## Planar Fourbar Linkages - Fundamental Design Parameters

### Stefan Gössner

*Department of Mechanical Engineering, University of Applied Sciences, Dortmund, Germany.*

**Keywords**: R3 chain; crank-rocker; rocker-crank; four-bar; fourbar; planar mechanism; linkage; Grashof; limit angles;

## Abstract

The planar fourbar mechanism is classified according to Grashof's rule. For the practically important crank-rocker mechanism six dependent design parameters are introduced. The relevant constraint equations are derived and discussed.

## Introduction

A fourbar linkage is the most simple mechanism having a mobility of one and revolute joints only. It is composed of four binary links - the bars - connected to each others by four revolute joints. With respect to the latter fact it is also called *4R mechanism*.

One of the four links is fixed and called *frame*. If a link connected to the frame can fully rotate, it is called *crank*. Otherwise it can only rotate between two angular limits and is called *rocker*. The floating link opposite to the frame is called *coupler* and functions as a *connecting rod* between its neighbour links.

## Design Parameters

The fourbar linkage is uniquely defined by its four link lengths

For successfully assembling the fourbar, the longest link length is required to be less than the sum of the other three link lengths.

A fourbar linkage is said to be *fully rotatable*, if its shortest link is able to perform a complete rotation. *Grashof*'s rule is satisfied then:

Programming hint:Most programming languages offer`min`

and`max`

functions. So by doubling the left sides of inequations (1) and (2) we can code easily

2*max(a,b,c,d) < a + b + c + d // (1) 2*min(a,b,c,d) + 2*max(a,b,c,d) < a + b + c + d // (2) Grashofwithout having to filter out the remaining link lengths on the right side of (1) and (2).

## Classification

Fourbar linkages can be classified regarding Grashof's rule (2). If there is a `less than`

relationship between the left and right side of equation (2), the fourbar linkage is fully rotatable. If there is a `greater than`

relationship, the linkage is not fully rotatable. In the special case of an `equal`

relationship, the linkage is able to fully rotate. It will take a pose at least twice during one revolution in which all four binary links are collinear. The linkage is said to *fold* in that configuration (Fig. 2).

Modler [3] and Mc.Carthy [2] have introduced more detailed classification parameters.

## Crank-Rocker

The most important type in practice is undoubtedly the crank-rocker, as it is converting a continuous rotation to a rotational oscillation.

We can identify the two limits on the rocker angle from the positions, where crank and coupler are collinear. Applying the cosine law to each triangle in Fig.2 we get the rocker angle limits

The associated crank angles can be determined the same way.

Much more of interest than the explicite angular limits are the angular ranges during one complete cycle.

The rocker's working angular range is named

During a complete cycle of the continuously rotating crank the forward movement of the rocker occupies the crank angle

So if *forth* and *back* motion.

## Extended Design Parameter Set

The degree of non-uniformity expressed by the ratio

So adding

Applying the cosine theorem twice for

Simplifying leads to

Using half angle identity

The half rectangular triangle

which can be resolved for

From triangle

which we can use to eliminate the dependency on

Finally we apply the cosine law to triangle

from where we can eliminate

When eliminating

With the design of a crank-rocker mechanism we can choose values for four out of six design parameters

$a, b, c, d, \\alpha, \\psi\_0$ . The missing two can be determined by two constraint equations from (6), (9) or (10).

Note: Please observe Grashof's rule ($a$ is shortest length) as well as the three special cases below !

## Case $\\alpha = \\psi\_0$

Here equations (9) and (10) reduce to the constraints

In addition to

## Case $\\alpha = \\frac{\\psi\_0}{2}$

Here equations (9) and (6) reduce to the constraints

In addition to

## Case $\\alpha = 0$

Here equations (9) and (6) reduce to the constraints

In addition to

The resulting fourbar is called *centric fourbar* and has interesting and mostly advantageous properties.

## Conclusion

From six meaningful design parameters for crank-rocker mechanisms only four are independent. The relevant constraint equations were derived. With those equations the two missing parameters can be calculated in general. Three special cases are discussed.

## References

[1] Gössner S., Mechanismentechnik – Vektorielle Analyse ebener Mechanismen, Logos, Berlin, 2016

[2] McCarthy J.M. et al., Geometric Design of Linkages, Springer, New York, 2010

[3] Modler K.H., Luck K., Getriebetechnik Analyse, Synthese, Optimierung, Springer, Berlin/Heidelberg, 1995

[4] Uicker, J.J. et. al., Theory of Machines and Mechanisms, Oxford Press 2011

[5] VDI-Richtline 2145, Ebene viergliedrige Getriebe mit Dreh- und Schubgelenken, Beuth Berlin, 1980.