Structural Dynamics (6).pdf

Chapter 8
Generalized Single-Degree-Of Freedom
Systems
§ 8-1 ments On SDOF Systems
• Assemblages of rigid bodies in which elastic
deformation are limited to localized spring elements
• Systems having distributed flexibility, in which the
deformations can continuous throughout the
structure.
1
8-2 Generalized Properties:
Assemblages of Rigid
„ All displacements and force can determined by the
Generalized displacement
2
Example E8-1
x
F (,)xt= F0 f () t
a mJ22,
A E H N
m1
k c k
c1 1 2 2
a 2a a l/2 l/2 l/2
F M
p I 2 FI 2
M
A I1 E H N Z
F
F D 2
S1 F
FD1 S 2
3
F M
p δ Z I 2 FI 2
M I1 Z
A F FD 2
S1 H N
E F
FD1 S 2
Zt&&() 4 ama (4)2 Zt && () 16
M ()tJ== = amZt2 && () F ()tFa= 8f ()t
J 114343aa p 0
Zt&&() 2()Zt&&
Mt()= J Ft()= m
J 223a I 223
Zt& () 3()Zt
Ft()= c Ft()= k
D114 S114
Zt()
F ()tcZt= & () Ft()= k
D 22 S 223
16 δ Z 2()2Zt&& δ Z Zt&&()δ Z Zt& ()δ Z
δWt()= − amZt2 &&() −m −J −c
34a 2 33 2 33aa 1 44a
3()3Ztδ Z Zt()δ Z 2
−cZt&()δ Z −k −k +8()F af tZδ= 0 4
2 1 44 2 33 0 3
am 4916Jc k
[(am++ m +21 )()( Z&& t ++ c )()( Z & t + k + 2 )() Z t − F af ()] tδ Z = 0
3922109a 2 16 169 3
am 4916Jc k
(am++ m +21 ) Z&& () t ++ ( c ) Z & () t + ( k + 2 ) Z () t = F af () t
3922109a 2 16 169 3
mZt∗∗&&()++ cZt & () kZt ∗() = F ∗() t
am 4 J
mam∗=++ m +2
392 9a 2 Generalized mass
c
cc∗= 1 +
16 2 Generalized Damping
9 k
kk∗=+2 Generalized stiffness
161 9
∗ 16
FFaft= 0 () Generalized load
3 5
Geometric stiffness of rigid member
a
aaeZ222=−() +
Z
02()=−aee −δ+ 2 ZZδ
a e
Z
δeZ= δ
a
ZZ7 Z
δWNeN==δδδ() + ZN = Z
N 43aa 12 a
∗ 97k2 N
kk=+−1 ∗ 97k2 N
16 9 12 kk= 1 +−=0
16 9 12
12 9 k2
Nk=+()= N cr
7161 9
δ Z
Z
A E N
H 6
Geometric stiffness of flexible member
11
dex=− d (d) x22 −(d) y =−−(11())d[1(1())]d y′′′ 2 x =−− y 2 x = ()d yx 2
& 22
1 L
e = ()dy′ 2 x
2 ∫0
L
δδeyyx= ′′d
∫0 A dy B
x