REPORT ON THE STUDY OF THE STEERING GEOMETRY OF A REAR AXLE

1. STEERING GEOMETRY OF THE

REAR AXLE

In this document, the geometric study of the steering of the rear axle is introduced.This study has the purpose of optimising the parameters that infl uence the steering and therefore securing the optimal one. It must accomplish the following requirements:

• It must allow a rotation of 55º of the insi-de wheel (Ψ1 objective=55º)

• It must be suffi ciently away from neutral• No interferences with the brake or with

the wheel rim• It must have a tolerable Ackerman error

However, it is necessary that the steering cylinder allows carrying out the required course and has the suffi cient force to sur-pass the resistant moment of the wheels. In order to accomplish the study, we started from a confi guration shown on an initial la-yout. Therefore, what we have is a rear axle

with the steering cylinder and the connec-ting rods in the rear of the vehicle.

The distribution of the cross pieces, the cylinder of the steering, the angular shift of the wheels and the direction of advance in forward gear of the engine are displayed schematically on the following image.

1.1. Study of the steering mechanismAny steering gear has the aim to assure that the vehicle should do the expected tur-ning with the minor slide possible.

Hydrostatic steering:For vehicles that need high forces in the steering, as for example four-wheel drive ve-hicles, hydrostatic steering is in use, that is to say, there is no mechanical link between steering wheel and wheels.

Thanks to the slight torque needed from the steering wheel, this type of steering makes them simple, safe and comfortable.

Pau Guarro i OliverAeronautical & Mechanical Engineer

Business Unit ManagerEngineering Department

SOLID ENGINYERIAwww.solid-enginyeria.com

It is possible to perceive that, so the steering converges, there is a necessary condition:Ψ1 > Ψ2

wb

Exterior - 2

Interior - 1

1 2

cylinder

dp

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1.5.6. Distances s1 and s2These distances are changeable for every geometric confi guration and are obtained subtracting d to the distance between pins.

s1=a·cos(φ

1)+b·cos(θ

1)

s2=dp-d-s1

2. STUDY OF PREVIOUS CASES

To obtain further information and to be able to validate the results that we obtain, be-fore the study of the cross piece we are involved in, there is accomplished a study of two cases of cross pieces that actually work and are validated.

2.1. Example 1It is a rear guideline and driving cross pie-ce. The cylinder is also in the rear of the vehicle. The parameters necessary for the study are:

Transmission Example 1a (mm) 122α (º) 110b (mm) 261d (mm) 582dp (mm) 1021

With the above data a model is created to simulate the mechanism in all its ope-rating range thus to be able to study the proportions of the rotation angles of the outside wheels with regard to the inside wheels.

Therefore, the model obtained for this case is shown on the following image:

The proportion of the interior angle (de-parture) to the exterior angle (arrival) in radians is shown on the graph below. We obtain:

Ackerman error measures the distance bet-ween the angles Ψ2 and Ψ2theoretical.

An Ackerman error zero would be ideal in any vehicle. Since this is not always possi-ble, it is necessary to reach a compromise. Still, it is necessary to bear in mind that for the type of vehicles for which the rear axle is designed (heavy vehicles); low speeds and uneven areas of work, Ackerman error can go second regarding optimisation prio-rities.

Therefore, it is necessary to defi ne what can be the admissible Ackerman error limit with the awareness of the aforementioned considerations and being able to compare results of machines already in functioning.

1.5. Variables of the Case Study mechanismThe confi guration studied here is the twin crank-connecting rod mechanism which is most used for these applications. It is dis-played on the following image:

1.5.1. Distance between pivots (dp)It is the distance between the joints on which the wheels spin. In this case:dp = 957.2mm

The value is extracted from the tread or dis-tance between slides subtracting the distan-ce of the pivoting point to the engine slide.

This way we obtain:dp=distance between slides– distance slide-pivot * 2 = 1290mm-166mm*2 = 957.2mm

1.5.2. Distance a and angle αThese two parameters are established by the geometry of the hydrostatic engine and they are always constant.For the engine in use, these values are spe-

cifi ed in the following table and obtained from the following fi gure:

a [mm] 90,1α [º] 81,1

1.5.3. Distance bDistance b is the length of the connecting rods that transmit the rectilinear movement of the steering cylinder towards the hydros-tatic engines. This parameter is changeable in the study.

1.5.4. Distance cThe parameter c indicates the distance of the cylinder centre line of the steering with regard to the line that joins both pins. Of all the considered parameters, with a and α defi ned, this one is the most infl uential on Ackerman error and on the necessary force of the cylinder rod.

With the studied confi guration, we obtain:c↑→ cylinder run↓c↑→ neutral is reached soonerc↑→ Ackerman error diminishes

1.5.5. Distance dDistance d is the length between joints of the cylinder rod. It is given by the distance between pins and the distances a and b.

Exte rior

Interior

Typical applications for this type of steering are those that cause inconvenience in con-ventional mechanics when devising com-pact designs between the steering wheel and the wheels. In addition, they are custo-marily more cost-effective.

This type of steering works the following way (see image):

The steering wheel is connected to the steering column and transmits torque to the steering unit.

This acts as a metering device providing the actuator cylinder with the volume of neces-sary oil proportionally to the steering wheel turn.

The steering unit offsets the cylinder auto-matically to neutral position when the tur-ning has fi nished.

During normal operation, the pump sets the pressure of the cylinder. In case of failure of the pump or the circuit, the mechanism becomes manually operated. In this case, torque to the steering wheel will be higher.

1.1.1. Mechanism 1A possibility of mechanism is the one known as Trapezium-Ackerman steering (see image):

This mechanism consists of a central rod connected to the wheels by means of the lateral steering arms.

Convergence condition:In this type of mechanism, to accomplish that the steering is convergent (Ψ1 > Ψ2)

it is necessary that the projection of the la-teral arms intersects at the level of another cross piece of the vehicle (in our case the front crosspiece). This is displayed on the following image:

We can also see on the image that the central rod is only parallel to the union line of the pivoting points of the wheels when Ψ1 = Ψ2 = 0º.

1.1.2. Mechanism 2Another type of mechanism used for stee-ring is the twin crank-connecting rod me-chanism (used in our case), that allows to place a hydraulic cylinder. It is displayed on the following fi gure:

As it is possible to perceive, this mecha-nism unlike others, incorporates interme-diate connecting rods that join the arms of steering columns with the central rod. In addition, unlike other mechanisms too, this one maintains at all time the central rod parallel to the imaginary union line of the pivoting points of both wheels.

Since it is a dif ferent mechanism from Mechanism 1, the convergence condition that must be fulfilled on Mechanism 1 in this case is not necessary to be res-pected.

Notwithstanding, to accomplish the condition of convergence, Ψ1> Ψ2 is also needed.

Exte rior

Interior

1.2. Modelling of the mechanismTo be able to accomplish the study of the mechanism, the data of the elements have been input in a Mechanism Analysis Progra-mme (see bibliography).

Five more solids have been defi ned as well as the mounting support, with the corres-ponding links between them (6 joints and 1 prismatic link), and a steady other one between the mounting support and the ground. An angular actuator has also been modelled.

1.3. Neutral – singular confi guration Mechanisms can achieve certain confi gura-tions called “singular” confi gurations. The-se confi gurations must be avoided, since they cause malfunctions of the mechanism and even cause the break.

Neutral is one of these “singular” confi gura-tions. In the mechanisms we study, it appears when the rod b lines up with the rod a.

Therefore, it will be necessary to design the steering to avoid this situation.

Actually, this study aims to optimise the geometry of the steering granting great importance to its neutral. When we are far from the neutral in the position of maximum rotation the “safety” of the steering will be defi ned.

1.4. Ackerman errorTheoretically, by any rotation angle of the vehicle, perpendicular lines to the steering of both wheels must cut in the same point as perpendicular lines to the steering of the front wheels.

Notwithstanding, due to the imprecision of the mechanism that triggers the rotation, perpendiculars to the back wheel steering do not concur at the same point, causing the skidding of any of the wheels during the turn.

a b

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We will also study the case of c negative:

Therefore, three options are presented:

3.1 CASE C>J (C=100MM)

According to the provided geometry and si-mulating for the cases of working angles, we obtain the following graph of the rotation angle of the outside wheel regarding the in-side wheel:

From the graph, we obtain:• fi Inside wheel max. = 55º = 0.96rad →

Failed.• Exterior angle > interior angle → diver-

gent steering In addition, for an inside wheel angle of 35º we obtain neutral.

It goes into neutral before the maximum ro-tation necessary.

3.2 CASE C<J

3.2.1 C=70MM

With this confi guration, we have:

From the graph, we obtain:• fi Inside wheel max. = 55º = 0.96rad →

Failed.• Exterior angle > interior angle → diver-

gent steering In addition, for an inside wheel angle of de 46º we have neutral.We are in neutral before the maximum rota-tion necessary.

3.2.2 C=40MM

From the graph, we obtain:• fi Inside wheel max. = 55º = 0.96rad →

fi Outside wheel = 0.93rad = 53º• Exterior angle < interior angle → conver-

gent steering → OK• Ackerman error > 10º

Simulating the mechanism in all its working range, we perceive that it does not encoun-ter any singular confi guration.

Therefore, making a graph of the advance of the piston regarding the angle offset by the inside wheel:

From the graph, we obtain:• fi Inside wheel max. = 55º = 0.96rad →

x = 72mm• fi Inside wheel = 1.11rad= 64º → x max

= 81mm

We are 9mm away from neutral.

3.2.3 C=20MM

We obtain the following model:

According to the provided geometry and si-mulating for the cases of working angles, we obtain the following graph of the rotation angle of the outside wheel regarding the in-side wheel:

From the graph, we obtain:• fi Inside wheel max. = 55º = 0.96rad → fi

Outside wheel = 0.76rad = 43.5º• Exterior angle > interior angle →

convergent steering → Ok• Ackerman error < 5º

Graph description (valid for all the graphs on this document):• The red curve indicates the ratio of the

angles.• The black straight line indicates the line

of points in which the angles are equal.• The blue straight line indicates the line of

points in which, for a certain interior an-gle, what exterior angle we should have so Ackerman error would be 0.

Therefore, for the case of Example 1 trans-mission, we have:• Exterior angle < interior angle → OK• Ackerman error < 5º• fi Inside Wheel max. = 45º = 0.78rad → fi

Outside Wheel = 0.58rad = 33º

To fi nish the study of this cross piece, we want to know how far we are from neutral. We will measure this parameter with the distance that has to run the piston from two positions. The initial position is the Ψ1 max and the fi nal position is the Ψ1 neutral (angle where there is the neutral).

Simulating the mechanism in all its working range, we perceive that it does not encoun-ter any singular confi guration.

Notwithstanding, the situation of the piston (x in mm) with regard to the inside angle is studied. We obtain the following graph:

From the graph, the following data are ob-tained:• fi Inside Wheel max. = 45º = 0.78rad → x

max. = 66mm• x =105mm for fi Inside Wheel = 1.75rad

= 100º

For x=105mm we have the following singu-lar confi guration:

Therefore, for the maximum rotation angle we are 39mm away from a singular confi -guration.

2.2. Example 2Rear steering cross piece with the cylinder at the rear of the vehicle.We have the following geometry:

Transmission Example 2a (mm) 100α (º) 90b (mm) 190d (mm) 700dp (mm) 1049

Modelling with bars, we obtain:

The proportion of the inside angle (departu-re) to the outside angle (arrival) in radians is shown on the graph below. We obtain:

From the graph, we obtain:• fi Inside wheel max. = 46º = 0.8rad → fi

Outside wheel = 0.58rad = 33º• Exterior angle < interior angle → OK• Ackerman error < 5º

Simulating the mechanism in full working range, we perceive that it does not encoun-ter any singular confi guration.

Notwithstanding, the situation of the piston (x in mm) with regard to the inside angle is studied. We obtain the following graph:

From the graph, we obtain:• fi Inside wheel max. = 46º = 0.8rad → x

max. = 62mm• x = 88mm for fi Inside wheel = 1.7rad

= 97º

For x = 88mm we have the following singu-lar confi guration:

We are 26mm away from a singular confi -guration.

3. CALCULATION OF THE STEERING

GEOMETRY OF CASE STUDY REAR AXLE

Once analysed two Example cases, the Case Study is now studied.We depart from an initial layout and obtain the following geometry (see image):

According to the geometry provided, we have:

Transmission Case Studya (mm) 90.1α (º) 81.1b (mm) TBDd (mm) 635dp (mm) 957.2wb (mm) 1740

For the Case Study we have:Ψ1 objective = 55ºAccording to Ackerman, to obtain a geome-try without sliding in the wheels, it would be necessary to have:Ψ2theoretical = 38.65ºThe length of the connecting rod (b) will depend on the placement of the steering cylinder (c).Depending on distance c, the cylinder re-mains either beyond the union line of the pi-voting points of the rods a and b or on the in-side, c>j or c<j (see image); with j = 89mm.

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Now, we calculate the necessary pressure.The steering cylinder will have to overcome the resistant moment (Mr) the wheel offers on rotating. This one is calculated for every wheel, bearing in mind that the maximum load that falls down on the axle must be even on both wheels. Condition of spin:In order that the steering is measured co-rrectly, it is required to determine the ne-cessary section of the cylinder so that the force that it generates will overcome the resistant moment of the wheels.

Obtaining of resistant moment:Depending on the criterion in use, some results or others are obtained. Therefore, different criteria will be used for obtaining the resistant moment (Mr) and the most restrictive will be chosen:

Criterion 1 – SOLID EnginyeriaThe force that opposes to the spin of every wheel is:

Where η is the coeffi cient of friction of the wheel with the ground.The resistant moment to the spin is obtai-ned from the following for every wheel:

Calculating, we obtain:

Criterion 2 – Ognibene

Criterion 3 – Danfoss

Mr = 0.05*C*B*f/(0.7*200*(1+E/B) daNmWith E and B in mm.C in daN With:

Calculating, we obtain:

Calculation of cylinder pressure:To determine the necessary pressure in the cylinder, we will apply the most unfavoura-ble case of the three criterions. This way, the cylinder must generate torque to sur-pass Mr=608.2Nmm in every wheel (see criterion 1). Decomposing the cylinder force for every rod, we obtain:

Therefore, we can see that the condition to accomplish is:Fm > Mr / aWithFm=Fb*cosγFb=Fcyl*cosβ→Fcyl=Fm/(cosβ*cosγ)

Calculating the force for two cases (straight and curve) we obtain:

Conclusion:It is perceived that the most critical case is the one with maximum wheel spin. In this case, a pressure of 140bar is necessary in the cylinder.

6. BIBLIOGRAPHY

PAM – ETSEIB – UPC. Mechanical Enginee-ring Department. Daniel Clos and Joan PuigSynthesis of Mechanisms. C.Riba – F.Civit. (TEM – UPC 1993)

2

1

dp

s2 d s1

2{+}

1{+}

2{-}

2{+

} 1

{-}

1{+

}

{-} {+}

Fc1

Fc2

FResis t

FResis t

r

a b

Mresist Mresist

c

Rod diameter [mm] 30.0Cylinder diameter [mm] 55.0Useful section [mm2] 1,669.0c 20.0r (mm) 166.4Max. machine load (kg) 3,450.0Rear axle distribution 0.27Rear axle load (N) 9,138.0Wheel load (N) 4,569.0µ 0.8F resistant (N) 3,655.2Resistant moment (Nmm) 608,226.3

Ognibene DanfossAxle load (daN) 913.8 913.8mu 0.15 0.8E (m) / (mm) 0.1664 166.4B (m) / (mm) 0.24 240E/B 0.69 N/AMr (daNm) 25.6 37.0Mr (Nmm) 256,027.8 370,042.7Mr*2 (motive - Nmm) 512,055.6 N/A

Data Straight ahead

55º Turn 43.5º Turn

a (mm) 90.1

β (º) 25.56 14.66 12.66

γ (º) 16.77 61.12 39.78

Fm (N) 6,750.6

Fcyl (N) 15,630.5 14.447,3 9.002,9

P necessary (Mpa)

9.4 8.7 5.4

P necessary (bar)

93.7 86.6 53.9 140.5

Simulating the mechanism in full working range, we perceive that it does not encoun-ter any singular confi guration.

Therefore, making a graph of the advance of the piston regarding the angle offset by the inside wheel:

From the graph, we obtain:• fi Inside wheel max. = 55º = 0.96rad →

x = 69mm• fi Inside wheel = 1.8rad =103º → x max

= 91mm We are 22mm away from neutral.

3.3 CASE C<0

3.3.1 C=-10MM

Model:Rotation angle of the inside wheel regar-ding the outside wheel:

From the graph, we obtain:

• fi Inside wheel max. = 55º = 0.96rad → fi Outside wheel = 0.61rad = 35º

• Exterior angle < interior angle → conver-gent steering → OK

• Ackerman error < 5º

Simulating the mechanism in all its working range, we perceive that it does not encoun-ter any singular confi guration.

Advance of the piston regarding the angle offset by the inside wheel:

From the graph, we obtain:• fi Inside wheel max. = 55º = 0.96rad →

x = 63mm• fi Inside wheel = 1.39rad= 80º → x max.

= 70mm We are 7mm away from a singular confi gu-ration.

4. SUMMARY TABLE AND CONCLUSIONSThe results of the different studied cases are presented in the table below.

In the table, we can perceive the most im-portant parameters of the study and com-pare some with others.

The options c=100 and c=70 are discarded because they are divergent (the exterior wheel spins more than the interior wheel).

Comparing the rest of the options, c=40, c=20 and c =-10, we can say the following observations:• The three options are convergent• None of the thre option have any singular

point (neutral)• c=20 has a better proportion to neutral.

It remains further from this position in the mechanism limit (22mm).

• c=-10 has the best Ackerman error (3.7º). However, c=20 is only 1º more (4.8º).

In Example 1 and Example 2 we are some 39mm and 26mm away respectively from neutral.

With regard to the Ackerman error in the maximum angle, with the chosen option it is around 5º, and both with Example 1 and Example 2 about 1º. However, we must rei-terate that having low Ackerman error is not critical in this type of applications due to the low speeds and the terrain to be used.Therefore, the conclusion of the study is that c=20 is chosen as the ideal option.

5. CYLINDER CALCULATIONIn order to chose the correct cylinder for the axle, we need to know two basic parameters:• Course• PressureAs seen on the study, we have selected the c=20 option. This implicates that the ne-cessary course of the cylinder is 138mm.

Case Study

Example 1 Example 2 c=100 c=70 c=40 c=20 c=-10

Ψ2 (º) 33 33 57 66 53 43.5 35

Ψ1 max (º) 45 46 35 46 55 55 55

Ψ1 required (º) 45 46 55 55 55 55 55

Converge? OK OK KO KO OK OK OK

a (mm) 122 100 90.1 90.1 90.1 90.1 90.1

α (º) 110 90 81.1 81.1 81.1 81.1 81.1

d (mm) 582 700 635 635 635 635 635

dp (mm) 1021 1049 957.2 957.2 957.2 957.2 957.2

neutral (mm) 39 26 <0 <0 9 22 7

wb (mm) 1640 1640 1740 1740 1740 1740 1740

Ψ2theoretical (º) 31.6 31.9 38.7 38.7 38.7 38.7 38.7

Ψ2theoretical (rad) 0.6 0.6 0.7 0.7 0.7 0.7 0.7

Ackerman error (º) 1.4 1.1 18.3 27.3 14.3 4.8 3.7

Cylinder course (mm) 132 124 N/A N/A 144 138 126

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