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90 Mutibody Systems Approach to Vehicle Dynamics
{n} e [n n n] T (3.29)
{n} e [n n n] T (3.30)
{Pn r } e [Pn Pn Pn] T (3.31)
The 15 equations for each part are:
d
{ Vn} 1 { Rn}
1
dt
d
{ n} e { n}
dt
(3.32)
(3.33)
{Pn r } e [B] T [I n ][B]{n} e (3.34)
m{An} 1 Σ{Fn A } 1 Σ{Fn C } 1 0 (3.35)
{Pn r } e T/{n} e Σ{Mn A } e Σ{Mn C } e 0 (3.36)
3.2.6 Basic constraints
e
Constraints are used to restrict the motion of parts. There are a number of
modelling elements that can be used to do this and the constraint may
restrict the absolute motion of a body relative to the ground or the relative
motion between interconnected parts. Constraints can be considered to be
of two types:
(i) Holonomic constraints are those which are dependent on restricting
displacement and result in algebraic equations.
(ii) Non-holonomic constraints are those where a velocity dependent motion
is enforced and result in differential equations.
There are a wide range of constraint elements available ranging from joint
primitives that can constrain combinations of individual degrees of freedom
between bodies through mechanical type joints and gear elements to higher
pair constraints such as those constraining a point to lie on a curve. The
examples shown here are restricted to those that are used to support the
examples used in this text.
A typical example of a joint primitive is the use of an inplane type of constraint
that restricts the motion of a point on one body to remain in a plane
on another body. A typical example of this is given later when the vertical
motion of individual suspension models is controlled by using a jack to
impart motion to the wheel centre as shown in Figure 3.12.
In this example the I marker is defined to belong to the wheel knuckle and
is located at the wheel centre. The J marker is defined to belong to the jack
part and will move vertically with that part according to other constraints
that control its motion. The orientation of the J marker must be defined
in this case so that the x- and y-axes define the surface plane of the jack in
which the I marker will be constrained to remain. For simplicity the jack in
Figure 3.12 is shown below the wheel but in actual fact it would be defined to
locate the xy plane of J at the wheel centre. The orientation of the I marker is
not important but the J marker must be defined as shown. In this case the z
direction of the J marker is parallel to the z-axis of the ground reference frame