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Kinematics and dynamics of rigid bodies 49
PQ
{Ac } 1
Z 1
{A t }
PQ 1
{A s }
PQ 1
Y 1
O 1
X |V 1 PQ
s | {l PQ } 1
Body 2
P
Body 3
{A p }
PQ 1
{ω 2 } 1 Q
{ω 3 } 1
{α 2 } 1
{α 3 } 1
Fig. 2.21
Relative acceleration with sliding motion
two more components of relative acceleration. This is best explained by
considering the situation shown in Figure 2.21 where point P is located on
Body 3 and point Q is located on Body 2. In this case Body 3 is constrained
to move and rotate with Body 2 but has an additional relative sliding degree
of freedom that allows it to move, relative to Body 2, along a slot with an
axis aligned with the two points P and Q. To simplify the understanding the
two bodies are assumed to rotate, as shown, about an axis passing through
point Q. Hence the relative acceleration vectors can be assumed to be acting
in the directions shown, as point P moves away from point Q. The angular
velocity and acceleration vectors for Body 2 and Body 3 will be the same
and either may be used in the subsequent formulations.
The first of the two new components is easy to comprehend and is associated
with the additional sliding motion. Since the direction of motion is
known to be constrained to act along the line PQ it is possible to define the
sliding acceleration, {A s PQ} 1 , using the magnitude of the sliding acceleration
|APQ| s factored with the unit vector {l PQ } 1 , acting along the line from
Q to P, as follows:
{A s PQ} 1 |A s PQ|{l PQ } 1 (2.119)
The second of the two new components, {A c PQ} 1 , is known as the Coriolis
acceleration and requires more detailed explanation. As a starting point we
can assume that both bodies are rotating as shown in Figure 2.21.
In deriving the Coriolis term consider first the formulation for the velocity
vector {V PQ } 1 . In addition to the component of velocity associated with
rigid body motion that would act in a perpendicular direction to the line PQ
there is an additional component of sliding velocity. This sliding component
can be defined using the magnitude of the sliding velocity |V PQ| s factored
with the unit vector {l PQ } 1 , acting along the line from Q to P. The
formulation of {V PQ } 1 now becomes
{V PQ } 1 { 2 } 1 {R PQ } 1 |V s PQ|{l PQ } 1 (2.120)