Short Answer:

The degrees of freedom (DOF) of a mechanism represent the number of independent movements it can perform. It shows how many input motions are required to determine the complete motion of the mechanism.
In simple terms, the degrees of freedom define how a mechanism or link can move — such as translation, rotation, or both. For example, a rigid body in a plane has three degrees of freedom: two translational and one rotational movement.

Detailed Explanation :

Degrees of Freedom of a Mechanism

The term degrees of freedom (DOF) is a fundamental concept in kinematics and mechanism design. It defines the number of independent parameters that determine the position and motion of a mechanism or a link. In simpler words, it tells how many different ways a mechanism can move. Understanding the degrees of freedom is very important for designing efficient mechanisms that can perform a desired motion correctly.

1. Meaning of Degrees of Freedom
   The degrees of freedom describe the number of independent movements a system can have. Each independent motion may be linear (translation) or angular (rotation).
   For example, a point moving along a straight line has one degree of freedombecause it can move only in one direction. Similarly, a body that can move forward and backward (translation in X), side to side (translation in Y), and rotate about an axis (rotation) has three degrees of freedomin a plane.
   In space, a rigid body can move in six independent ways — three translational and three rotational — hence it has six degrees of freedom.
2. Degrees of Freedom in Planar Mechanisms
   Most mechanisms used in mechanical systems are planar, meaning all their motions occur in one plane.
   For a planar mechanism:

• Each link can move in three independent ways — two translational and one rotational.
• When links are connected by joints, their motion becomes restricted, and the total degrees of freedom reduce.
  The formula used to calculate the degrees of freedom for a planar mechanism is known as Gruebler’s Equation or Kutzbach’s Criterion, given as:

Where:

•  = Degrees of freedom of the mechanism
•  = Number of links (including the frame)
•  = Number of lower pairs (revolute or prismatic joints)
•  = Number of higher pairs (such as cam or gear contact)

This formula helps to determine if a mechanism can move freely, is constrained properly, or is over-constrained.

3. Examples of Degrees of Freedom

• Single Link (Free Link): A single rigid body in a plane can move in three independent ways — hence it has 3 DOF.
• Four-Bar Mechanism: A simple four-bar linkage has four links and four revolute joints. Substituting in the equation:

This means it has one degree of freedom, which indicates that only one input motion is required to determine the motion of the entire mechanism.

• Slider-Crank Mechanism: Similar to the four-bar linkage, it also has one degree of freedom, which allows the conversion of rotary motion into linear motion.

4. Degrees of Freedom in Spatial Mechanisms
   Spatial mechanisms have movements in three-dimensional space.
   In this case, each link has six degrees of freedom— three translational and three rotational. The general formula for spatial mechanisms is:

Here, the coefficients differ because of the extra movements possible in space. Examples of spatial mechanisms include robotic arms and universal joints.

5. Importance of Degrees of Freedom in Mechanism Design
   The degrees of freedom play a crucial role in determining how a mechanism will work.

• If a mechanism has zero DOF, it means the structure is completely constrained and cannot move.
• If it has one DOF, it can perform the required motion with one input — this is the ideal condition for most mechanisms.
• If it has more than one DOF, it becomes under-constrained, leading to unwanted movements and instability.
  Hence, while designing a mechanism, the degrees of freedom must be carefully controlled to achieve smooth and predictable motion.

6. Factors Affecting Degrees of Freedom
   The degrees of freedom depend on several factors such as:

• Number of links in the mechanism.
• Type of joints used (revolute, prismatic, cylindrical, etc.).
• Type of pairs (lower or higher pairs).
• Geometric constraints or design limitations.
  By properly combining these factors, engineers can create mechanisms that perform specific functions like rotating, sliding, or oscillating.

Conclusion

The degrees of freedom of a mechanism describe the number of independent motions it can perform. It helps engineers analyze whether a mechanism will work as desired or if it will be over- or under-constrained. In mechanical systems, maintaining the correct degrees of freedom ensures smooth, stable, and precise operation. Therefore, understanding DOF is fundamental to the design and analysis of all types of mechanisms, from simple linkages to complex robotic systems.