real-time mbs formulations: towards virtual engineering

172 J. Cuadrado, M. Gonzalez, R. Gutierrez, M.A. Naya
a
11
= r ux 1
+ u
x1
; a
12
= r uy 1
+ u
y1
; a
13
= r uz 1
+ u
z1
,
a = r + u
a = r + u
p1 uxp xp
;
p2 uyp yp
;
p uzp zp
a
3
= r + u . (8)
The elastic displacements can be expressed as superposition of static and dynamic modes,
ns
∑
i=
1
i
i
nd
∑
*
u = Ω η + Ψ ξ , (9)
j=
1
j
j
where n s and n d are the number of static and dynamic modes respectively, Ω
i
are the static modes, η i
are their amplitudes, Ψ
j
are the dynamic modes, and ξ j
are their amplitudes. Either the amplitudes of
the static modes as well as those of the dynamic modes are considered as problem variables.
Therefore, the problem variables for a general flexible body when using the proposed modeling
technique are: the origin of the local reference frame, r
o
, the thee orthogonal unit vectors, a, b, and c,
which define de local axes, and the amplitudes of static and dynamic modes, η i
and ξ j
, respectively.
Substituting Eq. (9) into Eq. (7) leads to
v
*
⎡I
⎢
⎢M
= ⎢M
⎢
⎢M
⎢
⎣
I
b
b
11
I
p1
I
b
b
12
I
p2
I
b
b
13
I
p3
I
AΩ
M
M
M
AΩ
1
1
p
1
...
...
AΩ
M
M
M
AΩ
1
ns
p
ns
AΨ
AΨ
1
1
p
1
...
...
AΨ
M
M
M
AΨ
1
nd
p
nd
⎧ r&
0 ⎫
⎪
a&
⎪
⎪ ⎪
⎪ b&
⎪
⎤⎪
⎪
⎥⎪
c&
⎪
⎥
⎪ & η
⎪
1
⎥⎨
⎬ = Bq&
, (10)
⎥⎪
M
⎪
⎥⎪
& η ⎪
⎥
ns
⎦⎪
& ⎪
⎪ ξ1
⎪
⎪ M ⎪
⎪
&
⎪
⎪⎩
ξn
⎪
d ⎭
r
where Ω
i
is a 3 by 1 vector containing static mode i evaluated at node r,
containing dynamic mode j evaluated at node s, and
s
Ψ
j
is a 3 by 1 vector
b
ns
nd
ns
nd
11
= rux
1
+ ∑ Ωix
1η
i
+ ∑Ψ
jx1ξ
j
; bp
1
= ruxp
+ ∑ Ωixpηi
+ ∑Ψ
jxpξ
j
,
i=
1
j=
1
i=
1
j=
1
b
ns
nd
ns
nd
12
= ruy1
+ ∑ Ωiy1η
i
+ ∑Ψ
jy1ξ
j
; bp2 = ruyp
+ ∑ Ωiypηi
+ ∑Ψ
jypξ
j
,
i=
1
j=
1
i=
1
j=
1
b
= r
+
ns
∑
Ω
η +
nd
∑
Ψ
ξ
13 uz1
iz1
i
jz1
j
;
p uzp
izp i
jzp j
i=
1
j=
1
i=
1
j=
1
ns
∑
nd
∑
b
3
= r + Ω η + Ψ ξ . (11)
The vector appearing in the right-hand-side of Eq. (10), q& , is the vector of the derivatives
*
(velocities) of the body variables q, and then, substituting the value of v given by this equation into
Eq. (5), the kinetic energy of the body can be written as