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DEGREES OF FREEDOM
The motion of a body or mechanism is defined by the number of degrees of freedom it possesses. It can also be said that degrees of freedom is the minimum number of independent parameters which describe the motion of a body or of a mechanism without violating any constraint imposed on it. Degrees of freedom is very important for its true and full analysis of the dynamic systems found in various aspects of practical life.
PRACTICAL EXAMPLES FOR VARIOUS DEGREES OF FREEDOM
1. EXAMPLES SYSTEMS WITH ONE DEGREE OF FREEDOM
- A single mass spring system
- A disc and a shaft system
- A simple pendulum
- A crank has only one degree of freedom
- A four bar linkage has only one degree of freedom
2. EXAMPLES SYSTEMS WITH TWO DEGREES OF FREEDOM
- A two mass spring system
- Two rotor system where motion can be defined by ϴ1 and ϴ2
- A single mass connected to two different springs in perpendicular directions
3. EXAMPLES SYSTEMS WITH THREE DEGREES OF FREEDOM
- Skidding of an automobile
- A Hamiltonian system of three degrees of freedom with eight channels of escape
4. EXAMPLES 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.
5. EXAMPLE FOR THE INFINITE DEGREES OF FREEDOM
Continuous elastic members have an infinite number of degrees of freedom. For example a cantilever beam will have an infinite number of mass points. It will require infinite number of coordinates to specify the deflected beam configuration. Therefore a cantilever beam has infinite number of degrees of freedom.
DEFINITION: IT IS THE MINIMUM NUMBER OF INDEPENDENT VARIABLES
TO DEFINE A SYSTEM COMPLETELY.
IT IS NUMBER OF VARIABLES WHICH ARE FREE TO VARY IN A SYSTEM.
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. It is widely used
in robotics and kinematics.
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 links (bodies) connected with each other with one or more constraints.
When linked are joined, these make a sliding pair or a rotating pair. Whenever two links form a sliding or a rotating pair, there is a loss of two degrees of freedom. In addition if one of the links is ‘fixed’(OR its position is fully specified,) then further three degrees of freedom are lost.
TO FIND THE OVERALL NUMBER OF DEGREES OF FREEDOM
- Each added link contributes three degrees of freedom
- Each pair connection reduces the total by two degrees of freedom.
- A fixed link will FURTHER reduce the total by three degrees of freedom.
STATICALLY INDETERMINATE STRUCTURES
If you get the degrees as negative, NEGATIVE Degrees of freedom means some member (or link) is redundant in terms of assembly of a structure. Structures with redundant members (negative number of degrees of freedom) are called statically indeterminate structures. They are very difficult to analyze than the simple determinate structures. However a redundant member can carry a load in the structure.
DEGREES OF FREEDOM FOR A GENERAL KINEMATIC CHAIN
Equation for the degree of freedom
F = 3n—3j-3
Where F is the overall number of degrees of freedom for the kinematic chain,
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. The number of degrees of freedom in an assembly of links can give valuable information as to the forms of motion (or lack of motion) in the assembly
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 chain i.e. a mechanism|
|4.||> 1||Unconstrained kinematic chain or a mechanism with