Chapter 17 Planar Kinematics of Rigid Bodies

 

 

 

 

 

 

 


Rigid body motions:

Translation: Orientation of the body never changes. Every line segment on the body remains parallel to its original direction.

1)      The movements of the points in a rigid body are parallel.

2)      Every point has same velocity and acceleration

3)      Motion of one point completely describes the motion of the entire rigid body. We generally choose the motion of the center of mass to denote the motion of the rigid body.

Example:

 

 

 

 

Pure Rotation: The rigid body rotates about a fixed axis

1)      All points are rotating about same fixed axis.

2)      No two points have same linear velocity or linear acceleration although all of them have same angular velocity and angular acceleration.

Example:

 

 

 

 

General Motion:         Translation        +         Rotation

 

 

Example:

 

 

 

 

Exercise: Determine the type of motion of each moving component in following figure.

 

 

 

 

 

 

 

Pure Planar Rotation in Cartesian Coordinates:

 


Translation:

 

 

 

Rotation:

 

 

 

 

Given angular velocity ω and angular acceleration α of a rigid body, the linear velocity  and linear acceleration  of a point  in the rigid body are:

 

 

 

 

 

 

Example: A rigid bar is rotating about a fixed axis O with angular displacement . The distance from axis O to the end point P is R=1m. Pleas find the linear velocity and acceleration of point P at t=1s. 

 

 

 

Exercise: A gear with radius r is rotating about its axis with angular velocity ω and angular acceleration α. Please find the linear velocity and acceleration of the points shown.

 

 

 



General Motion: Rotation + Translation

Relative motion:

 

 

 

 

Exercise: In stationary water, the cruising speed of a ferry is 10 km/hr. The distance between two ports A and B is 24km. The speed of the water is 2km/hr going from A to B. How long it will take to travel from A to B and back from B to A.

 

 

 

 

 

 

 

 

 


Exercise: A box is sliding along a smooth surface at 5m/s. What is the velocity of A relative to B?

 

 

 

 


Exercise: A bar is rotating around a fixed axis with a constant angular velocity rad/s. R=1m. What is the relative velocity of point A compare to point B.

 

 

 

 

 

 

 

 

 

 

 

 

General motion is always decoupled to the sum of translation and rotation about a moving axis. The axis is always chosen at the center of mass, geometric center, or a pivot point.

Example:

 

 

Summery to analyze general motion:

1)      Find the point of rotating center, (which may not be unique or constant).

2)      Analyze the translation of the point

3)      Analyze the rotation of the rigid body about the point.

 

 

 

 

 

 

 

 

 

 

Example: A rigid bar is rotating about axis O with angular displacement . The axis O is mounted on a rigid body sliding along a surface with displacement m. The distance from axis O to the end point P is R=1m. Pleas find the linear velocity and linear acceleration of point P at t=1s. 

 

 

 

Exercise: A wheel with radius r is rotating without slipping on the ground. The angular velocity of the wheel is ω and angular acceleration is α. Please find the linear velocity and acceleration of the points shown.

 

 

 



Example: Given a linkage system as shown, wAB = 10 rad/s and aAB=1 rad/s2, both counter-clockwise. Find the angular velocity wCD and the angular acceleration aCD.

 

 


 
Common Procedure to solve rigid body kinematics:

  1. Establish coordinate system, set the directions of x, y and z.
  2. Draw a kinematic diagram of the body. Select the rotating axis, such as a pivot or center of mass. For example, we select a point A.
  3. Indicate the translating velocity of axis point , angular velocity and angular acceleration of the rigid bodies. If any of them is unknown, assume the positive direction.
  4. Indicate the unknown point, e.g. B, its velocity , and
  5. Apply the equations of general motion: 

  1. Apply any other geometric constraints
  2. If necessary, repeat steps 1) – 6)
  3. Solve the equations to get the unknown variables. If any solution yields a negative answer, it simply indicates the assumption of direction is opposite to the real one.

 

 

Common geometric restrictions that are helpful to solve rigid body motion problem:

  1. Rigid body

 

 

  1. Connected point

 

 

 

  1. Sliding along a horizontal surface

 

 

 

  1. Sliding along a inclined surface

 

 

 

 

 

Exercise: The common configuration of a reciprocating engine is the slider-crank mechanism. , . If the crank OB has a constant clockwise rotation speed of ω = 1500 rpm, determine for the position where θ = 60˚,

1)      The velocity of the piston A and the angular velocity of the connecting rod AB.

2)      The acceleration of the piston A and the angular acceleration of rod AB.

Example: The 3m long ladder AB slides along a corner. At the instant, θ = 30˚ and the lower end A is moving to the right at a constant speed of 1m/s. Determine the velocity, angular velocity, acceleration and angular acceleration of upper end B.