4569846498

Multibody systems simulation software 113
F Iy
y
M Iy
F Jy
F Jx
M Iz
M Ix
J
y
M Jy
M Jx
x
z
M Jz
F Jz
I
z
L
x
F Iz
F Ix
Fig. 3.35
Massless beam element
The forces and moments applied to the I marker are related to the displacements
and velocities in the beam using the beam theory equations
presented in matrix form as:
⎡ F
⎢
F
⎢
⎢ F
⎢
⎢
M
⎢M
⎣⎢
M
Ix
Iy
Iz
Ix
Iy
Iz
⎤ ⎡k11
0 0 0 0 0 ⎤ ⎡dx
L⎤
⎡c c c c c c
⎥ ⎢
0 k
k
⎥
d
⎥ ⎢
22 0 0 0 ⎢ ⎥ ⎢
26
⎥
y c c c c c c
⎢ ⎥ ⎢
⎥ ⎢ 0 0 k33 0 k35
0 ⎥ ⎢ dz
⎥ ⎢c c c c c c
⎥ ⎢
⎥ ⎢ ⎥ ⎢
⎥ ⎢
0 0 0 k44
0 0
⎥ ⎢
rx
⎥ ⎢
c c c c c c
⎥ ⎢ 0 0 k k ⎥
35 0 55 0 ⎢ r ⎥ ⎢
y c c c c c c
⎦⎥
⎣⎢
0 k26 0 0 0 k66
⎦⎥
⎣⎢
rz
⎦⎥
⎣⎢
c c c c c c
11 21 31 41 51 61
21 22 32 42 52 62
31 32 33 43 53 63
41 42 42 44 54 64
51 52 52 53 55 65
61 62 63 64 65 66
⎤ ⎡ vx
⎤
⎥ ⎢
v
⎥
⎥ ⎢
y
⎥
⎥ ⎢ vz
⎥
⎥ ⎢ ⎥
⎥ ⎢
x ⎥
⎥ ⎢
⎥
y
⎦⎥
⎣⎢
z ⎦⎥
(3.51)
The terms d x , d y and d z in (3.51) are the x, y and z displacements of the
I marker relative to the J marker measured in the J marker reference frame.
The terms r x , r y and r z are the relative rotations of the I marker with respect to
the J marker measured about the x-axis, y-axis and z-axis of the J marker. It
should be noted here that the rotations in the beam are assumed to be small
and that large angular deflections are not commutative. In these cases, typically
when deflections in the beam approach 10% of the undeformed
length the theory does not correctly define the behaviour of the beam. The
terms V x , V y , V z , x , y and z are the velocities in the beam obtained as
time derivatives of the translational and rotational displacements.
The stiffness and damping matrices are symmetric. The terms in the stiffness
matrix are given by:
K
EA
L
K
12EI
EI
EI
zz
6
12
zz
yy
K
K
L ( 1 P ) L ( 1
P ) L ( 1
P )
11 22 3 26 2 33 3
y
y
z