Vibration Analysis for Fault Diagnosis of Cycloidal Gearbox Using Wavelet Transform

For years, the cycloidal reducer  has proven its performance in a variety of applications, from robotics to wastewater treatment. Along with involute gearing, the cycloidal mechanism has its place in the mechanical power transmission world. It is well known for its high torque density, high integer ratio per stage, and shock load handling capabilities. Its poke-yoke assembly refrains from introducing any errors during assembly and allows the unit to operate with low maintenance. Despite these advantages, the reducer may have a variable angular speed at its output shaft, in large part due to manufacturing imperfections. A constant transmission ratio during gear meshing can be attained with involute gearing by following standardized methods established in the gearing industry . Considering the manufacturing deviations and loading conditions, one could optimize the macro and micro geometries of the involute gearing and get a fairly controlled transmission error. In the case of cycloidal gearing, however, the same may not hold. The steadiness of the ratio, or the output shaft's angular velocity, will vary with the cycloidal gearbox manufacturer. Small dimensional deviations in rotating components, such as the cycloidal disc and rollers, will influence the variations in the output shaft rotation. The torque load on the output shaft also induces the variation. Such scenarios have the potential to negatively influence traditional vibration-based fault diagnosis methods.

Vibration analysis has long been sought for condition monitoring and fault diagnosis of rotating equipment or machinery. Conventional methods utilize analyzing the time spectrum and frequency spectrum with fast Fourier transform (FFT), or in many instances both. The raw vibration data or signal recorded by the accelerometer generates the time spectrum with no additional signal processing. The time waveform is useful in identifying modulation and pulses in the signal. When frequencies of the rotary components in the gearbox are known, the same signal can be processed with FFT to obtain the frequency spectrum and then locate high-energy generating or out-of-sync components. The wide acceptance of FFT hinges on the assumption that the processed signal is stationary, meaning all the frequencies in the signal exist in all instances, and they do not vary with time. The stationary signal, hence, does not depend on the data sampling period or recording instances (Ref. 2). This assumption fails when the angular speed of the gearbox shafts fluctuates due to transmission error and load fluctuations, and the signal becomes non-stationary, or time-dependent. It is worth mentioning that the involute and cycloidal gearing differ in their primary failure modes. In involute gearing, the common failure is tooth breakage. In (epi)cycloidal gearing, as the cycloidal disc lobes (equivalent to the gear teeth) experience less shear force with a much higher contact ratio, they tend to break less. The common maintenance issues for a cycloidal gearbox would be a worn-out cycloidal disc, eccentric bearing, or damage to the ring gear housing pins. Therefore, the conventional way of fault diagnosis may not effectively capture such subtle changes. The FFT-processed signal has no time (or phase) related information in it. The nonstationary signal raises a need to know when an event has taken place on a timeline, along with its appearance on the frequency scale. Analyzing the separate graphs of time and frequency side-by-side may not be intuitive. If one tries to put time and frequency on one graph, it is well known that Heisenberg’s Uncertainty Principle would allow resolution of the signal for one of the components (time or frequency), but not both. To overcome this limitation, various time-frequency analysis methods were developed by sacrificing some resolution in both components. Among those methods are the short-time Fourier transform, Wigner-Ville Distribution, and wavelet transform (WT) (Ref. 3). Typical graphs of FFT and WT are shown in Figure 2 for comparison purposes. Each line in the FFT graph represents a frequency with a certain amplitude, whereas in the WT Scalogram, the brightness of each band signifies the strength of the frequency component in a specific frequency range seen on the Y-axis, at a certain time.

Since its introduction in 1982, the WT method has gained more interest in vibration measurements (Ref. 4). The French word wavelet means a “small wave.” In WT signal analysis, a wavelet function (or a predetermined signal) is blended with the signal to be analyzed using convolution to extract necessary information. This information about the signal could capture singular events that the Fourier Transform treats as anomalies and ignores. For instance, an event of tooth loss of the involute gear appears as a singularity in the time spectrum, but it is hard to capture in the frequency spectrum. In the frequency spectrum, it shows up in sidebands to the gear mesh frequency. The sideband, however, truly attributes to the frequency or amplitude modulation and not necessarily to the defect. In other instances, the singularities are buried in the vibration data and impossible to identify in any of the above-mentioned spectra. These events or singularities are also defined as fractal features. The term fractal was coined to denote geometries and patterns that could not be defined with Euclidean geometry (Ref. 5). The simplest way to comprehend fractals is with Taylor series.

where pi is a non-integer number quantifying the local singularity of the function f(t) at t=ti, with ah coefficient. Here, the non-integer exponent is referred to as a singularity exponent, and due to this exponent, the function approximates the singular behavior, which would be ignored when an integer exponent is used. Another important aspect of WT is that it generates coefficients that are associated with the singularities embedded in the signal. The signal could be a monofractal, i.e., having a single coefficient to represent all singularities, or multifractal. The popular applications of multifractal analysis are modeling turbulent flow of gases, stock markets, image processing, and filtering heartbeat signals to display on an electrocardiogram or ECG (Refs. 6,7,8). In this paper, WT will be used to detect fractals embedded in the vibration data to demonstrate its benefit in distinguishing a cycloidal reducer fault, which otherwise would go undetected.