Modeling of Bevel Gear Tooth For Gate Valve

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World Journal of Technology, Engineering and Research, Volume 2, Issue (27) 374-387 Contents available at WJTER World Journal of Technology, Engineering and Research Journal Homepage: www.wjter.

Keywords A B S T R A C T Bevel gear Gate valve Mathematical modeling Bevel gears are mostly used where need is to transmit the power between intersecting shafts in right angles.

1 World Journal of Technology, Engineering and Research, Volume 2, Issue (27) Contents available at WJTER World Journal of Technology, Engineering and Research Journal Homepage: Modeling of Bevel Gear Tooth For Gate Valve Avishkar Bhoskar a*, Rajeshwar Janunkar b, Mahesh Patil c a,b,c Automobile Engineering Department, Dhole Patil College of Engineering, Pune, India Keywords A B S T R A C T Bevel gear Gate valve Mathematical modeling Bevel gears are mostly used where need is to transmit the power between intersecting shafts in right angles. In this paper the main focus is on mathematical modeling of bevel gear for gate valve application so as to define a unique way the tooth of gear. Comparison between the results of mathematical modeling and theoretical modeling is to be done so as to have the validation of the work. A mathematical model presents a simplified version of something. The comparison results show the accuracy of modeling which is very much closer to the theoretical modeling. 27 WJTER All rights reserved. 374

I. INTRODUCTION Mathematical modeling is a branch of mathematical logic or a discipline, which helps us in shaping the real life problems into mathematical models and then solving them accordingly.

2 I. INTRODUCTION Mathematical modeling is a branch of mathematical logic or a discipline, which helps us in shaping the real life problems into mathematical models and then solving them accordingly. Here mathematical modeling of bevel gear is done so as to have the details of it. As matter of fact mathematical modeling is not a new subject. It has been there since ancient ages. Scientists, engineers, statisticians, astronomers have been studying a good number of variety of problems through mathematical models. In the generalized sense for every process, the problem is modeled into mathematical equations be called mathematical modeling. Thus there is hardly any area of study and research, which escapes from this definition. Gate valves are widely used. A gate valve, also known as a sluice valve, is a valve which opens by lifting a round or rectangular gate/wedge out of the path of the fluid. The distinct feature of a gate valve is the sealing surfaces between the gate and seats are planar, so gate valves are often used when a straight-line flow of fluid and minimum restriction is desired. Fig. - Hand Operated Gate Valve. II. MATHEMATICAL MODELING OF BEVEL GEAR According to usage, mathematical models are classified into descriptive models and optimization models. In our case we are using descriptive modeling. 375

Fig. 2- Tooth of a bevel gear The different terms used for denoting the gears parameters are as follow, g, g - Constant parameters in modeling; g - Height of the head modeling; g - Height of the foot

3 Fig. 2- Tooth of a bevel gear The different terms used for denoting the gears parameters are as follow, g, g - Constant parameters in modeling; g - Height of the head modeling; g - Height of the foot of modeling; S - Surface of the F flank; - Reference system linked to flank F ; X, Y, Z, O - Axes of co-ordinates;, - Transfer matrix from the system g ing ; O M - Vector radius, with projectionso M, O M, O M,of a current point on the surface S, reported to the reference system ; O N - Perpendicular vector with projections,,, attached to a vector radius O M. The whole system that is going to be modeled is under descriptive method and a part of gear that is tooth of gear is generalized. O M A [(3 2B )g q sinα q cosα q X Y Z () Where:- A and B are generalized coefficients; A Takes the value ±,as relation () refers to right side or left side of rack; B Takes the value or 2, as per reference plan X O Z overlaps with the summary plan of the tooth or the tooth gap; g Represents the width of tooth rack measured in median plan (π ); α Represents the pressure angle/standardized gearing. 376

4 We have, g tooth thickness of pinion; g ; Considering, q 2mm ; q 25mm ; α pressureangle 2 ; A,B ; O M [(3 2 ) sin2 2cos2 25 O M X Y Z X Y Z Perpendicular vector,o N attached to vector () it is expressed by following relation: O N A gα (2) O N π.395 To manufacture the involute-bevel gears having the half angle δ (fig. 3) and inclined teeth with angle are taken into account the following. Standardized rack, Fig. is fixed in relation to the half-finished part such a way that the median plan (π ) to be parallel with the generatrix OM of division cone shown in fig- 3. (Csibi et al. 27, 28; Herciu et al. 28) 377

5 Fig.3- Rack s reference positioning of bevel pinion. In case of inclined teeth, the rack is rotated in relation with the half-finished part thus axis O Z, Fig. 4. to close the angle with generatrix OM. Origin O of the system it s considered placed in the median plan of the gear. As a consequence, the parameters q and q, of the surface S will be limited by the teeth height and the gear width that is manufactured. As in this case the bevel gear is of straight type thus the angle is obviously zero. For better understanding of the formation of the bevel gears with some angle greater than zero which means its not a straight one. So as to have this the following generalization method is useful to understand it thoroughly. For shifting from the system in, based on the scheme from Fig. 4, it s used the transfer matrix T as it follows: Fig. 4- Generalized reference element (Tooth of Pinion). 378

6 cos (3) In this case as it is straight bevel gear the angle and δ faceangle 6.5, During the generation teeth phase, the rack is expressed by the vectors: OX and ON that can be expressed by relations: ON A cosβ A sinβsinδ + gα cosδ A sinβsinδ gα cosδ ONr sin sin π cos6.5 ON cos sin sin π cos (4) Also, they contain all the feature parameters for involute-bevel gears with straight-teeth ( ) or for spur gears ( ). Here the basic idea is to consider the angle for the straight bevel gear as it becomes zero and there it doesn t play any role ( Ioan Ardelean et al. 24). In the process of generation, the half-finished gear executes a rotation motion, and the rack with side a translation motion in direction of axis O X, Fig.5. The relative instantaneous position between the systems and (linked to the rack, respectively the gear) is defined by the parameter by the transfer matrix: T cos ( + ) sin cos ( + ) + (5) 379

7 In which the is the addendum modification coefficient of order i with unitary module is the division circle given by the known formula as follow; /2 ; where,z number ofteethon gears; the inclined angle of teeth; m- module4mm ; 5,.435, Fig. 5 Gears generation scheme (bevel gear pair) T for The side of teeth gear is expressed by the help of matrix product [T ] [ ], obtained position vector: O X ( γ )cosγ + ( + r )sinγ (γr X )sinγ + (Y + r )cosγ Z 38

8 O X O X (,, ) (6) To reduce the number of parameters to tow, the vector (6) is attached the condition of the general gearing:., in which the is the relative velocity vector in the contact points between the surface. In the present case, the gearing condition can be developed using matrix method obtaining: N V N (T γ X )( + ) + (X r ) (7) Solution of equation that defines the parameter for expression (9), is: [X + ( + )] (8) ( ) Taking into account that the chosen perpendicular vector is invariant toward translation, it results that the perpendicular in the point of contact, on the side is determined with Eqs. (4), (5) and (8), thus: N T N N (9) 38

9 N Equations (5) and (9) together with (8), define in a unique way the teeth of gear. III. THEORETICAL DESIGN OF BEVEL GEAR PAIR A) Input parameter: Number of teeth on pinion 2 Number of teeth on gear 52 Module m 4mm Pressure angle Ψ 2 Shaft angle 9 B) Detail dimensions of bevel gear pair: ) Reference diameter(d): Diameter of pinion m mm Diameter of gear m mm 2) Pitch angle(δ): Pitch angle of pinion tan ( ) tan ( ) Pitch angle of gear tan ( )tan ( ) 77.6 Cross check, Shaft angle ) Outer cone distance(r): RR R cosec R cosec mm 4) Addendum modification coefficient(k ): K.46 [ ( ) ] 382

10 K.46 [ ( ) ] K.435 K -K K ) Tooth width modification coefficient(k ): The value of 2K tanψ is., 2K tanψ. 2 K tan2. K.5 K - K K -.5 6) Equivalent tooth modification coefficient( K ): K K +K K -K ) Addendum(h ) : h m (+K ) 4 (+.435) 5.74mm h m (+K ) 4 (-.435) 2.26mm 8) Dedendum(h ): h m (.2-K ) 4( ) 3.6mm h m (.2-K ) 4( ) 6.54mm 9) Face width(b): b b b.3 R mm ) Dedendum angle(x ): x tan ( ) tan (.. ).642 x tan ( )tan (. ) ) Root angle(δ ): δ δ - x

11 δ δ - x ) Face angle of blank(δ ): δ δ +x δ δ +x ) Outside diameter of blank(d ): d d +2 h cosδ cos mm d d +2 h cosδ cos mm 4) Apex to crown distance(h ): h R cosδ - h sinδ 6.73cos sin mm h R cosδ - h sinδ 6.73 cos sin mm 5) Circular tooth thickness at the reference diameter(s): s m ( π +2K tanψ) 4 ( π tan2 ) 7.99mm s m ( π +2K tanψ) 4 ( π tan2 ) 4.58mm 6) Virtual spur gear diameter(d ): d d secδ 48 sec mm d d secδ 28 sec mm 7) Virtual spur gear teeth(z ):. Z 2.3 Z ) Choral tooth thickness(g ): g d sin( rad) sin(. rad) 7.86mm. g d sin( rad) 925 sin(. rad) 4.56mm 9) Choral height(h ): h h mm 384

12 h h mm 2) Constant chord(g ): g s cos Ψ 7.98 cos 2 7.4mm g s cos Ψ 4.57 cos 2 4.3mm 2) Constant chord thickness(h ): Ψ. h h mm Ψ. h h mm. IV. COMPARISON BETWEEN THEORETICAL AND MATHEMATICAL MODELING. Results obtained from the theoretical calculations and mathematical modeling is compared here so as to get the clear view of the work. Below is the chart given showing the results. Table- Comparison between theoretical and mathematical modeling. Sr. No. Design Parameters Theoretical modeling Mathematical modeling. Pitch radius 24mm 24mm Tooth angle( ) Addendum coefficient Tooth thickness mm 7.74mm 5. Face angle( ) Reference Pitch radius 24.43mm mm V. CONCLUSION The aim of this paper is to generate the mathematical model of the bevel gear pair for gate valve application. After been through all the detailed comparison between two genres of methods viz. mathematical and theoretical, it depicted from the contemplation of all the facts, that both the values are commensurable. It has been delineated with the computation that the reference pitch radius, tooth thickness, tooth angle are coequal. 385

13 REFERENCES [] Avishkar Bhoskar, Dr. Sanjay Yadav, "Design of bevel gear for gate valve", international journal of research in engineering and technology, volume 5, issue 2, 26. [2] M. Li, H. Y. Hu, Dynamic analysis of spiral bevel geared rotor bearing system, Journal of Sound and Vibration (23). [3] Faydor L. Litvin, Alfonso Fuentes, Qi Fan, Robert F. Handschuh, Computerized design, simulation of meshing, and contact and stress analysis of face-milled formate generated spiral bevel gears, Mechanism and Machine Theory 37, 22. [4]John Argyris, Alfonso Fuentes, Faydor Litvin, Computerized integrated approach for design and stress analysis of spiral bevel gears, Computer methods in applied mechanics and engineering 22. [5] Jixin Wang, Long Kong, Bangcai Liu, The Mathematical Model of Spiral Bevel Gears - A Review, Journal of mechanical engineering,24. [6] M. Kolivand, H. Ligata, G. Steyer, D. K. Benedict, J. Chen Actual Tooth Contact Analysis of Straight Bevel Gears, Journal of Mechanical Design, ASME, 25. [7] Ning Ma and Wenji Xu Mathematical modeling for finishing tooth surfaces of spiral bevel gears using pulse electrochemical dissolution, International Journal of Advance Manufacturing Technology, Springer-Verlag London Limited 2. [8] Ioan Ardelean, Mircea Bara, Sergiu-Dan Stan and Sorin Besoiu, Research Regarding Modelling of Involute-Bevel Gears, The th IFToMM International Symposium on Science of Mechanisms and Machines, Mechanisms and Machine Science 8, DOI:.7/ _9, Springer International Publishing Switzerland 24. [9] Wern-Kueir Jehng, Computer solid modeling technologies applied to develop and form mathematical parametric tooth profiles of bevel gear and skew gear sets Journal of Materials Processing Technology 22 (22). [] Vladimir I. Medvedev, Andrey E. Volkov, Marina A. Volosova, Oleg E. Zubelevich, Mathematical model and algorithm for contact stress analysis of gears with multi-pair contact Mechanism and Machine Theory 86, Elsevier, 25. [] Isamu Tsuji, Hiroshi Gunbara, Kazumasa Kawasaki Tooth Contact Analysis and Manufacture on Multitasking Machine of Large-Sized Straight Bevel Gears With Equi-Depth Teeth,International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, ASME,

14 [2] Csibi VI, Herciu D, Sudrijan M (27) Tool for precise grinding of crown gears. 2th IFTOMM World Congress, Besancon, pp 8 2. [3]Csibi VI, Herciu D, Sudrijan M (28) Scula pentru rectificarea precisa a rotilor dintate frontale. Brevet de Inventie OSIM RO 2729 B. Int, C. B24 B/ (26). BOPI nr. 3/28. [4] Herciu D, Csibi VI, Pop RO, Sudrijan M (28) Tool for grinding of face gears. Hungarian Technical Review, pp 9 95, Cluj-Napoca, ISSN [5] N.Mohan Raj, M. Jayraj, Design of Contact Stress Analysis in Straight Bevel Gear, International journal of computational engineering research, vol-3, issue-4, april

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