The deployment dynamics of a simplified solar sail quadrant consisting of two Euler–Bernoulli beams and a flexible membrane are studied. Upon prescribing the in-plane motion and modeling the tension field based on linearly increasing stresses assumed on the attached boundaries, the coupled equations of motion that describe the system's transverse deflections are obtained. Based on these equations and their boundary conditions (BCs), deployment stability is studied by deriving simplified analytic expressions for the rate of change of system energy. It is shown that uniform extension and retraction result in decreasing and increasing energy, respectively. The motion equations are discretized using expansions in terms of “time-varying quasi-modes” (snapshots of the modes of a cantilevered beam and a clamped membrane), and the integrals needed for the resulting system matrices are rendered time-invariant via a coordinate transformation. Numerical simulation results are provided to illustrate a sample deployment and validate the analytic energy rate expressions.

References

References
1.
Lang
,
W. E.
, and
Honeycutt
,
G. H.
,
1967
, “
Simulation of Deployment Dynamics of Spinning Spacecraft
,” National Aeronautics and Space Administration, Washington, DC, Report No.
TN D-4074
.https://ntrs.nasa.gov/search.jsp?R=19670039596
2.
Hedgepeth
,
J. M.
,
1970
, “
Dynamics of a Large Spin-Stiffened Deployable Paraboloidal Antenna
,”
J. Spacecr. Rockets
,
7
(
9
), pp.
1043
1048
.
3.
Cloutier
,
G. J.
,
1968
, “
Dynamics of Deployment of Extendible Booms From Spinning Space Vehicles
,”
J. Spacecr. Rockets
,
5
(
5
), pp.
547
552
.
4.
Cherchas
,
D. B.
,
1971
, “
Dynamics of Spin-Stabilized Satellites During Extension of Long Flexible Booms
,”
J. Spacecr. Rockets
,
8
(
7
), pp.
802
804
.
5.
Hughes
,
P. C.
,
1972
, “
Dynamics of a Spin-Stabilized Satellite During Extension of Rigid Booms
,”
C.A.S.I. Trans.
,
5
(
1
), pp.
11
14
.
6.
Lips
,
K. W.
, and
Modi
,
V. J.
,
1978
, “
Transient Attitude Dynamics of Satellites With Deploying Flexible Appendages
,”
Acta Astronaut.
,
5
(
10
), pp.
797
815
.
7.
Mote
,
C. D.
, Jr.
,
1972
, “
Dynamic Stability of Axially Moving Materials
,”
Shock Vib. Dig.
,
4
(
4
), pp.
2
11
.
8.
Wickert
,
J. A.
, and
Mote
,
C. D.
, Jr.
,
1988
, “
Current Research on the Vibration and Stability of Axially-Moving Materials
,”
Shock Vib. Dig.
,
20
(
5
), pp.
3
13
.
9.
Wang
,
P. K. C.
, and
Wei
,
J.-D.
,
1987
, “
Vibrations in a Moving Flexible Robot Arm
,”
J. Sound Vib.
,
116
(
1
), pp.
149
160
.
10.
Bergamaschi
,
S.
, and
Sinopoli
,
A.
,
1983
, “
On the Flexural Vibrations of Arms With Variable Length: An Exact Solution
,”
Mech. Res. Commun.
,
10
(
6
), pp.
341
344
.
11.
Stylianou
,
M.
, and
Tabarrok
,
B.
,
1994
, “
Finite Element Analysis of an Axially Moving Beam—Part I: Time Integration
,”
J. Sound Vib.
,
178
(
4
), pp.
433
453
.
12.
Behdinan
,
K.
,
Stylianou
,
M. C.
, and
Tabarrok
,
B.
,
1997
, “
Dynamics of Flexible Sliding Beams Non-linear Analysis—Part I: Formulation
,”
J. Sound Vib.
,
208
(
4
), pp.
517
539
.
13.
Wang
,
P. K. C.
, and
Wei
,
J.-D.
,
1994
, “
Correction and Remarks on ‘Vibrations in a Moving Flexible Robot Arm
',”
J. Sound Vib.
,
172
(
3
), pp.
413
414
.
14.
Park
,
S.
,
Yoo
,
H. H.
, and
Chung
,
J.
,
2013
, “
Eulerian and Lagrangian Descriptions for the Vibration Analysis of a Deploying Beam
,”
J. Mech. Sci. Technol.
,
27
(
9
), pp.
2637
2643
.
15.
Leech
,
C. M.
,
1970
, “
The Dynamics of Beams Under the Influence of Convecting Inertial Forces
,” Ph.D. thesis, University of Toronto, Toronto, ON.
16.
Tabarrok
,
B.
,
Leech
,
C. M.
, and
Kim
,
Y. I.
,
1974
, “
On the Dynamics of an Axially Moving Beam
,”
J. Franklin Inst.
,
297
(
3
), pp.
201
220
.
17.
Wang
,
L. H.
,
Hu
,
Z. D.
,
Zhong
,
Z.
, and
Ju
,
J. W.
,
2009
, “
Hamiltonian Dynamic Analysis of an Axially Translating Beam Featuring Time-Variant Velocity
,”
Acta Mech.
,
206
(
3–4
), pp.
149
161
.
18.
Wang
,
P. K. C.
,
1990
, “
Stabilization and Control of Distributed Systems With Time-Dependent Spatial Domains
,”
J. Optim. Theory Appl.
,
65
(
2
), pp.
331
362
.
19.
Stylianou
,
M.
, and
Tabarrok
,
B.
,
1994
, “
Finite Element Analysis of an Axially Moving Beam—Part II: Stability Analysis
,”
J. Sound Vib.
,
178
(
4
), pp.
455
481
.
20.
Yuh
,
J.
, and
Young
,
T.
,
1991
, “
Dynamic Modelling of an Axially Moving Beam in Rotation: Simulation and Experiment
,”
ASME J. Dyn. Syst., Meas., Control
,
113
(
1
), pp.
34
40
.
21.
Niemi
,
J.
, and
Pramila
,
A.
,
1987
, “
FEM-Analysis of Transverse Vibrations of an Axially Moving Membrane Immersed in Ideal Fluid
,”
Int. J. Numer. Methods Eng.
,
24
(
12
), pp.
2301
2313
.
22.
Koivurova
,
H.
, and
Pramila
,
A.
,
1997
, “
Nonlinear Vibration of Axially Moving Membrane by Finite Element Method
,”
Comput. Mech.
,
20
(
6
), pp.
573
581
.
23.
Shin
,
C.
,
Chung
,
J.
, and
Kim
,
W.
,
2005
, “
Dynamic Characteristics of the Out-of-Plane Vibration for an Axially Moving Membrane
,”
J. Sound Vib.
,
286
(
4-5
), pp.
1019
1031
.
24.
Cherchas
,
P. C.
, and
Gossain
,
D. M.
,
1974
, “
Dynamics of a Flexible Solar Array During Deployment From a Spinning Spacecraft
,”
C.A.S.I. Trans.
,
7
(
1
), pp.
10
18
.
25.
Cherchas
,
P. C.
,
1973
, “
Coupled Bending-Twisting Vibrations of a Single Boom Flexible Solar Array and Spacecraft
,”
C.A.S.I. Trans.
,
6
(
1
), pp.
56
60
.
26.
Hughes
,
P. C.
,
1976
, “
Deployment Dynamics of the Communications Technology Satellite—A Progress Report
,”
ESRO Symposium on Dynamics and Control of Non-Rigid Spacecraft
, pp.
335
340
.
27.
Janković
,
M. S.
,
1979
, “
Deployment Dynamics of Flexible Spacecraft
,” Ph.D. thesis, University of Toronto, Toronto, ON.
28.
Hughes
,
P. C.
, and
Garg
,
S. C.
,
1973
, “
Dynamics of Large Flexible Solar Arrays and Application to Spacecraft Attitude Control System Design
,” University of Toronto Institute for Aerospace Studies, Toronto, ON, Canada, Report No. 179.
29.
Shaker
,
F. J.
,
1976
, “
Free-Vibration Characteristics of a Large Split-Blanket Solar Array in a 1-g Field
,” National Aeronautics and Space Administration, Washington, DC, Report No. TN D83-7576.
30.
Weeks
,
G. E.
,
1986
, “
Dynamic Analysis of a Deployable Space Structure
,”
J. Spacecr. Rockets
,
23
(
1
), pp.
102
107
.
31.
Seffen
,
K. A.
, and
Pellegrino
,
S.
,
1999
, “
Deployment Dynamics of Tape Springs
,”
Proc. R. Soc. London A: Math., Phys. Eng. Sci.
,
455
(
1983
), pp.
1003
1048
.
32.
Walker
,
S. J. I.
, and
Aglietti
,
G.
,
2004
, “
Study of the Dynamics of Three-dimensional Tape Spring Folds
,”
AIAA J.
,
42
(
4
), pp.
850
856
.
33.
Oberst
,
S.
, and
Tuttle
,
S.
,
2018
, “
Nonlinear Dynamics of Thin-Walled Elastic Structures for Applications in Space
,”
Mech. Syst. Signal Process.
,
110
, pp.
469
484
.
34.
Shirasawa
,
Y.
,
Mori
,
O.
,
Miyazaki
,
Y.
,
Sakamoto
,
H.
,
Hasome
,
M.
,
Okuizumi
,
N.
,
Sawada
,
H.
,
Furuya
,
H.
,
Matsunaga
,
S.
,
Natori
,
M.
, and
Kawaguchi
,
J.
,
2011
, “
AIAA
Paper No. 2011-1890.
35.
Zhao
,
J.
,
Tian
,
Q.
, and
Hu
,
H.-Y.
,
2013
, “
Deployment Dynamics of a Simplified Spinning IKAROS Solar Sail Via Absolute Coordinate Based Method
,”
Acta Mech. Sin.
,
29
(
1
), pp.
132
142
.
36.
Tian
,
Q.
,
Zhao
,
J.
,
Liu
,
C.
, and
Zhou
,
C.
,
2015
, “
Dynamics of Space Deployable Structures
,”
ASME
Paper No. DETC2015-46159.
37.
Shabana
,
A. A.
,
1997
, “
Flexible Multibody Dynamics: Review of Past and Recent Developments
,”
Multibody Syst. Dyn.
,
1
(
2
), pp.
189
222
.
38.
Melnikov
,
V. M.
, and
Koshelev
,
V. A.
,
1998
,
Large Space Structures Formed by Centrifugal Forces
,
Gordon and Breach Science Publishers
,
Philadelphia, PA
.
39.
Miranker
,
W. L.
,
1960
, “
The Wave Equation in a Medium in Motion
,”
IBM J. Res. Develop.
,
4
(
1
), pp.
36
42
.
40.
Barakat
,
R.
,
1968
, “
Transverse Vibrations of a Moving Thin Rod
,”
J. Acoust. Soc. Am.
,
43
(
3
), pp.
533
539
.
41.
Wickert
,
J. A.
, and
Mote
,
C. D.
, Jr.
,
1989
, “
On the Energetics of Axially Moving Continua
,”
J. Acoust. Soc. Am.
,
85
(
3
), pp.
1365
1368
.
42.
Zhu
,
W. D.
, and
Ni
,
J.
,
2000
, “
Energetics and Stability of Translating Media With an Arbitrarily Varying Length
,”
ASME J. Vib. Acoust.
,
122
(
3
), pp.
295
304
.
43.
Yang
,
X.-D.
,
Zhang
,
W.
, and
Melnik
,
R. V. N.
,
2016
, “
Energetics and Invariants of Axially Deploying Beam With Uniform Velocity
,”
AIAA J.
,
54
(
7
), pp.
2181
2187
.
44.
Wickert
,
J. A.
, and
Mote
,
C. D.
, Jr.
,
1990
, “
Classical Vibration Analysis of Axially Moving Continua
,”
ASME J. Appl. Mech.
,
57
(
3
), pp.
738
744
.
45.
Zhu
,
W. D.
,
2000
, “
Vibration and Stability of Time-Dependent Translating Media
,”
Shock Vib. Dig.
,
32
(
5
), pp.
369
379
.
46.
Lin
,
C. C.
,
1997
, “
Stability and Vibration Characteristics of Axially Moving Plates
,”
Int. J. Solids Struct.
,
34
(
24
), pp.
3179
3190
.
47.
Vatankhahghadim
,
B.
, and
Damaren
,
C. J.
,
2018
, “
Solar Sail Deployment Dynamics
,”
Fifth Joint International Conference on Multibody System Dynamics
, Lisbon, Portugal, June 24–28
48.
Zhang
,
L.
, and
Zu
,
J. W.
,
1999
, “
Nonlinear Vibration of Parametrically Excited Viscoelastic Moving Belts, Part I: Dynamic Response
,”
ASME J. Appl. Mech.
,
66
(
2
), pp.
396
402
.
49.
Wickert
,
J. A.
, May
1992
, “
Non-Linear Vibration of a Travelling Tensioned Beam
,”
Int. J. Non-Linear Mech.
,
27
(
3
), pp.
503
517
.
50.
Behdinan
,
K.
, and
Tabarrok
,
B.
,
1997
, “
Dynamics of Flexible Sliding Beams Non-Linear Analysis—Part II: Transient Response
,”
J. Sound Vib.
,
208
(
4
), pp.
541
565
.
51.
Zhang
,
L.
, and
Zu
,
J. W.
,
1999
, “
Nonlinear Vibration of Parametrically Excited Viscoelastic Moving Belts, Part II: Stability Analysis
,”
ASME J. Appl. Mech.
,
66
(
2
), pp.
403
409
.
52.
Wu
,
K.
, and
Zhu
,
W. D.
,
2014
, “
Parametric Instability in a Taut String With a Periodically Moving Boundary
,”
ASME J. Appl. Mech.
,
81
(
6
), p.
061002
.
53.
Öz
,
H. R.
,
Pakdemirli
,
M.
, and
Boyacı
,
H.
,
2001
, “
Non-Linear Vibrations and Stability of an Axially Moving Beam With Time-Dependent Velocity
,”
Int. J. Non-Linear Mech.
,
36
(
1
), pp.
107
115
.
54.
Ghayesh
,
M. H.
, and
Amabili
,
M.
,
2013
, “
Steady-State Transverse Response of an Axially Moving Beam With Time-Dependent Axial Speed
,”
Int. J. Non-Linear Mech.
,
49
, pp.
40
49
.
55.
Wu
,
J.
,
Shao
,
M.
,
Wang
,
Y.
,
Wu
,
Q.
, and
Nie
,
Z.
,
2017
, “
Nonlinear Vibration Characteristics and Stability of the Printing Moving Membrane
,”
J. Low Freq. Noise, Vib. Active Control
,
36
(
3
), pp.
306
316
.
56.
Jenkins
,
C. H.
, and
Leonard
,
J. W.
,
1991
, “
Nonlinear Dynamic Response of Membranes: State of the Art
,”
ASME Appl. Mech. Rev.
,
44
(
7
), pp.
319
328
.
57.
Jenkins
,
C. H.
,
1996
, “
Nonlinear Dynamic Response of Membranes: State of the Art - Update
,”
ASME Appl. Mech. Rev.
,
49
(
10S
), pp.
S41
S48
.
58.
Shin
,
C.
,
Chung
,
J.
, and
Yoo
,
H. H.
,
2006
, “
Dynamic Responses of the In-Plane and Out-of-Plane Vibrations for an Axially Moving Membrane
,”
J. Sound Vib.
,
297
(
3–5
), pp.
794
809
.
59.
Li
,
Q.
,
Ma
,
X.
, and
Wang
,
T.
,
2011
, “
Reduced Model for Flexible Solar Sail Dynamics
,”
J. Spacecr. Rockets
,
48
(
3
), pp.
446
453
.
60.
Timoshenko
,
S.
,
1934
,
Theory of Elasticity
,
McGraw-Hill
,
New York
.
61.
Lanczos
,
C.
,
1986
,
The Variational Principles of Mechanics
,
Dover Publications
,
Mineola, NY
, Chap. 4.
62.
McIver
,
D. B.
,
1973
, “
Hamilton's Principle for Systems of Changing Mass
,”
J. Eng. Math.
,
7
(
3
), pp.
249
261
.
63.
Timoshenko
,
S.
, and
Woinowsky-Krieger
,
S.
,
1940
,
Theory of Plates and Shells
,
McGraw-Hill
,
New York
.
64.
Kane
,
T. R.
,
Likins
,
P. W.
, and
Levinson
,
D. A.
,
1983
,
Spacecraft Dynamics
,
McGraw-Hill
,
New York
.
65.
Shames
,
I. H.
, and Dym, C. L.,
1985
,
Energy and Finite Element Methods in Structural Mechanics
,
Hemisphere Publishing
,
Philadelphia, PA
, Chap. 3-D.
66.
Newmark
,
N. M.
,
1959
, “
A Method of Computation for Structural Dynamics
,”
ASCE J. Eng. Mech. Div.
,
85
(
3
), pp.
67
94
.
67.
Hassanpour
,
S.
, and
Damaren
,
C. J.
,
2018
, “
Linear Structural Dynamics and Modal Cost Analysis for a Solar Sail
,”
AIAA
Paper No. 2018-1434.
68.
Choi
,
M.
,
2015
, “
Flexible Dynamics and Attitude Control of a Square Solar Sail
,”
Ph.D. thesis
, University of Toronto, Toronto, ON.https://tspace.library.utoronto.ca/handle/1807/69253
69.
Janković
,
M. S.
,
1976
, “
Lateral Vibrations of an Extending Rod
,” University of Toronto Institute for Aerospace Studies, Toronto, ON, Technical Report No. 202.
You do not currently have access to this content.