Arbitrary vehicle steering characteristics with changing ratio rack and pinion transmission

Abstract

In order to achieve arbitrary steering characteristics at vehicles, a steering mechanism was developed, in which changing ratio rack and pinion connections have been applied. In contrary to a regular steering mechanism where only a single rack is used, two racks were applied in order to make arbitrary characteristics possible. The turning wheel’s required motion functions had to be defined first, thereafter could we determine the changing ratio rack and pinion geometry which produces this motion. First, a simplified two-dimensional mockup was created in order to study the difficulties and possibilities of a real construction. Later, a fully functional assembly was designed and manufactured to make further experiments possible.

Introduction

This article describes the major steps of our research in advanced steering mechanisms. The problem what we primarily wanted to solve was the possibility of creating arbitrary steering characteristics, applying changing ratio rack and pinion connection. This article demonstrates a method which can be used at such problems. First, a two-dimensional (2D) model was created in order to prove the correctness of our theories and study the construction possibilities of an actual design. Later, a fully functional prototype was calculated, designed, and manufactured, that will be mounted into a Formula Student ¹ vehicle for further testing. The feedbacks of the tests will allow us to go further in our research. It is important to mention that our goal was not to define the ideal steering characteristic, but to make any characteristic implementable easily.^2,3

Describing the problem

At steering, the front wheels have a special movement. In an idealized case, where the vehicle’s weight and the tire deformation are negligible, all axes of wheels should determine a section point that defines the vertical cornering axis. This is achieved by the Ackermann geometry.^3,4 The conventional rack and pinion mechanisms that are often used at this problem always have an error, which correlates with the magnitude of steering. In practice, one of the wheels is turned a bit more or less compared to the ideal Ackermann geometry that results a different driving stability. This is called pro-Ackermann or anti-Ackermann.^2,3

The major problem with conventional steering mechanisms is that a previously defined characteristic is in most cases not flawlessly implementable. The conventional steering systems consist of several four bar mechanisms. The movements of such mechanisms can be optimized by rearranging the geometry, but does not have an exact solution for most problems. Cam-follower mechanisms or changing ratio gears however make exact solution possible for most cases. Researches at Budapest University of Technology and Economics (BME), Department of Machine and Industrial Product Design (GT3) enable the application of gears or rack–pinion connections with changing ratio. These rarely applied machine parts make possible to create a steering mechanism with arbitrary characteristics. Not like the conventional solutions, where just one rack is applied, we use separate racks at each wheel.^4–6 These racks are connected to two non-circular gears that are fixed on the steering wheel’s axis. These separate changing ratio connections enable arbitrary positioning of each wheel at a specific steering wheel angle, so arbitrary steering characteristic can be implemented.

After a special calculation method was developed, we created a simple 2D model of a steering mechanism. For experimenting purposes later a fully functional model was constructed for the electric racing vehicle of BME Motion team at Formula Student. ¹ The prototype was manufactured by Dendrit Kft., ⁷ where several special manufacturing techniques were applied. The steering mechanism prototype implements a theoretically error-free Ackermann for a given suspension geometry. In this experimental project, we did not take the tire deformation and slip angle ³ into account, because the weight of the car is relatively small (~250 kg). A bigger lateral force on the vehicle causes slipping at the tires at small slip angles. ³

2D calculations

In order to prove the correctness of our theory and to test the calculation method, we worked out a simple 2D problem (Figure 1). The problem is not realistic and does not consider the dynamic effects at a real working situation. In order to define the (rolling) pitch curves of the rack and pinion, the S(φ) function of rack position is needed. ⁸ In s(φ), φ represents the angular position of the steering wheel. At changing ratio gears, the axis distance is also needed for defining the geometry. That is necessary because the function of the transmission is dimensionless and so the sizes are not defined (just the ratio). Although in this case, the function of transmission (determined by s(φ)) has already length dimension, so it defines not just the ratio but the size of the geometry as well. The problem of rack and pinion connections can be considered as a pair of non-circular gears with infinite axis distance.

Figure 1.

2D sketch of the models kinematic.

For the definition of s(φ), the following three steps have to be carried out:

1. Definition of R(φ) function of cornering radius where the desired steering characteristic is to be considered.

2. Definition of Ψ(φ) function of wheel position according to the geometry.

3. Definition of s(φ) function of rack position that is required for the rack and pinion pitch curve geometry.

Definition of R(φ) function of cornering radius

At most cases of the classical hinge mechanisms which are applied at specific problems, there is no exact solution. These problems are often solved by iteration and optimization, where the initial geometry of the topology is modified.

At gears or rack and pinion connections where changing ratio is applied or even at cam drive mechanisms, the exact geometry is determined by the definition of the desired motion. This is important at this specific problem, because unlike the conventional steering mechanisms, the steering characteristic (so the R(φ) function of cornering radius) has to be defined at the first place. At a classical mechanism, the characteristic is the result of the geometry, while here it is the opposite.

In order to make this first calculation more simple, we recreated the steering characteristic of tricycle with vertical steering axes in the front. The old fashioned horse carriages also had the same steering principle. The angle position of the virtual middle wheel is γ, the axis distance is a and the cornering radius is R (Figure 2(a)).

Figure 2.

(a) Sketch of the virtual wheel and (b) its cornering radius function.

The function of cornering radius can be defined by equation (1)

$$R(\gamma )=\frac{a}{\tan (\gamma )}$$

(1)

In order to express the R(φ) function, we need to determine the correlation between the φ angle position of the steering wheel and γ angle position of the virtual middle wheel. For this, we need to define an R_min minimal cornering radius value and a φ_range angle range of the steering wheel.

γ virtual wheel position will vary between [−γ₀, γ₀], where γ₀ has the value defined by equation (2)

$${\gamma }_0=\mathrm{atan}\left(\frac{a}{R_{\min }}\right)$$

(2)

Because the steering wheel turns in angle range φ $[(-{\phi }_{\mathrm{range}}/2),({\phi }_{\mathrm{range}}/2)]$ , the position of virtual wheel can be described by equation (3)

$$\begin{array}{c}\multicolumn{1}{c}{\gamma \left(\phi \right)=2\frac{{\gamma }_0}{{\phi }_{\mathrm{range}}}·\phi }\\ \multicolumn{1}{c}{=2\frac{\mathrm{atan}\left(\frac{a}{R_{\min }}\right)}{{\phi }_{\mathrm{range}}}·\phi }\end{array}$$

Therefore, the R(φ) function of cornering radius can be defined by equation (4) (Figure 2(b))

$$\begin{array}{c}\multicolumn{1}{c}{R(\phi )=\frac{a}{\tan (\gamma (\phi ))}}\\ \multicolumn{1}{c}{=\frac{a}{\tan (2\frac{\mathrm{atan}(\mathrm{a}/R_{\min })}{{\phi }_{\mathrm{range}}}·\phi )}}\end{array}$$

$$s_0=\frac{b}{2}-(r·\sin (\alpha -\frac{\pi }{2})+\sqrt{l^2-{(c-r·\cos (\alpha -\frac{\pi }{2}))}^2})$$

(5)

$$s(\phi )=\frac{b}{2}-\left(r\cdot \sin \left(\alpha -\psi (\phi )-\frac{\pi }{2}\right)+\sqrt{l^2-{\left(c-\cos \left(-\psi (\phi )-\frac{\pi }{2}\right)\right)}^2}+s_0\right)$$

(6)

Definition of Ψ(φ) function of wheel position

The expression Ψ(φ) defines the correlation between the angle positions of the “real” front wheel and the steering wheel. For defining this function, a sketch of the deflected geometry is needed. This can be seen in Figure 3. Because R(φ) is known, the function Ψ(φ)can be defined using equation (5)

$$\psi (R(\phi ))=\mathrm{atan}\left(\frac{a}{R(\phi )-(b/2)}\right)$$

(7)

Figure 3.

Sketch used for definition of the cornering radius function.

Definition of s(φ) function of rack position

The function s(φ) defines the correlation between the linear position of the rack and the angle position of the steering wheel (or the pinion). This function defines the pitch curves of the changing ratio rack and pinion. ⁸

As the first step, the distance between the vehicles symmetry plane and the racks hinge axes (s₀) is defined (equation (6)). For this, we used the sketch in Figure 4(a) and (b). As the next step, the motion function of the rack is determined using equation (7).

Figure 4.

(a) Initial position of the mechanism and (b) deflected position of the mechanism.

Defining the exact geometry of rack and pinion pitch curves

The method used for rack and pinion pitch curves geometrical calculations is similar to the one, which is used at a pair of non-circular gears.^8,9 The rack can be considered a cogwheel that has infinite big pitch diameter and thus zero angular velocity (equations (8) and (9))

$$v=r·\omega$$

(8)

$$r=\frac{v}{\omega }=\frac{v}{0}\to \infty$$

(9)

In equations (8) and (9), the angular velocity of the pinion is ω, the circumferential velocity is v and the radius of pitch curve is r.

One of the angular velocities will be 0, so it is pointless to use the conventional i = r₂/r₁ formula for function of transmission because it always will result infinite. We defined a $\hat{i}$ (equation (10)) function that determines the correlation between the instantaneous angular velocity of the pinion and the linear velocity of the rack

$$\hat{i}=\frac{v}{\omega }=\frac{r(\phi )·\omega }{\omega }=r(\phi )$$

(10)

So if we know the angular position of the pinion and the linear position of the rack, the geometry can be determined by equations (11)–(13)

$$\hat{i}=\frac{v}{\omega }=\frac{(\partial s/\partial t)}{(\partial \phi /\partial t)}=\frac{\partial s}{\partial \phi }=r(\phi )$$

(11)

$$f_1=\left[\begin{array}{c}\hfill X_{p1}\hfill \\ \multicolumn{1}{c}{Y_{p1}}\end{array}\right]=\left[\begin{array}{c}\hfill r(\phi )\cos (\phi )\hfill \\ \multicolumn{1}{c}{r(\phi )\sin (\phi )}\end{array}\right]$$

(12)

$$f_2=\left[\begin{array}{c}\hfill X_{p2}\hfill \\ \multicolumn{1}{c}{Y_{p2}}\end{array}\right]=\left[\begin{array}{c}\hfill s(\phi )\hfill \\ \multicolumn{1}{c}{-r(\phi )}\end{array}\right]$$

(13)

After generating the geometry of pitch curves, the entire rack and pinion geometry can be determined, using the geometrical parameters of the tooth profile (Figure 5(b)). These calculations are detailed in the following sections.

Figure 5.

(a) Steps of the rolling of the tool-gear on the rolling curve and (b) the resultant set of lines.

Rolling a cylindrical tool-gear on the pitch curve

In order to generate the proper gear geometry, we simulate the actual cutting process. Cylindrical tool-gear geometry can be used to generate convex and concave gears as well. The geometry of the virtual cutting tool is similar to a conventional cogwheel. This geometry can be easily defined analytically.^8,9 The radius of the tool gears pitch circle cannot be larger than the smallest concave radius of the f₁ and f₂ pitch curves. To simulate the cutting process of real gear manufacturing, we need to carry out coordinate system transformations. The transformation of the ξ-η system, containing the tool gears geometry, consists of two main steps (equation (17)) (Figure 5(a)):

• Rotation with (φ = α+φ) around the origo of ξ-η, where α (equation (15)) is the angle of the tangent in the current point of pitch curve and φ(p) = s(p)/r is the angle of the rolling rotation of the tool-gear.

• s(p) is the length of the pitch curve (equation (14)).

• p is an independent parameter.

• Offset of the center of ξ-η into the Pm point (equation (16))

$$s(p)=\int \sqrt{{\left(\frac{\partial X_p(p)}{\partial p}\right)}^2+{\left(\frac{\partial Y_p(p)}{\partial p}\right)}^2}\mathrm{dp}$$

(14)

$$\alpha (p)=\arctan \frac{\partial Y_p(p)(\partial p)}{\partial X_p(p)(\partial p)}$$

(15)

$$P_m=\left[\begin{array}{c}\hfill X_p(p)+(r+x_p)·\sin (\alpha (p))\hfill \\ \multicolumn{1}{c}{Y_p(p)+(r+x_p)·\cos (\alpha (p))}\end{array}\right]$$

(16)

$$\begin{array}{l}E=T_{mxy}\left(T_{\alpha \phi }{f}_q(q)\right)=\left[\begin{array}{ccc}1& 0& m{X}_p(p)\\ 0& 1& m{Y}_p(p)\\ 0& 0& 1\end{array}\right]\\ \left(\left[\begin{array}{ccc}\cos (\alpha (p)+\varphi (p))& \sin (\alpha (p)+\varphi (p))& 0\\ -\sin (\alpha (p)+\varphi (p))& \cos (\alpha (p)+\varphi (p))& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\xi }_q(q)\\ {\eta }_q\left(q\right)\\ 1\end{array}\right]\right)\end{array}$$

(17)

Using this method, we can place standard tooth profile even on concave pitch curves. As result, we get a set of lines. The inner envelope of this geometry will define the gears (Figure 5(b)).

Defining the envelope

Based on the previous sections, a set of lines can be defined for a general concave changing ratio gear. To get the final geometry, we need to find the inside envelope of these lines.^10,11

For the definition of the envelope, we worked out a method that is suitable for solving this kind of special problem. This method consists of numerical calculations that results in a many sided polygon that fits in the gear’s envelope. The resolution is arbitrary. The method is based on calculating the section point of two arbitrary line segments on a plane.

First, we need to find the starting point of the envelope which will be the first point of the final geometry. Then, we draw a loop around this point from the inside to the outside of the envelope. The first intersection between the loop and the line set will become the next point of the final geometry. In order to simplify the calculation, the loop consists of linear segments. This process continues until we reach the endpoint of the envelope.

Intersection of two arbitrary line segments

AB and CD line segments are given with their A, B, C, and D points. The equation of these lines can be given by the formula (equations (18) and (19)), where p₁, and p₂ are independent parameters, and their values change from 0 to 1

$$\mathrm{AB}=\left[\begin{array}{c}\hfill A_x\hfill \\ \multicolumn{1}{c}{A_y}\end{array}\right]+p_1(\left[\begin{array}{c}\hfill B_x\hfill \\ \multicolumn{1}{c}{B_y}\end{array}\right]-\left[\begin{array}{c}\hfill A_x\hfill \\ \multicolumn{1}{c}{A_y}\end{array}\right])$$

(18)

$$\mathrm{CD}=\left[\begin{array}{c}\hfill C_x\hfill \\ \multicolumn{1}{c}{C_y}\end{array}\right]+p_2(\left[\begin{array}{c}\hfill D_x\hfill \\ \multicolumn{1}{c}{D_y}\end{array}\right]-\left[\begin{array}{c}\hfill C_x\hfill \\ \multicolumn{1}{c}{C_y}\end{array}\right])$$

(19)

If we make these two equations equal, we get an equation system. Using this, we can determine the p₁ and p₂ parameters (equations (20) and (21)). If we put these values into the original equations, we get the intersection point

$$p_1=-\frac{A_y-C_y+((D_y-C_y)/(D_x-C_x))·(C_x-A_x)}{B_y-A_y-((D_y-C_y)/(D_x-C_x))·(B_x-A_x)}$$

(20)

$$p_2=\frac{A_y-C_y+((B_y-A_y)/(B_x-A_x))·(C_x-A_x)}{D_y-C_y-((B_y-A_y)/(B_x-A_x))·(D_x-C_x)}$$

(21)

By evaluating these equations, we can find out whether the intersection falls between the AB and CD points (0 ≤ p₁ ≤ 1 and 0 ≤ p₂ ≤ 1). If any of the lines is vertical (so Ax = Bx or Cx = Dx), then we get a 0/0 type equation. In order to handle this case, we need to use modified equations.

If AB is vertical and CD is not, the equation of p₂ changes (equations (22)–(24))

$$\mathrm{AB}=\left[\begin{array}{c}\hfill A_x\hfill \\ \multicolumn{1}{c}{A_y}\end{array}\right]+p_1(\left[\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{B_y}\end{array}\right]-\left[\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{A_y}\end{array}\right])$$

(22)

$$\mathrm{CD}=\left[\begin{array}{c}\hfill C_x\hfill \\ \multicolumn{1}{c}{C_y}\end{array}\right]+p_2(\left[\begin{array}{c}\hfill D_x\hfill \\ \multicolumn{1}{c}{D_y}\end{array}\right]-\left[\begin{array}{c}\hfill C_x\hfill \\ \multicolumn{1}{c}{C_y}\end{array}\right])$$

(23)

$$p_2=\frac{A_x-C_x}{D_x-C_x}$$

(24)

If CD is vertical and AB is not, the equation of p₁ changes (equations (25)–(27)).

$$\mathrm{CD}=\left[\begin{array}{c}\hfill C_x\hfill \\ \multicolumn{1}{c}{C_y}\end{array}\right]+p_2(\left[\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{D_y}\end{array}\right]-\left[\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{C_y}\end{array}\right])$$

(25)

$$\mathrm{AB}=\left[\begin{array}{c}\hfill A_x\hfill \\ \multicolumn{1}{c}{A_y}\end{array}\right]+p_1(\left[\begin{array}{c}\hfill B_x\hfill \\ \multicolumn{1}{c}{B_y}\end{array}\right]-\left[\begin{array}{c}\hfill A_x\hfill \\ \multicolumn{1}{c}{A_y}\end{array}\right])$$

(26)

$$p_1=\frac{C_x-D_x}{B_x-D_x}$$

(27)

The first point of the envelope can be found by drawing an AB line perpendicular to the start segment of the pitch curve. In the next step, we check for intersections between this AB line and the set of lines. If we start the line from the “inside” of the gear, the smallest p₁ parameter will define the first point of the envelope.

Further points of the envelope are determined using the above-mentioned loop of straight lines around the current point. We defined a clockwise searching direction for the loop, so the right side of the envelope will be the inside and the left will be the outside area. Around the current point, the loop is drawn counterclockwise, line by line. By drawing each loop-line, we check for intersections with the set of lines. If any found, we add the first (lowest p₁ grade) intersecting point to the final geometry. Then, we repeat the process until the endpoint is reached (Figure 6(a)).

Figure 6.

(a) Draft of the envelope calculation, (b) problems of the method, and (c, d) the geometry of the searching lines.

The main disadvantage of this procedure is that in some cases the searching loop will not find the next intersection. It happens if the envelope line has a turn of near 180°. Typical error cases are demonstrated in Figure 6(b). It is important to note that in practice these are rare scenarios in standard gear geometries. Other disadvantage of this method is that it is not able to detect sharp edges. These edges will be chamfered (Figure 6(b)).

The correctness and quality of the resulting envelope highly depend on the shape and size of the searching loop. Our goal was to define a searching loop that can reliably find the next intersection on the envelope and has less effect on the edges. It is beneficial to build up this loop with as few sections as possible as it reduces the computation time. After testing numerous various forms, we have finally chosen a brick-shape made of three sections (Figure 6(c)). With this shape, the next point will be closer if the envelope has a greater angular turn. This increases the chamfering effect at the sharper edges, while producing fewer points when the angle is obtuse.

The f value (Figure 6(c)) is the resolution parameter which can be set according to the actual problem. The g and h parameters define the rate of the elongation and back-shifting.

Practice showed that setting g to 0.2 and h to 0.1 gives the best results. The f resolution is to be set based on the manufacturing procedure and the expected precision. To calculate the points of the searching loop, we used the sketch showed in Figure 6(d).

α is the angle of the last envelope segment. It can be determined by the following expression

$$\alpha =a\tan \left(\frac{P_y(n)-P_y(n-1)}{P_x(n)-P_x(n-1)}\right)$$

where

$$\mathit{P}(n)=\left[\begin{array}{c}\hfill P_X(n)\hfill \\ \multicolumn{1}{c}{P_Y(n)}\end{array}\right]$$

describes the coordinates of the last envelope point. In order to make the calculations faster, we determined the individual coordinates of the searching loops I , J , K and L points (equations (28)–(31)) instead of applying a rotation matrix

$$\mathit{I}=\left[\begin{array}{c}\hfill P_X(n)-g·f·\sin (\alpha )-h·f·\cos (\alpha )\hfill \\ \multicolumn{1}{c}{P_Y(n)-g·f·\cos (\alpha )+h·f·\sin (\alpha )}\end{array}\right]$$

(28)

$$\mathit{J}=\left[\begin{array}{c}\hfill P_X(n)-g·f·\sin (\alpha )+f·\cos (\alpha )\hfill \\ \multicolumn{1}{c}{P_Y(n)-g·f·\cos (\alpha )-f·\sin (\alpha )}\end{array}\right]$$

(29)

$$\mathit{K}=\left[\begin{array}{c}\hfill P_X(n)+g·f·\sin (\alpha )+f·\cos (\alpha )\hfill \\ \multicolumn{1}{c}{P_Y(n)+g·f·\cos (\alpha )-f·\sin (\alpha )}\end{array}\right]$$

(30)

$$\mathit{L}=\left[\begin{array}{c}\hfill P_X(n)+g·f·\sin (\alpha )-h·f·\cos (\alpha )\hfill \\ \multicolumn{1}{c}{P_Y(n)+g·f·\cos (\alpha )+h·f·\sin (\alpha )}\end{array}\right]$$

(31)

Knowing the coordinates, the intersections can be calculated using the previously described method. To speed up the calculations, we look for the next envelope point, only in a smaller region around the previous one. This region is defined by a circle, with a specific R radius. This radius should be as low as possible, but high enough to contain at least one endpoint of each intersecting line. As there are much less points inside this region, also less calculation and processing are required. In the applied condition (expression (32)), R is the radius of the range

$$\mathit{V}=\left[\begin{array}{c}\hfill V_X(i)\hfill \\ \multicolumn{1}{c}{V_Y(i)}\end{array}\right]$$

is the indexed coordinate of the set of lines and

$$\mathit{P}(n)=\left[\begin{array}{c}\hfill P_X(n)\hfill \\ \multicolumn{1}{c}{P_Y(n)}\end{array}\right]$$

contains the coordinates of the envelopes last point

$$\begin{array}{c}{R}^2>{(V_X(i)-P_X(n))}^2\hfill \\ +{(V_Y(i)-P_Y(n))}^2\hfill \end{array}$$

(32)

Only those i indexed points will be used in the calculation that satisfies the described condition (expression (32)). These above described calculations have to be repeated in a cycle, until the finishing condition is satisfied. This condition is defined by the endpoint, which is an individually calculated point on the envelope at the end of the pitch curve. This point is on the normal, sets on the endpoint of the pitch curve. If the currently calculated point of the envelope is inside a defined small range around the endpoint, the finishing condition is satisfied, the cycle stops and the endpoint is added to the envelope polygon.

First test prototype

In order to examine the details of operation and prove the correctness of the principles, a plastic model was produced. We used the small computer numerical control (CNC) laser cutting machine of the department for this task. The applied material was a 3-mm-thick polycarbonate sheet. This mockup does not contain the wheels, just the mechanism (Figure 10(a)). Before manufacturing the model, we used the Creo computer-aided design (CAD) system to verify the principle that in every steering wheel position, all four axes have just one section.

Three-dimensional (3D) problem

After the 2D calculations and mockup were successfully completed, we decided to create a fully operational version as well. For request of the BME Motion team of Formula Student ¹ competition, a steering mechanism prototype was constructed that has the same principle changing ratio rack and pinion connection. For experimental purposes, the mechanism implements error-free Ackermann for the given suspension geometry. In this experimental project, we did not take the tire deformation and slip angle into consideration, because the weight of the car is relatively small (∼250 kg). A bigger lateral force on the vehicle causes slipping of the tires at small slip angles. ³

Buildup of the kinematic

At creating the geometry of this real mechanism, we needed to go through the same steps described in the previous sections. At this case, the biggest issue was the definition of the s(φ) function of rack position, because unlike the previously interpreted 2D problem, we have a 3D ball joint mechanism (Figure 7). The rotational steering axes of the wheels are not vertical, but they have a general orientation. The axes of the steered wheels and the rotational steering axes do not cross each other. These factors made our calculations more difficult.

Figure 7.

Phantom parts generating the projected axes on the ground level.

Problem of the non-horizontal axes of steered wheels

The two axes of the vehicles rear wheels are parallel to the ground so they are horizontal (in normal operation and forward drive). Because the two front wheels turning axes (defined by the upper and lower ball joints of the upright) are not vertical, the axes will not have a horizontal position by cornering, only in forward drive and normal operation. ³ It means that the axes will never have sections at cornering because their arrangement is skew. At our calculations, we considered that the cornering center of the vehicle will be at the intersection of projected axes lines on the ground.

These kinematic calculations were more difficult to handle analytically, so we decided to apply a numerical analysis. For this, we used the integrated mechanism analysis module of the PTC CREO CAD system that was also used for the 3D design and implementation of the vehicle. We used the CAD model of steering kinematic created by the BME Motion team. Later, we created and added some phantom elements and parts to the assembly in order to model the projected axes’ lines on the ground plane (Figure 7).

As shown in Figure 7, a phantom line part was added to the model which’s endpoint has a constant b/2 distance from the vehicles symmetry plane. This part connects to another one that consists of two vertical lines. Both vertical lines are connected to the horizontal line part on the ground and to the wheel axis. The ψ axis position of this part and the ax distance of rear axis (Figure 7) change during steering. Using these values, the magnitude of the cornering radius can be determined.

Definition of R(φ) function of cornering radius

For defining this R(φ) function, we use the same method as in the 2D problems. For this, the minimal cornering radius R_min and the angle range of steering wheel φ_range have to be defined.

This function can be arbitrarily defined if the following rules are taken into account:

• In order to make the axes intersect each other in one point, the R(φ) function must have a rotational symmetry. If we rotate it around the origo by 180°, it must remain the same (equation (33))

  $$R(\phi )=-R(-\phi )$$

  (33)

• If φ → 0 from the negative side, the function must give R → −∞;

• If φ → 0 from the positive side, the function must give R → −∞;

• On the negative side of φ, the function must be strictly monotone decreasing. Because of the first rule, it means that also on the positive side it will be strictly monotone decreasing.

The used function is equation (34)

$$R(\phi )=\frac{a}{\tan (2\frac{\mathrm{atan}(\mathrm{a}/R_{\min })}{{\phi }_{\mathrm{range}}}·\phi )}$$

(34)

It is important to mention that if we make a similar construction as the 2D mockup, the φ_range angle range cannot exceed 360° because the applied gears are not continuous. In practice however, this value should not be bigger than 270° because of the manufacturing methods, geometry, and other construction factors. Some methods exist that allow the extension of this angle range but it is not detailed in this article.

Definition of s(φ) function of rack position

In order to define the s(φ) function of rack position, we need the R(φ) function of cornering radius and another function R(s) which defines the correlation between R cornering radius and s rack position. This correlation is quite difficult, so as mentioned before, we used a numeric method instead of the analytic solution.

According to the kinematic sketch seen in Figure 7, the magnitude of R cornering radius can be expressed by equation (35)

$$R(s)=a_x(s)·\tan (\psi (s))+\frac{b}{2}$$

(35)

The values for a_x(s) and Ψ(s) are generated by the applied CAD system and exported in discrete table format. A motion analysis was created in which the wheel was moved between the two deflected end positions. The geometrical data were acquired during this motion.

The function s(φ) is determined by interpolation using the functions R(φ) and R(s). First, we went through all points of the discretely defined R(φ) function and determined the correspondent R(s) value. The s(φ) function was defined by coupling the correspondent φ and s values (Figure 8(a)). In order to enhance the precision, we used spline interpolation method, which is a built in feature of the applied MATLAB software.

Figure 8.

(a) Determining the s(φ) function and (b) correction of the s(φ) function.

The applied interpolation method became imprecise at φ → 0 location, where the magnitude of R was near to infinite. This has arithmetical reasons. For solving this problem, we sampled the s(φ) function at many point but the problematic φ → 0 area was avoided. Later, a spline curve was laid on these points and that provided us a continuous function (Figure 8(b)).

First actual rack and pinion geometry

After implementing the calculations, the first rack and pinion geometry was created. This enabled us to consider what kind of construction solutions are to be applied. We recognized that the magnitude of ratio alternation is quite big, which is caused by the arbitrarily chosen R(φ) function of cornering radius. This is unfavorable for the size, weight, and also for the loads of the construction. Our goal was to enhance the kinematic in order to reduce the size and weight of these moving parts.

Enhanced kinematic

A method has been developed which enables the rack to be much smaller. One side of the rack would be almost linear. The other side would be responsible for correcting the error of the linear geometry on the other, oppositely placed rack. For this, we created a special enhanced R_e(φ) function of cornering radius which is the closest to the original (conventional) steering characteristic, but satisfies the four rules listed in section “Definition of R(φ) function of cornering radius.” For the original R_o(φ) function of cornering radius, a conventional single rack steering mechanism was applied.

For determination of the enhanced R_e(φ) function, we made a kinematic analysis in the applied CAD system, in order to get the original (conventional, single rack) R_o(φ) function of cornering radius. This conventional function does not satisfy the condition that the function must have a rotational symmetry. The enhanced R_e(φ) function was created from the originally measured R_o(φ) function, by mirroring the [−∞, 0[ range (36). After performing the same calculations, we have got a better geometry (Figure 9)

$$R_e(\phi \in [-\infty ,0[)=R_o(\phi )$$

and

$$R_e(\phi \in ]0,\infty ])=-R_o(-\phi )$$

(36)

Figure 9.

(a) Original and (b) enhanced geometry.

Mechanical analysis of the teeth

The mechanical analysis of the changing ratio rack and pinion transmission occurs rarely in the engineering practice, so applicable analytical formulas are hardly available. There are two basic methods used at similar problems. At the first method, the most critical place of the geometry is determined using different load cases. By considering the local geometric properties of the teeth at the critical point, a conventional cylindrical substitute gear is calculated. The conventional mechanical inspection can be carried out on this gear using the conventional formulas of tooth break and Hertz stress.^12,13

The second method that we also applied is the finite element analysis.^14,15 We used Ansys Workbench at this problem. After defining the loads, boundary conditions and the proper contact constraints between the surfaces, several analyzes were carried out in order to locate the critical places. Then, we made the mesh denser at these places in order to increase the precision. Both results were compared and verified. The applied materials were chosen by considering these results.

In contrary to the conventional steering mechanisms, in this construction, the load of the teeth is much bigger. The reason of this is that the gears transmit not just the steering wheels torque but also the force between the two steered wheels. In a conventional mechanism, this force is transmitted by the single rack part. ² This fact had to be considered by the mechanical analysis.

Finished assembly

The prototype that will be mounted in the racing vehicle is manufactured by the team’s main sponsor, the Dendrit Kft. ⁷ Modern manufacturing and thermal treatment methods were applied in order to solve the difficulties. The racks consist of two welded parts with different material properties. The teeth were manufactured by wire spark cutting from a hard alloy, while the beam was milled from a softer one. There are more CNC technologies that can be applied for special gear cutting.^13,16,17 We applied wire spark because it is relatively cheap, precise, and effective for 2D gears. The racks were hardened and the gears nitrided. In order to reduce or eliminate the backlash at abrasive wear, the construction allows precise adjusting of the rack positions (Figure 10(b)). After mounting the steering assembly in the vehicle, several tests will be carried out. This experimental steering system will be compared to a conventional (single rack) construction. If the reliability of the construction is proven, more arbitrary characteristics (with different racks and pinions) will be tested. Stability, driving experience and noise, vibration, harshness (NVH) properties will be compared under different speed and track conditions.

Figure 10.

(a) 2D mockup and (b) the finished prototype.

Conclusion

In this article, we summarized the first steps of our research in advanced steering mechanisms. After working out a method for creating arbitrary steering characteristics, an experimental prototype has been constructed. This prototype creates error-free Ackerman ³ in a light-weight racing vehicle, for experimental purposes. After finishing and assembling the car, this steering mechanism will be tested frequently. The feedbacks of the tests will allow us to go further in our research and make more precise and better steering characteristics possible.