Instantaneous Center for Bodies in Plane Motion

Key Concept: A rigid body moving in a plane has a point, call it I, with zero velocity for an

instant of time, t, or for all time. It is called the instantaneous center of zero velocity. i.e. The instantaneous center for all time for a crank is the pivot point about which the crank rotates. At

any instant of time, if the tires of a car roll on the pavement without slipping, the point of contact

of the tire with the pavement is the instantaneous center of zero velocity.

In a Nut Shell: The instantaneous center provides a convenient geometrical way to find the angular

speed of a body. Note: In plane motion, each point in a body moves with a velocity perpendicular to

the line joining the instantaneous center, I to that point.

The strategy is to construct perpendiculars to known velocities to find the perpendicular distance,

IA and IB, between the instantaneous center and the known velocities using proportional geometry v_A/IA = v_B/IB.

Case 1: Rolling cylinder – point of contact, I, is the i.c.

Case 2: Crank – pin support, I, is the i.c. for all time

Case 3: Rotating link – i.c. lies between A and B

Case 4: Rotating link – i.c. lies below A and B

See the figures below for each case.

Recall the relative velocity equation in kinematics is

v_A = v_B + ω x r_BA and v_A = v_I + ω x r_IA

So v_A = ω x r_IA and v_B = ω x r_IB Use v_A = (r_IA) ω or v_B = (r_IB) ω to find ω.

i.e. ω = v_A /(r_IA) or ω = v_AB /(r_IB) Click here for examples.

Note: The concept of instantaneous center of zero velocity does not carry over to accelerations.

i.e. There is no instantaneous center of zero acceleration except for a fixed point.

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Copyright © 2019 Richard C. Coddington
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