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Modelling and analysis of suspension systems 215
Z 1
{A D } 1
{A t D} 1
O
X 1 Y 1
1
B
{A p {V D } 1
D } 1
A
{ 3 } 1
D
{ 3 } 1 Body 3
Fig. 4.72 Angular and translational acceleration and velocity vectors for the
upper wishbone
be used to find the translational accelerations at points within the system.
An example of this is shown for the upper wishbone in Figure 4.72.
Referring back to the earlier velocity analysis 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 acceleration of Body 3 will act along the axis
of rotation through AB. The components of this vector would adopt signs
consistent with a positive rotation about this axis as shown in Figure 4.72.
When setting up the equations to solve an acceleration analysis it will be as
before 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 acceleration vector { 3 } 1 could, for example, be represented
as follows:
{ 3 } 1 f 3 {R AB } 1 (4.168)
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 .
It also follows that since point A is considered fixed with an acceleration
{A A } 1 equal to zero that the absolute acceleration {A D } 1 of point D
can be found from a consideration of the triangle law of vector addition
giving
{A D } 1 {A DA } 1 (4.169)
Once the angular accelerations of Body 3 have been found, together with
the known angular velocities, it follows now that the translational acceleration
{A D } 1 of, for example, point D can be found from
{A D } 1 {A p D } 1 {A t D } 1 (4.170)
where the centripetal acceleration {A p D } 1 is given by
{A p D } 1 { 3 } 1 {{ 3 } 1 {R DA } 1 } { 3 } 1 {V D } 1 (4.171)
and the transverse acceleration {A t D } 1 is given by
{A t D } 1 { 3 } 1 {R DA } 1 (4.172)