4569846498

Multibody systems simulation software 89
a term for the momenta P j associated with motion in the q j direction, and a
term C j to represent the constraints:
T
Pj
∂ ∂q˙ j
C
j
n
∑
i1
∂i
∂q
This results in the equation:
T
Ṗj
∂
Qj
Cj
∂q
0
j
(3.17)
(3.18)
(3.19)
By way of example consider first the equations associated with the translational
co-ordinates. The generalized translational momenta {Pn t } 1 for the
part can be obtained from:
d
{ An} 1 { Vn}
1
dt
(3.20)
{Pn t } 1 T/{Vn} 1 m{Vn} 1 (3.21)
d
{ Pnt} m{ An}
dt
j
1 1
i
(3.22)
This results in {An} 1 as the acceleration of the centre of mass for that part.
It should also be noted that the kinetic energy is dependent on the velocity
but not the position of the centre of mass, T/{Rn} 1 is equal to zero. We
can now write the equation associated with translational motion in the
familiar form:
m{An} 1 Σ{Fn A } 1 Σ{Fn C } 1 0 (3.23)
where {Fn A } 1 and {Fn C } 1 are the individual applied and constraint reaction
forces acting on the body. The rotational momenta {Pn r } e for the part can
be obtained from:
{Pn r } e T/{n} e [B] T [I n ][B]{n} e (3.24)
We can now write the equations associated with rotational motion in
the form:
{Pn r } e T/{n} e Σ{Mn A } e Σ{Mn C } e 0 (3.25)
{Pn r } e [B] T [I n ][B]{n} e (3.26)
In this case {Mn A } e and {Mn C } e are the individual applied and constraint
reaction moments acting about the Euler-axis frame at the centre of mass
of the body. Introducing the equation above for the rotational momenta
introduces an extra three variables and equations for each part.
The 15 variables for each part are:
{Rn} 1 [Rnx Rny Rnz] T (3.27)
{Vn} 1 [Vnx Vny Vnz] T (3.28)