What is Kutzbach criterion?

Short Answer:

The Kutzbach criterion is a formula used to determine the degree of freedom (DOF) or mobility of a mechanism. It tells how many independent motions are possible in a system made of links and joints. The criterion helps in understanding whether a mechanism is movable, constrained, or over-constrained.

In simple terms, the Kutzbach criterion gives the number of possible movements that can occur in a mechanism. It is mainly used in kinematic analysis and synthesis to design mechanisms with the required motion. This criterion is very important in studying planar mechanisms like four-bar linkages.

Detailed Explanation :

Kutzbach Criterion

The Kutzbach criterion is an essential concept in kinematics of machinery that determines the mobility or degree of freedom (DOF) of a mechanism. It provides a mathematical method to find how many independent inputs or motions are required to define the position of every link in a mechanism. Understanding the DOF is important because it indicates whether a mechanism can perform its intended motion effectively without being locked or having extra unwanted movements.

Meaning of Degree of Freedom

Before understanding the Kutzbach criterion, it is important to know what degree of freedom means. The degree of freedom (DOF) is the number of independent parameters or movements needed to describe the position of a mechanism or a body.
In two-dimensional motion, each link can have three independent movements — two translational motions (along x and y axes) and one rotational motion. Therefore, each link in a planar mechanism has three degrees of freedom.
When the links are connected by joints, some motions are restricted, and the total DOF of the mechanism reduces.

Kutzbach Criterion Formula

For a planar mechanism, the Kutzbach criterion is given by the following formula:

Where,

  •  = Degree of freedom of the mechanism
  •  = Number of links (including the fixed link)
  •  = Number of lower pairs (like revolute or sliding joints)
  •  = Number of higher pairs (like cam or gear contact)

This formula helps in finding whether the mechanism will have the desired motion or not.

Explanation of Terms

  1. Links:
    Links are the rigid parts of the mechanism that are connected by joints. One link is always fixed to serve as a reference frame. The other links are movable and transmit motion.
  2. Joints or Pairs:
    Joints connect two or more links and allow motion between them. They are classified into:

    • Lower pairs: Have surface contact such as turning (revolute) and sliding (prismatic) joints.
    • Higher pairs: Have point or line contact such as cam and follower, or gear tooth contact.
  3. Mobility (F):
    The value of F obtained from the formula shows how many independent motions are possible.

    • If F = 1, the mechanism has one degree of freedom (simple mechanism like four-bar linkage).
    • If F = 0, the mechanism is completely constrained and cannot move.
    • If F > 1, the mechanism has multiple independent motions.
    • If F < 0, the mechanism is over-constrained (it may not move properly).

Example of Kutzbach Criterion

Let us consider a four-bar chain mechanism as an example.
It has:

  • Number of links,
  • Number of lower pairs,
  • No higher pairs,

Now applying the formula:

 

So, the four-bar mechanism has one degree of freedom. This means only one input motion is needed to define the motion of the entire mechanism.

Importance of Kutzbach Criterion

  1. Helps in Design:
    It helps designers to create mechanisms with the correct number of degrees of freedom. For example, a single-input mechanism must have one degree of freedom.
  2. Checks Mobility:
    It helps to check if a mechanism is properly constrained, under-constrained (too many movements), or over-constrained (no movement).
  3. Avoids Design Errors:
    Using this criterion avoids making mechanisms that are locked or have unwanted extra motions.
  4. Used in All Mechanism Types:
    It is applicable to different mechanisms like planar linkages, cams, and gear systems.

Extended Formula for Spatial Mechanisms

For three-dimensional (spatial) mechanisms, the mobility is calculated using the Gruebler’s equation, which is a general form of Kutzbach’s criterion. It is given as:

Here, 6 represents the total possible motions (three translations and three rotations) in 3D space, and 5 is the number of motions restricted by each lower pair in space.

Conclusion

The Kutzbach criterion is a key principle in mechanism design and analysis. It determines how many degrees of freedom a mechanism has and whether it can perform the required motion. By applying this criterion, engineers can design mechanisms that are properly constrained and perform efficiently. It ensures that the motion of each link is predictable and suitable for the required mechanical function. The Kutzbach criterion, therefore, forms the foundation of modern kinematic design and is widely used in analyzing linkages, cams, and gear trains.