This paper examines the spectrum and cepstrum content of vibration signals taken from a helicopter gearbox with two different configurations (3 and 4 planets). It presents a signal processing algorithm to separate synchronous and nonsynchronous components for complete shafts’ harmonic extraction and removal. The spectrum and cepstrum of the vibration signal for two configurations are firstly analyzed and discussed. The effect of changing the number of planets on the fundamental gear mesh frequency (epicyclic mesh frequency) and its sidebands is discussed. The paper explains the differences between the two configurations and discusses, in particular, the asymmetry of the modulation sidebands about the epicyclic mesh frequency in the 4 planets arrangement. Finally a separation algorithm, which is based on resampling the order-tracked signal to have an integer number of samples per revolution for a specific shaft, is proposed for a complete removal of the shafts harmonics. The results obtained from the presented separation algorithms are compared to other separation schemes such as discrete random separation (DRS) and time synchronous averaging (TSA) with clear improvements and better results.

Vibration signals originating from a helicopter transmission gearbox represent a rich source for monitoring its health. Many failures that occur in rotating components such as gears and bearings often show their signature in the vibration signal and can be well detected at early stages. Monitoring these vibrations often requires an extensive interpretation by a trained diagnostician, due to the complexity of such systems [

Signals are mixtures of different sources. For successful handling and interpretation of signals, analysts often need to separate these different sources and process them separately. One of the most successful ways of interpreting signals is the use of Fast Fourier Transformation (FFT), which transforms the signal from the time domain into the frequency domain by using sines and cosines as base functions for the signal decomposition. FFT requires the transformed signal to be stationary; that is, it has some statistical parameters which do not change with time. For nonstationary signals (have time dependent statistics), the use of time-frequency presentation such as the spectrogram (short time-frequency analysis), the wavelets, the winger vile transform, and so forth is commonly used. Stationary signals are mainly composed of deterministic (discrete) components and random components. Random components contain all nonstationary signals in addition to any nondeterministic part. Deterministic components are those which can be expressed as a series of discrete sinusoidal signals (thus they are predictable and periodic). Deterministic component can be interchangeably referred to as discrete signals. They generally fall into two main categories:

Periodic (cyclic): they are composed of sinusoids whose frequencies are all integer multiples of some fundamental frequency like the shaft speed in rotating machinery; The multiples of the fundamental frequencies are known as harmonics, with the fundamental being the first harmonic; periodic components can also manifest themselves as sidebands around a carrier frequency in the case of a modulated signal (e.g., a gearbox signal where the shaft speed (low frequency) modulates the gear mesh frequency (high frequency))

Quasi-periodic: they have at least two frequency components that are not rationally related and thus never repeated themselves exactly

An example of amplitude modulation and sidebands.

Carrier signal

Modulating signal

Modulated signal in time

Modulated signal in frequency domain

For example, gear mesh frequency is a fixed multiple of rotating speed and is displayed as a distinct peak in the spectrum. If there are defects associated with the gearing (physically related) such as gear eccentricity or excessive wear that generate a synchronous force variation, a disturbance will accompany the gear mesh frequency. The high frequency gear mesh is the carrier. The low frequency gear defects will appear as “sidebands” on either side of the gear mesh peak. The spectral distance from the peak to the side band peak is equal to frequency modulation or the rotating speed of the defective gear. Multiple gear defects are reflected as an extra set of sidebands.

Helicopter transmission gearboxes have been investigated in detail to understand the patterns of baseline frequencies and failure indicators that can be detected by monitoring vibrations [

In general, interpreting the vibration signal transmitted through those gearboxes often requires more than the traditional inspection of the time signal and/or its frequency content. This is due mainly to the existence of a large number of rotating components, all of which contribute and mix in different ways [

Another important aspect which is presented in this paper is the separation of the gear mesh frequencies and shafts harmonics form the spectrum, using the information from only one tachometer, to enable further processing of the signal to detect gears and bearings faults. The results obtained from this analysis are compared to time synchronous averaging (TSA) processing and discrete random separation (DRS) [

When using TSA, the signal is resampled into the angular domain (rather than the temporal domain) to have the same number of samples for each shaft revolution. This removes any speed fluctuation from the signal. The shaft harmonics, now called orders, become locked to the shaft rotation and appear as discrete components in the frequency domain (with no order tracking, the higher harmonics usually smear and become broad). The ensemble average for all the rotations is calculated to give the so-called “synchronous average,” which represents one shaft rotation and captures the deterministic part of the signal. If the synchronous average is subtracted from the signal, the result will be a residual, which contains noise, nonstationary signals, and any nondeterministic signal. TSA requires the presence of a tachometer signal for the order tracking stage.

DRS works well in the cases of slight (small) speed variation but requires a number of parameters to select and leaves notches in the signal at the locations of the discrete components. A linear transfer function (similar to the H1 transfer function estimate in modal analysis) is generated between the signal and a delayed version of it (using FFT). This gives a value of 1 for the discrete frequency bins and 0 elsewhere. This filter (amplitude of the transfer function) is then used to filter the signal and separate out the discrete components. The filtration is all based on the efficient FFT methods and thus the processing is computationally fast. The two main parameters required are the filter length and the amount of delay to capture the deterministic signals. There are a number of recommendations and a visual examination is usually required to set up these values to give the required separation. This method blindly removes all the discrete components from the whole frequency bandwidth and leaves notches in the spectrum at the location of these frequencies.

This paper is organized as follows. In Section

Although it is not necessary to be an expert in gear system design, it is essential to fully understand the power flow through the gears, the rate of rotation of each component as a function of input shaft speed, the number of teeth on each gear, and the placement/identification of bearings. Due to the variety of components, gear system frequencies typically populate a wide portion of the spectrum from less than shaft speed (tooth repeat frequencies) to multiples of gear mesh frequency.

Gear mesh frequency is defined as the number of teeth on the gear multiplied by the rotational speed of the shaft. Gear mesh is the key parameter to monitor, as any anomaly in the transfer of power through the gears will be reflected at this frequency.

Figure

Bell 206 gears, speed, and data acquisition parameters.

Parameter | Value | Reduction ratios |
---|---|---|

Shaft input speed (Hz) | 100 | |

Number of stages | 2 | |

Number of teeth of the bevel pinion | 19 | First-stage reduction ratio: |

Number of teeth of the bevel gear | 71 | |

Number of teeth of the sun gear | 27 | Second-stage reduction ratio: |

Number of teeth of the planet gear | 35 | |

Number of teeth of the ring gear | 99 | |

Overall reduction ratio: | ||

Carrier (arm) output speed (Hz) | 5.73 | |

Sampling frequency (Hz) | 51200 | |

Length of records | 30 seconds |

Bell 206 transmission with 3 planets.

Shaft speeds and gear mesh frequencies and their calculations for the 3- and 4-planet arrangements are presented in Table

Shaft and gear mesh frequencies.

Frequency of interest (Hz) | Three planets | Four planets | Relationships and calculations | |
---|---|---|---|---|

Stage | Input shaft frequency | 100.00 | 100.00 | |

Pinion/bevel mesh frequency | 1900.00 | 1900.00 | Number of teeth of the bevel pinion times input shaft frequency | |

Bevel gear shaft speed | 26.76 | 26.76 | Input shaft frequency/1st-stage reduction ratio | |

| ||||

Stage | Carrier frequency (output arm speed) | 5.73 | 5.73 | Input shaft frequency/overall reduction ratio |

Epicyclic mesh frequency | 567.71 | 567.71 | Number of teeth of the ring gear times carrier frequency | |

Planet pass frequency | 17.20 | 22.94 | Number of planets times the carrier frequency |

High quality gears are designed to transfer power by a combined sliding and rolling motion from one gear to the next as smoothly, quietly, and efficiently as possible. This statement means that, starting from the proper metal, the gear teeth are cut to precise dimensions about the geometric center. The tooth surfaces are ground smooth and any imperfections are removed. Each gear is properly fitted to a straight shaft to eliminate eccentricity. The shafts are spaced to optimize the tooth engagement. Any errors in gear manufacture or assembly and/or deterioration will result in a disturbance at gear mesh frequency with adjacent sidebands reflecting once per revolution modulation caused by pitch abnormalities in one or both gears. Since perfection is an asymptotic endeavor, gear systems always display some gear mesh activity. Thus, the presence of the frequency is acceptable and should be apparent. The amplitude increases as the load on the gearing increases. Hence, unless the load is steady, monitoring the amplitude can be a bit tricky. However, an unexplained and significant increase in amplitude at gear mesh frequency is cause of concern.

A large increase in sideband amplitude suggests that something is changing in the geometry and is also cause of concern. The relative amplitude of sideband to gear mesh peak is a good parameter to watch. Scrutinize the waveform for periodic impacts that relate to rotational speed of the gears. Since bearing failure will permit unexpected shaft displacement thereby upsetting gear engagement, always be on the alert for bad bearings. Often, bearing failure precedes gear damage. Increased impacting from deterioration in the transfer of power may excite gear natural frequencies.

A comparison between the three planets and four planets frequency contents from the front accelerometer is shown in Figure

(a) Power spectrum density (PSD) of 3-planet arrangement. (b) Power spectrum density of 4-planet arrangement.

Zoom in PSD showing the epicyclic mesh frequency and its side bands: (a) 3 planets and (b) 4 planets (epicyclic mesh suppressed).

The number of planets in the system affects the planet pass frequency (number of planets times the carrier speed) and may also cause an asymmetry of the modulation sidebands about the epicyclic mesh frequency (carrier speed times the number of teeth on the ring gear). In some cases, this may also cause complete suppression of the component at the epicyclic mesh frequency [

Bell-206-B4 gearbox modulation sidebands (assuming planets are evenly spaced).

Sidebands | | |
---|---|---|

−5 | 0 | (99 − 5)/4 (NI) |

−4 | 0 | (99 − 4)/4 (NI) |

− | | |

−2 | 0 | (99 − 2)/4 (NI) |

−1 (561.97 Hz) | 0 | (99 − 1)/4 (NI) |

0 (567.71 Hz) | 0 | (99 − 0)/4 (NI) |

| | |

2 | 0 | (99 + 2)/4 (I) |

3 | 0 | (99 + 3)/4 (NI) |

4 | 0 | (99 + 4)/4 (NI) |

| | |

0: suppressed.

To see if this agrees with the actual frequency content of the signal, a zoom-in around the fundamental epicyclic for the 3- and 4-planet arrangements is plotted and presented in Figure

For the four-planet arrangements, there is suppression of the epicyclic mesh frequency. The suppression of the epicyclic mesh frequency is quite obvious; however, the sidebands with the highest dB values (lower and upper sidebands, with a maximum dB at the 1st lower sideband at 562 Hz) do not agree with the predictions of Table

A further useful summary of these discussed observations can be seen clearly through inspecting the real cepstrum of the two signals as shown in Figure

Real cepstrum for the three- and four-planet arrangements: (a) 3 planets and (b) 4 planets.

In order to extract and then remove the harmonics related to each shaft (input, intermediate, and output shafts), three tachometers are required if the shafts are independent, for example, aero engines. In such cases, the angular resampling process should be repeated to allow the order tracking of the speeds of the three shafts, and their harmonics could be removed by subtracting the synchronous average after each resampling step. Order tracking (angular resampling) involves resampling the signal at equal intervals of shaft rotation rather than equal time intervals. This removes any speed fluctuation so that the harmonics of the shaft are genuinely discrete frequencies. This enables their removal by performing synchronous averaging of the order-tracked signal and subtracting it from the latter. Order tracking can be performed by phase demodulating a tacho or shaft encoder signal and using the mapping between the shaft angle and time to make interpolations in either direction. The process includes the use of cubic spline interpolation of the vibration signal to calculate the values at the required sample points, based on the tachometer signal.

In gearboxes, only one speed reference signal is provided, which is usually sufficient for removing the harmonics of the three related shafts.

A synchronous averaging separation algorithm [

This algorithm works by resampling the order-tracked signal to obtain an integer number of samples per revolution for a specific shaft. This enables the removal of the shaft harmonics without much disruption of the vibration signal. As there are only three main shafts in the gearbox arrangement (input, intermediate, and output shafts), this will provide a quick yet efficient way of removing all the harmonics of the three shafts (complete removal of shaft/gear related components). The removal of the harmonics of a specific shaft can be achieved by one of two methods. The first is by finding the synchronous average and subtracting it (repeated periodically) from the signal. The second is by truncating the signal to an integer number of revolutions (preferably a power of 2) as described next and setting the lines corresponding to the harmonics of that shaft (after FFT analysis) to the mean value of the adjacent frequencies. To avoid treating the negative frequency components, it is recommended to set them to zero after the FFT step and double the positive frequency components and then take the real part of the resulting analytic signal in the time domain. Both methods give the same results but with less processing time when using the synchronous averaging method.

The steps included in the algorithm are listed below and presented schematically in Figure

Order track the raw signal based on the input shaft tacho. Ensure that the number of samples per revolution of the input shaft (

Find an integer number of periods (

Truncate the signal to (Nfft) samples (preferably a power of 2), which is equivalent to

Take the Fast Fourier Transform (FFT) of the truncated signal.

Remove synchronous frequencies related to the input shaft by setting the frequency lines of

Perform an Inverse Fast Fourier Transform (IFFT) on the resulting frequency content obtained in step (5) back to the time domain. If the negative frequencies were set to zero, the real part of the signal should be obtained.

Resample so that there is an integer number of samples (power of 2) for each revolution of the intermediate shaft (this can be achieved by working out the gear ratio and using that to resample the signal to give an integer number of samples per revolution of the intermediate shaft).

Repeat steps (4) to (6) to remove the harmonics related to the intermediate shaft.

Resample the obtained signal so that there are an integer number of samples (power of 2) for each revolution of the output shaft (use gear ratio to resample the signal so that an integer number of samples per revolution of the output shaft are achieved).

Repeat steps (4) to (6) to remove the harmonics related to the intermediate shaft.

Signal separation algorithm.

Removing the harmonics of the input shaft speed: dark: before the removal; light: after the removal.

Zooming-in around the pinion mesh frequency: dark: before the removal; light: after the removal.

Note the circled parts of Figures

Power spectrum showing the effectiveness of the discrete removal algorithm.

Zoom-in (0–2000 Hz): light: signal before processing; dark: signals obtained through different approaches (synchronous averaging subtraction and harmonic removals in the frequency domain).

The end result of steps

(a) Raw signal [order tracked]. (b) Harmonics of input shaft removed. (c) Harmonics of intermediate shaft removed. (d) Harmonics of output shaft removed.

Note the disappearance of the discrete components from the spectrum in the processed signal as illustrated in Figure

Power spectrum of (a) order-tracked signal and (b) signal processed using the new algorithm.

Figure

Power spectrum of (a) order-tracked signal, (b) residual signal obtained by setting the shaft’ related harmonics to the mean of adjacent (noise) lines, (c) residual signal obtained by subtracting the synchronous averages after resampling, and (d) DRS residual (removing discrete components using DRS).

This paper has compared the effect of changing the number of planets of Bell 206 helicopter planetary gearbox on the modulation sidebands around the epicyclic mesh frequency. The planet pass frequency strongly modulates the epicyclic mesh in the 3-planet arrangement. In the case of the 4 planets, the epicyclic mesh frequency and its harmonics are suppressed (other sidebands appear strongly). McFadden and Smith’s model [

The author declares that there is no conflict of interests regarding the publication of this paper.

Data used in this paper was provided by Australian Defence Science and Technology Group (DSTG).