1、 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 STD-AGHA 7SFTMSL-ENGL 1995 b87575 0004807 737 Determination of the Dynamic Gear Mesh
2、ing Stiffness of an Acetal Copolymer Connie P. Marchek Fe 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 paramete
3、r for designing piastic gearing, thus the objective of this research was to determine the dynamic gear meshing stiffness of an acetal copoiymer, specincaily CelconB M90. A thmreticaimodel wasdevelopedtosimulatethetoIsionaivibrationandresonanceofanoperafinggearpair. Theresonant speed of the torsional
4、 system was determined expehentay. Using the theoretical model, it was possible to detemine the dynamic gear meshing stifmess bm the experhenmi resonant speed. The dynamic gear meshing stiffness was compared to the values caicuiated from available empbicai formulas. Disclaimer The conclusions and op
5、inions expsed in this research are those of the writer and do not necessarily represent the position of Worcester 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 Associ
6、ation Aiexandria, Virginia, 22314 1500 King street, suite 201 October, 1995 ISBN 1-55589453-7 - STD-AGHA 75FTMSL-ENGL 1775 111 b87575 0004808 873 Preface This paper, researched and conducted at Hoechst Celanese Advanced Materials Group (Wood Dale, Illinois) with the assistance of Packer Engineering
7、(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 suc
8、cessfully 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
9、 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 Mari
10、beth 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 a
11、ddition, 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
12、 the difficult times. iv 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
13、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 Table of Contents . Disclaimer 11 . Abstract 111 . Preface iv . 1. Mathematical Models 1 1.1 Nomenclature. 2 1.2 TheHolzerMethod 3 1.3 Predi
14、cted 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 Insmen
15、tation 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
16、 Correlation 26 . . . . - 9 - . . References 30 vi 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 of Figures and Tables Torsional System . 1 Cantilever Beam of Uniform Cross Section - - - - - - - -
17、 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 To
18、rsional 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
19、at Resonant . 34 vii 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
20、 shafts. The objective of this theoretical model is to determine the dynamic gear meshing stiffness at the experimental resonant speed. Driven Gear r m Inertia Disk d Driving ,o . +: Gear and Inertia Disk Fig. 1.1 Torsional System To simpliQ the mathematical model, the motor, gears and inertia disks
21、 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 st
22、iffness is assumed to be constant. The effects of backlash and mean torque were not considered in this model. 1 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) j3, = angular di
23、splacement 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 th
24、is instance is 7.6, vi and yz are auxiliary values of 0.75, E; and E; are dynamic elastic moduli of the driving and Magnetic Sensor (Shaft #i) _ Fig. 2.3 Instrumentation Diagram 15 STD-AGHA 95FTMS1-ENGL 1995 Ob87575 OOOq827 725 Fig. 2.4 Instrumentation 2.3 Procedure After the instrumentation was set
25、 up to monitor the experiment, the following is a summary of the procedure used to determine the experimental resonant speed of the torsional system. The plastic gears were attached to the shafts and an inertia disk secured on the end of the driven shaft. To monitor the speed of the gear, the magnet
26、ic sensor was located perpendicular to the thin metal gear. The motor was programmed to run an automatic speed sweep. While the experiment ran, the X-Y-Y recorder plotted the operating speed versus the torsional amplitude of the system. The experimental resonant speed was 16 STD-ALMA 75FTMSL-ENGL 17
27、75 Ob87575 0004828 bbL determined from the recorded plot. The resonant speed occurred where the torsional amplitude, or measured alternating torsional motion of the shafts, reached a maximum. To prove the dynamic gear meshing stiffness for a pair of acetal copolymer gears was constant, the experimen
28、tal procedure was repeated using three disks of different mass inertias. When the rotating inertia disk was changed, a new torsional vibration problem existed. Since all other elements in the apparatus remained the same, a change in inertia load shifted the systems resonant speed. As one would expec
29、t, an increase in inertia decreased the resonant speed. For each inertia disk, the dynamic gear meshing stiffness was determined from the experimental resonant speed. The dynamic gear meshing stiffness was constant for a pair of acetal copolymer gears. 17 3. Results and Discussion 3. i Experimental
30、Results 3.1.1 Physcal Observations A few interesting phenomena occurred while running the experiment. As the torsional system approached resonance, the surface temperature of the gear mesh increased at a rapid rate and to a high temperature (220 OF). When the operating speed increased above resonanc
31、e, the surface temperature of the gear teeth decreased to just above ambient temperature. Plastic chips came off the gear mesh as the system went through resonance. By comparing Fig. 3.1 and Fig. 3.2, it can be seen that considerable gear wear occurred during these experiments. In addition, chatter
32、or squeaking of the gears was heard as resonance was approached. 18 STD-AGMA 95FTMSL-ENGL L775 = Ob87575 O004830 2LT Fig. 3.1 Gear Prior to Experiment L Fig. 3.2 Gear Wear after Experiment 19 STD-ALMA 75FTMSL-ENGL 1795 = b87575 OOLi83L L5b = p-1 11/2 26 3.1.2 Experimental Resonant Speed 830 The expe
33、riments were successful in determining the fundamental resonant operating speed of the torsional system. The X-Y-Y Recorder plotted the operating speed (rpm) versus torsional amplitude (degrees peak to peak) for both shafts of the system. The plot from experiment #28 is shown in Appendix A. As the s
34、ystem approached resonance the torsional amplitude increased. Resonance was indicated when the torsional amplitude reached a maximum. The experimental resonant speed for each plot is summarized in Table 3.1. 860 1 Run# /I InertiaDisk Il Experimental Resonant Speed 850 820 13 14 15 16 17 1 114 Il 750
35、 II 860 850 850 850 I201 II 920 II 1 880 Table 3.1 Experimental Results 20 STD-AGMA 95FTMSL-ENGL 1995 Ob87575 0004832 O92 The plot from experiment #28 shows that in this operating range, the natural frequency of the apparatus was excited at speeds of 820 and 1570 rpm. A spectrum analysis of this exp
36、eriment, shown in Appendix A, provided the necessary information to conclude that the fundamental natural frequency of the system was excited by different harmonics. The spectrum analysis indicated that the fundamental natural frequency, approximately 13 hz., was excited at both speeds. Equation 3.1
37、 demonstrates the different orders of excitation or harmonics that were excited at 820 and 1570 rpm. fhz, = where: n is the order of excitation or harmonic 60 (820rpm)n At 820 rpm: 13hz.= thus, nr 1 W“;/,“) and (i 570rpm)n At 1570rpm: 13hz.= thus, nz 0.5 60( Segin 1 In conclusion, a first order and
38、a half order harmonic excited the systems natural frequency at operating speeds of 820 rpm and 1570 rpm, respectively. 21 STD-AGMA 75FTMSZ-ENGL 1795 = b87575 OOOY833 T27 3.1.3 Universal Joint The purpose of the universal joint was to induce a twice per revolution (i.e. second harmonic) excitation in
39、to the torsional system via the driving shaft. The spectrum analysis of experiment #28 indicated the natural frequency was dominated by the first and half order harmonics. A reason for no second order harmonic effect, was a result of the large inertia difference between the disk mounted near the mot
40、or and the motors rotor. Compared to the motor inertia, the inertia disk was the dominating or controlling mass. In retrospect, the torque variations from the universal joint altered the speed of the electric motor, not the driving shaft. The excitation source produced first and half order harmonics
41、. Machine run out of the disks, shah, and hubs, pillow block ball bearings, and misalignment of the shafts were all capable of producing first and half order harmonics. 3.2 Theoretical Results 3.2.1 The Holzer Method To use the Holzer Method, the moment of inertia of the masses and the stiffhess of
42、the shafts were calculated using basic engineering principles. A more accurate method of representing the torsional system was to calculate an equivalent moment of inertia for the motor, gears and disks, which included the inertia effect of the hubs and shafts. Table - 3.2 shows the calculated mass
43、moment of inertia and stiffness for the components of the experimental apparatus. 22 STD-AGMA 95FTMSL-ENGL L775 Obfl7575 0004834 9b5 - Apparatus Component Moment of Inertia Equivalent Moment I of Inertia (in-lh-s2 ) (in-lb-s ) Universal Joint Thin Metal Gear Hub 1“ Inertia Disk Im 0.73542 Inertia Hu
44、b Shaft # 1 0.00036 (I 0.00067 Driven Gear 0.00006 Shaft #2 0.00038 i i /4“ Inertia Disk i 1/2“ Inertia Disk 0.901 1 1 1 .O8 i 33 Torsional Stiffness knll (n-lxJd) Table 3.2 Mass Moment of Inertia and Torsional Stiffness A TK Solver 7 program was used to implement the Holzer Method to iterate Equati
45、ons 1.1-1.13 for the systems lowest natural frequency and respective eigenvector. These equations were iterated for different values of dynamic gear meshing stiffness. Table 3.3 includes the dynamic gear meshing stiffness and mode shape determined for each experimental resonant speed. The dynamic ge
46、ar meshing stiffness was the dominating stifhess in the system because the maximum relative angular displacement occurred across the gear mesh. Since the fundamental natural frequency of the system was dominated by the meshing stiffness, it was possible to determine the dynamic gear meshing stiffnes
47、s from the lowest natural frequency. 23 STD-AGHA 95FTMSL-ENGL 1795 = Ob87575 O004835 AT1 M * Thisvaiue Experimental Theoretical Angular Resonant Stiffness Displacemer Speed Motor km, f (rpd Pm (rad) 1 I 1 1 3310 1 3390 1 910 3420 I 920 3500 1 890 3270 1 890 3270 1 880 3190 1 910 3420 1 of torsional
48、dynamic gear meshing other experiments. Angular Displacement Driving Gear 0.978 0.979 0.979 0.977 0.976 0.975 0.977 0.980 0.974 0.975 0.975 0.975 0.974 0.971 0.971 0.972 0.972 0.973 0.971 Angular Xsplacemen Driven Gear P, (rad) -0.655 -0.657 -0.657 -0.654 -0.653 -0.652 -0.654 -0.793 -0.787 -0.787 -0
49、.787 -0.787 -0.787 -0.986 -0.985 -0.987 -0.987 -0.988 -0.986 Angular bisplacemen1 Load Pi (rad) -0.679 -0.679 -0.679 -0.679 -0.679 -0.679 -0.679 -0.814 -0.814 -0.814 -0.8 I4 -0.8 14 -0.814 -1.016 -1.016 -1.016 -1.016 -1.016 -1.016 - - - Average Theoretical Stimiess k“, ( 3370 3340 3350 = dynamic gear meshing stiffness is not considered in the average torsional stiffness, since the result from this experiment is a deviation from the Table 3.3 Calculated Dynamic Gear Meshing Stifmess Based on Experimental Resonant Speeds 24
