Dynamics of a flexible extendible beam

Dynamics o f a Flexible Extendible Beam

by

Marinos Costa Stylianou B.A.Sc., University o f Toronto, 1982 M .A.Sc., University o f Toronto, 1985

A Dissertation Submitted in Partial F ulfill ’ -ent o f the Requirements for the Degree or

DOCTOR OF PHILOSOPHY

in the l.Vparimeni of Mechanical Engineering

/J J ' , '■ i > We accept this dissertation as conforming

/ to the required standard

Dr. B. Tabarrok, Supervisor (Mechanical Engineering)

Dr. A. Doige, Departyhent Member (Mechanical Engineering)

Dr. S. Dost, Department Member (Mechanical Engineering)

Dr. D. Hoffman,1 (Vutside Member (Computer Science j

Dr. D.B. cTierchas, External Examiner (Mechanical Engineering,

University o f British Columoia)

© M ARINO S COSTA S T Y LIA N O U , 1993

University o f Victoria

A ll rights reserved. Dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author.

AB STR AC T

A x ia liy -m o v m g materials arise in problems associated with spaeeeral't antennas, pipes conveying fluid and telescopic robotic manipulators. Flexible extendible beams are a special d a s ; o f ax ially -m o vin g materials, in which the axially-m o ving material is modelled as a slender beam and (he mechanism o f elastic deform a­ tion >s transverse bending.

I la m ib o n ’s principle is used to derive the governing differential equation o f motion and system invariant properties o f a llcxibio extendible beam protruding from a rigid wall w ith prescribed extrusion profile. The mass o f the system is not constant and the general analytical solution to the equation o f motion is not known. In this study, numerical solutions are obtained using liiiite-eloment analysis. How ever, instead o f follow ing the obvious (but cumbersome) approach o f using lixed-si/.e elements and increasing their number, in a step­ wise fashion, as mass elements enter the domain o f interest, a more elegant approach is follow ed wherein the num ber ol elements is fixed, while the sizes o f the elements change with time. To this end, a variable- dom ain beam finite element whose size is a prescribed function o f time is formulated.

The accuracy o f this variahle-doma.'.i beam element is demonstrated through the tim e-iiitegralion o f equa­ tions o f motion using various extrusion profiles. Additional verification is performed by the evaluation o f the system's invariant quantities, comparison with a special analytical solution, and the dynam ic stability analy­ sis o f pipes conveying iluid. The effects o f wall flexibility, tip mass, and high-frequency axial-m otion pertur­ bations to the transverse response o f the flexible extendible beam are also examined. In order to gain a deeper insight into the mechanics o f this system, the dynamic stability characteristics o f the flexible extend­ ible beam are also in vest i gated using various extrusion profiles. The effect.- o f physical dam ping, lip mass, lip support and w ali flexib ility on the stability characteristics o f this system are exam ined.

The power and versatility o f this linile-elem eiu formulation is demonstrated in a simulation o f an extruding flexible extendible beam which carries a tip mass and protrudes from a Ilexible envelope beam which imparls three- iiu'ensional rigid-body rotations to the system.

Examiners:

Dr. if. Tabiurok, Supervisor (Mechanical Engineering)

--- ;—f ... i--- ^... . " Dr. A. Doige, Department Member (Mechanical Engineering)

Dr. S. Dost, Department Member (Mechanical Engineering)

Dr. D. Hoffman, Cluix'tde Member (Computer Science)

Dr. D.B. Cherchas, External Examiner (Mechanical Engineering, University o f British Columbia)

Table of Contents

Tab le o f Contents... lii

List o f Tables...vi

List o f Figures... vii

Nomenclature... x

A cknow ledgm ents... xv

D edication... xvi

Chapter 1: Intro du ctio n...I

1.1 Problem Description and H is to ry ... 1

1.2 O bjective s... ‘I 1.3 Thesis O rganisation... 3

1.4 Synopsis... A

Chapter 2: Derivation of the Differential Equations

and the System Invariant Properties... X

2.1 H am ilton’s P rin ciple... X 2.2 A Note on Conserved Q uantities...12

2.2.1 Conserved Quantities in Second-Order Functionals with tw o Indcpcnden* Variables... 13

2.2.2 T he Flexible Extendible B e a m ... 15

2.2.3 Energy Considerations... 2.3 Concluding Remarks... 24

5.1 Element Eagrangian Function... 25 5.2 l-inite-Flcm cnt Discretization...27 5.5 Finite-Element Liquations...31 3.4 Concluding Remarks... 32

Chapter

4 : T im e I n t e g r a t io n o f S y s te m E q u a t io n s ... 33 4.1 Verification Examples... 33

4.1.1 Experiments and Simulations by Yuh and Young ( 1 9 9 1) ... 34

4.1 2 Energy Considerations... 4 0 4 . 1.3 Conserved Q uantities... 42

4 . 1.4 Parabolic Extrusion... 45

4.2 A Flexible Extendible Beam with a Tip Mass...48

4.3 A Nested A xia lly-M o vin g Beam... 50

4.4 Active Vibration Suppression Using High Frequency Perturbations... 55

4.5 Concluding Remarks... 57

Chapter

5:

Stability Analysis...59

5.1 Dynamic S ta b ility ...59

5.2 Constant-Velocity Extrusion... 62

5 .2 .1 The Simplest C a s e ... 63

5.2.2 Effect o f Physical D a m p in g ...68

5.2.3 Effect o f a Flexible Envelope B e a m ... 71

5.2.4 Effect o f a Tip M ass... 76

5.2.5 Effect o,' T ip S u p p o rt...80

5.3 Constant-v ciocity Retraction... 84

5.3.1 The Simplest C a s e ...84

5.3.2 Effect o f Physical D a m p in g ... 88

5.3.3 Effect o f W all Flexibility. T ip Mass. T ip support... 92

5.4 Constant-Acceleration Extrusion and R etraction ... 92

5.5 Parabolic Extrusion...94

5.6 Pipes Conveying F lu id ... 97

5.6.1 Pipes Supported at Both End:;... 97

3.6 2 Cantilever Pipes...100

5.7 Concluding Remarks...101

C hapter 6: A Spatial Flexible Extendible Beam

104 6.1 Finlte-Eleniv.n» Discretization... 10 4 6.2 Velocity Distribution Along the Flexible Extendible

B eam ... ... 1 0 7 6.3 Complementary Kinetic Energs ... ... 1 0 8

6.4 Flexural S fa in Energy... ...112

6.5 Energy Due co Longitudinal Loads... ...113

6.6 Finite-Element Er,nations... ... 114

6.7 S im u latio ns... ... 116

6 . 7 .1 A Conslanl-Lenglh Beam in R o ta tio n ... 6 .7 .2 Two-dim ensional Experiments and .Simulations by Yuli ...117

and Y o u n g ... ...118

6.7 .3 A Spatial Flexible Extendible Beam with a Ti p M ass... ... 17 s 6.8 Concluding Remarks... I -8

Chapter 7: Closing Comments...

...120

References...

...131

Appendices...

...! 35

A .I: Second-Order Functionals with two Independent Variables.... ...1 35 A .II: The Flexible Extendible Beam... ...130

B .I-B .V .: Components o f System M atrices... 141 145 C.I: Newmark M eth od... ... 146

C .II: Procedure for Adaptive Time-Step Size... ... 140

D.I: Experimental Results by Yuli and Young... ... 151

D.1I: Simulation Results by Yub and Young... ...152

F .I-E .V III: Components o f Invariant Matrices and Vectors... 153 158 F: Stability Analysis by Option, (b) o f Section 5 .2 ... ...150

G:

Cantilever Pipes... ... 162

List of Tables

Table 4.1: Parameters for experimental test cases, Yuh and Young (1991)... 35

Table 4,2: Parameters for simulation cases Yuh and Young (1991)...37

Table 4.3: Parameters for vibration suppression... 55

Table 5 .1: Critical non-dimensional velocities versus number o f elements... 67

Table 5.2: Effect o f physical damping on the critical velocities...68

Table 5.3: Effect of the flexible envelope on the critical ''elocities...73

Tabi<- 5.4: Effect o f a tip mass on the critical velocities...76

Table .‘>.5: Effect of tip constraints on the critical velocities... 81

Table 5.6: Critical non-dimensional velocities... 84

'Fable 5.7: Effect o f physical damping on the critical velocities...88

'Fable 5.8: Effect o f axial-acceleration on the critical extruding velocities...93

Table 5.9: Effect o f axial-acceleration on the critical ietracting velocities...94

'Fable 5.10: Critical values for sim ilarity parameter U...94

Table 5 .11: Simply-supported pipe, |3 = 0.1... 99

'Fable 5.12: Clamped-clamped pipe, (3 = 0.1... 99

'Fable 5.13: Clamped-clamped pipe, (3 = 0.5... 99

Table 5.14: Cantilever pipe, (3 = 0.6... 100

Fable 6.1: Geometric and material properties o f constant-length lin k ... 117

'Fable 6.2: Parameters for experimental test cases, Yuh and Young (1991)...118

Table 6.3: Parameters for simulation cases, Yuh and Young (1 9 9 1)... 120

^HT & 'm 1 10

List of Figures

Figure 1.1: A flexible extendible beam...2

Figure 2.1: The flexible extendible beam... 6

Figure 2.2: Configuration space o f the flexible extendible beam...17

Figure 3.1: Variable-domain beam elements... 2o Figure 3.2: Element coordinate system (ECS)... 27

Figure 3.3: Nodal variables o f a beam finite element... 28

Figure 4 1: l o t case # 1. Constant-velocity retraction... .35

Figure 4.2: Test case # 2. Constant-velocity extrusion...16

Figure 4.3: Test case # 3. Constant-acceleration retraction... 16

Figure 4.4: Test case ^ 4. Constant-acceleration extrusion...17

Figure 4.5: Simulation 4 1. Low-frequency extrusion... 18

Figure 4.6: Simulation # 3. Low-frequency retraction...16

Figure 4.7: Simulation # 5. High-frequency extrusion... 36

Figure 4.8: Simulation # 7. High-frequency retraction... 46

Figure 4.9: Test case #1 hotal energy during constant-velocity retraction...a I Figure 4.10- Test case # 2. Total energy during constant-velocity extrusion... 4 1 Figure 4.11. Right- and left-hand sides o f Eq. (4.6) with a four-element model... 43

Figure 4.12: Right- and left-hand sides o f Eq. (4.7) with a lour-elcment model... 44

Figure 4.13: Error (RHS - LHS) for Eq. (4.7) with a four-element model...44

Figure 4.14: Error (RHS - LHS) for Eq. (4.7) with a ten-element model...45

Figure 4.15: First, sim ilarity mode... 46

Figure 4. ,o: Second sim ilarity mode... 46

Figure 4.17: Tip-deflection history o f the sim ilarity sim ulation... 47

Figure 4.18: Test case # 4 with a tip mass. Constant-acceleration cxmi'sion... 46

Figure 4.19: Simulation #7 with a tip mass. High-frequency retraction... 46

' igure 4.20: A nested axially-m oving beam...50

1 igure 4.21: Geometric and material properties o f the beams...5 1 Figure 4.22: Nested extruding right-hand tip ...53

Figure 4.24: Retracting left-hand lip with active transverse acceleration... 54

Figure 4.25: Te.it case # I with high-frequency perturba’ion (retr, . don)...56

Figure 4.26: 'Test case # 2 with high-frequency perturbation (extrusion)... 57

Figure 5 .1: Real and imaginary components of the first and second eigenvalues. 64 Figure 5.2: Real and imaginary components o f the third and fou th eigenvalues..65

Figure 5.3: Real and imaginary components o f the fifth and sixth eigenvalue... 66

Figure 5.4: Kffeci o f heavy damping on the first and second eigenvalues...69

[■igure 5.5: Fffect o f heavy damping on the third and fourth eigenvalues... 70

'■'igure 5.6- A nested flexible extendible beam... 72

[•'igure 5.7: Fffect o f the s tiff envelope on the first and second eigenvalues... 74

[•'igure 5.8: Effect o f the s tiff envelope on the third and fourth eigenvalues... 74

Figure 5.9: Fffect o f the soft envelope on the first and second eigenvalues... 75

6 igure 5.10: Effect o f the soft envelope on the third and fourth eigenvalues... 75

[•igure 5.11: .Effect o f a 50% tio mass on the first and third eigenvalues...77

Fig re 5 .12: Fffect o f a 50% tip mass on the second and fourth eigenvalues... 78

Figure 5.13: Simply-supported tip. First and second eigenvalues...81

[•'igure 5.14: Simply-supported tip. Third and fourth eigenvalues... 82

[•'igure 5.15: Ciamped tip. First and second eigenvalues... 83

[•’igure 5.16: Clamped tip. Third and fourth eigenvalues...83

Figure 5.17: Real and imaginary components o f the first and second eigenvalues..86

Figure 5 .18: Real and imaginary components of the third and fourth eigenvalues..87

Figure 5.19: Real and imaginary components of the fifth and sixth eigenvalues. ...87

Figure 5.20: Effect o f heavy damping on the first and second eigenvalues. ... 89

Figure 5.21: Effect o f heavy damping on the third and fourth eigenvalues... 90

Figure 5.22: Effect o f heavy damping on the fifth and sixth eigenvalues ... 91

Figure 5.23: Real and imaginary components o f the first and second eigenvalues..96

Figure 5,24. Real and imaginary components o f the third and fourth eigenvalues..96

Figure 5.25: A pipe coi .eying fluid... ...98

Figure 6.1: A spatial flexible extendible beam... 105

Figure 6.2: Nodal variables o f a beam finite element in the .vr-plane... 106

Figure 6.3: T ip response o f a constani-length lin k in rotation...118

Figure 6.4: Test case # 5. Constant-velocity ret; action and rotation...119

Figure 6.5: Test case # 6. Constant-velocity extrusion and rotation... 120

Figure 6.6: Simulation # 2. Low-frequency extrusion and rotation... 121

Figure 6.7: Simulation # 4 . Low-frequency retraction and rotation... 121

Figure 6.8: Simulation # 6. High-frequency extrusion and rotation...122

Figure 6.9: Simulation # 8. High-frequency retraction and rotation... 122

Figure 6.10: Simulation # 8. Initial system configuration... 123

Figure 6 .11: Simulation # 4. Comparison to a quasi-static form ulation...12 ! Figure 6.12: A nested spatial flexible extendible beam carrying a tip mass 126

Figure 6.13: Geometric and material properties of the beams... 12b Figure 6.14: Tip-deilection in y-direction... 127 Figure b. 13: T ip-deflection in z-Mrection... 128

Nomenclature

u consiunt acceleration o f beam extru.-acn/retraction

A beam cross sectional area

\Uj\ element invariant matrices ( Appendix E)

\ Aj \ system in va ria n t m atrices (Eq. (4,6))

{/;,■} clement invariant vectors (Appendix E)

| r / | elem ent d yn a m ic d a m p in g m a trix (Eq. (3.24). (6.57))

|('21 elem ent d y n a m ic d a m p in g m a trix (Eq. (3.25), (6.57))

I n i e lem ent d y n a m ic d a m n in g m a trix (Eq. (6.57))

K (’(/l elem ent and system e q u iva le n t d a m p in g m a trix (Eqs. (3.30), (5.26), (6.60))

(I, location o f beam’s tip with respect to ECS (Fig. (3.1), Eq. (3.5))

!•'. Young’s modulus

K( t ) total energy (Eqs. (2.58), (4.5))

H„, / / , y , HXl, 11 xx components of the Hamiltonian vectors (Eqs. (2.23), (2.37))

|A| d e m o n stiffness m a m '. (Eq. ( v ; n . to.-t’ D

|A /| elem e.n d yn a m ic stiffness m a trix (Hq. ( L 2 2 ). m 5 /))

[As 1 clem ent axial effects m a trix (F q. ( C ( < o . ’ n

[ A j| elem ent d yn a m ic rtiffiv .'s s m a trix (F q . (n.5M)

\ k j ] elem ent axia l effects m a trix (Hq. (0.5 'n

IA5 I elem ent axia l effects m a trix (I'q . (n .v t))

[Aa J elem ent d y n a m ic stifln e ss m a trix (Hq. tn.e '))

[ A7 1 elem ent d y n a m ic stiffness m a trix (Hq. (o.c7))

|A,tf| elem ent d y n a m ic stiffness m a trix (Hu ((> 5 /)

[A ^ /!, \ K cc/\ elem ent and system e t|iiiv a le n t stiffness m a trix (Fqs, ( t <1), (5.27). (6.00))

l ( t ) elem ent length (F ig. (.1 1 ))

L ( I ' length o f p ro tru d in g part o f beam (F ig s. ( U ), (-1.2 0 ), ( 5 .0 ), (n.i.1))

Lfi total length o f beam (Fig. (5 .1))

L[t length o f envelope beam tF ig s. (4 .2(>), ( 5.0 ), (o .l? ))

L j lo ca tio n o f ECS w ith respect to SC’S (F ig . C CI ))

L 0 in itia l length o f p ro tru d in g part o f beam

Lp length o f pipe (F ig . (5.25))

L system Lagrangian density fu n ctio n

L system Lagrangian

L j elem ent Lagrangian

m tran sla tio n a l in e rtia o f lum ped mass

//,, n x com ponents o f o u tw a rd -n o rm a l u n it-v e c to r (F ig. (2 .2 ))

| A7 j = i NJ , [_ Nj „ shape-functions vectors in .vy and .vr planes

| N j spatial shape-fun ctions ve cto r

P ,, P x m o d ifie d generaliseo ...om enta (Eqs. (2.15), (2.25))

(/•,.}, \ i\ , ) p o sitio n vectors (Eq. (6 .1 1))

{ /• /} , { /-2 } elem ent d y n a m ic load vectors (Eq. (6.57))

R tl, Rt f , R\'i< K x x hyperm om enta (Eqs. (2.16), (2.25))

S b o u n d in g curve (F ig . ( 2 .2 ))

(.S’, ) s im ila rity mode ve cto r (F igs. (4.14), (4.15))

I tim e

*

T total c o m p le m e n ta ry k in e tic energy o f system (E q. (6.17))

Tj * co m p le m e n ta ry k in e tic energy o f beam (E q. (6.18); *

Tm co m p le m e n ta ry k in e tic energy o f tip mass (Eq. (6.34))

[ T j), | 7 \ | e q u iv a le n t-flo w tip fo rce m atrices (A p p e n d ix G )

Uj n o n -d im e n sio n a l v e lo c ity o f beam e x tru s io n /re tra c tio n and o f flu id flo w in pipes (Eq. (5.12), (5.25))

r constant v e lo c ity o f beam e x tru s io n /re tra c tio n

V’s. fle x u ra l strain energy (Eq. (6.45))

\ 'l energy due to lo n g itu d in a l loads (Eq. (6.48))

{ V ) , ( Y l i p\ v e lo c ity ve cto r (Eqs. (6.16), (6.35))

.V p o s itio n along the elem ent w ith respect to ECS (F ig . (3.2))

p o s itio n along the beam w ith respect to STS (F ig . (4.2))

e :;v,ent spatial nod a l-va ria h le s ve cto r (Hq. (6 .(1))

system spatial nod a l-va ria b le s ve cto r

transverse displacement o f element in HCS (Fig. ( V.1))

transverse displacement of beam in SC'S (Fig. ( V' ))

element nodal deflections in .vv-plane (Hq. (4.9))

elem ent nod a l-va ria b le s ve cto r in .vv-plane (Hq. ( U 7 ))

system nodal-variables vector in XH-plane

elem ent noda l-va ria b le s ve cto r in v:-plane (Hq. (6.4))

e lem ent nodal re fle c tio n s in .v:-plaue (Hq. (6.5))

beam c o n trib u tio n s to elem ent m atrices (Hqs. (6.24) to (<..<i)))

beam contribution.') to elem ent loads (Hqs. (6.42), (6.4)))

tip mass c o n trib u tio n s to elem ent m atrices (Hqs. (6.47) to (6.42))

tip mass c o n trib u tio n s to elem ent loads (Hqs. (6,44), (6 44))

p ro p o rtio n a l-d a m p in g c o e ffic ie n ts (Hq (4.2))

in e rtia ratio fo r pipes (Eq. (5.25))

a p e rtu rb a tio n param eter (Eq. (4.16))

a s im ila rity param eter (Eq. (4.9))

eigenvalues (E q. (5.7))

m odal d a m p in g c o e ffic ie n ts (Eq. (5.8))

m aterial d e n sity

<1»/, d >2 nodal slopes in .vv-plane (Eq. (6.5))

0) c irc u la r frequency

(cn,.), {o}(,| ro ia tio n a l-v e lo c ity vectors (Eqs. (6.13))

LI region (Fig. (2.2))

Acronyms

EC'S Element Coordinate System

LHS, RHS Left- and Right-Hand Side

SCS System Coordinate System

Operators

8 variational operator

-[■ total time derivative

clt

r r * )

-B , f total differentiation as defined in Section 2.2.1

( , p a rtia l d iffe re n tia tio n w ith respect to .v and r

B~ , B p a rtia l d iffe re n tia tio n w ith respect to .v and /

d.x a t

, ° p a rtia l d iffe re n tia tio n w ith respect to .v and t

A CK NO W L EPGEMENTS

I thank my supervisor, Professor B. Tabarrok, for his guidance, encouragement, and moral support during the course o f this research. The financial assistance received from Profes­ sor B. Tabarrok (through the Natural Sciences and Engineering Research Council o f ('an ad a and Science Council o f British Columbia) and the University o f Victoria >s gratefully acknowledged. I also thank Dr. C. Konzelman for his numerous useful suggestions.

M y gratitude also goes to my fam ily whose saciilices made it possible for me to pursue a career in engineering.

For her love and patience.

Chapter 1

Introduction

1.1

Problem Description and History

The subject o f this investigation is the dynamics o f a flexible extendible beam. To pul this subject into perspective, Fig. (1.1) shows a particular physical system involving a flexible extendible beam. This particular system is comprised o f a beam protruding from a stationary rigid wall, The tip o f the beam may support a tip mass as shown. Note in particular that the length o f the protruding part is a prescribed function o f time.

The flexible extendible beam problem falls under the broad topic o f axially-m oving solid continua. A xially-m oving materials arise in problems associated with cable tram­ ways, spacecraft antennas, band saws, magnetic tape drives, belts, and chains. Since these systems have one large dimension (along the axis o f motion) and two smaller ones, they are usually analysed as one-dimensional string or beam problems. In some applications, however, account must be taken of a second large dimension. For instance, the deployment o f solar arrays in space applications requires m odelling the array as a m oving membrane or plate. A recent survey of these problems is given by W icker and M ote (1988).

A related problem is that of pipes conveying fluid. Literature on pipes conveying fluids is very extensive and an excellent survey o f these problems is given by Pa'i'doussis and Workman f 1991). From a materials perspective, these problems fall w ith in the domain o f solid mechanics, although the flow aspect of these problems gives them a flavour o f fluid mechanics. For instance, in the motion of a computer tape between two reels, it is not practical to follow the individual particles o f tape in time, since some tape particles leave (while others enter) the domain o f interest. One must then use the Eulerian wi n­ dow over the domain o f interest. If the tracking o f particles is relinquished, then con­ servation o f mass w ill not be automatically satisfied. That is, new mass elements enter the domain and, depending on the boundary conditions, some mass elements may leave the domain, with the result that the mass in the domain may change with time. In these problems, the late at which mass elements enter and leave the domain L, pre­ scribed.

Figure 1.1: A flexible extendible beam.

Most investigations o f axially-m oving beams deal with beams supported at two points, and it is the transverse motion o f the beam within the span that is o f interest (W icker and Mote (1988)). If the beam is assumed to be axially rigid, then under these

condi-3

tions, the mass o f the system within the domain o f interest is conserved for small amplitude motion. For the problem o f the flexible extendinle beam in Fig. (.1.1). mass is not conserved, as new mass elements enter the domain o f interest (the protruding part o f the beam).

A derivation o f the non-linear, coupled longitudinal and transverse equations o f m otion of the flexible extendible beam has been provided by Tabarrok, Leech, anil K im (1974) through Newton’s Second Law. A fter linearisation, a sim ilarity solution was obtained. In addition, it was shown that for a constant axial velocity, oscillatory motions dominate the response during the initial stage o f deployment and that, at least w ith in linear theory, the transverse deflection becomes unbounded with time. Their findings were confirmed by simulations using the assumed-modes technique. The assumed-modes technique was also employed in an investigation by ( ’hercli.is and Gossain (1974) o f the dynamics o f a large flexible solar array as it deploys from a spinning spacecraft. Several investigators have a ls' examined the stability o f beams under harmonic longitudinal motion for beams o f constant length (Flmaraghy and Tabarrok, (1975)) and variable length (Zajaczkowski and Lipinski (1979), Zajnc/.- kowski and Yam:.da (1980)). Regions o f stability and instability in the excitation amplitude and frequency parameter space were identified. Although such excitation does not occur in most band-and-wheel systems, many robotic and mechanism compo­ nents execute periodic axial motions.

Recently, flexible extendible beams have gained prominence due to new applications in the area o f robotics, specifically in the modeling o f flexible links traveling through prismatic joints. Wang and Wei (i9 8 7 ) use a modified Galerkin method to s o lw the equation o f motion o f an axially-m oving beam. However, their derivation o f the gov­ erning equation through Newton’s Second Law leaves out certain terms and lead.-, to an incorrect expression for the total energy o f the system. Yuh and Young (1991) i.sc the assumed-modes method and compare their simulation results to those obtained through experiments. However, their simulation results for cases involving rigid-body angular accelerations are not plausible on physical grounds, and are in disagreement 'vith results obtained in this stiuy. Buffinton (1992) also uses the assumed-modes technique to model the moving beam as an unconstrained body, and treats the beam’s finite number o f supports as kinematical constraints. Kim and Gibson (1991) use the finite-element approach to model a sliding flexible link. However, the derivation of the

complementary kinetic energy of the sliding flexible link, outlined by K im (1988,1, also leaves out certain terms.

1.2

Objectives

Reseurchers have successfully used global methods, such as the assumed-modes method, to obtain solutions for the axially-m oving beam protruding from a rigid wall. The purpose o f this study is to investigate the dynamics o f the flexible extendible beam under more general configurations. In particular, we investigate the effects o f the follow ing factors on the dynamic response and stability characteristics o f the flexible extendible beam:

1. Change o f tip support.

2. Addition o f a tip mass.

3. Wall fle xib ility

In the case o f the assumed-modes approach, different sets o f mode-functions must be used depending on the tip supports. Moreover, in the case o f more complicated config­ urations, mode-functions may not be available in closed form , making it necessary to numerically generate them and then tc numerically differentiate them to obtain their derivatives, resulting in considerable deterioration o f accuracy. In the case where the wall is not rigid, methods such as the assumed-modes method generally lose accuracy i f the dynamic stiffness o f the wall and the flexible extendible beam are o f the same magnitude. The capability to model such configurations is critical in this work if the dynamics o f flexible links traveling through flexible prismatic joints are to be accu­ rately modelled.

In the present study, we use the finite-element method *nd take advantage o f the greater fle x ib ility o f this versatile approach. Finite-element form ulations are norm ally carried out for fixed-size domains. However, in this study, we develop elements with tim e-varying domains. In this case the number of elements used to model the extend­ ible beam remains fixed, but their sizes change in a prescribed fashion.

5

1.3

Thesis Organisation

In Chapter 2, the equation o f motion and the consistent boundary conditions for the flexible extendible beam are derived through application o f Hamilton's principle. Then, by considering a second-order functional with two independent variables, we examine the notion o f conserved quantities in a general way. Next, we examine the extremum conditions o f the problem at hand in search o f its specific conserved quanti ties.

The equation o f motion o f the flexible extendible beam is reasonably complex and does not admit a closed-form solution. Accordingly, in Chapter ?> we derive the gov­ erning equations o f motion through finite-element discretization with a variable domain beam element. This element accounts for the coupling o f axial and transverse effects and changes length in a prescribed fashion so that a fixed number o f elements can be used to model the flexible extendible beam.

In Chapter 4, we integrate the finite-element equations in time. We first consider sim ­ ple configurations for which results are available in the literature through either special closed-form solutions, simulations, and/or experiments. The adequacy o f the linile- element models and the accuracy o f the solution procedure are f urther tested by com­ puting the invariant quantities derived in Chapter 2. The generality o f the variable- domain element is demonstrated in examples which are not readily analysed by the at sumed-modes approach. For instance, we examine the case o f ifc addition o f a 'ip mass and the case o f a flexible link moving through a flexible prismatic jo in t under the influence o f a transverse acceleration field. Finally, we look at how high-frequency axial-m otion perturbations can be used to suppress transverse oscillations o f flexible extendible beams.

Chapter 5 is devoted to the stability analysis o f the flexible extendible beam. We begin w ith a description o f the stability analysis procedure used. Next, w- define a non- dimensional system-characteristic parameter with respect to which the stability analy­ sis results are reported. Then, starring from the simplest case o f the flexible extendible beam protruding from a rigid wall, we examine the effect o f changes o f some o f its pertinent parameters, such as the axial-motion profile, physical damning, mass d istri­ bution, tip support, wall rigidity, etc. The sim ilarity solution obtained by Tabarrok et

a l ( 1974) is used in .his chapter for purposes o f comparison with the finite-element results. Finally, we demonstrate the versatility o f the variabie-domain beam element by analysing the stability o f the related problem o f pipes conveying fluid and compare our results to those found by PaTdoussis (1966, 1974, 1975, 1991).

In Chapver 6, we derive the governing equations o f motion for a spatial flexible extendible beam carrying a tip mass. Various planar simulations are performed (w ith ­ out a tip mass), and results obtained are compared to those found by Gaultier (1990), and Yuh and Young (1991). In out last example, we perform a simulation o f a flexible extendible beam (with a tip mass) partly nested in a flexible envelope beam and under­ going a spatial rigid body motion.

Finally, in Chapter 7 we summarise our work and suggest areas for future research.

1.4

Synopsis

The dynamical system o f a flexible extendible beam is considered in this dissertation. The dynamic response o f the system in the configuration shown in Fig. (1.1), where the w all is rigid and the beam is o f uniform mass distribution (without the tip mass), has been the subject o f previous investigations. In this study, we extend this investigation and we develop a spatial variabie-domain beam element that enables us to study the dynamic and stability characteristics o f this configuration as well as o f more c o m p li­ cated configurations. Our modeling and solution procedures are presented ana tested by comparing our results to:

1. Experimental and simulation results for the dynamic response o f flexible extend­ ible beams obtained by Yuh and Young (1991).

2. Experimental and simulation results for the stability analysis o f pipes conveying fluid by PaTdoussis (1966, 1974, 197.5, 1991).

3. The sim ilarity solution fo r the flexible extendible beam obtained by Tabarrok et al. (1974).

Further testing was performed by the evaluation o f certain invariant quantities inherent in the system.

7

A substantial part of this work involved the development o f a special-purpose linite element program for the simulation o f spatial flexible extendible beam problems. A detailed descriDtion o f our software development approach, based on principles o f software engineering, is beyond the scope o f this work. However, a brief outline o f this approach may be found in Appendix H.

Chapter 2

Derivation of the Differential Equations

and the System Invariant Properties

A flexible extendible beam is depicted in Fig. (2.1). The system is comprised o f a beam protruding from a rigid wall., where the length o f the protruding part is a prescribed function o f time. Such axial motion may be brought about by an applied root force or through application o f a uniform acceleration field, such as gravity. Both o f these cases are o f interest and w ill be investigated. In this chapter, we derive the equation o f motion for the flexible extendible beam through M am ikon’s principle. In addition, we examine the notion o f conserved quantities for this mechanical system.

2.1

H am ilton ’s Principle

H a m ilto n ’s principle requires that

9

A

were L is the Lagrangian. For tin. flexible extendible beam the l.agrangian may be expressed as (Tabarrok et al. (1974))

i . t t i

L( i ) ~ L n

where pA is the material density p rr unit length, " / is the flexural rigidity, I.g is the total length o f the beam, L ( t ) is the instantaneous length o f the protruding part o f the beam (a given function o f time), and Y(t,X) is the transverse displacement o f the beam as a function o f time t and position X (Fig. (2.1)). Partial derivatives o f Y with respect to t and X are denoted by corresponding subscripts on Y. Likewise, the time rate of change o f position X o f points along the beam is denoted by X,t which equal I. for the axially rigid beam considered here.

*

I R i g i d j j Wal l i V A

i

Y( t . X) X 1, (1)

Figure 2 .1: The flexible extendible beam.

The four terms in the Lagrangian correspond (in order) to the transverse complemen­ tary kinetic energy, the potential (strain) energy, the potential energy o f the axial load

(due to possible axial acceleration), and the complementary longitudinal kinetic energy (a prescribed quantity). It is important to note that the term accounting fo r the potential energy o f the axial load is relevant only for the case where the axial motion o f the beam is due to a root-applied force; in the case where the beam’s axial motion is caused by a uniform acceleration field, such as gravity, there w ill be no axial force in the beam and this term w ill not appear in the Lagrangian. Since time appears exp licitly in the expression for the Lagrangian, the flexible extendible beam is a rheonomous system.

To ensure that the variation between (/ and r? hi Eq. (2.1) applies fo r the same aggre­ gate in the continuum, an arbitrarily large length o f beam L B is considered, so that

Lr > L ( t ) for ( 2 > t > t {. (2.3)

Since L(t) - L B < 0, and since Y(t,Xj and all its derivatives are prescribed for.v < 0 (corresponding to the part o f the beam enveloped within the rigid wall in Fig. (2.1)), then the variation o f the Lagrangian L vanisi.es for.v < 0. In addition, since L is pre­ scribed, the variation o f the last term in Eq. (2.2) vanishes as well. Consequently, the variational statement becomes

h /) _ ^2,4)

5 j

J

[ i p A O ' ^ - X ^ ) 2 - ~ F J Y t x + i p A { L - X ) L Y x~ / , ( > * ■

clXclt = 0.

C a rry in g o u t the indicated v a ria tio n o f Y(t.X) (d e ta ils are g ive n in the second p a rt o f A p p e n d ix A ) one obtains:

in t h e d o m a h (0 < X < L U ) , l > 0 )

pA ( Y„ + 2 L Y lX + L 2Yxx + LYX) + EI YXXXX + p Act \ { L - X ) Yx \ x = 0, (2.5) at the boundaries

| p/W/ ( L - A') Yx + E I Y XXX ] 8 i / = 0, at X = 0 and X = L,

| £ / r v x l 5 K v = (), at X = 0 and X = L, (2,6)

w here we have in tro d u ce d a new v a ria b le a a cco rd in g to the cause o f the a x ia l m o tio n such that

11

a = L for extrusion/retraction due to a root-applied force, ^ a = 0 for extrusion/retraction due to a uniform acceleration field.

Equation (2.5) expresses th<_ dynamic equilibrium for the flexible extendible beam of Fig. (2.1). The differential operator is non-autonomous and does not admit a general closed-form solution. The non-autonomy arises from the time-dependent coefficients in the differential equation. Equations (2.6) provide the consistent boundary conditions

fo r this system. Specifically, the first one indicates that at the ends o f the protruding part o f the beam either

8F = 0, i.e. deflection is prescribed,

or (2.S)

p A u ( L - X ) Y x + EI YXXX = 0, i.e. shear fore vanishes.

Likewise, the second boundary condition indicates that at the ends o f the protruding beam either

8K^ = 0, i.e. slope is prescribed,

or (2.6)

EI YXX = 0, i.e. bending moment vanishes. The mechanical system o f Fig. (2.1) requires the follow ing kinematic and force bound ary conditions:

at the root (i.e. at X = 0)

Y - Yx = 0, (2.10)

and at the tip (i.e. at X = L ( t ))

Yxx = Yxxx ~

(21l)

There is no known general analytical solution to Eq. (2.5). However, Taba.rok et al. y 1974) obtained a sim ilarity solution for a special case of the flexible extendible beam problem. This sim ilarity solution w ill be discussed and used in subsequent sections for verification purposes.

2.2

A Note on Conserved Quantities

C o n s e rv a tio n law s p la y an im p o rta n t role in m echanics. T he best k n o w n o f these are the co n se rva tio n s o f mass, energy and m om entum . In the fle x ib le e x te n d ib le beam it is e v id e n t that the length o f the p ro tru d in g beam changes w ith tim e and hence mass is n o t conserved. L ik e w is e we can surm ise that since the elem ents o f mass b rin g (o r take) w ith them certain am ounts o f k in e tic energy, the total energy in the d o m a in o f interest w ill n o t be conserved. N evertheless, it is w o rth e n q u irin g i f there are other q u a n titie s in the system that are conserved in tim e. O ne m ay e q u a lly lo o k fo r q u a n titie s w h ic h do n o t change w id i p o sitio n . Such in v a ria n t q u a n titie s are n o t s e lf-e v id e n t in th is system and by id e n tify in g them one can shed new lig h t on the u n d e rly in g stru ctu re o f the g o v ­ e rn in g equatio ns fo r the system . G iv e n that the p ro b le m is governed by a v a ria tio n a l statem ent, it makes sense to e xam ine the e xtre m u m c o n d itio n s o f the p ro b le m in search o f conserved quantitie s.

2.2.1

Conserved Quantities in Second-Order Functionals with

two Independent Variables

T h e cla ssica l statem ent o f conservation o f energy, nam e ly ( T + V ) = constant, is n o r­ m a lly d e riv e d fro m a fu n c tio n a l w ith firs t-o rd e r d e riv a tiv e s in tim e . F o r the fle x ib le e x te n d ib le beam p ro b le m we have seen that the p e rtin e n t fu n c tio n a l in c lu d e s higher- o rd e r d e riv a tiv e s . To exam ine this case, co n sid e r the ro llo w in g general fu n c tio n a l

T h e e x tre m is in g fu n c tio n Y( t , X) is governed (details are g ive n in A p p e n d ix A ):

in the d o m a in Q by the E u ler-Lagra nge equation o f this fu n c tio n a l

(2 . 12)

13 and a lo n g the b o u n d a ry S by rB L ' BY, B i dj -l d t -l d Y n ~ B L .BX. B L BY,_{i x}) n . + ( BY x r 9 1B L r B i_{3 1} . Bl _{^ YXi} . B X . B Y _XX x S>'= 0 , (2.14) B L b l _ u i n BY_{I X} 0, B L B L ,1, + y y - n x ) b Y x = 0 . 0 1 xx

w h e re [ / ; , / ; VJ is the o u tw a rd -n o rm a l u n it-v e c to r to the b o u n d in g curve S o f the re g io n Q sh o w n in F ig . (2.2 ). A t this ju n c tu re it is w o rth u n d e rsco rin g the m e a n in g o f v a rio u s d iffe re n tia l operators. T h e operators B / B Y, B / B YX , and B / B Y f are p a rtia l d e riv a tiv e s , th a t is, f o r these operation s a ll variables, save those w ith respect to w h ic h th e d iffe re n tia tio n is bein g pe rfo rm e d , are held fixe d . O n the o th e r hand, the operators

B / B t and B / B X are, f o r all intents and purposes, total d e riv a tiv e s e xce p t that fo r the

fo rm e r X is held fix e d and fo r the latter t is fixed. To m a in ta in c la rity in the m e aning o f these operators we enclose them in brackets [ | w heneve r a total d e riv a tiv e , in the sense ju s t m e n tio n e d , is meant.

I f w e n ow d e fin e som e m o d ifie d generalised m om enta as1

P,

3 £ 3TT B L B_' .Bt. B r BX P<x> B Yx r i i Bt (2.15) * x r B : BX.. R XX' w h e re = B L " ~ BY, . ' R v , —_XI B L_{pj y '} r j r Xt R B L ix ~ pj y~' r j r iX R vv = ,BL ™ ~ B Y ,_{x x} (2.16)

and w ill be referred to as the hyperm om enta , the E u le r-L a g ra n g c equ a tio n (Hq. (2.13) m a y be w ritte n as

r 3

1

r a i

L

dt. J x {

P,

Px

d_L ?)Y in Q . (2.17)

E v id e n tly , when Y does not appear in the expression fo r the L a g ra n g ia n d e n s ity fu n c ­ tio n L, then the n o tio n of conservation o f momentum takes the fo rm

r i i

.

3

/ . F 3X.

Px

= 0, in Q . (2.18)

It is in te re s tin g to see that in th is case the conservation o f m o m e n tu m corre sp o n d s to the m o d ifie d m o m e n tu m v e c to r P x

J

being d ive rg e n ce -fre e . To d e v e lo p the co n s e rv a tio n equatio ns fo r the cases when t an d /o r X are absent ( e x p lic itly ) fro m the L a g ra n g ia n d e n s ity fu n c tio n , le t us determ ine the total d e riv a tiv e s (as d e fin e d e a rlie r) o f L w ith respect to t and X. T h u s we w rite

F .dt. r_ 3 / a x d L d L d L d L d L Ft , + w t " + aF^ “ , + w r x x, + Fy^ x x < ’ _ d L d L d L d L d L d L d X + d Y x + dY, l X + d Yu l l X + dYx X X + d Y x x x x x d L , (2.19)

N o w , s o lv in g fo r ^-p Yt in Eq. (2.13) and using it in the firs t o f Eqs. (2.19), we m a y w rite the fo llo w in g F dt. ( - L + P l Y, + R l l Yll + R l X YtX) + d 1

Lax.

d L (2 .20) ( ^X^t + R x F u + R XX^iX^ ■ d L .

S im ila rly , s o lv in g fo r ^ p Yx in Eq. (2.13) and using it in the second o f Eqs. (2.19) we can d e riv e the fo llo w in g equation

| ] L r s + R „ YXI + R,X YXX) +

d L ~dX'

15

Equations (2.20) and (2.21) suggest a generalisation o f the Hamiltonian function. To this end we write these equations as1

H n [ £ ] [ & ] ] It " X, t A 11X X = - d L dt d L dx w h e re we defin e

H „ * - L + P , Y , + R „ Y„

+

R,x Y,x.

I I ,

v =

P, r ,

+

R „ Yx ,

+

R , x Yx x .

H x , = h Y, + W n * R U Y,X-

« x x

- -

L + P x Yx

-I-

R Xl YXl

+

R xx Yx x

7

Here we see that the absence o f t and X from the Lagrangian density function L leads to the Hamiltonian vectors [/• /„ H XlJ and [ H l X

I I

xx j , respectively, being

divergence-free.

2.2.2

The Flexible Extendible Beam

F o r purposes o f u sin g the general results d e rive d in the last section, let us express the fu n c tio n a l v a ria tio n fo r the fle x ib le e x te n d ib le beam under a u n ifo rm acceleration Held as (see Eq. (2.4))

r o A 2 . .2 2 F I i

8 j C—

(Y,

+ 2

L Y , Y X

+

L Yx )

- ^ Y xx ) d i l = 0 . (2.24)

£2 ~

w h e re w e have in d ica te d the (t , X) d o m a in by Q. In this case, the h y p c rm o m e n in (2 .io ) and m o d ifie d m om e n ta (2.15) are g ive n by

Rji = 0, RiX = o,

^ X t = ^ X X = ~ ^ ^ X X ' (2.25)

P,

= p

A ( Y t + LYx ) ,

Px

=

P

A L ( Y , + L Y X ) +

R I Y X X X .

Upon substituting

P t

and

PX

in the Euler-Lagrange equation ( E q . (2.17)) the governing equation becomes

p A ( Y tl + 2L Y t f + L YXX + LYX) + E I Y XXXX - 0, (2.26) which agrees with Eq. (2.5) for the case o f a uniform acceleration field (i.e. a = 0 in Eq. (2.5)).

For the flexible extendible beam, Y is absent from £ (E q , (2.24)), hence, the modified momentum veciur is divergence-free. That is, from Eq. (2.18)

" r d i r d i Ld'- J x ]

P,

Px

0, in Q. (2.27)

Let us examine the consequence o f the modified momentum vector being divergence- free. From Green’s theorem we have that

Q

" f f l 1

- d

U /

_ d X . J

P,

A

Px

d a

i t

" i ni p, A

Px

(IS = 0, (2.28)

where [ /?, n J is the outward-normal unit-vecior to the bounding curve 5 o f the region £2 shown in Fig. (2.2). Before expanding the terms in the line integral we need to evaluate the outward-normal unit-vectors from the region Q . For three o f its sides the outward-normal unit-vectors are self-evident. An outward nonnal-vector (Fig. (2.2)) at any point along the curved side may be expressed as [ - L 1J where L is the slope o f the curve. Therefore, the outward-normal unit-vector on this side may be expressed as

I

(2.29) 1 f L

17

x A

i . ( o !

\ i .I.

n

Figure 2.2: Configuration space o f the flexible extendible beam.

N o w , e x p a n d in g the lin e in te g ra l o f Eq. (2.28) we see that the d ive rg e n ce -fre e m o d i- fie d -m o m e n tu m v e c to r im p lie s that

L ( l 2) /,

J ^ « U +j

( / 1 ^ ' / .2 ( - LP, + Px ) ( * j ( \ + L ) ) ( - d l ) ^ (1 + L )

,

+ A- H i ) _(2.10)

J

(~Pt ) {-clX)\i = i + \ { - P x )dt

H i . ) i. X - 0 0.

The term ^ (1 + L ) appears in the calculation o f the outward-normal unit-vector o f the curved edge and in the Jacobian o f the transformation from the straight line / to the curved line L( t j and therefore conveniently cancels out.

Now, considering an infinitesimal change in /, namely letting

/ 1 = t and t 2 = t + A t,

E q. (2.30) m a y be w ritte n as fo llo w s

d dt

Hi )

J

P,dX

= Px \X s l ) - i - L P . + P x )

_{X = H i ) '} (2.32)

which, upon substitution o f the generalised momenta from Eqs. (2.23) becomes

d

dt

H i )

J p A \ Y , + LYx ) d X (p A L { Y , + LYX } + E I Y x x x ) \ XsQ

( - p A L { Y l + LYx } + p A L { Y , + LYX } + E I Y XXX) \x = L u ) .

Upon making use o f the follow ing boundary conditions

(2.33) K = Yy = 0,

xx " 1xxx

= 0 , at at X = 0. X = L ( t ) , (2.34)

and sim plifying, Eq. (2.33) becomes

d dt L A I ) J p A { Y t + LYx } d X . o

= ( E I Y XXX) x = o

(2.35)

Noting that the expression in braces ( ) corresponds to the transverse velocity o f points along the beam, the above equation states that the rate o f change o f transverse momentum is equal to the shear force at the base o f the protruding part o f the beam.

Recall that since time appears explicitly in L through the prescribed function L(t), the Hamiltonian vector [/- /„ H XlJ is not divergence-free. That is, from Eq. (2.22)

(2.36) ~H n

1

d t. [ & i 11

" x .

= - p A L \ ( Y I + LYX ) YX \ ,

H lX = p /\ (Y, + L Y X )YX ,

( 2 . 3 7 )

H Xl = p A L ( Y , + LYX) YI + EI YXXXYI - E 1 Y XXYIX .

D/1 ,2 2 2 FI 1

" x x d Yx x + EI YXXXYX .

In the case o f the flexible extendible beam, Ru and RlX vanish in Eqs. (2.2 U, Ivnce, //,, corresponds to the classical definition o f the Hamiltonian. Using Green's ‘ ..eorem in the region Q , w ith the bounding curve S, and the outward-normal unit-vectors as shown in Fig. (2.2), we see that the modified Hamiltonian vector | l l n //.., j im plies that L {i2) _{( \} I 12 (1 +L~) [ .2 ( - L H u + H Xl ) ( - >

1 ( 1

+ L ) ) d t X l A D _{( 2 U K )}

j

( - / / „ ) (-cI X) | +

j

( - H Xl ) dt\ = J -p /1 /! \ ( Y , + LYX ) Yx \ d i l . 1 /. £2 L U . )

F ollow ing the same procedure as that employed with the modified momentum vector, we may express Eq. (2.38) as

d_ dt ■ H D

J

f / „ dX L 0 Xi>_{' X H i )} ( 2 W ) J p A L | (Y, + L Y X ) Yx \ d i l . £2

We w ill now examine the above expression in the context o f different forms o f the given function L ( / ) . I f the length o f the beam is constant ( L ( t ) = L ( i ) = 0), then

Thus in this special case the total energy is conserved. If, on the other hand, the beam is moving axially at a constant velocity (L (t) = ra n d L ( t ) = 0 ) , Eq. (2.39)

becomes a statement o f evolution o f the modified Hamiltonian-component H „ in time, namely

£

dt - H I )

| H „ ‘I *

= - L

. 0 f Yi + r Yx ) 2_ (2.41) x = n n

The expression in brackets on the right-hand side w ill be recognised as the com ple­ mentary kinetic energy per unit length. Since this is a positive-definite term, one can see that changes in H u are governed by the sign o f

L.

That is, if the beam is extruding (i.e. L is positive) H u decreases, whereas if the beam is retracting (i.e. L is negative) H u increases.

Finally, for the general case where neither L , nm- L are equal to zero, Eq. (2.39) becomes

£

dt ■Hi )

J

H „

= - L - 0 p A { Y, + LYx }

x = n n

_(2.42) l p A L \ { Y , + L Yx ) Y x \ dCl. n

Another invariant result is obtained by noting thatX is absent (e xplicitly) from L. Hence the modified Hamiltonian vector [ H lX ^ x x \ ‘s divergence-free. That is, from Eq. (2.22)

.dt. M .

H_{I X}

(2.43)

H_XX

Once again, using Green’s theorem, we can assert that the divergence-free modified Hamiltonian vector \_HlX ^ x x \ implies that

L ( t z) /, 1 H - . v ' « U „ + l <2 V { - L H IX + H XX) ( - V ( 1 +/ - ) )<//

a / O + L ' ) J

0 | ( -r fX ) = f + | ( - h„ ) « / , ! < ! (i = o. "f-X I. { D _(2.-14) M M

F ollow ing the same procedure as that employed with the modified momentum vector, we may express Eq. (2.44) as

d_ dt

■Li t )

| H IX clX (2.45)

which, upor. substitution o f Eqs. (2.34) and (2.37), becomes

<L

dt ■L{t )

J p

A { Y , + LYX } Y X dX El 2_{4 h x )} X = {) + Pv { r , + i . r x .(2.46) X I. ( I )

This equation states that the rate o f change of the component of the transverse momen­ tum in the tangential direction is equal to the difference between the potential energy per unit length at the base o f the protruding beam and the transverse complementary kinetic energy per unit length at the tip o f the protruding beam.

The governing equation for the flexible extendible beam, Eq. (2.26), can also be obtained from any one o f (2.27), (2.36), and (2.43), by simply carrying out the d iffe re n ti­ ations and sim p lifyin g the resulting expressions.

The correctness o f Eqs. (2.35), (2.40), (2.41), (2.42), and (2.46) can be confirmed by direct differentiation o f their left-hand sides. Since this d'fferenlialion is with respect to time and the upper lim it o f the integral is also a function of time, one must use the Leibniz rule to carry out the differentiation. For example, carrying out the differentiation o f the left-hand side o f Eq. (2.46) we have

d dt L i t )

J p

A { Y , + L Y X ] Yx d X H i )

J

p A \ ( Y „ + i Y x + 2 L Y Xl ) Y x + { Y , Y Xt ) ) d X + (2.47) p A L ( Y t + L Y x ) Y x \ X = L U )

S ince the lo w e r lim it o f the in te g ra l is fix e d , the a d d itio n a l term o f L e ib n iz in te g ra tio n vanishes at th a t lim it. B u t fro m the g o v e rn in g equation (2.26) we have that

P A ( Y II + L Y X + 2 L Y , X ) = - E I Y x x x x - p A L Y x x

w h ic h when substituted in Eq. (2.47) yie ld s,

£ d t L i t )

J p

A { Y 1 + L Y x } Y x d X L i t )

j

+ p * W

‘I X +

(2.48) (2.49) pa L (y i + l yx ) yx \x = L{ n N o tin g that oA L ' Y Y - — '^ L 2— ‘ xx x ~ ^2 dX

Eq. (2.49) can be s im p lifie d to

i Y* p A 2 and PA Y t YXl = ^ ( K , ) i i dt L ( t )

J p

A { Y t + L Y x } Y x d X L i t )

=

J

( - E / Y x x x x Yx ) d X + p^4 2 ( Y, + L Y x ) _{X = L { t )} (2.50) (2.51)

23 L d ) L ( l ) I (D

J

{ - E l Y xx x x Yx ) dX = -

J

^ ( E l Y xxxYx ) dX +

J

( EI Yx x x Yx x ) dX . (152) or L ( t ) 1. (1)

j l - £ ' W

V " = - ( £ ' 1' m V i r ,+ I <

k

' W

.

y

.

y

>‘«

0 0

U s in g the b o u n d a ry c o n d itio n s in Eqs. (2.34), the above expression s im p lifie s to 53)

L(i) L i t )

J

( - E l Y x x x x Yx ) dX =

J

( EI Yx x x Yx x ) d X . (2.54) F in a lly , n o tin g that

E I Y x x x Yxx ~

2

(f x

^Yxx^ ’ Eq. (2.54) fu rth e r s im p lifie s to

L ( i )

f i - n r x x x x Y'x ) d x = - E± r xx

X 0

S u b s titu tin g Eq. (2.56) in to Eq. (2.51) we obtain

£ d t L ( I ) | p A { Y , + LYx } Yx dX 1.55) (2.56) .. ' -I Y :

x

L i n 2

xx

(2.57) X 0 w h ic h is id e n tic a l to Eq. (2.46).

2.2.3

Energy Considerations

T h e to ta l energy o f the fle x ib le e xte n d ib le beam fo r the special case o f co n sta n t axial v e lo c ity m ay be expressed as

Hi )

This expression for total energy is in disagreement with the simpler expression fo r the total energy given by Wang and Wei (1987, Eq. (13)). These authors fail to take proper account o f the time dependency o f X ( X t = L) through the total time derivative o f Y in the complementary kinetic energy. Due to the time-dependent domain o f the flexible extendible beain, the total energy o f the svstem is time dependent. In fact, if the above differentiation is carried on;, we get the follow ing expression fo r the rate o f change o f total energy

2.3

Concluding Remarks

We have derived the governing differential equation and the associated boundary con­ ditions o f the flexible extendible beam from H am ilton’s principle. By not making a particular distinction between time and space variables we have developed other con­ servation statements and we have illustrated them for the flexible extendible beam problem. Such invariant forms are manifestations o f inherent symmetries in the system heing studied. Thev are therefore o f Intrinsic interest on physical grounds and they provide useful checks for soundness o f numerical algorithms devised to solve the sys­ tem equations as pointed out by Kane and Levinson (1988). The invariant statements w ill be used in subsequent sections for verification purposes.

I f the differential equation and the boundary conditions o f a physical problem are well posed, that is, i f the problem is properly modelled, then one can expect to find solu­ tions fo r the governing differential equation. The fact that the boundary conditions and the differential equation for the problem at hand emerge from a single variational statement ensures the consistency o f these equations. However, these equations are reasonably complex and do not admit closed form solutions. Accordingly one must resort to a numerical procedure fo r solution. In the next chapter, we address this matter and devise Hnite-element procedures for solution o f the problem at hand.

(2.59) p A L ^ Y , * - LYX) 2 + L { Y t Yx ) )

25

Chapter 3

Variable-Domain Beam Elements

In this work we use the finite-element method to obtain solutions for the axia lly-m o v­ ing beam. Finite-element formulations are normally carried out for lixed-si/.e domains. For time-dependent domains it is possible, in principle, to use lixed-si/.e elements and to nerease/decrease their number, in a step-wise fashion, as a function o f time. Such an approach, though straightforward in concept, is d ifficu lt to implement in practice. It requires a very large number o f small elements if reasonable smoothness in the results is to be obtained, and since the system’s number of degrees o f freedom changes con­ tinuously, special programing considerations are necessary. In this chapter, we fo llo w a more elegant approach by developing elements with time-varying domains. In this case the number of elements remains fixed, but their sizes change in a prescribed fash­ ion.

3.1

Elem ent Lagrangian Function

In order to develop the finite element equations for the variable domain beam element, let us divide the protruding part o f the beam o f Fig. (2.1) into /; elements o f equal length 1(0 as shown in Fig. (3.1). The Lagrangian for the i'1' element is then given by

H I )

L, =

J

(v, + X ,yx ) 2 - l2E / y.xA + ^ P A ( L - L - x ) L y ^ d x + ^ p A l L , (3.1)

o

1

w h e re the lo w e r case sym bols x and y correspond to the elem ent c o o rd in a te system (E C S ), and Lt locates ihe ECS as shown in F ig. (3.2). A s tim e e vo lve s, elem ents e lo n ­ gate (o r co n tra ct). L e t us co n sid e r a p o in t A along the neutral axis o f the i'1' ele m e n t as show n in F ig . (3.2). T h is p o in t’s lo ca tio n , w ith respect to the in e rtia l fra m e AT, is g ive n by

X = L; + x. (3.2)

Solving fo r the element coordinate x and differentiating with respect to time, we obtain

x t = X t —

Lj.

(3.3) l i i (j i d W a l l X 1(1) .-1

A

T

Y(i.X) ! I, (I.) X d . (I ) l ‘(t) h

Figure 3 .1: Variable-domain beam elements.

However, since X, = L, the rate o f change with respect to time o f the element coordi­ nate .v is given by

27

It is convenient at this point to introduce a new parameter dr shown in F ig .1 Cl), defined for each element as follows

d. = L - L i = L ( 1 - ( / - \ ) / n ) i - 1 ,2 ,.,./;. (3.5) Using the above definition, the expression for the Lagrangian of the i '1' element (Eq. (3.1)) becomes / ( ; ) r T l 2 1 o I - V I 1 .2 J I ^ p A (y, + ( h y x) - ^ EI y ~x + 2- p /\ {clr . \ ) [ . \ \ ilx + ^ p AI I . . (3.(1) /, ( I ) H \ x ' “

,i

!l( r )

Figure 3.2: Element coordinate system (EC'S).

3.2

Finite-EIem ent Discretization

The transverse deflections o f the flexible extendible beam are modelled by a cubic polynom ial as follow s

where the r ’.v are functions o f time only and .v is the distance along the element as shown in Fig. (3.3). The slope distribution is given by

y x ( t , x ) = Cj + 2c2x + 3 c y 2. (3.8)

\

», I S j 0,

x 0 x I.

Figure 3.3: Nodal variables o f a beam finite element.

Referring to Fig, (3.3), the nodal deflection and slope variables are:

v(r, 0) = v,, yx (t, 0) = 0 , ,

y (/, /) = y 2, y v (/, /) = 0 2 , (3.9)

When Eqs. (3.9) are used in Eqs. (3.7) and (3.8), we can write the resulting system o f equations as

I d [q] = {V},

(3.10)

where