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Modelling and analysis of suspension systems 205
Z 1
{V D } 1
O 1 Y1
X B
1
Body 3
A
{ 3 } 1 D
Fig. 4.69
Angular and translational velocity vectors for the upper wishbone
velocities at points within the system. An example of this is shown for the
upper wishbone in Figure 4.69.
Referring back to the earlier treatment in Chapter 2 we can remind ourselves
that as the suspension arm is constrained to rotate about the axis AB,
ignoring at this stage any possible deflection due to compliance in the suspension
bushes, the vector { 3 } 1 for the angular velocity of Body 3 will act
along the axis of rotation through AB. The components of this vector
would adopt signs consistent with producing a positive rotation about this
axis as shown in Figure 4.69.
When setting up the equations to solve a velocity analysis it will be desirable
to reduce the number of unknowns based on the knowledge that a particular
body is constrained to rotate about a known axis as shown here. The
velocity vector { 3 } 1 could, for example, be represented as follows:
{ 3 } 1 f 3 {R AB } 1 (4.103)
Since { 3 } 1 is parallel to the relative position vector {R AB } 1 a scale factor
f 3 can be introduced. This reduces the problem from the three unknown
components, x 3 , y 3 and z 3 of the vector { 3 } 1 , to a single unknown f 3 .
Once the angular velocities of Body 3 have been found it follows that the
translational velocity of, for example, point D can be found from
{V DA } 1 { 3 } 1 {R DA } 1 (4.104)
It also follows that since point A is considered fixed with a velocity {V A } 1
equal to zero that the absolute velocity {V D } 1 of point D can be found from
a consideration of the triangle law of vector addition giving
{V D } 1 {V DA } 1 (4.105)
A consideration of the complete problem indicates that the translational
velocities throughout the suspension system can be found if the angular
velocities of all the rigid bodies 2, 3, 4 and 5 are known. Clearly the same
approach can be taken with the lower wishbone, Body 2, as with the upper
wishbone using a single a scale factor f 2 to replace the three unknown
components, x 2 , y 2 and z 2 , of the vector { 2 } 1 . Finally a consideration
of the boundaries of this problem reveals that while points A, B, E, F and J
are fixed the longitudinal velocity, V Px and the lateral velocity V Py at the
contact point P remain as unknowns.