Short Answer:

Grashof’s law is a fundamental rule in kinematics used to determine whether a link in a four-bar chain will make a complete revolution or not. It relates the lengths of the four links of a mechanism. According to Grashof’s law, the sum of the shortest and longest links must be less than or equal to the sum of the remaining two links for continuous rotation to occur.

This law helps engineers predict the type of motion (rotary or oscillatory) that a mechanism will produce. If the condition is not satisfied, none of the links can rotate completely, and the mechanism will only produce rocking or oscillating motion.

Detailed Explanation :

Grashof’s Law

Grashof’s law is an important concept in the study of mechanisms and kinematics of machines. It is used to determine the relative motion between links in a four-bar chain mechanism. The law provides a simple geometric relationship among the lengths of the four links that helps predict whether a particular link can make a complete revolution or will only oscillate.

In a four-bar chain mechanism, four rigid links are connected by four turning pairs. These links can be named as follows:

• Link 1: Fixed link or frame
• Link 2: Input link (crank)
• Link 3: Coupler or connecting rod
• Link 4: Output link (rocker)

To analyze their motion, Grashof’s law compares the lengths of these links.

Statement of Grashof’s Law

According to Grashof’s law:

“In a four-bar chain, the sum of the shortest (S) and the longest (L) links is less than or equal to the sum of the other two links (P and Q) for at least one link to make a complete revolution relative to the other links.”

Mathematically, the law can be expressed as:
S + L ≤ P + Q

Where,
S = length of the shortest link
L = length of the longest link
P and Q = lengths of the other two links

This relationship determines the type of motion possible in the mechanism.

Explanation of Grashof’s Condition

If the Grashof’s condition (S + L ≤ P + Q) is satisfied, the mechanism is known as a Grashof mechanism. In such cases, at least one link can rotate completely relative to the other links. Depending on which link is fixed, the motion of other links changes.

If the Grashof’s condition is not satisfied (S + L > P + Q), then no link will be able to make a complete rotation. In this case, the mechanism is called a non-Grashof mechanism, and the motion of all moving links is limited to oscillation or rocking.

This simple mathematical relationship allows designers to predict the possible motions without building or testing the actual mechanism.

Types of Mechanisms Based on Grashof’s Law

1. Double Crank Mechanism (Drag-Link Mechanism)
   When the shortest link is adjacent to the fixed link, both links connected to it can rotate completely. This arrangement is called a double crank mechanism.
   Example: Coupling rod mechanism in locomotives.
2. Crank and Rocker Mechanism
   When the shortest link is adjacent to the fixed link but only one link can rotate completely while the opposite link oscillates, the mechanism is called a crank and rocker mechanism.
   Example: Reciprocating pump and internal combustion engine valve mechanism.
3. Double Rocker Mechanism
   When the shortest link is opposite to the fixed link, none of the links can rotate completely. Both adjacent links to the fixed one act as rockers.
   Example: Certain linkage arrangements in presses or shaping machines.
4. Non-Grashof Mechanism
   When the Grashof condition is not satisfied, all links are restricted to oscillation. No continuous rotation is possible.

Practical Example

Consider a four-bar chain where the lengths of the links are:

• Link 1 = 200 mm
• Link 2 = 80 mm
• Link 3 = 150 mm
• Link 4 = 250 mm

Here,
S = 80 mm (shortest link)
L = 250 mm (longest link)
P = 150 mm, Q = 200 mm

Now check Grashof’s condition:
S + L = 80 + 250 = 330 mm
P + Q = 150 + 200 = 350 mm

Since 330 < 350, Grashof’s condition is satisfied. Therefore, at least one link in this four-bar chain can rotate completely, and the mechanism will be a Grashof mechanism.

Importance of Grashof’s Law

1. Predicts Motion Type: It helps determine whether a mechanism will produce continuous rotation or limited oscillation.
2. Simplifies Design Process: Designers can easily analyze link lengths without building physical models.
3. Ensures Smooth Motion: Proper link proportions as per Grashof’s law ensure smooth motion and prevent interference between parts.
4. Used in Many Machines: Grashof’s principle is used in the design of engines, pumps, presses, couplings, and robotic linkages.
5. Enhances Efficiency: Mechanisms designed following this law transmit motion and power effectively with minimal friction and vibration.

Violation of Grashof’s Law

If S + L > P + Q, the mechanism does not satisfy Grashof’s law. This results in a non-Grashof mechanism, where continuous rotation is not possible. The motion becomes limited, and the mechanism acts as a double rocker mechanism. Although such mechanisms are not used for continuous motion, they are useful in cases where only oscillating or limited angular motion is required.

Conclusion

Grashof’s law defines a simple relationship between the lengths of links in a four-bar chain that helps determine whether continuous rotation is possible. When the sum of the shortest and longest links is less than or equal to the sum of the remaining two links, the mechanism is called a Grashof mechanism, capable of full rotation. This law plays a vital role in designing various mechanical systems like engines, pumps, and robotic arms, ensuring that the required motion is achieved efficiently and reliably.