getview vol.2 no.8 august 2012

Committee of the Global Engineers & Technologists Review Chief Editor

Ahmad Mujahid Ahmad Zaidi, MALAYSIA Managing Editor Mohd Zulkifli Ibrahim, MALAYSIA

Editorial Board

Dr. Arsen Adamyan Yerevan State University ARMENIA

Assoc. Prof. Dr. Gasham Zeynalov Khazar University AZERBAIJAN

Assistant Prof. Dr. Tatjana Konjić University of Tuzla Bosnia and Herzegovina BOSNIA and HERZEGOVINA

Assistant Prof. Dr. Muriel de Oliveira Gavira State University of Campinas (UNICAMP) BRAZIL

Assoc. Prof. Dr. Plamen Mateev Sofia University of St. Kliment Ohridsky BULGARIA

Dr. Zainab Fatimah Syed The University of Calgary CANADA

Assistant Prof. Dr. Jennifer Percival University of Ontario Institute of Technology CANADA

Prof. Dr. Sc. Igor Kuzle University of Zagreb CROATIA

Assoc. Prof. Dr. Milan Hutyra VŠB - Technical University of Ostrava CZECH

Prof. Dr. Mohamed Abas Kotb Arab Academy for Science, Technology and Maritime Transport EGYPT

Prof. Dr. Laurent Vercouter INSA de Rouen FRANCE

Prof. Dr. Ravindra S. Goonetilleke The Hong Kong University of Science and Technology HONG KONG

Assoc. Prof. Dr. Youngwon Park Waseda University JAPAN

Prof. Dr. Qeethara Kadhim Abdulrahman Al-Shayea Al-Zaytoonah University of Jordan JORDAN

Prof. Yousef S.H. Najjar Jordan University of Science and Technology JORDAN

Assoc. Prof. Dr. Al-Tahat D. Mohammad University of Jordan JORDAN

Assoc. Prof. Dr. John Ndichu Nder Jomo Kenyatta University of Agriculture and Technology- (JKUAT) KENYA

Prof. Dr. Megat Mohamad Hamdan Megat Ahmad The National Defence University of Malaysia MALAYSIA

Prof. Dr. Rachid Touzani Université Mohammed 1er MOROCCO

Prof. Dr. José Luis López-Bonilla Instituto Politécnico Nacional MEXICO

Assoc. Prof. Dr. Ramsés Rodríguez-Rocha IPN Avenida Juan de Dios Batiz MEXICO

Dr. Bharat Raj Pahari Tribhuvan University

NEPAL

Prof. Dr. Abdullah Saand Quaid-e-Awam University College of Eng. Sc. & Tech. PAKISTAN

Prof. Dr. Naji Qatanani An-Najah National University PALESTINE

Prof. Dr. Anita Grozdanov University Ss Cyril and Methodius REPUBLIC OF MACEDONIA

Prof. Dr. Vladimir A. Katić University of Novi Sad SERBIA

Prof. Dr. Aleksandar M. Jovović Belgrade University SERBIA

Prof. Dr. A.K.W. Jayawardane University of Moratuwa SRI LANKA

Prof. Dr. Gunnar Bolmsjö University West SWEDEN

Prof. Dr. Peng S. Wei National Sun Yat-sen University at Kaohsiung. TAIWAN

Prof. Dr. Ing. Alfonse M. Dubi The Nelson Mandela African Institute of Science and Technology TANZANIA

Assoc. Prof. Chotchai Charoenngam Asian Institute of Technology THAILAND

Prof. Dr. Hüseyin Çimenoğlu Instanbul Technical University (İTÜ) TURKEY

Assistant Prof. Dr. Zeynep Eren Ataturk University TURKEY

Dr. Mahmoud Chizari The University of Manchester UNITED KINGDOM Prof. Dr. David Hui University of New Orleans

USA

Prof. Dr. Pham Hung Viet Hanoi University of Science VIETNAM

Prof. Dr. Raphael Muzondiwa Jingura Chinhoyi University of Technology ZIMBABWE

Dear the Seeker of Truth and Knowledge Journal writing. It is rather like keeping a diary; some consider, but it is actually we have to take far more seriously. Therefore, you might prefer to present your unpublished work as a poster or a talk, as a easier way rather than writing it through scientific manuscripts forms. However, the 5 times nominated Nobel Prize, Fritz Schaefer, who have published 1300 scientific publications said, “if it isn’t published, it doesn’t exist.”. This is means, that only through a publication, they will elevate it to the status of a legitimate; completed project worthy of discussion. By the increasing competition for the publication of scientific research that has led to an increased emphasis on determining the perceived "quality" or "status" of a specific journal, then scientists, like everyone else, want to publish papers in journals, especially in where their work is likely to have the highest impact. On this, over the past two decades there has been a marked shift in the way scientific journals are published and disseminated; shifted the reliance from the print versions of most journals; everything is now available on-line. Many journals are available from more than one source, and sometimes one source is free while the other is restricted-access and very expensive. All of this would be a matter of the interest, were it not for the now pervasive and inappropriate practice to the quality of an individual's research. Considering on this, journals like GETview are also certainly only want to publish original research that will have a significant impact and therefore it necessary to explain how your paper differs from previous work, why your paper is important, and what new insights it presents. This journal has been nearly two years in the making. It was a natural outgrowth of the expansion and it was decided that the journal should be published as an open-access online journal. It was also decided that we would endeavour to publish twelve issues a year. Since the GETview is also an online initiative designed to provide a platform for the disciplines of the engineering and technology sciences where students and professionals alike can engage in provoking and engaging explorations of knowledge that push the boundaries of disciplinary lines, by opening space for cross-disciplinary discussions, this journal hopefully can inspires an intersectional investigation and consideration of the issues that scholars in the early part of the 21st century recognize as most compelling in our changing world. This is due to throughout engineering and technology science have had profound impacts, positive and negative, on humankind, other species and the environment. Hence, in an ongoing effort to acquaint our readers with the prominent scholars making up the editorial board that advises and serves the Getview, we are honoured to provide the independent's evidence-based and authoritative information also the advice concerning engineering, technology, and science to policy makers, professionals, leaders in every sector of society, and the public at large. Certainly, involving yours; with the interest and expertise, through paper submitted and published in the Getview. Prof. Ahmad Mujahid Ahmad Zaidi, PhD. Chief Editor The Global Engineers and Technologists Review

©PUBLISHED 2012 Global Engineers and Technologists Review GETview ISSN: 2231-9700 (ONLINE) Volume 2 Number 8 August 2012 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, electronic, mechanical photocopying, recording or otherwise, without the prior permission of the Publisher. Printed and Published in Malaysia

Vol.2, No.8, 2012 1. DESIGN AND ANALYSIS OF 5KW SAVONIUS ROTOR BLADE WIDODO, W.S., CHIN, A.C., HAERYIP SIHOMBING and YUHAZRI, M.Y. 9. INTEGRATING SPUR GEAR TEETH DESIGN AND ITS ANALYSIS WITH G2

PARAMETRIC BÉZIER-LIKE CUBIC TRANSITION AND SPIRAL CURVES YAHAYA, S.H., ALI, J.M., YAZARIAH, M.Y., HAERYIP SIHOMBING and YUHAZRI, M.Y. © 2012 GETview Limited. All right reserved

CONTENTS

ISSN 2231-9700 (online)

GLOBAL ENGINEERS & TECHNOLOGISTS REVIEW www.getview.org

G.L.O.B.A.L E.N.G.I.N.E.E.R.S. .& .-.T.E.C.H.N.O.L.O.G.I.S.T.S R.E.V.I.E.W 1

WIDODO1, W.S., CHIN2, A.C., HAERYIP SIHOMBING3, and YUHAZRI4, M.Y.

1, 2, 3, 4 Faculty of Manufacturing Engineering Universiti Teknikal Malaysia Melaka

Hang Tuah Jaya, 76100 Durian Tunggal, Melaka, MALAYSIA [email protected]

3iphaery@ utem.edu.my [email protected]

1.0 INTRODUCTION The developments of renewable energy especially wind energy become widely since 1973 due to the oil crisis issues. This view has been supported by Peter et al., (2008) who states that the oil price is forecasted to be raised in the future and thus wind energy is an alternative energy sources. The wind turbine is a device that utilizes wind energy to generate mechanical or electrical power. According to Manwell et al., (2009), there are two types of wind turbine: Horizontal Axis Wind Turbine (HAWT) and Vertical Axis Wind Turbine (VAWT). HAWT are most commonly known type of wind turbine and it operates parallel to the direction of the wind whereas VAWT rotor is operated perpendicular to the direction of wind. The two most common design principle of VAWT is Savonius type and Darrieus type. The paper focuses on the Savonius type rotor blades design and analysis. Savonius type rotor blade is a simple wind turbine that operates based on drag concept. According to Akwa et al., (2011) Savonius rotor can be designed with two or three blades, in single stage or multi-stages. The working principle of Savonius rotor is resembled to a cup anemometer. The low efficiency of VAWT limited its use in large power production. However, VAWT has several advantages over HAWT that make it widely use in another sector such as water pumping system. The most apparent advantage of VAWT is it can operate in all wind direction and thus are built without using any yaw mechanism (Halsey, 2011). Other advantages included low noise and simplicity.

In this project, a VAWT is designed to produce 5 kW power output. As the performance of VAWT is relatively low, it becomes necessary to explore the effect of number of blades and sizing of the rotor blades in the design stage. The Savonius wind turbine is a drag type VAWT where utilizes the drag force for its operation. D’Alessandro et al., (2009) highlighted that the aerodynamic theories developed for lift type wind turbine (HAWT and Darrieus wind turbine) cannot be applied for Savonius rotor. According to Islam et al., (2005) the flow pattern around the Savonius rotor blade is characterized by flow phenomena that produce pressure differences between the concave and convex surfaces of the blades which will induce to aerodynamic force and torque. Islam et al., (2005) further explained that the feature of flow phenomena including high turbulence, unsteadiness and flow separation. Figure 1 shows the normal drag force, FN acts perpendicular on the blade surface whereas tangential drag force, FT acts along tangential direction on each blade. Both FN and FT equations have been stated by Islam et al., (2005) as follow:

FN = ∆ PS sin ∅ (1) FT = ∆ PS cos ∅ (2) Where the △P represents the pressure difference between the concave and convex surfaces of the blade

and the S represent the chord length.

ABSTRACT

This paper presents the design and analysis of the Savonius rotor blade to generate 5 kW power output. The relevant design parameters and theories were studied in this paper and used to determine related design geometry and requirements of the Savonius rotor blade. The Savonius rotor was designed with the rotor diameter of 3.5 m and the rotor height of 7 m. The 3D model of Savonius rotor blade was created by using SolidWorks software. Computational Fluid Dynamics (CFD) analysis and structural Finite Element Analysis (FEA) are presented in this paper. CFD analysis was performed to obtain the pressure difference between concave and convex region of the blade while FEA was done to obtain the structural response of the blade due to the wind load applied in term of stresses and its displacements. Keywords: Savonius, Rotor Blade, CFD, FEA.

DESIGN AND ANALYSIS OF 5 KW SAVONIUS ROTOR BLADE

Global Engineers & Technologists Review, Vol.2 No.8 (2012)

© 2012 GETview Limited. All rights reserved

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Figure 1: Forces acting on a blade of two blades Savonius rotor, (Islam et al., 2005). 2.0 THEORY AND DESIGN Wind power, Pw is defined as the multiplication of mass flow rate, ρAV and the kinetic energy per unit mass, ½ V2 (Musgrove, 2010). The wind power is denoted by the equation of:

Pw = ½ ρAV2 (3)

Hayashi, et al. (2004) found that the swept area for Savonius Wind Turbine is calculated by multiplication of rotor diameter, D and the rotor height, H. The larger the swept area, the larger the power generated.

A = D.H (4)

The wind power in Equation (3) represents the ideal power of a wind turbine, as in case of no aerodynamic or other losses during the energy conversion processes. However, as stated by Manwell et al., (2009) there is not possible for all energy being converted into useful energy. The ideal efficiency of a wind turbine is known as Betz limit. According to the Betz limit, as supported by Musgrove (2010) there is at most only 59.3 % of the wind power can be converted into useful power. Some of the energy may lose in gearbox, bearings, generator, transmission and others (Jain, 2011). The maximum power coefficient, Cp for Savonius rotor is 0.30. Hence, the Cp value used in this project is 0.30 and the power output, P with considering the power efficiency is:

P = 0.15 ρAV2 (5)

Wind speed is the major element that affects the power output. The three wind speed parameters involve in this project is cut-in speed, rated wind speed and cut-out speed. Jain (2011) stated that the three wind speed parameters related to the power performance are as follow:

Vcut-in = 0.5 Vavg (6) VRated = 1.5 Vavg (7) Vcut-out = 3.0 Vavg (8)

All these parameters depend on the value of average wind speed. The average wind speed, Vavg was found at 7 m/s. Table 1 summarizes the value of these three wind speed parameters.

Table 1: Value of cut-in speed, rated wind speed, and cut-out speed.

Wind Speed Parameter Equation Calculation Cut-in speed, Vcut-in Vcut-in = 0.5 Vavg 3.5 m/s

Rated wind speed, VRated VRated = 1.5 Vavg 10.5 m/s Cut-out speed, Vcut-out Vcut-out = 3 Vavg 21 m/s

Aspect ratio is a crucial criterion to evaluate the aerodynamic performance of Savonius rotor. Johnson (1998) suggests the Savonius rotor is designed with rotor height twice of rotor diameter and this lead to better stability with proper efficiencies.

AR = H/D (9)

Tip speed ratio, λ is defined as the ratio of the linear speed of rotor blade ω.R to the undisturbed wind speed, V (Solanki, 2009). ω is the angular velocity and R represent the radius revolving part of the turbine. The maximum tip speed ratio that Savonius rotor can reach is 1. Manwell et al., (2009) write that high tip speed ratio



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improves the performance of wind turbine and this could be obtained by increasing the rotational rate of the rotor. λ = ω.R /V (10) According to Manwell et al., (2009) solidity is related to tip speed ratio. A high tip speed ratio will result in a low solidity. Musgrove (2010) defines solidity as the ratio of blade area to the turbine rotor swept area. For VAWT, the solidity is defined as σ = nd /R (11) Where n is the number of blades, d is the chord length or can be defined as the diameter of each half cylinder, and R is radius of wind turbine. Many researchers have proved that the higher the number of blades, the higher the performance of most wind turbine. However, Saha et al., (2008) and Zhao et al., (2009) found that the two-bladed Savonius rotor has higher performance than three-bladed Savonius rotor. Referring to Saha then the two-blade rotor has been chosen for this project. Table 2 shows the design parameters used in this research.

Table 2: Summary of design parameters.

Parameter Value Power Generated 5 kW

Swept Area 23.5 m2 Rated Wind Speed 10.5 m/sec

Aspect Ratio 2 Tip Speed Ratio 1.0

Solidity 2.114 Diameter – Height 3.5 m – 7 m Number of Blade 2

The design of the Savonius rotor blade is shown in Figure 2, while Table 3 shown the detail dimension of the Savonius rotor blade. The material proposed for the Savonius rotor blade in this paper is E-glass fiber. Since the manufacturing process and material cost are not considered in this paper, the material is chosen based on the material properties. Table 3 summarizes the general properties of the E-glass fiber and also rotor blade for this research.

Figure 2: Design of Savonius rotor blade.

Table 3: Summaries of rotor blade design and the material properties of E-glass fiber.

Parameter Value Swept Area, A 23.5 m2

Rotor Diameter, D 3500 mm Rotor Height, H 7000 mm

End Plate Diameter, Df 3850 mm Chord Length, d 1850 mm

Overlap Distance, e 200 mm Blade Thickness, t 10 mm

End Plate Thickness, tf 50 mm Density 1.7e3 – 2e3 kg/m3

Young’s modulus 27.2 – 39.4 GPa Poisson’s ratio 0.07 – 0.11

Tensile strength 217 – 520 Mpa Compressive strength 276 – 460 MPa



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3.0 SIMULATION AND ANALYSIS In this paper, two kinds of simulation and analysis were done i.e. Computational Fluid Dynamics (CFD) Analysis and Structural Analysis by using SolidWorks Flow Simulation and SolidWorks Structural Simulation/Cosmo.

3.1 Computational Fluid Dynamics (CFD) Analysis The purpose of this simulation is to obtain the pressure different between the concave and convex surface. The pressure difference between the concave and convex blade surface of the Savonius rotor induced drag force that turns the blade. The pressure difference was obtained by implementing Computational Fluid Dynamics (CFD) analysis on SolidWorks Flow Simulation. The two flow types in this paper were external flow and internal flow analysis. Both analyses were static analysis. The engineering goals for the internal analysis and the external analysis are two Surface Goals and four Global Goals. The two Surface Goals are dealing with total pressure, for both concave and convex surface. The four Global Goals are deal with total pressure, velocity, normal force and force. 3.1.1 External flow analysis The flow type of Savonius rotor blade is considered as external flow since it involves a solid model which is fully surrounded by the flow. The fluid flow is not bounded by outer surface, but bounded only by the Computational Domain boundaries. The Computational Domain is firstly defined to 7m x 7m x 9 m, which means that the Savonius rotor is enclosed by this region and the volume is fixed within this fluid flow field as shown in Figure 3. The Computational Domain can be set in different size, however, the bigger the Computational Domain, the longer the meshing and solution time.

Figure 3: Computational domain for Savonius rotor in external flow analysis. After the input data is ready, the model then is entering the meshing process. The meshing is viewed through a cut plot as shown in Figure 4 (a). The fluid is experienced separation when it passes through the blade and this region is considered as high-gradient flow region. The mesh control is set to be finer in this region to obtain better solution accuracy.

(a) (b) Figure 4: External flow distribution (a) Meshing and (b) pressure.

The pressure distribution around the Savonius rotor is viewed by a contour cut plot from the top view. The contour cut plot display the higher pressure region and lower pressure region as red and blue color respectively. From Figure 4(b), the pressure is high near the concave surface and is low near the convex surface. The maximum and minimum pressures are 101.469 kPa and 101.206 kPa respectively. The flow pattern is viewed by a flow trajectory is shown in Figure 5.



5

Figure 5: Flow pattern of external flow (ISO view and top view).

3.1.2 Internal flow analysis In this analysis, the wind enters the wind tunnel with the size of 10m x 25m x 8m, flow through the Savonius rotor blade, and then exits through the outlet that is set to environmental conditions. The Savonius rotor blade is placed in the middle of wind tunnel. In internal analysis, the Computational Domain is automatically enveloped the model wall, which is the wind tunnel size for this paper. The lids are used to apply boundary conditions which introduce to inlet velocity and outlet condition as shown in Figure 6. The lid thickness for an internal analysis is usually not important for the analysis. However, the lid should not be so thick until the flow pattern is affected downstream in some way. If the lid is created to be too thin, this will make the number of cells to be very high. For most cases the lid thickness could be the same thickness used to create the neighboring wall (Dassault System, 2011).

Figure 6: Setup boundary condition for internal flow analysis.

Figure 7 shows the meshing of the model and pressure distribution along the wind tunnel, observed from the top. The maximum pressure is indicated by the red color region whereas the minimum pressure is in the blue color region. Figure 7 also shows that the concave surface experience higher pressure than convex surface. The maximum and minimum pressure found in this analysis are 101.511 kPa and 101.189 kPa respectively. The flow trajectory pattern of the internal flow analysis is shown in the Figure 8.

Figure 7: Meshing and pressure distribution result of the internal flow.

Figure 8: Flow pattern of internal flow (ISO view and top view).



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3.1.3 Comparison of external and internal flow analysis Both external and internal flow analysis shows different results. From the view of pressure distribution, both analyses have shown that the pressure is higher on the concave surface and lower on the convex surface. This has verified the research statement by Sargolzaei and Kianifar (2007). However, the pressure differences of both analyses are not same. Internal flow analysis has higher pressure differences than external flow analysis, which are 322.15 Pa and 262.65 Pa respectively. This explained by the velocity vector in Figure 4 and Figure 7. Since the Savonius rotor was set at static position, when the wind reached on the blade, it was not turn as in real world condition and hence blocked the wind flow. From Figure 4, the velocity vector plot shows that the wind which blocked by the Savonius rotor blade are tend to flow in the direction that bias from the wind flow in the external flow analysis. The wind was not completely flow through the rotor blades. However, the wind was completely passed through the Savonius rotor blades as shown in Figure 7. This is because the Savonius rotor is bounded by the wind tunnel wall. The air which blocked by the blade is reflected to the wind tunnel wall and then passes through the Savonius rotor blades. Hence, in the static analysis, the result of internal flow analysis is more accurate and precise. Table 4 shows the comparison summary of the external and internal flow analysis.

Table 4: Comparison of external and internal flow analysis.

Aspect External Flow Analysis Internal Flow Analysis Pressure Differences 262.15 Pa 322.15 Pa High Pressure Region Concave surface Concave surface Low Pressure Region Convex surface Convex surface

Vector Plot

Flow Pattern

The wind flow is blocked by the Savonius rotor blades and bias from the wind direction. Since there is no wall boundary, the wind flow out from the Computational Domain. The wind is not completely flow through the rotor blades.

The wind flow is blocked by the Savonius rotor blades and bias. However, the Savonius rotor is bounded by the wall of wind tunnel. Hence, the wind is completely flow through the rotor blades.

3.2 Structural Analysis The structure of the Savonius rotor blade is analyzed using FEA static method by SolidWorks Simulation/Cosmo software. Since the two Savonius rotor blades are symmetry, the analysis is performed only on one blade. The FEA result is interpreted in three criteria: stress, deformation, and factor of safety. First step of FEA analysis is assigned material to the blade model where E-glass fiber was the material chosen. Then the fixed constraints/ fixtures are applied on the top, bottom and center of the shaft, as shown in Figure 9. The fixtures constrained all translational and all rotational degrees of freedom. Therefore, the blade is stay in a static and fixed position. The load for this analysis is force with 595.98 N obtained from the aerodynamic analysis and equally distributed on the concave blade surface.

Figure 9: Boundary condition of the rotor blade.

Figure 10 shows the meshing of the model by using tetrahedral shape mesh elements and also depicts the FEA result of the model which presents the stress distribution over the rotor blade structure.



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Figure 10: Meshing and stress distribution of the rotor blade.

The maximum and minimum Von Mises stress for the Savonius rotor blade are 6.6052 x 105 Pa and 5.0534 x 10-2 Pa respectively. The result is acceptable because the maximum Von Mises stress is much lower than the yield strength of the material and provide the minimum Factor of Safety as 787.26. Figure 11 shows the deformation of the model under the given load, the maximum displacement is 2.94 mm at the edge of the blade. The deformation is acceptable because it is small in relation to the overall size of the blade structure.

Figure 11: Deformation of the rotor blade. 4.0 CONCLUSION The paper has investigated the elements that contribute to the design and analysis Savonius rotor blade. The blade was design by using SolidWorks software. CFD analysis was performed in order to obtain the pressure different between the concave and convex surface of the rotor blade. The force induced to the blade was calculated from aerodynamic analysis. The structural feasibility was analyzed by Finite Element Analysis method to obtained the maximum deformation and stress experienced by the rotor blade.

From the CFD analysis, it is found that the concave blade region experience high pressure while the convex blade region experience low pressure for two blades Savonius rotor. The maximum pressure and minimum pressure from internal flow analysis were 101.511 kPa and 101.189 kPa respectively. The high pressure region produces 595.98 N of drag force that spinning the Savonius wind turbine.

The maximum deformation of the Savonius rotor blade was 2.94 mm. This deformation is acceptable because it is relatively small compared to the whole blade model. The maximum Von Mises stress obtained from the FEA was 6.6052 x 105 Pa with the minimum Factor of Safety was 787.26. This concluded that the Savonius rotor blade was safe enough to withstand the aerodynamic force on the turbine.

The presented study was limited by the software and computer capability. In real world condition, when the air flows through the blade, it will induce a force to turn the rotor blade. However, the SolidWorks software is unable to perform the CFD analysis while the blades are turning (dynamic condition). Therefore, only the static CFD analysis was performed in this paper, for both external and internal flow analysis. ACKNOWLEDMENT Authors would like to thanks to the Universiti Teknikal Malaysia Melaka for its financial support of the Vertical Axis Wind Turbine Project under short term grant PJP/2011/FKP (7A)/S00874.



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REFERENCES [1] Peter, C., Sheets, B. and Tablar, S. (2008): Emerging Technologies in the Wind Turbine Market, The Palace

Hotel, San Francisco, CA. 19-20 November 2008. [2] Manwell, J.F., McGowan, J.G. and Rogers, A.L. (2009): Wind Energy Explained UK, John Wiley & Son Ltd. [3] Akwa, J.V., Junior, G.A.S. and Petry, A.P. (2012): Discussion on the Verification of the Overlap Ratio Influence

on Performance Coefficients of a Savonius Wind Rotor Using Computational Fluid Dynamics. Renewable Energy, Vol.38, pp.141-149.

[4] Halsey, N. (2011): Modeling the Twisted Savonius Wind Turbine Geometrically and Simplifying its Construction. Oregon Episcopal School.

[5] D’Alessandro, V., Montelpare, S., Ricci, R. and Secchiaroli, A. (2010): Unsteady Aerodynamics of a Savonius Wind Rotor: A New Computational Approach for the Simulation of Energy Performance. Energy, Vol.35, pp. 3349-3363.

[6] Islam, M.Q., Hasan, M.N. and Saha, S. (2005): Experimental Investigation of Aerodynamic Characteristics of Two, Three and Four Bladed S-Shaped Stationary Savonius Rotors. The Proceeding of International Conference on Mechanical Engineering 2005, 28-30 December 2005, Dhaka, Bangladesh.

[7] Musgrove, P. (2010): Wind Power, UK: Cambridge University Press. [8] Hayashi, T., Yan, L. and Suzuki, K. (2004): Wind Tunnel Tests on a Three-stage Out-phase Savonius Rotor.

[Online]. Available at: http://www.2004ewec.info/files/231400_tsutomuhayashii_01.pdf [Accessed on 13 November 2011].

[9] Jain, P. (2011): Wind Energy Engineering, McGraw-Hill, New York. [10] Johnson, C. (1998): Practical Wind-Generated Electricity. [Online]. Available at: http://mb-

soft.com/public/wind.html [Accessed on 24 April 2012]. [11] Solanki, C.S. (2009): Renewable Energy Technologies: A Practical Guide for Beginners, PHI Learning Private

Limited, New Delhi, pp.103-106. [12] Saha, U.K., Thotla, S. and Maity, D. (2008): Optimum Design Configuration of Savonius Rotor Through Wind

Tunnel Experiments. Journal of Wind Engineering and Industrial Aerodynamics, Vol.96, pp.1359-1375. [13] Zhao, Z.Z., Zheng, T., Xu, X.Y., Liu, W.M. and Hu, G.X. (2009) Research on the Improvement of the

Performance of Savonius Rotor Based on Numerical Study Sustainable Power Generation and Supply, The Proceeding in SUPERGEN’09, Nanjing, 6-7 April 2009.

[14] Dassault System (2011): Productivity: Experience the Benefits of SolidWorks. [15] Sargolzaei, J. and Kianifar, A. (2007): Estimation of the Power Ratio and Torque in Wind Turbine Savonius

Rotors Using Artificial Neural Networks. International Journal of Energy, Vol.1, No.2.

GLOBAL ENGINEERS & TECHNOLOGISTS REVIEW www.getview.org

G.L.O.B.A.L E.N.G.I.N.E.E.R.S. .& .-.T.E.C.H.N.O.L.O.G.I.S.T.S R.E.V.I.E.W 9

YAHAYA1, S.H., ALI2, J.M., YAZARIAH3, M.Y., HAERYIP SIHOMBING4, and YUHAZRI5, M.Y.

1, 4, 5 Faculty of Manufacturing Engineering Universiti Teknikal Malaysia Melaka

Hang Tuah Jaya, 76100 Durian Tunggal, Melaka, MALAYSIA [email protected] 4iphaery@ utem.edu.my [email protected]

2, 3 School of Mathematical Sciences

University of Science Malaysia Minden, 11800, Penang, MALAYSIA

[email protected] [email protected]

1.0 INTRODUCTION Gears are the most widely used elements in both applications such in consumer and industrial machineries. As we know, gear types may be grouped into five main categories, namely, spur, helical, rack and pinion, worm and bevel. As referred to in (Babu and Tsegaw, 2009; Bradford and Guillet, 1943; Higuchi and Gofuku, 2007), these are the most common curves used for the gear tooth profiles. These curves are developed based on the approximation theory, for instance; the development of involute curves has used a Chebyshev approximation (Higuchi and Gofuku, 2007). Besides, the tracing point method has also been applied along the path (shape) design of involute curves (Margalit, 1995; Reyes et al., 2008; Yeung, 1999). Therefore, several methods and concepts have been employed in the generation of involute curves. With reference to the above evidence, it shows that this involute curve is not directly produced and is shown as the approximated (inexact) curves. For these reasons, we propose the theoretical developments of the exact (or known as the transition) curves using the parametric function.

Mathematically, parametric function is a method to define a relation between the independent (free) variables. Previously, Ali (1994) and Ali et al., (1996) explored the parametric of Bézier-like cubic curve using Hermite interpolation. They are, however, only focused on the function developments, while the scope of designs through the proposed curves is not touched upon. Therefore, in this study, we use the Bézier-like cubic curve approach as the degree three (cubic) polynomial curves that allow the inflection points. This is due to the approach is suitable for G2 (curvature) blending application curves and also contains the shape parameters which can control the shape of the curve (Ali, 1994; Ali et al., 1996; Walton and Meek, 1999). As compared to the cubic Bézier curves, the shape parameters are not automatically included (Rashid and Habib, 2010; Habib and Sakai, 2008). Figure 1 shows the method of designing the transition curves will follow the five cases of clothoid templates as was identified by Baass (1984) and successfully used in highways or railways design. These templates are crucial to determine the design parameters in order to ensure the comfort and safety of road users (Baass, 1984). The templates are 1) straight line to circle, 2) circle to circle with C transition, 3) circle to circle with an S transition, 4) straight line to straight line and 5) circle to circle where one circle lies inside the other with a C transition (Walton and Meek, 1999; Baass, 1984; Walton and Meek, 1996). Figure 2 shows the profiles

ABSTRACT

An involute curve (or known as an approximated curve) is mostly used in designing the gear teeth (profile) especially in spur gear. Conversely, this study has the intention to redesign the spur gear teeth using the transition (S transition and C spiral) curves (also known as the exact curves) with curvature continuity (G2) as the degree of smoothness. Method of designing the transition curves is adapted from the circle to circle templates. The applicability of the new teeth model with the chosen material, Stainless Steel Grade 304 is determined using Linear Static Analysis, Fatigue Analysis and Design Efficiency (DE). Several concepts and the related examples are shown throughout this study. Keywords: Spur Gear Profile, S-Transition Curve, C-Spiral Curve, FEA, DE.

INTEGRATING SPUR GEAR TEETH DESIGN AND ITS ANALYSIS WITH G2 PARAMETRIC BÉZIER-LIKE CUBIC TRANSITION AND

SPIRAL CURVES



10

design where the third and fifth templates are similar to the involute curves. In this study, these two templates with the application of Bézier-like cubic curve function are therefore chosen to redesign a spur gear teeth profile. By using the new method, the curve is directly generated with the significant increases in accuracy, and also the actual spur gear profile can be produced through this method.

This study will continue to find out the applicability of the new profile generation and the gear material through Stress-Strain Analysis, Fatigue Analysis and DE. FEA is the most common tools for the analysis while Stainless Steel Grade 304 is chosen for the selection of material in this study. This gear material is often selected for gear application since it has an aesthetic appearance, ease of fabrication and good in impact resistance. The scheme in Stress-Strain Analysis known as a Linear Static Analysis is used to determine many of the physical structures such in the stress, force and displacement distributions (Westland, 2006; Sapto and Safarudin, 2008). Many of the studies use this linear scheme to determine the structural applicability between the involute profiles and the gear material (Feng, 2011; Gurumani and Shanmugam, 2011; Reagor, 2010). However, there is no related study for the proposed curves. The retrieved values from Linear Static Analysis will be further explored in Fatigue Analysis and DE where both are strongly connected to the safety factor (Moultrie, 2009; Khai et al., 2007; Niederstucke et al., 2003; Firth and Long, 2010). Incidentally, the descriptions of DE in the design area are still less-decrypted. Figure 3 shows one of the applications using Linear Static Analysis. The next section will explain the nomenclature used in this study.

Figure 1: Railways route design modeling using transition curves application.

Figure 2: Third (left) and fifth (right) cases of circle to circle templates. (Baass, 1984; Walton and Meek, 1996; Meek and Walton, 1989).

Figure 3: Automotive crankshaft in Linear Static Analysis. 2.0 NOMENCLATURE Consider the notations such as symbol and unit in the nomenclature, as shown in Table 1. This nomenclature is the basic foundation of Linear Static Analysis (M Britto, 2005).

It is noticed that the unit of these input and output parameters will be based on the system of units that are applied in the solid model. Table 1 is extensively used for the analysis purposes. In the next section, we will explore the description method of the designing of spur gear using S transition and C spiral curves.



11

P3

P2P1

P0

Table 1: Some of the parameters that are commonly used in the system of units

System of Units Input Parameters Output Parameters

Length, ζ Force, ƒ Mass, η Density, ρ Displacement, δ Force, ƒ Stress, υ

1 m N kg kgm-3 m N Pa

2 mm N mg mgmm-3 mm N MPa

3 ft Ibf slug slugft-3 ft Ibf psf

4 in Ibf Ibf.s2in-1 Ibf.s2in-4 in Ibf psi

3.0 METHOD OF DESCRIPTIONS

3.1 Bézier-Like Cubic Curve as a Function Bézier-like cubic curve (also known as the cubic alternative curve) is a new basis function in the field of Computer Aided Geometric Design (CAGD). Bézier-like basis function simplifies the process of controlling the curve since it has only two shape parameters, λ0 and λ1 to control or change the shape of the curve such in Figure 4, compared to the cubic Bézier curve for which the control points need to be adjusted (Farin, 1997). Bézier-like cubic curve is formulated by applying the form of Hermite interpolation given by (Ali, 1994; Ali et al., 1996; Ahmad, 2009).

10 ,)()()()()( 33221100 tPtPtPtPttz (1)

with,

))1)(2(1()( ,)1()(

)1()( ),)2(1()1()(

12

32

12

2010

20

tttttt

tttttt

(2)

where 3210 ,,, PPPP are the control points, and )(

3)(

2)(

10,,),( tttt are Bézier-like cubic basis

functions. Hence, Bézier-like cubic curve can be written as

312

22

112

0002 ))1)(2(1()1()1())2(1()1()( PttPtttPtPtttz (3)

Figure 4: Distribution of Bézier-like cubic curves with the different values of λ0 and λ1.

In the next subsection, we will explain the control points identification in Bézier-like cubic curve based on the circle to circle template (Figure 2) 3.2 Third Case of Circle Templates as an S-Shaped Transition Curve The cubic Bézier curve had been predominantly used in curve design, as shown in studies by Habib and Sakai (2003) and Walton and Meek (1999). The similarity between this function and Bézier-like cubic curve is that both functions have four control points. By referring to Habib and Sakai (2003), the control points are stated as



12

}cos,sin{* },sin,{cos*}cos,sin{* },sin,{cos

32113

01*000

kPPrcPhPPrcP

(4)

where, c0 and c1 are the centre points of circles ψ0 and ψ1, r0 and r1 are the radii of circles ψ0 and

ψ1, h and k are the length or norm of 01, PP and 23 , PP while α is an angle's circles measured

anticlockwise. These control points are unique only in the case of the S-shaped transition curve. The improvements are made in Eq. (4) to increase the degree of freedom and its applicability such

}cos,sin{* },sin,{cos*}cos,sin{* },sin,{cos

32113

01*000

kPPrcPhPPrcP

(5)

with the angles in the circle {ψ0, ψ1} denoted as α and β which are measured anticlockwise. The

other parameters remain the same as in Eq. (4). The value of h and k will be calculated by using the curvature continuity (G2 continuity) shown as

10

1)1( ,1)0(r

tr

t (6)

By applying these theories, an S-shaped curve is demonstrated in Figure 5.

Figure 5: An S-shaped transition curve produced by using G2 Bézier-like cubic curve.

3.3 Fifth Case of Circle Templates as a C-Shaped Spiral Curve The fifth case will produce a type of C-shaped transition curve as a single spiral (Baass, 1984; Walton and Meek, 1996; Habib and Sakai, 2005). The curve architecture in transition and spiral curves is totally different from the segmentations used. As shown in Figure 6, three segments are needed to design C-shaped transition curve whereas only two segments are used in C-shaped spiral curve.

Figure 6: Transition (left) and spiral (right) curves architecture applied in cubic Bézier curve (Walton et al., 2003).

We now identify the control points as in (Habib and Sakai, 2003)



13

}cos,{sin* },sin,{cos*}cos,sin{* },sin,{cos*

32113

01000

kPPrcPhPPrcP

(7)

Where the notations are the same as in Eq. (4). The definitions in Eq. (7) will produce a C-shaped

transition curve which is related to the second case (Walton and Meek, 1999; Baass, 1984 ; Walton and Meek, 1996). Modifications of Eq. (7) are needed for fifth case template, and therefore, we have

}cos,sin{* },sin,{cos*}cos,{sin* },sin,{cos*

32113

01000

hPPrcPkPPrcP

(8)

where {c0, r0, β} is the centre point, radii and angle of the circle, ψ0 while the centre point, radii and

angle in the circle, ψ1 represented as {c1, r1, α}. Parameter h and k are the length or norm of 23 , PP

and 01, PP . In order to design C spiral curve, the segments used in Eq. (8) should be reduced (Walton et al., 2003) by assuming that

21 PP (9)

Then, in Eq. (9) either h or k can be eliminated by using a dot product. If the parameter chosen is k, the expression will be dotted with a vector, }sin,{cos where

]sin[]cos[*})sin,{cos)(( 0101

rrcck (10)

The modified control points that satisfy the fifth case template (C spiral curve) are given by

}sin,{cos*

}cos,{sin* }sin,{cos*

113

01

000

rcPkPPrcP

(11)

In order to join C spiral curve with the circles, we will need to apply the curvature continuity such

10

1)1( ,1)0(r

tr

t (12)

This continuity is also used to determine the shape parameters, λ0 and λ1 in Eq. (3). A generated C-

shaped spiral curve is visualized in Figure 7.

Figure 7: C-shaped curve in the fifth case of circle templates using G2 Bézier-like cubic curve.

Both curve theories will be applied to redesign a profile (tooth) of the spur gear. Currently, an involute or evolute curve is always used in designing this spur tooth profile.



14

4.0 SPUR GEAR DESIGN

4.1 Modeling of Spur Gear Traditional spur gears have the teeth which are straight and parallel to the axis of the shaft that conveys the gear (Mott, 2003). Normally, the teeth have an involute form where this form can be acted as in contacting the teeth (Mott, 2003). Figure 8 shows a schematic and related terminology for spur gear

Figure 8: Some of the terms in spur gear (Price, 1995).

Based on Figure 8, an inner circle and an outer circle can be drawn, and smaller circles can be fitted within their boundaries, as shown in Figure 9. This resulting structure will be used to design a profile (tooth) of the spur gear.

Figure 9: A new basis model of designing a spur gear tooth. 4.2 Single Tooth Design uses an S-Shaped Transition Curve By applying the segmentation of divisions, two segments are divided to create this single tooth. This is because an S-shaped curve has the beginning point at the tangent of base circle and ending at the tangent of the outside circle. For the first segment, the inputs are c0 = {-0.398, 0.689}, c1 = {0, 0.795}, α = 0.6667π radian, β = 0.5π radian, r0 = r1 = 0.206 are used, while for segment two, the inputs are c0 = {0.398, 0.689}, c1 = {0, 0.795}, α = 0.3333π radian, β = 0.5π radian and other parameters remain the same as in segment one. Then, h and k are determined to be 0.2779 and 0.3780 in accordance with the inputs and theories above. Notice that, the value of h and k is similar in both segments. Figure 10 shows the design of the tooth and its segmentation using an S-shaped transition curve.

Figure 10: Single tooth of spur gear using an S-shaped transition curve (left) and its segmentation (right). 4.3 Single Tooth Design uses a C-Shaped Spiral Curve In this case, Figure 9 is modified to suit the elements in designing a C-shaped spiral curve (Subsection 3.3). The small circles are drawn at the intersection of two circles. For this curve, it needs four segments to design a single tooth of the spur gear. Segment one consists of c0 = {-0.398, 0.689}, c1 = {-0.247, 0.725}, α = 0.05556π radian, β = 0.6667π radian, r0 = 0.206 and r1 = 0.05. The values of k and shape parameters, λ0, λ1 are computed by using the Eqs. (3), (10), (11), and (12). We found that k = 0.1431 whereas λ0 and λ1 are equal to 2.7695 and 0.3619, respectively. For segment two, the inputs are c0 = {0, 0.795}, c1 = {-0.247, 0.725}, β = 1.5π radian while the rest is exactly the same as in segment one. This time, k and shape parameters, λ0, λ1 are found to be 0.2449, 2.1279 and 0.5343, respectively. These two segments have a



15

CATIA V5(Develop the solid model of the spur

gear)

CATIA V5(All coordinates are joi ned using spl ine package)

CATIA V5(Coordinates definition in

accordance to the tooth profile in Wolfram Mathematica 7.0)

Wolfram Mathematica 7.0

(Apply the curve theories in designing the tooth

profile of the spur gear )

symmetrical curve which can be defined as mirroring or balancing to segment three and four. Thus, segment three and four have the similar shape to segment one and two, respectively (Du Sautoy, 2009). Figure 11 shows the design of the tooth and its segmentation using this C spiral curve. Next, we will apply the single tooth designs using both developed curves in order to design a solid model of the spur gear.

Figure 11: Single tooth of spur gear using a C-shaped spiral curve (left) and its segmentation (right). 4.4 S Transition and C Spiral as the Shaped Curves in Solid of Spur Gear In Computer Aided Geometric Design (CAGD), the surface design technique is commonly used where this technique has several design and surface properties to be considered (Bloor et al., 1995), difficult to extrude as a solid model and unsuitable in Computer Aided Analysis (CAE) purposes. To solve these problems, the integration of Mathematical and Computer-Aided Design (CAD) may be applied. Wolfram Mathematica 7.0 and CATIA V5 are both selected as the integrated software of the designing of the solid model of the spur gear. The following process flow diagram is used to show the integration of both software and its applications in spur gear design (Figure 12). Two models of six gear teeth are developed with the outside and shaft diameters as well as the gear thickness are 40.04 mm, 5.672 mm and 4 mm (Figure 13).

Figure 12: Integrated process flow diagram used in designing a solid model of the spur gear.

Figure 13: Solid of spur gear using S transition (left) and C spiral (right) curves.

We will extend these solid design structures to analyze its applicability of the new tooth profiles with the gear material using Linear Static Analysis.

5.0 LINEAR STATIC ANALYSIS

5.1 Fundamental of Linear Static Analysis Most of the structural problems can be treated as a linear static problem if and only if the several



16

PatranProcesses: Import 3D model,

meshing process, material selection, locate the load and boundary

conditions.

NastranProcesses: Create the Stiffness or

Square Matrix, calculate the displacement, stress

and force.

PatranProcesses: Plot the contour

graph of the stress, di splacement and force

distributions.

assumptions are made such as linear behavior, elastic material and static loads (Ziaei Rad, 2011). The scenario of the assumptions is shown in Figure 14. One of the applicabilities of Linear Static Analysis is to find out the structural response of the bodies' spinning with the effect of the velocities or accelerations are constants since the applied loads are static (do not change with time). This analysis measures the responses of the displacement, strain (deformation), stress and reaction force. The linear static problem can be simplified using the expression of the linear matrix system such

}{}]{[ FxK (13) where,

[K] is the structural (assembled) stiffness matrix or square matrix of order (n x n), {x} are the

unknown parameters (displacement, strain and stress) of order (n x 1) while {F} is the loading in the system represents the matrix of the order (n x 1). For the solution of {x}, the matrix solver that will be used is FEA. The next subsection will explain the FEA solver and its structural modeling.

Figure 14: Linearity and elastic material assumptions in the static analysis (Hamburger, 2000). 5.2 FEA Solver and Its Modeling MSC Nastran & Patran is the FEA software chosen as the preferred solution for Linear Static Analysis. The following process flow diagram is the FEA processes that will be applied using this software, as shown in Figure 15 (Sapto and Safarudin, 2008; Daryl, 2006).

Figure 15: Process flow diagram of MSC Nastran & Patran and its FEA application.

5.2.1 CAD models and its meshes The CAD or solid models (Figure 13) will be imported to MSC Patran software as the starting process of the Linear Static Analysis (FEA). The selection of the element model in FEA is very crucial and will be determined the overall process of the analysis. The gear structures (Figure 13) are the Three-Dimensional (3D) geometry solid models that reveal the complex structures. Solid element models are widely used to analyze the complex elements such as structural components and loading conditions and also the estimation of the stress levels (Fellipa, 2009; Entrekin, 1999). The element topology that is chosen for the structure (Figure 13) is Tetrahedral-10 (Tet-10) that represents a 3D solid triangle with four planar faces and ten nodes (Fellipa, 2009; Fellipa 2011), as shown in Figure 16. The Tet-10 or second order tetrahedral element is commonly used for its ability to mesh almost any solid regardless of its complexity (Entrekin, 1999; Said et al., 2012). The resulting of the applied Tet-10 is shown in Table 2 and Figure 17.

Figure 16: Tet-10 element's topology (Sapto and Safarudin, 2008; Fellipa, 2011).



17

Table 2: Tet-10 and its mesh data structures

Solid of the spur gear (S transition) Solid of the spur gear (C spiral)

Global Edge Length Nodes Elements Global Edge Length Nodes Elements

1.0551234 34897 22448 1.011111 34854 22668

Figure 17: Tet-10 and its spur gear meshing of the applied curves, S transition (left) and C spiral (right). 5.2.2 Boundary and loading conditions The next process is the setting of the boundary and loading conditions on the solid model (Figure 17). Boundary condition can be referred as the external load on the border of the structure (Feng, 2011). As this case, the boundary condition is applied on the gear shaft (Figure 17) where in this shaft, the displacement does not happen (is equivalent to zero) (Nikolić et al., 2012; Lee, 2009). This is the important setting to simulate the gear transmission (Feng, 2011). Many of the references used Torque (T) as the applied load (loading condition) for their analysis and application (Feng, 2011; Nikolić et al., 2012; Lee, 2009). In order to find out the new tooth profiles and its applicability (Figure 13), the suitable load to be applied is Pressure (P) which this load is set on the contact stress of both spur gear models (Figure 17) (Gilbert Gedeon, 1999; Wang et al., 2011). The critical region of the contact stress is shown in Figure 18 (Rameshkumar et al., 2010). Figure 19 shows the setting of both conditions in the models (Figure 17). One of the conditions, P is applied as 50 MPa

Figure 18: The critical region (two dots along the blue line) of the contact stress in the tooth profile (Rameshkumar et al., 2010).

Figure 19: Spur gear model using C-spiral curve with its boundary conditions and load, P = 50 MPa.



18

5.2.3 Material selection for spur gear Stainless Steel Grade 304 also known as AISI 304 is widely used in the range of applications such as a gear material (Zehe et al., 2002). This grade of material is the type of Austenitic Stainless Steels for the highly contained of Chromium and Nickel. These Steels have high ductility and ultimate tensile strength and also low yield stress. Figure 20 shows one of the cases of Austenitic Stainless Steels when compare to a Typical Carbon Steel. It can be seen clearly that the stability region is occurred in the types of Austenitic while the Carbon type has an instability diagram (Figure 20). The mechanical properties of the AISI 304 are presented in Table 3. These properties will be used in this static analysis.

Figure 20: Stress-strain behavior of the Austenitic and Carbon Steels. After applying the above settings on the models (Figure 19), it is ready to analyze using MSC

Nastran & Patran software (Figure 15). The processor of Intel ® Core ™ 2 Duo CPU with the RAM, 3.49 GB is used to operate the software. The resulting of this process will be discussed in the next subsection.

Table 3: AISI 304 and its characteristics (Peckner and Bernstein, 1977)

Modulus of Elasticity

Yield Strength

Ultimate Strength

Poisson Ratio

Density Damping

Coefficient 195 GPa 215 MPa 505 MPa 0.29 8 gcc-1 0.003

5.2.4 Simulation results and its safety factor The discussion began with the presentation of the generated results using Linear Static Analysis. These results of the proposed models will be compared with the existing model. The existing model (EM) is directly designed using the graphical properties in CATIA V5 with the dimensions remained the same. Table 4 shows the maximum Von Mises stress (υmax) and maximum displacement (δmax) distributions among the models through the repeated loads. Hence, the safety factor (Sf) or also known as a design margin (Michalopoulos and Babka, 2000) can be derived as in Eq. (14) (Clifford et al., 2008; Shenoy, 2004) and where the yield strength of AISI 304 can be found in Table 3.

max

f

304 AISIof StrengthYieldS (14)

Table 4: The result distributions of δmax, υmax and Sf in the applied models

P (MPa)

EM S Transition C Spiral δmax

(mm) υmax

(MPa) Sf δmax

(mm) υmax

(MPa) Sf δmax

(mm) υmax

(MPa) Sf

10 5.93E-3 61.3 3.507 5.99E-3 61.9 3.473 5.77E-3 62.5 3.440 20 1.19E-2 123 1.748 1.20E-2 124 1.734 1.15E-2 125 1.720 30 1.78E-2 184 1.168 1.80E-2 186 1.156 1.73E-2 187 1.150 40 2.37E-2 245 0.878 2.40E-2 248 0.867 2.31E-2 250 0.860 50 2.97E-2 307 0.700 3.00E-2 309 0.696 2.89E-2 312 0.689

Figure 21 depicts the contour plot of determining the presented values of υmax and δmax (Table 4) for the applied P = 30 MPa. It is clearly shown in the plot legend that δmax approximates to 1.73E-2 at the node, 34450 while the value of υmax is equal to 187 MPa at the node, 27631. The υmax



19

(red color) occurs in the gear shaft because of the model (Figure 17) is analyzed in the steady state (static) behavior (Boulos et al., 1997). The resulting of υmax (P = 30 MPa) is far below the yield strength of AISI 304. Therefore, Sf approximates to 1.15 (Table 4). Failure occurs if and only if Sf < 1 or Sf has the negative sign (Michalopoulos and Babka, 2000). All gear models in Table 4 are safe to be used if P 30 MPa whereas the models are in failure modes if P > 30 MPa. These inequalities show that the new teeth profiles (S transition and C spiral curves) have the same applicability (strength) when compare to the EM. The findings in Table 4 will be used to predict when the failure (fatigue) modes start to occur among the gear models. This fatigue prediction will be discussed in the next section.

Figure 21: The example of findings {δmax, υmax} for the applied P = 30 MPa in C spiral gear model.

6.0 FIRST-ORDER NEWTON INTERPOLATING POLYNOMIAL AS A FATIGUE PREDICTOR Fatigue can be described as a failure mode that happened on the structure design which this structure is affected by the implications of repeated or varying loads, fluctuating loads and rapidly applied loads (Pun, 2001). Fatigue also can be occurred by the different physical mechanisms such as low-cycle and high-cycle fatigue (Pun, 2001). As referred to in failure mode above, let assume that the fatigue starts to occur if Sf = 1. Due to this Sf, the related load will be predicted using the Newton’s Interpolating Polynomial. The general form of this polynomial such (Chapra and Canale, 2010).

))...()((...))(()()( 110102010n nn xxxxxxbxxxxbxxbbxf (15)

Where, n is referred to as the nth-order polynomial, b0, b1, b2 and bn are the polynomial coefficients and

the set of (n+1) data points represented by {xi, f(xi)} with i = {0, 1, 2, 3,…, n}. The polynomial coefficients also can be shown as (Chapra and Canale, 2010).

],,[ ];,,[];,[);( 0110122 011 00 xx,...,xxfbxxxfbxxfbxfb nnn (16)

And the bracketed functions are evaluated using the Finite Divided Differences where (Chapra and

Canale, 2010).

0

02111011

02

0112012

01

0101

],...,,[],...,,[],,[

;][],[

],,[;)()(

],[

xx

xxxfxxxfxx,...,xxf

xx

x,xfxxfxxxf

xx

xfxfxxf

n

nnnnnn

(17)

We can see clearly in Table 4 that the fatigue of the models starts to occur in the range of P [30, 40] that

allows only two data points. Therefore, the First-Order Newton Interpolating Polynomial or known as Linear Interpolation is the suitable polynomial to be applied. Let P is denoted as x-axis while Sf is equal to f(x) or y-axis. The first-order or linear form (n = 1) is given as

)()( 0101 xxbbxf (18)

With b0 and b1 are calculated using Eqs. (16) and (17). The algorithms of using this linear interpolation

include four steps such as: Set x0, x1, f(x0) and f(x1); Calculate b0 and b1; Substitute x0, b0 and b1 into f1(x); Solve x



20

for f1(x) = 1. All calculated values of x among the models (Table 4) are fully depicted in Table 5. It shows that the models are in safety mode if P 35 MPa while the fatigue may start occurring when P > 35 MPa.

Table 5: The Initial loads of P for the fatigue mode

EM S Transition C Spiral 35.7931 MPa 35.3979 MPa 35.1724 MPa

With the reference of Table 5, the values in this table will be used to determine the Design Efficiency (DE) for the proposed models. Further discussion of DE will be explained in the following section.

7.0 DESIGN EFFICIENCY OF THE PROPOSED DESIGN MODELS Design Efficiency (DE) or Sensitivity Analysis (SA) relates to the design quality (Aas, 2000; Azarian et al., 2011) that can achieve the stated objectives of the certain studies such in this study, to find out the applicability of the new teeth design with AISI 304 as the selected material. Generally, the levels of DE in the engineering field are almost 85 % and above. Hence, 85 % can be assumed as the benchmark of the improvements that will be made (Gupta et al., 2006). Let make the assumptions in Table 5 that EM equivalents to the true value, xt (input) whereas S Transition and C Spiral are represented as the approximation values, xa (output). Thus, DE can be expressed as (Khai et al., 2007).

%100)(t

a xx

InputOutputDE (19)

After applying Eq. (19), DE of an S Transition approaches to 99% while for C Spiral method approximates

to 98%. Both results have shown that DE almost achieved 100% of the design effectiveness. With reference to the above mentioned benchmark (85%), the proposed designs are the acceptable methods of designing the spur gear teeth with AISI 304 as the selected material. 8.0 CONCLUSIONS AND FUTURE WORK This study has proposed the designing of spur gear teeth using S-shaped transition and C-shaped spiral curves (directly produced). Circle to circle templates have successfully been applied in both curves design. The solids of spur gear have been generated by the integration between Mathematical and CAD software. The applicability of the proposed design and the material, AISI 304 is measured using Linear Static Analysis, Fatigue Analysis and DE. First-order Newton interpolating polynomial can be employed as a fatigue predictor for all design models. As referred to in Table 4, all models are in safety mode if P 30 MPa while in failure mode if P > 30 MPa. Generally, fatigue starts to occur when P > 35 MPa in all design models whereas the models are safe to use in the related applications if P 35 MPa. The applicability (strength) of the proposed design, S and C curves are the same when compare to EM as shown in the above analyses. The new design methods, S and C curves are the acceptable methods of designing the spur gear teeth as both methods have presented DE greater than 85 % of the design effectiveness. In future, this study will continue in the analysis of dynamic and acoustic features such as normal, frequency and transient modes and also in the noise test. ACKNOWLEDMENT This research was supported by Universiti Teknikal Malaysia Melaka under the Fundamental Research Grant Scheme (FRGS). The authors gratefully acknowledge everyone who contributed helpful suggestions comments. REFERENCES [1] Ali, J.M. (1994): An Alternative Derivation of Said Basic Function. Journal of Sains Malaysiana, Vol.23, No.3,

pp. 42-56. [2] Ali, J.M., Said, H.B. and Majid, A.A. (1996): Shape Control of Parametric Cubic Curves. SPIE, Fourth

International Conference on Computer-Aided Design and Computer Graphics, Zhou, J. (Ed), Vol.2644, pp.128-133.

[3] Ahmad, A. (2009): Parametric Spiral and its Application as Transition Curve. Ph.D Thesis, Universiti Sains Malaysia, pp.19-22.

[4] Aas, E.J. (2000): Design Quality and Design Efficiency; Definitions, Metrics and Relevant Design Experiences. International Symposium on Quality of Electronic Design, 20-22 March, San Jose, CA, USA.



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[5] Azarian, A., Vasseur, O., Bennaï, B., Jolivet, V., Claeys-Bruno, M., Sergent, M. and Bourdon, P. (2011): Sensitivity Analysis of Interference Optical Systems by Numerical Designs. Journal de la Société Française de Statistique, Vol.152, No.1, pp.118-130.

[6] Babu, V.S. and Tsegaw, A.A. (2009): Involute Spur Gear Template Development by Parametric Technique using Computer Aided Design. African Research Review, Vol.3, No.1, pp. 1-6.

[7] Bradford, L.J. and Guillet, G.L. (1943), Kinematics and Machine Design, John Wiley & Sons, New York. [8] Baass, K.G. (1984): The Use of Clothoid Templates in Highway Design. Transportation Forum, Vol.1, pp.47-

52. [9] Bloor, M.I.G., Wilson, M.J. and Hagen, H. (1995): The Smoothing Properties of Variational Schemes for

Surface Design. Computer Aided Geometric Design, pp.381-394. [10] Boulos, P.F., Wood, D.J. and Lingireddy, S. (1997): Pressure Vessels and Piping Systems-Shock and Water

Hammer Loading. Encyclopaedia of Life Support Systems (EOLSS), American Water Works Association, UNESCO.

[11] Clifford, K.H., Bibeau, T.A. and Quinones, A.Sr. (2008): Finite Element Analyses of Continuous Filament Ties for Masonry Applications: Final Report for the Arquin Corporation. SAND2006-3750, Sandia National Laboratories Albuquerque, California, USA.

[12] Chapra, S.C. and Canale, R.P. (2010): Numerical Methods for Engineers, 6th Edn. McGraw-Hill, New York. [13] Du Sautoy, M. (2009): Symmetry: A Journey into the Patterns of Nature, HarperCollins, UK. [14] Daryl, L. (2006): A First Course in Finite Element Analysis, Brookscole, Singapore. [15] Entrekin, A. (1999): Accuracy of MSC/Nastran First and Second Order Tetrahedral Elements in Solid

Modeling for Stress Analysis. MSC Aerospace Users’ Conference Proceedings. [16] Farin, G. (1997): Curves and Surfaces for Computer Aided Geometric Design, A Practical Guide, 4th Edition,

Academic Press, New York. [17] Fellipa, C. (2009): Solid Element: Overview. Advanced Finite Element Method for Solid, Plates and Shells

(ASEN 6367), 5 March, University of Colorado, Boulder, USA. [18] Fellipa, C. (2011): The Quadratic Tetrahedron. Advanced Finite Element Method for Solid, Plates and Shells

(ASEN 6367), 7 May, University of Colorado, Boulder, USA. [19] Feng, X. (2011): Analysis of Field of Stress and Displacement in Process of Meshing Gears. International

Journal of Digital Content Technology and its Applications, Vol.5, No.6, pp.345-357. [20] Firth, A. and Long, H. (2010): A Design Software Tool for Conceptual Design of Wind Turbine Gearboxes.

Europe’s Premier Wind Energy Event, 20-23 April, Warsaw, Poland. [21] Gilbert Gedeon, P.E. (1999): Gears in Lubrication of Gears and Bearings. EM 1110-2-1424, 28 February,

Stony Point, New York. [22] Gupta, S., Tewari, P.C. and Sharma, A.K. (2006): TPM Concept and Implementation Approach. Maintenance

World, 21 August, pp.1-18. [23] Gurumani, R. and Shanmugam, S. (2011): Modeling and Contact Analysis of Crowned Spur Gear Teeth.

Engineering Mechanics, Vol.18, No.1, pp.65-78. [24] Habib, Z. and Sakai, M. (2003): G2 Planar Cubic Transition between Two Circles. International Journal of

Computer Mathematics, Vol.8, pp.959-967. [25] Habib, Z. and Sakai, M. (2005): Circle to Circle Transition with a Single Cubic Spiral. The Fifth IASTED

International Conference Visualization, Imaging and Image Processing, 7-9 September, Benidorm, Spain, pp. 691-696.

[26] Habib, Z. and Sakai, M. (2008): G2 Cubic Transition between Two Circles with Shape Control. Journal of Computational and Applied Mathematics, Vol.223, pp.133-144.

[27] Hamburger, M. (2000): Stress, Strain and the Physics of Earthquake Generation. G141 Earthquakes and Volcanoes, 25 October, Indiana University, USA.

[28] Higuchi, F. and Gofuku, S. (2007): Approximation of Involute Curves for CAD-System Processing. Engineering with Computers, Vol.23, pp.207-214.

[29] Khai, N.M., Ha, P.Q. and Oborn, I. (2007): Nutrient Flows in Small-Scale Peri-Urban Vegetable Farming Systems in Southeast Asia-A Case Study in Hanoi. Agriculture, Ecosystems and Environment, Vol.122, No.2, pp.192-202.

[30] Lee, C.H. (2009): Non Linear Analysis of Spur Gear using Matlab Code. M.Sc. Thesis, California Polytechnic State University.

[31] Margalit, D. (1995): History of Curvature. Concept of Curvature, Brown University, USA. [32] M Britto, A. (2005): Patran Beginner’s Guide. Department of Engineering, 15 April, University of Cambridge,

UK. [33] Meek, D.S. and Walton, D.J. (1989): The Use of Cornu Spirals in Drawing Planar Curves of Controlled

Curvature. Journal of Computational and Applied Mathematics, Vol.25, pp.69-78. [34] Michalopoulos, E. and Babka, S. (2000): Evaluation of Pipeline Design Factors. GRI 00/0076, The Hartford

Steam Boiler Inspection and Insurance Company, Hartford, USA. [35] Mott, R.L. (2003), Machine Elements in Mechanical Design Fourth Edition, Prentice Hall, New Jersey.



22

[36] Niederstucke, B., Anders, A., Dalhoff, P. and Grzybowski, R. (2003): Load Data Analysis for Wind Turbine Gearboxes. Final Report of Enhanced Life Analysis of Wind Power Systems, Hamburg, German.

[37] Nikolić, V., Dolićanin, Ć. and Dimitrijević, D. (2012): Dynamic Model for the Stress and Strain State Analysis of a Spur Gear Transmission. Strojniški Vestnik-Journal of Mechanical Engineering, Vol. 58, pp.56-67.

[38] Peckner, D. and Bernstein, I.M. (1977): Handbook of Stainless Steels, McGraw-Hill, New York. [39] Price, D. (1995): Spur Gears. Mechanical Engineering Design Topics, 9 August, GlobalSpec. [40] Pun, A. (2001): How to Predict Fatigue Life. Design News, 17 December, Lexington, USA. [41] Rameshkumar, M., Sivakumar, P., Sudesh, S. and Gopinath, K. (2010): Load Sharing Analysis of High-

Contact-Ratio Spur Gears in Military Tracked Vehicle Application. Gear Technology, pp.43-50. [42] Rashid, A. and Habib, Z. (2010): Gear Tooth Designing with Cubic Bézier Transition Curve. Graduate

Colloquium on Computer Sciences (GCCS), Vol.2, No.1, 16 June, Lahore, Pakistan. [43] Reagor, C.P. (2010): An Optimal Gear Design Method for Minimization of Transmission Error and Vibration

Excitation. Ph.D Thesis, Pennsylvania State University, USA. [44] Reyes, O., Rebolledo, A. and Sanchez, G. (2008): Algorithm to Describe the Ideal Spur Gear Profile. The

World Congress on Engineering, II, 2-4 July, London, UK. [45] Said, M.R., Yuhazri, M.Y. and Lau, S. (2012): The Finite Element Analysis on Hexagonal Ring under Simple

Lateral Loading. Global Engineers and Technologists Review, Vol.2, No.2, pp.1-7. [46] Sapto, W.W. and Safarudin, M. (2008): Finite Element Analysis of Wide Rim Alloy Wheel. National

Conference on Design and Concurrent Engineering, 28-29 October, Melaka, Malaysia, pp.407-411. [47] Shenoy, P.S. (2004): Dynamic Load Analysis and Optimization of Connecting Rod. M.Sc. Thesis, University of

Toledo. [48] Walton, D.J. and Meek, D.S. (1999): Planar G2 Transition between Two Circles with Fair Cubic Bézier Curve.

Computer Aided Design, Vol.31, pp.857-866. [49] Walton, D.J. and Meek, D.S. (1996): A Planar Cubic Bézier Spiral. Journal of Computational and Applied

Mathematics, Vol.72, pp.85-100. [50] Walton, D.J., Meek, D.S. and Ali J.M. (2003): Planar G2 Transition Curves Composed of Cubic Bézier Spiral

Segments. Journal of Computational and Applied Mathematics, Vol.157, pp.453-476. [51] Wang, P.Y., Fan, S.C. and Huang, Z.G. (2011): Spiral Bevel Gear Dynamic Contact and Tooth Impact Analysis.

Journal of Mechanical Design, Vol.133, No.8, pp.084501-1-084501-6. [52] Westland, A. (2006): Aerospace Structural Analysis, Jesmond Engineering Limited, East Yorkshire, UK. [53] Yeung, K.S. (1999): Analysis of a New Concept in Spline Design for Transmission Output Shafts. BEASY

Publications, Southampton, UK. [54] Ziaei Rad, S. (2011): Bar Linear Static Analysis. Finite Element Method, Isfahan University of Technology,

Iran. [55] Zehe, M.J., Gordon, S. and McBride, B.J. (2002): CAP: Computer Code for Generating Tabular

Thermodynamic Functions from NASA Lewis Coefficients. NASA/TP–2001–210959/REV1, Glenn Research Center, Ohio, USA.

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