 11near transformat1on 

[ElS. 

EI 

g + 4 d1ag (~l' ~2' ... ) g 

pi 

(3-43) 

In Sect10n 3.1 we expanded the structural deformat10n 1n terms of the 

e1genfunct10ns Sk(n) and arr1ved d1rectly at the uncoupled system (3-24). 

Here we have expanded the structural deformat10n 1n terms of the e1genfunct10ns s~(n) and, as would be expected, the modal coord1nate equat10ns (3-39) do not uncouple. A transformat1on was requ1red to arr1ve at 

the uncoupled set (3-43). It 1S reasonable to expect that as the number 

of reta1ned modes s~(n) 1S 1ncreased the f1n1te system (3-43) should 

approach the 1nf1n1te system (3-24). More spec1f1cally 

~k - \ 

E.l - [ETl~l 

~2 - [ETl~2 

E.3 - [ETl~3 

E.4 - [ETl~4 

49 
CHAPTER 4 

A MODEL OF THE SHUTTLE ORBITER WITH DEPLOYED PAYLOAD 

4.1 Mot10n Equat10ns 1n Terms of C1amped-T1p Body E1genfunct10ns 

In th1S sect10n we der1ve equat10ns of motion for the veh1c1e 

presented 1n F1gure 4.1. A large class of shuttle deployed payloads can 

be approx1mated 1n th1S fash10n, spec1f1ca1ly, as long slender beams w1th 

attached t1P bod1es. The orb1ter and t1P body are assumed r1g1d wh1le 

the beam 1S allowed to undergo small, elast1c transverse bendlng. The 

attachment p01nt of the beam to the orb1ter 1S located arb1trar1ly w1th 

respect to the mass center of the orb1ter, but 1n keep1ng w1th other 

sect10ns of th1S report, the mass center of the t1P body 1S located along 

the t1P tangp.nt of the beam. (It should be noted that the d1ff1culty 

d1scussed 1n Section 2.2 lS st111 present due to aX1al loads exerted by 

the orb1ter on the beam. This compl1cat1ng affect w111 be 19nored 1n the 

present analys1s.) In order to fully understand the 1nteract10n between 

the flight control system and the flex1ble body dynam1cs, all rig1d body 

mot10n and bend1ng w111 be restr1cted to the orb1ter p1tch plane. Hence 

there are essent1ally three r1g1d body degrees of freedom--two trans lat10nal and one rotat10nall and an 1nf1n1te number of elastic degrees of 

freedom. Slnce the fl1ght control system lS pr1mar1ly concerned w1th 

orb1ter attitude control we analyt1cally el1m1nate the translat10nal 

coord1nates so that the f1nal set of d1fferent1al equat10ns of mot10n 

only 1nvolve orb1ter att1tude and elast1c degrees of freedom. 

The att1tude control system exerts a net force !o and moment ~ on 

the orb1ter. There lS an external torque ~ act1ng on the t1P body and 

external force f appl1ed at 1tS mass center perpend1cular to the neutral 

-p 

aX1S of the beam. 

50 
o 

Yo 

$~XO o 0 

0-- ---

INERTIAL FRAME 

ORBITER BODY FRAME FIXED AT ORBITER MASS CENTER (E90' 

BODY FRAME ATO CANTILEVERED AT BEAM TIP, PARALLEL TO /¥ 

xl AXIS PASSES THROUGH TIP BODY MASS CENTER ($t' 0 

PITCH ANGLE OF ORBITER MEASURED FROM xI TO Xo 

F~gure 4.1. Planar dynam~cs model of Shuttle Orb~ter 

w~th deployed payload. 

Bas~c Mass and Geometr~c Parameters 

mo - orb~ter mass 

IO - orb~ter moment of inert~a about ~ts mass center perpend~cular to 

plane of mot~on 

a - vector from orb~ter mass center to beam attachment po~nt 

p - mass per un~t length of beam 

t - beam length 

51 
c - t1P body mass center offset 

mt - t1P body mass 

I - t1P body moment of 1nert1a about 1tS mass center perpend1cular to 

t 

plane of mot1on 

It w111 prove conven1ent to def1ne the follow1ng quant1t1es. 

m = 

Pr1mord1al Mot1on Equat10ns 

mass of beam plus t1P body 

total veh1cle mass (4-1) 

d1mens1onless mass rat10s 

locat10n of mass center of beam + t1P body 

relative to 0 under no deformat10n 

(4-2) 

F1gure 4.2 1S the free body d1agram of two sub-bod1es assoc1ated 

w1th the veh1cle be1ng analyzed: the orb1ter, and the composite body of 

beam + t1P body. ! and Q are respect1vely the 1nterbody force and moment 

exerted by the beam on the orb1ter at the attachment po1nt O. 

We have the translat10nal equat10ns 

o meAD !o + F 

f - F ~ 

(4-3) 

o 1 where ~ and ~ are the 1nert1al accelerat10ns of the orb1ter mass center 

and beam + t1P body mass center respect1vely. The correspond1ng rotational equat10ns are 

52 
o /-r 

G 

~o :;0 

F~gure 4.2. Free body d~agrams. 

(4-4) 

-G + 9 + b x F + (_i + c - b + 0 ) x f 

-=p - -p -p 

(4-5) 

o 1 where ~, ~ are the angular momenta of the orb~ter and beam + t~p body 

about the~r respect~ve mass centers. The vector b locates the mass center of the deformed beam + tip body relat~ve to 0 and ~s ~llustrated 

along w~th i, 0 and _c ~n F~gure 4-3. - -p 

If ~ denotes the ~nert~al accelerat~on of the attachment po~nt 0, 

then we have the s~mple cond~t~ons 

53 
fj" 1 

O~;:~~~~----Q--------------~------------------~~Xl 7' 

F1gure 4-3. Deformed beam + t1P body - vector geometry. 

Equat10n (4-5) 

the beam + t1P 

(4-6) 

1 

~ 

1 can be rewr1tten 1n terms of ~ - t~e angular momentum of 

body about O. US1ng the second of results (4-3) and (4-6) 

and the trans1at10n theorem for angular momentum we arr1ve at the more 

useful form 

-Q - m1~ x ~ + g + (~+ c + 0 ) x f -v --p -p-p (4-7) 

Ignor1ng completely the affect ofax1a1 acce1erat10n on the beam + 

t1P body, the e1ast1C deformat10n of the system 1n F1gure 4.3 1S 1dent1-

cal to the problem cons1dered 1n Sect10n 3.1. Indeed, 1f u(x,t) denotes 

54 
the elast~c deflect~on of the beam along the y ax~s of3fl the govern~ng 

part~al d~fferent~al equat~on ~s g~ven by (3-1) where aO(t) ~s now the 

component of ~ along the Yl ax~s and ~o(t) ~s the ~nert~al angular accelerat~on of sri perpend~cular to the plane of mot~on. By def~n~t~on of the 

body frame sri' the geometr~c boundary cond~t~ons (3-2) apply. The natural boundary cond~t~ons are g~ven by (3-3) and (3-4) w~th the follow~ng 

replacements 

m -+-

J -+-

00 

It was shown that u(x,t) = ~ L Pk(t)Sk{n) w~th the modal coord~nate 

k=l 

equat~ons g~ven by (3-24). 

El~inat~on of Interbody Force ~ from Rotat~onal Equat~ons 

Add~ng the translat~onal equations (4-3) and ~nvok~ng (4-6) we can 

show 

d 2 d 2 -1 

1-10 -2 ~ - 1-11 --2 E. + m (E:o + f ) dt dt -p 

(4-8) 

Insert~ng th~s express~on ~nto the f~rst of (4-3) we obta~n 

F (4-9) 

wh~ch expresses the ~nterbody force ~n terms of k~nematic quant~t~es and 

the external forces on the veh~c1e. 

55 
The orb~ter rotat~onal mot~on equat~on (4-4) can now be wr~tten 

d 0 

dt ~ 

and the rotat~onal mot~on equat~on (4-7) becomes 

d 1 

dt !!o 

Angular Momentum Calculat~ons 

(4-10) 

(4-11) 

- - - Let ~, ], k be un~t vectors along the x, y and z axes of ~ (srI) . and let the plane of mot~on be the xy plane. If S ~s the orb~ter p~tch 

rate then 

= ) ] (4-12) 

If u(x,t) ~s the elast~c deflect~on of the beam and we neglect the term . 

Su then the angular momentum of the beam about 0 has a nonzero component 

along z given by 

The z component of the angular momentum of the t~p body about 0 ~s 

I [8 + "a2~ (Q"t)] + m [(Q, + 8 + c) x d (_Q, + 8 + _C)]. k 

t at oX t - -p - dt --p 

56 
We have 

Q,+o +c 

-p (c + Q,) ~ + [U(Q"t) + c ~~ (Q"t)]] 

From th1s po~nt onward terms of the order structural deflect~on x angular 

rate w~ll be dropped as well as any der~vat~ves of such quant~t~es. It 

then can be shown that the z component of the angular momentum of the t~p 

body about 0 ~s g~ven by 

Add~ng these two contr~but~ons ~t follows that 

(4-13) 

where J O ~s the ~nert~a of the beam + t~p body about 0 ~n the undeformed 

state 

(4-14) 

D~fferent~at~ng (4-12) I (4-13) ~t follows that 

(4-15) 

57 
~ h1) . k 

dt -0 

[

a

2 

c) ~ (i,t) + C 

at 

(4-16) 

Vector Geometry 

The mot10n equat10ns (4-10), (4-11) call for var10US geometr1c 

vectors and the1r t1me der1vat1ves. Recall that b lS the pos1t10n vector 

of the mass center of beam + t1P body relat1ve to O. 

US1ng the ser1es expans10n for u(x,t) and def1n1t1on (3-25) for the modal 

parameter u 3k we can show 

b (4-17) 

where u (t), the Sh1ft 1n the mass center of the beam + t1P body due to 

c 

elast1c deformat1on, lS given by 

It follows that 

u (t) 

c 

d 

dt £. 

.- .. -U

C (t)81 + (b1 8 + u c

)] 

(Slnce u lS of order structural deflectlon) c 

58 

(4-18) 
It follows that 

a x 

a x 

b x 

b x 

d 2 

-b 2 - dt 

a = 

d 

-a dt -

d 2 

= 

--a 

dt2 -

d 2 

-- a 2 -

dt 

d 2 

-b 2-

dt 

d 2 

-- a 2 - dt 

d 2 

--b 

2 - dt 

J.n § (§) 

o I 

{-a - - O)T 

2

8, a 1

8, 

-2 -a 8 - a l

8 

2 

8 

-2 a

l - a

2

8 

0 

59 

(4-19) 

(4-20) 

(4-21) 

(4-22) 
WrJ.t~ng 

a x ~ 

a x f 

-p 

b x F 

~ 

= 

f 

-p w~(~) o 1 

(4-23) 

Rotat~onal Mot~on Equat~ons - Expanded Form 

Us~ng the above results w~th (4-15) allows us to wr~te (4-10) ~n 

the form 

·2 

+ ~la2F01(t) + ~Oa1fp(t) - m~0~la2b16 

(4-24) 

Here GO(t) ~s the component of ~ perpend~cu1ar to the plane of mot~on 

(only nonzero component by assumpt~on) and G (t) ~s the correspond~ng z 

component of G. 

60 
t 

+ J 

0 

+ mt(t 

S~m~larly w~th the a~d of (4-16) we can wr~te (4-11) ~n the form 

a

2

u px ---2 (x/t) dx + It 

at 

[ ,2 + c) --1 (tit) + c 

at 

a

3

u 

at

2 (t ,t) 

ax 

,3 ] 2 u (tit) 

at ax 

2 •• - mj.l b u 

1 1 c 

(4-25) 

Us~ng the ser~es expans~on for u(x/t) and def~n~t~on (3-25) for the modal 

parameter u 4k we can show that 

t 

a

2

u a

3

u J px --- (x/t) dx + It 

at

2 (t I t) 

0 at

2 

ax 

[" '2

3u (i,tl] 00 

3 .. + mt(t + c) ~ (tit) + c pt E u

4kPk (t) 

at at ax k=l 

Introduc~ng th~s result ~nto (4-25) and add~ng w~th (4-24) el~~nates the 

~nterbody torque G (t) z 

61 
00 

+ (~Oal + t + c - ~lbl)fp(t) 

(4-26) 

As d1scussed above, the modal coord1nates Pk(t) are governed by 

- .. (3-24) w1th wO(t) = e and aO(t) be1ng the component of ~ along the 

Yl aX1S. From (4-8) w1th (4-19) and (4-21) 

The modal coord1nate different1al equat10n assumes the form 

g (t) f (t) 

u 11 

_P_-+ u 21 

_P_-

pt 3 pt2 

u 31 

(F 02 + f ) a 2 -2 

n2

p (t) mt + U31~0 - e -

P t 1 1 

1, 2, 3, ... ) 

(4-27) 

Note the coup11ng of the modal coord1nate equat10n w1th all the elast1c 

coord1nates as well as orb1ter p1tch. Introduce the follow1ng notat10n 

62 
= 

(k 1,2,3, ... ) 

Equat~ons (4-26), (4-27) can now be wr~tten 

00 

F 01 (t) g (t) 

---+ -p--

pt

2 

pt 3 

(

a b ) f (t) 

+ J.l

O 0

1 + 1 + c* - J.l --.! -p-- N 1 t pt2 

g (t) f (t) 

U -p-- + U -p-- 1i 3 pt 2~ pt2 

u3~ 

rnt (F 02 (t) + fp (t» 

~2 62 _ r/2p (t) + U3~J.lO N ~ ~ 

(~ 1, 2, 3, •.. ) 

(4-28) 

(4-29) 

(4-30) 

Equat~ons (4-29) and (4-30) are the f~nal set of d~fferent~al equat~ons 

of rnot~on ~nvolv~ng orb~ter p~tch and elast~c coord~nates of the beam. 

The coeff~c~ents of the accelerat~ons are constant but (4-29) conta~ns a 

t~e dependent coeff~c~ent through the term F01 (t)uc(t) wh~le (4-30) has 

a nonl~near term in s. 

63 
Matr1x-Vector Form of Mot10n Equat10ns 

Go (t) (a1 + b 1 )F02 (t) a 2 F01 (t) g (t) 

Plo (t) - ---- ~1 t 2 + ~1 T 2 + -p--

pt 3 pt pt pt 3 

+ (1 + c* + 

a

1 b 1 ) fp (t) 

~ -- ~1 T --2- o t pt 

PI (t) 

1 

g (t) f (t) U

~+ 3 P 1 ( u h pi,3 u217 - mt (F02 t) + fp(t» 

(1 = 1, 2, 3, ... ) 

(4-31) 

The genera11zed forces Pio(t) and ~ (t) are funct10ns only of the external 

forces and moments act1ng upon the veh1cle. Let 

u (t) 

c F 01 (t) 0 

~1 t pi,2 u 31 

0 a

2 8

2 

N + ~ - u (t) 32 0 o i, (4-32) 

0 u 33 

64 
We have the symmetr~c general~zed mass matr~x A 

a

OO all a

12 a

l3 

all (1 - .Q.& u

2 

) .Q.& pQ, 

m 31 - m u31 u 32 - -;- u 31 u 33 

A a 12 

.Q.& (1 - .Q.& u

2 

) .2.! - m u 32u 31 m 32 - m u 32u 33 

a

13 

_.Q.& u

33u

31 

.2.! (1 - .2.! u

2 

) m - m u33u32 m 33 

(4-33) 

and st~ffness matr~x 

[K] (4- 34) 

In terms of the above notat~on Eqs. (4-29), (4-30) are 

- .. = fj(t) + N (ti X, x) 

4.2 Mot~on Equat~ons ~n Terms of Clamped-Free E~genfunct~ons 

We treat here the same veh~cle as ~n Section 4.1; the only d~fference be~ng the set of e~genfunct~ons used to expand the elast~c d~splacement ~n the beam. Here we use the t~p-bodyless e~genfunctions s~(n) 

descr~ed ~n Sect~on 3.3. For the same reasons as g~ven there we expect 

that, for the same degree of truncat~on, the results w~ll be ~nfer~or to 

those obta~ned by use of the mot~on equat~ons of the prev~ous sect~on. 

The results of the present sect~on ~n conJunct~on w~th our prev~ous analys~s can serve as a bas~s for numer~cal conf~rmat~on of th~s statement. 

65 
Throughout th~s sect~on ~t w~ll be understood that the elast~c 

deflect~on ~n the beam ~s expanded ~n terms of the set {s~(n)} 

u (x,t) 

No confus~on should ar~se between the modal coord~nates Pk(t) and those 

appear~ng ~n Sect~on 4.1. V~rtually all results prev~ously obta~ned 

rema~n ~ntact except ~n those ~nstances where expl~c~t use was made of 

the ser~es expans~on for u(x,t). We l~st sequent~ally those equat~ons ~n 

Sect~on 4.1 wh~ch requ~re mod~f~cat~on due to a sw~tch of the modal bas~s 

from {sk(n)} to {s~(n)}. 

(1) Equat~on (4-18) ~s replaced by 

u (t) 

c = (V3k ~s g~ven by Eq. (3-41» 

(2) Equation (4-26) - second term on left hand side ~s replaced 

by 

(3) The modal coord~nates are now governed by Eq. (3-39) w~th 

2 pt .. 1 "2 (~Oal - ~lbl)6 - --- E V P + - (F02 + fp) - ~Oa26 m k 3k k m 

6. Thus Equat~on (4-27) ~s replaced by 

66 
a 2 02 EI 

- e - - "Op (t) + V3~1l0 .II. 4 pt ~ ~ 

where M~k ~s g~ven by Eq. (3-40). 

(4) Equat~on (4-28) ~s replaced by 

(k 1, 2, 3, ... ) 

(5) Equat~on (4-30) ~s replaced by 

a 2 02 EI 

+ V II - e - -4 "Op (t) 3~ 0 .II. pt ~ ~ 

(w~th al~ g~ven by (4) above). 

(6) The second def~n~t~on (4-31) ~s replaced by 

(7) 

g; (t) 

~ 

In Eq. (4-32) use the expression for u (t) g~ven by (1) 

c 

above and replace second vector w~th 

67 
A 

aDO 

a 11 

= a 12 

a 13 

(8) Def~n~t~on (4-33) ~s replaced w~th 

all a 12 a 13 

(M11 pt ) - -;- V3lV3l (M12 pt ) - -;- V3l V32 (M13 pt ) --;-V3lV33 

( M2l pt ) - -;- V32V3l (M22 pt ) --;-V32V32 (M 23 - m V ~ ) 32V33 

(M3l - pt V V ) m 33 31 (M32 - ~t V33V32 ) (M33 - ~t V33V33 ) 

(9) Def~n~t~on (4-34) ~s replaced by 

[K] 

Note: 

EI 0 0 0 --4 d~ag(O, A

1 , A2 , A3 , ... ) 

pt 

these are the squares of the natural frequenc~es of a un~form clamped beam w~thout 

t~p body. 

4.3 D~fferent~al Equat~ons of Motion - F~rst Order Form 

In the last two sect~ons equat~ons of mot~on were der~ved for the 

veh~cle dep~cted ~n F~gure 4.1. These equat~ons were of the form of a 

system of second order d1fferent1al equat10ns and are not d~rectly su~table for digital implementat~on. d 2-

x S1nce these equat~ons are l1near ~n --2 

dt 

they can be transformed ~nto an equ1valent set of f1rst order d1fferent~al equat10ns. In effect~ng tills transformation the "system natural 

frequencies" reveal themselves and the cpt10n of further modal truncat10n 

(beyond the truncat10n of elast1c degrees of freedom 1n the beam) 'can be 

adopted. 

68 
Our mot~on equat~ons can be wr~tten 

d

2 EI 

[A] --2 ~ + 4 [B]~ 

dt p£, 

&r(t) + !!(t; 

. ~, ~) (4-35) 

where 

If we follow the procedure ~n Sect~on 4.1 the A are the e~genvalues of ~ 

the problem for a clamped beam w~th t~p body (Equat~on 3-5) wh~le ~f we 

follow the procedure ~n Sect~on 4.2 they are the e~genvalues appropr~ate 

to a clamped beam w~thout t~p body (Equat~on 3-32). 

The free v~brat~on of the system assoc~ated w~th (4-35) leads to 

the e~genvalue problem 

[B]:!. = ~ [A]:!. 

s and the "system natural frequenc~es" w are given by 

s 

w = 

(4-36) 

[B] ~s symmetr~c, pos~t~ve-se~def~n~te and [A] ~s symmetr~c, pos~t~ve 

def~n~te. All e~genvalues are therefore nonnegat~ve. Spec~f~cally ~ = 0 

T ~s a s~mple eigenvalue w~th e~genvector (1, 0, 0, ... , 0) correspond~ng 

to r~g~d body rotat~on. Th~s e~genvalue problem has a full set of l~ne1 d d :!. (1) , :!.(2) , (n+l) d 1 ar y ~n epen ent e~genvectors ... , :!. an we can a ways 

arrange that they be orthonormal relat~ve to [A], ~.e. 

69 
Def~ne 

[z] v (n+1) ] (4-37) 

If we make the l~near transformat~on x = [z]~ ~n (4-35) we arr~ve 

at 

. 

~) ) 

(4-38) 

Let 

1. (t) ~. ~t~~ ~(t) 

and label the e~genvalues so that ~1 = o. 

The system (4-38) can now be wr~tten as 

(4-39) 

70 
CHAPTER 5 

FORTRAN PROGRAMS 

Two separate FORTRAN computer programs have been created for the 

purpose of numer1cally 1mplement1ng the mot10n equat10ns of Sect10ns 4.1 

and 4.3. Complete l1st1ngs of these programs accompan1ed by annotated 

sample 1nput and output data are prov1ded 1n Append1ces Band C. Each 1S 

extens1ve1y commented throughout and 1n most 1nstances the FORTRAN var1-

able names are mnemon1cally s1m1lar to correspond1ng analyt1cal quant1-

t1es. Where relevant, reference 1S made to spec1f1c equat10ns of the 

report. 

The program of Append1x B computes the e1genvalues, modal parameters and modal parameter 1dent1t1es of a cant1levered beam w1th t1P 

body. Note that these computat10ns are performed 1n (IBM) quadruple 

preC1S10n due to the numer1cal sens1t1v1ty of the transcendental expresS10ns 1nvolved. The roots of the character1st1c equation (Eq. 2-19) are 

estlffiated by 1ncrernent1ng the parameter Sand search1ng for s1gn changes 

1n the left-hand-s1de. Root estlffiates are then 1mproved by NewtonRaphson 1terat1on. The modal parameters of Eq. 3-25 are computed and 

together w1th the roots of Eq. 2-19 (ra1sed to the fourth power) wr1tten 

to a d1sc data set to be used by the mot10n equat10n program (descr1bed 

below). The part1al sums and asymptot1c values of the modal parameter 

~dent~t~es (Eqs. 3-26 thru 3-31) are evaluated and output. Th1s 1nformat10n serves to check the calculat10ns of the modal parameters and can 

1nd1cate "m1ssed" roots of the character1st1c equat10n by show1ng poor 

convergence. 

71 
The program of Append~x C numer~cally synthes~zes and ~ntegrates 

the mot~on equat~ons for the planar dynam~cs of an elast~c beam w~th 

r~g~d bod~es at each end. The formulat~on ~plemented ~s that for wh~ch 

the beam deflect~on ~s expanded as a ser~es of e~genfunct~ons appropr~ate 

to a clamped beam w~th t~p body (as per Sect~on 4.1). The e~genvalues 

and modal parameters generated by the above program are read and checked 

for cons~stency with the NAMELIST ~nput data. Note that the ~n~t~al 

cond~t~ons on the att~tude angle and ~ts rate may be spec~f~ed arb~trarily as ~nput, wh~le those of the modal coord~nates and the~r t~me der~vat~ves are ~nternally set to zero. The algebra~c e~genvalue problem of 

Eq. 4-36 ~s solved v~a the double prec~s~on IMSL subrout~ne EIGZS. The 

"system natural frequenc~es" are output and are helpful for select~on of 

an ~ntegrat~on t~e step. External forces and torques on the "Orb~ter" 

and t~p body are spec~f~ed through subrout~nes ORBFOR and TPBFOR respect~vely. The mot~on equat~ons ~n f~rst order form, Eq. 4-39 are ~ntegrated us~ng the Adams Method w~th th~rd order d~fferences. 

72 
APPENDIX A 

EIGENFUNCTION NORMALIZATION FOR CANTILEVERED BEAM WITH TIP BODY 

In Sect10n 3.1 extenS1ve use was made of the e1genfunctions Sk(n) 

def1ned by the boundary value problem (3-5). In th1S append1x we formulate a procedure to obta1n these norma11zed e1genfunct10ns from a set of 

non-norma11zed e1genfunct1ons. Certa1n def1n1te 1ntegrals 1nvo1v1ng the 

e1genfunct10ns are evaluated to fac111tate computat10n of the modal parameters given 1n Eqs. (3-25). 

Recall that for the case of a s1mple beam w1th no t1P body the 

norma11zed clamped-free e1genfunct1ons are g1ven by 

where Sk 1S a root of 

cosh S cos S + 1 0 

and the norma11zat10n cond1t1on 1S 

1 

To obta1n a set of e1genfunct10ns for the case of a beam w1th tip 

body we can, 1n the d1scuss1on preced1ng Eqs. (2-17), (2-18), choose the 

73 
constant c

4 = 1. (So c 2 = -1). Invok~ng the boundary cond~t~on (2-18) 

the result~ng unnormal~zed e~genfunct~on ~k(n) ~s g~ven by 

(A.1) 

where now Sk are the roots of Eq. (2-19) and the parameter Yk ~s g~ven by 

Y 

2 3 m*c*S (cosh S - cos S) + J*S (s~n S + s~nh S) - cos S - cosh S 

= 2 3 m*c*S (s~n S - s~nh S) + J*S (cos S - cosh S) + s~n S + s~nh S 

where subscr~pt k ~s deleted for clar~ty. (Note that for m* 

th~s e~genfunct~on ~s already normal~zed) . 

J* = 0 

Us~ng the e~genfunct~on (A.1) we read~ly evaluate the fo110w~ng 

~ntegrals 

1 

1 

f ~2 (n) dn 

o 

f ~(n) dn 

o 

1 1 + (s~nh S cosh S + s~n S cos S) 2S 

1 S (s~nh S cos S + cosh S s~n S) 

2 

+ 1- [s~nh S cosh S - s~n S cos S 2S 

- 2(s~n S cosh S - cos S s~nh S)] 

+ y (s~n S - s~nh S)2 S 

1 S [s~nh S - s~n S + y(cosh S + cos S - 2)] 

74 

(A.2) 

(A.3) 
1 

! n1j!(n) dn 

o 

1 S [slnh S - Sln S + y(cosh S + cos S)] 

1 

+ -- [2 - cosh S - cos $ - y(sin $ + slnh $)] 

S2 

(A.4) 

We can now evaluate the square of the norm of the elgenfunctlon 1j!k(n) 

wlth respect to the lnner product In (3-6) 

(A.5) 

The normallzed elgenfunctlons are glven by 

75 
ISN 0002 

ISN ()003 

ISN 0004 

ISN 0005 

ISN 0006 

ISN 0007 

ISN 0008 

ISN 0009 

ISN 0011 

ISN 0012 

ISN 0013 

ISN 0014 

ISN 0015 

ISN 0016 

ISN 0017 

ISN 0018 

ISN 0019 

ISN 00'0 

ISN 0021 

ISN 0022 

ISN 0023 

ISN 0024 

APPENDIX B 

"EIGENVALUE"-MJDAL PARAMETER 

FORTRAN PROGRAM LISTING 

C *.*****~***. .** *.* ••• ***t.**.******.**.* ••••• **· •• **~ •• *.** ••• * •• *00000100 

C. THIS PROGRAM COMPUTES THE EIGENVALUES, MODAL PARAMETERS ANO '00000110 

C' MODAL PARAMETER IDENTITIES OF A CLAMPED BEAM WITH TIP BODY '00000120 

C ~ THE EIGENVALUES AND MODAL PARAMETERS WHICH ARE NECESSARY INPUT '00000130 

C' TO THE PLANAR DYNAMICS PROGRAM ARE WRITTEN TO A DISC FILE '00000140 

C' (WRITTEN BY JOEL STORCH & STEPHEN GATES C SOL BASED UPON '00000200 

C. C SOL REPORT # R-1629 MAY 1983 ) ~00000300 

C *******t***.** •• *.***.** ••• *** ••• ********.***t****~* •• **t*t •••• *.****.00000400 

C 00000500 

C NOTE ARRAYS DIMENSIONED FOR A MAXIMUM OF 50 EIGENVALUES 00000510 

C 00000520 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

c 

C 

C 

C 

2 

3 

4 

5 

6 

C 

C 

C 

C 

IMPLICIT REAL'16(A-H,D-Z) 00000600 

REAL'16 LAM,MSTAR,JSTAR 00000700 

DIMENSION LAM(50),Ul(50),U2(50),U3(50),U4(50) 00000900 

NAMELIST /INPUT/ MSTAR,JSTAR,CSTAR,NF 00000910 

00000918 •••••••••••••••••••••• INPUT - OUTPUT FILES 

FILE #5 NAMELIST INPUT FILE 

·························00000926 

FILE #6 PRINTED OUTPUT FILE 

00000934 

00000942 

FILE #8 DISC OUTPUT FILE FOR EIGENVALUES & MODAL PARAMETERS 00000950 

00000958 •••••••••••••••• DESCRIPTION OF /INPUT/ LIST ITEMS 

"MSTAR" MASS RATIO (OF EQ 2-19) 

"JSTAR" INERTIA RATIO (OF EQ 2-19) 

"CSTAR" OFFSET RATIO (OF EQ 2-19) 

··················00000966 

00000374 

00000982 

00000983 

"NF" NUMBER OF EIGENVALUES & MODAL PARAMETERS TO BE COMPUTED 

00000984 

00000985 

** ••• **. .**  ••••• ****.***.** •••••••• **.******* ••••• ·······****·*··00000986 

PROMPT FOR NAMELIST INPUT DATA 

WRITE(5,l ) 

FORMAT(lH ,5X,'INPUT DATA',/) 

READ(5,INPUT) 

IF(NF LE 50) GO TO 3 

WRITE(6,2) NF 

STOP 

FORMAT(lHO,5X,I3,' MODES REQUESTED MAXIMUM ALLOWABLE IS 50') 

WRITE(6,6) 

WRITE(6,4) MSTAR,JSTAR,CSTAR 

FORMAT(lHO,5X,'MSTAR=',F8 4,4X,'JSTAR=',F8 4,4X,'CSTAR=',F8 4) 

WRITE(6,5) NF 

FDRMAT(lH ,5X,'NF=',I3) 

FORMAT(lHl,15X,'DATA FROM NAMELIST INPUT') 

CALL EIGEN(NF,MSTAR,JSTAR,CSTAR,LAM,Ul,U2,U3,U4) 

OUTPUT EIGENVALUES AND MODAL PARAMETERS (NOTE ARRAY LAM CONTAINS 

ROOTS OF EQ 2-19 RAISED TO THE FOURTH POWER) 

WRITE(6,13) 

WRITE(6,11 ) 

WRITE(8) NF,MSTAR,JSTAR,CSTAR 

DO 10 N=l,NF 

76 

00000990 

00000993 

00000996 

00001000 

00001100 

00001200 

00001300 

00001400 

00001500 

00001600 

00001610 

00001700 

00001800 

00001810 

00001820 

00001830 

00001900 

00001910 

THE00001920 

00001930 

00001940 

00001950 

00002000 

00002100 

00002200 
ISN 0025 

ISN 0026 

ISN 0027 

ISN 0028 

ISN 0029 

ISN 0030 

ISN 0031 

ISN 0032 

ISN 0033 

ISN 0034 

ISN 0035 

ISN 0036 

ISN 0037 

ISN 0038 

ISN 0039 

ISN 0040 

ISN 0041 

ISN 0042 

ISN 0043 

ISN 0044 

ISN 0045 

ISN 0046 

ISN 0047 

ISN 0048 

ISN 0049 

ISN 0050 

ISN 0051 

ISN 0052 

ISN 0053 

ISN 0054 

ISN 0055 

ISN 0056 

ISN 0057 

ISN 0058 

ISN 0059 

ISN 0002 

ISN 0003 

ISN 0004 

ISN 0005 

ISN 0006 

ISN 0007 

ISN 0008 

ISN 0009 

ISN 0010 

ISN 0011 

ISN 0012 

ISN 0013 

ISN 0014 

ISN 0015 

ISN 0016 

ISN 0017 

ISN 0018 

10 

11 

12 

13 

C 

C 

C 

20 

21 

22 

23 

C 

C 

C 

30 

C 

C 

C 

WRITE(8) LAM(N),Ul(N),U2(N),U3(N),U4(N) 

WRITE(6,12) N,LAM(N),Ul(N),U2(N),U3(N),U4(N) 

FORMAT( lHO, T4,'N', T12, 'LAM', T27, 'Ul', T40, 'U2', T49, 'U3', T66, 'U4') 

FORMAT(lHO,T3,I2,T8,G12 5,T24,F9 4,T36,F9.4,T47,F9 5,T60,G13 5) 

FORMAT(lHO,/,17X,'"EIGENVALUES" & MOOAL PARAMETERS') 

MODAL PARAMETER IDENTITIES 

WRITE(6,20) 

FORMAT(lHl.35X,'MODAL PARAMETER IOENTITIES') 

WRITE(6, 23) SUM1~0 000 SUM2~0 000 

SUM3~0.000 

SUM4~0 000 

SUM5=0 000 

SUM6=0 000 

DO 22 N=l,NF SUM1~SUM1+U3(N)"2 

SUM2~SUM2+U4(N)'*2 

SUM3~SUM3+U3(N)'U4(N) 

SUM4=SUM4+Ul(N)*'2!LAM(N) 

SUM5=SUM5+Ul(N)'U2(N)/LAM(N) 

SUM6=SUM6+U2(N)**2!LAM(N) 

WRITE(6,21)N,SUM1,SUM2.SUM3,SUM4.SUM5.SUM6 

FORMAT(lHO,5X,I2.6(G13 5.2X» 

CONTINUE 

FORMAT(lHO,42X,'PARTIAL SUMS',!!/!/!) 

EXACT SUMS 

SUM1-l +MSTAR 

SUM2-1 /3 +MSTAR+uSTAR+2 'MSTAR*CSTAR 

SUM3=.5+MSTAR*(1 +CSTAR) 

SUM4=1 

SUM5= 5+CSTAR 

SUM6=1 /3.+CSTAR*(1.+CSTAR) 

WRITE(6.30) SUM1.SUM2,SUM3,SUM4,SUM5.SUM6 

FORMAT(1HO,!/////.8X,6(G13 5.2X» 

STOP 

END 

FUNCTION CHARDT(AlF.MSTAR.uSTAR.CSTAR) 

THIS FUNCTION COMPUTES THE EOUIVALENT OF EO 2-19 

IMPLICIT REAl*16(A-H.0-Z) 

REAl"16 MSTAR.uSTAR 

S=OSIN(AlF) 

C=OCOS(AlF) 

SH=OSINH(AlF) 

CH=OCOSH(ALF) 

C1=MSTAR*ALF 

C2~Cl'CSTAR'AlF 

C3~uSTAR'AlF**3 

A11~C2"(C-CH)-C3*(S+SH)+C+CH 

A12~C~"(S-SH)+C3'(C-CH)+S+SH 

A21~C1"(CH-C)+C2'(S+SH)-S+SH 

A22~C1'(SH-S)+C2*(CH-C)+C+CH 

CHARDT~Al1'A22-A12'A21 

RETURN 

END 

77 

00002300 

00002400 

00002500 

00002700 

00002710 

00002800 

00002900 

00003000 

00003100 

00003200 

00003210 

00003300 

00003400 

00003500 

00003600 

00003700 

00003800 

00003900 

00004000 

00004100 

00004200 

00004300 

00004400 

00004500 

00004600 

00004700 

00004800 

00004810 

00004900 

00005000 

00005100 

00005200 

00005300 

00005400 

00005500 

00005600 

00005700 

00005800 

00005900 

00006000 

00006100 

00006200 

00006210 

00006220 

00006230 

00006300 

00006400 

00006500 

00006600 

00006700 

00006800 

00006900 

00007000 

00007100 

00007200 

00007300 

00007400 

00007500 

00007600 

00007700 

00007800 
ISN 0002 

C 

C 

C 

C 

ISN 0003 

ISN 0004 

ISN 0005 

C 

C 

C 

C 

C 

ISN 0006 

ISN 0007 

ISN 0008 

ISN 0009 1 

C 

C 

C 

ISN 0010 

ISN 0012 

ISN 0013 

ISN 0014 2 

ISN 0015 

C 

C 

C 

ISN 0016 

ISN 0017 3 

ISN 0018 

ISN 0019 

ISN 0020 

ISN 0021 

ISN 0022 

lSN 0023 

1St! 0024 

ISN 0025 

ISN 0026 

ISN 0027 

ISN 0028 

ISN 0029 

ISN 0030 

ISN 0031 

1St! 0032 

ISN 0033 

ISN 0034 

ISN 0035 

SUBROUTINE EIGEN(NF,MSTAR,JSTAR,CSTAR,lAM,Ul,U2,U3,U4) 

THIS SUBROUTINE COMPUTES "NF" EIGENVALUES (ROOTS OF EO 2-19) AND 

MODAL PARAMETERS (EO'S 3-25) OF A CLAMPED BEAM WITH TIP BODY 

IMPLICIT REAl*16(A-H,O-Z) 

REAl*16 MSTAR,JSTAR,lAM(NF),lEFT 

DIMENSION Ul(NF),U2(NF),U3(NF),U4(NF) 

ESTIMATE ROOT OF CHARACTERISTIC EOUATION (EO 2-19) 

N IS THE NUMBER OF ROOTS FOUND 

N=O 

STEP= 1 00-02 

lEFT=O 000 

RIGHT=lEFT+STEP 

COMPARE THE SIGNS OF THE FUNCTION "CHAROT" AT THE TWO POINTS 

IF(CHAROT(LEFT,MSTAR,JSTAR,CSTAR)fCHAROT(RIGHT,MSTAR,JSTAR,CSTAR) 

1 lE 0 000) GO TO 2 

LEFT=RIGHT 

GO TO 1 

N=N+l 

ALF=RIGHT 

IMPROVE ROOT ESTIMATE WITH NEWTON-RAPHSON ITERATION 

NIT=O 

S=OSIN(AlF) 

C=OCOS(ALF) 

SH=OSINH(AlF) 

CH=OCOSH(ALF) 

C1-MSTAR*ALF 

C2=CPCSTAR*AlF 

C3=JSTAR*ALF**3 

All=C2*(C-CH)-C3 t (S+SH)+C+CH 

A12=C2*(S-SH)+C3*(C-CH)+S+SH 

A21=Cl*(CH-C)+C2*(S+SH)-S+SH 

A22=Cl*(SH-S)+C2*(CH-C)+C+CH 

AllP=MSTAR*CSTAR*(2 000*AlF*(C-CH)-ALF**2*(S+SH»-C3*(C+CH) 

1 -3 000*JSTAR*ALF**2*(S+SH)-S+SH 

A12P=MSTAR*CSTAR*(2 000*AlF*(S-SH)+AlF**2*(C-CH»-C3*(S+SH) 

1 +3 OOO*JSTAR*AlF**2*(C-CH)+C+CH 

A21P=Cl*(S+SH)+C2*(C+CH)+2 OOO*MSTAR 

1 *CSTAR*AlF*(S+SH)-C+CH+MSTAR*(CH-C) 

A22P=MSTAR*(SH-S)+Cl*(CH-C)+C2*(SH+S) 

1 +2 OOO*MSTAR*CSTAR*ALF*(CH-C)-S+SH 

F=All*A22-A12*A21 

FP=AllP*A22+All*A22P-A12P*A21-A12*A21P 

OA=f/FP 

AlF=AlF-OA 

78 

0000790(' 

00007910 

00007920 

00007930 

00007950 

00008000 

00008100 

00008200 

00008320 

00008330 

00008340 

00008400 

00008500 

00008600 

00008700 

00008800 

00008900 

00009000 

00009100 

00009200 

00009300 

00009400 

00009500 

00009600 

00009700 

00009800 

00009900 

00010000 

00010100 

00010200 

00010300 

00010400 

00010500 

00010600 

00010700 

00010800 

00010900 

00011000 

00011100 

00011200 

00011300 

00011400 

00011500 

00011600 

00011700 

00011800 

00011900 

00012000 

00012100 

00012200 

00012300 

00012400 

00012500 
ISN 0036 

ISN 0038 

ISN 0039 

ISN 0041 

ISN 0042 

ISN 0043 

ISN 0044 

ISN 0045 

ISN 0046 

ISN 0047 

ISN 0048 

ISN 0049 

ISN 0050 

ISN 0051 

ISN 0052 

ISN 0053 

ISN 0054 

ISN 0055 

ISN 0056 

ISN 0057 

ISN 0058 

ISN 0059 

ISN 0060 

ISN 0061 

ISN 0062 

ISN 0063 

ISN 0064 

ISN 0065 

ISN 0066 

ISN 0067 

ISN 0068 

ISN 0069 

ISN 0070 

ISN 0071 

ISN 0072 

ISN 0073 

ISN 0074 

ISN 0075 

ISN 0071 

ISN 0078 

ISN 0079 

IF(OABS(DA/ALF) LE 1.00-09) GO TO 6 00012600 

NIT=NIT+l 00012700 

IF(NIT GT. 10) GO TO 4 00012800 

GO TO 3 00012900 

4 WRITE(6,5) NIT,N,DA,ALF 00013000 

5 FORMAT(lHO,5X,'NEWTON RAPHSON ITERATION FAILED TO CONVERGE IN',lX,00013100 

1 12,' ITERATIONS. ROOT H ',I2,2X,'DA=',E13 5,2X,'ALF=',E13 5) 00013200 

6 S=OSIN( ALF) 00013300 

C=OCOS(ALF) 00013400 

SH=OSINH(ALF) 00013500 

CH=OCOSH(ALF) 00013600 

C1=MSTARtALF 00013700 

C2=Cl*CSTAR*ALF 00013800 

C3=uSTAR*ALF*t3 00013900 

A11=C2t(C-CH)-C3 t (S+SH)+C+CH 00014000 

A12=C2*(S-SH)+C3 t (C-CH)+S+SH 00014100 

LAM(N)=ALF**4 00014200 

BETA=-A11/A12 00014300 

T1=ALF*(SH+S+BETA*(CH-C» 00014400 

T2=CH-C+BETA*(SH-S) 00014500 

T3=(SH-S+BETA*(CH+C-2.000»/ALF 00014600 

T4=(SH-S+BETA*(CH+C»/ALF-(CH+C-2 000+BETA*(S+SH»/ALF**2 00014700 

C 00014800 

C CALCULATE MODAL PARAMETERS WITH NON NORMALIZED EIGENFUNCTIONS 00014900 

C 00015000 

C 

C 

C 

C 

C 

C 

C 

U1(N)-T1 00015100 

U2(N)=T2+CSTAR*Tl 00015200 

U3(N)=T3+MSTAR*T2+MSTAR*CSTAR*T1 00015300 

U4(N)=T4+MSTAR*(1 000+CSTAR)*T2+(MSTAR*CSTAR+uSTAR)*T1 00015400 

00015500 

CALCULATE INTEGRAL OF SOUARE OF EIGENFUNCTION 00015600 

00015700 

C1=1 000+BETA*'2 00015800 

C2=BETA**2-1 000 00015900 

T5=(C1 t CH+2 000tBETA*SH)*SH/(2 OOO*ALF) 00016000 

T6=C2*SH*C/ALF-Cl*CH*S/ALF-2.000*BETA*SH*S/ALF 00016100 

T7=St(2 000*BETA*S-C2*C)/(2.000*ALF)+1 000 00016200 

VALINT=T5+T6+T7 00016300 

00016400 

CALCULATE NORM OF EIGENFUNCTION & MODAL PARAMETERS FOR NORMALIZED 00016500 

EIGENFUNCTIONS 00016600 

00016610 

P2=VALINT+MSTAR*T2**2+uSTAR*Tl**2+2 000*MSTAR*CSTAR*T1 t T2 00016700 

P=OSORT(P2) 00016800 

U1(N)=U1(N)/P 00016900 

U2(N)=U2(N)/P 00017000 

U3(N)=U3(N)/P 00017100 

U4(N)=U4(N)/P 00017200 

IF(N .EO NF) RETURN 00017300 

LEFT=RIGHT 00017400 

GO TO 1 00017500 

END 00017600 

79 
OUTPUT FROM "EIGENVALUE"-MODAL PARAMETER PROGRAM 

DATA FROM NAMELIST INPUT 

MSTAR- 2 0000 JSTAR- o 0280 CSTAR- o 1000 

NF- 10 

"EIGENVALlJtS" & MODAL PARAMETERS 

N LAM Ul U2 U3 U4 

0310 0 9087 o 6760 56911 6540 

2 143 31 -4 8354 -0 1266 0 52240 14854 

3 1220 0 6 0703 -0 0027 o 29800 505870-01 

4 5231 5 -4 9666 o 0552 o 22042 259090-01 

5 16775 3 5599 -0 0608 0 17072 150010-01 

6 42936 -2 6385 o 0551 0 13693 951230-02 

7 93095 2 0584 -0 0485 0 11354 650020-02 

8 178940+06 - 1 6739 o 0427 o 09673 470290-02 

9 314510·06 4044 -0 0380 o 08415 355330-02 

10 516170+06 - 1 2066 o 0341 o 07442 277650-02 

MODAL PARAMETER IDENTITIES 

PARTIAL SUMS 

N N N N 2 N N 2 

2: 2 2: 2 2: 2: u1k 2: u1ku

2k L u2k u3k u4k u3ku4k 

N k-l k-l Ak Ak Ak k-l k-l k-l k-l 

2 4621 2.7356 2 5953 80094 59581 44322 

2 2 7350 2 7577 2 6729 96409 60008 44333 

3 2 8238 2 7602 2 6879 99429 60007 44333 

4 2 8724 2 7609 2 6936 99901 60002 44333 

5 2 9015 2 7611 2 6962 99976 60001 44333 

6 2 9203 2.7612 2 6975 99993 60000 44333 

7 2 9332 2 7613 2 6983 99997 60000 44333 

8 2 9425 2 7613 2 6987 99999 60000 44333 

9 2 9496 2.7613 2 6990 99999 60000 44333 

10 2 9552 2 7613 2 6992 1 0000 60000 44333 

ASYMPTOTIC VALUES 

1/3 +m' 

1 +m' + J' + 2m'c' 1/2 + m' +m°c' 10 1/2 + c' 1/3 + c' +co2 

30000 2 7613 2 7000 0000 60000 44333 

80 
ISN 0002 

ISN 0003 

ISN 0004 

ISN 0005 

ISN 0006 

ISN 0007 

ISN OOOS 

ISN 0009 

ISN 0010 

APPENDIX C 

PLANAR DYNAMICS FORTRAN PROGRAH LISTING 

C 00000100 

C'*****'**·***"***--**·-*******' --·** ***'**·-***''*""**"**"-***00000200 

C' THIS PROGRAM SOLVES THE EQUATIONS OF MOTION FOR THE PLANAR DYNAMICS '00000300 

C* OF A FLEXIBLE BEAM WITH RIGID BODIES ATTACHED TO EACH END THE -00000400 

C* BEAM DEFORMATION IS EXPANDED IN TERMS OF THE MODES APPROPRIATE TO '00000500 

C* A CLAMPED BEAM WITH TIP BODY ONLY THE ATTITUDE AND BENDING COOR- '00000600 

C* DINATES ARE CALCULATED (WRITTEN BY JOEL STORCH & STEPHEN GATES .00000700 

C* DASED UPON C SOL REPORT # R-1629 MAY 19S3 ) 'OOOOOSOO 

C** •• ** •• ·**********·*.*·*******~*·**·*·*********·*·** ·*********··***···00000900 

C 

C NOTE ARRAYS DIMENSIONED FOR A MAXIMUM OF 20 CLAMPED-TIP BODY MODES 

00001000 

00001100 

00001200 

00001100 

00001400 

00001500 

00001600 

00001700 

00001800 

00001900 

00002000 

00002100 

00002200 

00002300 

00002400 

00002500 

c 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

IMPLICIT REAL*S(A-H,O-Z) 

REAL'16 LAM,Ul,U2,U3,U4,MSTAR,JSTAR,CST 

REAL'8 M,L,MO,IO,MT,IT,Ml,MUO,MU1,JO,NA 

DIMENSION LAM(20),Ul(20),U2(20),U3(20),U4(20),A1V(20),AS(231), 

1 BS(231),Z(21,21),EV(21),WK(483),FV(21),CN5(20),CN6(20),CN7(20), 

2 CN8(20),FO(2),FA(21),NA(21),Wl(21),W2(21),Y(42),YDOT(42) 

COM~DN /DIM/ NF,NFP1,NSZ 

COMMON /EXTFC/ CN1,CN2,CN3,CN4,CN5,CN6,CN7 

COMMON /NLKTC/ CN9,U3,CN8 

COMMON /STATE/ X(21),XDOT(21) 

NAMELIST /INPUT/ MO,IO,A1,A2,L,PHD,C,MT,IT,NF,EI,THETA,THETAD, 

1 DT,TSTOP,TPRT 

.** •• _ •••• -•••• INPUT - OUTPUT FILES 

FILE #5 NAMELIST INPUT FILE 

**"**'**'·*"-"'*""""'-'*00002600 

FILE #6 PRINTED OUTPUT FILE 

FILE #8 INPUT FILE OF EIGENVALUES & MODAL 

TIP BODY 

00002700 

00002800 

PARAMETERS FOR BEAM WITH 00002900 

00003000 

00003100 

* •••• ·······*····DESCRIPTION OF /INPUT/ LIST ITEMS··················00003200 

"MO" ORBITER MASS 

"10" ORBITER MOMENT OF INEPTIA ABOUT ITS MASS CENTER 

00003300 

00003400 

00003500 

PERPENDICULAR TO PLANE OF MOTION 00003600 

"A1","A2" VECTOR FROM ORBITER MASS CENTER TO BEAM ATTACHMENT POINT 00003700 

(EXPRESSED IN ORBITER FRAME) 

"L" BEAM LENGTH 

"RHO" MASS PER UNIT LENGTH OF BEAM 

"EI" BENDING STIFFNESS IN BEAM 

IIC" TIP BODY MASS CENTER OFFSET 

"MT" TIP BODY MASS 

"IT" TIP BODY MOMENT OF INERTIA ABOUT ITS MASS CENTER FOR AXIS 

PERPENDICULAR TO PLANE OF MOTION 

"NF" NUMBER OF CANTILEVERED-TIP BODY MODES TO BE RETAINED 

FOR EXPANSION OF BEAM ELASTIC DISPLACEMENT 

"THETA" ORBITER INITIAL ATTITUDE (DEG ) 

"THETAD" ORBITER INITIAL ATTITUDE RATE (DEG /SEC) 

"DT" INTEGRATION TIME STEP 

"TSTOP" FINAL TIME 

"TPRT" TIME INTERVAL BETWEEN CONSECUTIVE PRINTOUTS 

81 

00003800 

00003900 

00004000 

00004100 

00004200 

00004300 

00004400 

00004500 

00004600 

00004700 

00004800 

00004900 

00005000 

00005100 

00005200 

00005300 
ISN 0011 

ISN 0012 

ISN 0013 

ISN 0014 

ISN 0015 

ISN 0016 

ISN 0017 

ISN 0018 

ISN 0019 

ISN 0020 

ISN 0021 

ISN 0022 

ISN 0023 

ISN 0024 

ISN 0025 

ISN 0026 

ISN 0027 

ISN 0028 

ISN 0029 

ISN 0030 

ISN 0031 

ISN 0032 

ISN 0033 

ISN 0034 

ISN 0035 

ISN 0036 

ISN 0038 

ISN 0039 

ISN 0040 

ISN 0041 

ISN 0043 

ISN 0044 

ISN 0045 

ISN 0046 

ISN 0048 

ISN 0049 

ISN 0050 

ISN 0051 

ISN 0053 

ISN 0054 

ISN 0055 

ISN 0056 

ISN 0058 

ISN 0059 

ISN 0060 

ISN 0061 

ISN 0062 

ISN 0063 

ISN 0064 

C 

READ(5,INPUT) 

WRITE(6.7) 

WRITE(6,l) MO,IO 

FORMAT(lHO,'MO ~ ',lPE17.8,5X,'IO ',lPE17.8) 

WRITE(6,2) MT,IT 

2 FORMAT(lH ,'MT = ',lPE17 8,5X,'IT ,. ',lPE17.8) 

WRITE(6.3) Al,A2 

3 FORMAT(lH ,'A1 = ',lPE17 8,5X,'A2 ',lPE17 8) 

WRITE(6,4) L,C 

4 FORMAT(lH ,'L" ',lPE178,5X,'C R ',lPE17 8) 

WRITE(6,5) RHO,EI 

5 FORMAT( lH ,'RHO =' ,lPE17 8,5X.'EI ' ,lPE17 8) 

WRITE(6,6) DT,TPRT 

6 FORMAT(lH ,'DT " ',lPE17 8,5X,'TPRT =',lPE16 8) 

WRITE(6,8) THETA,THETAD 

7 FORMAT(lHl,13X,'DATA FROM NAMELIST INPUT') 

8 FORMAT(lH ,'THETA e',lPE15 8,3X,'THETAOOT "'lPE15 8) 

C 

C TEST FOR DATA CONSISTENCY WITH VALUES ON DISC DATA SET 

C 

10 

11 

12 

13 

14 

15 

16 

17 

C 

READ(8) NFT,MSTAR,JSTAR,CST 

Dl=DABS(DBLEQ(MSTAR)-MT/(RHO"L» 

D2·DABS(DBLEQ(JSTAR)-(IT~MT·C**2)/(RHO·L··3» 

D3=DABS(DBLEQ(CST)-C/L) 

Dl=01/0BLEQ(MSTAR) 

02=02/0BLEQ(JSTAR) 

D3=03/0BLEQ(CST) 

IER=O 

IF(NFT GE NF) GO TO 11 

IER=l 

WRITE(6,10) NF,NFT 

FORMAT(lHO.5X,'" * * FATAL ERROR * " .',2X,I3.' CANTILEVERED'. 

1 ' TIP-BOOY MODES REQUESTED DATA ON DISC ONLY FOR',I3, 

2' MODES') 

IF(Ol LE 00100) GO TO 13 

IER=l 

WRITE(6,12) MSTAR 

FORMAT(lHO,5X," • • FATAL ERROR * • ·',2X,'MSTAR(OISC)R',G13 5) 

IF(02 LE 00100) GO TO 15 

IERR 1 

WRITE(6,t4) JSTAR 

FORMAT(lHO,5X,'* * * FATAL ERROR* " " ',2X,'JSTAR(OISC)=',G13 5) 

IF(03 .LE 00100) GO TO 17 

IER"l 

WRITE(6,16) CST 

FORMAT(lHO,5X,'· • * FATAL ERROR" * * ',2X,'CSTAR(DISC)=',G13 5) 

IF(IER NE 0) STOP 

C COMPUTE CONSTANT QUANTITIES 

C 

R5=EI/(RHO*L**4) 

FCc 1591549*OSORT(R5) 

Ml=RHO*L+MT 

M=MO~M1 

MUO=MO/M 

MU1=Ml/M 

81=( 5*RHO*L**2+MT*(L+C»/Ml 

82 

00005400 

00005500 

00005700 

00005800 

00005900 

00006000 

00006100 

00006200 

00006300 

00006400 

00006500 

00006600 

00006700 

00006710 

00006720 

00006800 

00007000 

00007010 

00007100 

00007200 

00007300 

00007400 

00007500 

00007600 

00007700 

00007800 

00007900 

00008000 

00008100 

00008200 

00008300 

00008400 

00008500 

00008600 

00008700 

00008800 

00008900 

00009000 

00009100 

00009200 

00009300 

00009400 

00009500 

00009600 

00009700 

00009800 

00009900 

00010000 

00010100 

00010200 

00010300 

00010400 

00010500 

00010600 

00010700 

00010800 

00010900 

00011000 
ISN 0065 

ISN 0066 

ISN 0067 

ISN 0068 

ISN 0069 

ISN 0070 

ISN 0071 

ISN 0072 

ISN 0073 

ISN 0074 

ISN 0075 

ISN 0076 

ISN 0077 

ISN 0078 

ISN 0079 

ISN 0080 

ISN 0081 

ISN 0082 

ISN 0083 

ISN 0084 

ISN 0085 

ISN 0086 

ISN 0087 

ISN 0088 

ISN 0089 

ISN 0090 

ISN 0091 

ISN 0092 

ISN 0094 

ISN 0095 

ISN 0096 

ISN 0097 

ISN 0098 

ISN 0099 

ISN 0100 

ISN 0101 

ISN 0102 

ISN 0103 

ISN 0104 

ISN 0106 

ISN 0107 

ISN 0108 

ISN 0109 

ISNOll0 

C 

JO=RHO*L**3/3 000~IT~MT*(C~L)**2 

Rl=RHo*L/M 

R2=Al/L 

R3=A2/L 

R4=Bl/L 

CSTAR=C/L 

AOO=(IO~JO)/(RHO*L**3)~MUO*MU1*(R2'*2~R3·*2~2 

1 -MU1'*2*R4**2/Rl 

000*R2*R4)/Rl 

00011100 

00011200 

00011300 

00011400 

00011500 

00011600 

00011700 

00011800 

00011900 

C READ & PRINT EIGENVALUES AND MODAL PARAMETERS FOR BEAM~TIP BODY 

C 

00012000 

00012100 

00012110 

00012200 

00012300 

00012400 

20 

21 

22 

23 

24 

C 

WRITE(6,23) 

WRITE(6,21) 

DO 20 N=I,NF 

REAo(8) LAM(N),Ul(N),U2(N),U3(N),U4(N) 

WRITE(6,22) N,LAM(N),Ul(N),U2(N),U3(N),U4(N) 

FORMAT(IHO,T4,'N',TI2,'LAM',T27,'Ul',T40,'U2',T49,'U3',T66,'U4') 

FoRMAT(IH ,T3,I2,T8,G12 5,T24,F9 4,T36,F9 4,T47,F9 5,T60,G13 5) 

FORMAT(IHO,/, l1X, 'EIGENVALUES & MODAL PARAMETERS FROM DISC FILE') 

WRITE(6,24) 

FoRMAT(IHO,/,7X,'SYSTEM NATURAL FREOUENCIES') 

00012500 

00012600 

00012800 

00012810 

00012820 

C COMPUTE TERMS All 

C 

00012830 

00012900 

00013000 

00013100 

00013200 

00013300 

00013400 

00013500 

00013600 

00013700 

00013800 

00013900 

00014000 

00014100 

00014200 

00014300 

00014400 

00014500 

00014600 

00014700 

00014800 

00014900 

00015000 

00015100 

00015200 

00015300 

00015400 

00015500 

00015600 

00015700 

00015800 

00015900 

00016000 

30 

C 

C 

C 

C 

40 

41 

50 

51 

C 

C 

C 

DO 30 K=I,NF A1V(K)=MUO*R2*U3(K)~U4(K)-MU1*R4*U3(K) 

STORE "A" IN SYMMETRIC STORAGE MODE - "AS" 

STORF "K" IN SYMMETRIC STORAGE MODE - "BS" 

AS( 1)=AOO 

N=l 

DO 41 I=1,NF 

N=N~1 

AS(N)=A1V(I) 

DO 40 J=I,I 

N=N~l 

AS(N)=-Rl*U3(J)*U3(I) 

IF(I EO J) AS(N)=AS(N)~l 

CONTINUE 

CONTINUE 

BS( 1)=0 000 

N=1 

DO 51 1= " NF N=N~1 

BS(N)=O 000 

DO 50 J= I, I N=N~1 

BS(N)=O 000 

IF(I EO J) BS(N)=LAM(I) 

CONTINUE 

CONTINUE 

000 

GET EIGENVALUES AND EIGENVECTORS FOR 2,3, 

NFP1=NF~1 

NSZ=2*NFP1 

DO 66 N=2,NFPl 

83 

,NFP 1 DEGREES OF FREEDOM00016100 

00016200 

00016300 

00016400 

00016500 
ISH 0111 

ISH 0112 

ISH 0114 

ISH 0115 

ISH 0116 

ISH 0117 

ISN0118 

ISNOl19 

ISH 0120 

ISN 0121 

ISN 0122 

ISN 0123 

ISN 0124 

ISN 0125 

ISN 0126 

ISN 0127 

ISN 0128 

ISN 0129 

ISH 0130 

ISN 0131 

ISH 0132 

ISH 0133 

ISN 0134 

ISN 0135 

ISN 0136 

ISN 0137 

ISH 0138 

ISN 0139 

ISN 0140 

ISH 0141 

ISN 0142 

ISN 0143 

ISN 0144 

ISN 0145 

ISN 0146 

ISN 0147 

ISN 0148 

ISN 0149 

ISH 0150 

ISN 0151 

ISH 0153 

ISN 0154 

CALL EIGZS(BS,AS,N,I,EV,Z,21,WK,IER) 

IF(IER EO. 0) GO TO 62 

WRITE(6,61) N,IER 

61 FORMAT(IHO,5X,'ERROR IN EIGENVALUE EXTRACTION MATRIX OROER~',I2, 

13X,'IER=',I3) 

STOP 

62 WRITE(6,63) N 

63 FORMAT(IHO,I2,' DEGREES OF FREEDOM IN EIGENVALUE PROBLEM') 

00 64 I=I,N 

WS=FC*OSQRT(EV(I» 

64 WRITE(6,65) I,WS 

65 FORMAT(IH ,2X,'MOOE',I2,2X,'SYSTEM FREQUENCY(HZ )=',GI3 5) 

66 CONTINUE 

C 

C NORMALIZE EIGENVECTORS 

C 

DO 70 N= I ,NFPI 

70 CALL NORM(AS,NFP1,Z(I,N» 

C 

C CALCULATE CONSTANTS IN MOTION EOUATIONS 

C 

00 80 1=I,NFPI 

80 FV(I)=R5 t EV(I) 

CN1=RHO*L**3 

CN2=MUI*(AI+BI)/CNI 

CN3=MU1"A2/CNI 

CN4=(1 ODO+CSTAR+MUOtR2-MU1*R4)/(RHO*Lt*2) 

CN9=1 ODO/(M*L) 

DO 90 I=I,NF 

CN5(1)=Ul(I)/CNI 

CH6(I)=U2(I)/(RHO*L**2) 

CN7(I)=U3(I)*CN9 

90 CN8(I)=MUOtR3 t U3(I) 

C 

C SET INITIAL CONDITIONS 

C 

c 

T=O 000 

CPRT=O 000 

x( 1)=TmU* 0174532 

XOOT(I)=THETAO* 0174532 

C INITIAL DEFORMATION AND RATE ARE SET TO ZERO 

C 

00 100 I=I,NF 

x(I + 1) =0 .000 

100 XOOT(I+I)=O.OOO 

C 

C CALCULATE Y AT T=O 

C 

00 112 I"I,NFPl 

II "I 

WI(I)·O 000 

W2(I)=0 000 

DO III u= I • NFP 1 

uu=u 

IF( I GE. u) GO TO 110 

II=u 

uu=I 

84 

00016600 

00016700 

00016800 

00016900 

00017000 

00017100 

00017200 

00017300 

00017500 

00017600 

00017700 

00017800 

00017900 

00018000 

00018100 

00018200 

00018300 

00018400 

00018500 

00018600 

00018700 

00018800 

00018900 

00019000 

00019100 

00019200 

00019300 

00019400 

00019500 

00019600 

00019700 

00019800 

00019900 

00020000 

00020100 

00020200 

00020300 

00020400 

00020500 

00020600 

00020700 

00020800 

00020900 

00021000 

00021100 

00021200 

00021300 

00021400 

00021500 

00021600 

00021700 

00021800 

00021900 

00022000 

00022100 

00022200 

00022300 

00022400 
ISN 0155 

ISN 0156 

ISN 0157 

ISN 0158 

ISN 0159 

ISN 0160 

ISN 0161 

ISN 0162 

ISN 0163 

ISN 0164 

ISN 0165 

ISN 0166 

ISN 0167 

ISN 0168 

ISN 0169 

ISN 0170 

ISN 0171 

ISN 0172 

ISN 0173 

ISN 0174 

ISN 0175 

ISN 0176 

ISN 0177 

ISN 0178 

ISN 0179 

ISN 0180 

ISN 0181 

ISN 0182 

ISN 0183 

ISN 0184 

ISN 0185 

ISN 0186 

ISN 0187 

ISN 0188 

ISN 0189 

ISN 0190 

ISN 0191 

ISN 0192 

ISN 0193 

ISN 0195 

ISN 0196 

ISN 0197 

ISN 0198 

110 LL=II'(II-l)/2+JJ 

Wl(I)=Wl(I)+AS(LL)'X(J) 

111 W2(I)=W2(I)+AS(LL)*XOOT(J) 

112 CONTINUE 

DO 114 I=l,NFPl 

Y(I)=O 000 

Y(NFP1+I)=0 000 

DO 113 J=1,NFPl 

Y(I)=Y(I)+Z(J,I)*W1(J) 

113 Y(NFP1+I)=Y(NFP1+I)+Z(J,I)*W2(J) 

114 CONTINUE 

WRITE (6, 115) 

115 FORMAT(1HO,l,20X,'TIME RESPONSE') 

C 

C CALCULATE EXTERNAL FORCES ON ORBITER AND TIP BODY 

C 

120 CALL ORBFOR(T,FO,GO) 

CALL TPBFOR(T,FP,GP) 

CALL EXTF(FO,GO,FP,GP,FA) 

C 

C CALCULATE NON-LINEAR TERMS 

C 

CALL NLKT(FO,NA) 

DO 131 I=1,NFPl 

W1 (I )=0 000 

DO 130 J=l,NFPl 

130 Wl(I)=W1(I)+Z(J,I)"(FA(J)+NA(J» 

131 CONTINUE 

C 

C CALCULATE ·YOOT" 

C 

DO 140 I=l,NFPl 

YOOT(I)=Y(NFP1+I) 

YDOT(NFP1+I)=Wl(I)-FV(I)'Y(I) 

140 CONTINUE 

C 

C INTEGRATE DIFFERENTIAL EQUATIONS IN FIRST ORDER FORM 

C 

CALL ODESLV(NSZ,Y,YDOT,DT) 

T=T+DT 

CPRT=CPRT+DT 

DO 151 I=l,NFPl 

x(I )=0 000 

XDOT (I ) =0 000 

DO 150 J=1,NFP1 

X(I)=X(I)+Z(I,J)*Y(J) 

150 XOOT(I)=XOOT(I)+Z(I,J)*Y(NFP1+J) 

151 CONTI NUE 

C 

THETA=X(1)*5729578 

THETAD=XDOT(1)'5729578 

C PRINT OUTPUT 

C 

IF(CPRT LT TPRT) GO TO 170 

CPRT=O 000 

WRITE(6,160) T 

160 FORMAT(1HO,'TIME =',F7 3) 

WRITE(6,161) THETA 

85 

00022500 

00022600 

00022700 

00022800 

00022900 

00023000 

00023100 

00023200 

00023300 

00023400 

00023500 

00023510 

00023520 

00023600 

00023700 

00023800 

00023900 

00024000 

00024100 

00024200 

00024300 

00024400 

00024500 

00024600 

00024700 

00024800 

00024900 

00025000 

00025100 

00025200 

00025300 

00025400 

00025500 

00025600 

00025700 

00025800 

00025900 

00026000 

00026100 

00026200 

00026300 

00026400 

00026500 

00026600 

00026700 

00026800 

00026900 

00027000 

00027100 

00027200 

00027300 

00027400 

00027500 

00027600 

00027700 

00027800 

00027900 

00028000 
ISN 0199 

ISN 0200 

ISN 0201 

ISN 0202 

ISN 0203 

ISN 0204 

ISN 0205 

ISN 0206 

ISN 0207 

ISN 0208 

ISN 0210 

ISN 0211 

ISN 0002 

ISN 0003 

ISN 0004 

ISN 0005 

ISN 0006 

ISN 0007 

ISN 0008 

ISN 0009 

ISN 0010 

ISN 0002 

ISN 0003 

ISN 0004 

ISN 0005 

ISN 0006 

ISN 0002 

ISN 0003 

ISN 0004 

ISN 0005 

ISN 0006 

ISN 0007 

ISN oooa 

ISN 0009 

ISN 0010 

ISN 0011 

161 

162 

163 

164 

165 

C 

170 

C 

C 

C 

C 

C 

10 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

10 

WRITE(6,165) THE TAD 

FORMAT(IH ,'THETA ~',IPEI7 8,' DEG') 

WRITE(6,162) 

FORMAT(IHO,' MODAL COORDINATES',10X,'DERIV MODAL COORDINATES') 

DO 163 I=2,NFPl 

u:I-l 

WRITE(6,164) u,X(I),XDOT(I) 

FORMAT(IH ,I2,2X,1PE17 a,1ox,1PE17 a) 

FORMAT(IH ,'THETADOT -',IPEI7 8,' DEG/SEC') 

If(T GE TSTOP) STOP 

GO TO 120 

END 

SUBROUTINE ORBFOR(T,fO,GO) 

THIS SUBROUTINE CALCULATES THE EXTERNAL FORCE "FO" ON THE ORBITER 

(IN ORBITER FRAME) AND NET MOMENT "GO" PERPENDICULAR TO PLANE OF 

MOTION AT TIME "T" 

IMPLICIT REAL*8(A-H,O-Z) 

DIMENSION fO(2) 

FO(I):O 000 

FO(2):0 000 

GO=O 000 

FDRMAT(IH ,3X,'FO:',2EI3 5,3X,'GO~',EI3 5) 

RETURN 

END 

SUBROUTINE TPBFOR(T,F,G) 

TIllS SUBROUTINE CALCULATES THE EXTERNAL FORCE "F" ON THE TIP BODY 

(ACTING TRANSVERSE TO BEAM NEUTRAL AXIS) AND MOMENT "G" 

PERPENDICULAR TO PLANE Of MOTION AT TIME "T" 

f:O 000 

G:O 000 

RETURN 

END 

SUBROUTINE EXTF(FO,GO,FP,GP,F) 

THIS SUBROUTINE ASSEMBLES THE VECTOR OF EXTERNAL FORCES "F" GIVEN 

BY EO (4-31) THE ORBITER FORCE AND MOMENT "FO","GO" AS WELL AS THE 

FORCE ANO MOMENT ON THE TIP BODY "FP","GP" ARE INPUT 

IMPLICIT REAL*8(A-H,O-Z) 

DIMENSION FO(2),F(NFP1) 

COMMON IDIMI NF,NFP1,NSZ 

COMMON IEXTFCI CN1,CN2,CN3,CN4,CN5(20),CN6(20),CN7(20) 

F(I)=Go/CN1-CN2*FO(2)+CN3*FO(I)+GP/cNl+CN4*FP 

DO 10 I:l,NF 

F(I+l)=CN5(1)*GP+CN6(1)*FP-CN7(1)*(FO(2)+FP) 

RETURN 

END 

86 

00028010 

00028100 

0002a200 

00028300 

00028400 

00028500 

00028600 

00028700 

00028710 

00028800 

00028900 

00029000 

00029100 

00029200 

00029300 

00029400 

00029500 

00029600 

00029700 

00029800 

00029900 

00030000 

00030100 

00030200 

00030300 

00030400 

00030500 

00030600 

00030700 

00030800 

00030900 

00031000 

00031100 

00031200 

00031300 

00031400 

00031500 

00031600 

00031700 

00031800 

00031900 

00032000 

00032100 

00032200 

00032300 

00032400 

00032500 

00032600 

00032700 

00032800 

00032900 

00033000 
ISN 0002 

ISN 0003 

ISN 0004 

ISN 0005 

ISN 0006 

ISN 0007 

ISN 0008 

ISN 0009 

ISN 0010 

ISN 0011 

ISN 0012 

ISN 0013 

ISN 0014 

ISN 0015 

ISN 0016 

ISN 0017 

ISN 0018 

ISN 0002 

ISN 0003 

ISN 0004 

ISN 0005 

ISN 0006 

ISN 0007 

ISN 0008 

ISN 0009 

ISN 0010 

ISNOOll 

ISN 0012 

ISN 0013 

ISN 0014 

ISN 0015 

ISN 0016 

ISN 0017 

ISN 0018 

ISN 0019 

ISN 0002 

ISN 0003 

ISN 0004 

ISN 0005 

c 

C 

C 

C 

10 

20 

C 

C 

C 

C 

C 

20 

10 

30 

C 

C 

C 

C 

C 

C 

C 

C 

SUBROUTINE NLKT(FO,N) 

THIS SUBROUTINE ASSEMBLES THE VECTOR OF NON-LINEAR TERMS "N" 

GIVEN BY EO (4-32) THE EXTERNAL FORCE ON THE ORBITER "FO" IS INPUT 

IMPLICIT REAL*8(A-H,O-Z) 

REAL*8 N 

REAL*16 U3 

COMMON /NLKTC/ CN9,U3(20),CN8(20) 

COMMON /STATE/ X(21),XDDT(21) 

COMMON /DIM/ NF,NFP1,NSZ 

DIMENSION FO(2),N(NFP1) 

N(l)=O 000 

DO 10 K=1,NF 

N(1)=N(1)+DBLEQ(U3(K»*X(K+1) 

N(1)=N(1)*CN9*FO(1) 

THD2=XDOT(1)'*2 

DO 20 K=l,NF 

N(K+l)=CN8(K)*THD2 

RETURN 

END 

SUBROUTINE NORM(AS,N,X) 

THIS SUBROUTINE NORMALIZES THE EIGENVECTOR "X" WITH 

SYMMETPIC POSITIVE DEFINITE MATRIX 'A' OF 

SYMMETRIC STORAGE MODE AS THE VECTOR "AS" 

IMPLICIT REAL'8(A-H,0-Z) 

DIMENSION AS(l),X(l) 

S=AS( 1 )*X( 1 )H2 

NC=l 

DO 10 I=2,N 

IM1=I-1 

DO 20 J=1,IMl 

NC=NC+1 

S=S+2 ODO*AS(NC)*X(I)*X(J) 

NC=NC+l 

S=S+AS(NC)*X(I)'*2 

CONTINUE 

S=DSQRT( S) 

DO 30 1= 1,N 

x(I )=x(I )/S 

RETURN 

END 

SUBROUTINE ODESLV(N,Y,DERIV,H) 

IMPLICIT REAL*8(A-H,O-Z) 

ORDER "N" 

RESPECT TO THE 

STORED IN 

DIMENSION OERIV(N),Y(N),DERIVO(42),BD1(42,2),BD2(42,2),BD3(42) 

OATA INTF/1/,Cl/0 0/,C2/0 0/,C3/0 / 

THIS SUBROUTINE INTEGRATES THE FIRST ORDER SYSTEM OF ORDINARY 

DIFFERENTIAL EQUATIONS "DY/DT=DERIV" BY THE ADAMS METHOO 

USING THIRD ORDER DIFFERENCES 

N- SIZE OF SYSTEM 

Y- VECTOR OF INITIAL VALUES ON INPUT "yo IS OVERWRITTEN 

WITH THE NEW SOLUTION 

H- STEP SIZE 

87 

00033100 

00033200 

00033300 

00033400 

00033500 

00033600 

00033700 

00033800 

00033900 

00034000 

00034100 

00034200 

00034300 

00034400 

00034500 

00034600 

00034700 

00034800 

00034900 

00035000 

00035100 

00035200 

00035300 

00035400 

00035500 

00035600 

00035700 

00035800 

00035900 

00036000 

00036100 

00036200 

00036300 

00036400 

00036500 

00036600 

00036700 

00036800 

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ISN 0006 

ISN 0008 

ISN 0009 

ISN 0010 

ISN 0011 

ISN 0012 

ISN 0013 

ISN 0014 

ISN 0015 

ISN 0016 

ISN 0017 

ISN 0018 

ISN 0019 

ISN 0020 

ISN 0021 

ISN 0022 

ISN 0023 

ISN 0024 

ISN 0025 

ISN 0026 

ISN 0027 

ISN 0028 

ISN 0029 

ISN 0030 

ISN 0031 

ISN 0032 

ISN 0033 

ISN 0034 

ISN 0035 

ISH 0036 

ISN 0037 

ISN 0038 

ISN 0039 

ISN 0040 

ISN 0041 

ISN 0042 

ISN 0043 

ISN 0044 

C 

IF(N LE 42) GO TO 10 

WRITE(6,12) N 

STOP 

12 FORMAt(lHO,5X,'ERROR IN SUBROUTINE **OOESLV** CALLED 

lSIZE =',13,' EXCEEDS DIMENSION SIZE OF ARRAYS') 

10 GO TO(1000,2000,3000,4000),INTF 

C 

C FIRST CALL TO ROUTINE - EULER INTEGRATION 

C 

1000 DO 20 I=l,N 

20 OERIVO(I)=OERIV(I) 

INTF=2 

GO TO 5000 

C 

C SECOND CALL TO ROUTINE - FIRST ORDER DIFFERENCES 

C 

2000 DO 30 l=l,N 

B01(I,l)=OERIV(I)-OERIVO(I) 

B01(I,2)=B01(I,l) 

30 OERIVO(I)=OERIV(I) 

Cl= 5 

INTF=3 

GO TO 5000 

C 

C THIRD CALL TO ROUTINE - SECOND ORDER FIFFERENCES 

C 

3000 DO 40 l=l,N 

B01(I,2)=OERIV(I)-OERIVO(I) 

B07(I,l)=B01(I,2)-B01(I,l) 

B02(I,2)=B02(I,l) 

OERIVO(I)=OERIV(I) 

40 B01(I,l)=B01(I,2) 

C 

INTF=4 

C2=5 0/12 0 

GO TO 5000 

C ADAMS METHOD WITH 3RO ORDER DIFFERENCES 

C 

4000 DO 50 I=l,N 

B01(I,2)=OERIV(I)-OERIVO(I) 

B02(I,2)=B01(I,2)-B01(I,l) 

B03(I)=B02lI,2)-B02(I,l) 

OERIVO(I)=OERIV(I) 

B01(I,l)=B01(I,2) 

50 B02(I,l)=B02(I,2) 

C3=3 0/8 0 

GO TO 5000 

C 

C UPDATE VECTOR 'V' 

C 

5000 DO 60 I=l,N 

WITH STATE 

60 Y(I)=Y(I)+H*(OERIV(I)+Cl*B01(I,2)+C2*B02(I,2)+C3*B03(1» 

RETURN 

END 

88 

00038700 

00038800 

00038900 

00039000 

00039100 

00039200 

00039300 

00039400 

00039500 

00039600 

00039700 

00039800 

00039900 

00040000 

00040100 

00040200 

00040300 

00040400 

00040500 

00040600 

00040700 

00040800 

00040900 

00041000 

00041100 

00041200 

00041300 

00041400 

00041500 

00041600 

00041700 

00041800 

00041900 

00042000 

00042100 

00042200 

00042300 

00042400 

00042500 

00042600 

00042700 

00042800 

00042900 

00043000 

00043100 

00043200 

00043300 

00043400 

00043500 

00043600 

00043700 

00043800 

00043QOO 

00044000 

00044100 
EXAMPLE PROBLEM PARAMETERS 

rno 98739.5 kg 

IO 9769869.5 kg-rn 2 

rnt 875.32 kg 

1400.512 kg-rn 2 

It 

a

1 2.0 rn 

a 2 0.0 rn 

£, 20.0 m 

c 2.0 rn 

p = 21. 883 kg/rn 

EI 353520.0 N-rn 2 

rn* 2.0 

J* 0.028 

c* 0.1 

NAMELIST INPUT DATA 

&INPUT MO=98739 5.10=9769869 5,A1=2 ,A2=0 ,L=20 ,RHO=21 883,El=353520 ,C=2 , 

MT=875 32,IT=1400 512,NF=3,THETA=0 ,THETAO=O ,OT= 01,TSTOP=1 ,TPRT= 02,&ENO 

89 
OUTPUT DATA FROM PLANAR DYNAMICS PROGRAM 

4 x 104 N for all t > 0 

All un~ts are metr~c (MKS) 

DATA FROM NAMELIST INPUT 

MO 

MT 

A1 

9 873950000+04 

8 753200000+02 

2 000000000+00 

2 000000000+01 

2 188300000+01 

1 000000000-02 

00 

10 

IT 

A2 

9 769869500+06 

1 400512000+03 

o 0 

L = 

RHO = 

OT = THETA 

C 

EI 

TPRT 

2 000000000+00 

3 535200000+05 

2 000000000-02 

THETAOOT = 0 0 

EIGENVALUES & MODAL PARAMETERS FROM DISC FILE 

N 

1 

2 

3 

LAM 

1 0310 

143 31 

1220 0 

U1 

o 9087 

-4 8354 

6 0703 

SYSTEM NATURAL FREQUENCIES 

U2 

o 6760 

-0 1266 

-0 0027 

2 DEGREES OF FREEDOM IN EIGENVALUE PROBLEM 

MODE 1 SYSTEM FREQUENCY(HZ): 0 

MODE 2 SYSTEM FREQUENCY(HZ)= 531060-01 

3 DEGREES OF FREEDOM IN EIGENVALUE PROBLEM 

MODE 1 SYSTEM FREQUENCY(HZ)= 0 

MODE 2 SYSTEM FREOUENCY(HZ)= 531060-01 

MODE 3 SYSTEM FREQUENCY(HZ)= 60600 

4 DEGREES OF FREEDOM IN EIGENVALUE 

MODE 1 SYSTEM FREOUENCY(HZ ). 

MODE 2 SYSTEM FREQUENCY(HZ )= 

MODE 3 SYSTEM FREQUENCY(HZ )= 

MODE 4 SYSTEM FREQUENCY(HZ ): 

TIME RESPONSE 

TIME = 0 020 

THETA' 3 518391500-05 OEG 

THETADOT = 4 691188670-03 OEG/SEC 

PROBLEM 

o 

531060-01 

60600 

1 7669 

U3 

1 56911 

o 52240 

o 29800 

MOOAL COORDINATES 

1 -1 111891280-06 

OERIV MODAL COORDINATES 

-1 482521710-04 

2 -1 23251~140-07 -1 643349520-05 

3 -4 933956080-08 -6 578608100-06 

TIME = 0 040 

THETA = 1 7591~099D-04 OEG 

THETADOT = 9 382316970-03 OEG/SEC 

MODAL COORDINATES 

t -5 559381600-06 

2 -6 154680680-07 

3 -2 440258990-07 

DERIV MODAL COORDINATES 

-2 964948560-04 

-3 276707300-05 

-1 281842280-05 

90 

U4 

1 6540 

14854 

505870-01 
End of Document