Page 431 - Marks Calculation for Machine Design

P2: Sanjay
P1: Shibu/Rakesh
Brown˙C10
Brown.cls
−→
where v C = absolute velocity of point C, meaning relative to ground
−→ January 4, 2005 15:34 MACHINE MOTION 413
v B = absolute velocity of point B, meaning relative to ground
−→
v C/B = velocity of point C relative to point B, as if point B is fixed
These three velocity vectors are shown graphically in Fig. 10.5,
B w rod
Rod
v
2 C/B
v B
C v = v slider
C
v C/B
v B
FIGURE 10.5 Vector velocities on the connecting rod.
where the vector triangle at point C represents the relationship given by Eq. (10.1).
−→
Based on the definition of the velocity ( v C/B ), the velocity of point C relative to point B
as if point B is fixed, has a magnitude given by Eq. (10.2) as
v C/B = L BC ω rod (10.2)
and its direction is perpendicular to the line connecting points B and C of length (L BC ). The
direction of the angular velocity (ω rod ) will either be clockwise (CW) or counterclockwise
(CCW), determined from the vector equation defined by Eq. (10.1).
If an xy coordinate system is added, along with angles (φ) and (β) defining the directions
of (v B ) and (v C/B ), respectively, then Fig. 10.5 becomes Fig. 10.6.

y
b
B w
rod
Rod
v B f 2 v C/B
C v = v slider
C
v
v B C/B
x
FIGURE 10.6 Vector velocities on the connecting rod.

Using Fig. 10.6, the vector equation in Eq. (10.1) can be separated into two scalar
equations. One equation will represent the relationship between the velocity components in
the x-direction, and the other equation will represent the relationship between the velocity
components in the y-direction, respectively, as
x: v C = v B cos φ + v C/B sin β (10.3)
y: 0 =−v B sin φ + v C/B cos β (10.4)
where the velocity (v C ) has a horizontal component, but its vertical component is zero.
Setting the velocity (v C ) equal to the velocity of the slider (v slider ) and substituting for
(v C/B ) from Eq. (10.2) in Eqs. (10.3) and (10.4) gives
x: v slider = v B cos φ + (L BC ω rod ) sin β (10.5)
y: 0 =−v B sin φ + (L BC ω rod ) cos β (10.6)