28 3 Vol.28 No.3 2015 6 Journal of Vibration Enginering Jun.2015
– *
1 1 1 2 (1. 116024;
2. 116024)
: ;
:;;;
: O 3 1 3 .7 : A : 1 0 0 4 -4 5 2 3 (2 0 1 5 )0 3 -0 3 3 7 -0 8
DOI:10.16385/j.cnki.isn.1004-4523.2015.03.001
[1-12]
Kane[1] : [2-6] ;
* :2013-12-26; :2014-06-12
: (11372057)
[7 ] [8] Shi P [9]
[10] [11]
338 28
[12] () [1 3 -1 6 ]
1 1 .1
()=d()/ds
1
1 pe = (EA+EI)ds (6)
{g 1 g 2 } {e1e2}e1 0 {t 1t 2 }
1
Fig.1A plane beam element undergoes large range of
motion 1s
12
= 1 + 1 .2
(4 )
F = EA;M = EIz (5) F ;M ;EA EIz
l
0
() s =N
()
T ;
=[1 2]N=[0 1-]8
( ) ()
= s/l (7 )
= N 9 (7 ) (9 )
7
(1 ) r1 s ds
(6 )
pe = ()TK (10)
K
EAl 0 0
EIl/3EIl/6 (11) //
r = r 1 +
K = 0 0
: = d s / d s 1 ; d s
(7)
d/ds= (1+)t1 (2)
x=e1=(1+)cos
(3) [-]
EIl 6EIl 3 1 .3
(13) w=e2 = 1+sin
N= 0 (2 2)l/2 2l/2 {()
s
s= Nds=N ()
0
( )
12
0l
[]
: = T ; g (A B ) = A T B + B T A ;
3 :- 339
(12)
= 1 + N (1 4 ) (3 ) s
s
( ) x= 1+ cosds
0
s(15) ()
^^
H=N1 N2 N2 0 0 (21)
l2 -1N2=2 3-2 () ()
cos 1 + 0 sin 1 + 0
A= () ()(23)
ss
ds- (w x )ds
10
18 s =sinc =cos=1+ 0:x0 =0w0 = 00 =0;(14)(15) x w
1
/2
s
0l ()
=(x w)T =Hl (ll)
(19) l=xlcwls (20)
T
() ()
Dl= (l
0 0 0 N2 ^N2 ^2^
N l1 N
1 =( ) 2 =
s
r(s) r A (22)
[ ]
sT +cNl ll
013 T()
0
w=1+ sinds
:
0 :sin cos 1 2 / 2 (3 )
////() w x = (w s ) (x s )= tan 16
[]
(15) x = s +
I= 0 -1
=1+
sin 1 + 0 cos 1 + 0 () ()
0
(22)
r(s)=r1+ 1AI+A (24)
02 w = w/x ds
0
2
( )
(17)
() rs=r1+1AI+A-1A+21AI 25
0
()
(19)
(12) s (1 5 )
(15) xw
x(0)=x0x(0)=c ;x(l)=xl =x(l)=c
=Hl=HDl;=HDl+HDl (26)
T ClT -Snl
cT sN
– l ;
Dl =
SlT +Cnl
ll
-(gSnlT +Cnl) TT
– s (N T )+c N N
g l l l l) (27)
T() (gCnlT -Snl)
T() T
(cg Nl T -sNlNl)
ll
T = 1 0 0
() () ;() () w0=w0w0=s wl=wlwl=s
lll
; ; TSl = sdsSnl = sNdsSnl = sN Nds
000
lll
Cl = cdsCnl = cNdsCnl = cN Nds
(28)
(26)
() () rs =r1 +AIHl 1 +AHDl 29
() () rs =r1+AIHl1+AHDl 30
000 ;;T
(26) (25)
rs =r1+ 1AIHl+AHDl+Aar 31
340 28
() ar =HDl- 21Hl+21IHDl 32
1 1
NH = Hd;NHH = HTHd;
(14)
0
0
l T
() () pa = [r r+Iz]ds 39
Iz
fqg= I mg (50)
(30)~ (34)
AINHl
(AN D )Tm (51)
1T q1 q q g f=Hlg
p a = (q ) (M q + M + F ) +
[13]
T q1
( ) (M q + M + F )
Mr Mr M q q = [
l
Mq = [
Fq = ()
r M M
0
Mr
;Mq = (Mq)T (41) ]
(10)
M Fr
(42)
pd = ()Tfd (53)
F
fd =K (54) 2
2 .1
r ; T
M = mI M = m l N H H l + mIz
(40 )
; p d = E A + E I z d s 5 2
()() ]
11
;T;() NN = NdNNN = NNd 45
1 .5
; () = 1 + N = 1 + N 33
0
1
0
() = 1 + N 34
r q
r1 r2
12 2
TT NHIH = HIHd
0
ll
IAIHll 1 .4
() m1 +m2 +l AIHl f2 +lp
B =
q = q = (35)
l
( )T ( )T ()
1 2
(14)(29)(31)(33)(34)
pf =[ri fi+imi]+ r psds i=1 0
; ;
q2 =q1+q2 =Bq1+Qq2 =Bq1+Q+
(46) p(s);fimi
(30)(33)
AHll Aalr
==(37)
l
f1+f2+ pds
N l
0
fq =
T T()T T ()
(36)
pf =(q1)Tfq+()Tf (47)
[]
AHlDl [0 1 ] [Nl ]
0
( ) ; ( ) p = AIH Tpdsp = AH Tpds
;Q =
(38)
TTT
(48) f=m2Nl+DlAHl f2+Dlp 49
0
[ ] ()
0
T
0
; r;r ;
T
M = m D l N H H D l + m I z N N N (4 3 )
M = mAINHl M = mANHDl
T
M = mlNHIHDl +mIzNN .
r2
( )
( )
F = m A N H D l N H l 1 + 2 I N H D l 1
T( 2 ) F = m l N H I H D l N H I H l 1 + 2 N H H D l 1
F = m D lT N H H D l N H H l 21 + 2 N TH I H D l 1
( ) (44)
36
3 :- 341
n
(10) (40)
(47) (55)
1
1 q0
1T 001 0 0 ()
:
(1) ;
(2) L
:L 1 n L (i ) i 1; 0
2
2 Fig.2 Regularly labeling of beam elements
:L = [0 1 2 ]
L
(3 6 ) i
; ;
q i1 = q 10 + i q i1 = iq 10 + i q i1 = iq 10 + i + i
q0 Mq0+M+F+ ()
T 01 () ()
(M q0 + M + F )= 0 59
M
i Mi i
n
00 Tq
=
M = i Mii+MiTi M = M
i=1 n
q ; 0 ()
( 0 )T [Tq q T(q
0
T q q
i=1 n
M = i Mii +MiTi +Ti ()
Mii +
Mi Ti
i=1
)]
(60)
F=i Mii+Fi-fi + [()
n
0 Tqqq
F= i Mii+Fi-fi
() n()
i=1
Tqqq 61
i=1
Tq
T i ( M i i + F i + K i i f i ) ]
(59)
M00 M0q10 Fi0
()
(55)
3
3 [10-1114-15]
3 .1
M L = 10 m u v 3
5
=0.1
4
i =L(i)+L(i);i =BL(i)L(i)+QL(i)TL(i) {;
i =BL(i)L(i)i =BL(i)L(i)+L(i)
: T T T T ; =1 2 n
() Ti i:i =Ti ;
0 =0;0 =0;0 =0;0 =0; {;;;;
(57) 0 =IB0 =I0 =0Q0 =0T0 =0
2 .2
:
n nnn
pia + pie = pif i=1 i=1 i=1
(58)
(56)
[ + =0 62
]
0
M M Fi
q10
342
28 : L = 10 m A = 1 m 2
3 Fig.3 Cantilever beam undergoes bending load
I =510-4 m4E=2.8107 Pa z
= 1 .2 k g / m 3
6 Fig.6 A flexible beam on a rotating base
(t ) 6 t2 15 2 2t
cos 10 t 15 s [(2)(15 )]
+
(t)=152
6t 45t > 15 s 5
(63) 30s7
4 Fig.4 Large deformation of the cantilever under bending
moments
M =2EIz/L
5
5
Fig.5The tip displacements of the cantilever with an
end bending load
5
3 .2
[14-15] 6[14]
7 Fig.7 Displacement curves at the fre end
7: 5 .1 4 1 0 4 m 0 [14-15]
3 :- 343
3 .3
8
8 Fig.8 The flexible pendulum
[10 -11 ]
CPU [10]
4
:
(1) ;
(2) ;
(3)
:
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P a A = 2 .5 c m 2 I z = 0 .1 3 c m 4 = 27 6 6 .6 7 kg/m3
5 [10] Matlab ode23tb : AbsT=0.1 RelT=
0 .0 1 2 .5 s C P U 1
1 CPU Tab.1 The CPU time comparison betwen absolute cordinate
method and the proposed method
C P U 9
[10 -11 ]
9 Fig.9 The tip deformation of the flexible pendulum
[10] 2 2 5 .0 5 s
6 .2 9 s
344 28
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Rigid-flexible dynamics analysis of a large deformation beam element based on interpolation of strains
ZHANG Zhi-gang1 QI Zhao-hui1 WU Zhi-gang12 (1.State Key Laboratory of Structural Analysis for Industrial EquipmentDalian University of Technology
Dalian 116024China; 2.School of Aeronautics and AstronauticsDalian University of TechnologyDalian 116024China)
Abstract:The dynamic stifeningphenomenon in flexible multi-body system dynamics is due to the deformation coupling. The first-order approximation model has ben sucesfuly applied in the rigid-flexible coupling modeling of smal deformation. Howeverit is found necesary to consider more deformation coupling efects in lager deformation cases.In this paperthe beam bending curvature and axial strain are selected as the element parameters.Then the recursion formulations are obtained
for the kinematic parameters of two end nodes of an element based on theories of large deformation and finite rotation.A planar beam element used for large deformation rigid-flexible dynamics analysis is proposedwhich can automaticaly take into acount the dynamic stif fening terms.Final lythe validity and ef fectivenes s of the proposed algorithm are verified through some nu- merical examples which involve the large deformations and rigid-flexible dynamics of beams.
Key words:rigid-flexible coupling;beam element;strain interpolation;dynamic stifening
: (1984 ) :15326175369;E-mail:[email protected] : (1964 ) E-mail:[email protected]
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