What are the inversions of a four-bar chain?

Short Answer:

The inversions of a four-bar chain are obtained by fixing different links of the same four-bar mechanism one at a time. Since there are four links, we can get four possible inversions. Each inversion gives a different mechanism having unique motion characteristics. The most common inversions are the crank-rocker mechanism, double-crank mechanism, and double-rocker mechanism.

In every inversion of a four-bar chain, the relative motion between the links changes, which results in different types of output motions. These inversions are used in many machines like engines, pumps, and couplings for transmitting motion and power effectively.

Detailed Explanation :

Inversions of a Four-Bar Chain

A four-bar chain is a closed kinematic chain that consists of four rigid links connected by four turning pairs (hinges). In this mechanism, one link is kept fixed while the other three links move relative to it. The motion of the links depends on which link is chosen as the fixed one.

The term inversion of a four-bar chain means obtaining different mechanisms by fixing different links one at a time. Since there are four links, theoretically four inversions are possible. However, only three of these inversions are practically used in machines. The type of motion produced in each inversion depends on the relative lengths of the links and the position of the fixed link.

The four-bar chain is named as follows:

  • Link 1: Fixed link (frame)
  • Link 2: Crank or input link
  • Link 3: Coupler or connecting link
  • Link 4: Rocker or output link

By applying Grashof’s Law, we can determine whether a link will rotate completely or oscillate. According to the law:

The sum of the shortest (S) and the longest (L) links must be less than or equal to the sum of the remaining two (P + Q) for continuous rotation to occur.

If this condition is satisfied, at least one link (usually the shortest) will rotate completely. Depending on which link is fixed, we get different inversions of the four-bar chain.

  1. First Inversion – Double Crank Mechanism (Drag-Link Mechanism)

In this inversion, the shortest link is adjacent to the fixed link. Both the links connected to the fixed link can make a complete revolution. Therefore, this arrangement is called a double crank mechanism or drag-link mechanism.

Features:

  • Two links rotate completely.
  • It is mainly used for continuous rotary motion between two parallel shafts.
  • The coupler acts as the connector transmitting motion between the two cranks.

Example:
The coupling rod of a locomotive engine, which connects the two driving wheels, works on this principle. It ensures that both wheels rotate together in a synchronized manner.

  1. Second Inversion – Crank and Rocker Mechanism

In this inversion, one of the links adjacent to the shortest link is fixed. The shortest link acts as the crank, which rotates completely, and the opposite link acts as the rocker, which oscillates back and forth.

Features:

  • The crank rotates continuously.
  • The rocker swings through a limited angle.
  • It converts rotary motion into oscillating motion.

Examples:

  • Pumping mechanisms in reciprocating pumps.
  • Steam engine valve gear.
  • Hand-operated water pumps.

This inversion is very common in machines where controlled oscillating output is required from a rotating input.

  1. Third Inversion – Double Rocker Mechanism

In this inversion, the shortest link is not connected to the fixed link. The link opposite to the shortest one is kept fixed. In this case, none of the links can rotate completely; both links connected to the fixed link only oscillate. Therefore, it is called the double rocker mechanism.

Features:

  • No link rotates completely.
  • Both connected links act as rockers.
  • The coupler transfers motion between the two rocking links.

Examples:

  • Coupling mechanisms in certain types of engines.
  • Rocker arm mechanisms used in internal combustion engines.

This inversion is mainly used where limited angular motion is needed for both connected links.

  1. Fourth Inversion (Rarely Used)

When the coupler link (link 3) is fixed, we get the fourth inversion. This type of inversion is rarely used because the relative motion between the other links becomes complex. It is not practically useful in most machines. However, in theoretical analysis, it represents the fourth possible case of inversion in a four-bar chain.

Practical Importance of Inversions

The concept of inversion is very useful in mechanism design. By fixing different links of the same four-bar chain, engineers can create entirely different machines from a single basic arrangement. This flexibility helps reduce design complexity and cost.
For example:

  • In engines, the crank and rocker inversion converts rotary motion of the crankshaft into oscillating valve motion.
  • In locomotives, the double crank inversion helps in transmitting motion to both wheels.
  • In tools and presses, the double rocker inversion provides limited, controlled movement.

Thus, each inversion has its own significance depending on the type of motion required in the machine.

Conclusion

The inversions of a four-bar chain are formed by fixing different links one at a time. Out of the four possible inversions, three are practically important — the double crank mechanism, crank and rocker mechanism, and double rocker mechanism. Each inversion provides a unique motion pattern, such as full rotation, oscillation, or rocking motion. These mechanisms are widely applied in engines, pumps, and other mechanical systems. The study of inversions helps engineers design effective machines for converting motion and transmitting power efficiently.