LIST OF SYMBOLS
T Kinetic energy of the system; X Generalized coordinate vector; M Inertia or mass matrix;
First order time derivative of X;
Q Vector that includes external and velocity dependent inertia force;
ĭi Geometrical constrain equations; A Jacobian matrix of constrain equations; h1, h2, …, hp p independent displacement modes of
rigid-body movement;
Row vector consists on these combined FRHI¿FLHQWVLQ(T
ȡ Length density of the active cable; T(ș) Internal force vector;
f(ș) Friction force vectors in the contact point; N(ș) Pressure force vectors in the contact point;
, Velocity and acceleration of cable length variety; r Radius of the pulley;
Į Half angle between the two active cable element nearby the point; Ȝ Direction cosine of the cable element;
ȝ '\QDPLFIULFWLRQFRHI¿FLHQW
șk , șk–1 Start and end angle according to contact region; Ȝij 'LUHFWLRQFRVLQHRIWKHXQLSOHWVLM
INTRODUCTION
Deployable truss structures have been applied in many applications, such as solar arrays, masts and antennas (Meguro et al., 2003) that have small-stowed volumes during launch, and are deployed by certain means to assume LWV SUHGHWHUPLQHG VKDSH DFFXUDWHO\ LQ RUELW 7UDGLWLRQDOO\ a deployable truss structure consists of a large number of struts and kinematic pairs, which are simple, such as revolute MRLQWVVOLGLQJKLQJHV7DNDPDWVXDQG2QRGDIJHDUV and pantograph struts (Cherniavsky et al.7KLVW\SHRI deployable structure has many advantages, including lighter weight, higher precision, smaller launch volume, and higher UHOLDELOLW\IRUGHSOR\PHQW
7KHUH LV D ODUJH DPRXQW RI SXEOLVKHG OLWHUDWXUH RQ FRQVWUDLQHGULJLGDQGÀH[LEOHERG\G\QDPLFVZKLFKIRFXVHG on building the dynamic model, solving the differential HTXDWLRQV DQG NLQHPDWLF VLPXODWLRQ 0DQ\ VWXGLHV KDYH
Kinematic Analysis of the Deployable Truss Structures for Space
Applications
Xu Yan1*, Guan Fu-ling1, Zheng Yao1, Zhao Mengliang1,2 Zhejiang University, Hangzhou – China
2Shanghai JiangNan Architectural Design Institute – China
Abstract: Deployable structure technology has been used in aerospace and civil engineering structures very popularly. This paper reported on a recent development of numerical approaches for the kinematic analysis of the deployable truss structures. The dynamic equations of the constrained system and the computational procedures were summarized. The driving force vectors of the active cables considering the friction force were also formulated. Three types of macroele-ments used in deployable structures were described, including linear scissor-link element, multiangular scissor element, and rigid-plate element. The corresponding constraint equations and the Jacobian matrices of these macroelements were formulated. The accuracy and ef¿ciency of the proposed approach are illustrated with numerical e[amples, including a double-ring deployable truss and a deployable solar array.
Keywords: Kinematic analysis, Deployable truss structures, Macroelements, Deployable solar array.
5HFHLYHG $FFHSWHG
DXWKRUIRUFRUUHVSRQGHQFH[\]V#]MXHGXFQ
only addressed simple beams and rigid bodies, which are QRWFRPSOLFDWHGGHSOR\DEOHVWUXFWXUHV%D\RDQG/HGHVPD SUHVHQWHGDQHZLQWHJUDWLRQPHWKRGIRUFRQVWUDLQHG PXOWLERG\ G\QDPLFV )LVHWWH DQG 9DQHJKHP XVHG the coordinate partitioning method of constrained Jacobian PDWUL[ WR DQDO\]H WKH PXOWLERG\ V\VWHP$Q RUWKRQRUPDO tangent space method for constrained multibody systems ZDVSURSRVHGE\%ODMHULQZKLFKWKHLQGHSHQGHQFH basis vector of the tangent space of constrained surface QHHGVWREHFDOFXODWHG2UWKRJRQDOPDWUL[WULDQJXODUL]DWLRQ 45 GHFRPSRVLWLRQ .LP DQG 9DQGHUSORHJ VLQJXODUYDOXHGHFRPSRVLWLRQ6LQJKDQG/LNLQVDQG GLIIHUHQWLDEOHQXOOVSDFHPHWKRG/LDQJDQG/DQFH ZHUHXVHGWRREWDLQVXFKEDVLVYHFWRUV6RPHDXWKRUVKDYH investigated the mechanism characteristics of deployable truss and tensegrity structures in their literatures (Calladine DQG 3HOOHJULQR <RX %DHet al. (2000) SURSRVHG DQ HI¿FLHQW LPSOHPHQWDWLRQ DOJRULWKP IRU UHDO WLPHVLPXODWLRQRIWKHPXOWLERG\YHKLFOHG\QDPLFVPRGHOV Newton chord method was employed to solve the equations RIPRWLRQDQGFRQVWUDLQWV8VLQJWKH¿QLWHSDUWLFOHPHWKRG )30 <X DQG /XR SUHVHQWHG D PRWLRQ DQDO\VLV approach of deployable structures based on the straight- and DQJXODWHGURGKLQJHV.LQHPDWLFDQGG\QDPLFDQDO\VLVDQG control methods of the hoop truss deployable antenna were LQYHVWLJDWHGE\/L
7KLV SDSHU UHSRUWV D UHFHQWO\FRQGXFWHG HIIRUW WKDW systematically addressed a kinematic analysis method of deployable truss structure based on macroelements, in which WKHIULFWLRQIRUFHZDVFRQVLGHUHG
EQUATIONS OF MOTION CONSIDERING DRIVING CONDITIONS
Dynamic equations for the constrained system
7KHG\QDPLFHTXDWLRQVIRUWKHFRQVWUDLQHGV\VWHPDQG WKHFRPSXWDWLRQDOSURFHGXUHVDUHVXPPDUL]HGLQWKLVVHFWLRQ For the deployable spatial structure, the dependent Cartesian coordinates are used as generalized ones for the dynamic HTXDWLRQ7KHPDVVDQGYHORFLW\RIWKHVWUXWVDUHUHGXFHGWR two revolute joints at the two ends, and the kinetic energy of WKHV\VWHPLVGH¿QHGDV(T
T=21X MXoT o
where
X: is the generalized coordinate vector, and MLVWKHLQHUWLDRUPDVVPDWUL[
7KH¿UVW/DJUDQJHHTXDWLRQLVSUHVHQWHGDV(T
dX dtd X T
X T
Q 0
T 2 2
2 2
- - =
o
^ hc m (2)
where
Q: is the vector that includes the external and velocity dependent inertia force,
LVWKH¿UVWRUGHUWLPHGHULYDWLYHRI X
%\VXEVWLWXWLQJ(TLQWRWKHG\QDPLFHTXDWLRQFDQEH GHWHUPLQHGDV(T
dXT^MXp-Qh=0
(3)
%HFDXVH WKHUH DUH PDQ\ FRPSOLFDWHG FRQVWUDLQV LQ WKH deployable truss structures, the vector dX, according to the JHQHUDOL]HGFRRUGLQDWHYHFWRULVGHSHQGHQW&RQVLGHULQJDOO types of constrains of the entire structures, the geometrical FRQVWUDLQHTXDWLRQVDUHIRUPXODWHGDV(T
; , , , X 0 i 1 2 s
i g
U^ h= =
Since all constrains of the deployable structure are constant with time t during the deployment process, the derivative of WKHFRQVWUDLQWHTXDWLRQVSURYLGHWKH-DFRELDQPDWUL[(T
AXo =0 (5)
where
A: is the corresponding Jacobian matrix of constraint HTXDWLRQV
7KHYHORFLW\HTXDWLRQ(TLVVROYHGE\WKHJHQHUDOL]HG LQYHUVHPDWUL[PHWKRG(T
Xo =ao1h1+ao2h2+gaophp=Hao where
h, h2, … , hp: are p independent displacement modes of the rigid-body movement,
Xp =Hap-A AX+ o o
%\ UHSODFLQJ (TV DQG LQWR WKH G\QDPLF HTXDWLRQ (TWKHIROORZLQJ¿QDOG\QDPLFHTXDWLRQVDUHREWDLQHG (TVDQG
X H
H MHT H MA AHT H QT 0 a
a a
=
- + - =
o o
p o o
With the initial condition:
,
Xt=0=X X0 ot=0=Xo0
When the initial displacement and velocity vectors are known, the vector t=0LQ(TFDQEHREWDLQHG%\WKHVHLQLWLDO YDOXHV 1HZPDUN¶V PHWKRG LV HPSOR\HG WR VROYH WKH ¿QDO G\QDPLFHTXDWLRQV(TWKHUHIRUHGLVSODFHPHQWYHORFLW\ and acceleration of the deployable process in each time step FDQDOVREHGHWHUPLQHG
7KHQXPHULFDODSSURDFKFRPSXWDWLRQDOSURFHGXUHLVYHU\ simple and can be summarized as follows:
DOOLQLWLDOLQSXWGDWHVRIWKHHQWLUHVWUXFWXUHDQGQXPHULFDO simulation, such as the coordinates of the joints, structural topology, constraint conditions, driving mechanisms, boundary conditions and the length of time step, and so on, are provided;
DW DQ\ WLPH VWHSn, the mass matrix and driving force vectors are formed;
WKH-DFRELDQPDWUL[ DQGWKH¿UVWRUGHUGHULYDWLYHRIWKH Jacobian matrix are formulated;
WKHJHQHUDOL]HGLQYHUVHPDWUL[DQGEDVLVYHFWRURIQXOO space of are determined;
LIPDWUL[ has full column, the rank will be estimated, if so go to the ninth step;
WKHGLIIHUHQWLDOG\QDPLFHTXDWLRQVDUHFDOFXODWHGDQGWKH displacement, velocity, and acceleration of all joints are obtained;
XSGDWHWKHSRVLWLRQVRIDOOMRLQWVDQGFKHFNZKHWKHUWKH ORFNHGFRQGLWLRQVRIWKHVWUXFWXUHLVVDWLV¿HGLIWKH\DUH not, go to the second step; otherwise, go to the ninth step; LIWKHSHULRGLVOHVVWKDQWKHHQGLQJWLPHJRWRWKHVHFRQG
step, otherwise, the analysis should be stopped;
WKHDQDO\VLVLVVWRSSHGDQGWKHRXWSXWGDWHVDUHUHDG\IRU SRVWSURFHVV
Active cable driving and friction
7KHSXUSRVHRIWKLVVHFWLRQLVWRIRUPXODWHWKHGULYLQJ force vectors of the active cables, which forms the term Q in WKHG\QDPLFHTXDWLRQV'ULYHHQHUJ\RIWKHGHSOR\DEOHWUXVV VWUXFWXUHFRPHVIURPWKHHOHFWULFDOPRWRU:KHQWKHDFWLYH cables are driven by the motor, the cable length becomes VKRUWHUDQGWKHVWUXFWXUHLVGHSOR\HG7KHGULYLQJIRUFHVLQ the active cables will become smaller after it loops over the pulley, so the Coulomb friction law is employed to consider WKHIULFWLRQEHWZHHQWKHDFWLYHFDEOHVDQGWKHSXOOH\)LJ 7KHSUHWHQVLRQIRUFHRIWKHDFWLYHFDEOHLQWKHIUHHHQG is assumed as T&RQVLGHULQJWKHIULFWLRQIRUFHVEHWZHHQWKH active cable and the pulley in the joints, driving forces of the cables in each deployable element are T2, …, Tk, …, TnLQWXUQ 7KHUHIRUHWKHUHDUHT > … > Tk > … > Tn.
7KHHODVWLFGHIRUPDWLRQVRIWKHDFWLYHFDEOHVDUHLJQRUHG and a microarc element ds in the contact point between the FDEOHDQGWKHSXOOH\LVDQDO\]HGDVVKRZQLQ)LJ7KH length density of the active cable is ȡ and the internal force vector is denoted as T(ș7KHIULFWLRQDQGSUHVVXUHIRUFH vectors in the contact point are denoted as f(ș) and N(ș), UHVSHFWLYHO\ 7KH HTXLOLEULXP HTXDWLRQV RI WKH PLFURDUF
(a) contact domain
HOHPHQWDUHREWDLQHGDVLQ(T
*
T d sind N ȡdsr
i+ i i= i +
^ h ^ h lo2
cos
T^i+dih di-T^ih-f^ih=ȡdslp
where
and : are the velocity and acceleration of cable length variety, r: is the radius of the pulley,
Į: is the half angle between the two active cable elements QHDUE\WKHSRLQWDVVKRZQLQ)LJ
)LJXUH $FWLYHFDEOHVLQWZRDGMDFHQWGHSOR\DEOHHOHPHQWV
7KHQWKHYHORFLW\DQGDFFHOHUDWLRQRIFDEOHOHQJWKYDULHW\ and ZHUHIRUPXODWHG)RUWKHDFWLYHFDEOHHOHPHQWij in a GHSOR\DEOHHOHPHQWWKHOHQJWKFDQEHH[SUHVVHGDV(T
Xi Xj X X l
T
i j 2 1
- - =
^ h^ h
6 @
'LIIHUHQWLDWLQJ(TIRUWKH¿UVWDQGVHFRQGWLPHV (TVDQGZHUHGHWHUPLQHG
l X X X
X
j i
i
j
m m m
= - =
-o o o o
o
^ h 6 @) 3
where
Ȝ = 1l (Xj – Xi)7LVWKHGLUHFWLRQFRVLQHRIWKHFDEOHHOHPHQW
7KH9LVFRXVDQG&RXORPEIULFWLRQODZVFDQEHFRPELQHG into equations of motion, being the second one employed:
f^ih=ȝN^ih
where ȝLVWKHG\QDPLFIULFWLRQFRHI¿FLHQW %\XWLOL]LQJWKH¿UVWHTXDWLRQRI(TVDQG(TLV the result: 7KHQWKHUHLV(T
(TLVGLYLGHGE\dș and the limit is gotten by dșĺ.
%\XVLQJ(TLW\LHOGV(T
d dT
T prl pl2
i =n + -n
p o
^ h
7KHVWDUWDQGHQGDQJOHVDFFRUGLQJWRFRQWDFWUHJLRQDUH provided as șk and șk ±7KHGH¿QLWHLQWHJUDOUHVXOWRI(TLV ZULWWHQDV(T
T
T T ȡrl ȡl
dT d k 1 k k k 2 1 n n i i i + - = - -p o ^ h
#
#
6ROYLQJVXFKHTXDWLRQVLWZLOO\LHOG(TT ȡrl ȡl T ȡrl ȡl
e
k k
2
1 2 k1 k
n n
n n
+
-+
-= n i i
- - -p o p o ^ ^ ^ h h h
7KHUHLVșk ± – șk = ʌ – 2ȕ, and the relation of the internal
force of cable in two adjacent deployable elements can be REWDLQHG(T
(20) T ȡrl l
T ȡrl l e
k k
2
1 2 2
n nȡ
n nȡ
+
-+
-= n r b
- -p o p o ^ ^ ^ h h h %\XWLOL]LQJ(TWKHIULFWLRQIRUFHEHWZHHQWKHDFWLYH cable and the pulley in the joint jLVREWDLQHG(T
fk-1=Tk-1-Tk
7KHJHQHUDOL]HGGULYLQJIRUFHRIDGULYLQJFDEOHHOHPHQWLV DV(T
Qce Q Qie je
T
=" , (22)
JACOBIAN MATRICES OF THE MACROELEMENTS
In this section, some deployable macroelements are investigated, and the corresponding Jacobian matrices are
IRUPXODWHG7KHHTXDWLRQVRIWKHFRQVWUDLQWVDQGWKH-DFRELDQ
matrices for a constant distance constraint on the members, the position constraint of the sleeve element, the angle constraint of the revolute joints and of synchronize gears have
EHHQSUHVHQWHGLQPDQ\UHIHUHQFHV)RUWKHVDNHRIEUHYLW\
the formulations are not fully developed here, therefore see
Nagaraj et al.=KDRDQG*XDQIRUGHWDLOV
Linear scissor-link element
/LQHDUVFLVVRUOLQNHOHPHQW6/(LVDW\SHRIIXQGDPHQWDO
macroelement in the deployable truss structures, where two pairs
RIVWUXWVDUHFRQQHFWHGWRHDFKRWKHUDWDSLYRWSRLQW2WKURXJK DUHYROXWHMRLQWVKRZQLQ)LJ,WDOORZVWZRSDLUVRIVWUXWV
to rotate freely around the axis perpendicular to their common
SODQHEXWUHVWUDLQVDOORWKHUGHJUHHVRIIUHHGRP$WWKHVDPH WLPHWKHLUHQGSRLQWVDUHKLQJHGWRWKHRQHVRIRWKHUHOHPHQWV $VFDQEHVHHQLQ)LJWKHDQJOHEHWZHHQOLQNVil and oi is
GH¿QHGDVWKHGHSOR\DQJOHĮRI6/(:KHQWKHPDFURHOHPHQW LVGHSOR\HGWKLVGHSOR\DQJOHEHFRPHVODUJHU
7KHUHDUHWZRFRQVWDQWGLVWDQFHFRQVWUDLQWHTXDWLRQVRIWKH
uniplets ij and lk, and the Jacobian matrix is formulated as
(T
A
0 0
0 0
e ijT ijT
kl T
kl T 1
m m
m m
=
-= G (23)
where the direction cosine of the uniplets ij is
Ȝij = L ij
1
(Xj – Xi), like the klXQLSOHWV
It is assumed that the length of oi, ok are a and the length of oj, ol are k*a7ZRSDLUVRIXQLSOHWVDUHFRQQHFWHGWRHDFK
other at the point o 'XULQJ WKH GHSOR\PHQW SURFHVV WKH
relative position of the connection point oLVLQYDULDEOH7KH
FRQVWUDLQWHTXDWLRQLVDV(T
X X
a k a
X X X
a k a
X
1 1
j i k l l
i- + + = - + + ^
^ ^ ^
h
h h h
7KHXSSHUHTXDWLRQLVVLPSOL¿HGWRWKHIROORZLQJIRUP (T
kXi+Xj-kXk-Xl=0 (25)
Differentiating it with respect to X, therefore the Jacobian matrix is obtained:
Ae kI I kI I
2=" 3#3 3#3 - 3#3 - 3#3,
%HFDXVH ¿YH UHYROXWH MRLQWV DUH FRSODQDU GXULQJ WKH
deployment process, the planar equation is the constraint
HTXDWLRQVRIWKHPDFURHOHPHQW(T
rij#rik :ril=0
^ h
When differences are compared with respect to X, the
-DFRELDQPDWUL[LVREWDLQHG(T
Ae r r r r r r r r r r il jk ij ik ik il ij il ij ik
3="^ # - # h1#3 ^ # h1#3 ^ # h ^ # h1#3,
)RUWKHSODQDU6/(WKHURZQXPEHURIWKH-DFRELDQPDWUL[
Ae = Ae 3
Ae 3
Ae 3
LV
Planar multiangular scissor element
Planar multiangular scissor element, as illustrated in
)LJLVDQRWKHUW\SHRIPDFURHOHPHQWXVHGLQWKHGHSOR\DEOH
truss structures, in which the uniplets ij and lk are not aligned at an intermediate point O
%DVHGRQWKHFRQGLWLRQRIWZRFRQJUXHQWWULDQJOHVWKH
constraint equations of the planar multiangular scissor
element include:
IRXUWUXVVPHPEHUVio, jo, ko and lo of the macroelement are
considered and there are four constant distance constraint equations;
WZR GXPP\ WUXVV PHPEHUVij and lk are added to the
macroelement; thus, there are two constant distance constraint equations;
¿YH SRLQWV DUH FRSODQDU DQG WKH SODQDU HTXDWLRQ LV constraint, which can be formulated by the same method DVDOUHDG\PHQWLRQHG
For the planar multiangular scissor element, the row
number of the Jacobian matrix AeLVHLJKW
Rigid-plate element
Planar rigid-plate element is a type of macroelement used
LQVRODUDUUD\V$IRXUQRGHULJLGSODWHHOHPHQWijkl is shown in
)LJZKLFKLVFRQQHFWHGWRRWKHUPHPEHUVDWFRUQHUSRLQWV WKURXJKDUHYROXWHMRLQW
7KHPDVVSURSHUW\RIWKHPDFURHOHPHQWLVUHGXFHGWRD ¿QLWHQXPEHURISRLQWV7KHULJLGSODWHHOHPHQWLVVXEVWLWXWHG ZLWKDQHTXLYDOHQWJULGRIYLUWXDOULJLGVWUXWV7KHIUHHGRP degree of the element is analyzed: the degree of freedom of
four spatial points i, j, k, lLV$IWHUWKHOHQJWKFRQVWUDLQWVRI
six struts are appended, the total degree of freedom becomes VL[ZKLFKLVHTXDOWRWKDWRIWKHULJLGSODWHHOHPHQW
Six struts ij, jk, kl, li, ik and jl of the element are considered, DQGWKHUHDUHVL[FRQVWDQWGLVWDQFHFRQVWUDLQWHTXDWLRQV)RU the planar-plate element, the row number of the Jacobian
matrix AeLVVL[
)LJXUH 5LJLGSODWHHOHPHQW
NUMERICAL EXAMPLES
Double-ring deployable truss
A type of double-ring deployable truss based on quadrilateral elements is investigated for large-size mesh
DQWHQQDVWKHIXOOGHSOR\HGIROGHGFRQ¿JXUDWLRQRIZKLFKDUH VKRZQLQ)LJ
7ZR DGMDFHQW GHSOR\DEOH HOHPHQWV RI WKH RXWHU DQG LQQHU ULQJ DUH VKRZQ LQ )LJ ZKLFK DUH TXDGULODWHUDO HOHPHQWVZLWKDGLDJRQDOVOHHYHHOHPHQW7KHDFWLYHFDEOHV SDVVWKURXJKWKHGLDJRQDOVOHHYHV:KHQWKHDFWLYHFDEOH is deployed by a motor and becomes shorter, the diagonal VOHHYHV$(DQG&(FRQWUDFW7KHQWKHGHSOR\DEOHHOHPHQWV DUHGHSOR\HG:KHQWKHOHQJWKVRIWKHGLDJRQDOVOHHYHV$( DQG&(DUHHTXDOWRDGHVLJQHGYDOXHWKHGHSOR\PHQWRIWKH HOHPHQWVLVVWRSSHG
Several planar truss elements can make a closed loop by DUUDQJLQJWKHPVLGHE\VLGH7KHSODQDUWUXVVHOHPHQWVDUHDOVR XVHGLQFRQQHFWLQJWKHLQQHUDQGRXWHUORRSV7KHVWUXFWXUH topology is determined and the major design parameters of the WUXVVDUHLQFOXGHGWKHVLGHQXPEHURIWKHULQJLVWKHWUXVV DSHUWXUHLVPWKHWUXVVKHLJKWLVPWKHOHQJWKRIWKH FKRUGVWUXWVLQWKHRXWHUULQJLVPDQGWKHOHQJWKRIWKH FKRUGVWUXWVLQWKHLQQHUULQJLVPP
DGHSOR\HGFRQ¿JXUDWLRQ EIROGHGFRQ¿JXUDWLRQ )LJXUH 'RXEOHULQJGHSOR\DEOHWUXVV
)LJXUH 'HSOR\PHQWSURFHVVRIGHSOR\DEOHWUXVVHOHPHQWV
7KHDFWLYHFDEOHVORFDWHGLQWKHPRGHODUHVKRZQLQ)LJ 7KH HODVWLF GHIRUPDWLRQV RI WKH DFWLYH FDEOHV DUH QHJOHFWHG DQGDFRQVWDQW1GULYLQJIRUFHRIWKHDFWLYHFDEOHVLQDOO FRQ¿JXUDWLRQVLVDVVXPHG7RGHSOR\WKHWUXVVVXFFHVVIXOO\WKH G\QDPLFIULFWLRQFRHI¿FLHQWȝ between the active cables and the SXOOH\LVUHTXLUHGWREHVPDOOWKHUHDOYDOXHRIZKLFKLVGLI¿FXOW WREHPHDVXUHG,WLVJLYHQWREHDVPDOOGHIDXOWYDOXHLQ WKLVVLPXODWLRQZRUN7KHOHQJWKRIWLPHVWHSLQWKHVLPXODWLRQLV
VHFRQG7KHWUXVVVWDUWVWRPRYHE\DFWLYHFDEOHVDQGUHDFKHV DVWDWLF¿QDOFRQ¿JXUDWLRQDWODVWZKLFKLVVKRZQLQ)LJ When the lengths of the diagonal sleeves are equal to a designed value, the locked constraint conditions of the sleeves work and WKHVLPXODWLRQVWRSV,WWDNHVVHFRQGVWREHIXOO\GHSOR\HG 7KHFRRUGLQDWHYDULDWLRQVRIMRLQWVDQGLQWKHGHSOR\DEOH WUXVVDUHVKRZQLQ)LJZKLFKFDQGHVFULEHWKHYDULDWLRQVRI FRQ¿JXUDWLRQVGXULQJWKHVWUXFWXUDOPRWLRQLQGHWDLOV
DW VHFRQG EW VHFRQGV FW VHFRQGV
GW VHFRQGV HW VHFRQGV
)LJXUH0RWLRQSURFHVVRIWKHGHSOR\DEOHWUXVV
)LJXUH $FWLYHFDEOHV )LJXUH1XPEHULQJRIWKHWUXVVMRLQWV
(a) x axes coordinate (b) y axes coordinate
,Q WKH IXOO\ GHSOR\HG FRQ¿JXUDWLRQ WKH FRRUGLQDWHV RI SRLQWVDQGDUHUHVSHFWLYHO\ DQG +RZHYHU WKH UHIHUHQFH FRRUGLQDWHV RI WKH GHSOR\HG FRQ¿JXUDWLRQ
which are obtained from the geometric equation of this
FRQ¿JXUDWLRQDUH DQG7KHPD[LPXPHUURULV 7KHUHIRUHWKHPHWKRGFDQVLPXODWHWKHPRWLRQEHKDYLRURI WKHGHSOR\DEOHWUXVVHIIHFWLYHO\DQGDFFXUDWHO\
Deployable sail arrays
$VFDOHPRGHORIGHSOR\DEOHVDLODUUD\EDVHGRQ6/(V LVSUHVHQWHGLQ)LJ7KHVDLODUUD\FRQVLVWVRIHLJKW6/( PDFURHOHPHQWVDQGHLJKWULJLGSODWHPDFURHOHPHQWV7KH6/(
macroelements are dynamic machines during deployment and
VXSSRUWVWUXFWXUHDIWHUGHSOR\PHQW7KHULJLGSODWHHOHPHQWV DUHVDLODUUD\GHFNDQGWKHVHYHUWLFDOVWUXWVVXFKDVDQG DUHVOHHYHHOHPHQW
7KHUHDUHWZRDFWLYHFDEOHVLQWKHVDLODUUD\LQZKLFKFDQ EHVHHQLQ)LJ2QHRIWKHPZKLFKLV¿UPO\FRQQHFWHGWR MRLQWUXQVRYHUDSXOOH\DWMRLQW]LJ]DJVGRZQWKH6/( IROORZLQJWKHURXWHVKRZQLQWKH¿JXUHLWUXQVRYHUDSXOOH\DW
each kink) and, after passing over a pulley at joint 2, is connected
WRWKHPRWRUL]HGGUXPORFDWHGEHORZWKHEDVH7KHRWKHUDFWLYH FDEOHLVDUUDQJHGE\WKHVDPHPHWKRG7KHHODVWLFGHIRUPDWLRQ RIWKHDFWLYHFDEOHVLVQHJOHFWHGDQGDFRQVWDQW1GULYHIRUFH RIWKHDFWLYHFDEOHVLQDOOFRQ¿JXUDWLRQVLVDVVXPHG
)L[HGERXQGDU\FRQGLWLRQVDUHXVHGDWMRLQWVDQGDQG WKHWUDQVODWLRQDORQJ[GLUHFWLRQRIMRLQWVDQGDUH¿[HG7KH
VWUXFWXUHLVGHSOR\HGIURPWKHQHDUO\ÀDWFRQ¿JXUDWLRQĮ =
WRWKH¿QDORQHĮ = DVVKRZQLQ)LJ7KHPDVVRIHDFK
UHYROXWHMRLQWLVNJDQGWKDWRIMRLQWVLQWKHULJLGSODWH
PDFURHOHPHQWVLVNJ$WLPHVWHSdt s is used in
WKHVLPXODWLRQ7KHPRWLRQEHKDYLRURIWKHVDLODUUD\VWUXFWXUH ZDVVLPXODWHGE\WKHVDPHSURJUDP7KHVDLODUUD\VWDUWVWR PRYHE\DFWLYHFDEOHVDQGUHDFKHVDVWDWLF¿QDOFRQ¿JXUDWLRQ DVVKRZQLQ)LJ:KHQWKHOHQJWKVRIWKHYHUWLFDOVOHHYHV
are equal to a designed value, the locked constraint conditions
RIWKHVOHHYHVZRUNDQGWKHVLPXODWLRQVWRSV
7KH GHSOR\PHQW WLPH RI WKH VWUXFWXUH LV VHFRQGV 7KH FRRUGLQDWH YDULDWLRQV RI SRLQW LQ WKLV PRGHO ZLWK WLPHDUHVKRZQLQ)LJZKLFKFDQGH¿QHWKHYDULDWLRQVRI FRQ¿JXUDWLRQVGXULQJWKHVWUXFWXUDOPRWLRQLQGHWDLOV
,Q WKH ¿QDO FRPSOHWHO\ GHSOR\HG FRQ¿JXUDWLRQ WKH FRRUGLQDWHV RI SRLQWV DQG DUH UHVSHFWLYHO\ DQG
)LJXUH 6FDOHPRGHORIWKHGHSOR\DEOHVDLODUUD\
DW VHFRQGV EW VHFRQGV FW VHFRQGV
GW VHFRQGV HW VHFRQGV
x-co
o
rd
in
a
te (
m
)
Time (s) 4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0 20 40 60 80 100 120 140
y
-c
o
o
rd
in
a
te (
m
)
Time (s) 0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
0 20 40 60 80 100 120 140
(a) x axes coordinate (b) y axes coordinate )LJXUH&RRUGLQDWHYDULDWLRQVRIMRLQW
ZKLOHWKHUHIHUHQFHFRRUGLQDWHVRIWKHVH SRVLWLRQVRQWKHGHSOR\HGFRQ¿JXUDWLRQZKLFKDUHREWDLQHG IURP WKH JHRPHWULF HTXDWLRQ RI WKLV FRQ¿JXUDWLRQ DUH DQG7KHPD[LPXP HUURULV7KHUHIRUHWKHPHWKRGFDQVLPXODWHWKHPRWLRQ EHKDYLRURIWKHGHSOR\DEOHVDLODUUD\HIIHFWLYHO\
CONCLUSIONS
7KLVSDSHUSUHVHQWVDQXPHULFDODSSURDFKIRUNLQHPDWLF DQDO\VLV RI WKH GHSOR\DEOH WUXVV VWUXFWXUHV 7KH GULYLQJ
forces of active cables are combined with the equations of
PRWLRQ7KHIULFWLRQEHWZHHQDFWLYHFDEOHVDQGWKHSXOOH\LV FRQVLGHUHG7KUHHNLQGVRIPDFURHOHPHQWVXVHGLQGHSOR\DEOH VWUXFWXUHV DUH GHVFULEHG 7KH FRUUHVSRQGLQJ FRQVWUDLQW
equations and Jacobian matrices of the macroelements are
IRUPXODWHG$GRXEOHULQJGHSOR\DEOHWUXVVDQGDGHSOR\DEOH VRODUDUUD\VWUXFWXUHDUHVHOHFWHGDVQXPHULFDOH[DPSOHV7KH
deployment process and dynamic parameters at each time step can be simulated for evaluating the deployment behaviors of
WKHVWUXFWXUH5HVXOWVRIWKHQXPHULFDOVLPXODWLRQVKRZWKDW
the capabilities of this method in the motion analysis are of
FRPSOH[GHSOR\DEOHWUXVVVWUXFWXUHV7KHRULJLQRIWKH¿QDO
error in the time step of the simulation is too large to stop
WKHVLPXODWLRQQHDUE\WKHUHIHUHQFHFRQ¿JXUDWLRQ7RDFKLHYH KLJKHUVLPXODWLRQDFFXUDF\WKHWLPHVWHSQHHGVWREHVPDOOHU %DVHG RQ WKH UHVHDUFKHV LQFOXGHG LQ WKLV SDSHU IXWXUH
works are suggested: the reliability analysis of the deployment process can be researched; and deployment control of the
GHSOR\DEOHWUXVVDQWHQQDVQHHGVWREHLQYHVWLJDWHG
REFERENCES
%DH '6et al., 2000, “An explicit integration method
for real time simulation of multibody vehicle models”,
&RPSXWHU0HWKRGVLQ$SSOLHG0HFKDQLFVDQG(QJLQHHULQJ 9ROSS
%D\R(DQG/HGHVPD5³$XJPHQWHG/DJUDQJLDQDQG
mass-orthogonal projection method for constrained multibody
G\QDPLFV´1RQOLQHDU'\QDPLFV9ROSS
%ODMHU : ³$Q RUWKRQRUPDO WDQJHQW VSDFH PHWKRG
for constrained multibody systems”, Computer Methods In
$SSOLHG0HFKDQLFV$QG(QJLQHHULQJ9ROSS
&DOODGLQH&5DQG3HOOHJULQR6³)XUWKHUUHPDUNVRQ ¿UVWRUGHULQ¿QLWHVLPDOPHFKDQLVPV´,QWHUQDWLRQDO-RXUQDORI 6ROLGVDQG6WUXFWXUHV9ROSS
&KHUQLDYVN\ $*et al., 2005, “Large deployable space
antennas based on usage of polygonal pantograph”, Journal of
$HURVSDFH(QJLQHHULQJ9ROSS
)LVHWWH3DQG9DQHJKHP%³1XPHULFDOLQWHJUDWLRQ
of multibody system dynamic equations using the coordinate partitioning method in an implict Newmark scheme”,
.LP66DQG9DQGHUSORHJ0-³45GHFRPSRVLWLRQ
for state space representation of constrained mechanical
G\QDPLFV\VWHPV´-RXUQDORI0HFKDQLVPV7UDQVPLVVLRQV DQG$XWRPDWLRQLQ'HVLJQ9ROSS
/L7-³'HSOR\PHQWDQDO\VLVDQGFRQWURORIGHSOR\DEOH VSDFHDQWHQQD´$HURVSDFH6FLHQFHDQG7HFKQRORJ\9RO SS
/LDQJ&*DQG/DQFH*0³$GLIIHUHQWLDEOHQXOO
space method for constrained dynamic analysis”, Journal
RI0HFKDQLVPV7UDQVPLVVLRQVDQG$XWRPDWLRQLQ'HVLJQ 9ROSS
0HJXUR$et al., 2003, “Key technologies for high-accuracy
ODUJHPHVKDQWHQQDUHÀHFWRUV´$FWD$VWURQDXWLFD9RO SS
1DJDUDM%3et al.³.LQHPDWLFVRISDQWRJUDSKPDVWV´
0HFKDQLVPDQG0DFKLQH7KHRU\9ROSS
1DJDUDM %3et al. ³$ FRQVWUDLQW -DFRELDQ EDVHG
approach for static analysis of pantograph masts”, Computers
DQG6WUXFWXUHV9ROSS
6LQJK 53 DQG /LNLQV 3: ³6LQJXODU YDOXH
decomposition for constrained dynamical systems”, Journal
RI$SSOLHG0HFKDQLFV9ROSS
7DNDPDWVX.$DQG2QRGDI-³1HZGHSOR\DEOHWUXVV
concepts for large antenna structures or solar concentrators”,
-RXUQDORI6SDFHFUDIW9ROSS
<RX=³'HSOR\DEOHVWUXFWXUHRIFXUYHGSUR¿OHIRU VSDFHDQWHQQDV´-RXUQDORI$HURVSDFH(QJLQHHULQJ9RO SS
<X< DQG /XR< ³0RWLRQ DQDO\VLV RI GHSOR\DEOH VWUXFWXUHVEDVHGRQWKHURGKLQJHHOHPHQWE\WKH¿QLWHSDUWLFOH
method”, Proceedings of the Institution of Mechanical
(QJLQHHUV 3DUW * -RXUQDO RI $HURVSDFH (QJLQHHULQJ 9ROSS
=KDR 0/ DQG *XDQ )/ ³.LQHPDWLF DQDO\VLV RI
deployable toroidal spatial truss structures for large mesh antenna”, Journal of the International Association for Shell
