Mechanisms and Degree of Freedom (DOF)

Definition of a Mechanism

• A mechanism is defined as a combination of interconnected links or rigid bodies capable of movement.

Degree of Freedom (DOF)

• Definition: The minimum number of independent coordinates or variables to fix the configuration of a mechanism.
• Keywords: Configuration, independent coordinates.
• To calculate DOF, one link of the chain must be grounded.

Calculation of DOF

1. Planar Kinematic Chains

• A rigid body in a plane has 3 degrees of freedom:
  • Movement along X-axis
  • Movement along Y-axis
  • Rotation in the plane
• Example with two bodies:
  • Each body has 3 DOF.
  • Total for two bodies = 6 DOF.

Grounding a Link

• Grounding one body means it loses all degrees of freedom.
• Calculation formula:
  • For two rigid bodies, if one is grounded:
    • DOF = (Total DOF of bodies - 1) + (DOF of kinematic pairs)
    • Kinematic pair reduces DOF (depending on type of pair).

2. Generalized Calculation for Planar Mechanism

• Formula:
  • F = 3 * (NL - 1) - 3 * NJ + ΣFi
• Where:
  • NL = Number of links
  • NJ = Number of joints or kinematic pairs
  • Fi = Degree of freedom of each joint.

3. Spatial Mechanisms

• Rigid body in space has 6 degrees of freedom (3 translation + 3 rotation).
• Formula:
  • F = 6 * (NL - 1) - 6 * NJ + ΣFi

Key Concepts

• A mechanism must have at least 1 degree of freedom to be considered movable.
• A structure has 0 degrees of freedom (i.e., it does not move).
• A negative degree of freedom indicates an over-constrained structure.

Examples of DOF Calculation

1. Example 1: Structure with 6 links and 8 joints - Total DOF = 0 (indicates structure).
2. Example 2: Steering wheel mechanism with 4 links, 4 joints - Total DOF = 1 (expected).
3. Example 3: Crimping tool with 4 links and 4 joints - Total DOF = 1 (expected).
4. Example 4: Surgeon’s tool with 4 links and 4 joints - Total DOF = 1 (expected).
5. Example 5: Scissors with 6 links and 7 joints - Total DOF = 1 (expected).
6. Example 6: Transfer device with 8 links and 10 joints - Total DOF = 1 (expected).

Summary

• The lecture covered the definition and calculation of the degree of freedom for mechanisms, focusing on planar and spatial kinematic chains, and provided examples for better understanding.