Proceedings, Bicycle and Motorcycle Dynamics 2013 Symposium on the Dynamics and Control of Single Track Vehicles, 11 - 13 November 2013, Narashino, Japan
Vertical Dynamic postural model for Semi-Active Mountain Bike rider L.S. Lima Faculty of Mechanical Engineering Politecnico di Milano Via La Masa 1, 20158, Milano, Italy e-mail:
[email protected]
ABSTRACT The main objective of this present work is to present a systematic approach to representing the interaction between the parameters comprising both primary (tires) and secondary (fork/linkage) suspensions of a Mountain Bike and its rider, over rough terrain. Proper dimensioning of the system allows for an ergonomically suitable riding position without penalizing the general compliance of the system or concentrating muscle efforts on one single region of rider´s body. In-plane kinematics is used to determine instant frame linkage leverage ratio, which in turn interacts with the spring-damper unit´s rheological [1] parameters, resulting in frame rate, which is then associated in series with tire´s properties to determine reduced suspension characteristic values. Rider position is considered to be standing and not pedaling, and its variation from initially determined posture is here named as rider offset. Maximum rear, front, up and down variation with respect to zero offset position define the space within which the rider is allowed to position with respect to the front triangle. By neglecting nonsuspended masses, the model aims to maintain the uncoupled condition (1) between front and rear suspensions so that both work independently [2] and vehicle pitch motion is ruled by the phase between front and rear vertical motions. Such aim is tracked by means of variation in rider posture. Differently from previously presented rider models [3], rider body´s stiffness is represented by 4 torsional springs positioned at the rider´s knees, hips, shoulders and elbows. Variations on angular position between rider´s body segments are obtained, according to the maneuvers being performed,
and serve as input for a muscle-skeletal model [4], used in the determination of activation level on muscles in the human body. Such information allows the development of Mountain Bike Rider specific exercises that prioritize determined muscles and optimize training. Rider is here named semi-active for being able to vary only posture with respect to the bicycle, but no rider-input forces are considered. Though, rider-joint optimal stiffness distribution can be evaluated for each instant of a determined maneuver, so that a rider´s optimal posture and muscle activation level can be obtained for either comfort and/or endurance purpose or for minimal time riding performance evaluation. Besides, once the rider´s inertial and compliance data is added to the formulation, a complete integration of vehicle´s and rider´s compliances can be obtained, and the same previous condition (1) for uncoupling front and rear suspension behaviours can be evaluated, so that mass distribution is complemented by rider posture and suspension parameters can be integrated to the corresponding desired rider stiffness distribution. By ignoring rider input forces, we conclude that further optimization of riding performance can be done, as well as higher level of specific training is required to fully understand the behaviour and output of top-level riders.
Keywords: compliance, training.
Integrated suspension, rider posture, ergonomy, specific
1 INTRODUCTION
2 Compliance levels
It is very intuitive that when we hear “bicycle suspension”we associate the idea to telescopic forks or 4-bar linkage systems, restricting the whole concept to only one part of the compliant system. In that sense, the concept of compliance is used here instead, so we can include all the components being deformed or articulated in order to absorb, transmit and dissipate the input coming from terrain irregularities or maneuvers performed by the rider. Compliance allows forces generated by irregularities to be partially dissipated and transmitted in proper directions, other than those that could affect the forward momentum of the moving system, but not completely. No compliant system is able to absorb irregularities to an extent that no penalty is felt in terms of rolling efficiency, so that getting as close as possible to ideal compliance is the goal here. And when we say ideal, we should also understand that ideal is not the same for every situation. For example, in downhill mountain bike racing, where classification is defined by fractions of seconds and the runs rarely exceed 7 minutes in time, comfort is not a priority, so that “ideal” is driven by minimum time. For marathon endurance mountain bike racing instead, classification times are not as sharply close to each other and the race runs go for hours, so that “ideal” is better related to comfort, as to provide the minimum energy expenditure due to absorbing terrain irregularities. In that sense, understanding the purpose of compliance and its effects on rider performance is important in the system optimization, and benefits from its own complexity so it can be fine tuned to very specific combinations of intrinsic parameters and surrounding conditions. Understanding the interaction and effects of such elements can make crucial difference to those who require the most out of such systems, but not only, it is part of the duties of those who work in designing, tuning and fitting the right vehicle for the right place and a specific biotype under particular conditions. Compliance happens by means of interaction, so in this work, we intend to identify the potentials and effects of each of the compliance levels, so we can represent such interaction mathematically, and control it in order to go a step further in creating faster and more comfortable human-bicycle-terrain systems.
The first step in order to understand compliance, is to divide it according to the nature of each level. Following the nomenclature used in railway vehicles, compliance levels are here defined as:
primary: deformation of tyres
secondary: wheels vertical movement
third: rider-body vertical movement
Wheels are here considered to be fully rigid, once radial deformations are small enough to be negleted, when compared to the amount of deformation happening at other components. primary Travel (m) Moving mass (Kg)
0,05 ~0
secondary
third
0,15
0,4
1,6
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Table 1. Representativity of compliance levels
(Values for 2.25 section tyre, all-mountain bicycle type, and a 70 Kg rider) In order to cause the minimum alteration to forward momentum, a compliant system should be able to absorb disturbances in a way that the least possible mass is moved verticaly, but since travel is limited and forces are transmitted from one level to another, all levels end up being activated to different travel amounts. And a balanced system should activate diferent levels proportionaly to the mass being moved, and the total available travel of each component. Here lies the core of this work, which is that of tuning suspension parameters so that one suspension level works in accordance with each other and with the whole system itself. But, for example, tyre inflation pressure, which defines a tyre´s radial stiffness, also influences traction performance, so that compliance cannot be taken as a general rule, and other factors will influence the final decision for parameters definition.
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2.1 Tyre compliance
2.1.1 Radial stiffness
Tyre being the primary element of contact between the complete system and the ground, plays a key role in the absorption of highfrequency and low-amplitude excitations, by means of its elastic deformation. In order to characterize tyre properties that are representative for system compliance, one single variable is sufficient, being it Radial Stiffness the force resistance to variation in the distance between the wheel axle and the point of contact between the outer diameter of tyre and the ground, starting from unloaded/undeformed condition on flat ground. In this work, only deformations on the vertical plane are considered. In order to dimension compliance levels, the first step is to identify is the travel range, representing how much linear deformation is accepted at that level. Here, for tyre compliance travel, we assume that the maximum accepted deformation is that when upper and lower inner walls of tyre´s carcass touch each other. Such assumption is due to the fact that any deformation beyond that point will result in an exponential increase in radial stiffness, with a consequent sensation that the wheel is directly touching the ground, which brings discomfort and possible damage to system´s components, mainly the tyre itself and the wheel. Besides, the recent introduction of tubeless tyres has allowed the use of much lower inflation pressures, making the definition of such limits still more valuable.
The first indicated value for radial stiffness [6] was measured statically at kt=150000 N/m for 688 kPa and used as referece for later work [7], where static deflection was further studied for one single combination of tyre and inflation pressure and a range of different contact surface geometries, allowing the estimation of forces generated by tyre deformation over bumps and hollows of differente sizes. [8] proposes values for cornering and camber stiffnesses, but no radial estimations, while [9] presents maximum and minimum values that are in good agreement with the previous. Table 2. Previously presented values for radial stiffness
Kt type
[6]
[7]
[9]
150000
22.727
125.787
150.000
178.165
MTB
road
n.d.
In that scenario we see that a reference value for stiffness can be infered from previous work [10], with the appropriate correction for different kinds of tyres. But still, tyre radial stiffness depends mostly on air pressure, in the sense that an experiment [11] evaluating sensitivity of radial stiffness to inflation pressure was needed, once varying air pressure in the tyre is an extremely simple procedure.
Figure 2. Tyre radial stiffness sensitivity to inflation pressure (non SI units)
Figure 1. Tyre maximum accepted radial deformation ( lt )
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Such experiments were performed at the Bicycle Dynamics Laboratory from TUDelft, by James Dressel [12], with a 700X28c tyre, showing good agreement with previously proposed values, and a strong dependence of radial stiffness on inflation pressure. Such information is of essencial importance in harmonizing stiffness properties at different system levels, which allows one to benefit from the whole capacities of available travel. In order to implement such values to our MSC ADAMS model, different strategies were tried, and that which worked best was one of modelling wheel and tire as two concentric rings on the vertical plane, connected by springs having a parametrized stiffness coefficient ki , so that by attributing different values to this coefficient simulates variation in tyre inflation pressure.
2.2 Fork/linkage compliance The subsequent level of compliance in our analysis, earlier defined here as secondary, is that represented by the vehicle suspension system, being it either the front fork or the articulation/linkage system used in the rear wheel. Such analysis is of virtual importance, once the introduction of suspension systems to mountain bicycles not only increased the capabilities of using it over previously unconceiveable rough terrain, but also, it raised prices, complexity and maintenance requirements to a whole new level, which can many times create an erroneous belief that an enormous amount of suspension on your bike is essential for the use of it in general, which may stop many potential enthusiasts from attempting an approach to such practice. The first concept concerning linkage suspension is Leverage Ratio (LR), which is calculated instantaneously by the ratio involving the amount of travel at the wheel and the amount of linear advance of the spring-damper unit. (1)
Figure 3. Spring-wheel used to implement and vary radial stiffness.
Once inflation pressure will influence other characteristics of the final vehicle, and tendency is that one will try to use as low of a pressure as one can, it was considered important to determine the minimum inflation pressure in order to guarantee that carcass inner walls won´t touch. Such hypothesis should then be based on the maximum force input a tyre could receive, and the chosen condition was a landing maneuver with 80% of total weight being delivered to one wheel only, and for a 100kg rider landing on flat ground from 2 meter-high. Such information is very important in order to guarantee the integrity of the system and the safety of the ride, but no requirement is made to tyre manufacturers to include minimum inflation pressure in tyre information, the same way they are required to include a maximum value for it.
Such parameter makes it very simple to relate movement, forces and velocities between the wheel axle and the damping device. In order to compare different frame linkage designs, 16 different frame models have been modeled on the vertical plane and kinematically analysed. Such comparison showed that values around 3 are common practice in the industry, and higher values are typical for the initial stages of travel, in order to improve compliance sensitivity, with a decrease along travel aiming to avoid bottom-out
Figure 4. Leverage ratio absolute values (mm/mm) for 16 different commercially available frames.
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2.2.1 Spring-Damper Unit
2.2.2 Reduced Suspension
A core element of a suspension system is its spring-damper unit. The versatility of settings made possible by different construction strategies and the ability to change its properties by means of valve control make it very much adaptable to different kinds of bicycles, riders and terrain. Spring work can be done by means of a coil spring or an air spring. Coil springs show a more linear behaviour along compression when compared to air springs, which also suffer from output behaviour variation due to the increase in air temperature with extended use in time. Air springs are lighter. The primary function of the spring is to store the energy received from the linkage. The damping work is then performed by the passage of hydraulic fluid through restrictive orifices, generating viscous friction that aims to dissipate part of the stored energy in the spring, in the form of heat. Spring work is ruled by position variation, while damping forces generated by the hydraulic circuit depend on the velocities at which the movement happens [13].
Once linkage properties are known (LR inst) and the properties of the spring-damper unit are know, we are able to determine characteristic values for Reduced Suspension [14] which simplify very much the dynamic analysis on the vertical plane, once both linkage and fork are represented by a single spring each, and only the vertical movement of the wheels (zs) is considered. For the rear spring-damper unit (shock), reduced values for spring and damping are:
Figure 5. Non-linear spring behaviour
Figure 6. Velocity-dependent damping forces for 3 different valve settings.
(2)
(3)
At this point we can say we have enough information to determine the dynamic output of the vehicle itself. Tyre properties and suspension properties are described, so that combining them with each body´s inertial parameters we can write inertia, stiffness and damping matrixes that properly related will represent the dynamic behaviour of the vehicle without the rider [15]. In this case, unsprung masses are not neglected, and the importance of such parameter can be analysed. Such tool can be useful to adjust a series of geometric, static and dynamic properties of the vehicle according to the preferences for a determined situation. Natural frequencies can be calculated too, like wheel hop ressonance, and vertical modes of the suspended mass [14]. Such assumptions consider no rider, or a rider rigidly attached to the vehicle´s chassis, by simply adding the mass of the rider to the main frame. Good aproximation can be achieved for some situations. In mountain biking though, the movement of the rider over the bicycle is representative to an extent that further refinement is needed, and assuming the rider rigidly attached to the frame will neglect most of a rider´s capabilities to adapt to terrain. It is simple to understand that differently from motorcycles, here, the mass of the rider represents nearly 80% of the total mass of the system, and according to Table 1, variations in travel regarding the rider are twice more significant than the sum of tire deformation and suspension travel.
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2.3 Rider Compliance The third level of compliance present in our system is that performed by the articulated movement of rider´s limbs. As mentioned in Tab.1, the most representative travel range is that performed by rider compliance and also in terms of moving mass, that is the movement with the biggest potential to influence system behaviour as a whole. In that sense, special attention is dedicated to represent the influence of rider position and its characteristic parameters that serve to us in compliance analysis. The rider position considered is that of standing and not pedaling, and the reason for that is because this is the position at which maximum compliant travel range and posture variation is possible, being the typical strategy for overcoming rough sections where rider positioning defines the output of the system.
Mass and Inertia data [19] used in this work has been recalculated in order to meet the assumptions made in colapsing different body segments as single bodies.The attachment of rider´s hands to the handlebar has alson been chosen to be a revolute joint, and the reason for this is that a very common mistake by less experienced riders is that of bending the wrist upwards or downwards. As a first analysis, we see that the force and torque amplitude evaluated at the hand-handlebar interface during a free-fall landing maneuver is reduced with the use of a revolute joint. That leads us to conclude that for each maneuver there will be an optimal grip positioning of the hands, in order to avoid stress at the wrist, being it the most common fatigue-related injury due to mountain bike riding.
Figure 7. Compliant rider model
Our rider model is a simplification of the human body and is composed of 6 solids, with right and left legs and arms being colapsed into one single body for each of them, once the analysis happens on the vertical plane only. Lower legs comprise both feet and shanks into one single solid, which is attached by means of a revolute joint to the axle where pedal cranks intersect the bicycle frame. The revolute joint at this point is equivalent to that of pedals. Such assumption neglects the articulation and compliant properties of ankles, and also pedal positioning, being the latter most important for maneuvers out of the vertical plane. In order to represent stiffness generated by muscle activity, 4 torsional springs have been used at the knees, hips, shoulders and elbows, and these will be the four variables defining the attitude of the rider towards an obstacle.
Figure 8. Comparison between rigid and revolute handlebar attachment
The reduction of lower-leg limbs into one single body simplifies the model to a convenient extent, without compromissing the validation of results. Since both right and left legs are also colapsed into one single body. Due to the fact that pedal position on one side is all the time the opposite along the pedal cycle with respect to the other side, assuming both legs being positioned at the center of the circle described by the pedal trajectory does not compromise inertial analysis.
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The envelope within which rider´s upper-body is able to position with respect to the bicycle has been measured and validated with the multi-body ADAMS model. Weight distribution can be considered as the key relation in order to understand the influence of rider position on the dynamic behaviour of the system. By changing weight distribution, the functioning of front and rear suspensions can be coupled or uncoupled, which means that by being coupled, the functioning of one suspension will influence the other and vice versa. Such interaction can be avoided, so that both work independently, and the condition for such behaviour is given by eq. 4 [14], where m is the total mass of the system, p is the wheelbase and Iyg is the moment of inertia around the y axis, (4)
Figure 10. Amount of compliant travel performed by each level.
The complete system multibody model also allows us to extract valuable information, such as angular variation at each of the rider´s joints. Such information can be fed into a muscular-skeletal model [21] which is able to identify the muscle firing pattern, which details the activation level os muscles of the human body according to external input.
b represents the distance between the system´s center of mass and the point of contact between tha rear tyre´s contact point and the ground. And this is the variable affected by variation in rider position, here called rider offset. Its influence on weight distribution is seen in Fig. 9:
Figure 11. Angular variation at rider body´s articulations. 3. CONCLUSIONS Figure 9. Effect of rider offset on weight distribution.
Rider offset is performed both horizontally and vertically. Vertical variation is followed by consequent variation in angular stiffness in order to adapt to discontinuities of the ground plane. The amount each compliance level performs in the total movement of the system while passing over 2, 5 and 8cm-high bump can be seen in Fig. 10, in good agreement with theory, once heavier parts present smaller variations in vertical movement.
Investigations on the vertical plane have little representativity in literature when compared to lateral dynamics. Though, much of the perception one gets while riding a mountain bike over rough terrain happens on the vertical plane. A model relating different levels of compliance allows full balanced benefit from each level and also permits to study the interaction between them. The integration of a system model and a muscleskeletal model can avoid injury due to effort concentration on one single part of rider´s body and also serve as valuable information in developing specific training according to one´s biotype and desired riding conditions.
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