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.

Fig.1: Fourbar Linkage

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 a,b,c,da, b, c, d. There are only two restrictions regarding those values.

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

lmax<l2+l3+l4l\_{max} < l\_2 + l\_3 + l\_4(1)

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:

lmin+lmax<l3+l4l\_{min} + l\_{max} < l\_3 + l\_4 (2)

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) Grashof

without 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).

Fig.2: Classification of Fourbar Linkages

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.

Fig.2: Limit positions of the Crank-Rocker

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

cosψmax=c2+d2−(b+a)22cdandcosψmin=c2+d2−(b−a)22cd.\\cos\\psi\_{max} = \\frac{c^2 + d^2 - (b+a)^2}{2cd} \\quad and \\quad \\cos\\psi\_{min} = \\frac{c^2 + d^2 - (b-a)^2}{2cd}\\,.(3)

The associated crank angles can be determined the same way.

cosφmax=−c2+d2+(b+a)22(b+a)dandcos(φmin−180∘)=−c2+d2+(b−a)22(b−a)d\\cos\\varphi\_{max} = \\frac{-c^2 + d^2 + (b+a)^2}{2(b+a)d} \\quad and \\quad \\cos(\\varphi\_{min}-180^{\\circ}) = \\frac{-c^2 + d^2 + (b-a)^2}{2(b-a)d}(4)

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

Fig.3: Angular regions of rocker and crank while moving unidirectional

The rocker's working angular range is named ψ0\\psi\_0 and corresponds to the crank's angular range φ0\\varphi\_0 during tracking motion, when crank and rocker move unidirectional (counterclockwise in Fig.3). On their way back they run in opposite directions.

ψ0=ψmax−ψmin ,φ0=φmin−φmaxandα=φ0−180∘\\psi\_0 = \\psi\_{max} - \\psi\_{min} \\space, \\quad \\varphi\_0 = \\varphi\_{min} - \\varphi\_{max} \\quad and \\quad \\alpha = \\varphi\_0 - 180^\\circ(4)

During a complete cycle of the continuously rotating crank the forward movement of the rocker occupies the crank angle 180∘+α180^\\circ+\\alpha, while the backward movement belongs to crank angle 180∘−α180^\\circ-\\alpha (Fig. 3). Using this we can define the time ratio tf/tbt\_f/t\_b of the rocker moving cycle

tftb=180∘+α180∘−α.\\frac{t\_f}{t\_b} = \\frac{180^\\circ+\\alpha}{180^\\circ-\\alpha}\\,.(5)

So if α\\alpha is not 0°, the crank needs different times for the rocker's forth and back motion.


Extended Design Parameter Set

The degree of non-uniformity expressed by the ratio tf/tbt\_f/t\_b as well as the angular range ψ0\\psi\_0 of the rocker are very meaningful design parameters from the engineer's point of view.

So adding ψ0\\psi\_0 and α\\alpha to the four link lengths a,b,c,da, b,c,d, we now have six design parameters. There must be two constraint equations between them, since only four parameters are sufficient to uniquely define a crank-rocker mechanism.

Fig.4: Geometric relations between six design parameters

Applying the cosine theorem twice for ΔA0B1B2\\Delta A\_0 B\_1 B\_2 and ΔB0B1B2\\Delta B\_0 B\_1 B\_2 with respect to common side s0s\_0 gives (Fig.4)

(b+a)2+(b−a)2−2(b+a)(b−a)cosα=c2+c2−2c2cosψ0(b+a)^2 + (b-a)^2 - 2(b+a)(b-a)\\cos\\alpha = c^2 + c^2 - 2c^2\\cos\\psi\_0

Simplifying leads to

a2+b2−(b2−a2)cosα=c2(1−cosψ0)a^2 + b^2 - (b^2 -a^2)\\cos\\alpha = c^2(1-\\cos\\psi\_0)

Using half angle identity cosx=cos2x2−sin2x2\\cos x = \\cos^2\\frac{x}{2} - \\sin^2\\frac{x}{2} finally results in a first constraint equation between the five parameters a,b,c,α,ψ0a,b,c,\\alpha,\\psi\_0.

a2cos2α2+b2sin2α2=c2sin2ψ02a^2\\cos^2\\frac{\\alpha}{2} + b^2\\sin^2\\frac{\\alpha}{2} = c^2\\sin^2\\frac{\\psi\_0}{2}(6)

The half rectangular triangle ΔB0B1B2\\Delta B\_0 B\_1 B\_2 in Fig.4 shows relations between the angles ψ0,β\\psi\_0, \\beta and γ\\gamma

sin(β+γ)=cosψ02andcos(β+γ)=sinψ02,sin(\\beta+\\gamma) = \\cos\\frac{\\psi\_0}{2}\\quad and \\quad cos(\\beta+\\gamma) = \\sin\\frac{\\psi\_0}{2}\\,,

which can be resolved for cosγ\\cos\\gamma using angle addition identities

cosγ=sinψ02cosβ+cosψ02sinβ.cos\\gamma = \\sin\\frac{\\psi\_0}{2}\\cos\\beta + \\cos\\frac{\\psi\_0}{2}\\sin\\beta\\,.(7)

From triangle ΔA0B1B2\\Delta A\_0 B\_1 B\_2 we can extract the trigonometric relations

s0sinβ=(b−a)sinαand(b−a)cosα+s0cosβ=b+awiths0=2csinψ02,s\_0\\sin\\beta = (b-a)\\sin\\alpha\\quad and\\quad (b-a)\\cos\\alpha + s\_0\\cos\\beta = b+a\\quad with\\quad s\_0 = 2 c\\sin\\frac{\\psi\_0}{2}\\,,

which we can use to eliminate the dependency on β\\beta in expression (7)

2ccosγ=b+a−(b−a)cosα+(b−a)sinαtanψ02.2 c \\cos\\gamma = b + a - (b-a)\\cos\\alpha + (b-a)\\frac{\\sin\\alpha}{\\tan\\frac{\\psi\_0}{2}}\\,. (8)

Finally we apply the cosine law to triangle ΔA0B0B2\\Delta A\_0 B\_0 B\_2

(b+a)2+c2−2(b+a)ccosγ=d2,(b+a)^2 + c^2 - 2(b+a)c\\cos\\gamma = d^2\\,,

from where we can eliminate cosγ\\cos\\gamma using expression (8), yielding another relation between the six design parameters

(b2−a2)sin(ψ02−α)=(d2−c2)sinψ02.(b^2-a^2)\\sin(\\frac{\\psi\_0}{2}-\\alpha) = (d^2-c^2)\\sin\\frac{\\psi\_0}{2}\\,.(9)

When eliminating cc from equation (9) with help of equation (6), we get another relation for the five parameters a,b,d,α,ψ0a,b,d,\\alpha,\\psi\_0 with a nice similar shape to (6).

a2cos2(ψ02−α2)+b2sin2(ψ02−α2)=d2sin2ψ02a^2\\cos^2(\\frac{\\psi\_0}{2}-\\frac{\\alpha}{2}) + b^2\\sin^2(\\frac{\\psi\_0}{2}-\\frac{\\alpha}{2}) = d^2\\sin^2\\frac{\\psi\_0}{2}(10)

With the design of a crank-rocker mechanism we can choose values for four out of six design parameters a,b,c,d,α,ψ0a, 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 (aa is shortest length) as well as the three special cases below !


Case α=ψ0\\alpha = \\psi\_0

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

a2−b2=d2−c2anda=dsinψ02a^2 - b^2 = d^2 - c^2\\quad and\\quad a = d\\sin\\frac{\\psi\_0}{2}(11)

In addition to ψ0\\psi\_0 and α=ψ0\\alpha=\\psi\_0 any combination of two link lengths can be chosen, except [a,d][a,d], since the missing length's bb and cc couldn't be determined then.

Case α=ψ02\\alpha = \\frac{\\psi\_0}{2}

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

c=danda2+b2tan2α2=4c2sinα2c = d\\quad and\\quad a^2 + b^2\\tan^2\\frac{\\alpha}{2} = 4c^2\\sin\\frac{\\alpha}{2}(12)

In addition to ψ0\\psi\_0 and α=ψ02\\alpha=\\frac{\\psi\_0}{2} any combination of two link lengths can be chosen, except [c,d][c,d], as the missing length's aa and bb couldn't be determined then.

Case α=0\\alpha = 0

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

b2−a2=d2−c2anda=csinψ02b^2 - a^2 = d^2 - c^2\\quad and\\quad a = c\\sin\\frac{\\psi\_0}{2}(13)

In addition to ψ0\\psi\_0 and α=0\\alpha=0 any combination of two link lengths can be chosen, except [a,c][a,c], as the missing length's bb and dd couldn't be determined then.

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.