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Kinematics and dynamics of rigid bodies 47
{ A }
d
{ } R
R
d t
{ } d
2
1
d t
{ 2
} { }
PQ 1 1 PQ 1 PQ 1
Since it is known that
d
d { RPQ
t
} { VPQ
}
1 1
(2.103)
(2.104)
d
dt { }
{ }
2 1 2 1
(2.105)
We can therefore write
{A PQ } 1 { 2 } 1 {V PQ } 1 { 2 } 1 {R PQ } 1 (2.106)
Since it is also known that
{V PQ } 1 { 2 } 1 {R PQ } 1 (2.107)
This leads to the expression
{A PQ } 1 { 2 } 1 {{ 2 } 1 {R PQ } 1 } { 2 } 1 {R PQ } 1 (2.108)
The acceleration vector {A PQ } 1 can be considered to have a centripetal
component {APQ} p 1 and a transverse component {APQ} t 1 . This is illustrated
in Figure 2.20 where one of the arms from a double wishbone suspension
system is shown.
In this case the centripetal component of acceleration is given by
{APQ} p 1 { 2 } 1 {{ 2 } 1 {R PQ } 1 } (2.109)
Note that as the suspension arm is constrained to rotate about the axis NQ,
ignoring at this stage any possible deflection due to compliance in the suspension
bushes, the vectors { 2 } 1 for the angular velocity of Body 2 and
{ 2 } 1 for the angular acceleration would act along the axis of rotation through
NQ. The components of these vectors would adopt signs consistent with
producing a positive rotation about this axis as shown in Figure 2.20.
When setting up the equations to solve a velocity or acceleration analysis it
may 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
Z 1
{A PQ } 1
O 1
X
Y 1 1
p N
{APQ } 1
Q
{ω 2 } 1
{α 2 } 1 Body 2
t
{A PQ } 1
{V PQ } 1
P
Fig. 2.20
Centripetal and transverse components of acceleration vectors