The **Grashoff’s law** states that: *In a flat mechanism with four articulated bars with one of them fixed, at least one of the bars will be able to make a complete turn, provided that the sum of the shortest bar and the longest bar is less than or equal to the sum of the other two.*

There are five flat four-bar or link mechanisms that obey Grashof’s law (An example is shown in Figure 1). In order for the bars or links of the mechanisms that comply with the law to be able to turn completely, it is necessary that in a real arrangement, each bar is occupying different parallel planes.

Grashof’s law is a simple rule that allows designing a mechanism in which full rotation is required, either because a motor will be connected or, on the contrary, because an oscillatory movement is to be transformed into a rotary one, in such a way that it is mathematical. and physically viable.

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## borderline cases

Suppose that the four articulated bars have the following lengths ordered from smallest to largest according to:

*s > p > q > l*

Grashof’s law establishes that for at least one bar or link to complete one revolution or turn, the condition must be met:

*s + l <= p + q*

This inequality has the following implications:

– The only bar or link that can make full revolutions with respect to another is the shortest bar.

– If the shortest bar makes complete turns with respect to another, then it will also make complete turns with respect to all the others.

### types of movement

The movement of the articulated quadrilateral that complies with Grashof’s law can be of the following types:

– Double turn or crank, if the shortest bar is the fixed one and the adjacent bars make complete turns.

– Turn and swing, if the short bar is adjacent to the high bar.

– Double seesaw, provided that the shorter bar is opposite the fixed one.

When the equality in Grashof’s formula is fulfilled, then we are in the limit case in which the sum of the shortest bar with the longest is equal to the sum of the other two.

In this case, the mechanism can adopt a configuration in which the four bars are aligned. And it is in this position, the non-fixed joints can go indifferently in one direction or the other, causing the mechanism to get stuck.

Mechanisms that meet the Grashof condition are more reliable and suffer less stress in their joints and links, as they are further from the limiting case of equality.

## Mechanisms that comply with Grashof’s law

We will denote the consecutive articulations with A, B, C and D, then:

– A and B are fixed pivots.

– AB = d1 (high bar)

– BC= d2

– CD= d3

– AD= d4

**– Double crank mechanism**

Rods b2 and b4 rotate completely and Grashof’s law holds:

d1+d3 <= d2+d4.

**– More mechanisms that obey Grashof’s law**

Next, the characteristics of other mechanisms that comply with Grashof’s law are named and described:

**Crank-rocker mechanism**

It is fulfilled d2 + d3 <= d1 + d4

The shorter bar d2 rotates completely and the opposite bar d4 does a seesaw movement.

**double rocker mechanism**

– Fixed bar AB is higher than the opposite bar CD and fulfills that:

d1 + d3 <= d2 + d3

– For the shortest bar (the opposite of the horizontal bar), it is capable of making a complete turn.

**Articulated parallelogram mechanism**

– Rods AD and BC are of equal length and always parallel.

– On the other hand, the bars AB and CD are of equal length and always parallel.

– In the case of the opposite bars, they have the same length and d1 + d2 = d3 +d4 is fulfilled, according to Grashof’s law.

– Finally, bars AD and BC rotate completely in the same direction.

**Articulated Anti-Parallelogram**

– Rods AD and BC are of equal length and not parallel.

– For bars AB and CD, they must be of equal length and not parallel.

– On the other hand, the opposite bars have the same length, two of them are crossed.

– In this mechanism the following condition must be met:

* d1 + d2 = d3 +d4*

– The rotation of bars AD and BC is complete but in opposite directions.

## Applications

Mechanisms that comply with Grashof’s law have multiple applications:

### Crank-rocker mechanism

It is applied to the treadle sewing machine, useful in places where there is no electricity, in which the treadle makes a rocking or seesaw movement, which is transmitted to a wheel connected by a pulley to the sewing machine.

Another example to mention is the windshield wiper mechanism. In this, a motor is connected to the crank bar that makes complete turns, transmitting a rocker movement to the bar that moves the first brush in the system.

Another application of the crank-rocker mechanism is the rocker for pumping oil from the ground.

A motor is connected to the crank that rotates completely and transmits the movement to the pumping head or rocker.

### Articulated parallelogram mechanism

This mechanism used to be used to connect the wheels of steam locomotives, so that both wheels rotate in the same direction and at the same speed.

The main characteristic of this mechanism is that the bar that connects both wheels has the same length as the separation of their axes.

The pantograph is a drawing instrument used to copy and enlarge images. It is based on a four-bar mechanism, in which there are four joints that form the vertices of a parallelogram.

### Articulated anti parallelogram mechanism

It is the mechanism used in the tennis ball throwing machine, where the wheels that drive and throw the ball are required to rotate in opposite directions.

## References

Clemente C. Virtual laboratory of a crank-rocker mechanism. Degree work in mechanical engineering. University of Almeria. (2014). Retrieved from: repositorio.ual.es

Hurtado F. Grashof’s Law. Retrieved from: youtube.com

Mech Designer. Kinematics Grashof criteria. Retrieved from: mechdesigner.support.

Shigley, J. Theory of machines and mechanisms. McGraw Hill.

We are F1. Four-bar mechanism analysis. Retrieved from: youtube.com

UNAM. Development of a four-bar mechanism for use in teaching. Recovered from: ptolomeo.unam.mx

Wikipedia. Four-bar linkage. Retrieved from: en.wikipedia.com

Wikipedia. Grashoff’s law. Recovered from: en.wikipedia.com