A Multibody Model for the Simulation of Bicycle ... - Semantic Scholar

Vehicle System Dynamics 2002, Vol. 37, No. 1, pp. 3±28

0042-3114/02/3701-003$16.00 # Swets & Zeitlinger

A Multibody Model for the Simulation of Bicycle Suspension Systems MATTHIAS WAECHTER1, FALK RIESS1,2 and NORBERT ZACHARIAS1

SUMMARY The paper describes a two-dimensional mathematical model for the motion of a bicycle-rider system with wheel suspensions. It focusses on the prediction of vibrational stress on the rider due to uneven track. The model was evaluated by comparing its predictions with measuring data concerning weighted accelerations on the human body, depending on various bicycle designs and road surfaces. For the intended purpose the predictions for vibrational stress and vibrational behaviour are suf®ciently precise, and the model turns out to be adequate for designing and developing bicycle suspensions.

1. INTRODUCTION It is well known that vibrational stress on vehicle riders does not only affect riding comfort, but impairs the rider's capability of perception and reaction as well (which is quite signi®cant in road traf®c), and can even cause damage to the rider's health [1, 2]. Thus, the compromise between the minimization of vibrational stress and the reduction of wheel load variation is a most important aim in automobile research and development [3]. For bicycles the dicussion about suspension systems only began a few years ago, primarily aiming at a better performance of mountain bikes. Suspensions on everyday bicycles with the objective to reduce vibrational stress recently have appeared on the bicycle market. The intention of the development of the model discussed in this paper was investigating the physics of bicycle suspensions. Although some models with 1

Physics Department, Carl von Ossietzky University, D-26111 Oldenburg, Germany. Corresponding author. Tel.: 49-441-798-3540; Fax: 49-441-798-3990; E-mail: falk.riess @uni-oldenburg.de 2

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different complexity and degree of explanation have been published [4±8], and a method for the quantitative determination of harmful oscillations has been presented [9], no attempt has been made to our knowledge to predict vibrational stress on cyclists depending on bicycle parameters and road surface. In the following we introduce the concept of the mathematical model, its variables and the equations of motion, in Section 2. Section 3 describes the empirical validation of the model with recent measurements. 2. THE MATHEMATICAL MODEL FOR BICYCLE SUSPENSIONS The mathematical model was developed with the intention of describing bicycles with wheel suspension. The bicycle-rider system is represented by a mechanical system of four rigid bodies linked together by rotational and linear joints, springs, and dampers. The rigid bodies represent the parts of the bicycle-rider system, i.e., the rider, the front and rear swingarm systems (including wheel, derailleur and other accessories), and the central frame. The spring and damper elements represent tires, saddle or seat, hand-arm system and the springs and dampers of the suspension systems. A sketch of the geometry is shown in Figure 1. As mentioned above, the rider is modeled as one rigid body formed by his or her trunk. It is pivoted on top of the saddle at the ischial tuberosities and suspended at the shoulder by a spring-damper element representing the handarm system (for recumbent bicycles the shoulder is supported by the backrest of the seat). The masses of legs and feet were considered as being part of the rider's trunk. This assumption seems plausible as the masses acting on the pedals do not exceed 12% of the whole body mass [15]. The rider's mass, moment of inertia, and location of center of gravity are derived from data of parts of the human body arranged in a typical riding position (Section 3.1). The hand-arm system is modeled as a linear spring combined with a linear viscous damper. The tires are treated as linear springs and linear viscous dampers and are allowed to leave the ground if the distance between wheel hub and ground level is greater than the given wheel radius. Characteristic curves and the location of the spring and damper elements of the wheel suspensions can be chosen at will.

MULTIBODY MODEL FOR BICYCLE SUSPENSIONS

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Fig. 1. Scheme of the Mathematical Model. Centres of gravity of the four rigid bodies (numbers 1±4) are indictaed by , pivot points of the swingarms and the rider's trunk by . The numbers in grey circles mark components of the model as follows: 1. rider; 2. central frame; 3. front swingarm system and 4. rear swingarm system (both including wheel); 5. handarm system; 6. saddle; 7. suspension springs and dampers; 8. tire springs and dampers.

The torque at the bottom bracket axle is controlled in order to keep the riding speed constant (which must be given as one of the parameters). Thus, the in¯uences of the forces in the power train are taken into consideration according to gear ratio and location of the rear swingarm pivot. The parameterization of bicycle geometry is ¯exible enough to ®t standard bicycles as well as a wide variety of recumbents and other unusual bicycle types. The front fork geometry can also be adapted to describe a telescopic fork as well as a swingarm [10]. Thus, the coordinates of the crucial parts of the bicycle-rider system can be calculated for everyday use as well as in extreme situations. As a further result the vibrational stress on the rider can be derived from these data. 2.1. Variables and Parameters of the Model The state of the bicycle-rider system is described by two-dimensional position vectors pointing to the centres of gravity of the four parts of the system (described above), together with their angles in the x-z-plane.

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The seven elements of the state vector ~ q zr ; r ; xcf ; zcf ; cf ; fs ; rs

1

of this system are de®ned as follows (Fig. 1): zr r xcf zcf cf fs rs

^ ^ ^ ^ ^

de¯ection of the saddle (or seat) spring angle between the rider's trunk and the central frame horizontal location of the central frame's center of gravity vertical location of the central frame's center of gravity angle between the central frame and the reference frame of the road ^ angle between the front swingarm and the central frame ^ angle between the rear swingarm and the central frame

These seven quantities form a complete set of generalized coordinates which ± together with their time derivatives ± completely de®ne the state of the system at any given time. The time evolution of the system is described by the development of these coordinates. The geometry of the system is de®ned by a number of reference points, such as the front and rear swingarm pivot, the saddle or seat (being the `pivot' of the rider's trunk), the handlebars, the wheel axles, and the mounting locations of the spring and damper elements. Further parameters are the masses and moments of inertia of the four parts of the system, the moments of inertia of the wheels and the stiffness and damping parameters of all spring and damper elements. Additionally, the wheel radii, the gear ratio of the drive train and the riding speed are needed. The parameters of rolling and wind resistance may also be given. Vibrations are induced by the unevenness of the road surface which must be given as a longitudinal pro®le. Numerous pro®les of typical road surfaces in West German bicycle traf®c have been measured and prepared for calculation [11] (Section 3.1). 2.2. Equations of Motion The equations of motion of the bicycle-rider system have been derived in the form of Lagrange's equations ([12], p. 141). The correct application of this formalism automatically takes into account given constraints and at the same time guarantees that all interactions between forces are included.


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Let T be the kinetic energy and U the gravitational potential of the system, then its Lagrange function L is given by LT ÿU

2

with T and U being composed as follows: 4 X mi _ 2 i 2 fw 2 rw 2 ~ T '_ fw '_ ; x i '_ i 2 2 2 2 rw i1 Ug

4 X

xi ~ mi ~ ez :

3 4

i1

Here ~ xi denote the position vectors of the four parts of the system (Fig. 1), i their moments of inertia, and '_ i their angular velocities compared to the reference frame of the road. The quantities fw , rw , '_ fw , and '_ rw symbolize the moments of inertia and angular velocities of the front and rear wheel. The angles 'i are composed from the angular generalized coordinates q2 r , q5 cf , q6 fs , and q7 rs . For example, the angle of the front swingarm system compared to the reference frame of the road is given by xi is described with the 'fs q5 q6 . The evolution of the position vectors ~ help of initial values (containing the geometry and initial position of the bicycle-rider system) and the seven generalized coordinates (Section 2.1). As an example the position vector of the front swingarm system is given: ~ x cf x fp ex zcf~ ez Dcf ~ xfs xcf~ 0; fp Dcf fs ~ 0; fs :

5

The rotation matrices D in the above equation are used to rotate anticlockwise the vector they act on by the angle . The vector ~ x cf 0; fp describes the initial position of the front swingarm pivot measured from the central frame's center of gravity, whereas ~ x fp 0; fs points from the front swingarm pivot to the initial position of the front swingarm's center of gravity. All forces except gravity will be summed up to one vector of generalized forces. This makes it possible to de®ne arbitrary functions for spring stiffness and (e.g., frictional) damping forces that may not necessarily possess a potential. A set of seven equations of motion is derived from Lagrange's equation ([12], p. 141) d @L @L _ t; Fk ~ q; ~ q; dt @ q_ k @qk

k 1 . . . 7;

6

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M. WAECHTER ET AL.

where ~ F is the vector of the seven generalized forces concerning the seven generalized coordinates. As we also include the time dependent forces (induced by the drive train and the road surface) into the vector of generalized forces ~ F, the Lagrangian function itself does not explicitly depend on time, and, consequently, can be regarded as L L~ q; ~ q_ . Switching to matrix notation (A and C) we can now write the equation of motion (6) as A~ q C~ q_ ÿ ~ B~ F

7

where Akj

@ @L ; @ q_ j @ q_ k

Bk

@L ; @qk

Ckj

@ @L ; @qj @ q_ k

k; j 1 . . . 7:

8

In this form ([12], p. 146) the system can be integrated numerically with the help of standard methods like the Runge±Kutta algorithms published in [13]. Due to the geometry of the system the quantities A, ~ B, C, and ~ F are functions of all variables and some elements additionally depend on time as a result of the uneven road surface and the drive train forces. Recalling the example of the vector ~ xfs in Equation (5) one can see that after deriving the components of Equation (7) the resulting system of equations is coupled and _ depends nonlinearly on ~ q and ~ q. The matrix A, representing masses and moments of inertia of the system, unfolds according to Equation (8) as 4 X mi @ @ _2 i @ @ 2 ~ '_ Akj x 2 @ q_ j @ q_ k i 2 @ q_ j @ q_ k i i1

fw @ @ 2 rw @ @ 2 '_ '_ : 2 @ q_ j @ q_ k fw 2 @ q_ j @ q_ k rw

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Here, especially '_ fw and '_ rw are explicitly time dependent due to the contact between tires and road surface. The vector ~ B contains gravity forces and is developed from Equation (8) as 4 X mi @ _2 i @ 2 ~ '_ x Bk 2 @qk i 2 @qk i i1

4 X fw @ 2 rw @ 2 @~ xi mi ~ : '_ fw '_ rw ÿ g ez 2 @qk 2 @qk @q k i1

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9

In this equation the ®rst line contains the derivatives of the components of T, the second contains those of U (Equations (3) and (4)). The matrix C, according to its de®nition in Equation (8), takes the following form: 4 X mi @ @ _2 i @ @ 2 ~ '_ Ckj x 2 @qj @ q_ k i 2 @qj @ q_ k i i1

fw @ @ 2 rw @ @ 2 '_ fw '_ : 2 @qj @ q_ k 2 @qj @ q_ k rw

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The vector ~ F sums up all generalized forces, in detail the spring and damper forces (including those induced by the tires), the forces in the drive train, and the external resistances (air drag and rolling resistance). Its k th element contains the contributionsPof all impressed forces ~ F to the generalized 4 ~ xi =@qk if qk is a translational, or coordinate P4 qk as Fk i1 Fi @~ Fk i1 Mi @'i =@qk , if qk is an angular coordinate ([12], p. 131). Here ~ F is Fi is the force and Mi the torque acting on the i th body. The derivation of ~ outlined in Appendix 3.5. 3. EMPIRICAL VALIDATION OF THE MODEL 3.1. Methods In order to validate the mathematical model its predictions concerning the vibrations were compared with the results of numerous measurements (a detailed description of the results can be found in [11] and [14]). The empirical basis for the comparison is a set of extensive measuring data concerning weighted accelerations on the human body, depending on bicycle design, rider mass, and road surfaces the majority of which were cycle tracks. The vibration measuring system consisted of a set of acceleration transducers mounted on the handlebars and the seat. Data were ampli®ed and recorded on a DAT tape recorder carried in a rucksack [11]. Road surfaces were measured as a set of ten to twenty longitudinal pro®les in parallel, with a distance of 1 cm from each other, for each surface. The pro®les were obtained from a laser distance sensor which was moved along a precisely aligned aluminum rod, controlled by a pocket computer. Longitudinal resolution was 1.04 mm, vertical error was less than 0.5 mm [14].

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M. WAECHTER ET AL.

The measuring length of our pro®lometer is roughly 20 m. This length is the result of a compromise: For simulation purposes the track length ought to be as long as possible whereas the distance was limited due to construction and handling problems of the measuring device. In addition, the characteristic bicycle parameters (Section 2.1) had to be determined. Moments of inertia of the bicycle parts were measured with a compound pendulum method. The locations of the centers of gravity and the inertial properties of the human bodies were derived from data reported in [15] using the mass and height of the test rider. Spring stiffness and damping constant for the hand-arm system were taken from [6] and linearly adjusted to the rider's mass. Spring and damper properties of the wheel suspensions were determined with commercial testing machines. They are shown in Figures 2 and 3. Spring and damper properties of the tires were measured in free oscillation experiments as described in [10]. The values of the parameters are listed in Appendix 3.5. Vibration measurement data consist of time series of the acceleration at the handlebars and the saddle of the bicycle. From these data frequency spectra are obtained which can be transformed into weighted vibration intensities (`bewertete Schwingstaerken', also called K-values) according to ISO 2631 [1] and VDI 2057 [2] with the help of a weight function (Fig. 4, see also [11]). The physical dimension of the K-values is m=s2 , whereas there is no unit at the vertical axes of Figures 6 and 7 according to VDI 2057. For the following reasons any comparison between the vibration measurement and the respective time series of the simulation must be carried out using mean values in the form of frequency spectra and K-values: Vibration measurements were taken over the longest possible distance of a track surface to collect as much data as possible for the respective surface. As a consequence, the distance used for acceleration measurement is about ten times the length used for the surface pro®le (20 m), and it is not possible to ®nd the vibration data which belong precisely to the surface track. So the acceleration data taken closest to the surface track were chosen for the comparison. As an example of a detailed comparison we selected four of the surfaces which are very common in everyday bicycle traf®c: old asphalt layer, concrete pavement, brick pavement, and cobblestone pavement. Two bicycles with a rather different geometry and construction were selected: The Ostrad Adagio is a long wheel base recumbent with swingarm suspension and elastomer springs on both wheels. The Radical is a city bicycle with 20 in wheels and swingarm suspension, elastomer springs and hydraulic dampers on both


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Fig. 2. Load-de¯ection curves for the Ostrad Adagio rear elastomer spring and both front rubber spring elements. Measured data points are drawn as symbols, the interpolated function as solid line.

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M. WAECHTER ET AL.

Fig. 3. Characteristic curves for the elastomer spring and hydraulic damper elements of the Radical. Measured data points are drawn as symbols, the interpolated function as solid line.


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Fig. 4. Weight Functions according to ISO 2631 and VDI 2057.

wheels. Figure 5 shows a picture of each bicycle. A more detailed description can be found in [11]. 3.2. Results 3.2.1. Comparison of K-Values 3.2.1.1. Ostrad Adagio. Figure 6 shows measured and simulated data for the Adagio on the four surfaces; the estimated measuring error is marked. For each of the four surfaces one pair of simulated and one pair of measured Kvalues (Section 3) are drawn as vertical bars in logarithmic scale, according to the human perception. Simulation and measurement correspond well for the asphalt layer. For the concrete pavement the simulated value at the handlebars is greater than the measured one, whereas the correspondence for the seat is good. On the brick pavement the calculation for the handlebars turns out to be too high, but for the seat the calculated value is smaller than measured. For the cobblestone pavement simulation and measurement correspond rather well at the handlebars, but on the other hand the K-value for the seat is again smaller than the measurement.

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Fig. 5. Ostrad Adagio (a) and Radical (b) bicycle equipped for acceleration measurement.


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Fig. 6. Comparison of K-Values for Ostrad Adagio. Riding speeds were 6.39±6.94 m / s for concrete stones 6.94±7.22 m / s for old asphalt layer, 6.67±6.94 m /s for brick stones, and 5.27± 6.53 m / s for cobblestone pavement.

The overall result for the Adagio is: The simulated K-values for the handlebars are satisfactory or too high, whereas the calculated values for the seat are satisfactory or too small, compared to the measured values. Consequences will be drawn in Sections 3.3 and 3.5. 3.2.1.2. Radical. Measured and simulated K-values for the Radical including estimated errors are shown in Figure 7. On the old asphalt surface measurement and simulation correspond rather well, the simulated value for the saddle being a little too high. For the concrete and brick surface the correspondence is very good for the saddle as well as for the handlebars. On the cobblestone pavement both values are calculated too low, and the difference is higher for the saddle. It can be summarized that the correspondence between calculation and measurement is rather good for the Radical, except for very uneven cobblestone pavement. 3.2.2. Comparison of Acceleration Spectra In Figures 8 and 9 examples of simulated acceleration spectra are compared to those from the respective measurements. There is one diagram for the spectra

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M. WAECHTER ET AL.

Fig. 7. Comparisons of K-values for Radical. Riding speeds were 6.67±7.22 m / s for concrete stones, 6.94±7.5 m / s for old asphalt layer, 6.67±7.08 m / s for brick stones, and 6.67 m / s for cobblestone pavement.

at the handlebars and one for the seat. The graphs show the frequency response of the acceleration in the range of 0 to 50 Hz. This interval contains all signi®cant parts of the acceleration, as our measurements have shown. The dimension of the y-axis is acceleration amplitude per frequency interval (m / s2 / Hz). The measuring data have been multiplied by the weight function (which is shown in Fig. 4), so that all spectra are represented according to ISO 2631 and VDI 2057. As mentioned the riding track of the vibration measurement is close but not identical to the measured surface pro®le. Therefore the acceleration spectra cannot be expected to agree perfectly. Moreover, the pro®le length limits the spectral resolution and causes some scatter. In addition to the acceleration spectra in Figure 10 we show amplitude spectra of the respective road surfaces. The frequency values on the horizontal axis are derived from the wavenumbers of the surface pro®les and the average riding speed of the acceleration measurement. The vertical axis shows relative units of pro®le height per frequency band (m / Hz). 3.2.2.1. Ostrad Adagio. For the seat the acceleration spectra of the Adagio (Fig. 8) show good correspendence between simulation and measurement


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Fig. 8. Comparison of acceleration spectra for Ostrad Adagio on old asphalt surface. Riding speed was 6.94±7.22 m / s. Estimated natural frequencies of the suspension (rear: 2.3 Hz, front: 6.8 Hz, Section 3.4) are marked by arrows.

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M. WAECHTER ET AL.

Fig. 9. Comparison of acceleration spectra for Radical on concrete stone pavement. Riding speed was 6.67±7.22 m / s. Estimated natural frequencies of the suspension (rear: 3.1 Hz, front: 4.6 Hz, Section 3.4) are marked by arrows.


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Fig. 10. Amplitude spectra of concrete stone pavement and old asphalt layer surface. Riding speeds are according to the vibration measurements. Please note the logarithmic scale of the vertical axis.

within the most important frequency range between 4 and 8 Hz (Fig. 4). For higher frequencies the acceleration is calculated remarkably lower than measured. For the handlebars we have also a rather good correspondence within the important range (which is between 8 and 16 Hz); in most cases this is also true for higher frequencies. In this case the model tends to calculate the acceleration too high. 3.2.2.2. Radical. The spectra of the Radical (Fig. 9) show ± especially for the most important frequency range from 4 to 8 Hz ± a rather satisfying correspondence between simulation and measurement for the saddle. Looking at higher frequencies the calculated accelerations are too small. In contrast to this the calculated and the measured values for the handlebars correspond rather well for the whole frequency range. 3.3. Discussion The comparison of measurement and simulation shows a satisfactory result, with the frequency weighting according to VDI 2057 applied. In the most

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M. WAECHTER ET AL.

important frequency regions for handlebars and saddle (or seat) the simulation corresponds rather well to the measurement; this is also proven by the Kvalues derived from the spectra. Especially the relative differences between the measured vibrational stress values of the bicycle-rider systems under investigation are reliably predicted. However, there are discrepancies which reveal possible limitations of the model and ± even more fundamentally ± which raise doubts whether the bicycle-rider system can be described completely. In the acceleration spectra of the saddle or seat the difference between measurement and simulation is noticeable mostly in the region above approximately 10 Hz. This is probably a result of the simpli®ed model of the human body which has been used and which does not contain any degree of freedom for the vibration characteristics of the trunk. The vibrational properties of the human body seem to have a bigger in¯uence than initially supposed. In contrast to motorized vehicles the rider's body represents the biggest part of the total mass of the system and may be important for its vibrational characteristics as well. The human body shows a very complex vibrational behaviour if one takes into account the high number of oscillating masses and the interaction between them [16]. Furthermore, the body reacts actively to vibrational stimuli. At the handlebars the differences between measured and simulated acceleration amplitudes are smaller than at the saddle or seat. For the Ostrad the model tends to calculate the acceleration too high at the handlebars. The reason for this might be found in the fact that the model does not assume a static force from the arms on the handlebars for recumbent bicycles. For the Radical there is no such tendency. Because of the upright position the model assumes a damping force resulting from the static force of the arms on the handlebars. This result seems to match with reality. For high frequencies above approximately 40 Hz the accelerations in all calculations are higher than in the respective measurements. Possible reasons for this deviation may be the simpli®ed tire model used in the simulation or structural vibration of the frame which is not described by our model. The difference, however, does not impair the intended purpose, namely to predict vibrational stress on human bodies. If the weight function is applied the mentioned frequency range is rather insigni®cant as can be seen in Figures 8 and 9.


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3.4. Linear Analysis Taking the deviations from the measurement in some frequency ranges seriously two methods of simpli®ed analysis were applied in order to localize possible sources of error. 3.4.1. Natural Frequencies of Front and Rear Suspension Using a simpli®ed model for the front and rear half bicycle system in the con®gurations used for the simulations, natural frequencies for the suspension system were estimated. The bicycle-rider system is regarded as two separate point masses each combined with a linear spring on which the (front or rear) total wheel load is acting. A factor de®ning the spring travel per wheel elevation was determined for the reference con®guration when only static load is present. Combined with the local spring stiffness the effective suspension stiffness at the wheel was derived. Considering the tire stiffness natural frequencies of these simpli®ed systems were calculated. For the Ostrad Adagio the results are 6.8 Hz (front) and 2.3 Hz (rear) and for the Radical 4.6 Hz and 3.1 Hz, respectively. The static load points are marked in Figures 2 and 3, the system parameters are given in Table 3 in Appendix B. For the Ostrad Adagio the estimated rear natural frequency is close to a peak in the simulated acceleration spectrum at the seat as shown in Figure 8, while there is no correspondence at the front. It seems plausible that there is no substantial excitation of the seat acceleration sensor caused by front wheel vibration. The simulated saddle spectrum of the Radical shows two peaks close to the estimated front and rear natural frequencies; they are not visible in the spectrum at the handlebars due to the weight function. In order to determine the natural frequencies of the swingarm-wheel systems the swingarm body is considered as oscillating between tire and suspension spring. The horizontal distances between wheel hubs and swingarm pivots are given in Table 3 in Appendix B. Using the effective suspension stiffnesses derived above the results of the calculations are 57 Hz (front swingarm) and 35 Hz (rear swingarm) for the Ostrad, and 56 Hz and 34 Hz for the Radical, respectively. While the front resonance frequencies can be neglected as they are beyond 50 Hz there are acceleration peaks at about 30±35 Hz in Figures 8 and 9. They are clearly visible for the Radical due to a peak in the excitation spectrum of the concrete stone pavement (Fig. 10).

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M. WAECHTER ET AL.

To sum up these ®ndings it can be stated that the numerical results of the linearization are satisfactorily con®rmed by the simulation model. 3.4.2. Modal Analysis For modal analysis the dynamic equations for the bicycle-rider system were numerically linearized and a simpli®ed set of equations of motion was derived as M~ q K~ q ~ 0

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where M is the matrix of masses and moments of inertia and K the matrix of spring stiffnesses. For the bicycle con®gurations used the natural frequencies and the modes of this linear model are listed in Table 4 in Appendix C. The mode vectors are normalized with respect to their largest element, and the elements are given in the same order as stated in Section 2.1. The ®rst mode with a very low frequency ( 1 Hz) for both bicycles is mainly associated with the xcf coordinate (horizontal location of the central frame's center of gravity, Section 2.1) where no restoring force is present. The highest natural frequency (> 1 kHz) cannot be uniquely worked out, but it is situated well beyond the interesting range of 1±50 Hz. If we focus on the interpretation of the Ostrad values only two modes can be found in the relevant frequency range (at 5 and 10.6 Hz) which cannot be retrieved from the spectra. These modes are mainly associated with the front swingarm angle fs whereas the other amplitudes remain relatively small. Obviously, the construction of the Adagio as a long wheel base recumbent with under seat steering prevents the seat region from being in¯uenced by front wheel vibration to a large amount. In contrast to the Ostrad four of the Radical modes (one of them twofold) lie between 1 and 50 Hz. While the ®rst one can be identi®ed in the saddle spectrum the others are less clearly visible. These modes are associated with the fs and rs coordinates and seem to be suppressed by the high amount of damping introduced by the hydraulic suspension dampers, especially at higher frequencies. The results of the modal analysis, as far as they are signi®cant for the empirical validation of the model, show that the reason for differences between simulation and measurement will probably not be found in the mathematical model.


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4. CONCLUSIONS With the correspondence between measurement and simulation as shown above the mathematical model for the bicycle-rider system turns out to be adequate for the design and development of bicycle suspensions. For the intended purpose the prediction for vibrational stress and vibrational behaviour is suf®ciently precise. Further improvements of the model should include more advanced models of the human body and the bicycle tires. A more detailed evaluation of the model could be carried out by extending the length of the measured surface pro®les signi®cantly and by performing acceleration and pro®le measurement on exactly the same track of the road surface. ACKNOWLEDGEMENTS Part of this work was ®nancially supported by Stiftung Industrie Forschung (Foundation for Industrial Research), Cologne/Germany. Essential improvements of the paper were initiated by the remarks of the referees. REFERENCES 1. ISO: Guide for The Evaluation of Human Exposure to Whole-Body Vibration. International Standard 2631, International Organisation for Standardization, 1985. 2. VDI: Einwirkungen mechanischer Schwingungen auf den Menschen (Effects of Mechanical Vibrations to Humans). VDI-Richtlinie 2057, Verein Deutscher Ingenieure, Duesseldorf, 1987. 3. Mitschke, M.: Dynamik der Kraftfahrzeuge (Motor Vehicle Dynamics), Part B, Schwingungen (Vibrations), 3rd ed., Springer Verlag, Berlin, 1997, Chapter 2. 4. Wang, E.L. and Hull, M.L.: A Model for Determining Rider Induced Energy Losses in Bicycle Suspension Systems. Vehicle System Dynamics 25 (1996), pp. 223±246. 5. Wang, E.L. and Hull, M.L.: Minimization of Pedaling Induced Energy Losses in Off-road Bicycle Rear Suspension Systems. Vehicle System Dynamics 28 (1997), pp. 291±306. 6. Wong, M.G. and Hull, M.L.: Analysis of Road Induced Loads in Bicycle Frames. Journal of Mechanisms, Transmissions, and Automation in Design 105 (1983), pp. 139±145. 7. Bossel, D.: Ueber Stock und Stein. Welche Kraefte wirken beim Ueberfahren von Hindernissen auf einen Zweiradrahmen? (Forces Induced to a Bicycle Frame by Crossing Obstacles). Diploma Thesis, Mechanical Engineering Department, University of Kassel, 1993. 8. Gross, E.: Betriebslastenermittlung, Dimensionierung, strukturmechanische und fahrwerkstechnische Untersuchungen von Mountainbikes (Loads, Dimensioning, Structural Mechanics

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9. 10. 11. 12. 13. 14.

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and Chassis Investigations for Mountain Bikes). Fortschritt-Berichte, Series 12, No. 308, VDI-Verlag, Duesseldorf, 1997. Pivit, R.: Vibrational Stress on Cyclists. Human Power 7:2 (1988), pp. 4±6. Waechter, M.: Modellierung der Bewegungsgroessen bei Fahrraedern mit gefederten Laufraedern (Modelling of the Motion Quantities for Bicycles with Wheel Suspension). Diploma Thesis, Faculty of Physics, Carl-von-Ossietzky University, Oldenburg, 1994. Waechter, M., Zacharias, N. and Riess, F.: Measurement and Simulation of the Vibrational Stress on Cyclists. Proceedings of the 3rd European Seminar on Velomobile Design, August 1998, Roskilde, Denmark, ISBN 87-987188-0-0. D'Souza, F.A. and Garg, V.K.: Advanced Dynamics: Modeling and Analysis. Prentice-Hall, Englewood Cliffs, New Jersey, 1984. Press, W.E., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P.: Numerical Recipes in C. 2nd ed., Cambridge University Press, Cambridge, UK, 1992, Chapter 16. Waechter, M., Zacharias, N. and Riess, F.: Abschlussbericht des Forschungsprojektes ``Simulation des Schwingkomforts gefederter Fahrraeder'' (Final Report of the Research Projekt ``Simulation of the Vibrational Comfort of Bicycles with Suspension''). Faculty of Physics, Carl-von-Ossietzky University, Oldenburg, 1999. Pheasant, S.: Bodyspace: Anthropometry, Ergonomics and Design. Taylor & Francis, London and Philadelphia, 1986, Chapter 6. Anon.: Humanschwingungen (Human Body Vibrations). Bruel & Kjaer, Naerum, Denmark, 1989.

APPENDIX A Generalized Forces ~ F Saddle and hand-arm system forces ~ Fsaddle and ~ Farm are acting on ~ xr , their torques Msaddle and Marm acting on 'r are given by multiplication with the fraction of the trunk length perpendicular to the respective force. , which is derived The drive train forces can be divided into the chain force Fch from the actual bottom bracket torque and the chainwheel radius, and the Fch rsp =rrw with the sprocket and rear wheel radii rsp propulsive force Fpro and rrw . Fpro is acting on~ xrs with its direction being the unit vector of the rear hub acting on 'rs is given by Fpro multiplied with the velocity~ evrh. The torque Mpro . distance between wheel hub and swingarm pivot, situated perpendicular to ~ Fpro ~ ~ The tire forces Fft and Frt point in vertical direction and contain spring and xrs . The torques Mft and Mrt are damper forces of the tires. They act on ~ xfs and ~ given by the force multiplied by the horizontal distance between wheel hub and swingarm pivot, and they act on 'fs and 'rs . Suspension forces are converted to torques Mfs and Mrs acting on the swingarm angles fs and rs .

25


Summarized the vector ~ F takes the following form. Each element is preceded by the respective generalized coordinate. 9 ÿ @~ xr > > Fsaddle ~ Farm zr : F1 ~ > > @zr > > > ÿ > @' r > > r : F2 Msaddle Marm > > @r > > > > @~ x rs > > xcf : F3 ~ Fpro > > > @xcf > = @~ x @~ x @~ x fs rs rs ~ ~ ~ 13 zcf : F4 Fft Frt Fpro > @zcf @zcf @zcf > > @' > @'fs > rs > > cf : F5 Mft Mrt Mpro > > @cf @cf > > > > @' @' fs fs > > fs : F6 Mft Mfs > > @fs @fs > > @' > > @' rs rs > ; rs : F7 Mrt Mpro Mrs @rs @rs For clarity external resistance forces have not been shown as they do not contribute to vibrational stress. They can be derived repeating the same procedure as above. APPENDIX B Parameter Values Table 1. Parameter values for the Ostrad Adagio and Radical bicycle, part I. Ostrad Adagio Rider Mass Moment of inertia Distance seat ± shoulder Distance seat ± center of gravity (CoG) (x) Distance seat ± CoG (z) Distance seat ± shoulder (x) Distance seat ± shoulder (z) Distance shoulder ± handlebars Hand-arm system damping constant Hand-arm system spring stiffness

96.0 kg 10.897 kgm2 0.612 m ÿ0.040 m 0.308 m ÿ0.306 0.530 ± ± ±

Radical 96.0 kg 10.897 kgm2 0.612 m 0.273 m 0.147 m 0.385 m 0.475 m 0.611 m 680.0 Ns/m 30.5 kN/m

26

M. WAECHTER ET AL.

Table 1. (continued) Ostrad Adagio Central frame (including all accessories) Mass Moment of inertia Angle of seat back support Length of seat back support Distance bottom bracket ± CoG (x) Distance bottom bracket ± CoG (z) Distance CoG ± rear swingarm pivot (x) Distance CoG ± rear swingarm pivot (z) Distance CoG ± front swingarm pivot (x) Distance CoG ± front swingarm pivot (z) Distance CoG ± seat (x) Distance CoG ± seat (z) Distance CoG ± handlebars (x) Distance CoG ± handlebars (z) Distance CoG ± rear spring mounting hole (x) Distance CoG ± rear spring mounting hole (z) Distance CoG ± front spring mounting hole (x) Distance CoG ± front spring mounting hole (z) Seat backrest spring stiffness Seat backrest damping constant Seat spring stiffness Seat damping constant

11.5 kg 3.04 kgm2 60.0 0.400 m ÿ0.680 m ÿ0.100 m ÿ0.854 m ÿ0.167 m 0.116 m ÿ0.353 m ÿ0.884 m 0.045 m ÿ0.884 m ÿ0.015 m ÿ1.013 m ÿ0.014 m 0.137 m ÿ0.320 m 30.0 kN/m 1000.0 Ns/m 40.0 kN / m 1000.0 Ns / m

Radical 8.530 kg 1.25 kgm2 ± ± 0.100 m 0.250 m ÿ0.044 m 0.153 m 0.466 m ÿ0.032 m ÿ0.299 m 0.740 m 0.423 m 0.705 m ÿ0.137 m 0.325 m 0.518 m 0.195 m ± ± 46.7 kN / m 1000.0 Ns / m

Table 2. Parameter values for the Ostrad Adagio and Radical bicycle, part II. Ostrad Adagio Rear swingarm system (including wheel and accessories) Mass 5.85 kg Moment of inertia1 0.184 kgm2 Distance pivot ± center of gravity (x) ÿ0.374 m Distance pivot ± center of gravity (z) ÿ0.162 m Distance pivot ± wheel hub (x) ÿ0.460 m Distance pivot ± wheel hub (z) ÿ0.208 m Distance pivot ± spring mounting hole (x) ÿ0.146 m Distance pivot ± spring mounting hole (z) ÿ0.030 m Rear wheel radius 0.250 m Rear wheel moment of inertia1 0.054 kgm2 Rear tire stiffness 200.0 kN/m Rear tire damping constant 250.0 Ns/m

Radical 3.65 kg 0.065 kgm2 ÿ0.300 m ÿ0.150 m ÿ0.370 m ÿ0.196 m ÿ0.179 m 0.012 m 0.250 m 0.060 kgm2 134.0 kN/m 272.0 Ns/m

27


Table 2. (continued) Ostrad Adagio

Radical

Front swingarm system (including wheel and accessories) Mass 2.350 kg Moment of inertia1 0.018 kgm2 Distance pivot ± center of gravity (x) 0.095 m Distance pivot ± center of gravity (z) 0.005 m Distance pivot ± wheel hub (x) 0.140 m Distance pivot ± wheel hub (z) ÿ0.022 m Distance pivot ± spring mounting hole (x) 0.030 m Distance pivot ± spring mounting hole (z) 0.013 m Front wheel radius 0.250 m Front wheel moment of inertia1 0.054 kgm2 Front tire stiffness 200.0 kN / m Front tire damping constant 250.0 Ns / m Chainwheel teeth Sprocket teeth

2.300 kg 0.012 kgm2 0.093 m 0.002 m 0.148 m ÿ0.008 m 0.068 m 0.014 m 0.250 m 0.060 kgm2 134.0 kN / m 272.0 Ns / m

52 15

52 16

Rear elastomer damping factor2

0.15

±

Front elastomer damping factor2 Average riding speed

0.05 7.08 m / s

± 6.94 m / s

Characteristic curves of the suspension spring and damper elements are shown in Figures 2 and 3. The wheel is allowed to rotate independently from the swingarm. 2 Elastomer damping forces are commonly regarded as proportional to the spring force: Damping Force ± (Damping Factor) (Spring Force). 1

Table 3. Suspension Parameters of both bicycle con®gurations. Ostrad Spring de¯ection (mm) Effective spring stiffness (kN/m) Spring de¯ection per wheel elevation Effective stiffness at the wheel (kN/m) Wheel load (N) Horizontal distance pivot-hub (m)

Radical

Front

Rear

Front

Rear

4.65 248

59.8 74.6

6.96 111

13.2 70.4

0.232

0.285

0.457

0.437

57.4

21.2

50.6

30.8

241 0.141

895 0.500

424 0.148

667 0.382

Suspension Parameters of both bicycle con®gurations with only static loads present.

28

M. WAECHTER ET AL.

APPENDIX C Modes and Natural Frequencies of the Linearized System Table 4. Modes and natural frequencies of the linearized model. Ostrad Adagio Frequency (Hz) Mode Vector zr r xcf zcf cf fs rs

0.15

1.30

5.00

10.56

51.18

55.69

(2603)

ÿ0.00 ÿ0.00 ÿ1.00 0.00 ÿ0.00 0.01 0.01

ÿ0.10 ÿ0.16 0.66 ÿ0.35 0.20 1.00 ÿ0.94

ÿ0.04 ÿ0.08 ÿ0.19 ÿ0.02 ÿ0.14 1.00 0.12

0.13 ÿ0.00 ÿ0.01 ÿ0.13 ÿ0.02 1.00 ÿ0.20

ÿ0.03 0.08 ÿ0.02 0.01 ÿ0.08 ÿ1.00 0.62

0.03 ÿ0.01 ÿ0.00 ÿ0.04 0.00 0.10 ÿ1.00

ÿ0.04 0.12 ÿ0.06 ÿ0.01 ÿ0.18 1.00 ÿ0.18

Radical Frequency (Hz) Mode Vector zr r xcf zcf cf fs rs

0.19

1.91

1.91

16.06

38.59

45.72

(3374)

ÿ0.00 0.00 1.00 ÿ0.00 0.01 ÿ0.04 ÿ0.02

0.04 0.16 0.25 0.06 0.33 1.00 0.44

0.04 0.16 0.25 0.06 0.33 1.00 0.44

ÿ0.18 0.00 0.01 0.16 0.01 ÿ1.00 0.35

0.00 ÿ0.07 0.04 0.03 0.08 ÿ0.37 1.00

ÿ0.02 0.07 ÿ0.04 ÿ0.01 ÿ0.07 1.00 ÿ0.33

0.06 ÿ0.26 0.14 0.05 0.27 ÿ1.00 0.35