DEGREES OF FREEDOM CLASS NOTES FOR MECHANICAL ENGINEERING

DEGREES OF FREEDOM CLASS NOTES FOR

MECHANICAL ENGINEERING

Body motion is controlled by number of

degrees of freedom it possesses. Degrees

of freedom is the minimum number of

independent parameters to describe a

motion without violating any constraint

imposed on it. Degrees of freedom does

analysis of the dynamic systems in practical life.

Fig. Six Degrees of Freedom

NUMBER OF DEGREES OF FREEDOM

(i) 6 = For an unconstrained SINGLE BODY moving in space
(ii) N = any number for inter connected rigid bodies moving in space

FOR A SINGLE RIGID BODY MOVING IN SPACE

It has 6 degrees of freedom as described below:
(j) Three degrees of freedom for transnational motion along x, y and z axis (left and right of x axis + up and down along y axis + forward and backward along z axis)
(iii) Three degrees of freedom for rotational motion about x, y and z axis (Clockwise and anticlockwise about x axis + Clockwise and anticlockwise about y axis + Clockwise and anticlockwise about z axis).Thus maximum degrees of freedom = 6 (for an unconstrained SINGLE RIGID body moving in SPACE).

SINGLE RIGID BODY MOVING IN ONE PLANE

Thus maximum degrees of freedom = 3 (for an unconstrained body in a single PLANE=Two degrees for the motion of translation and one for rotational motion).
Degrees of freedom = 6 – number of constraints (for a constrained single body in SPACE)
Degrees of freedom = 3 – number of constraints (for a constrained single body in a single PLANE)
Thus any type of constraint reduces the degrees of freedom.

DYNAMIC SYSTEM OF INTERCONNECTED BODIES

These can have any number of degrees of freedom. A dynamic system with several interconnected bodies would have combined degrees of freedom (DOF). It will be the sum of the DOF of the individual bodies, less the internal constraints these have among their relative motion. A dynamic system containing a number of connected rigid bodies will have more degrees of freedom than that for a single rigid body. Thus the degrees of freedom are the number of independent parameters which describe the dynamic system of inter connected rigid bodies in space and it can be ANY NUMBER (even much greater than six for a single rigid moving body in space).

TYPES OF CONSTRAINTS

(i) For a planar motion

A simple support reduces one degree of freedom.
A hinge support reduces two degrees of freedom
A fixed joint reduces three degrees of freedom

(ii) For a space motion

A hinge support reduces FIVE degrees of freedom
A slider reduces FIVE degrees of freedom

EMPIRICAL FORMULA TO FIND THE DEGREES OF FREEDOM

Kutzbach empirical formula is used to find the degrees of freedom for a certain body or a mechanism having

PLANE MOTION ONLY

F = 3 (N – 1) – 2P1 – P2
Where
F = number of degrees of freedom
(N – 1) = Number of moving links
N = total number of links out of which one link must be a fixed link
P1 = Number of simple joints or lower pairs which restrict one degree of freedom
P2 = Number of higher pairs which restrict two degrees of freedom
If there is only simple joints or lower pairs in a mechanism, then P2 = 0
And then F = 3 (N – 1) – 2P1

DETERMINATION OF P1

It is found by the following relation in case of a mechanism with different types of links
P1 = 0.5 ( 2N2 + 3N3+ 4N4 + 5N5 + ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙)
Where
N2 is number of binary links
N3 is number of tertiary links
N4 is number of quadrilateral links
And so on

TYPES OF JOINTS IN SINGLE PLANE MECHANISM

There are many types of joints in a single plane mechanism. Revolute joint ( have only one degree of freedom). Prismatic joint ( have only one degree of freedom).Rolling contact joint ( have only one degree of freedom).Cam /gear joint ( have two degrees of freedom). In analyzing a planar mechanism after considering the types of joints present, three types of situations can arise. These will be
(i) F≥1 The mechanism will have ‘F’ degrees of freedom
(ii) F = 0 The mechanism is static and can be analyzed with the laws of statics.
(iii) F <1 The mechanism is statically cannot be analyzed and is thus statically indeterminate.

DEGREES OF FREEDOM FOR VARIOUS PLANAR

MECHANISM

These are given in the Table below:
TABLE: Degrees of Freedom For Planar Mechanisms

Sr. No.	Type of Mechanism	Total number of Links	Number of Lower Pairs	Number of Higher Pairs	Degrees of Freedom	Remarks
1.	Three bar	3	4	0	-2	Non working mechanism and is indeterminate
2.	Three bar	3	3	0	0	Static Mechanism
3.	Three bar	3	2	1	1	Moving mechanism with constraints
4.	Four bar	4	4	0	1	Moving mechanism with constraints
5.	Four bar	4	5	1	-2	Non working mechanism and is indeterminate
6.	Five bar	5	5	0	2	Moving mechanism less constraints
7.	Six bar	6	8	0	-1	Non working mechanism and is indeterminate

(a)  Systems with one degree of freedom

1. A single mass spring system

2. A disc and a shaft system

3. A simple pendulum

4. A crank has only one degree of freedom

5. A four bar linkage has only one degree of freedom

(b) Systems with two degrees of freedom

1. A two mass spring system

2. Two rotor system where define motion by ϴ₁ and ϴ2

3. A single mass connected to two different springs in perpendicular directions

(c) Systems with three degrees of freedom

1. Skidding of an automobile

2. A Hamiltonian system of three degrees of freedom with eight channels of escape

(d) Systems with six degrees of freedom

(i) A ship in a sea has six degrees of freedom.

(ii) A flying aircraft has six degrees of freedom.

(iii) Any body or set of bodies in space will have six degrees of freedom.

(e) Systems with infinite degrees of freedom

 Continuous elastic members, like a cantilever beam, have an infinite number of degrees of freedom.   It will require infinite number of coordinates to specify the deflected beam configuration. .

DEFINITION

(i) IT IS THE MINIMUM NUMBER OF INDEPENDENT VARIABLES

TO DEFINE A SYSTEM COMPLETELY.

(ii) IT IS NUMBER OF VARIABLES WHICH ARE FREE TO VARY IN A SYSTEM.

(iii) IT IS NUMBER OF INDEPENDENT MOTIONS A BODY CAN HAVE TO DEFINE A SYSTEM COMPLETELY.

BASIC DEGREES OF FREEDOM FOR A SYSTEM UNDER

NO CONSTRAINT

Six degrees of freedom =  three degrees translation +  three degrees of rotation

SYMBOL FOR THE DEGREES OF FREEDOM

‘ df ‘

PRACTICAL APPLICATION OF THE DEGREES OF

FREEDOM

Degrees of freedom are an important concept in mechanics. Robotics and kinematics use these widely.

 DIFFERENCE BETWEEN A STRUCTURE AND A

MECHANISM

STRUCTURE : WHICH HAS NO MOTION

MECHANISM: WHICH HAS MOTION

RELATIVE POSITIONS OF THE LINKS IN A MECHANISM

DEPENDS ON

• Link dimensions

• Position of any one particular link

A mechanism has connected links connected with one or more constraints.

                             Joining of links make a sliding pair or a rotating pair. There is a loss of two degrees of freedom in a sliding/rotating pair. One fixed link causes loss of three degrees of freedom.

 POINTS CONSIDERED FOR FINDING DEGREES OF FREEDOM

1. Each added link contributes three degrees of freedom

2. Each pair connection reduces two degrees of freedom.

3. A fixed link reduces three degrees of freedom.

STATICALLY INDETERMINATE STRUCTURES

 Negative degrees of freedom means some link  is redundant in a structure. Statically indeterminate structures have redundant members. Their analysis differs from that of simple determinate structures.

DEGREES OF FREEDOM FOR  A GENERAL KINEMATIC

CHAIN

Equation for the degree of freedom

F = 3n—3j-3

Where F is number of degrees of freedom

n is the number of links (including the fixed link),

j is the number of sliding or rotating pairs with one fixed link.

NOTE: The above equation for finding the number of degrees of freedom does not consider the geometric

dimensions of the links involved in a kinematic chain. Degrees of freedom in an assembly give valuable information about forms of motion.

TABLE : Degrees of freedom in linkage assemblies

Sr. No.	Number of degrees of freedom , F	Type of assembly
1.	< 0	Statically indeterminate structure
2.	=0	Statically determinate structure
3.	=1	constrained kinematic mechanism
4.	> 1	Unconstrained kinematic chain or a mechanism with Several inputs

NUMBER OF DEGREES OF FREEDOM

One particle in space has Degrees of freedom = 3

A body has Degrees of freedom in space =6

 Analysis of a particle and a rigid body are different.

Single plane degrees of freedom

Rigid body in one plane has three degrees of freedom=One rotational + two transnational

Single rigid body in space has 6 degrees of freedom= 3 rotational + 3 transnational

There are  two degrees of freedom about each axis. It depends on the number and types of constraints.  Motions of translation are SURGING, HEAVING AND SWAYING respectively.  Rotational motions are YAW, PITCH AND ROLL respectively.

Translation degrees of freedom

These are freedom of movement of a rigid body in a three dimensional space.  The body is free to change positions in three translation, namely,

(i) Perpendicular axis as  moving forward/backward (surge),

(ii) Moving up/down (heave)

(iii)  Left/right motion (sway)

Rotational degrees of freedom

The body has three degrees of freedom of rotation about the three perpendicular axis. These are yaw (normal axis), pitch (lateral axis), and roll (longitudinal axis). Total number of degrees of freedom of an independent body is six.

COMPOSITE MECHANISM

 A composite mechanism contains two or more two bodies.  These are now no longer independent bodies. Join these to each other. Now it is a constrained body. A constrained body has lesser degrees of freedom. It becomes a kinematic pair. Kinematic constraints control the degrees of freedom.

KINEMATIC CONSTRAINTS FOR DEGREES OF FREEDOM

The motion of the independent rigid bodies can be controlled with kinematic constraints. A Kinematic constraint  between two or more rigid bodies decreases the degrees of freedom for the connected rigid bodies. Main classification of kinematic pairs falls in three categories. However these three categories are further sub divided as described below:

1. Depends on the type  of contact between the elements making the kinematic pair 

(a)Lower pair

 Kinematic pair having only one degree of freedom is called a lower pair.  A slider joint which allows only translation in one direction such as a cylinder and a piston. A rigid body in a plane has three degrees of freedom. A lower pair in a plane motion reduces two degrees of freedom. Net degree of freedom for such a system will be only one. These are of three types.

(i) Plane pair: A plane pair will keep the two surfaces together. It will have two degrees of freedom one translational and one rotational. Thus three degrees of freedom will be reduced. It is also called E- pair.

(ii) Revolute pair

 A revolute pair keeps together the axes . It will have only one degree of rotation. Degrees of freedom reduces by five.

(iii) Prismatic pair

 keeps the two axes aligned. It does not permit rotational motion at all. There will be only one translational motion. The degrees of freedom reduces by five

(b) Higher pair

 Kinematic pair has more than one degree of freedom. There is a single point or single line contact in a higher pair. Higher pair are Ball bearing, Disc cam and follower, ball and socket joint.

2.Depends on type of mechanical contact between the elements forming the kinematic pair

(a) Self closed pair

 There is a direct mechanical contact in the pair. There is no external force in this.

(b) Forced  closed pair: It is a closed kinematic pair if it has mechanical contact due to an external force. For example : ball and roller bearings

3. Depending upon the type of relative motion between the elements of kinematic pair

(a) Sliding pair: When there is a sliding contact for each element in the pair.

For example : piston inside a cylinder, square bar in a square hole and a spur gear drive.

(b) Rolling pair: When one element has rolling motion with respect to the other element in the pair. For example: wheel rolling on a road

(c) Turning pair: When one link has  turning motion relative to the other link in the pair.

For  example:   shaft in a bearing

(d) Screw pair

 A screw pair keeps the two axes aligned. It permits only relative screw motion between the elements forming the pair. There will be only one motion which is partially translational and partially rotational.  There will be only one degree of freedom. Five degrees of freedom are lost. Example: Bolt and a nut

(e) Cylindrical pair

 Keeps aligned the axes of the two rigid bodies. There will only one rotational and one translational degree of freedom. There will be only two degrees of freedom. Thus four degrees of freedom are lost. It is a R-pair. Example: A solid cylindrical bar inside a hollow shaft.

(f) Spherical pair

 A spherical pair keeps the two centers of the spheres together. It reduces three degrees of translation. Thus there will be only three rotational degrees of freedom. It is a s-pair.

Degrees of Freedom Formula

(i) Single variable Samples

 Degrees of freedom formula for single variable sample with sample size N is expressed as sample size minus one.

(ii)  Two-variable samples

With the Chi-square test with R number of rows and C number of columns is expressed as the product of a number of rows minus one and number of columns minus one.