A New Helicopter Transmission Model for Condition-based 

Maintenance Technologies Using First Principles 

by David B. Stringer, Pradip N. Sheth, and Paul E. Allaire 

ARL-TR-4984 September 2009 

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Army Research Laboratory 

Cleveland, Ohio 44135 

ARL-TR-4984 September 2009 

A New Helicopter Transmission Model for Condition-based 

Maintenance Technologies Using First Principles 

David B. Stringer 

Vehicles Technology Directorate, ARL 

Pradip N. Sheth and Paul E. Allaire

University of Virginia, Charlottesville, Virginia, 22904 

Approved for public release; distribution unlimited. 
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A New Helicopter Transmission Model for Condition-based Maintenance 

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14. ABSTRACT 

This report presents a new helicopter transmission model in support of condition-based maintenance (CBM) initiatives. The 

model uses the first principles approach to analyze the rotating shafts and gear couplings using rotor dynamic analysis 

techniques. The helicopter transmission is presented, followed by a discussion of the finite element formulation to include gear 

mesh modeling methodologies. System natural frequencies and mode shapes are discussed, along with the model’s validation 

procedure. The model’s primary purpose is to simulate multiple operating conditions to include faults. One such fault 

condition simulates the wear of gear tooth contact surfaces. The nature of the fault and its modeling techniques are discussed. 

Simulation results are presented in the time domain using a conventional vibration diagnostic parameter. A new diagnostic 

parameter, the FE2, is also introduced for detecting faults in the frequency domain. Additional considerations and further 

research points are also discussed. The results demonstrate the potential of physics-based, mathematical models in providing a

key technology enabler for improving health monitoring and prediction capabilities of CBM strategies. 

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Thermal, simulation, silicon carbide, power 

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iii

Contents

List of Figures iv

List of Tables iv

Acknowledgments v

1. Introduction 1

2. OH-58 Helicopter and Transmission 2

3. Transmission Model 3

3.1 Description and Model Assembly ...................................................................................3

3.2 Simulation Setup .............................................................................................................6

4. Model Validation 8

5. Eigen Analysis Results – Mode Shapes 9

6. Dynamic Simulation Results with Fault Detection 10

7. Additional Considerations 14

8. Conclusions 15

9. References 16

Appendix. Nomenclature 21

Distribution List 22 
iv

List of Figures

Figure 1. OH-58 transmission schematic (left) and disassembled (right). .....................................3

Figure 2. Finite element model component layout. ........................................................................5

Figure 3. Sample data from OH-58 transmission test stand. ..........................................................7

Figure 4. Experimental versus model frequency. ...........................................................................8

Figure 5. Shaft mode shapes –3,230 Hz. ........................................................................................9

Figure 6. Gear tooth with wear. ....................................................................................................11

Figure 7. Spiral-bevel gear loading conditions. ............................................................................11

Figure 8. (a) Normal and (b) worn vibration response. ................................................................13

Figure 9. Kurtosis diagnostic parameter results. ...........................................................................13

Figure 10. Frequency spectrum comparison. ................................................................................14

Figure 11. FE2 diagnostic parameter. ...........................................................................................14

List of Tables

Table 1. Transmission model parameters. ......................................................................................6

Table 2. Loading pattern Fourier coefficients...............................................................................12
v

Acknowledgments 

Dr. Dave Lewicki and Dr. Brian Dykas of the Army Research Lab’s Vehicle Technology 

Directorate, Cleveland, Ohio, were particularly helpful in providing test data, transmission 

information, and assistance in testing set-up on ARL/NASA’s 500 hp Transmission Test Stand. 
vi

INTENTIONALLY LEFT BLANK
1

1. Introduction

The paradigm shift to condition-based maintenance (CBM) of vertical lift aircraft requires 

advanced methods of health and condition monitoring, characterized by high detection rates and 

low false alarms, using recorded signals from sensors placed at key locations throughout the 

aircraft. Furthermore, not only must the condition monitoring equipment detect any faults, it 

should also determine which component has those faults based upon the physical signature. A 

key technology enabler is a fast, reliable source of information to compare against real-time or 

near real-time aircraft data. This capability requires a database of operating conditions, 

simulating both normal and abnormal operations. Ultimately, it requires a method of modeling 

the dynamics of the system.

This is particularly relevant in rotorcraft drive trains. Helicopters rely on the integrity of their 

drive train(s) for their airworthiness and autorotational capability. Indeed, drive system 

emergencies represent the most critical in-flight emergency, after fire, in many rotary-wing 

platforms. A study of civilian rotorcraft accidents between 1960 and 1997 concluded that 

transmission malfunctions caused or contributed to approximately 13 accidents per year (1). 

This was most recently reiterated in a recent accident off the coast of Scotland, caused by the 

catastrophic failure of the aircraft’s main rotor gearbox (2). 

Physics-based models are an acknowledged shortcoming in rotorcraft health monitoring and 

diagnostic methodologies (3). The current state-of-the-art is characterized by an overwhelming 

focus on experimental testing of faults, fault seeding, and detection methodologies using 

vibration signatures and condition indicators (CIs). The literature groups current research into 

the following loose classifications: (1) stochastic and statistical detection methodologies (4–8), 

(2) empirical testing and results (9–11), (3) condition indicators, threshold values, algorithms, 

and algorithm effectiveness (12–16), (4) the types of faults (17–25), and (5) vibration signatures 

and signal processing (26–35). These classifications are not meant to provide rigid divisions in 

the research; indeed, some research crosses into two or more of these classifications. 

The overarching enabler of transmission effectiveness is gear technology. Drive trains rely on 

gear quality and health for their integrity and function. The dynamics associated with geared 

motion are complex. Gears greatly influence the vibration characteristics of a mechanical system 

and contribute significantly to the noise, component fatigue, and personal discomfort prevalent in 

rotorcraft.

The dynamic analysis of gears dates back to the 1950s. Models of various gear-mesh 

interactions permeate the literature (36, 37). By and large, these studies dealt with dual shafts 

coupled by a single gear pair, or multiples shafts coupled by gear pairs using the same types of 

gears, i.e., spur, helical, bevel, etc. 
2

What has not been adequately captured, however, is the dynamic modeling of a helicopter or 

gearbox consisting of multiple shafts connected by multiple gears of different configurations. 

With the exception of one study in the late 1980’s (38), work in this area has been relatively 

overlooked. The complexity of the transmission dynamics and transmission paths has 

traditionally made extraction and interpretation of fault detection data difficult (39). However, 

interest in physics-based models has increased in recent years, specifically in terms of predicting 

the noise path through a rotorcraft transmission (40). 

This report presents a finite element model of a helicopter transmission in support of health 

monitoring, diagnostics, and other CBM initiatives. It includes a detailed description of the 

model and system parameters, method of validation, summary of results simulating both normal 

and abnormal operating conditions, and finally, a demonstration of fault detection capability 

using both a conventional and a new diagnostic parameter. 

2. OH-58 Helicopter and Transmission

The model configuration is based upon the 500 hp Transmission Stand jointly used by the Army 

and NASA at the NASA Glenn Research Center in Cleveland, Ohio. The OH-58 helicopter is a 

single-engine, light helicopter used for observation and reconnaissance. More recognizable are 

its civilian counterparts, the Bell 406, which are familiar to most as news and traffic helicopters 

reporting over major metropolitan areas in the United States. The current transmission mounted 

in the stand belonged to the C-model aircraft. The stand has also tested the A-model 

transmission (10). The D-model is the most current platform in use by the Army.

The OH-58 transmission provides speed and torque reduction through two stages. The first stage 

consists of a spiral-bevel configuration. The near-horizontal input shaft holds a 19-tooth spiralbevel pinion which drives a 71-tooth spiral-bevel gear on a near-vertical intermediate shaft. The 

directional change in shaft rotation totals 94 degrees and occurs in this first stage. 

The second stage uses a planetary gear configuration. The intermediate shaft from the first stage 

also contains a 27-tooth sun gear. The sun gear drives three or four 35-planet planetary gears, 

depending upon aircraft model. The planets mesh with a stationary 99-tooth ring gear splined to 

the top of the transmission casing. The rotating planet gears drive the carrier, which is attached 

to the output shaft. The output shaft rests within the intermediate shaft. Various roller and ball 

bearings hold the shafts and planets in place. The total reduction ratio is 17.44:1. Figure 1 

depicts the transmission of the A-model aircraft.
3

[ ] Mq Cq Gq Kq F { }   +++ = [ ]{ } [ ]{ } [ ]{ } ( ) { } t

Figure 1. OH-58 transmission schematic (left) and disassembled (right). 

3. Transmission Model

3.1 Description and Model Assembly

The model uses a lumped parameter, finite element formulation of the familiar equations of 

motion. 

 (1)

The subcomponents of the model consist of the three shafts, the stationary ring gear, and the 

planet gears. In other words, these five subcomponents have their own matrix elements for 

satisfying equation 1 above (41). The spiral-bevel pinion, spiral-bevel gear, sun gear, and planet 

carrier are assumed to be point masses acting at the appropriate nodes along their respective 

shaft. The ring and planet gears also act as point masses at independent nodes not associated 

with any shaft. The input, intermediate, and output shafts are designated as Shafts A, B, and C, 

respectively.

Coupling of subcomponents occurs through gear-mesh interactions. The model interprets these 

interactions as linear stiffness matrices. While gear loading itself is inherently nonlinear, the 

model maintains the nonlinearities on the right hand side of equation 1 as forcing functions. 

Although a nonlinear model is possible with minor modifications, the linear treatment is 

maintained for ease of analysis.

Many gear-mesh formulations are available in the literature. However, the fundamental 

characteristic of the spiral-bevel mesh matrix used here is its twelve degrees-of-freedom (dof’s), 

corresponding to each of the six dof’s at the two coupling nodes. Although spiral-bevel gear 

geometry and force transfer are some of the most complex in the gearing industry, two 
4

approaches have been used in this model. The first uses the method of Tregold’s 

Approximation, modeling the two spiral-bevel gears as equivalent spur gear sets (42). This 

assumption enables the use of a 12-dof spur and helical gear mesh matrix (43). The second and 

more accurate approach rigorously develops the mesh matrix based upon vector calculus of the 

gear geometry (44). Both techniques have been used with comparable results. The results 

presented here use the more rigorous mesh matrix development. 

The epicyclic stage of the transmission is more complex due to its multiple components and the 

orbital motion of the planets. For convenience, the system model implements an existing lumped 

parameter epicyclic model from the literature (45, 46). Each subcomponent is placed at its 

appropriate shaft node in the model as discussed previously. 

Figure 2 presents a comprehensive illustration of the transmission model by subcomponent. The 

nodes on the three shafts each have the conventional six dof’s. The nodes of each shaft were 

placed at key locations as dictated by shaft geometries, gear-mesh, and bearing locations. Shafts 

A and B each have eight nodes. Shaft C consists of 13 nodes. The total number of dof’s for the 

shafts is 174. The epicyclic components each have three dof’s – two in translation and one in 

rotation (torsional). The ring and planets therefore total 3+3n dof’s, where n is the number of 

planet gears. The carrier and sun gear also contain three dof’s, which are expanded into the 

traditional 6-dof matrix at their respective nodes. The total number of dof’s in the model is 

177+3n. 

Some final comments regarding the model setup. Note that each shaft has a non-uniform cross 

sectional area. The model assumes an equivalent cross-sectional area for the inner and outer 

diameters of each shaft. It neglects transmission error within the gear meshes. Lastly, the 

transmission casing itself is neglected in this analysis. The system parameters follow in table 1.
5

Figure 2. Finite element model component layout.
6

Table 1. Transmission model parameters. 

Material Parameters Shaft A: Input Shaft

Young’s Modulus (E) (N/m2) 2.05E11 Length (m) 0.150

Shear Modulus (G) (N/m2) 8.0E10 Outer Diameter (eq) (dso) (m) 0.050

Poisson’s Ratio () 0.29 Inner Diameter (eq) (dsi) (m) 0.033

Density () (kg/m3) 7850

Shaft B: Intermediate Shaft Shaft C: Mast Output Shaft

Length (m) 0.154 Length (m) 0.460

Outer Diameter (eq) (dso) (m) 0.073 Outer Diameter (eq) (dso) (m) 0.057

Inner Diameter (eq) (dsi) (m) 0.058 Inner Diameter (eq) (dsi) (m) 0.021

Spiral-Bevel Pinion (Shaft A) Spiral-Bevel Gear (Shaft B)

Mass (kg) 0.47 Mass (kg) 2.24

Moment of Inertia (It) (kg m2) 1.4E-4 Moment of Inertia (It) (kg m2) 1.5E-4

Polar Moment of Inertia (Ip) (kg m2) 2.8E-4 Polar Moment of Inertia (Ip) (kg m2) 3.0E-4

Diameter (dg) (m) 0.062 Diameter (dg) (m) 0.228

Pitch Angle (Γ) (deg) 15.3 Pitch Angle (Γ) (deg) 78.7

Pressure Angle (αn) (deg) 20 Pressure Angle (αn) (deg) 20

Spiral Angle (ψ) (deg) 30.0 Spiral Angle (ψ) (deg) 30.0

Orientation Angle (φ) (deg) 0.0 Orientation Angle (φ) (deg) 0.0

Average Mesh Stiffness (Kg) (N/m) 1.0E8 Average Mesh Stiffness (Kg) (N/m) 1.0E8

Number of Teeth 19 Number of Teeth 71

Carrier Sun

Mass (kg) 5.43 Mass (kg) 0.40

Moment of Inertia (It) (kg m2) 0.0492 Moment of Inertia (It) (kg m2) 5.841E-4

Diameter (dg) (m) 0.1768 Diameter (dg) (m) 0.0774

Pressure Angle (αn) (deg) 20 Pressure Angle (αs) (deg) 24.6

Bearing Stiffness (kbc) (N/m) 1.0E8 Bearing Stiffness (kbs) (N/m) 1.0E8

Torsional Bearing Stiffness (kcu) (Nm/rad) 100 Torsional Bearing Stiffness (ksu) (Nm/rad) 100

Number of Teeth 27

Ring Planet

Mass (kg) 2.35 Number of Planets 4

Moment of Inertia (It) (kg m2) 0.0567 Mass (kg) 0.66

Diameter (dg) (m) 0.275 Moment of Inertia (It) (kg m2) 0.0015

Pressure Angle (αr) (deg) 24.6 Diameter (dg) (m) 0.1003

Bearing Stiffness (kbr) (N/m) 1.0E15 Pressure Angle (αn) (deg) 20

Torsional Bearing Stiffness (kru) (Nm/rad) 1.0E15 Bearing Stiffness (kbr) (N/m) 1.0E8

Number of Teeth 99 Number of Teeth 35

Duplex, Triplex, and Roller Bearing

kxx, kyy (N/m) 1.0E9

3.2 Simulation Setup

The basic function of the model is to provide a response to a given set of inputs, such that the 

output approximates experimental results. These inputs represent both normal and abnormal 
7

[ ]{ } [ ]{ } [ ]{ } [ ]{ } { } { } ( ) { } j j j( )

1 1 1

ee e b b bb b t i NT t i NT j t

b i i sb b

i i j

∞ ∞ ∞ Ω ⋅ ⋅Ω ⋅ ⋅Ω ± ⋅Ω

−

= = =

Mq Cq Gq Kq T F    +++ = + + ∑ ∑∑ F +

operating conditions. The inputs are modeled by forcing functions on the right hand side of 

equation 1. It is here that gear dynamics become readily apparent. The dynamics of the spiralbevel and gear-mesh frequencies and their harmonics dominate the transmission response (10). 

This is most easily illustrated by a data sample from the OH-58 Transmission Test Stand. The 

test conditions were an input speed of 6060 rpm, torque of 350.1 N-m, and full mast loading. 

Figure 3 depicts the response from an accelerometer near the ring gear at the left trunnion mount. 

The gear-mesh frequency harmonics of both gear stages are identified.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (Hz)

Normalized Magnitude

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (Hz)

Normalized Magnitude

Spiral-bevel mesh

Planetary mesh

Figure 3. Sample data from OH-58 transmission test stand.

The data in figure 3 show 22 total gear-mesh harmonics for the OH-58C transmission in the 

spectrum between 0 and 10,000 Hz, clearly illustrating the gear-mesh contributions to the overall 

dynamics of the system. This significantly contributes to the complexity of the modeling 

problem, since each harmonic represents an individual model input. This leads to another 

dilemma, specifically, how to determine the amplitudes of the respective forcing functions, since 

no closed form methodology exists. Similarly, the gear-mesh contributions also result in 

sideband frequencies, which must be addressed as they have also been used to detect fault 

conditions (47). 

In light of these considerations, the force vector of equation 1 must consist of several terms to 

accurately simulate the dynamics of the system. This may or may not be feasible, and the results 

must be closely scrutinized to determine the sensitivity of the model output to the inputs. A 

representative simulation equation therefore might resemble equation 2. 

 (2)

where the terms on the right hand side represent the torque vector, the spiral-bevel gear-mesh 

harmonics, and the spiral-bevel sideband harmonics, respectively. The ellipsis in equation 2 
8

represents other multiple inputs required to model the system dynamics, such as the planetary 

gear-mesh harmonic contributions. 

4. Model Validation

Preliminary model validation occurred by using traditional impact testing procedures to identify 

natural frequencies of the four-planet transmission configuration in the ARL/NASA Test Stand. 

Three accelerometers at different locations on the casing recorded the response from successive 

impacts at four different impact locations. Four impacts occurred at each location, resulting in 

48 test samples. From this data, the experimental natural frequencies were extracted.

Figure 4 compares the model’s natural frequencies (x-axis) determined from eigenvalue analysis 

to those frequencies identified from the experimental data (y-axis), and plots them with respect to 

the correlation line. The correlation line is the 45° dotted line representing an exact 1:1 

comparison. In figure 4, note the close proximity of all predicted data points to the correlation 

line. Additionally, figure 4 identifies four bands of unpredicted frequencies. These represent 

frequencies measured during experimental testing with no corresponding prediction in the model. 

They were highlighted and placed on the correlation line. The current hypothesis suggests that 

they are frequencies associated with the transmission casing, which the model currently neglects. 

Future incorporation of the casing into the model will prove or disprove this hypothesis. 

Nevertheless, these results provide a significant level of confidence to the model in its current 

configuration. 

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Finite Element Model Frequency (Hz)

Experimental Frequency (Hz)

Predicted Frequencies

Unpredicted Frequencies

Correlation Line

Unpredicted Frequency Bands

Figure 4. Experimental versus model frequency.
9

5. Eigen Analysis Results – Mode Shapes

Initial model analysis solved the eigenvalue problem of the undamped system, accounting for 

gyroscopic effects, using the traditional state-space formulation. Since gyroscopic effects are a 

function of rotational speed, the nominal speed of the input shaft was set at 6,060 rpm, matching 

the speed of the transmission input from the test environment and the aircraft (10). 

The eigenvalues and eigenvectors represent the system natural frequencies and mode shapes, 

respectively, as a function of shaft operating speed. Since the frequencies have already been 

addressed, the mode shapes are the objectives of this discussion. Frequency variation with 

operating speed is not significant in this analysis and is neglected. 

The lateral-torsional vibration coupling that occurs in mode shapes of geared systems has been 

well documented since the 1970s (48). Epicyclic mode shapes result in a different coupling and 

have also been characterized (45, 46). Figure 5 depicts the coupled mode shapes of the three 

shafts corresponding to a frequency of 3,230 Hz. The upper three charts show the lateral 

displacements of (a) Shaft A, (b) Shaft B, and (c) Shaft C. Likewise the lower three charts show 

the torsional displacement among the three shafts, respectively. The dotted vertical lines in the 

lower charts correspond to the location of the gear mesh at each shaft. In the case of Shaft C 

(figure 5.c), the vertical line corresponds to the location where the planet carrier connects to the 

mast output shaft. 

0

50

100

-1

0

1

-1

-0.5

0

0.5

1

X Shaft A Location (mm)

Y

0 20 40 60 80 100 120 140 0

0.2

0.4

0.6

0.8

1

1.2

Shaft A Location (mm)

Torsional Mode Shape

0

50

100

150

-1

0

1

-1

-0.5

0

0.5

1

X Shaft B Location (mm)

Y

0 50 100 150 0

0.2

0.4

0.6

0.8

1

1.2

Shaft B Location (mm)

Torsional Mode Shape

0 100 200 300 400

-1

0

1

-1

-0.5

0

0.5

1

X Shaft C Location (mm)

Y

0 100 200 300 400 0

0.2

0.4

0.6

0.8

1

1.2

Shaft C Location (mm)

Torsional Mode Shape

0

50

100

-1

0

1

-1

-0.5

0

0.5

1

X Shaft A Location (mm)

Y

0 20 40 60 80 100 120 140 0

0.2

0.4

0.6

0.8

1

1.2

Shaft A Location (mm)

Torsional Mode Shape

0

50

100

150

-1

0

1

-1

-0.5

0

0.5

1

X Shaft B Location (mm)

Y

0 50 100 150 0

0.2

0.4

0.6

0.8

1

1.2

Shaft B Location (mm)

Torsional Mode Shape

0 100 200 300 400

-1

0

1

-1

-0.5

0

0.5

1

X Shaft C Location (mm)

Y

0 100 200 300 400 0

0.2

0.4

0.6

0.8

1

1.2

Shaft C Location (mm)

Torsional Mode Shape

a) b) c)

Figure 5. Shaft mode shapes –3,230 Hz. 
10

Figure 5 illustrates the lateral-torsional mode coupling of the three shafts. The mode shapes are 

planar, another feature indicative of coupling (43). Although this study presents only one 

illustrative mode shape, it is worth mentioning that all observed lower mode shapes exhibited 

this same pattern, namely, coupled and planar. The mode shapes of the ring and planets have 

been omitted for brevity. However, these results illustrate the need for further study and 

classification of the mode shapes of such a complex system.

6. Dynamic Simulation Results with Fault Detection

The dynamic simulations examine two conditions. The first represents a transmission under 

“normal” operating conditions. The second condition seeds a fault in the spiral-bevel mesh, 

simulating gear-tooth surface wear. Wear is a gradual process, characterized by the mild 

deterioration of the mating surface due to sliding action between teeth. It can accelerate the 

likelihood of fatigue failures (25) and negatively impact other drive system components (24). 

Figure 6 shows a gear experiencing a worn condition. 

Due to its gradual nature, the likelihood of serious damage from worn gears is quite low. 

Nevertheless, this fault was selected because it is one of the most overlooked faults (23), and it is 

also one of the easiest to model for demonstration purposes. 

In the mathematical model, the wear condition is captured by an altered loading pattern within 

the gear-mesh (23). Such a pattern is presented in figure 7. The solid line represents a normal 

gear loading pattern. The dotted line simulates the worn condition. These loading conditions 

occur at the pitch point in the gear mesh. One will also note that due to the gradual process, wear 

affects more than one gear tooth simultaneously. The chart in figure 7 therefore implies that all 

gear teeth have undergone wear. This decreases the model complexity substantially.

The normal loading pattern of figure 7 is found using a conventional finite element, contact 

analysis solver. As such, these loading patterns do not exist in closed form. They must be 

approximated by curve fit. Afterwards, a transformation procedure converts them into the 

appropriate force vectors. 

The first step in this simulation was to approximate a curve to the loading pattern representing 

normal operation of the spiral bevel gear mesh. The Fourier series of equation 3 is appropriate. 
11

( ) { } ( ) { } ( ) 1 1

cos sin n n b b b b

n n

t a n NT t b n NT t

∞ ∞

= =

F = ∑ ∑ ⋅ ⋅Ω ⋅ + ⋅ ⋅Ω ⋅

Figure 6. Gear tooth with wear.

0 0.5 1 1.5 2 2.5

x 10-3

-1

-0.5

0

0.5

1

time (sec)

normalized load

normal

worn

0 0.5 1 1.5 2 2.5

x 10-3

-1

-0.5

0

0.5

1

time (sec)

normalized load

normal

worn

Figure 7. Spiral-bevel gear loading conditions.

 (3) 
12

Equation 3 represents the periodic loading of the gear teeth at harmonics of the spiral-bevel, 

gear-mesh frequency, (n·NTb·Ωb), with the Fourier coefficients as the gear-mesh harmonic 

amplitudes. The input operating speed of 6,060 rpm (101 Hz) results in 5 gear-mesh harmonics 

between 0 and 10,000 Hz. The Fourier coefficients are then approximated by the curve-fit 

software. Table 2 provides the Fourier coefficients for both the normal and damage conditions 

used in figure 7. Finally, these loading patterns when resolved into two 6-dof force vectors, 

represent the forces and moments acting at the appropriate nodes on Shafts A and B. 

Table 2. Loading pattern Fourier coefficients.

The system response was calculated for the force and moment vectors of each harmonic as 

indicated in equation 2. The superposition of these responses results in the total dynamic 

response. For this study, the forcing functions include only the force and moment vector of the 

applied torque, and the vectors of the spiral-bevel gear harmonics. It does not include any gearmesh sidebands or planetary gear-mesh contributions. Figure 8 depicts the lateral response, 

measured at the input node of Shaft A, for both (a) normal and (b) worn conditions. 

The difference in the two vibration patterns is clearly noticeable. However, the presence of this 

worn condition (figure 8.b) did not produce a drastic change in the vibration pattern. This is not 

unexpected, given the gradually progressive nature of the fault. This also indicates that the 

traditional vibration diagnostic parameters may not be effective in detecting this type of fault.

One such parameter is the signal kurtosis, presented in figure 9. It is a statistical measure,

calculated from a sample of accelerometer signals. A normal signal condition returns a kurtosis 

value of approximately 3. A fault condition results in a higher value, with a threshold of 7 being 

a clear indicator of damage. Figure 9 shows the kurtosis value for a string of 2,048 data points. 

The first 1,024 data points have no fault. The second half of the data points represents the 

introduction of the fault condition. Figure 9 shows only a slight increase in the kurtosis value. 

Although the increase is detectable, it does not come near the threshold value. Another method 

of fault detection is necessary for this condition. 

The frequency domain provides such an alternative. Using the FFT algorithm, the model 

provided both responses in the frequency domain, presented in figure 10. The differences 

normal worn

n an bn an bn

1 –402 –3.26 –402 –4.25

2 –49.04 –0.795 –100 –0.7

3 –2.44 –0.059 –2 –0.02

4 –15 –5 –9 –6 

5 –10 –3 –15 –8 
13

2

2 100 damage normal

normal

s s

FE

s

    − = ⋅        

between the two conditions are readily apparent. Quantifying these differences resulted in the 

development of a simple metric, designated FE2. In the most basic of terms, the FE2 is the 

square of the percentage error between frequency signals, defined by equation 4. The square in 

equation 4 increases the detection of faults by eliminating smaller peaks (<1), while magnifying 

larger peaks.

(4)

Figure 8. (a) Normal and (b) worn vibration response.

200 400 600 800 1000 1200 1400 1600 1800 2000 0

1

2

3

4

5

6

7

8

9

10

data point

kurtosis value

200 400 600 800 1000 1200 1400 1600 1800 2000 0

1

2

3

4

5

6

7

8

9

10

data point

kurtosis value

Figure 9. Kurtosis diagnostic parameter results.

-1

-0.5

0

0.5

1

amplitude (g's)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 -1

-0.5

0

0.5

1

time (sec)

amplitude (g's)

-1

-0.5

0

0.5

1

amplitude (g's)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 -1

-0.5

0

0.5

1

time (sec)

amplitude (g's)

a)

b)
14

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

frequency (Hz)

amplitude (g's)

normal

damage

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

frequency (Hz)

amplitude (g's)

normal

damage

Figure 10. Frequency spectrum comparison.

This utility of this metric is demonstrated in figure 11. The FE2 was calculated using the FFT 

data points from the two curves in figure 10. Comparing the FE2 metric of figure 11 with the 

kurtosis parameter of figure 9, FE2 provides a much stronger indication for this particular 

damage condition. However, this may not always be the case. Further investigation and 

calibration of the FE2 threshold is required. 

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

frequency (Hz)

FE2

FE2

Threshold

Figure 11. FE2 diagnostic parameter.

7. Additional Considerations

The worn condition presented here did not seem to indicate a drastic fault condition, which leads 

to a discussion on the different levels of damage. This implies that some a priori knowledge is 
15

required on the differences between light, intermediate, and severe fault levels and how to model 

each of them. This may not be necessary in more distinct faults such as pitting, spalling, 

cracking, etc., where the defect has a more pronounced signature. A more severe wear condition 

may produce a more drastic result.

Another consideration is the ability to determine a standard fault signature for a given condition. 

These signatures will most likely be statistical or “average” patterns, requiring several iterations 

of testing and modeling scenarios to establish, which leads to the third consideration. The 

damage thresholds of the diagnostic parameters, especially the FE2 parameter, require 

calibration to ensure accuracy.

Finally, the instrumentation dynamics associated with epicyclic gear vibration must be 

addressed. An accelerometer attached to the casing near the ring gear measures the vibration of 

the epicyclic gear train. As each planet passes the accelerometer during a rotation, the sensor 

detects an increase in the vibration level. Thus, for n planets, a n/rev amplitude modulation is 

produced. The resulting measured vibration signal is therefore a convolution of the actual 

vibration of the gear set and the modulation caused by the planets moving past the accelerometer. 

The current model does not capture this effect.

8. Conclusions 

This report has presented the results of a preliminary study into the feasibility of using a physicsbased helicopter transmission model as a method of fault prediction and detection in support of 

CBM initiatives. Two simulations, one each for normal and damage operating conditions were 

presented using a lumped parameter, finite element model. The time and frequency domain 

responses were calculated for each and compared. Damage detection was illustrated using both a 

traditional diagnostic parameter and a new frequency-based parameter, FE2, although several 

iterations of testing and simulation are required to establish new parameter thresholds. The 

model itself requires further development and testing, to include sensitivity studies, to ensure a 

valid representation of the system dynamics.

Maturity of this technology will result in increased accuracy and confidence in fault detection 

and diagnosis, characterized by higher detection rates and fewer false alarms. It can serve as the 

foundation for incorporating transmission models into vibration and health monitoring systems, 

provide predictive capability prior to testing, and provide a useful tool for testing and evaluating 

condition indicator effectiveness. 

From a CBM standpoint, this type of technology can aid in improving operational readiness, cost 

savings, and maintenance downtime. 
16

9. References

1. Harris, F.; Kasper, E.; Iseler, L. U.S. Civil Rotorcraft Accidents, 1963 through 1997; NASA 

TM 2000-209597, USAAMCOM TR-00-A-006; 2000. 

2. Aviation Accident Investigation Board (U.K.), Initial Accident Report EW/C2009/04/01, 

2009. 

3. Samuel, P.; Pines, D. A Review of Vibration-Based Techniques for Helicopter Transmission 

Diagnostics. Journal of Sound and Vibration 2005, 282, 475–508. 

4. Lebold, M.; McClintic, K.; Campbell, R.; Byington, C.; Maynard, K. Review of Vibration 

Analysis Methods for Gearbox Diagnostics and Prognostics. Proceedings of the 54th

Meeting of the Society for Machinery Failure Prevention Technology, MFPT, Haymarket, 

VA, 2000, 623–634. 

5. Khanmohammedi, S.; Alizadeh, G.; Giasi, A. Stochastic Modeling of Gearbox via Genetic 

Algorithm. Proceedings of the 15th IASTED International Conference, edited by M. Hamza, 

IASTED, Calgary, Alberta, 2006, 484–488. 

6. Zakrajsek, J.; Townsend, D.; Decker, H. An Analysis of Gear Fault Detection Methods as 

Applied to Pitting Fatigue Failure Data. Proceedings of the 47th Meeting of the Mechanical 

Prevention Group, Compiled by H. Pusey and S. Pusey, Vibrational Institute, Willowbrook, 

IL, 1993, 199–208. 

7. Garga, A.; Elverson, B.; Lang, D. Fault Classification in Helicopter Vibration Signals. 

Proceedings of the American Helicopter Society 53rd Annual Forum, AHS, Alexandria, VA, 

1997, 1316–1323. 

8. Swansson, N. Application of Vibration Signal Analysis Techniques to Signal Monitoring. 

Proceedings of the Conference on Lubrication, Friction, and Wear in Engineering 1980. 

Barton, ACT; Institution of Engineers, Melbourne, Australia, 1980, 262–267. 

9. Dykas, B.; Krantz, T.; Decker, H.; Lewicki, D. Experimental Determination of AH-64 

Apache Tailshaft Hanger Bearing Vibration Characteristics with Seeded Faults. Proceedings 

of the American Helicopter Society 65th Annual Forum on Disc [CD-ROM], AHS, 

Alexandria, VA, 2009. 

10. Lewicki, D.; Coy, J. Vibration Characteristics of OH-58A Helicopter Main Rotor System; 

NASA TP-2705, ARL TR-86-C-42; 1987. 
17

11. Keller, J.; Grabill, P. Vibration Monitoring of UH-60A Main Transmission Planetary 

Carrier Fault. Proceedings of the American Helicopter Society 59th Annual Forum on Disc

[CD-ROM], AHS, Alexandria, VA, 2003. 

12. Dempsey, P.; Handschuh, R.; Afjeh, A. Spiral Bevel Damage Detection Using Decision 

Fusion Analysis; NASA TM-2002-211814, ARL-TR-2744; 2002. 

13. Schwartz, B.; Jones, D. Quadratic and Instantaneous Frequency Analysis of Helicopter 

Gearbox Faults. Mechanical Systems and Signals Processing 2000, 14 (4), 579–595. 

14. Dempsey, P.; Keller, J.; Wade, D. Signal Detection Theory Applied to Helicopter 

Transmission Diagnostic Thresholds. Proceedings of the American Helicopter Society 65th

Annual Forum on Disc [CD-ROM], AHS, Alexandria, VA, 2009. 

15. Dempsey, P.; Lewicki, D.; Decker, H. Investigation of Gear and Bearing Fatigue Damage 

Using Debris Particle Distribution; NASA TM 2004-212883, ARL-TR-3133; 2004. 

16. Dempsey, P.; Lewicki, D.; Decker, H. Transmission Bearing Damage Detection Using 

Decision Fusion Analysis; NASA TM 2004-213382, ARL-TR-3328; 2004. 

17. Hess, A.; Hardman, W. Seeded Fault Testing in Support of Mechanical Systems Prognostic 

Development. Aerospace Conference 2002 Proceedings, Vol. 6, IEEE, Washington, D.C., 

2002, 2995–3004. 

18. Blunt, D.; Keller, J. Detection of a Fatigue Crack in a UH-60A Planet Gear Carrier Using 

Vibration Analysis. Mechanical Systems and Signals Processing 2006, 20, 2095–2111. 

19. Wu, S.; Zuo, M.; Parey, A. Simulation of Spur Gear Dynamics and Estimation of Fault 

Growth. Journal of Sound and Vibration 2008, 317, 608–624. 

20. McFadden, P. Detecting Fatigue Cracks in Gears by Amplitude and Phase Demodulation of 

the Meshing Vibration. Journal of Vibration, Acoustics, Stress, and Reliability in Design

1986, 108, 165–169. 

21. Lewicki, D.; Dempsey, P.; Heath, G.; Shanthakumaran, P. Quantification of Condition 

Indicator Performance on a Split Torque Gearbox. Proceedings of the American Helicopter 

Society 65th Annual Forum on Disc [CD-ROM], AHS, Alexandria, VA, 2009. 

22 Choy, F.; Polyshchuk, V.; Zakrajsek, J.; Handschuh, R.; Townsend, D. Analysis of the 

Effects of Surface Pitting and Wear on the Vibrations of a Gear Transmission System. 

Tribology International 1996, 29 (1), 77–83. 

23 Flodin, A.; Andersson, S. Simulation of Mild Wear in Spur Gears. Wear 1997, 207, 16–23. 

24. Krantz, T.; Oswald, F.; Handschuh, R. Wear of Spur Gears Having a Dithering Motion and 

Lubricated with a Perflourinated Polyether Grease; NASA TM 2007-215008, ARL TR4124; 2007. 
18

25. Kuang, J.; Lin, A. The Effect of Tooth Wear on the Vibration Spectrum of a Spur Gear Pair. 

Journal of Vibration and Acoustics 2001, 123, 311–317. 

26. Ma, J.; Li, C. On Localized Gear Defect Detection by Demodulation of Vibrations - A 

Comparision Study. Manufacturing Science & Engineering Division, Proc. ASME 1995 Int. 

Mech. Engr. Congress and Exposition 1995, 2 (1), 565–576. 

27. Forrester, B. A Method for the Separation of Epicyclic Planet Gear Vibration Signatures. 

Proceedings of the 3rd International Conference on Acoustical and Vibration Surveillance 

Methods and Diagnostic Techniques, 1998, 539–547. 

28. Wang, W.; McFadden, P. Early Detection of Gear Failure by Vibration Analysis I. 

Calculation of the Time-Frequency Distribution. Mechanical Systems and Signals 

Processing 1993, 7 (3), 193–203. 

29. Wang, W.; McFadden, P. Early Detection of Gear Failure by Vibration Analysis II. 

Interpretation of the Time-Frequency Distribution Using Image Processing Techniques. 

Mechanical Systems and Signals Processing 1993, 7 (3), 205–215. 

30. Barry, D. Fast Calculation of the Choi-Williams Time-Frequency Distribution. IEEE 

Transactions on Signal Processing 1992, 40 (2), 450–454. 

31. Trendafilova, I. Vibration-Based Damage Detection in Structures Using Time Series 

Analysis. Journal of Mechanical Engineering Science 2006, 223 (3), 261–272. 

32. Ma, J.; Li, C. Gear Defect Detection Through Model Based Wideband Demodulation of 

Vibrations. Mechanical Systems and Signals Processing 1996, 10 (5), 653–665. 

33. McFadden, P.; Smith, J. An Explanation for the Asymmetry of the Modulation Sidebands 

About the Tooth Meshing Frequency in Epicyclic Gear Vibration. Proceedings of the 

Institution of Mechanical Engineers, Part C, Vol. 199, No. 1, 65–70. 

34. McFadden, P.; Smith, J. A Signal Processing Technique for Detecting Local Defects in a 

Gear from the Signal Average of the Vibration. Proceedings of the Institution of Mechanical 

Engineers, Part C, Vol. 199, No. 4, 287–292. 

35. Wu, B.; Saxena, A.; Patrick, R.; Vachtsevanos, G. Vibration Monitoring for Fault Diagnosis 

of Helicopter Planetary Gears. Proceedings of the 16th IFAC World Congress on Disc [CDROM], IFAC, Laxenburg, Austria, 2005. 

36. Özgüven, H.; Houser, D. Mathematical Models Used in Gear Dynamics. Journal of Sound 

and Vibration 1988, 121, 383–411. 

37. Blankenship, G.; Singh, R. A Comparative Study of Selected Gear Mesh Interface Dynamic 

Models. Proceedings of the 6th International Power Transmission and Gearing Conference, 

ASME, New York, NY, 1992, 137–146. 
19

38. Choy, F.; Townsend, D.; Oswald, F. Dynamic Analysis of Multimesh-Gear Helicopter 

Transmissions; NASA TP-2789; 1988. 

39. McFadden, P. Examination of a Technique for the Early Detection of Failure in Gears by 

Signal Processing of the Time Domain Average of the Meshing Vibration. Mechanical 

Systems and Signals Processing 1987, 1 (2), 173–183. 

40. Dickey, Beth. NASA Announces Pennsylvania Aeronautics Research Awards. 25 Nov 

2008, NASA Press Release 08-312e. 

41. Stringer, D. Geared Rotor Dynamic Methodologies for Advancing Prognostic Modeling 

Capabilities in Rotary-Wing Transmission Systems. Ph.D. Dissertation, Mechanical and 

Aerospace Engineering Dept., University of Virginia, Charlottesville, VA, 2008. 

42. Yoon, K. The Combined Mesh Stiffness Characteristics of Straight and Spiral Bevel Gears. 

Master’s thesis, North Carolina State University, Mechanical Engineering Dept., 1989. 

43. Stringer, D.; Sheth, P.; Allaire, P. Gear Modeling Methodologies for Advancing Prognostic 

Capabilities in Rotary-Wing Transmission Systems. Proceedings of the American 

Helicopter Society 64th Annual Forum on Disc [CD-ROM], AHS, Alexandria, VA, 2009. 

44. Litvin, F. Theory of Gearing; NASA RP-1212, AVSCOM TR 88-C-035, 1989. 

45. Lin, J.; Parker, R. Analytical Characterization of the Unique Properties of Planetary Gear 

Free Vibration. Journal of Vibration and Acoustics 1999, 121, 316–321. 

46. Parker, R.; Lin, J. Modeling, Modal Properties, and Mesh Stiffness Variation Instabilities of 

Planetary Gears; NASA CR 2001-210939, ARL-CR-462; 2001. 

47. Mobley, R. An Introduction to Predictive Maintenance, 2nd ed.; Butterworth Heinemann: 

New York, 2002, pp. 306–309. 

48. Mitchell, L.; Mellen, D. Torsional-Lateral Coupling in a Geared, High-Speed Rotor System; 

ASME Paper 75-DET-75; 1975. 
20

INTENTIONALLY LEFT BLANK. 
21

Appendix. Nomenclature

an,bn = Fourier coefficients

b = subscript, spiral-bevel gear pair

[C] = general damping matrix

{F(t)} = force and moment vector 

[G] = gyroscopic matrix

i, j = indices

j = imaginary component

[K] = stiffness matrix

kmesh = mesh stiffness 

[M] = mass matrix

NT = number of gear teeth

n = number of planets; n

th harmonic 

{q} = generalized displacement vector

sb = subscript, sideband 

{T} = torque vector 

t = time

Ω = rotational speed
22

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