Mohammad I. Kilani Dynamics and Vibrations Kinematics of Rigid Bodies

Mohammad I. Kilani

Dynamics and Vibrations

Kinematics of Rigid Bodies

What is a Rigid Body?

A body is considered to be a rigid body when the

distance between any two points on it remain

unchanged as the body moves.

A result of the above definition is that the angle

between any two lines on the body does not change

as the body moves. This is because the angle between

any two lines can be seen as an angle in a triangle

containing the two lines. Since the lengths of the

sides triangle do not change, the angles do not change

either.


Note that a rigid body contains an infinite number of

particles. However, the above observations on rigid body

motion allows determining the position of any point on the

body if the position of one reference point and the angular

orientation of one reference line on it are known.

For example, the figure shows the location of all points on

the body if reference point A is located at the origin and

reference line AB is aligned with the x- axis (orientation = 0

degrees)

ABO

A

B

O


In the previous example, three scalar quantities were

needed to fully define the location of a body in the

plane. Those were the x- and y- coordinates of point

A and the angular orientation of line AB.

The degrees of freedom (DOF) for a system is the

number of independent coordinates that need to be

specified to completely describe the configuration of

the system in space. A rigid body in plane motion has

DOF = 3.

ABO

A

B

O


The independent coordinates for determining the location of a

rigid body is not unique. For example, one could choose the x-

and y- coordinates of point B and the x- coordinates of point A.

Another choice could be the x- coordinates of point A, the y-

coordinates of point B, and the orientation of line AB.

There is an infinite number of possible choices for the

coordinates needed to determine the location of the body.

However, regardless of the choice, there number is always the

same, and once they are given, the position of any point on the

body can be determined.

ABO

A

B

O

Analytical Determination of the location of points

Knowing the location of the reference point A, and the

orientation of the reference line AB, the location of a point C on

a body can be determined from the relative position equation

below:

In using the above equation, note that the angle γ and the

magnitude of the vector rc/a

does not change as the body

moves.

ACAC rrr

ABO

A

B

O

rA

rC/A

γ

Example:Determination of the location of a point on a rectangle

The rectangle shown is 3 m long and 1 m wide. Determine the

location of point C if point A is located at point (1,1) and the line

AB is oriented at 90 degrees from the x- axis

)4,0(

)3,1()1,1(

)43.108sin10,43.108cos10()1,1(

43.1081043.189010

3

1tan9013

90

122

C

C

C

AC

AC

ACAC

ACAC

r

r

r

r

r

rr

rrr

O

AB

C

A

BC

rA

rC/A γ

O


The lengths of the sides AB and AC right angle triangle are 2 m and 1.5 m, respectively.

Determine the coordinates of the apex A if point B is located on the y- axis and point C is

on the x-axis, and the line BC makes 45 degrees with the horizontal

77.12

5.2

77.12

5.2

)2

5.2,2

5.2(),0()0,(

)2

5.2,2

5.2(

455.2

455.12 22

B

C

BC

BCBC

BC

BC

BC

BCBC

y

x

yx

rrr

r

r

r

rrr

O

C

O

A B

C

A

B

C

98.1,28.0

)98.1,28.0()77.1,0(),(

)98.1,28.0(

87.812

2

5.1tan452 1

yx

yx

rrr

r

r

r

rrr

AA

BABA

BA

BA

BA

BABA


The lengths of the sides AB and AC right angle triangle are 2 m and 1.5 m, respectively.

Determine the coordinates of the apex A if point B is located on the x- axis and point C is on

the y-axis, and the line AB makes 45 degrees with the horizontal

O

AC

O

A B

C

A

B

C

77.12

5.2

77.12

5.2

)2

5.2,2

5.2()0,(),0(

)2

5.2,2

5.2(

225.2

2255.12 22

B

C

BC

BCBC

BC

BC

BC

BCBC

x

y

xy

rrr

r

r

r

rrr

98.1,49.1

)98.1,28.0()0,77.1(),(

)98.1,28.0(

87.2612

2

5.1tan2252 1

AA

AA

BABA

BA

BA

BA

BABA

yx

yx

rrr

r

r

r

rrr

TYPES OF MOTION

Three Dimensional Motion

A rigid body free to move within a

reference frame will, in the general

case, have a simultaneous combination

of rotation and translation.

In three-dimensional space, there may

be rotation about any axis and

translation that can be resolved into

components along three axes.

Plane Motion

In a plane, or two-dimensional space, rigid body

motion becomes a combination of simultaneous

rotation about one axis (perpendicular to the

plane) and also translation resolved into

components along two axes in the plane.

Planar motion of a body occurs when all the

particles of a rigid body move along paths which

are equidistant from a fixed plane

Translation

All points on the body describe

parallel (curvilinear or rectilinear)

paths.

A reference line drawn on a body in

translation changes its linear

position but does not change its

angular orientation.Rectilinear Translation

Curvilinear Translation

Fixed Axis Rotation

The body rotates about one axis that has no motion with

respect to the “stationary” frame of reference. All other

points on the body describe arcs about that axis. A

reference line drawn on the body through the axis

changes only its angular orientation.

When a rigid body rotates about a fixed axis, all the

particles of the body, except those which lie on the axis of

rotation, move along circular paths

General Plane Motion

When a body is subjected to

general plane motion, it

undergoes a combination of

translation and rotation, The

translation occurs within a

reference plane, and the rotation

occurs about an axis

perpendicular to the reference

plane.

DEGREES OF FREEDOM (DOF) OR MOBILITY

Definition of the DOF

The number of degrees of freedom (DOF)

that a system possesses is equal to the

number of independent parameters

(measurements) that are needed to

uniquely define its position in space at any

instant of time.

Note that DOF is defined with respect to a

selected frame of reference.

xA

yA

xB

YB

θB

DOF of a Rigid Body in a 2D Plane

If we constrain the pencil to always remain in the plane of the

paper, three parameters are required to completely define its

position on the paper, two linear coordinates (x, y) to define

the position of any one point on the pencil and one angular

coordinate (θ) to define the angle of the pencil with respect to

the axes.

The minimum number of measurements needed to define its

position is shown in the figure as x, y, and θ. This system o has

three DOF.

DOF of a Rigid Body in a 2D Plane

Note that the particular parameters chosen to define the

position of the pencil are not unique. A number of alternate

set of three parameters could be used.

There is an infinity of sets of parameters possible, but in this

case there must be three parameters per set, such as two

lengths and an angle, to define the system’s position because

a rigid body in plane motion always has three DOF.

DOF of a Rigid Body in 3D Space

If the pencil is allowed to move in a three-dimensional space,

six parameters will be needed to define its position. A

possible set of parameters that could be used is three

coordinates of a selected point, (x, y, z), plus three angles (θ,

φ, ρ).

Any rigid body in a three-dimensional space has six degrees of

freedom. Note that a rigid body is defined as a body that is

incapable of deformation. The distance between any two

points on a rigid body does not change as the body moves.

ϕ

θρ

DOF of Mechanisms

Rotation about a Fixed Axis


Since a point is without dimension, it cannot have angular

motion. Only lines or bodies undergo angular motion. For

example, consider the body shown and the angular motion of

a radial line r located within the shaded plane.

At the instant shown, the angular position of r is defined by

the angle θ, measured from a fixed reference line to r



PPP

PPP

PrPP

PPrPPP

PPPPrPP

OPP

OPOP

rra

rra

erera

errerra

rererererv

vv

vvv

2

2

2

/

/

2

Relative Motion Analysis: Velocity

ABAB

ABAB

/

/

vvv

rrr

Relative Motion Analysis: Acceleration

ABABAB

ABABAB

rABABAB

ABAB

ABAB

ABAB

/2

/

//

//

/

/

/

rrαaa

rωωrαaa

aaaa

aaa

vvv

rrr

Problem 16-2

Problem 16-2

Problem 16-6

Problem 16-6

Problem 16-9

Problem 16-9

Problem 16-18

Problem 16-18

Problem 16-29

Problem 16-29

Problem 16-29

Problem 16-29

Problem 16-42

Problem 16-42

Problem 16-45

Problem 16-45

Documents

Mohammad I. Kilani Dynamics and Vibrations Kinematics of Rigid Bodies