A mechanism is shown below. The number of tertiary links and the DOF, respectively are:
Concept:
Kutzback equation for DOF is given by
DOF = 3(n  1)  2j  h
where n = Number of links, j = Number of joints, h = Number of higher pairs.
Calculation:
Given:
From fig.
n = 10, j = 12, h = 0
DOF can be calculated as
DOF = 3(n  1)  2j  h
DOF= 3(10  1)  (2 × 12)  0
∴ DOF = 3
The number of tertiary links are 3 as shown below.
The degrees of freedom of a plane mechanism as shown in the figure is:
Concept:
Grubler's Criterion:
For plane mechanisms, the following relation is used to find the degree of freedom:
F = 3(N  1)  2P_{1}  1P_{2}
where F = degree of freedom., N = total no. of links in mechanisms, P_{1 }= no. of pairs having one degree of freedom (lower pair) and P_{2 }= no. of pairs having two degrees of freedom (higher pair).
Calculation:
Given:
N = 8, P_{1} = 10 (for 7 links 7 turning pair and slider has 1 sliding and 2 turning pairs) and P_{2} = 0.
F = 3(N  1)  2P_{1}  1P_{2}
∴ F = 3(8  1)  (2 × 10)  0
∴ F = 1
The degree of freedom of the mechanism shown in the figure is
Concept:
From Gruebler's criterion, we have Degree of freedom (F) is given by:
F = 3(N  1)  2P_{1}  1P_{2}  F_{r}
where, N = number of links, P_{1} = number of pairs with 1 degree of freedom, P_{2} = number of pairs with 2 degrees of freedom, F_{r} = number of redundant pairs.
Calculation:
Given:
We have N = 4, P_{1} = 3, P_{2} = 1, F_{r} = 1
where F_{r} = Number of redundant kinematic pair.
Redundant kinematic pair:
The main purpose of the roller follower is to reduce friction. It is not playing any role in the transfer of relative motion. Oscillation of follower ϕ is the function of rotation of cam θ.i.e ϕ = f(θ ). Hence we can say that there is a redundant pair in this mechanism. After considering the redundant pair we shall get a degree of freedom.
F = 3(N  1)  2P1  1P2  F_{r}
F_{r} = 1, Kinematic pair between roller and cam.
F = 3 × (4  1)  (2 × 3)  (1 × 1)  1 = 1
Hence, the degree of freedom of the mechanism is 1.
If we neglect the count of redundant kinematic pair
Degree of freedom (F) is given by:
F = 3(N  1)  2P1  1P2
So, F = 3 × (4  1)  (2 × 3)  (1 × 1)
F = 2
So the answer will come 2, which is wrong,
Since there is a redundant link present in the system.
Explanation
Degree of freedom(DOF) – Degree of freedom of plane mechanism is defined as the number of inputs or independent coordinates needed to define the configuration or position of all the links of mechanism with respect to a fixed link.
Kutzback equation of degree of freedom,
DOF = 3 × (L – 1) – 2j – h
Where L = number of links in the mechanism, j = Binary joint or lower pair, h = higher Pair
DOF of the epicyclic gear train
L = 4
j = 3 Between ( 1 and 2, 1 and 4, 3and 4)
h = 1 Between 2 and 3
DOF = 3 × (4 – 1) – 2 × 3 – 1 = 2
Alternate Method
Degree of freedom is the number of inputs required to produce a unique output.
In Epicyclic gear train, there are two inputs required to produce a unique output i.e. one at arm and other at sun gear or at ring gear.
Input 1:
If sun is fixed, then input is given to Ring gear Or If ring gear is fixed, then input is given to sun gear.
Input 2:
Input is given to Arm.
Hence, (3) is the correct answer.
The number of degrees of freedom of the linkage shown in the figure is
Concept:
Gruebler's equation: This equation is used to find out the degree of freedom of a planner mechanism.
F = 3(n1)  2j  h
n = Number of links
j = Number of revolute pair or binary joint
h = Number of higher pairs
Calculation:
Number of links, N = 6
Total number as binary joints, j = 7
h = 0
F = 3 (N1) – 2j
= 1514 = 1.Explanation:
Degree of freedom(DOF):
Degree of freedom of plane mechanism is defined as the number of inputs or independent coordinates needed to define the configuration or position of all the links of mechanism with respect to a fixedline.
For a body moving freely in space the position and orientation of a rigid body in space are defined by three components of translation and three components of rotation, which means that it has six degrees of freedom.
Additional Information
Each particle that makes up a mechanical system, can be located by three independent variables labelling a point in space.
You can choose any particle in the rigid body to start with and move it anywhere you want, giving three independent variables needed to specify its location.
Choosing a second particle, you choose another set of three independent variables to specify its location, the obvious being spherical coordinates with the origin at the first particle. The first constraint is that the radius is a constant, leaving two remaining independent variables.
Choosing a third particle, you have complete freedom to rotate it by any angle about the axis through the first and second particles giving just one degree of freedom, the other two variables constrained.
For the remaining (N  3) particles, all three coordinates are constrained.
Therefore, the total number of degrees of freedom for a rigid body is 3 + 2 + 1 = 6, with 0 + 1 + 2 + 3(N  3) = (3N  6) constraints.
Degree of freedom
Dof = [∑(freedoms of points)]  (number of independent constraints),
Dof = (3N)  (3N  6) = 6Concept:
Degree of Freedom: The degree of freedom (DOF) of a mechanical system is the number of independent variables required to define the position or motion of the system.
For a simple mechanism, the degree of freedom (F) is given by the Gruebler’s criterion:
F = 3 (n  1)  2j  h
Where j = number of revolute joints
n = number of links
h = number of higher pairs
Calculation:
Given:
n = 8, j = 10, h = 0
F = 3 (8  1)  2 × 10  0
F = 21  20 = 1
Concept:
Degree of Freedom: The degree of freedom (DOF) of a mechanical system is the number of independent variables required to define the position or motion of the system.
For a simple mechanism, the degree of freedom (F) is given by the Grubler’s criterion:
F = 3 (n  1)  2j  h
Where j = number of revolute joints
n = number of links
h = number of higher pairs
Calculation:
Given, n = 8, j = 9, h = 0
F = 3 (8  1)  2 × 9  0
F = 21  18 = 3
Concept:
For a simple mechanism, the degree of freedom (F) is given by the Grubler’s criterion:
F = 3 (n  1)  2j  h
Where j = number of revolute joints
n = number of links
h = number of higher pairs
For a constrained planar mechanism, DOF is 1
⇒ 1 = 3 (n  1)  2j  2
⇒ 3n  2j = 6
⇒ 2j = 3n  6
The RHS of the equation should be positive and even number,
Put n = 3
2j = 3 × 3  6 = 3 odd
put n = 4
2j = 3 × 4  6 = 6 even
So, the minimum value of n should be 4
Explanation:
Degree of Freedom
What is the total number of links and binary joints in the mechanism as shown in the figure?
Explanation:
There are four links, three binary joints and one higher pair, i.e. L = 4, j = 3 and h = 1
Degree of freedom:
n = 3 (L – 1) – 2 j – h
∴ n = 3 (4 – 1) – 2 × 3 – 1 = 2
Match the items in columns I and II
Column I 
Column II 
(P) Higher kinematic pair 
(1) Grubler’s equation 
(Q) Lower kinematic pair 
(2) Line contact 
(R) Quick return mechanism 
(3) Euler’s equation 
(S) Mobility of a linkage 
(4) Planer 

(5) Shaper 

(6) Surface contact 
Explanation:
Lower pair: When the two elements of a pair have a surface contact when relative motion takes place and the surface of one element slides over the surface of the other, the pair formed is known as lower pair. Automobile Steering gear, Sliding pairs, turning pairs and screw pairs are the examples of lower pairs.
Higher pair: When the two elements of a pair have a line or point contact when relative motion takes place and the motion between the two elements is partly turning and partly sliding, then the pair is known as higher pair. A pair of toothed gearing, belt and rope drives, ball and roller bearings and cam and follower are the examples of higher pairs.
Grubler’s equation
The Grubler’s criterion is applied to mechanisms with only a single degree of freedom joints where the overall mobility of the mechanism is unity.
n = 3 (l  1)  2j  h
Shaper is a reciprocating type of machine tool in which the ram moves the cutting tool backward and forwards in a straight line. It is intended primarily to produce flat surfaces. These surfaces may be horizontal, vertical, or inclined.
In a shaper, the rotary motion of the drive is converted into reciprocating motion of the ram by the mechanism housed within the column or the machine.
In a standard shaper, metal is removed in the forward cutting stroke, while the return stroke goes idle and no metal is removed during this period.
The shaper mechanism is so designed that it moves the ram holding the tool at a comparatively slower speed during the forward cutting stroke, whereas during the return stroke it allows the ram to move at a faster speed to reduce the idle return time.
This mechanism is known as the quick return mechanism.
Explanation:
The general expression for the number of degree of freedom in a plane mechanism having n links, j simple hinge joints, h number of higher pair and F_{r} redundant degree of freedom is:
N = 3(n – 1) 2j – h  FrA double – parallelogram mechanism is shown in the figure. Note that PQ is a single link. The mobility of the mechanism is
Concept:
Mobility of the mechanism = Degree of Freedom = [3(l  1)  2j  h]  F_{r}
where, l = no. of links, j = no. of the binary joint, h = no. of higher pairs, F_{r} = redundant link
Here the middle link is acting as a dummy link.
Calculation:
Given:
l = 4, j = 4, h = 0, F_{r} = 0
DOF = [3(4  1)  2 × 4  0]  0 = 9  8 = 1
So degree of freedom of double – parallelogram mechanism is one.
Number of degrees of freedom for the shown figure is
Concept:
Kutzbach equation:
The actual degree of freedom = maximum degree of freedom  restricted degree of freedom
F = 3 × (N  1)  2 × J  H
Where N = number of links, J = number of binary motion pairs, and H = number of higher pairs
Calculation:
Given:
No of links (N) = 5, No of binary pairs (J) = 5, and No of higher pairs (H) = 1
F = 3 × (N  1)  2 × J  H
F = 3 × (5  1)  2 × 5  1
∴ F = 1
Additional Information
Chain: Assembly of several links and pairs which is able to transfer relative motion is known as chain
Mechanism: If any of the links is fixed in the chain and it is able to transfer the relative motion with or without transformation is known as a mechanism
Kinematic mechanism: The mechanism which has a degree of freedom equal to 1 is called the kinematic mechanism
Explanation:
The degree of freedom (DOF) of a mechanical system is the number of independent parameters that define its configuration.
The position and orientation of a rigid body in space are defined by three components of translation and three components of rotation, which means that it has six degrees of freedom.
Consider the following statements:
I. The degree of freedom for lower kinematic pairs is always equal to one.
II. A ball and socket joint has three degrees of freedom and is a higher kinematic pair.
III. Oldham’s coupling mechanism has two prismatic pairs and two revolute pairs.
Which of the statements given above is/are correct?
Concept:
Oldham’s coupling mechanism:
Lower kinematic pairs:
Spherical pair
Concept:
Gruebler's criteria for planar mechanism:
DOF = 3(n  1)  2P
where n = no of links, P = No of lower pairs(turning pairs)
Calculation:
Given:
n = 4, P =4
DOF = 3(4  1)  2 × 4 = 1
Concept:
Degree of freedom (DOF): The minimum number if independent variables which are required to define the position of system.
When we put the long cylinder in free space, then we have 6 DOF i.e. 3 translation and 3 rotation.
3 translation: along xaxis, yaxis, zaxis and
3 rotation: about xaxis, yaxis, zaxis
But when we put the cylinder on V block, 4 DOF arrested (2 translation and 2 rotation)
So, by formula, DOF = 6  restraints
∴ DOF = 6  4 = 2
DOF of sphere on plate, DOF = 6  1 = 5
Explanation:
In kinematics:
Degree of Freedom: The degree of freedom (DOF) of a mechanical system is the minimum number of independent variables required to define the position or motion of the system.
For a simple mechanism, the degree of freedom (F) is given by Grubler’s criterion:
F = 3 (n  1)  2j  h
Where, j = number of revolute joints, n = number of links, h = number of higher pairs
Additional Information
If F > 1 ⇒ more than 1 input required to obtain a definite motion of links
if F = 1 ⇒ Only 1 input required to obtain a definite motion of links ⇒ Kinematic chain
If F = 0 ⇒ It has zero mobility ⇒ Frame
If F < 0 ⇒ It is a redundant frame.