Page 435 - Marks Calculation for Machine Design

P2: Sanjay
P1: Shibu/Rakesh
15:34
January 4, 2005
Brown.cls
Brown˙C10
417
MACHINE MOTION
As was the case for the velocity analysis, the acceleration of point B on the crank is
the same as that for point B on the left end of the connecting rod, and the acceleration
of point C on the right end of the connecting rod is the same as the acceleration of the
slider. However, the acceleration of point B on the crank has two components, one in the
same direction as the velocity (v B ) and the other is directed toward point A as shown in
Fig. 10.9.
f
B
L AB w 2 crank
B
2
w
L AB crank L AB crank
a
a
L AB crank
a crank a
A w crank B
FIGURE 10.9 Components of the acceleration at point B.
The acceleration in the direction of the velocity (v B ) is called the tangential acceleration
t
(a ) and its magnitude is given by Eq. (10.13) as
B
t
a = L AB α crank (10.13)
B
n
and the acceleration toward point A is called the normal acceleration (a ) and its magnitude
B
is given by Eq. (10.14) as
n 2
a = L AB ω (10.14)
B crank
The magnitude of the total acceleration (a B ) is therefore given by the Pythagorean
theorem as

t 2 n 2 2 2 2

a B = a + a = (L AB α crank ) + L AB ω (10.15)
B B crank
Note that even if the angular acceleration (α crank ) is zero, there is still an acceleration
n
(a B ) equal to the normal acceleration (a ) and given by Eq. (10.14).
B
Also, note that the acceleration of the slider (a slider ) shown in Fig. 10.8 is opposite to
the direction of its velocity (v slider ), meaning the slider is slowing down. This is consistent
with the orientation of the slider-crank linkage defined by the angle (φ).
Similar to Eq. (10.1), there is a relationship between the accelerations at each end of the
connecting rod in Fig. 10.8, given by the vector equation.
−→ −→ −→
a C = a B + a C/B (10.16)
−→
where a C = absolute acceleration of point C, meaning relative to ground
−→
a B = absolute acceleration of point B, meaning relative to ground
−→
a C/B = acceleration of point C relative to point B, as if point B is fixed