Identification and Modeling of a Mountain Bike Front Suspension Subsystem Equipped with a Telescopic Fork and Tire Damping

Type of the Paper: Conference Paper

Identification and Modeling of a Mountain Bike Front Suspension Subsystem Equipped with a Telescopic Fork and Tire Damping

Noah Schoeneck^1*, James Sadauckas², and Mark Nagurka³

¹Vehicle Measurements Group, Harley-Davidson Motor Company, Yucca AZ 86438, USA; noah.schoeneck@harley-davidson.com

²Trek Performance Research, Trek Bicycle Corporation, Waterloo, WI, 53594, USA; jim_sadauckas@trekbikes.com; ORCID 0000-0002-6055-9047;

³Professor Emeritus, Department of Mechanical Engineering, Marquette University, Milwaukee, WI 53233, USA; mark.nagurka@marquette.edu

*corresponding author

Name of Editor: Jason Moore

Submitted: 25/03/2024
Revised: 25/03/2024

Accepted: 25/03/2024

Published: 08/04/2024

Citation: Schoeneck, N., Sadauckas, J. & Nagurka, M. (2023). Identification and Modeling of a Mountain Bike Front Suspension Subsystem Equipped with a Telescopic Fork and Tire Damping. The Evolving Scholar - BMD 2023, 5th Edition.

This work is licensed under a Creative Commons Attribution License (CC-BY).

Abstract:

A key component in the mountain bike industry is the telescopic front suspension, which offers performance advantages when traversing obstacles, rough terrain, and high impact landings. Despite the popularity of telescopic forks in the market, their modeling details and assumptions have received limited attention in the literature. This paper presents a system identification and modeling approach that promises a deeper understanding of the dynamic behavior of mountain bikes with telescopic front suspensions. The mountain bike front suspension subsystem is modelled initially using the classic quarter car model with the suspension and tire both included as second-order systems, each with spring and damper elements in a Kelvin-Voigt arrangement stacked in series. The boundary and initial conditions of the subsystem are modified from the classic quarter car model to capture the impact and loss of ground contact condition. The paper then incrementally increases the complexity of the quarter car model based on a parameterization study of the fork and tire. Simulation results are compared to data from an impact sled test of a telescopic mountain bike front suspension subsystem. The correlation between the quarter car model response and the test data varies with the complexity of the model and its parameters suggesting the importance of including key parameters in models of mountain bike front suspension subsystems.

Keywords: Mountain Bike Suspension, Tire Damping, System Identification, Quarter Car, Simulation

Introduction

Modeling and simulation of vehicle suspension subsystems have been essential to the design of high performance vehicles. The automotive industry has used various versions of the quarter car model as a first approach in vehicle suspension design. Despite the similarities between automotive and mountain bike suspensions in function and utility, the mountain bike telescopic front suspension subsystem has received little attention in the literature at the subsystem level. Lumped parameter mass-spring-damper models for the mountain bike tire and for the mountain bike telescopic front suspension have been used as part of a broader full vehicle model with minimal focus spent on the complexity of the subsystem parameters.

Through the years the classic two degree-of-freedom quarter car model has been met with both criticism and praise in predicting vehicle suspension performance. One concern is the classic quarter car model used in automotive studies is a linear model. Automotive suspension springs can be represented as piecewise linear stiffnesses combining multiple linear spring rates (i.e., stiffnesses) or have nonlinear characteristics. Automotive suspension dampers often exhibit nonlinear behavior. Dedicated hydraulic circuits for compression and rebound damping can further be divided as having low and high speed behavior. A second concern with classic quarter car models from literature is their inability to model the system when the tire loses contact with the ground. Typically, the classic quarter car model predicts relatively low amplitude displacements from an equilibrium position, but this is inadequate for an impact study.

Pneumatic tires have complicated stiffness behaviors, typically represented by dynamic and static stiffnesses. A survey of automotive literature suggests a long-running debate on whether to include tire damping within the model. Those advocating for the exclusion of the effects of tire damping have purported, but not necessarily shown, that these effects have only a small influence on the overall system response and thus can be neglected. Those including it generally do so without particular attention to its measurement or estimation except for a few specific studies (Acosta, 2020, Mahr, 2011, Levitt, 1991). This paper demonstrates that for relatively flexible tires, such as those of a mountain bike with minimal carcass plys, limited tread rubber, and relatively low inflation pressures, encountering relatively large deformations with respect to their aspect ratio under impact, both dynamic stiffness and damping can influence system response.

A feature that is often included in mountain bike telescopic forks, whether coil or air sprung, is the ability to adjust the suspension preload. For a coil, the preload is the amount the spring is displaced when installed into the fork or shock plus any additional compression the user adds via the preload adjuster. The preload allows the equilibrium position of the suspension under the steady-state load of vehicle and rider, also known as sag, to be set independently from the spring rate chosen for ride characteristics. When properly set, the preload serves the important task of balancing the available suspension travel between compression, to resist bottoming at maximum stroke, and rebound, to avoid topping the suspension at full extension. Despite its practical implications, preload is not included in many classical quarter car models (Lot, 2021).

Suspension bottoming presents a variety of concerns. It can lead to significant loading with the road input directly transferred to the chassis and, from a structures and durability perspective, should be avoided. In addition, it can lead to large chassis accelerations that contribute to rider discomfort. For these reasons shock absorbers often include a “bump stop” or “jounce bumper”, i.e., a stiff elastomer in parallel to the main spring to ease the transition to end stroke. Topping events are of less concern as the loads transmitted to the chassis are lower, but such events may still affect vehicle road holding, diminish rider comfort, and possibly even damage suspension components. Motorcycle and mountain bike suspensions often include a top out spring to help prevent damage to the suspension internal components and provide a better riding experience. The top out spring also helps control the wheel and keeps the tire in contact with the ground. It is another suspension element that is not typically included in the classic quarter car model.

This paper investigates the parameters and accuracy of the quarter car model as applied to a mountain bike front telescopic suspension subsystem. It describes the experimental measurement of the stiffness and damping parameters of both the telescopic fork and the tire. Then, starting from a classic two degree-of-freedom linear quarter car model (denoted S1), a series of 14 simulations incrementally explore nonlinear stiffness (S2 & S3), additional spring parameters such as top out (S4 & S5) and preload S6 & S7), linear damping (S8), nonlinear damping (S9 & S10), as well as tire stiffness and damping (S11 through S14). Simulation results are compared to fork and tire subsystem test data collected using an impact test sled to help identify model elements that are critical for a given level of fidelity.

Model

Figure 1. Quarter car model.

Figure 1 shows a lumped-parameter quarter car impact model, where m_b is the mass of the body, m_w is the mass of the wheel assembly, k_s and b_s are the stiffness and damping coefficients of the suspension, respectively, k_t and b_t are the stiffness and damping coefficients of the tire, respectively, z_w is wheel displacement, z_b is the body displacement, and h₀ is the initial drop height for impact. This height is used to calculate the velocity at impact, v_i, which combined with z_b and z_w represent the initial conditions of the model.

The equations of motion for the model of Figure 1 are:

These coupled linear second-order equations are recast as state space equations that can be solved using MATLAB’s ode23s solver given numerical parameters.

Test Setup

A test sled was fabricated using 2.5 x 2.5 cm (1x1 inch) 16-gauge square steel tubing. The sled attaches to intermediate rails at two locations on both sides of the fixture for a total of four mounting locations. The intermediate rails each slide on two linear bearings, four bearings total, which ride on bearing rails that are attached to outer fixture supports. This setup constrains the sled to translate only in the vertical direction.

The normal load is set by adding weights to each side of the test sled. The weights are placed onto a threaded rod and secured using a threaded nut. The masses of the weights used range from 1 kg to 15 kg. The mass of the wheel, m_w, is 2.3 kg and the mass of the body, m_b, including the fork assembly and additional weights, is 53.2 kg for a combined normal load for this test of 544.5 N which represents an 85 kg rider and an 11 kg bicycle with 58% front 42% rear weight distribution (as if descending a ledge).

A winch was installed to the outer fixture support to facilitate lifting the loaded sled to the desired drop height. Once at the desired height the sled was then connected to a snap shackle release and the rope from the winch was removed. The snap shackle opens rapidly to release the sled. The tire then lands on the bottom of the fixture where a removable insert was secured to allow the use of different frictional surfaces; 3M^® Safety Walk was used for all data presented in this paper.

The test sled was designed to mimic a common impact event that a mountain bike and rider may experience while descending a trail. As such, the fork axis is aligned with the sled vertical axis. The sled is released from a nominal height of 95 mm. (By releasing from this height, the stroke of mid-priced coil fork used in this study achieves 80% of its available 120 mm of travel and the tire is displaced 40% of its maximum displacement.) After release, the tire fork system generally exhibits one bounce in which the tire leaves the ground in rebound and the fork experiences a topping event before regaining ground contact.

Figure 2. Test sled schematic (left) and actual test sled with front suspension subsystem installed (right).

A linear potentiometer is used to measure sled displacement with respect to the outer fixture support and zeroed when the sled is lifted such that the tire tread just contacts the ground. A second linear potentiometer is mounted on the fork to measure displacement of the fork and is zeroed with the fork at resting equilibrium with no weight. Tire displacement is calculated as the difference between the readings of the two potentiometers. The specifications of the potentiometers and data logger are as follows:

Linear potentiometers

• Range: 275 mm (Sled); 150 mm (Fork)

• Make: Penny & Giles

• Model: SLS190/0275/C/66/01/N (Sled); SLS190/0150/C/66/01/N (Fork)

Data logger

• Make: imc Dataworks 

• Model: CRONOS-XT logger with uni-8 amplifier 

Data is sampled at 1 kHz with an antialiasing filter applied and a 400 Hz roll-off frequency.

Parameter Identification and Simulation

Spring

The mountain bike telescopic front suspension was installed on a shock dynamometer to experimentally determine the stiffness and damping of the suspension. The suspension was stroked to the middle of the available travel before it was exercised. During the spring dynamometer test the fork was first compressed and then pulled in extension past both the main spring and top out spring. In this way, the top out spring rate was determined as the fork returned to zero on the compression stroke.

Figure 3. (a) Linear fit of fork compression and rebound spring rates, (b) bi-linear fit of fork compression spring rate,
(c) linear fit of fork top out spring rate, (d) progressive fit of compression spring rate.

Figure 3 displays four different approaches for fitting the force versus displacement data to determine the fork spring rate. The fork displacement zero point is taken with the fork hanging in its equilibrium position with no external loads applied. From this reference point, the top out spring is exercised when the displacement is negative, and the main spring is exercised when the displacement is positive. Figure 3a shows the characteristic for the main spring compression (denoted subscript SC). and spring rebound (denoted subscript SR). The top curve is the compression stroke, and the bottom curve is the rebound stroke. Figure 3b depicts the compression stiffness with a bi-linear fit given by the piecewise function (with subscripts SC1 and SC2 for spring compression 1 and 2, respectively):

Figure 3c indicates the characteristic stiffness of the top out spring. The slope is evident only when the suspension is extended, unloading the main spring as if the suspension is approaching a topping event.

Figure 3d shows the characteristic with a progressive fit to the compression stroke. To prevent bottoming events, a progressive suspension stiffness is often desirable for mountain bike suspensions. Air springs are used widely in mountain bikes as they allow the rider to adjust the spring rate and preload by adding air pressure and volume spacers. The suspension tested in this paper is a coil spring that exhibits a slightly progressive trend due to air trapped in both the spring and damper sides of the fork. The quarter car model is used to perform a parameterization study of the fork stiffness based on the values described above. For these first seven simulations (denoted S1 through S7), focused on spring parametrization, an average value for the fork damping coefficient and singular baseline (impact-only) values for the tire damping and spring rates were used (Sadauckas, 2023). The fork damping coefficient and tire parameters are investigated and discussed later in the paper.

Figure 4 shows the time histories of sled displacement, fork displacement, and tire displacement plotted for each simulation (blue lines) compared to the data measured on the sled (black dash-dot line). The tire impacts the ground at time zero and the fork and the tire compress as the sled continues to travel down on the bearing rails after impact. After the initial impact is absorbed by the system at Peak 1, the tire and fork begin to rebound with the tire eventually leaving the ground around 0.2s into the test. As the tire leaves the ground the suspension has a topping event after which the fork is fully extended (zero fork displacement). The sled then transitions back into freefall and the tire again impacts the ground leading to Peak 2 after which it remains in contact with the ground as the sled oscillates through Valley 1 and Peak 3 before eventually coming to rest.

Figure 4. Mountain bike quarter car simulation results varying fork spring parameters overlaid on sled, fork, and tire drop test displacement data for (a) linear and bi-linear stiffnesses, (b) bi-linear and progressive stiffnesses both including a top out spring, (c) bi-linear and progressive stiffness with top out spring and preload force.

Examining Figure 4a, the first simulation, S1 (blue dotted line) models the fork spring stiffness using a single linear parameter for rebound and compression. When compared to the sled test data (black dash-dotted line), this simulation exhibits the most overshoot in the sled, fork, and tire displacement. This subsystem model does not come to rest and there is poor correlation of the top out event at Valley 1. Simulation S2, shown by the long-dashed blue line, uses separate compression and rebound spring rates resulting in a slightly more accurate subsystem model. Overshoot is reduced and the system damps more quickly. Simulation S3 (solid blue line) which contains the bi-linear compression spring rate (solid line) starts to match the test sled data more closely at Peaks 1, 2, and 3.

In Figure 4b, the top out spring is introduced to the quarter car model in combination with the separate compression and rebound spring rates from Figure 3a and the progressive spring rate fit shown previously in Figure 3d. Both simulations (S4 and S5) suggest the top-out spring greatly improves the model fidelity around the initial fork topping event (between Peaks 1 and 2). The return of tire to ground also changes subtly prior to Peak 2.

Finally, the fork spring stiffness simulation results (S6 & S7) shown in Figure 4c include a preload force in the quarter car model with the same spring stiffness values and top out spring used in the previous plot. The results suggest additional improvements in following the experimental data.

Figure 5. Mountain bike quarter car simulation results showing fork displacement
with emphasis on topping event with various fork spring parameters.

Figure 5, focused on fork displacement, shows the benefits of increased fidelity of the fork stiffness parameterization (S2 & S3) as well as a top out spring (S4 & S5) and fork preload (S6 & S7) in greater detail. There is less overshoot in fork displacements at Peaks 1, 2, and 3 with the addition of the top out spring and preload parameters. The results shown in the zoomed area (circle) of the middle and right plots demonstrate the benefits of including the top out stiffness and preload in matching the topping event.

Damper

Figure 6 shows the fork damping force versus stroking velocity measured on the suspension dynamometer. From this data the damping coefficient can be approximated. The friction component can also be derived from the damping curves as evident by non-zero force values at the velocity origin. Various linear fits of damping coefficient for compression and rebound are shown. A single linear damping coefficient (green dotted line) was found taking an average of the compression and rebound coefficients. This single term average is often used in literature and textbooks. In this case it misses various nuances of the fork system measurements. As noted previously, this average damping value was used as the fork damping coefficient for all simulations in the fork spring rate parameterization study above.

Figure 6. Left: Mountain bike front suspension fork damper peak force versus velocity. Dynamometer measurements are shown as black points, average linear fit shown as green dashed line, and bi-linear fits for compression and rebound
blue solid and red dashed, respectively. Right: Force vs. velocity plots for four different friction models (Olsson, 1998).

Four friction models are shown in the right side of Figure 6: plot a) displays a Coulomb friction model, plot b) combines a Coulomb and viscous model, plot c) adds stiction to the Coulomb and viscous model, and plot d) shows a more realistic nonlinear friction model that captures the essence of plot c) (Olsson, 1998). A comparison between these friction models to the data collected on the shock dynamometer is shown for a deeper understanding of the type of friction forces (i.e., Coulomb, viscous, etc.) present in the fork. These classic friction models can be observed in the dynamometer damping data shown on the left of Figure 6. The rebound curve (negative sign convention) mimics the shape of plot d). The front suspension compression damping curve (positive sign convention) displays a similar shape to that in plot b). The ability to identify and model the friction forces acting in the telescopic fork plays an important role in modeling mountain bike longitudinal braking performance (Klug et al, 2019) and modeling impact events as shown in the following simulations.

The quarter car model was used to examine the parameterization of the fork damping. The simulation results for three different damping approximations are shown in Figure 7. The first of these simulations (S8) uses the average fork damping value (dotted red line), the second (S9) adopts the bi-linear approach with a separate fork damping coefficient for compression and for rebound (dashed red line), and the third simulation (S10) uses a lookup table (solid red line).

Figure 7. Quarter car model simulation results with varying fork damping and inclusion of fork friction.

Figure 7 suggests subtle differences in the displacement responses of the subsystem comparing the average fork damping (S8) and bi-linear damper (S9) simulations. The lookup table for damping which includes the frictional characteristics (S10) provides improvement in modeling the fork top out event and the displacement of the subsystem as it comes to a rest.

Figure 8 shows a more detailed view of the fork displacement from the fork damping parameterization study. The frictional force in the simulation with the fork damping coefficient lookup table (S10) reduces the ringing during the top out event (zoomed circle at left) and aides in better predicting the resting displacement (zoomed circle at right). In addition, there are improvements in the match of displacement responses at Peak 2, Valley 1 and Peak 3.

Figure 8. Fork displacement from Figure 7 with emphasis on top out and system coming to rest.

Tire

The tire stiffness and damping coefficients used here are based on impact and measured using a coefficient of restitution method (Sadauckas, 2023). This method can be used to identify a dynamic tire stiffness and damping coefficient for both impact and in-contact oscillatory behavior. For these simulations the tire stiffness and damping can be divided into two piece-wise regions based on the system response. The impact stiffness and damping are used when the subsystem is in free fall through the initial impact event, i.e., Peak 1 through topping. The in-contact values are then applied in the region when the tire is in constant contact with the ground and may still be oscillating. Previous studies found the tire stiffness by applying a force to the tire and measuring the displacement. This tire stiffness value was used in the parameterization study as a comparison to the coefficient of restitution approach. (Dressel, 2020).

The quarter car model was utilized for a tire parameterization study. Four different simulations were performed. The tire parameter simulations included the progressive fork spring rate, fork top out spring rate, fork preload, and the use of the lookup table for the fork damping force. The first of the tire parameterization simulations (S11) used the tire impact dynamic stiffness and damping coefficient, the second (S12) used both the impact and in-contact values, the third simulation (S13) used the impact stiffness but with tire damping removed from the model (to test the hypothesis of some automotive papers), and the fourth tire simulation (S14) used the static tire stiffness (which is nearly 30% lower than its dynamic stiffness) with the impact dynamic damping coefficient.

Figure 9 shows the simulation results overlaid on the experimental data from the test sled. The first two simulations (S11 & S12) produce similar tire displacement during the impact region of the model. There is a noticeable improvement in predicting the resting displacement when the in-contact tire spring rate and damping coefficient are included (S12, green dashed). The simulation with no tire damping (S13, green solid) exhibits slight ringing seen in the tire displacement and in the fork displacement. Using the tire static spring rate (S14, green dash-dot) predicts large overshoots in tire displacement at Peaks 1, 2, and 3. In addition to the overshoots in the tire displacement, the subsystem response peaks are adversely affected regarding phasing and amplitude and the subsystem continues to oscillate.

Figure 9. Quarter car model simulation results with varying tire spring rates and damping coefficients.

Discussion

The displacement of the peaks and valley from the parameter sweep simulations varying model complexity were compared to the test sled data and the differences in amplitude were calculated. The results of the amplitude errors are shown in Figure 10.

Figure 10. Sled, fork, and tire displacement amplitude errors. Each bar represents the error in predicting a key peak or valley in the response, with parameterization effects of fork stiffness (blue), damping (red), and tire parameterization (green) grouped accordingly. Positive error values overpredict the displacement; negative error values underpredict the displacement.

Regarding the fork spring parameterization, simulation S1 with a single linear rate for fork and tire spring rate and damping coefficients showed the most sled and fork displacement errors with displacement overshoot exceeding 25 mm in both directions. Incrementing across the fork spring rate simulations (S1 through S7) the sled and fork displacement errors decrease while the tire displacement errors increase as the parameterization fidelity of the fork spring stiffness increases. This highlights the tradeoffs of the different parameters within the subsystem. The incremental additions to fork stiffness (via the progressive rate and preload term) increase the force applied to the tire resulting in overshoot. This is most notable on the initial impact (see Peak 1 for simulations S3 through S7).

As for the fork damping coefficient parameterization study (S8 through S10), the largest reduction in error was seen in simulation S10 with the introduction of the fork damping lookup table (and frictional terms). The most significant improvement of this parameterization was seen in the fork and sled displacement error on Peak 2 and as the subsystem comes to rest.

Finally, in comparing tire parameterization (S11 through S14), simulation S14 the static tire spring rate (with nominal impact tire damping) exhibited the largest tire displacement error. The amplitude error at Peak 1 was four times higher than for the other three tire parameterizations. The increased tire error also has a negative effect on sled displacement error. Tire amplitude error is lower when using both the impact and in-contact dynamic spring rates (S12) and damping coefficients. The improvement was most notable when the tire remained in contact with the ground with amplitude error ranging from 0.1 mm to 1.5 mm compared to the error using only the impact dynamic stiffness and damping coefficient (S11) ranged from 1.0 mm to 2.2 mm. The exclusion of tire damping (S13) increased the tire error at Valley 1, Peak 3, and at rest and had a negative influence on the fork and sled displacements.

Conclusion

In this paper, the mountain bike telescopic front suspension subsystem was investigated using a quarter car model. Several methods were applied to identify the parameters for both the fork and tire ranging from linear, bi-linear, to nonlinear. Additionally, the model was extended to include other forces from a top out spring, preload forces, and friction forces in the fork. Predictions of fork displacement were more accurate using a bi-linear or progressive spring rate. Additionally, it was shown that including a top out spring in the model can have a positive effect for topping events and may be omitted if the subsystem being modeled does not leave the ground (or if the actual system being modelled is not equipped with this feature). The effects of friction on the fork were studied and shown to play an important role in modeling the fork displacement at low velocities as the subsystem is coming to rest and at topping events and helps to attenuate earlier peak overshoots. Tire stiffness and damping parameters were also studied. The model demonstrates that the use of an impact tire stiffness more closely matches the drop sled test data compared to simulations using the tire stiffness measured statically which produced large displacement amplitude error. Neglecting tire damping forces in the quarter car model resulted in an increase in displacement error and model stability of both the tire and the entire suspension subsystem response.

The lightweight, low load (relative to automotive, aerospace, or agriculture tires), and aggressive use case of mountain bikes, in conjunction with the sophistication of their suspension systems, make them particularly interesting to study in terms of in-plane dynamics. This paper builds on previous tire-only measurement and simulation studies by including measurement, modeling, and simulation of the front fork plus tire front suspension subsystem to quantify tradeoffs in the model parameterization for an impact use case.

Future work may consider adopting this subsystem-focused approach to parameter identification, model sensitivity analysis, and targeted parameterization based on use case prior to making broad assumptions about those constitutive elements within a larger, more complex vehicle or system model. This study was limited to a relatively simple coil sprung front suspension fork and can be extended to more performance-oriented fork specifications, settings, alternate tires, inflation pressures, test boundary conditions (such as drop height, rocks, or other terrain under the tire) and even a rolling wheel on a drum, treadmill, or otherwise.

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