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Multibody systems simulation software 93
{a J } 1
J
{d IJ } 1
I
Fig. 3.14 Inplane constraint element. (This material has been reproduced
from the Proceedings of the Institution of Mechanical Engineers, K2 Vol. 213
‘The modelling and simulation of vehicle handling. Part 1: analysis methods’,
M.V. Blundell, page 114, by permission of the Council of the Institution of
Mechanical Engineers)
force corresponding to this constraint can be represented by a scalar term
d . The reaction force on part i can be represented by the vector {a J } 1 d
with a moment given by {r I } 1 {a J } 1 d . Applying Newton’s third law again
the reaction force on part j can be represented by the vector {a J } 1 d . The
moment contribution to part j is given by ({r J } 1 {d IJ } 1 ) {a J } 1 d .
Expanding this again using the definition given for {d IJ } 1 in equation (3.38)
gives ({R i } 1 {r I } 1 {R j } 1 ) {a J } 1 d . In order to complete the calculation
the contribution to the term {Mn C } e in equation (3.36) a transformation
of the moments into the co-ordinates of the part Euler-axis frame is
needed.
For part i this would be achieved using [B i ] T {r I } i [A ij ]{a J } 1 d .
For part j this would be achieved using [B j ] T {a J } j [A j1 ]({R i } 1
[A 1i ]{r I } i {R j } 1 ) d .
The third basic constraint element constrains a unit vector fixed in one part
to remain perpendicular to a unit vector located in another part and is known
as the perpendicular constraint. The constraint shown in Figure 3.15 is
defined using a unit vector {a J } 1 located at the J marker in part j and a unit
vector {a I } 1 located at the I marker belonging to part i.
The vector dot (scalar) product is used to enforce perpendicularity as shown
in equation (3.41):
p {a I } 1 • {a J } 1 0 (3.41)
The constraint can be considered to be enforced by equal and opposite
moments acting on part i and part j. The constraint does not contribute
any forces to the part equations but does include the scalar term p in the
formulation of the moments. The moment acting on part i is given by
{a I } 1 {a J } 1 p Applying Newton’s third law the moment acting on part j
is given by {a I } 1 {a J } 1 p . The moments must be transformed into the
co-ordinates of the part Euler-axis frame.