Penn State Rotorcraft Center of Excellence

1. Penn State Rotorcraft Center of Excellence Task 2.3b: Rotorcraft Gear Optimization for Minimization of Vibration Excitation Principal Investigator: William D. Mark, Ph.D. Senior Scientist and Professor of Acoustics Student: Cameron P. Reagor Candidate for Ph.D. degree in Graduate Program in Acoustics

2. Introduction • Background • Gear noise is a principal contribution to the interior noise of rotorcraft. • Problem Statement • Develop analytical methods, computational algorithms, and software to minimize the vibration excitation caused by meshing gear pairs over a significant range of gear loadings. • Technical Difficulties • New innovative optimization methods and computational algorithms are required. • Accurate computations of tooth/gearbody stiffnesses are required. • Accurate modifications of tooth working surfaces are required.

3. Tradition Definition; u(1) = 0, u(2) = 0. Definition of Transmission Error, (x)The transmission error describes the amount meshing gear teeth come together, relative to their rigid, equispaced, perfect involute counterparts, as a function of the rotational position of the gears. Transmission Error is Principal Source of Vibration Excitation For use in gear system equations of motion; u(1) 0, u(2)  0.

4. Requirements for Zero Vibration Excitation Source of Transmission Error is Deviations of Tooth Working Surfaces (Under Loading) from Equispaced Perfect Involute Surfaces which includes • Geometric Deviations of the Unloaded Teeth and • Tooth Elastic Deformations. Zero Vibration Excitation requires • Non-fluctuating (Constant) Transmission Error  and • Non-fluctuating (Constant) Mesh Loading W.

5. Meshing Helical Gear Pair Upper Portion of Figure Shows Lines of Tooth Contact Within Nominal Zone of Tooth Contact in Plane of Contact of Meshing Pair of Single-Helical Gears (Lower Portion of Figure). Elliptically Shaped Region Illustrates Actual Region of Tooth Contact, Which is Dependent on Gear Mesh Loading W.

6. Relevance of Poisson Sum Formula and Uncertainty Principle for Fourier Transforms From sketch of lines of tooth contact: Total mesh loading where w (x) is loading on a single tooth pair. Fourier series coefficients p of total mesh loading W (x): where is Fourier transform of single tooth pair loading w (x): Therefore, we require for p= 1, 2, 3, ···.

7. Upper Right Portion of Figure Shows Line of Tooth Contact on Working Surface of a Single-Helical Gear Tooth. Elliptically Shaped Region on Tooth Working Surface Illustrates Actual Region of Tooth Contact, Which is Dependent on Gear Meshing Loading W.

8. Sketch Illustrating Modified Tooth Working Surface and Instantaneous Line of Contact Positions Normalized “roll distance” variable is s=x-j∆. Total roll distance contact span s under full loading shown is s = Q∆ = 3∆. Parameter Q is “total” contact ratio.

9. Tooth-Pair Loading Functions for Q=3 for Normalized Transmission Error

10. Tooth-Pair Loading Functions for Q=4 for Normalized Transmission Error

11. Tooth-Pair Loading Functions for Q=5 for Normalized Transmission Error

12. Mesh-Loading Normalized Fourier Transform/Coefficients for Q=3

13. Mesh-Loading Normalized Fourier Transform/Coeffcients for Q=4

14. Mesh-Loading Normalized Fourier Transform/Coeffcients for Q=5

15. Mesh-Loading Normalized Fourier Transform/Coeffcients for Q=3

16. Mesh-Loading Normalized Fourier Transform/Coeffcients for Q=4

17. Mesh-Loading Normalized Fourier Transform/Coeffcients for Q=5

18. Basic Computational Procedure At each line-of-tooth-contact location, s, on the tooth working surface: 1. Incrementally increase the transmission error (the same amount at all line-of-contact locations). 2. This yields the tooth-pair loading Wj(s) at that line of contact location, s. 3. Numerically solve for the increase in contact span that will yield this increased tooth-pair loading Wj(s). 4. When carried out for all line-of-contact locations, s, a region of tooth contact on tooth working surfaces for that transmission error and corresponding tooth-pair loading function Wj(s) has been determined. 5. Incrementally increase the transmission error by a constant amount and repeat the above procedure for all line-of-tooth contact locations.

19. Basic Method for Stepwise Computing Tooth Modifications from the Involute • We prescribe the constant transmission error, . • We have computed tooth-pair loading w(s) at each line of contact location, s. • We have a very accurate method of computing tooth-pair stiffness per unit length of line of contact. For incremental transmission error step , and linear incremental relief from the unmodified tooth region, the only unknown is 1, as shown above.

20. Accomplishments During Past Year • Developed “final” form of tooth-pair loading functions with optimized properties. • Developed optimum combinations of parameters that relate constant transmission error  to tooth-pair loading functions. • Developed method for minimizing vibration excitation from high-contact-ratio spur gears over a finite range of gear loadings. • Continued development of software to numerically implement optimized designs. • Under other funding, developed physics-based computation of vibration excitation in time and frequency domains caused by gear-tooth bending-fatigue damage for detection and prognosis of such damage.

21. Main Project Accomplishments • Developed “exact” tractable analytical formulation of optimum tooth-modification problem for minimizing vibration excitation from helical gears. • Developed family of tooth-pair loading functions, compatible with physically realizable tooth-stiffness properties, possessing frequency-domain maximum envelope attenuation, maximum “width” of zeros about tooth-meshing-harmonic locations, and maximum load-carrying capacities. • Developed physically realizable family of analytical formulas relating tooth-stiffness properties and (constant) transmission errors to above-described tooth-pair loading functions. • Developed systematic computational method for numerically computing optimized tooth-working-surface modifications to minimize transmission-error fluctuations and mesh-loading fluctuations over full range of mesh loadings. • Developed tooth-modification method for high-contact-ratio spur gears to minimize transmission-error fluctuations and mesh loading fluctuations over a finite range of mesh loadings. • Documentation and numerical implementation currently are underway.

22. Collateral Payoffs The developed theoretical relationships • provide a first-principles analytical formulation and understanding as a basis for future research and applications • provide a mathematical model for comparison with achieved vibration reductions • explain why “long” tip relief produces less vibration than “short” tip relief • explain applicability of “rule of thumb” vibration-reduction rule using “total contact ratio” Q=Qa+Qt versus “aggregate contact ratio” QaQt • explain why parabolic-like axial crowning under heavy loading should produce less vibration excitation than end relief modifications • explain why there is little basis for the commonly held belief and often-stated arguments for utilizing integer-contact-ratio helical gears when profile and lead modifications are applied, and so on.

23. Some Future Potential Gear Vibration Minimization and Related Work • Implement and test topological tooth-working-surface modifications for minimum vibration excitations. • Utilizing the developed theoretical framework, determine and test optimized non-topological modifications compatible with conventional gear finishing (grinding) methods. • Develop and test high-contact-ratio spur gear non-topological tooth modifications optimized over a finite range of tooth loadings. • Develop physics-based methodology for optimum detection and prognosis of gear damage utilizing existing capabilities for prediction gear-vibration excitations.

24. Related Efforts • CRI project for implementation of optimum designs in rotorcraft gears is underway. • NAVAIR “Seed Funding” project for “Physics-Based Model of Vibration in Damaged Gears” was completed. • Provided White Paper to FAA for “Physics-Based Algorithm and Software Development for Optimum Detection and Prognosis Rotorcraft Gear Damage.”

25. Schedule and Milestones Rotorcraft & Gear Firm Relationships Optimization Algorithms & Software Optimization Implementations Experimental Validations

26. Project Title: Rotorcraft Gear Optimization For Minimization of Vibration Excitation. PI: William D. Mark; Phone:(814)865-3922; Email: wdm6@psu.edu Project Number: PS 2.3b • Technical Difficulties • New innovative optimization methods and computational algorithms are required. • Accurate computations of tooth/gearbody stiffnesses are required. • Accurate modifications of tooth working surfaces are required. • Objectives • Develop analytical methods, computational algorithms, and software to minimize the vibration excitation cause by meshing gear pairs over a significant range of gear loadings. • 2004 Accomplishments • - Developed “final” form of tooth-pair loading functions with optimized properties. • - Developed optimum combinations of parameters that relate constant transmission error  to tooth-pair loading functions. • - Developed method for minimizing vibration excitation from high-contact-ratio spur gears over a finite range of gear loadings. • - Continued development of software to numerically implement optimized designs.

27. Project Number PS 2.3b Rotorcraft Gear Optimization for Minimization of Vibration Excitation PI: William D. Mark; Phone 814/865-3922; Email: wdm6@psu.edu Background: Gear noise is a principal contribution to the interior noise of rotorcraft. Research Objectives: Develop optimization methods, computational algorithms, and software to minimize the vibratory excitation of meshing helical and spur gear pairs for a range of gear-pair loadings and a constrained range of gear designs. Approach: (1) Develop capability to accurately compute the correct definition of tooth/gearbody stiffnesses including Hertzian (local contact) contributions. (2) Develop algorithms and software to modify involute tooth working surfaces to minimize vibration excitation for a prescribed range of gear-pair loadings. 2004 Accomplishments: (1) Developed “final” form of tooth-pair loading functions with optimized properties. (2) Developed optimum combinations of parameters that relate constant transmission error  to tooth-pair loading functions. (3) Developed method for minimizing vibration excitation from high-contact-ratio spur gears over a finite range of gear loadings. (4) Continued development of software to numerically implement optimized designs. Future Work: (1) Continue development of software to implement optimum modification of tooth working surfaces. (2) Work with rotorcraft company(s) to implement and test designs.