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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