Motion analysis of a bicycle rear derailleur during the shifting process

~

Pergamon

Mech. Mach. Theory Vol. 33. No. 4, pp. 365-378, 1998 © 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0094-114X(97)00045-1 oo94-114x/98 $19.oo + o.oo

PII:

MOTION ANALYSIS OF A BICYCLE REAR DERAILLEUR DURING THE SHIFTING PROCESS W. H. LAI and C. K. SUNG Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan 300, Republic of China

and J. B. WANG Department of Mechanical Engineering, Nan-Jung Institute of Technology and Commerce, Tainan, Taiwan 737, Republic of China (Received 22 November 1995; in revised form 3 March 1997)

Abstract--This paper presents a comprehensive theoretical and experimental investigation into the motion characteristic of the guide pulley of a typical rear derailleur during the shifting process. This characteristic can serve as a performance index of precise engagement and smooth shifting and can also be specified during the design stage. By observing the phenomenological behavior of the principal elements of the derailleur, a four-mode mathematical model was developed to illustrate the motion characteristics of the guide pulley during the shifting process. The equations governing the motion of the guide and tension pulleys were derived by using Lagrange's method. These equations were then employed to predict the motion characteristic of a derailleur as it shifted at different cycling speeds. Finally, an experimental rig was constructed and the motion of the guide and tension pulleys was measured using two encoders mounted on the shaft of each pulley for verification. This study provides a scientific basis for the systematic design of derailleurs and the related components, such as the spring constants of the B and P springs, the tooth profile of each sprocket, and the configuration of the changer mechanism which keeps the guide and tension pulleys moving along the designated path to guarantee precise and smooth shifting. © 1997 Elsevier Science Ltd. All rights reserved

NOMENCLATURE a~, a:, a3, a4--1engths of the chain in contact with the sprockets or pulleys b--distance between points H and E dz, dr--coefficients of rolling friction of the front and rear tires fi, fr--friction forces applied on the front and rear tires F--pedaling force exerted by the rider Fw--aerodynamic resistance applied on the bicycle h~--height where the aerodynamic resistance applies h~, h2, h3--horizontal and vertical positions of the center of gravity It, /,---mass moment of inertia of the flywheel and tire kw--radius of gyration of the fan Kcq-----equivalent spring constant of the shift-control cable K~, K~-spring constants of the B and P springs /--length of the crank of the bicycle Lo--total chain length l~, 12, 13, 14--1engths of the chain not in contact with the sprockets or pulleys 12o, 12~-lengths of the tangential lines from the target sprocket to the previous sprocket and guide pulley, respectively l:~--length of the chain in contact with the target sprocket m--mass of the bicycle including the rider Nw-normal contact force applied on the flywheel N~, Nr--normal forces applied on the front and rear tires, respectively Q¢, Q,~--generalized forces corresponding to the coordinates ~ and q~t r--radii of the guide and tension pulleys R, Rw-radii of the rear wheel and flywheel of the fixed-frame model R~, R:, R:*--radii of the front, previous and target sprockets Tj--kinetic energy generated by the front sprocket set, free wheel and tires T3--kinetic energy generated by the flywheel Vk--potential energy generated by the torsional springs B and P T2, V~--kinetic and potential energy of the guide pulley T~, V:--kinetic and potential energy of the chain 365

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To,, Vc~--kineticand potential energyof the changer mechanism To2, V~2--kineticand potential energyof the cage w--rotational speed of the fan c~--angular accelerationof the rear wheel fls--slope angle 0--rotation angle of the cage ~--rotation angle of the freewheel ¢0--rotation angle at which the shiftingprocess starts z--torque acting on the rear wheelof the bicycle •t, z~--preloads applied on the P and B springs zr--torsional resistance applied on the flywheel qb~--angle betweenthe lines OE and OA

1. INTRODUCTION Different from the chain drive of automotive power transmissions, the chain of the bicycle derailleur must have adequate clearance between every pair of bushing and pin in order to achieve excellent maneuverability. The design of the derailleur may directly influence the quality of precise engagement and smoothness during the shifting process. Although there are only a few components in the system, a particular matching behavior between chain and sprockets exists. This involves the phase difference between adjacent sprockets, and is also influenced by tooth profile, positioning of the guide pulley, and the tension in the chain. Different from the configuration of the chain of the regular chain drives, the chain of the bicycle derailleur must possess adequate clearances between the pin and hole. Since the R&D documents of the derailleur design are mostly classified, information can only be obtained from patent documents, commercial catalogs and non-academic magazines. The purpose of this paper is to provide industry with some theoretical background information and guidelines for derailleur design. Attention will be specifically focused on geometrical configuration and the motion characteristics of the guide and tension pulleys during the downshifting process. By observing the phenomenological behavior of the principal elements of the derailleur, a four-mode model is developed for describing the motion characteristics of the downshifting process. The downshifting process involves the following steps: the changer mechanism moves the guide pulley into a specific position; the guide pulley brings the chain onto the desired sprocket according to the designated path; the chain starts to seat onto the desired sprocket; finally, the entire chain is shifted to the desired sprocket to complete the shifting process. The equations of motion corresponding to the four modes are then derived by using Lagrange's equation. Computer simulation for predicting the motion of the guide and tension pulleys is then performed. Finally, an experimental rig is fabricated and the motion of the two pulleys is measured using two encoders mounted on the shafts of the pulleys to verify the validity of the proposed analytic method. 2. P H Y S I C A L M O D E L I N G OF T H E R E A R D E R A I L L E U R

Figure 1 presents a rear derailleur system consisting of the following components: freewheel with multiple sprockets, changer mechanism, shift-control cable, guide and tension pulleys, and a pulley cage. The shifting process is accomplished in four steps: 1. The rider moves the shifting lever to result in the changer mechanism positioning the guide-pulley in the desired location, so that the guide pulley can pull the chain into a position ready for shifting. 2. The guide pulley exerts a lateral force which moves the chain towards the desired sprocket until the downshifting point is reached. The downshifting points are defined as the loci of several hollowed out shapes called "pockets". When the chain reaches the nearest downshifting point, it fits snugly into the pocket and is then lifted onto the sprocket. 3. The cage rotates to adjust the free-chain length, defined as the sum of the parts of the chain that do not make direct contact with the sprocket. The tooth profile can be designed to permit smooth transfer of the chain to the desired sprocket. 4. The entire chain is transferred to the desired sprocket to complete the shifting process.

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367

mechanism

Fig. 1. The bicycle rear derailleur.

Because the shifting process of derailleur systems are rather complex, the following assumptions are made for simplifying the physical model so that the mathematical expressions of the system can be derived: 1. The entire shifting process is performed in the same plane. 2. The polygon effect on the motion of the chain can be ignored. 3. The flexibility of the chain, the teeth of the sprocket and the guide pulley can be omitted from the model. 4. The B and P springs of the derailleur operate within the linear range. 5. The rotary inertia of each pitch of the chain about its center of mass can be neglected [1, 2]. 6. The chain is tightly stretched by the actuation of the tension pulley during the shifting process• 7. There is no relative motion between the sprocket and chain because the center of a seated roller is coincident with the center of the sprocket seating curve•

Fig. 2. Physical model of the rear derailleur.

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I1. before Mi°de 2. from shiftingstarted to point I.

Mode I frompoints I~ to I

[

[ frompointsitoJ

[MI~e [ from points J to M

~

shiftingcomplete

] Fig. 3. The shiftingprocess.

Figure 2 presents the physical model of the rear derailleur. That is: 1. The spring constants of the B and P springs are represented by K~ and K2, respectively. Ken indicates the equivalent spring constant of the shift-control cable due to elastic deformation. 2. Except when the changer mechanism is in motion, it is considered as one link designated by link i, and the pulley cage is indicated by link j. 3. There is no relative motion between points, A, G, O, E and the frame. 4. The chain smoothly surrounds the sprocket as the polygon effect is neglected.

Mode

-- _ ~ ~ . ~ ~ " " ~

MllO~e~I IVo M Jd _e..2:.~=..~-:-- -- ~

km../~________~.._R~

Fig. 4. Definitionof the four-modemodel.

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3. A F O U R - M O D E MODEL FOR THE SHIFTING PROCESS

Figure 3 illustrates the procedures of the shifting process corresponding to the four-mode model. The motion of each mode may be described as follows:

3. I. Mode I configuration Mode I occurs at three different time instants as shown in Fig. 4: (1) at the moment before the shifting lever is moved; (2) at the moment when the shifting lever is moved but the chain has not engaged onto the desired sprocket, i.e. has not reached the shifting point; (3) at the time after the shifting process is completed. The length of link i at the first instant is not the same as link i at the second and third stages. Because the period between the time when the rider moves the shifting lever and the chain reaches the downshifting point can vary, and it does not affect the shifting motion, it is assumed that the time between them is zero, e.g. as soon as the rider moves the shifting lever, the chain starts downshifting immediately. 3.2. Mode H configuration As shown in Fig. 4, Mode II occurs from chain pitch S at position I0, e.g. as the chain just catches the desired sprocket, to position I, e.g. the common tangent of the guide pulley with a radius of r and the desired sprocket with a radius of R2*. The position of the guide pulley and the cage continuously changes in this mode, the guide pulley is dislocated and the common tangent is also new as the process of this mode is just completed. For clearly expressing the period of this mode, an alternative parameter Q is used to indicate that Mode II occurs from Q = n - sin-~ R2/R* to Q = n/2, where R2 is the radius of the previous sprocket. 3.3. Mode III configuration Mode III occurs as chain pitch S moves from position " I " to "J" where the extension of the common tangent of the front and rear sprockets R~ and R2 intersects with the pitch circle of the sprocket R2*. In this mode the chain is simultaneously engaged on both sprockets R2 and R*, and will escape from R2 at the instant that S just passes position "J". 3.4. Mode I V configuration Mode IV occurs when the chain pitch S moves from position "J" to " M " , where the common tangent of sprockets R, and R? intersects with the pitch circle of the sprocket R*. The shifting process completes at position " M " .

4. D E R I V A T I O N OF THE GOVERNING E Q U A T I O N S

Having developed the four-mode model of the derailleur system, the equations governing the motion of the guide and tension pulleys will be derived in this section. The rear wheel of a bicycle in actual riding condition is subjected to aerodynamic resistance, rolling resistance of tires, grade and acceleration resistance, etc. In this study, a fixed-frame model was constructed for the simulation of the actual riding condition and the design of the experimental rig. The governing equations were then derived based on the aforementioned four-mode model.

4.1. A fixed-frame model for the simulation of actual riding conditions Figure 5 illustrates the major forces acting on a bicycle. The coefficients of rolling friction of front and rear tires are represented by dt and d2;/~ is the slope angle; and the normal and friction forces acting on the front and rear tires are indicated by N~, jq, N2, and jr, respectively. Having defined all parameters which may be obtained from direct measurement, the actual riding conditions may now be expressed by the fixed-frame model shown in Fig. 6. The mass moment

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N2 Fig. 5. Forces acting on the bicycle.

of inertia of the flywheel, L, and the radius of gyration of the fan, kw, and the torsional moment, rr, and normal contact force, Nf, acting on the flywheel may be determined as follows: 1. The mass moment of inertia of the flywheel

R

Ir ~

= It

, . f mh3R "~ "5 mR 2 + (dx + a2)|h----Z-'E'.]

(1)

V'I ± "2/

where R and Rr are the radii of the rear wheel and the flywheel of the fixed-frame model, respectively; It and m are the mass moment of inertia of the tire and the mass of the bicycle including the rider; h~, h2 and h3 are the horizontal and vertical positions of the center of gravity with respect to the frame. 2. The torsional resistance acting on the flywheel

Zr = mgI(d~ + d2)(h2 c°s fls - ha sin fls) h~ + h2

q- sin

flsR - cos flsd2

where g is the gravitational acceleration.

..

Fan~ y w h c c l

(R,,T,)

Fig. 6. The fixed-framemodel.

F

1

(2)

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3. The radius of gyration of the fan, kw, employed for the simulation of the aerodynamic resistance acting on the flywheel [3]. kw~o2=Fw R

h.Lu: +

h,+h:

]

(3)

where co, Fw, and hw are the rotational speed of the fan, aerodynamic resistance applied on the bicycle, and the height that the aerodynamic resistance applies, respectively. 4. The normal contact force acting on the flywheel by the rear wheel

( mh3R ~ot Img(hz cos [3~- h3 sin [3~)- F~h,~1 Nf = mg cos fls - \ h, + h2] - " ~ + h2

(4)

where ~ is the angular acceleration of the wheel.

4.2. Governing equations In this analysis, four parameters vary during the shifting process as shown in Fig. 7. They are: b: the distance between points H and E, which varies in accordance with the configuration of the changer mechanism, ~b~: the angle between the lines OE and OA, due to the rotation of the guide pulley, 0: the rotation angle of the cage, ~: the rotation angle of the freewheel. Where only ~b~ and ~, representing two degrees of freedom, need to be determined during the shifting process as mentioned previously; b, as a design variable, may be designated at design stage; and 0 is a function of q~ and (, which may be resolved from the constraint condition that the total length of the chain remains constant. Except for the design variable b, the other three parameters, 0, ~b, and ¢, corresponding to the configuration of each mode, will be determined from the geometrical constraint and Lagrange's equations. 4.2.1. The governing equations of the Mode I configuration. The origin of the axis system is located at point O and OA is the horizontal axis. Other parameters are defined as shown in Fig. 7. O bo/,..~ ' T,~' 3 "

_

/1 a2 a!

12

r

/4 a4

Fig. 7. Definition of the parameters of Mode I configuration. MAMT 33/4--B

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The constraint condition of the constant total chain length, L0, is employed for resolving the angle ~ and may be written as

Lo = a~ + a2 + a3 + a, + l, + 12 + 13 + 14 =riO, dp~)

(5)

where a,, i = 1, 2, 3, 4, are the lengths of the chain in contact with the sprockets or pulleys; and L, i = 1, 2, 3, 4, are the lengths of the chain that are not in contact with the sprockets or pulleys. The equations governing the motion of the guide pulley and freewheel, 4~ and ¢, respectively, may be derived by employing Lagrange's equation

-dt - ~

--~=

Fl

d d~ ~

-

g r-r,----rr

E

~, = -r~-r~

where

L=T-V T = T~+ T2+ T3+ T4+ T¢~+ To2

V = Vk+ 1/1+ G + G , + Vc2 d II2+a3-Rffe~+63+q~) ] TI: the kinetic energy generated by the front sprocket set, free wheel and tires,

T3: the kinetic energy generated by the flywheel, Vk: the potential energy generated by the torsional springs, B and P, T2, VI: the kinetic and potential energy of the guide pulley, respectively, T4, V2: the kinetic and potential energy of the chain, Tel, VcI: the kinetic and potential energy of the changer mechanism, To2, Vo2: the kinetic and potential energy of the cage, F: the pedaling force exerted by the rider, 1: the length of the crank of the bicycle T: the torque acting on the rear wheel of the bicycle, Tt, "CS: the preloads acting on the P and B springs, respectively, II

a3

13

a4

Fig. 8. Definition of the parameters of Mode II configuration.

(6)

(7)

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373

4.2.2. The governing equations of the Mode H configuration. Figure 8 indicates the related parameters for the derivation of the governing equations of the Mode II configuration. Again, the total length of the chain must be constant and, in addition, the constraint equation for determining the angle, 0, is related to not only the angle, q~, but the rotation of the freewheel, ~. Lo = a, + a2 + as + a4 + l, + 120+/22 + ls + L =f(O, ok,, ~)

(8)

where 120and/22 are the lengths of the chain among the sprockets R2, R2* and guide pulley during shifting. The equations governing the motions of the guide pulley and the freewheel are derived again by employing Lagrange's equation

d'-tt O ~

- - ~ = FI

d (aL'~

- T~

- zd r, - z -~,

aL

(9)

(lo)

where L-T-V T = T j + T2+ T3 +/'4 + T~, + To2 V=Vk+

V,+ V2+ V~j+ V~2

f, _ O(l,2 + as)/r f2 = O(la + as)/r ~)1 4.2.3. The governing equations of the Mode III configuration. Figure 9 defines the parameters for the derivation of the governing equations of the Mode III configuration. Similarly, the total length of the chain is constant and the constraint equation for determining the angle, 0, is related to the angles, ~bl and ~. Lo = a, + a2 + a3 + m + A + l~o + l~, + h: + l, + h = f(O, 4'~, ~)

Ii 12o

l

13~.~ a3 a4( C'i.~r ~~

14

.---

Fig. 9, Definition of the parameters of Mode III configuration.

(ll)

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et al.

The equations governing the motions of the guide pulley and the freewheel are derived again by employing Lagrange's equation d aL 0L d--tt - ~ - - ~ = FI

-j';(zg +

R~ zt) -- z - -

(12)

d {~L'~ oL ~ , ~ 1 - ~ = - ~ ( ~ ' + ~')

(13)

where L=_T-V

T = TI + T2+ T3+ T . + Td + T¢2 V = Vk+ I/1+ 1/"2+ V~+ V~2

f3 = ~-~ [ 12z + a3 - Rz*(~) + 63 +

~ [122+a3-R2*(flp+O3+qO]

Y'=~

r

4.2.4. The governing equations of the Mode I V configuration. Figure 10 illustrates the parameters for the derivation of the governing equations of the Mode IV configuration. The constraint equation for determining the angle, 0, is identical to equation (11). Similarly, the equations governing the motions of the guide pulley and the freewheel are written dt \ 3 ¢ J - - ~ = FI ~ c o s (p, -p~*)

(14)

-)q(zg + z,) - z R---7

(15) where

L=T-V

T = T~ + T2+ T3+ T4+ T~ + T~2 V = V k + V I + V 2 + V ¢ I + Vo2

II al 121

a3

a4 l

~

Fig. I0. Definition of the parameters of Mode IV configuration.

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375

5. C O M P U T E R S I M U L A T I O N The above equations can be expressed in the following three forms:

I_,o = f(O,

\-~]

d-~ ~

~,,

~)

(16)

- - ~ = Q~

(17)

(18)

-~-~=Q~'

These equations are sets o f coupled non-linear equations and the solutions are almost not attainable. Fortunately, they can be simplified because the magnitude o f the terms involving the velocity ~ and acceleration q~'~is small in c o m p a r i s o n with that o f the terms containing the spring forces. In addition, the conservative-force terms are also relatively small. These terms can be neglected to simplify the governing equations. This analysis was directed towards determining the angular displacements o f the guide pulley and cage, 0 and (k~, versus the freewheel rotation, ¢, during the downshifting process. The results can reflect the shifting performance for the evaluation purpose. The solution procedure is as follows: 1. Designate a value ¢ and substitute it into equation (18) to result in an ordinary differential equation with the parameters 0 and q~. 2. C o m b i n e equation (18) with (17) to yield the values o f 0 and ~b~ corresponding to the designated value ~. 3. Repeat steps (1) and (2) for the different ~ values until the shifting process is completed. In this study, a S h i m a n o - D X type rear derailleur is adopted as an example for the c o m p u t e r simulation and experimental verification. The dimensions o f the derailleur are listed in Table 1.

Table 1. The dimensions of a Shimano-DX Parameter Radius of the front sprocket (24T), R~ Radius of the rear sprocket (30T), R2 Radius of the rear sprocket (24T), R2 Radius of the guide pulley (10T), r Radius of the tension pulley (10T), r Length of link I, b Initial angle of link i, q~o7 Length of line OE, b0 Angle between horizontal line and OE, ~0 Length of line AO, D~ Length of line BC, d3 Length of line BH, d4 Length of line HC, d5 Chain length, L0 Constant of the P spring,/('2 Constant of the B spring, /(i Equivalent spring constant of the shifting cable, K, Angle O Angle 0* Angle 0*

derailleur Dimension 48.65 60.75 52.68 20.55 20.55 62.04 245.78 29.62 110.6 423 86.14 19.64 87.38 1346 684 1274 2387 91.90 113.0 0

Unit mm mm mm mm mm mm ° mm ° mm mm mm mm mm g mm/° g mm/° g mm/° ° ° °

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W.H. Lai et

al.

6. E X P E R I M E N T A L S E T U P

The purpose of the experiment was two fold: to verify the feasibility of the four-mode model, and to check the accuracy of the simplified governing equations. In addition, the motion characteristic of the guide pulley can serve as a performance index reflecting the precise engagement and smooth shifting. Therefore, the experimental rig and instruments are designed in accordance with the aforementioned four-mode mathematical model and the fixed-frame model (see Fig. 11). Two encoders for measuring the rotation of the guide and tension pulleys were mounted on the shafts of the joints, respectively, where the B and P springs were located. A third encoder was installed on a friction-drive disk to record the freewheel rotation. Note that the size of the encoders must be small to avoid significant increase of the mass moment of inertia. The bicycle was mounted on a test stand specially designed for this experiment. The crank of the bicycle can be either driven by a servo motor or by an assistant. The total resistance applied on the rear wheel was generated by a flywheel and a fan that could be adjusted to simulate road conditions. The signal of the encoder was conditioned by an HP HCTL-2016 4X quadrature decoder, and then fed into a PC-386 micro-computer through a data-acquisition interface card. 7. RESULTS AND DISCUSSION Figures 12 and 13 present the analytical and experimental results of the rotation of the guide and tension pulleys with respect to the freewheel rotation angle, ~-¢0, that represents the angular displacement from which the shifting process starts. The chain is shifted from the 24-teeth sprocket to the 30-teeth sprocket while the front one is with 24 teeth. The shifting process that was theoretically divided into four modes is completed in 170° . Since the total length of the chain does not change, this results in the rotation of the guide and tension pulleys to absorb or release the chain pitches during the shifting process. In this experiment, the derailleur system that performs excellent shifting characteristics is selected from a number of different types of systems in order to verify the theoretical model that assumes smooth shifting. Any imperfection of the derailleur system that may arise from the ill-design, manufacturing or assembly in tooth profile, phase angle between adjacent sprockets or change mechanisms may result in a sharp change of the rotation of the guide and tension pulleys.

Fig. 11. Photograph of the experimentalsetup.

Motion analysis of a bicyclerear derailleur

377

66

! I

I m

I

l

I ! I I

I I I I

I

!

I

I

-~ o,

-~ :

T Expe~mentat low-speed ~ Experimentat high-speed

i

:

.~

~ Analysis

'l

0

50

100

-'-

150

200

~-~o(degrccs) Fig. 12. A comparison betweenthe analytical and experimentalresults on the motion of the guide pulley. In practice, corresponding to this situation the rider may feel uncomfortable at this moment. The correctness of the mathematical model can be clearly discovered from these two figures. Favorable agreement between theoretical and experimental results was obtained by observing the inclination and motion characteristics of the guide and tension pulleys. This illustrates the simplification of the governing equations provides adequately acceptable results for industrial design purposes. Ideally, the trajectories of the guide and tension pulleys should be smooth. However, the slope of these curves are discontinuous at the transitions among the four modes. The tooth profile and phase difference between adjacent sprockets may play the key roles for this imperfect phenomenon. Therefore, these two figures can be employed as an index for the shifting performance of the derailleur system. On the other hand, the proposed four-mode model can also be used during design stage. By assigning the trajectories of the guide and tension pulleys within each mode, the spring constants of the B and P springs, and the dimension of the cage can first be determined. The lengths

95 I .

.

.

.

90-

.

.

.

.

.

.

.

I

I

I I

I I

t

I

J

I

I I

I I 75 . . . . . . . . . . . . . a.

70

.~

.a . . . . . . . . . . . . ~. . . . . . . . . . . . 1 Experiment at l o w - s p e e d

C

:

~ Experiment at high-speed

*

*

~ Analysis

.J . . . . . . . . . . . .

,,,,,,,,,i,,,,,,,,,l,C',,i,',,, 0

50

,,,,,,',',,' 100

150

200

~-~o(degrees)

Fig. 13. A comparisonbetweenthe analyticaland experimentalresults on the motion of the tension pulley.

378

W.H. Lai et al.

of the links of the changer mechanism can then be calculated on the basis of the required position and orientation of the guide pulley. 8. CONCLUSION This paper presented a comprehensive theoretical and experimental investigation into the motion characteristic of the guide pulley of a typical rear derailleur during the shifting process. A four-mode model was developed to describe the phenomenological behavior of the principal elements. Equations governing the motion of the guide and tension pulleys with respect to the freewheel rotation were derived by employing Lagrange's equation. These equations were further simplified to facilitate computer simulation. In addition, a fixed-frame model was developed as the basis for the design of experimental rig. Comparisons between analytical and experimental results clearly illustrate the feasibility of the four-mode model and the correctness of the assumptions underlying the simplified equations. The motion of the guide and tension pulleys can serve as a performance index for indicating the status of precise engagement and smooth shifting. This study not only proposed a systematic approach for the analysis of the motion characteristics of the derailleur during the shifting process, but also provided a design methodology which involves and influences the design of the tooth profile and changer mechanism, and the development of a class of new products. Acknowledgements--This work was supported by the National Science Council of the Republic of China under grant

NSC82-0401-E007-045. This fundingis gratefullyacknowledged. REFERENCES

1. Veiko,N. M. and Freudenstein,F., Proceedings of the ASME 22nd Biennial Mechanisms Conference, Mechanical Design and Synthesis, DE-Vol. 46, 1992, 431. 2. Freudenstein,F. and Chen, C. K., ASME Journal of Mechanisms, Transmissions, and Automation in Design, 1990, 113(3), 253. 3. Eck, B., Fans--Design and Operation of Centrifugal, Axial-Flow and Cross-Flow Fans. Pergamon, Braunschweig, Germany, 1973.