AGMA-95FTMS1-1995.pdf
STD-AGMA 75FTMSL-ENGL 1795 211 0687575 0004806 TTII W 95FTMS1 Determination of the Dynamic Gear Meshing Stiffness of an Acetal Copolymer by: Connie P. Marchek American Gear Manufacturers Association TECHNICAL PAPER Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 05:28:03 MDTNo reproduction or networking permitted without license from IHS -,-,- STD-AGHA 7SFTMSL-ENGL 1995 üb87575 0004807 737 Determination of the Dynamic Gear Meshing Stiffness of an Acetal Copolymer Connie P. Marchek F e statements and opinions contained herein are those of the author and should not be construed as an official action or opinion of the American Gear Manufacturers Association. Abstract The dynamic gear meshing stiffness is an important parameter for designing piastic gearing, thus the objective of this research w a s to determine the dynamic gear meshing stiffness of an acetal copoiymer, specincaily CelconB M90. A thmreticaimodel wasdevelopedtosimulatethetoIsionaivibrationandresonanceofanoperafinggearpair. Theresonant speed of the torsional system was determined expehentaüy. U s i n g the theoretical model, it w a s possible to detemine the dynamic gear meshing stifmess b m the experhenmi resonant speed. The dynamic gear meshing stiffness w a s compared to the values caicuiated from available empbicai formulas. Disclaimer The conclusions and opinions expsed in this research are those of the writer and do not necessarily represent the position of W o r c e s t e r Polytechnic Institute or Hoechst Celanese, or any of its directors, officers, agents or employees with respect to the mafters discussed. Copyright O 1995 American Gear Manufactums Association Aiexandria, V i r g i n i a , 22314 1500 King street, suite 201 October, 1995 ISBN 1-55589453-7 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 05:28:03 MDTNo reproduction or networking permitted without license from IHS -,-,- - STD-AGHA 75FTMSL-ENGL 1775 111 üb87575 0004808 873 Preface This paper, researched and conducted at Hoechst Celanese Advanced Materials Group (Wood Dale, Illinois) with the assistance of Packer Engineering (Naperville, Illinois), was submitted to the American Gear Manufacturers Association and will be presented at the Fall 1995 Technical Conference. Sufficient knowledge was acquired through course work at Worcester Polytechnic Institute and experience at Hoechst Celanese and Packer Engineering, to successfully complete this research. I would like to thank God for giving me the courage and strength to continually accept the necessary risks to fulfill my dreams. I would like to express my appreciation to Hoechst Celanese Advanced Materials Group and Worcester Polytechnic Institute for providing me the opportunity to complete a graduate Thesis and Mechanical Engineering Degree. I would like to extend a special thanks to Professor W. W. Durgin for his continued guidance, assistance and support during my graduate career at Worcester Polytechnic Institute. Sincere thanks to Stuart Cohen and Maribeth Fletcher for providing the opportunity to complete a thesis at Hoechst Celanese. Many thanks to Michael J. Clemens of Packer Engineering for being a lifesaver. Your continued guidance, assistance, enthusiasm and overwhelming support made the completion of this research and thesis possible. In addition, I would like to thank the following people who provided me with support and assistance: Zan Smith, Ken Gitchel, ABNPGT and the Faculty at WPI. Many thanks to my special fiiends: Barb, Beth, Stimpy, Ira, Brian, Lucille, Ginny, Joe and Sandy. Thanks for standing by me during both the good and the difficult times. iv Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 05:28:03 MDTNo reproduction or networking permitted without license from IHS -,-,- Finally, I would like to thank my parents, Carlyle Jr. and Kathleen Marchek, and my brothers and sisters, Carleen, Kevin, Christie, Kelly, Kurt, Celee, Came, Kyle, Clare, Keith, Cheryl and Kenneth. You have given me the necessary support, encouragement and assistance making it possible for me to pursue my love of learning. You have made me realize I can achieve anything my heart desires. Thanks so much, you mean the world to me. V Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 05:28:03 MDTNo reproduction or networking permitted without license from IHS -,-,- Table o f Contents . Disclaimer 11 . Abstract 111 . Preface iv . 1. Mathematical Models 1 1.1 Nomenclature. 2 1.2 TheHolzerMethod 3 1.3 Predicted Dynamic Gear Meshing Stiffness - - - - - - - - - - - - . 6 1 -3.1 Cantilever Beam Theory Spotts 6 1.3.2 Hoechst Method 7 1.3.3 Cantilever Beam Theory . Tobe andTakatsu . 9 1.3.4 Cantilever Beam Theory - Nestorides . - - - - - - - - * 10 . . 2. ExperimentalMethod 12 . 2.1 Apparatus 13 2.2 Insmentation 14 2.3 Procedure 16 . 3. ResultsandDiscussion. 18 3.1 Experimental Results 18 3.1.1 Physical Observations 18 3.1.2 Experimental Resonant Speed 20 3.1.3 Universal Joint 22 3.2 Theoretical Results 22 3.2.1 The Holzer Method. 22 3.2.2 Predicted Dynamic Gear Meshing Stiffness - - - - . . 25 3.2.3 Correlation 26 . . . . - 9 - . . References 30 vi Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 05:28:03 MDTNo reproduction or networking permitted without license from IHS -,-,- Fig 1.1 Fig 1.2 Fig 1.3 Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 3.1 Fig. 3.2 Table 3.1 Table 3.2 Table 3.3 Table 3.4 Fig. A.l Fig. A.2 Fig. A.3 List o f Figures and Tables Torsional System . 1 Cantilever Beam of Uniform Cross Section - - - - - - - - Cantilever Beam of Variable Cross Section - - - - - - - - - - a - - - - 9 - 1 O Apparatus Schematic . 12 photograph of Apparatus . 1 3 Instrumentation Diagram. . 15 Instnimentation 16 Gear prior to Experiment . 19 Gear Wear after Experiment 19 Experimental Results . 20 Mass Moment of Inertia and Torsional Stiffness - - - - - - - Calculated Dynamic Gear Meshing Stiffness Based on Experimental Resonant Speeds * - * - - - - - 0 24 Predicted Dynamic Gex Meshing Stiffness 25 23 Experiment # 28 32 Spectrum Analysis of Experiment # 28 at Resonant Speed of820 rpm 33 Spectrum Analysis of Experiment # 28 at Resonant . 34 vii Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 05:28:03 MDTNo reproduction or networking permitted without license from IHS -,-,- STD-AGUA 75FTMSL-ENGL 1775 Ob87575 OOOYBL2 2T4 = 1. Mathematical Model The torsional system to be theoretically modeled consists of a parallel shaft gear apparatus depicted in Fig. 1.1. The system consists of a pair of plastic gears connected to a motor and inertia disk by steel shafts. The objective of this theoretical model is to determine the dynamic gear meshing stiffness at the experimental resonant speed. Driven Gear r ß m Inertia Disk ßd Driving ,o '. í + : Gear and Inertia Disk Fig. 1.1 Torsional System To simpliQ the mathematical model, the motor, gears and inertia disks are assumed to be the rotating inertias. The shafts and gear mesh function as the springs of the rotating system. This model is concerned with only the rotational motion of the gears. The gear teeth are assumed to maintain contact along the theoretical line of action and the dynamic gear meshing stiffness is assumed to be constant. The effects of backlash and mean torque were not considered in this model. 1 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 05:28:03 MDTNo reproduction or networking permitted without license from IHS -,-,- STD-ALMA 75FTMSL-ENGL 1?795 m Ob87575 0004833 130 m 1.1 Nomenclature p, = angular displacement of the driving gear (rad) p, = angular displacement of the driven gear (rad) j 3 , = angular displacement of the inertia disk (rad) p , = angular displacement of the motor (rad) ApgiSLg2= angular displacement between the driving and driven gears (rad) ADg2 is the tangential force (Ib), bk is the smallest tooth width (in), a is the pressure angle (degrees), cp is an auxiliary value, which in this instance is 7.6, vi and yz are auxiliary values of 0.75, E; and E; are dynamic elastic moduli of the driving and &ven gears respectively ('yn2 ). The linear stiffness of the gear tooth was determined using the following equation: (1.16) Rearranging Equation 1.15 and substituting into Equation 1.16, the linear gear tooth stiffness was determined as follows: (1.17) 7 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 05:28:03 MDTNo reproduction or networking permitted without license from IHS -,-,- The equivalent linear dynamic gear meshing stiffness was calculated by adding, in series, the gear tooth stiffhess of the driving and driven gears. Linear springs in series combine as follows: 1 1 1 -=-+- km, ktmih ktooth simpli fjmg, (1.18) Therefore, the linear dynamic gear meshing stiffness, determined by the Hoechst Method, was calculated to be: (1.19) 8 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 05:28:03 MDTNo reproduction or networking permitted without license from IHS -,-,- - - STD-AGHA SCFTMSL-ENGL L995 E Ob87575 0004820 370 D 1.3.3 Cantilever Beam Theory - Tobe and Takatsu The linear stifmess of a gear tooth was determined using the cantilever beam theory, as shown in Fig. 1.2 and developed in Equation 1.20 4. Fig. 1.2 Cantile7 :r Beam of Uniform Cross S 3EI I% - ktooth - xi cos2 p tion (1.20) where: the length of the tooth is I = 2.2m (in), rn is the tooth module (in), the height of the tooth is h = - (in), b is the gear face width (in), the modulus of the gear is ), m 2 1 12 the moment of inertia of the cross section is I = - bh3 ( i n ' ) , x, is the loading point (in), and the pressure angle is p (degrees). The equivalent linear dynamic gear meshing stiffness was calculated as shown: 3EI % n 2 x 0 3 cos2 p k“li = (1.21) 9 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 05:28:03 MDTNo reproduction or networking permitted without license from IHS -,-,- 1.3.4 Cantilever Beam Theory - Nestorides The linear flexibility of a gear tooth was determined using a cantilever beam of variable cross section, as shown in Fig. 1.3 and developed in Equation 1.22 5. P -7 h +-7 Fig. 1.3 Cantilever Beam of Variable Cross Section where: 6 is the tooth deflection (in), P is the applied load (Zb), L is the tooth face width (in), h is the total height of the triangle (in), the distance from the base of the triangle to the applied load is hp (in), B is the thickness of the tooth at the base of the triangle (in), Poisson's ratio of the material is v, and E is the modulus of the material (lxnz). The linear stiffhess of a gear tooth was determined using the following equation: (1.23) P ktooih - - 6 ' X n - 1 0 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 05:28:03 MDTNo reproduction or networking permitted without license from IHS -,-,- STDmAGMA 75FTMSL-ENGL 1975 Ob67575 0004822 143 Rearranging Equation 1.22 and substituting into Equation 1.23, the gear tooth stifmess becomes: The equivalent linear dynamic gear meshing stiffness was determined using Equation 1.25. 11 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 05:28:03 MDTNo reproduction or networking permitted without license from IHS -,-,- STD*AGMA 95FTflSL-ENGL 1775 = Ob87575 OOOV823 O A T 2. Experimental Method The gear testing apparatus, as shown in Figs. 2.1 and 2.2, was designed, machined and assembled. The apparatus was capable of driving the gear system through its torsional natural frequency. Torsional vibration within the system was monitored using the appropriate instrumentation. The apparatus was designed to ensure that the system's dominating link was the gear mesh. The natural frequency was primarily dependent on the gear meshing stifiess since the largest relative angular displacement occurred across the mesh. - _ Monitoring Instrumentation Ï7, -A I l :i c c Gear Inertia Disk Tñin Metai Gear Fig. 2.1 Apparatus Schematic 12 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 05:28:03 MDTNo reproduction or networking permitted without license from IHS -,-,- STD.AGMA 95FTMSL-ENGL 1995 0687575 0004624 TLb D Fig. 2.2 Photograph of Apparatus 2.1 Apparatus The experimental apparatus shown in Figs. 2.1 and 2.2, consisted of a pair of plastic gears connected to a motor and inertia disks by steel shafts. The plastic gears were injection molded from CelconB M90 6. The gear had an outside diameter of 2.4 inches and a total of 27 teeth. The steel shafts were cut from one inch diameter cold rolled round stock to a length of 11 inches. Compared to the stiffness of the plastic gear mesh, the shafts were considered rigid. Thus, the first torsional vibration mode was primarily dependent on the stiffness of the plastic gear mesh. The steel inertia disks used 13 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 05:28:03 MDTNo reproduction or networking permitted without license from IHS -,
