Planar Kinematics - Dymanics

There are three types of planar rigid body motion:

1. Translation- a line in the body stays parallel to the original orientation during the entire motion. All points on the body move with the same velocity and acceleration. Postion is defined as rB = rA + rB/A where rB and rA is the positions of points A and B on the body and rB/A is the
   
2. Rotation- the body rotates about a fixed axis such that points on the body will appear to move in a circle. Angular velocity is defined as the time rate of change in the position. Angular acceleration is the time rate of change in the angular velocity. For constant angular acceleration: ω = ω0 + αt where ω is the angular velocity, α is the constant acceleration and t is the time. The velocity is defined as v = ω r where r is the radius from the point of rotation to the point where the velocity is to be determined. The acceleration is the sum of the tangential acceleration (at = α r) and the normal acceleration (an = ω^2 r)
   
3. General Plane Motion- a combination of translation and rotation. The velocity for two points (A & B) can be defined as: VA = VB + VA/B or as VA = VB + ω x rA The relative velocity (VA/B is the effect of circular motion about the point B) is easier written as the cross product of ω x rA. The acceleration can be defined as:
   

aA = aB + (α x rA/B) + ω x (ω x rA/B)

For 2-D: α x rA/B is the tangential acceleration and ω x (ω x rA/B) = -ω^2 rA/B is the normal acceleration.


For two surfaces that are in contact with each other and don't slip (example: gears) the velocity will be the same for both surfaces at the point of contact. The angular velocity of both surfaces can be related by: rA ωA = rB ωB where r is the raduis and ω is the angular velocity. Similarly the tangential acceleration is: rA αA = rB αB where α is the angular acceleration.



Instantaneous Centers

This is for velocity analysis only. A point is picked that has zero velocity, a point that is the fixed point of rotation. The velocity equation now becomes: VB = ω x rA VA is no longer in the equation because it has zero velocity. Now the velocity of any other point on the rigid body can be found with respect to this point with zero velocity.

How to locate the instant center (IC):

• For a rolling disk it will be the point of contact between the surfaces
• For a body with two known velocities (and their directions) then draw lines perpendicular to the velocities. The point where these two lines intersect is the IC.

This can only be used for an instant in time because the IC will change as the body moves.

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