Kin Ematic s of Rigid B o d Ies

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Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

1 Instant Centers or Instantaneous Centers

2.003J/1.053J Dynamics and Control I, Spring 2007Professor Thomas Peacock

3/5/2007

Lecture 8

Kinematics of Rigid Bodies

Instant Centers or Instantaneous CentersPoint on a rigid body whose velocity is zero at a given instant Instantaneous : May only have zero velocity at the instant under consideration. Idea : If we know the location of an instant center in 2D motion and we know

the angular velocity of the rigid body, the velocities of all other points are easy to determine.

Figure 1: Rigid body where I is the instant center. Figure by MIT OCW.

vA = vI + r IA = 0 + r IA (1)

vA =

r IA

So body is undergoing rigid body rotation about I.

How to locate an instant center? 1. Draw lines in direction of motion of two points on a rigid body. 2. Draw perpendiculars to both these lines at points.


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Instant Centers or Instantaneous Centers 3

Figure 3: Rolling Disk. The disk rolls with angular velocity in the x-direction. Figure by MIT OCW.

Point I marks the contact with disk and ground: no slip, not moving for an instant because ground is still. Instant center.

vA = ez r IA = ez ( 2R cos45

ex 2R sin45 ey ) (2)

ex ey ez 0 0 = R 2 cos 45 R 2 sin 45 0

ex ( 2R sin 45 )ey ( 2R cos45

) = 2R sin45 ex + 2R cos45 ey = vA


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5 Plane Kinetics (Dynamics) Of Rigid Bodies

vC = 2 10 = CD 2

CD = 10rad/s (+ ez )

Of course, this is a nice geometry can easily become more complicated.

Good method for multibar linkages, and ladders against walls.

Plane Kinetics (Dynamics) Of Rigid Bodies

Recall that for a system of particles

1. Linear Momentum: d P = F dt ext d = ext 2. Angular Momentum: H B + vB

P Bdt

3. Work-Energy Principle: T 1 + V 1 ext = T 2 + V 2 ext (All external forces are potential or do no work or no relative motion in line joining centers of particles)

Now for rigid bodies:

Linear Momentum Principle

d P = F ext

dt

P = Mv c

M : Total Mass v : Velocity of center of mass c


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6 Plane Kinetics (Dynamics) Of Rigid Bodies

Angular Momentum Principle

Figure 5: Rigid body that rotates with angular velocity . Figure by MIT OCW.

H B = ( hB ) i = r i pi i i

= r i m i (vc + i ) i

= ( r C + i ) (vc + i ) (3) i

H B = m i (r C vC )+ (r C i )+ m i ( i vC ) + m i ( i i ) i i i i

i (r C i ) = 0 because m i i = 0 i m i ( i vC ) = 0 because m i i = 0

H B = r C P + m i i i i

i m i i i : Contains moment of inertia .

Kin Ematic s of Rigid B o d Ies

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