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
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W. D. Mark
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.
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Transmission Error is Principal Source
of Vibration Excitation
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.
Tradition Definition; u(1) = 0, u(2) = 0.

 ( x)  Rb(1)  (1)  Rb( 2) ( 2)
For use in gear system equations of motion;
u(1)  0, u(2)  0.
xR 
(1)
b
(1)
R 
( 2)
b
( 2)
 ( x)  Rb(1)   (1) u (1) ( x)  Rb( 2)   ( 2)  u ( 2) ( x)
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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.
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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 SingleHelical Gears (Lower Portion of Figure). Elliptically Shaped Region
Illustrates Actual Region of Tooth Contact, Which is Dependent on Gear
Mesh Loading W.
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Relevance of Poisson Sum Formula and Uncertainty
Principle for Fourier Transforms
From sketch of lines of tooth contact:

Total mesh loading W ( x)   w( x  j)
j  
where w (x) is loading on a single tooth pair.
Fourier series coefficients p of total mesh loading W (x):
1
 p  wˆ p /  

where wˆ ( g ) is Fourier transform of single tooth pair loading w (x):

wˆ ( g )   w( x)e i 2 g x dx .

ˆ ( p / ) for p= 1, 2, 3, ···.
Therefore, we require w
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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.
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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.
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Tooth-Pair Loading Functions for Q=3 for Normalized Transmission
Error   0.1,0.2,  ,1.0

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Tooth-Pair Loading Functions for Q=4 for Normalized Transmission
Error   0.1,0.2, ,1.0

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Tooth-Pair Loading Functions for Q=5 for Normalized Transmission
Error   0.1,0.2,  ,1.0

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Mesh-Loading Normalized Fourier Transform/Coefficients for Q=3
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Mesh-Loading Normalized Fourier Transform/Coeffcients for Q=4
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Mesh-Loading Normalized Fourier Transform/Coeffcients for Q=5
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Mesh-Loading Normalized Fourier Transform/Coeffcients for Q=3
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Mesh-Loading Normalized Fourier Transform/Coeffcients for Q=4
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Mesh-Loading Normalized Fourier Transform/Coeffcients for Q=5
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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.
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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.
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W. D. Mark
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-contactratio 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 bendingfatigue damage for detection and prognosis of such damage.
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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 toothstiffness properties and (constant) transmission errors to above-described toothpair 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.
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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 oftenstated arguments for utilizing integer-contact-ratio helical gears when
profile and lead modifications are applied, and so on.
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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.
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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.”
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Schedule and Milestones
Rotorcraft & Gear Firm Relationships
Establish Relationships
Optimization Algorithms & Software
Candidate Gear Types
Optimization Implementations
Experimental Validations
Tooth/Gearbody Compliance
Optimization Framework &
Algorithms
Optimization Software
Finalize Software
First (Spur) Gear
Optimization Solution
Second (Helical) Gear
Optimization Solution
First (Spur) Gear Validation
Second (Helical) Gear
Validation
2001
2002
2003
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2004
2005
2006
W. D. Mark
Project Title: Rotorcraft Gear Optimization For Minimization of Vibration Excitation.
Project Number: PS 2.3b
PI: William D. Mark; Phone:(814)865-3922; Email: wdm6@psu.edu
•
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.
Rotorcraft Firm
Relationships
•
- Developed “final” form of tooth-pair loading
functions with optimized properties.
Optimization Software
- Developed optimum combinations of parameters that
relate constant transmission error  to tooth-pair
loading functions.
Optimization
Implementation
- Developed method for minimizing vibration
excitation from high-contact-ratio spur gears over a
finite range of gear loadings.
Experimental
Validations
2001
2004 Accomplishments
2002
2003
2004
2005
2006
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- Continued development of software to numerically
implement optimized designs.
W. D. Mark
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.
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