modeling of constructional features of timing … · modeling of constructional features of timing...

Proceedings of PACAM XI 11th Pan-American Congress of Applied Mechanics Copyright © 2009 by ABCM January 04-08, 2010, Foz do Iguaçu, PR, Brazil

MODELING OF CONSTRUCTIONAL FEATURES OF TIMING BELTS MADE OF MATERIALS WITH MACROMOLECULAR STRUCTURES

Dudziak Marian, [email protected] Poznan University of Technology, POLAND Domek Grzegorz, [email protected] Kazimierz Wielki University in Bydgoszcz, POLAND Kołodziej Andrzej, [email protected] Higher Vocational State School in Kalisz Abstract. The work presents constructional features of transmission timing belts depending on materials used for their production. Design of composites and usage of new polymer materials allows for improvement of constructional properties of belts. Keywords: power transmissions belts, timing belts.

1. INTRODUCTION

Apart from natural through butadiene, nitrile, chloroprene and butyl, rubber, next to urethane elastomer is the most popular material used for the production of timing belts in connection with different kinds of fibers creating cord and covering of belts’ running side. Producers of timing belts usually choose from different kinds of materials by their own experience or practice while they manufacture universal or slightly sophisticated applications. Timing belts are made of polymer, that are neither subject to Hook’s elasticity law nor Newtonian viscosity law. Therefore they are called viscous-elastic materials. Deformation of teeth does not occur at the same time as tensioning, and going back to initial shape does not occur immediately after taking the force away. Initial state of a timing belt changes due to the arch of contact on a timing pulley in longitudinal and cross-section. The change of shape of longitudinal and cross-section influences belts’ work, especially the belts with non-uniform teeth such as Eagle or BAT types. For the belts with straight teeth the most important is change of longitudinal section. 2. MECHANICAL CHARACTERISTICS OF MATERIALS Different types of materials are used not only in production of timing belts but also in calculation and modeling of timing belts, where different dependences are employed. It is frequently observed in literature that many cases describe mechanical properties of timing belts in the aspect of capability function. W=C(I1-3) (1) When one assumes that material is homogeneous and isotropic, and C is material constant and I1=Grsgrs (2) Such model is called a new Hooke’s model and is sufficient for simple calculations and simulations. More complex model called Rivlin-Mooney’s, potential is a function of two material constants C1 and C2 and form equation: W= C1(I1-3) + C2(I2-3) (3) where:

)(21 2

12 mnrssnrm GGggII −=

(4) Ishihara, Hashitsume and Tatiana extended the model by nonlinear part. The already existing three constants were experimentally verified by Zahorski, who certified good correspondence between theory and experiment. In order to describe bouncy properties theory of hyper elastic materials if often applied. The model is frequently polynominal:

∑ −∑ +−−===

N

k

k

k

N

ji

jiij J

dIICW

1

2

1,21 )1(1)3()3( (5)

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with use of

23

22

211 λλλ ++=I

21

23

23

22

22

212 λλλλλλ ++=I

23

22

213 λλλ=I

For i=1, k=1 it is new Hooke’s model, for i=1,j=1 first Mooney-Rivlin’s model for i,j,k=1 it is modified Odgen’s model in ABAQUS system called superfoam. The model of material used for the analyses was assumed on the basis of mechanical characteristics that were formerly determined by Dudziak for polymer materials used in power transmission belts. From the above characteristics, the author selected the course of stress and deformation changes for the highest determined frequency of changes in the case of stabilized deformation without any stress. Hence, for some timing belts, it can be assumed that the tooth will represent the shape of the tooth space in the pulley.

? l

F

Fo

Fc

t

a b

c

d

e

∆F

lo ε

Figure 1. Diagram of dependences between circumferential force, preliminary tensioning force and elongation of belt’s element.

293K

313K

333K

ε

σ

Figure 2. Mechanical characteristic of belt material depending of number of deformation cycles.


One of the latest constructions are belts with non-uniform teeth on belt’s width. These constructions try to copy cylindrical gears in order to transfer their advantages to timing gears. Belts with BAT, Eagle, bevel and cone teeth are currently produced. The design of these gears was very problematic due to cross-force created in belt and often resulting in its lengthwise damage. The shape of a belt in cross-section was left unchanged and in case of BAT it is -AT, Eagle it is -HTD, bevel –CTD, cone – AT. The shape of pulley tooth space is the same as belts tooth profile and it does not consider teeth twisting that occurs on the arch of contact.

A problem that producers meet is the appropriate construction of belt that would consider characteristics of polymers used for production of timing belts. Internal friction of twisted teeth causes dissipation of much of the energy. The increase of temperature also influences mechanical characteristics of material which results in worse meshing on central part of the belt. Material with increased temperature is more deformed because of squeezing and bending. The result is stretching of the belt followed by deformation of teeth on belts’ width. The problem of deformation of timing belts teeth is meant to be solved by usage of composite fabrics, covering fabrics or by excluding central part of the belt from meshing, by putting a v-guide like in BATK belts type. On the other hand, producers do not take into consideration the problem of guide bending on the arch of contact. In this case deformation of the guide is significant as it is placed below the cord and exposed to squeezing stresses. Partially, it may be solved through an increase of backlash between belt’s and pulley’s teeth but it deteriorates conditions on a coupling . As an effect belts with non-uniform teeth do not transfer higher torque moment. Thanks to an increased covering ratio more teeth take part in coupling and no impact is made between heads of belt teeth and heads of pulley teeth. 3. TWO TYPES OF COOPERATION BETWEEN TIMING BELTS AND PULLEYS

So far research has been developed on the basis of observation of cross-section changes in timing belt’s tooth shape on the arch of contact in timing gears, where frictional contact between a timing pulley and a timing belt occurs. The material that took part in coupling was placed either above or beneath the cord. In case of gears with timing belts the most widespread is coupling shape and therefore analysis must also take into consideration the way how belt is supported by a pulley. Only the material below the cord takes part in coupling and is exposed to compression and cutting tensions. Bending of the belt, allows for observing the change of timing belt’s tooth shape in the zone of bending. The material squeezed below the cord is pushed out into the teeth, and a change of their parameters depending on teeth’s type is noticeable. In case of trapezoid teeth one may notice clear rounding of tooth’s sides and top. In the moment of coupling, timing belt is straight and bending occurs only on the arch of contact. In coupling with timing pulley, timing belt’s material is squeezed between the cord and the timing pulley surface. Therefore, in the analysis of change of tooth shape on the arch of contact, separate cases of belts supported by the top of pulleys teeth and belts supported by pulley’s tooth space must be taken into consideration.

Figure 3. Two types of cooperation between a belt and a pulley.


Figure 4. Characteristic place in gear with timing belt.

Figure 5. Teeth of RPP belt on the driving pulley.

1 2

3 4

5 6


Subsequent analyses were shown for successive stress conditions of a belt tooth on the arch of contact with the driving pulley for three consecutive belt types: HTD, RPP and STD. In spite of the fact that those belts are most often made of rubber, in order to make these and previous tests’ results comparison possible, the author assumed the same model of material as well as the same load type for each case of calculation. The calculation of stress distribution for the arch of contact with the driven pulley has been skipped. The phenomena observed in case of earlier calculations are certainly reflected in belts of that type. The influence of teeth shape for those belts on the stress distribution was recognised as an important factor. The stress distribution obtained for start meshing phase of the HTD belt make us expect significant pressure of a tooth side on a pulley side. It results in uneven operation of a gear, high acoustic emission as well as volume wear of a tooth side, whereas the rounded geometric form of a tooth side ensures better coupling with a pulley tooth during smaller loads. The first contact between a belt tooth and a pulley tooth will take place much later than in case of a transmission with trapezoid tooth shape. In the full meshing situation, the map shows large areas of small stress. That fact inclines us to ask the question whether the belt was made of right material and whether the load-bearing layer of higher elongation resistance should be used? The geometric form of a belt tooth marked with the STD symbol resembles the shape of the HTD belt tooth. This tooth is clearly rounded in order to improve the conditions of moment when it starts meshing with the pulley. It is confirmed by the analysis of phases of meshing start between the belt and the pulley. Contour lines of the highest stress cover the majority of the area and are oriented to the centre of a tooth. Stress images for subsequent positions of a tooth on the arch of contact show the lack of stress in the central part of a tooth. This means usefulness of research on the belt material. Solutions including composite material, which are applied fibres in KPS belts, are very advantageous in that case. There are fibres that are oriented crosswise to the load-bearing layer in a belt tooth of STD shape. Those fibres improved the crosswise stiffness of the belt and ensured stress transmission from the load-bearing layer deep into a belt tooth. The belt RPP can be perceived as a sort of modification of the shape HTD and STD. The notch in the belt tooth addendum is meant to improve the quality of the meshing start between a belt tooth and a pulley tooth. The analysis of drawings allows us to observe more distribution of stress than for STD belts. The contour lines of the highest stress are oriented into the tooth centre and there was no accumulation of stress near the load-bearing layer. Teeth being in full coupling show areas of very low stress. Belts of that type are also modified in order to increase usage of a tooth volume in load transmission. The advantage of that kind of belts consists in possibility to cooperate with such pulleys types like HTD and STD. One of the belt manufacturers has developed that advantage and performed researches on improvement of belt operating properties and the result of that work was a belt marked Omega. A belt tooth of that type has a slightly different geometric form than RPP belt and it makes possible the cooperation with such pulleys like HTD, STD and RPP. 4. CONCLUSIONS

The analysis of the coupling between timing belts and timing pulleys permitted to develop conclusions and belt design recommendations. They take into consideration those parameters of the timing belt transmission gear that influence the way of coupling in the transmission gear. 1. While developing new types of transmission gears with timing belts, it is necessary to take into account the

relationships between basic belt dimensions. To facilitate this, using dimensional analysis, the author determined coefficients that connect tooth surface, volume and pitch. For example, belts with higher surface coefficient and lower volume coefficient are characterised by lower change of momentary coupling value.

2. It is necessary to change the rule that the belt pitch is the same as the pulley pitch. Test of such timing belts like T, AT and ATP showed that despite different manner of cooperation between the timing belt and the pulley, the timing belt pitch should be higher than the pulleys’ pitch. The value of the belt pitch to the pulley pitch ratio is the function connected with mechanical characteristics of the belt. While designing a timing belt, the manufacturer should take into account the value of unit elongation of a belt that will be caused by initial stress and he should adjust the belt pitch value respectively. Using pitch correction in all types of transmission gears with timing belts shall improve operating parameters of those gears and shall enable timing belts users to increase reliability of those transmission gears.

3. It has been proved that the coupling is of shaped-and-frictional type and that friction coefficients for the belt and the pulley material as well as mechanical characteristics of the belt material are important. Using FEM and the results of tests, the author indicated to tooth shapes for which the stress condition under pressure are more even. Values of stress contour lines show local stress and dislocations as well as deformations of a tooth shape under the influence of load. Occurrence of permanent deformations of belt teeth after the belt has worked for some time after which deformations stabilise confirmed importance of geometric form of the pulley’s tooth space as regards geometric form of coupling between the timing belt and pulleys.

4. Studies of belt reliability in the operating conditions were used to confirm the following premises: - Volume wear takes place in places were theoretical analysis forecasted that the belt tooth will hit pulley tooth

addendum and that teeth will slip during meshing start,


- new design solutions of timing belts made possible much longer lifetime of a transmission gear or transmission of a

higher torque, - it has been experimentally proven that fractures of material after operation occurred in places where FEM

calculations forecasted stress concentration, - modification of material used for timing belts manufacturing results in improvement of the transmission gear

operation characteristics. 5. REFERENCES Domek G., Dudziak M., Mechanics of bending of timing belts with non straight teeth, Proceedings of the PACAM X

volume 12, s. 219-222, Cancun 2008. Domek G., Malujda I., Modeling of timing belts construction, , Wiley Inter Science PAMM Volume 7, Issue 1,

Date: December 2007, Pages: 4070045-4070046 Domek G., Kołodziej A.: „The surface condition of pulleys in use”. Machine Dynamics Problems 2006, Vol. 30, No 3,

p.72-78. Domek G., Leistungverluste in Zahnriemengetrieben, Antriebstechnik 12/2006, s. 30-31. Domek G., Krawiec P., Methods of Designing of Timing Belts Pulley, University Reviev, Vol 1, No.3 /2007,

Aleksander Dubcek University of Trencin, Izhevsk State Technical University s. 15-20. Dudziak M., Domek G., Change of belt’s tooth shape caused by bending and pressure on the pulley,

TRANSACTIONS of VŠB – Technical University of Ostrava, Metallurgical Series, Issue 1 / 2008, Vol. LI Dudziak M., Domek G., Model of load in timing belts, Proceedings of the PACAM X, volume 12, s. 215-218, Cancun

2008. Dudziak M., Domek G., Analisys of antitorque in gear with timing belts, Materials Engineering, Vol. 11, 2004, No.1,

Źilina (SK) ISSN 1335-0803, s. 121-124. Dudziak M., Domek G., Gear with timing belts in mechatronic drives, Machine Dynamics Problems 2004, Vol 28, No

Warsaw University of Technology, s. 83-88, ISSN 0239-7730. Dudziak M., Mielniczuk J., Nonclassicall material model in machine design, Publishing House Instytut Technologii

Eksploatacji, Radom 2001 Nagel T., Zahnriemengetriebe, Carl Hanser Verlag, Wien - Munchen 2008 Schafer F.H, Antriebsriemen, Huxaria Druckerei, Hoxter 2008 White J.R., De S.K., Rubber Technologiest’s Handbook, Rapra Technology LTD, Shropshire 2001 6. ACKNOWLEDGEMENT

The work reported in this paper was funded in part by the Ministry of Science and Higher Education, Poland. 7. RESPONSIBILITY NOTICE

The authors are the only responsible for the printed material included in this paper.

modeling of constructional features of timing … · modeling of constructional features of timing...

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