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How to calculate degree of freedom of a mechanism | Grubler's Rules

How to calculate degree of freedom of a mechanism | Grubler's Rules


What is Degrees of freedom (DOF): 

It is the number of independent coordinates required to describe the position of a body in space. A free body in space can have six degrees of freedom. i.e., linear positions along x, y and z axes and rotational/angular positions with respect to x, y and z axes.

In a kinematic pair, depending on the constraints imposed on the motion, the links may lose some of the six degrees of freedom.
Degree of Freedom of a mechanism
Degree of Freedom of a mechanism 
Possible Motions:

1.       Three translations along x, y and z axes.

2.       Three rotations about x, y and z axes.

So an object in free space has six degrees of Freedom.

A fixed object has zero degree of freedom.

Constrained Rigid Bodies

Rigid bodies and kinematic constraints are the basic components of mechanisms. A constrained rigid body system can be a kinematic chain, a mechanism, a structure, or none of these. The influence of kinematic constraints in the motion of rigid bodies has two intrinsic aspects, which are the geometrical and physical aspects. In other words, we can analyze the motion of the constrained rigid bodies from their geometrical relationships or using Newton's Second Law.

A mechanism is a constrained rigid body system in which one of the bodies is the frame. The degrees of freedom are important when considering a constrained rigid body system that is a mechanism. It is less crucial when the system is a structure or when it does not have definite motion.

Calculating the degrees of freedom of a rigid body system is straight forward. Any unconstrained rigid body has six degrees of freedom in space and three degrees of freedom in a plane. Adding kinematic constraints between rigid bodies will correspondingly decrease the degrees of freedom of the rigid body system. We will discuss more on this topic for planar mechanisms in the next section.

GRUBLER’S RULE

Degrees of freedom/mobility of a mechanism: It is the number of inputs (number of independent coordinates) required to describe the configuration or position of all the links of the mechanism, with respect to the fixed link at any given instant.

Grubler’s equation: Number of degrees of freedom of a mechanism is given by

F = 3(n-1)-2l-h. Where,

F = Degrees of freedom

n = Number of links = n2 + n3 +……+nj, where, n2 = number of binary links, n3 = number of ternary links…etc.

l = Number of lower pairs, which is obtained by counting the number of joints. If more than two links are joined together at any point, then, one additional lower pair is to be considered for every additional link.

h = Number of higher pairs

Examples of determination of degrees of freedom of planar mechanisms:
Grubler's Rules Mechanism Example
F = 3(n-1)-2l-h

Here, n2 = 4, n = 4, l = 4 and h = 0.
F = 3(4-1)-2(4) = 1
I.e., one input to any one link will result in definite motion of all the links.
Grubler's Rules Mechanism Example
 F = 3(n-1)-2l-h

Here, n2 = 5, n3 =1, n = 6, l = 7 (at the intersection of 2, 3 and 4, two lower pairs are to be considered) and h = 0.
F = 3(6-1)-2(7) = 1
Grubler's Rules Mechanism Example
F = 3(n-1)-2l-h

Here, n2 = 5, n = 5, l = 5 and h = 0.

F = 3(5-1)-2(5) = 2

I.e., two inputs to any two links are required to yield definite motions in all the links.

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