JUL FEB Mar 5 2005 2007 2008 29 captures 24 Jan 01 - 5 Feb 07

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The growing popularity of offroad bicycles has led many people to investigate the mechanics of off-road cycling in depth. In the search for more information about the loads that a bike must bear during vigorous riding, several people have worked toward developing models of the bicycle/rider system.^1-5 One very important component of any off-road bicycle model is the tires. Being an integral part of the suspension system, the tires attenuate impacts from the ground by distributing the impacts over a longer period of time. By conforming to the ground, tires also filter out much of the high-frequency, small-amplitude inputs associated with rough terrain. Both of these effects are important either to the design of suspension systems or to the determination of loads for structural design/testing purposes. Tires are also extremely important in terms of handling; however, that issue deals with the lateral characteristics of the tires and was not addressed in this study.

Thus, an accurate mathematical description of the force-deflection behavior of an off-road bicycle tire is needed. Since loading occurs predominantdy in the plane of the bicycle frame, the forces of interest are those due to the radial denections of the tires.

Because of the importance of tires in affecting the dynamics of ground vehicles, considerable work has been devoted to characterizing them. Both analytical ^6-8 and empirical models ^9,10 have been created that relate tire loads primarily to slip and camber. Mechanics of radial deformations have also been described ¹¹. Despite this body of work, little is applicable to describing the radial forces in bicycle tires traveling over rough terrain, because the emphasis has been on automobile tires traveling over a flat surface.

Research has also been devoted to bicycle tires per se. Because precious energy must be expended by the rider in propelling a bicycle, the majority of this work has concentrated on rolling resistance.¹²

Previous research on the subject of force-deflection behavior of bicycle tires is scarce. Wang and Hull ² measured the radial stiffness of a road bicycle tire and approximated it as linear (k=150000 N/m at 688 kPa). Wang and Hulls performed a similar test to evaluate the stiffness of an off-road bicycle tire. They found that the force-deflection relationship could be described by a power-law equation¹:

F = Axk

where F is the compressive tire force [N], x is the tire deflection [m], A is a constant that contained the pressure dependence, and k is a constant with a value of 1.4. However, neither of these papers investigated stiffness for nonflat surfaces, which is critical for off-road bicycles. The goal of this study was to investigate the relationship between tire stiffness and tire deflection for twodimensional (extruded) nonflat surfaces.

METHODS

Similar to Wang and Hull ⁵ a powerlaw equation was used to describe the force-deflection relationship ²:

F=AxB

where F is the compressive force [N], x is the compression [cm], and A and B are given by:

A = C₀ - D₀e ^(-E₀*Rb) for convex radii. (3a)
A = C₁ + D_le^(E₁*Rb) for concave radii. (3b)
B = C₂ - D₂e^(-E₂*Rb) for convex radii. (4a)
B = C₃ + D₃^(E₃*Rb) for concave radii. (4b)

Where C_i, D_i and E_i (i=0,1,2,3) are constants and Rb is the radius of curvature of the surface. To determine the coefficients in Equations 3 and 4, the values for A and B in Equation 2, which resulted in the lowest root-mean-squared-error (RMSE), were determined for each cylinder radius. Once the values for A and B were known for each cylinder radius, Equations 3 and 4 were used to determine the best C_i, D_i and E_i by minimizing the RMSE. From these equations it is possible to model the force response of a tire to terrain defined by the instantaneous radius of curvature of the ground in contact with the tire.

RESULTS

A= 550.9 - 337.8e^(-2519*Rb) for convex radii. (5a)
A = 550.9 + 7822.0e^(0.l398*Rb) for concave radii. (5b)
B = 1.422 - 0.2469e^(-01651*Rb) for convex radii. (6a)
B = 1.422 + 0.2128e^(0.0289*Rb) for concave radii. (6b)

The quality of the empirical models was best for coefficient A for the concave radii (Figure 1) and worst for coefficient B for the concave radii (Figure 2).Table 1 and Figure 3 show that the force as calculated by Equation 2 agreed very well with the data collected. For a convex bump the highest absolute error observed was 193 N for a 25.4 cm radius bump at 2.53 cm of deflection. For a convex bump the maximum error was 116 N (-50.8 cm radius, 1.9 cm deflection). While the magnitudes of the absolute errors are comparable, the percentage errors for the concave surfaces are substantially lower then those of the convex surfaces (4.2% versus 13.5% respectively) because the loads involved are typically higher.

DISCUSSION

As suggested by Gough¹¹, the compressive tire force can be divided into two components: structural and pneumatic. The pneumatic component is simply the force due to pressure in the tire. The structural component is due to elastic deformation of the tire casing (e.g., elongation) and the interaction between the tire and the surface (e.g., friction).

Unlike automobile tires, bicycle tires have virtually no stiffness without being inflated. Thus, it was originally thought that the structural component of the tire force would be negligible. If this were the case, then the force would be a function of the contact patch area alone for a given pressure. Based on this, it was initially decided that the contact patch area was the most likely single variable to produce an accurate model with a single independent variable.

However, after initial testing it was determined that the tire stiffness could not be modeled as a function of the contact patch area only for several reasons. First, the concept of a contact patch does not make sense for convex radii smaller than approximately 3.175 cm (1.25 in) since the tire can no longer conform to the surface. Take, for example, a knife edge where the theoretical contact patch would be nearly zero, yet the force would be finite.

The second problem with the contact patch model was due to the circumferential deformation of the tire (i.e., a structural load component). Close visual observation of the tire during a load cycle revealed that locally the tire not only deformed radially, but also deformed circumferentially. The local circumferential deformation occurred by local stretching of the tire and/or by pulling in additional material from outside the contact patch. The local stretching of the tire obviously would cause a structural load component not found in the contact patch model, as would pulling material in from outside the contact patch because of the sliding friction between the tire and the surface.

With the knowledge that one variable was not sufficient to model the stiffness of the tire, the tire compression (x) and surface radius of curvature (Rb) were chosen as the two independent variables for the empirical model. These two quantities were chosen because they are easy to measure during testing and can easily be calculated in a computer simulation at each time step.

Although the results indicate that the model can be used to accurately determine the force response behavior of an off-road tire to a realistic terrain function, there were some assumptions that were made so that this model could be produced. For example, based on the findings of Wang and Hull ⁵, it was assumed that the force response of a tire was not dependent on the deflection rate. Because of this finding, the deflection rate was not considered in this study.

It was also assumed that hysteresis was negligible. As can be seen in Figure 3A (flat surface), for some cases the loading force data were approximately 100 N (22.32 lbs.) greater than the corresponding unloading force. However, the hysteresis only represents approximately 6.5% of the total force (for the worst cases). Additionally, when the data were curve fit, the loading and unloading cycles were weighted equally. This essentially averaged the loading and unloading curves, thus reducing the errors due to neglecting hysteresis by a factor of two.

In addition to modeling assumptions, limitations of the results need to be addressed. First, because a particular wheel was used in the tests (32-hole, 3cross), that wheel's characteristics were inherently included in the tests results. According to Hopkins and Principle ¹⁴, wheel stiffness was in the range of 2xl0^6 N/m for a 3-cross, 32-spoke road wheel. The "maximum" stiffness (force-deflection at maximum deflection) found in these experiments represented that of the wheel/tire system and ranged from 2.3x10^4 N/m for a 1.27 cm (0.5 in.) radius to 1.7xl0^5 N/m for a -38.1 cm (-15 in.) radius. Since the wheel and tire act as springs in series, these values show that the "maximum" tire stiffness could have been underestimated by up to 8.8%. Because an offroad bicycle wheel is most likely to be stiffer than the road wheel due to the smaller diameter and shorter spokes, this error represents an upper bound. Additionally, since the maximum error occurs with concave bumps with radu close to the tire radius (i.e., when the tire has the largest stiffness), which is a situation that rarely arises, the wheel stiffness is inconsequential for the majority of the time. However, if a wheel with a different stiffness were used, then small errors could result, most notably for concave surfaces.

Not only are the results dependent on the particulars of the wheel, but they are also dependent on the tire. Off-road bicycle tires are available in a wide variety of tread patterns and are constructed from different materials. The complex mechanics associated with tire deformations on curved surfaces described earlier manifest as differences in load-deflection behavior for different tire constructs ⁵. Nevertheless, the bicycle tire chosen in the study was a popular, commercially available design, so that the results herein are certainly representative of tires used in practice. Moreover, the variations in load-deflection behavior that arise from different constructs are probably of secondary importance in the development of the bicycle/rider system model, because the main source of variability in such a model will be due to the rider characteristics.

Since a direct relationship between force and contact patch area was not found, the model was only developed to account for two-dimensional (extruded), cylindrical bump geometries. However, this does not mean that the results cannot be applied to a wide range of terrain. Many obstacles commonly encountered during off-road riding are nearly twodimensional, such as sticks, logs, tree roots and rain ruts. While rocks are not typically two-dimensional, many rocks may appear two-dimensional to the tire due to the relatively narrow tire width.

Furthermore, a constant radius of curvature can be used to approximate many shapes. Admittedly, this approximation requires a somewhat complicated approach. The radius of curvature can be found at any point for any continuous terrain function by taking the inverse of the second derivative of a parameterized description of the surface. If this radius of curvature function is continuous, then an average radius of curvature for the section of the terrain in contact with the tire can be used in Equation 2 to find the force response of the tire.

Physically, there tends to be a correlation between the radius of curvature and the gradient in the radius of curvature. That is, the radius of curvature tends to vary slowly for terrain functions with large undulations (e.g., woop-dee-doos) and, as the radius of curvature gets smaller, it is more likely to change faster (e.g., stutter bumps). Since the greatest inaccuracies in using an average radius of curvature will occur when it changes quickly, the model wil1 be more accurate for terrain with large radii bumps.

Although it would be desirable, the model developed cannot predict the responses to terrain, which indudes sharp edges and terrain with discontinuous radius of curvature functions. Nevertheless, the tire model produced by this study should prove to be very useful as part of the off-road bicycle system model.

Based on the above discussion, it appears that the model developed is capable of being used to predict the radial force-deflection behavior of an off-road bicycle tire. With this in mind, it is useful to interpret some of the more interesting aspects of the results. For example, the general shapes of the curves describing the coefficients A and B are not unexpected. Intuitively, for a convex surface and a given displacement, x, the force should decrease as the radius of curvature decreases because the contact patch decreases (Figure 1). For the same reason, for concave surfaces, the force should increase as the radius of curvature increases (approaches the radius of the wheel) for a given displacement.

Similarly, for a convex surface, as the radius of curvature decreases, the force should become insensitive to the displacement (i.e., the exponent B should decrease as in Figure 2). As an extreme, imagine a knife edge, for which the "average" stiffness would be very low because the force would not vary much as the displacement increased. For a concave surface whose curvature is near that of the wheel, it is expected that the force should asymptotically approach infinity (or at least some very large number) because a slight increase in x (displacement) causes a large increase in contact patch area (and thus, the pneumatic component of the load).

In comparing the results of the present study to those previously reported, Wang and Hull ⁵ found the value of the exponent (k in Equation 1) to be 1.4 when averaged over three different tires. For a flat surface, the exponent B was found to 1.422 in this study. Converting the results of Wang and Hulls to similar units, the constant of proportionality (A in Equation 1) was found to be 718.5 as compared to the value of 550.9 found in this study. The relatively large discrepancy can be attributed to the possibility of offset error in the displacement. Inspection of Figure 3A reveals that if the curve for the flat surface were shifted to the left by 1 mm, then the values would be quite similar (about 5% difference) to those in Ref. 5. The relatively good agreement in the exponent values (1.4 versus 1.422) supports this theory. The question then arises as to which study contained the displacement offset. The procedure used by Wang and Hulls did not include the ultrastiff fork used in the present study. Instead, at the start of each test the wheel was balanced vertically by applying a very light force that pinched the wheel between the upper and lower crossheads of the actuator. It is quite conceivable that a preload of 22 N existed, which would be enough to account for the offset.

CONCLUSIONS

• Provides accurate radial force-response behavior of the tire.
• Can be used for a wide range of offroad cycling terrain.
• Can be easily included in many existing bicycle models.

• Cannot be used for terrain with a discontinuous radius of curvature function (e.g., corners, edges and steps).
• Application to three-dimensional surfaces is complicated.
• Applicable to only a single type of tire and pressure.

REFERENCES

David F. Moore is a student of the Dept.of Mechanical Engineering at the University of California, Eric L. Wang, Ph.D., is an Assistant Professor of the Dept. of Mechanical Engineering at the University of Nevada, and M. L. Hull, Ph.D. is a Professor of the Dept. of Mechanical Engineering at the University of California.