real-time mbs formulations: towards virtual engineering

174 J. Cuadrado, M. Gonzalez, R. Gutierrez, M.A. Naya
Dynamic equilibrium can be established at time-step n+1 by introducing the difference equations
(17) and (18) into the equations of motion (15), leading to
4
∆t
( Φ + λ ) − Q + Mq&
ˆ 0
T
2 n+ 1
+ Φ
1 + 1 + 1 + 1
=
n+
n n n
n
Mq &
q
α . (19)
For numerical reasons, the scaling of Eq. (19) by a factor of ∆t 2 /4 seems to be convenient, thus
yielding
2
2
2
∆t T
∆t ∆t
Mq ( ) ˆ
n+ 1
+ Φq α Φ
1
+
1
−
1
+ = 0
n+
1 n+
λ
n+
Q
n+
Mq&
n
, (20)
4
4 4
or, symbolically f ( q n+1
) = 0 .
In order to obtain the solution of this nonlinear system, the widely used iterative Newton-Raphson
method may be applied
( q)
⎡∂f
⎢
⎣ ∂q
⎤
⎥ ∆q
⎦
i
i +1
= −
being the residual vector
[ f ( q)
] i
, (21)
2
∆t
T
T *
[ f ( q)
] = ( Mq&&
+ Φ αΦ
+ Φ λ − Q)
4
q
q
(22)
and the approximated tangent matrix
⎡∂f
⎢
⎣ ∂q
2
( q) ⎤ ∆t ∆t T
= M + C + ( Φ αΦ
K)
⎥
⎦
2
4
q q
+
, (23)
where C and K represent the contribution of damping and elastic forces of the system provided they
exist.
A closer look at the tangent matrix reveals that ill-conditioning may appear when the time-step
becomes small. It may be seen in Eq. (23) that K and the constraint terms are multiplied by ∆t 2 , C by
∆t and M is not affected by the step size. As a consequence when ∆t reaches small values, large roundoff
errors will occur. In fact, it has been demonstrated in [7] that for an index-3 differential-algebraic
equation, the tangent matrix has a condition number of order 1/∆t 3 . Consequently, the method is
bound to have round-off errors for step sizes smaller than 10 -5 , which lets a sufficient range for
solving practical problems.
The procedure explained above yields a set of positions q
n+ 1 that not only satisfies the equations
of motion (19), but also the constraint condition Φ = 0 . However, it is not expected that the
corresponding sets of velocities and accelerations satisfy Φ & = 0 and Φ &
= 0 , because these conditions
have not been imposed in the solution process. To overcome this difficulty, mass-damping-stiffnessorthogonal
projections in velocities and accelerations are performed. It can be seen that the projections
leading matrix is the same tangent matrix appearing in Eq. (23). Therefore, triangularization is avoided
and projections in velocities and accelerations are carried out with just forward reductions and back
substitutions.
*
If q& and q& & * are the velocities and accelerations obtained after convergence has been achieved in
the Newton-Raphson iteration, their cleaned counterparts q& and q& & are calculated from