± Instantaneous Center of Zero Velocity

Learning Goal:

To describe the angular and linear velocity of systems under translational and rotational motion using the instantaneous center of zero velocity.

The velocity of any point B in a rigid body can be described by

vB=vA+ω×rBA

where ω is the angular velocity, rBA is the relative position vector from point A to point B, and vA is the velocity of point A. Choosing point A to have zero velocity, vA=0, simplifies the equation to

vB=ω×rBA

The point chosen such that the velocity is zero is called the instantaneous center of zero velocity (IC). The IC lies along the instantaneous axis of zero velocity, which is perpendicular to the plane of motion. The intersection of the axis and plane of motion define the location of the IC. Because we chose point A as the IC, the equation becomes

vB=ω×rBIC

Point B revolves around the IC. Because of this circular motion the direction of vB must always be perpendicular to rBIC and vB's magnitude becomes vB=ωrBIC

Part A

The mechanism shown has lengths l1 = 3.70 ft and l2 = 0.800 ft . (Figure 1) At the instant shown, block C is sliding to the left. The magnitude of the block's velocity is vC = 1.00 ft/s , and θ = 47.0 ∘. For link BC, graphically locate its instantaneous center of zero velocity (IC) by drawing two vectors: one from the IC to point B and one from the IC to point C.

It may be easiest to draw the vectors longer than they need to be initially and then shorten them after the IC is located. The location, orientation, and length of the vectors will be graded.

 

Part B

At the instant shown, determine ωBC, the magnitude of the angular velocity of link BC. Use the same values from Part A: θ = 47.0 ∘, vC = 1.00 ft/s , l1 = 3.70 ft , and l2 = 0.800 ft .

Express your answer numerically in radians per second to three significant figures.

ωBC =

0.852

rad/s

 

Part C

At the instant shown, determine ωAB, the magnitude of the angular velocity of link AB.

Express your answer numerically in radians per second to three significant figures.

ωAB =

0.198

rad/s