CHAPTER 17
CHAPTER 16
PLANAR KINEMATICS OF
A RIGID BODY
Chapter Objectives
To classify the various types of rigid-body planar motion
To investigate rigid-body translation and show how to analyze motion about a
fixed axis
To study planar motion using an absolute motion analysis
To provide relative motion analysis of velocity and acceleration using a translating
frame of reference
To show how to find the instantaneous center of zero velocity and determine the
velocity of a point on a body using this method
To provide a relative motion analysis of velocity and acceleration using a rotating
frame of reference
1. Rigid-Body Motion
➢ Particles of a rigid body move along paths equidistant from a fixed plane
➢ Has 3 types:
1. Translation
- every line segment on the body remains parallel to its original direction during the motion
- rectilinear translation: path of motion – along equidistant straight lines
- curvilinear translation: path of motion – along curved lines which are equidistant
2. Rotation about a fixed axis
- all particles of the body (except those lie on the axis of rotation) move along circular paths
3. General plane motion
- undergoes a combination of translation and rotation
2. Translation
➢ Position:
- location of points A and B – defined from fixed x, y reference frame – using position vectors rA and rB
- x’,y’ coordinate system – fixed in the body where origin = A (base point)
- position of B with respect to A = relative position vector rB/A (r of B with respect to A)
- vector addition: rB = rA + rB/A
➢ Velocity:
- relationship between instantaneous velocities of A and B – obtained by taking the time derivative of the position equation:
vB = vA + drB/A/dt
Since drB/A/dt = 0 due to the magnitude of rB/A = constant, and vB = vA = absolute velocities,
( vB = vA
➢ Acceleration:
- time derivative of velocity equation:
aB = aA
- ( velocity and acceleration equation indicates that all points in a rigid body subjected to either rectilinear or curvilinear translation move with the same velocity and acceleration.
3. Rotation About a Fixed Axis
➢ Angular motion
- only lines or bodies undergo angular motion
- angular motion of a radial line r located within the shaded plane and directed from point O on the axis of rotation to point P
1. Angular position
- angular position or r = defined by angle (
- measured between a fixed reference line and r
2. Angular displacement
- defined by the change in the angular position, measured as a differential d(
- has a magnitude of d(, measured in degrees, radians or revolutions, where 1 rev = 2( rad.
- Since motion is about a fixed axis, direction of d( which always along the axis
- Direction – determine by the right hand rule
3. Angular velocity
- defined as the time rate of change in angular position, (, where ( = d( / dt +
- has a magnitude measured in rad/s
- direction – always along the axis of rotation where the sense of rotation being referred as clockwise or counterclockwise
- arbitrarily chosen counterclockwise as positive
4. Angular acceleration
- measures the time rate of change of the angular velocity
- magnitude: ( = d( / dt or ( = d2( / dt2 +
- direction – depends on whether ( is increasing or decreasing
- e.g.: if ( is decreasing, ( = angular deceleration, (direction – opposite to (
- by eliminating dt from the above equation,
( d( = ( d( +
5. Constant angular acceleration
- when angular acceleration of the body is constant,
( = (c
( + ( = (o + (c t
+ ( = (o + (o t + ½ (c t2
+ (2 = (o2 + 2(c (( - (o)
where (o = initial angular position
(o = initial angular velocity
➢ Motion of point P
- as rigid body rotates, point P travels along a circular path of radius r and center at point O.
1. Position
- defined by the position vector r, which extends from O to P
2. Velocity
- has a magnitude of [pic]
- since r = constant, [pic]
- since [pic]
- direction of v = tangent to the circular path
- magnitude and direction of v – accounted from:
[pic]
where rp: directed from any point on the axis of rotation to point P
- to establish the direction of v – right hand rule
- by referring to the figure,
since [pic]
3. Acceleration
- can be expressed in terms of its normal and tangential components:
where [pic]
- tangential components – represents the time rate of change in the velocity’s magnitude
- normal component – time rate of change in the velocity’s direction
- acceleration in terms of vector cross product:
since [pic]
- by referring to the next figure, [pic]
- applying right hand rule yields [pic]in the direction of at
- hence obtain [pic]
- magnitude: [pic]
Procedure for Analysis:
To determine velocity and acceleration of a point located on a rigid body that is rotating about a fixed axis:
a) Angular Motion
1. Establish positive sense of direction along the axis of rotation and show it alongside each kinematics equation as it is applied.
2. If a relationship is known between any two of the 4 variables (, (, ( and t, then a third variable can be obtained by using one of the following kinematics equation which relates all 3 variables:
[pic]
3. For constant angular acceleration, use:
[pic]
4. (,(, ( - determine from algebraic signs of numerical quantities.
b) Motion of P
1. Velocity of P and components of acceleration can be determine from:
[pic]
2. If geometry of problem is different to visualize, use:
[pic]
Note:
- rp – directed from any point on the axis of rotation to point P
- r – lies in the plane of motion P
- vectors – expressed in terms of its i, j, k components.
See Example 16.1 and 16.2.
5. Relative-Motion Analysis: Velocity
➢ General motion: combination of translation and rotation
➢ To view motions separately – use relative-motion analysis, involving 2 sets of coordinate axes
➢ Fixed reference – measures the absolute position of 2 points A & B on the body
➢ Translating reference – do not rotate with the body; only allowed to translate with respect to the fixed frame; origin – attached to the selected ‘base point’ A
• Position vector rA – specifies the location of ‘base point’ A
• Relative position rB/A – locates point B with respect to point A
• (by vector addition, position of B: rB = rA + rB/A
• Points A & B – undergo displacements drA & drB during an instant of time dt
• Consider general plane motion by its component parts:
- entire body – translates by drA – A moves to its final position and B to B’
- rotated about A by d( - B’ moves to its final position (relative displacement drB/A)
- displacement of B:
[pic]
• to determine the relationship between the velocities of points A and B – take the time derivative (divide displacement equation by dt):
[pic]
[pic]
• since vB/A also representing the effect of circular motion about A:
[pic]
A) Vector Analysis
1. Kinematics Diagram
• Establish the directions of the fixed x,y coordinates and draw a kinematics diagram of the body
• Indicate vA, vB, (, rB/A
• If magnitudes of vA, vB or ( are unknown, the sense of direction can be assumed
2. Velocity Equation
• To apply [pic], express the vectors in Cartesian vector form and substitute them into the equation.
• Evaluate the cross product and then equate the i and j components to obtain two scalar equations.
• If negative answer obtained for an unknown magnitude, (direction of vector – opposite to that shown on the kinematics diagram.
B) Scalar Analysis
1. Kinematics Diagram
• Draw a kinematics diagram to show the relative motion
• Consider body to be ‘pinned’ momentarily at base point A, magnitude: vB/A = ( rB/A
• Direction of vB/A – established from the diagram
2. Velocity Equation
• From equation vB = vA + vB/A, represent each vectors graphically by showing magnitudes and directions.
• Scalar equation – determine from x & y components of these vectors.
See Example:
-----------------------
Position
Planar motion
Displacement
Velocity
due to rotation about A
due to translation of A
due to translation & rotation
absolute velocities of points A & B
relative velocity vB/A
relative velocity of ‘B with respect to A’
velocity of base point A
velocity of point B
relative-position vector drawn from A to B
angular velocity of the body
velocity of base point A
velocity of point B
Procedure for Analysis
16.6
16.7
16.8
16.9
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