DEGREES OF FREEDOM-2

 

DEGREES OF FREEDOM-2

The motion of a body or mechanism is

defined by the number of degrees of

freedom it possesses. 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. 

  (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 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

(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. Further there is a loss of three degrees of freedom with fixing of one link.
TO FIND THE OVERALL NUMBER OF DEGREES OF FREEDOM
  1. Each added link contributes three degrees of freedom
  2. Each pair connection reduces the total by two degrees of freedom.
  3. A fixed link will FURTHER reduce the total by 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. 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
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
Thus, there is difference in analysis of a particle and a rigid body.

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. Thus, a body has 0 to 6 degrees of freedom. 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 more than one body.  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.

 

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