Automated Condition Monitoring of a Cycloid Gearbox

Abstract

While condition monitoring techniques have been developed for many gearbox types, there has been almost no research on condition monitoring of cycloid driver gearboxes. Cycloid gearboxes are used where high reduction ratios are needed in a single stage. While most gear designs are based on an involute subject to a sliding force, cycloid gear designs are subject to compression. As a result, cycloid gearboxes are quiet, have low backlash, and have large torsional stiffness. Because there is no typical pinion-gear pair in this gearbox, the calculation of the reduction ratio is non-standard. Further, as the eccentric bearing which drives the cycloid gears is in the rotating frame, the calculated fault frequency rates are not as expected. This paper describes the dynamics needed to identify cycloid gearbox fault features to achieve automated fault detection and alerting.

1. Introduction

There are few non-standard condition monitoring applications for unusual gearbox designs. While most reduction (example: epicyclical) gearboxes have well-understood dynamics, others, such as a Variator (continuously variable transmission) or cycloidal gearbox, have not been reported. This paper covers the dynamics, configuration, and some test observations of work done on a cycloidal gearbox. The analysis procedure applies to other, more standard gearboxes as well.

In a cycloidal gearbox, the drive uses an eccentric bearing that causes the gears’ center to rotate in a housing. The rotation orbit is reversed as the gear’s teeth are less than the housing’s diameter. The path of a fixed point of the gear traces a hypocycloid, which is fundamentally different from the circular motion of traditional gears.

Interest in the cycloidal gearbox is derived because they are used in many applications where low-cost drive motors are needed. For example, many conveyer belt systems (sorting, moving bulk media, slew drives) use cycloid drives. Typically, the drive itself is a relatively low-cost asset. However, the processes that drive unit support can have significant economic impacts if they fail. For example, one of the more significant courier delivery services has a distribution center with 3000 cycloid drives to move packages. The loss of a drive unit halts sorting, impacting up to $200,000 per day in revenues.

The cycloid drive for gearboxes allows for a high reduction ratio and zero or very low backlash. The cycloid gear design is based on compression vs. shear forces, where the contact is typically rolling vs. sliding. These features allow high shock load capacity, high torsional stiffness, and quiet operation. Single-stage ratios of more than 200:1 are possible.

The gearbox chosen for the test was an integrated induction motor and gearbox. This gearbox is rated for 0.75 kW, approximately 1 Hp drive. For 60 Hz operations, using a four-pole motor, the drive unit has a 100% input shaft rate of approximately 1795 rpm. The gearbox has a 51:1 reduction ratio.

2. State of the art for gearbox condition monitoring

There has been little documented work on the condition monitoring of cycloidal gearboxes. Chrochran and Bobak [1] describe the complexity of vibration analysis of cycloid gearbox using traditional spectrum analysis. They give information on calculating the cycloidal disc mesh frequency but do not describe a method for automated fault detection. There is no process given for bearing fault frequency indication.

Condition monitoring of motors or gearboxes has generally used spectrum analysis. The spectrum measures the magnitude of a frequency associated with the component fault frequencies, such as a shaft or gear mesh. The Fourier transform, used in spectrum analysis, is defined by cosines. The spectrum is good at measuring periodic sinusoids. However, many faults result in impacts that are not well measured by the spectrum. In recognizing this, R.M. Stewart [2] ushered in modern gearbox analysis using the Time Synchronous Average (TSA). The TSA, which controls for variance in speed, also performs as a comb filter that rejects nonsynchronous vibration features. The resulting time-domain signal reveals impact features. These features can be quantified via statics indicators, such as RMS, kurtosis, skewness, or crest factor [3, 4].

3. Vibration-based condition monitoring

Condition monitoring uses vibration sensors and configuration representing the drivetrain/motor to calculate condition indicators (CIs). These CIs are used to infer the current health of the component, and with the current component health, and the component threshold, an estimate the remaining useful life (RUL) can be estimated. The RUL (e.g., prognostics) allows the operator to better manage the asset by scheduling maintenance opportunistically. The goal, along with increased asset safety, is improved availability and more opportunities for revenue generation.

Some CIs have physical meaning. For example, a shaft imbalance is measured by the 1st shaft harmonic (SO1), typically as a velocity such as inches per second (IPS). ISO 10816 Vibration gives direction on these limits for various equipment types. Other fault conditions for shafts could include bending or coupling issues, which excite high harmonics. For these faults, there are no standard limits. Similarly, for components such as gears and bearings, the CIs have little physical meaning, and statistical or machine learning methods are used to set a threshold representative of a fault condition.

Acceleration, the second derivative of displacement, is a function of the shaft rate to the second power. Hence, acceleration from high-speed shaft tends to dominate simpler time-domain statics such as RMS. The vibration spectrum can give magnitude for a given shaft or gear mesh in the frequency domain. This is valid if the shaft rate is relatively stable RPM. However, for many systems, Fourier analysis, of say, the Gear mesh frequency is not necessarily a good fault indicator. For bearings, detection of the fault frequency (Cage, Ball, Inner, or Outer Race rate) is only possible close to failure. For these reasons, more advanced analyses are required.

Signal processing techniques such as the Time Synchronous Average (TSA) is used to control for variance in shaft rate and are the basis of gear condition indicators. As the TSA is a time-domain analysis, it is sensitive to impacts associated with, for example, a breathing crack. However, TSA does not work for bearing analysis. Bearings require other techniques because their motion depends on non-Hertzian contact resulting in slip (by definition, nonsynchronous). Additionally, due to the nature of bearing faults (e.g., we are measuring the effect of an impact inducing resonance in the bearing itself), successful fault detection requires careful consideration of parameter inputs necessary (e.g., envelope window) to perform the analysis.

3.1 Analysis based on the time synchronous average

Modern techniques for vibration diagnostics using the TSA were introduced in “Some Useful Data Analysis Techniques for Gearbox Diagnostics” [2]. In addition to the TSA, Stewart proposed several new gear fault condition indicators. These gear algorithms and subsequent new analyses by McFadden [3], Ma [5], and others are based on the functions operating on the TSA.

The model for vibration in a shaft in a gearbox was given in [2] as:

$$x\left(t\right)=\sum _{i=1}^k{X}_i\left(1+a_i\left(t\right)\right)\mathit{cos}\left(2\mathit{\text{πifm}}\left(t\right)+{\varphi }_i\left(t\right)\right)+b\left(t\right)$$E1

where:

X_i is the amplitude of the kth mesh harmonic.

FM(t) is the average mesh frequency.

a_i(t) is the amplitude modulation function of the i^th feature harmonic.

ϕi(t) is the phase modulation function of the i^th feature harmonic.

b(t) is additive background noise.

The mesh frequency is a function of the shaft rotational speed: FM = Nf(t), where N is the number of teeth on the gear and f(t) is the shaft speed as a function of time. As most drive motors are induction machines, the slip, and hence, motor speed, will change based on changes in torque over time. This will cause the resulting spectrum to be smeared in the frequency domain.

The vibration data can be resampled with a tachometer signal (such as a key phasor) and with the ratio from the key phasor to the shaft under analysis. The number of data points between one revolution and the next revolution is the same. The time-synchronous averaging (TSA), sums each point over the revolution, with the resampled data, then divides by the number of revolutions during the acquisition.

Since the radix-2 FFT is most used, the number of data points in one shaft revolution (n) is interpolated into m number of data points, such that:

• For all shaft revolutions n, m is larger than r (the number of samples in one revolution), and

• m = 2ceiling (log₂ (r))

The TSA acts as a comb filter, where the passband (comb) is each shaft harmonic. This removes nonsynchronous signals from the TSA. Operations on the TSA, such are RMS or magnitude of the first harmonics of the Fourier transform of the TSA, define various Condition Indicators (CIs). There are many potential CIs, which may include the second and third harmonics derived from the spectrum of the TSA, and other statistics such as: Peak to Peak, or kurtosis (Figure 1).

Figure 1.

Calculation of the TSA.

As there are gears associated with the input/output shaft, further analysis is performed on the TSA and the spectrum of the TSA. Some analyses are classified as gear specific, which use the number of teeth on the gear under analysis (FM0 [2], the AM/FM analysis [3], for example). Other non-gear-specific analyses are also performed, such as the residual or the energy operator (a time/frequency analysis). It should be noted that there are many implementations of gear analysis [4], and there is no single analysis that works for every gear fault type. In this implementation, the system generated 18 CIs for each gear (Figure 2).

Figure 2.

Operating on the TSA to generate gear condition indicators.

3.2 Basic gear analysis

In the residual gear analysis, the Fourier transform of the TSA is taken, and harmonics associated with shaft and gear mesh are zeroed, then the inverse Fourier transform is taken. In the frequency domain of the TSA, each index is a shaft harmonic. Hence if there are 31 teeth on a gear, this mean the 32 index in the frequency domain (index 1 is DC) is the 1st gear mesh harmonic. Removing this and 2nd/3rd gear mesh harmonics in the frequency domain removes these superimposed tones in the time domain. Without these known periodic signals in the time domain TSA signal, non-periodic features, such as the impact of a broken tooth, can be identified in the waveform.

Damage to a component change the measured instantaneous frequency will. The energy operator, developed by Ma [5], can quantify the amplitude and phase-modulated signal of a fault, whose product can measure the instantaneous frequency due to say, a scuff or cracked tooth. The energy operator is sensitive to torque, so statical indicators that reflect distribution “shape,” such as kurtosis and crest factor, can be used. The EO is given as:

$${\mathrm{\Psi }}_{\mathit{EO}}\left({\mathit{TSA}}_n\right)={\mathit{TSA}}_n^2-{\mathit{TSA}}_{n+1}\times {\mathit{TSA}}_{n-1}$$E2

The Narrowband analysis [3] uses the Fourier transform as a bandpass filter to remove all frequencies not associated with the gear mesh. The selection of the filter bandwidth is usually 25% of the gear tooth count. For example, if the TSA is length 1024, and the gear tooth count is 31, the filter bandwidth is 31/4 = 8, or [23 to 39]. After taking the Fourier transform of the TSA, the from DC to index (31–8) = 23, and index 39 to 512 are set zero (along with their conjugate). The Narrowband signal is then the real part of the inverse Fourier transforms.

The AM (amplitude modulation) Analysis is the envelope of the narrowband signal. Essentially, this is simply the magnitude of the Hilbert transform. Similarly, the FM (frequency modulation) Analysis is derivative (instantaneous frequency) of the argument of the Hilbert transform.

The TSA Fourier transform is used for miscellaneous analysis [2, 4, 5]. The Figure of Merit 0, for example, is a well know analysis [2] and is generally calculated as:

$$\mathrm{FM}0=\raisebox{1ex}{$\mathit{tsapeaktopeak}$}\!\left/ \!\raisebox{-1ex}{${\sum }_{i=1}^3{\mathit{GM}}_i$}\right.$$E3

GM_i is the i^th gear mesh harmonic. As the TSA is a time-domain signal, the peak-to-peak value is the maximum of the TSA time domain value minus the minimum of this TSA time domain value. In general, the peak to peak will increase over time as if there is a propagating cracked or soft tooth, while the gear mesh harmonics will remain constant. FM0 can be a powerful indicator of a crack or soft tooth.

The residual RMS is sensitive soft/crack tooth, as the residual of the TSA does not remove features associated with the impact of a breaking crack. The RSM of the TSA will is not as sensitive to these impact events. As such, the ratio of the residual RMS to the TSA RMS can be a helpful condition indicator, defining the energy ratio. If the residual signal is defined as r, and the TSA is tsa_i, then.

$$\mathit{er}=\sqrt{\frac{{\sum }_{i=1}^n{\left(r_i-\overline{r}\right)}^2}{n}}/\sqrt{\frac{{\sum }_{i=1}^n{\left({\mathit{tsa}}_i-\overline{\mathit{tsa}}\right)}^2}{n}}$$E4

The sideband level factor is defined as the sum of the first-order sideband amplitudes about the gear mesh, divided by the TSA rms [4]:

$$\mathit{SLF}=\raisebox{1ex}{${\mathit{TSA}}_{\mathit{gm}-1}+{\mathit{TSA}}_{\mathit{gm}+1}$}\!\left/ \!\raisebox{-1ex}{$\sqrt{\frac{{\sum }_{i=1}^n{\left({\mathit{tsa}}_i-\overline{\mathit{tsa}}\right)}^2}{n}}$}\right.$$E5

The ratio of the second gear mesh harmonic energy ratio to the first gear mesh harmonic energy defines the G2 analysis. Example analyses are found in the appendix, and a full description is given in [2, 3, 4, 5, 6].

3.3 Bearing envelope analysis

Bearing analysis is a separate processing flow. Bearings, as they are designed to be greased/oiled, have non-Hertzian contact. Typically, we observe a 1% slip in the calculated motion of the bearing components. Some bearings, when under thrust, will have changed their contact angle and pitch diameter, resulting in an increased fault rate by 2 to 3% [7, 8]. The bearing analysis is asynchronous but must also consider the non-stationarity of the shaft. To control for changing shaft rate, the vibration data can be resampled [8]. Bearing analysis uses this speed corrected signal for envelope analysis, which takes the spectrum of the demodulated signal and envelopes (absolute value of the Hilbert transform) the vibration data (Figure 3).

Figure 3.

Bearing analysis process flow.

The bearing analysis process returns seven CIs for each bearing, including the cage, ball, inner and outer race energies, the 1/rev spectral energy, the whip/whirl energy (for journal bearing analysis), and the kurtosis of the spectrum.

3.4 Health indicator paradigm

Intending to automate fault detection, we wish to use the calculated CIs to infer the health of a component [4]. Defining a health indicator (HI) assumes that CIs have some distribution. The HI is then a function of distributions. This allows a rigorously defined threshold setting process for a given false alarm rate. With that in mind, one can define the HI such that:

• The HI is scaled from 0 to 0.35, where 0.35 is the PFA (probability of false alarm). The PFA is set to say 10e-6, which is small,

• When the HI is greater than 0.75, the component is in warning. The probability of false alarm is then minimal for a nominal component.

• When the HI is greater than 1.0, the component is in alarm.

It is not claimed that the HI is a measure of failure. An HI based on the function of distributions develops evidence to reject the Null hypothesis: that the component is nominal. When the hypothesis is rejected, e.g., the HI is greater than 1.0, evidence suggests that the component is damaged. Hence, it allows for a proactive maintenance policy to restore the component to its nominal condition through repair. Proactive maintenance protects against cascading damage and reduces gearbox replacements.

The HI paradigm, from a maintainer perspective, is a stoplight-based threshold setting/alerting system: when a component is yellow, plan maintenance, and when the component turns red, do maintenance.

3.5 Controlling for the correlation between CIs

It is assumed that CIs have a probability distribution (PDF). Operation on the CI to build an HI is then a function of distributions. The norm of the CIs is the HI function used in this test:

$$\mathit{HI}=0.35/\mathit{\text{crit}}\sqrt{{\mathit{Y}}^T\mathit{Y}}$$E6

where Y is the whitened, normalized array of CIs, and crit, is the critical value.

Only if the CIs are independent and identical (e.g., IID) is (6) valid. For Gaussian distribution, subtracting the mean and dividing by the standard deviation will give identical Z distributions. Ensuring the independence of a vector of CIs is much more difficult. In Table 1, the correlation coefficients for 6 CIs used for gear fault analysis: most correlation values are statically significant. Hence preprocessing is needed to whiten the CIs for (6) to be valid.

ρij	CI 1	CI 2	CI 3	CI 4	CI 5	CI 6
CI 1	1	0.84	0.79	0.66	−0.47	0.74
CI 2		1	0.46	0.27	−0.59	0.36
CI 3			1	0.96	−0.03	0.97
CI 4				1	0.11	0.98
CI 5					1	0.05
CI 6						1

Table 1.

Correlation coefficients for the six CIs used in the study.

This correlation between CIs implies that for a given function of distributions to have a threshold that operationally meets the design PFA, the CIs must be whitened (e.g., de-correlated). The Cholesky decomposition was used as a whitening function, as the Cholesky decomposition of the Hermitian is always positive definite. If the inverse correlation matrix of the CIs is Σ⁻¹, then:

$${\mathit{LL}}^{\ast }={\mathrm{\Sigma }}^{-1},\text{then}\mathit{Y}=\mathit{L}x{\mathit{CI}}^T$$E7

Where L is a lower triangular, and L* is its conjugate transpose. Y is 1 to n independent CI with unit variance (one CI representing the trivial case).

3.6 Finding the critical value

The critical value is calculated by using the inverse cumulative distribution function for the HI. In this example, it was assumed that the CIs had Rayleigh PDFs, or through a simple transformation, made to approximate Rayleigh. This assumption was made because for magnitude-based CIs, it can be shown that the CI PDF is Rayleigh. In the case of Gear or Bearing CIs (where a DC offset biases magnitudes), the bias is removed to make CIs approximate Rayleigh.

The Rayleigh PDF has some nice properties. For one, Rayleigh distribution uses a single parameter, β, defining the mean μ = β*(π/2)^0.5, and variance σ² = (2 - π/2) * β². The PDF of the Rayleigh is: x/β²exp(x/2β²). When applying these equations to the whitening process, the value for β for each CI will be: σ² = 1, andβ = σ² / (2 - π/2)^0.5 = 1.5264.

The HI derived from (6), will have a Nakagami PDF [3]. The statistics for the Nakagami are η = n, and ω = 1/(2-π/2)*2*n, where n is the number of IID CIs used in the HI calculation.

4. The cycloid gearbox

The main components of the gearbox are the input shaft, input shaft support bearing, two eccentric bearings, the cycloid gears, the pin teeth-case, the pins, output rollers, output shaft, and the output support bearing. The ratio for the gearbox is given as:

$$\text{ratio}=\left(n_{\mathit{\text{teeth}}-1}\right)\times n_{\mathit{\text{pins}}}/\left(n_{\mathit{\text{teeth}}}-n_{\mathit{\text{pins}}}\right)$$E8

The test gearbox has a dual disc with 26 teeth and 51 pins.

4.1 Equations of motion and configuration

Configuration is driven by the equations of motions for the monitoring components. This consists of describing synchronous motion analysis of the shafts and gears and the asynchronous motion of the bearings.

The simple input/output gearbox design uses three bearings on the input shaft: bearing D (the eccentric bearing) and input shaft bearing C. Two bearings support the output shaft: bearing B and bearing A.

The shaft rate determines the bearing rate fault frequencies and:

• the number of rolling elements (b),

• the roller element diameter (d),

• the bearing pitch diameters (e), and

• the bearing contact angle (α).

The fault features are related to damage accumulated on the bearing itself.

There are typically six fault features calculated for the bearing associated with bearing elements: cage, ball, inner race, outer race. For mechanical looseness, the bearing may also generate signatures associated with whip/whorl (in the base spectrum) or a 1/revolution impact (tick) in the heterodyne analysis. The bearing feature rates are calculated as:

$$\mathit{\text{cage}}=0.5\left(1-d/e\ast cos\left(\alpha \right)\right)$$E9

$$\mathit{\text{ball}}=e/d\left(1-{\left(d/e\right)}^2\ast cos{\left(\alpha \right)}^2\right)$$E10

$$\mathit{\text{innerrace}}=b/2\left(1+\left(d/e\right)\ast cos\left(\alpha \right)\right)$$E11

$$\mathit{\text{outerrace}}=b/2\left(1-\left(d/e\right)\ast cos\left(\alpha \right)\right)$$E12

Because the outer rate of the eccentric bearing is in contact with the cycloid gear and the input shaft, the total rate seen by the bearing is the input shaft + output shaft. The eccentric bearing analysis was assigned to the input shaft. To capture the change in the relative motion of the bearing to the shaft, the bearing rates were corrected by 1 + 1/51 = 1.0196. This is used to determine the correct bearing rate fault features.

Shaft and gear analyses are based on the time-synchronous average, which requires an accurate ratio from the tachometer. The tachometer is used to resample the vibration data and correct for any changes in shaft rate. Gear analysis, and more importantly, gear mesh frequencies, is a function of the shaft rate and the number of teeth on the gear. In a traditional gearbox, an input shaft with a 29.23 Hz rate with 26 teeth would have a gear mesh frequency of 29.23 × 26 = 759.96 Hz. However, in the cycloid gear, the relative motion to the shaft is driven by the eccentric gear and the output shaft. The motion of the cycloid to the ring gear has, for each revolution, one extra gear mesh. The actual gear mesh frequency is then 789.19 Hz. For this reason, gear analysis is based on 27 teeth, not 26 teeth.

Normally, the ring gear analysis would usually be associated with the number of ring gears teeth. However, there are pairs of cycloid gears (of 26 teeth), resulting in a measured mesh of 51 × 2 or 102 mesh. The TSA spectrum and raw spectrum then show frequencies at 29.23/51 × 102 = 58.46 Hz. Due to the modulation of two-disc, there are sidebands at 102 +/−51 = 51 and 153, or 29.22 and 87.69 Hz.

Example CIs used for the analysis were: Residual RMS, Residual Kurtosis, Residual Crest Factor, Energy Ratio, Energy Operator Kurtosis, Energy Operator Crest Factor, Figure of Merit 0, Side Band Lifting Factor, Side Band Analysis, Narrow Band Kurtosis, Narrow Band Crest Factor, Amplitude Modulation RMS, Amplitude Modulation Kurtosis, Frequency Modulation RMS, Frequency Modulation Kurtosis, Gear Mesh Energy (reference [4], see appendix for Matlab © source code for these analyses).

The envelope analysis is based on the demodulation of high-frequency resonance from impact s(bearing envelope analysis is given in the appendix). Poor selection of a window results in poor envelope/bearing analysis. In general, techniques such as spectral kurtosis have been used to select envelop windows, but it is not easy without fault data. Alternatively, a simple calculation of the resonance can be performed.

Lord Rayleigh [9] equated kinetic energy at the mean position of a beam to strain energy at the maximum displacement on a ring with a similar nodal configuration. This can be used to estimate the resonance of a ring, such as a bearing. When evaluated, this equation seemed to underestimate the natural frequency of the bearing when tested. Timoshenko [10] further developed the concept of Rayleigh to calculate the natural frequency of a ring. Timoshenko teaches that for a ring with uniform mass, the exact shape of the mode of vibration consists of a curve which is a sinusoid on the developed circumference of the ring.

The natural frequencies are then:

$${\omega }_s=n\left(n^2-1\right)/\sqrt{n^2+1}\sqrt{\mathit{EI}/\mu R^4}$$E13

where:

μ is the mass per unit length,

EI is the bending stiffness (Youngs Modulus × Inertia).

R is the radius.

Window selection is based on the sample rate of the sensor. The sample rate also affects the length of the TSA:

$$\mathit{\text{TSAlength}}=2^{\mathit{\text{ceil}}\left({\mathit{log}}_2\left(\mathit{\text{SampleRates}}/\mathit{\text{ShaftRate}}\right)\right)}$$E14

Given the low output shaft rate of approximately 0.57 Hz, the measured acceleration will be low. For this reason, the acquisition length must be adequately long to capture perhaps 20 revolutions. Hence, a high sample rate taken over an extended period results in a large data set, which takes more time to process and download raw data (if needed).

For this reason, the sample rate of the output shaft was taken at 2930 sps for 60 seconds. As the output shaft rate is 0.57 Hz, this collects 34 revolutions. The TSA length is then 8192. For the input shaft, which is closer to 30 Hz, only 8 s of data were taken at 23438. This allows a Nyquist frequency of 1465 Hz for the output shaft and 11719 for the input shaft. From the model response of Eq. (13), the window for output shaft analysis was taken at 300 to 1300 Hz, which covers the small resonant mode at 1000 Hz. The window was taken from 9 to 11 kHz for the input shaft, covering the modal response at 10 kHz.

5. Test stand results

We ran the gearbox unit at approximately 50% load for 45 hours using a nominal gearbox. Acquisitions were taken every 5 min. This allowed us to collect healthy gearbox data from which we could set thresholds as per (6). After the initial test run to set thresholds, the gearbox was run at 150% torque load for 1 h. The high torque load was used to initiate a propagating fault. The gearbox was then run 100% (rated torque) until failure (e.g., the gearbox seized due to a failure of the output bearing). In general, vibration data indicated multiple damaged components because of the torque overload.

For example, clearing seen in Figure 4 is the step change due to the overload at time − 175 hours, followed by an increasing trend/imbalance in the input shaft. The imbalance in the input shaft was due to the eccentric bearing being damaged during 150% loading.

Figure 4.

Input shaft health. Step change occurs from 150% overload.

Surprisingly, while reflecting the damage initiation, the output bearing only began the trend to failure toward the end of the run (Figure 5).

Figure 5.

Output bearing health vs. time.

The envelope spectrum of the failed output bearing 1 day prior to failure (Figure 4) shows mechanical looseness. The mechanical looseness is seen at the 1/Rev. at 27 Hz. Additionally, the ball rolling elements and outer race were damaged. Note that the rolling elements and outer race fault are approximately 1% below the calculated rate due to slip (Figure 6).

Figure 6.

Output bearing envelope Spectrum.

Both the cycloid gear and ring gear also showed damage propagation. The cycloid gear shows in alarm level gear mesh 50 to 20 hours before failure. This was driven predominately by gear mesh energy, which is not usually a consistent indicator of damage (Figure 7).

Figure 7.

Cycloid gear heath vs. time.

Note that from 20 hours before failure, Residual RMS, Energy Ratio, and FM0 are sensitive to the impending fault. From this, it was learned that the best five indicators for the cycloid gear health: Residual RMS, Energy Ratio, FM0, AM Kurtosis, and Gear Mesh. This suggests that during the last 10 to 20 hours of the run, the cycloid gear experienced a second failure mode detected by the more traditional gear faults.

6. Conclusion

The cycloid gearbox has unique dynamics, requiring the correct ratios for the TSA and bearing rate calculation. The bearing analysis for the Cycloid gearbox is relatively standard. The eccentric bearing rate was multiplied by a correction factor to account for the rotating frame of the outer race. During the run to failure test, the eccentric bearing roller elements and the output bearing rolling elements were faulted. Toward the end of the run to failure test, the high level of damage (resonance energy) associated with the eccentric gear raised the noise floor of the envelope analysis, which contributed to the high HI level of all the bearings in the gearbox.

The cycloid gear itself posed a challenge in that the apparent gear mesh frequency is based on the gear’s eccentric behavior and not on the gear mesh frequency alone. The observed gear mesh frequency is the gear tooth +1 vs. gear tooth, multiplied by the shaft rate.

Standard analysis techniques used on other gearboxes for shaft/gear, based on the time-synchronous average, were used. Forbearing fault detection, the envelope analysis was found to work well. For all components, a generalized health indicator was used to measure when maintenance actions were used.

A.1 Example time synchronous average

Function [tsadata, navgs,rpm] = tsaLinearInterp( data, zct, sr, ratio, ppr)

%[tsadata, navgs,rpm]=tsaLinearInterp(data,zct,sr,ratio,ppr,navgs)

%Inputs:

%  data:  time domain data in g's

%  zct:  zero cross time

%  sr:   sample rate

%  ratio: gear ratio/pulse per revolution on the tach

%  ppr:  pulse per rev

%Output:

%  tsadata: time synchronous average data

%  navgs: the number of averages in the TSA

%  rpm: mean shaft rpm

%data = data - mean(data);

ndata = length(data);

dt = ndata/sr;  %sample length

rev = 0;

i = 1+ppr;

while zct(i) < dt && i < length(zct)-1

  rev = rev + 1;

  i = i + ppr;

end

% Define the number of averages to perform

navgs = floor(rev * ratio);

trev = zct(navgs*ppr) - zct(1);

rpm = navgs/trev*60*ratio;

% Determine radix 2 number where # of points in resampled TSA

% is at sample rate just greater than fsample

N=(2^(ceil(log2(60/rpm*sr))));

% now calculate times for each rev (1/ratio teeth pass by)

% resample vibe data using zero crossing times to interpolate the vibe

yy = zeros(1,N); %data to accumulate the resampled signal once per rev

ya = yy; %ya is the resample signal once per rev

iN = 1/N; %resample N points per rev

ir = 1/(ratio/ppr); %inverse ratio - how much to advance zct

tidx = 1; %start of zct index

 while (floor(zct(tidx)*sr) == 0)

   tidx = tidx + 1;

 end

  zct1 = zct(tidx);    %start zct time;

for k = 1:navgs

  tidx = tidx + ir;    %get the zct for the shaft

  stidx = floor(tidx)-1;  %start idx for interpolation

  dx = tidx - stidx;

  yo = zct(stidx);

  dy = zct(stidx+1)-yo;

  zcti = yo + dx*dy;    %interpolated ZCT

  dtrev = zcti - zct1;   %time of 1 rev

  dtic = dtrev*iN;     %time between each sample

  zct1c = zct1;

for j = 1:N

   cidx = floor(zct1c*sr);

   yo = data(cidx); y1 = data(cidx+1);

   x1 = zct1c*sr;

   xo = floor(x1);

   dx = x1-xo;

   dy = y1-yo;

   yaj = yo + dx*dy;     %simple linear interp

   ya(j) = yaj;

   zct1c = zct1 + j*dtic; %increment to the next sample

end

  zct1 = zcti;

  yy = yy + ya;             %accumulate the tsa per reve

end

tsadata = yy/navgs;         % compute the average

A.2 Example residual signal

  function [xres] = residualSignal(x, geartooth)

%[xres] = residualSignal(x, geartooth)

%Inputs:

%  x       :input TSA signal

%geartooth :array with number of teeth on a gear

%from Vercer

x = x(:)';

n = length(x);

n2 = n/2;

nHarmonics = 3;

X = rfft(x);                %real fft - no conjugate

X(1) = 0;                  %DC is removed

X(2) = 0;                  % SO1 is removed

X(3) = 0;                  % SO2 is removed

nGears = length(geartooth);

for j = 1:nGears

    crtGear = geartooth(j);

 for i = 1:nHarmonics

      indx = 1+crtGear*i;

 if indx < n2           %projection against running over the array

          X(indx) = 0;          %gear tooth meash are removed

 end

 end

end

 xres = irfft(X);         % residual signal from the inverses real fft

A.3 Example of the narrowband, AM and FM analysis

 function [nb,am,fm] = narrowband(x, gt, BW)

%[nb,am,fm] = narrowband(x, gt, BW)

% x is the TSA

% gt is the number of gear teeth and

% BW is bandwidth, usually 25% of gt.

%Output:

%    nb: narrow band signal

%    am: amplitude modulated signal

%    fm: phase modulated signal

 X = rfft(x);

 lw = gt-BW; %calculate the band pass indexes

hi = gt+BW + 2;

 X(1:lw) = 0; %idealized filter

X(hi:end) = 0;

 nb = irfft(X);

 n = length(nb);

n2 = n/2;

X = fft(x); %take the Hilbert Transform

X(1:n2) = X(1:n2) * 2;

X(n2:end) = 0;

h = ifft(X); %Analytic Signal

 % Amplitude Modulation signal - am

am = abs(h);

 % Phase Modulation signal - fm

arg = unwrap(angle(h)); %take the argument

fm = arg - (arg(end)-arg(1))*linspace(0,1,n); %take the derivate

A.4 Example of the bearing envelope analysis

function [env,dty] = envelope(data,dt,lowf,highf)

% [env,dty] = envelope2(data,dt,nfilt,lowf,highf);

%Inputs:

% data    :data vector, time domain

% dt      :sampling time interval

% lowf    :low frequency limit of bandpass filter

% highf    :high frequency limit of bandpass filter

%Outputs:

% env :Envelope of data

% dty      : decimated sample rate

 n = length(data);

dfq = 1/dt/n;

idxLow = floor(lowf/dfq);

idxHi = ceil(highf/dfq);

D = fft(data);

idx = idxHi-idxLow + 1;

 D(1:idx) = D(idxLow:idxHi);

D(idx+1:end) = 0;

data = abs(ifft(D));

bw = highf - lowf;

r = fix(1/(bw*2*dt));

env = data(1:r:n);

 dty = dt*r;   %calculate the decimated sample rate