Experimental Characterization of Wind Turbine Gearbox in Operation
The gearbox is one of the key subsystems in a geared wind turbine, as it must transfer the power from the low speed shaft connected to the rotor to the high speed shaft connected to the generator. As turbines become larger, more power can be generated, but consequently gearboxes with higher load capacity need to be designed. Gaining a deep knowledge into gearbox dynamics becomes of fundamental importance and more and more accurate and detailed noise and vibration measurements are demanded. When dealing with a machine in operating conditions with several rotating components and, in particular with a multi-stage transmission system, components are introduced in the signal that make the application of standard techniques such as Operational Modal Analysis (OMA) very difficult and in some cases almost impossible. For this reason, new techniques to tackle with these conditions have been investigated, such as Order Based Modal Analysis (OBMA). As suggested by its own name, this technique is a combination of Order Tracking and Operational Modal Analysis. On one hand, OMA is based upon the calculation of auto- and cross-powers and it works very well for most cases. On the other hand, OBMA is based upon the extracted orders during run-up or coast-down. During such events, the orders are sweeping through a certain frequency band which is useful for characterizing the dynamic behavior of the rotating structures.
KeywordsModal Parameter Time Domain Data Modal Assurance Criterion Experimental Modal Analysis Operational Modal Analysis
10.1 Introduction and Motivation
The current approach in industry is to qualitatively measure the vibrations generated by the dominant excitation sources as well as the overall sound power levels, so that the machine performance can be certified according to standards and/or customer specifications. However, as mentioned, by following this approach only qualitative information can be obtained and the root causes of high vibration levels or acoustic tonalities cannot be understood. Also, analyzing these data requires a lot of user-interaction, as each peak need to be independently analyzed. Experimental modal analysis techniques (Heylen et al. 2013) aim at identifying a system characteristic model based on input/output dynamic measurements. By identifying the origin of each of the phenomena of interest, it is possible to understand where corrective action should be applied increase the system NVH performances. Additionally, the identification of a system characteristic model will allow designer to compare experimental and numerical models using similar quantities. This approach will objectively quantify the agreement between the design assumptions and the behavior of the real structure.
However, during transient operations such as run ups and run downs, the rotational speed, and consequently its harmonics, is varying during the measurement. The different orders are then sweeping in a frequency band which is related to the minimum and maximum rotational speeds. As it happens during a sine sweep test, whenever one of these orders crosses a resonance, the response will increase accordingly. Orders can then be considered as representative of the system transfer (Fig. 10.1) and modal parameters identification techniques can be applied. The method was first introduced few years ago as Order based Modal Analysis (Janssens et al. 2006) and was further developed and validated recently. In this section, the concept of the method will be reviewed, with particular focus on the techniques required to extract orders that can be successfully used for modal analysis. Moreover, it will be positioned against standard Operational Modal Analysis techniques, showing the clear advantages in identifying reliable modal models of operating rotating machines and in particular for the wind turbine gearbox case.
As already discussed, the identification of operational modal model will allow not only to characterize the dynamic response of a structure in operating conditions, but will also provide means of validating, and subsequently updating, global numerical models developed with Multi Body and Finite Element tools, as demonstrated in Goris et al. (2013).
10.2 Order Tracking Techniques
Time domain sampling Fast Fourier Transform (FFT) order tracking;
Angle domain computed order tracking;
Time Variant Discrete Fourier Transform (TVDFT);
Vold-Kalman (VK) filter based order tracking.
10.2.1 Time Domain Sampling-Based Fast Fourier Transform Order Tracking
10.2.2 Angle Domain Resampling Order Tracking
The advantages of the resampling based order tracking are leakage free estimates of orders which fall on spectral lines as well as an order resolution which is constant in terms of width. On the other hand, also this method has several restrictions. Orders may only be tracked with reference to one rotating shaft and it is very difficult to distinguish among order which cross one another. Another limitation is due to the finite defined order resolution which makes very difficult the analysis of orders which do not fall on a spectral line.
10.2.3 Time Varying Discrete Fourier Transform (TVDFT)
The formulation can be extended in order to separate close or crossing orders through a secondary calculation. There can be a leakage error using the TVDFT with constant Δt sampled data because it is not guaranteed that the integer revolution values required for a constant order bandwidth analysis will fall on a Δt. If it is not the case, it will lead to a leakage error by performing the transformation over a non-integer number of revolutions. This error can be reduced by oversampling the data to finer Δt. The method retains most of the advantages of the resampling based order tracking and it can be implemented in a very efficient manner without having the computational load and complexity of the transformation from the time domain to the angle domain.
10.2.4 Vold-Kalman (VK) Filter-Based Order Tracking
Vold and Leuridan (1993) introduced an algorithm for high resolution, slew rate independent order tracking based on the concepts of Kalman filtering. The Vold-Kalman (VK) algorithm allows tracking multiple orders at the same time and it is able to decouple close and crossing orders. This method extracts the time history of the order as well as the estimate of the amplitude and the phase of the same order.
Applying Eq. (10.10) to all observed time samples will give a global system of over-determined equations for the desired waveform x(n) that can be easily solved with standard least square techniques. For the specific case of order tracking, the filtered waveform is most conveniently described in terms of amplitude and phase with respect to a reference channel such as the tachometer.
10.3 Order-Based Modal Analysis
Operational Modal Analysis (OMA) algorithms, such as Operational Polymax (Peeters et al. 2007), allow the identification of the modal parameters of a structure by taking into account only operational measurements. However, when applied to data acquired during transient phenomena, such as run ups and coast down, data need to be carefully interpreted. Input data for the modal parameter identification process are auto and cross spectra calculated from the complete time histories and assuming the excitation can be considered broad-band white noise. However, when spectra are calculated on run up data of rotating machineries, sharp peaks appears at fixed intervals. These peaks relate to the so-called “end-of-order” effect and originate from order components suddenly stopping at the maximum rpm. When using this spectra in standard OMA, they will be erroneously identified as physical poles from the algorithm; additional, they can hamper the accuracy in the estimation of modes at nearby frequencies.
Displacement orders are proportional to the squared rotation speed and, as a consequence, acceleration orders are proportional to the forth power of the same rotation speed. The main difference is that in the classical modal analysis the acceleration FRFs are proportional to the squared frequency axis.
Upper and lower residuals are complex, while in classical modal analysis they are real.
Participation factors are complex both in classical and order based modal analysis.
Methods such as Operational Polymax and Operational Polymax Plus are robust again these observations and they can be employed for estimating the modal parameters in case of rotating machineries by looking at the orders rather than at the spectra.
10.4 Dynamic Characterization of Operational Gearboxes
Shaker sine sweep during standstill and in stationary operating conditions at 1200 rpms.
Stationary operating conditions at 1200 and 800 rpms.
Run up from 200 to 1500 rpms with a speed of 5 rpms/s.
The operational measurements were all repeated under different torque loading (33 %, 66 % and 100 % of nominal torque).
As shown in Fig. 10.4 (right) an extensive grid of 250 points on both the gearboxes and the test rig was measured using tri-axial accelerometers. To measure all points, the whole test schedule was repeated 7 times roving the available sensors to cover the whole grid. To ensure the data form the different dataset could be compared, a set of 7 single axis accelerometers was always kept in the same positions. 3 optical sensors (zebra tape + laser) were respectively installed on the Low Speed Shaft and at the High Speed Shaft of each of the two gearboxes in the test rig.
With the objective of comparing the modal parameters obtained in different operating conditions, data measured during shaker excitation with the gearbox in standstill conditions have been analysed. After computing Frequency Response Functions, standard Experimental Modal Analysis methods were applied and a set of reference modal parameters obtained (Manzato et al. 2015). A similar processing was also performed on the data collected applying the sine sweep via the shaker during stationary operations. However, although some of the modes could still be identified, it was concluded that the shaker were not powerful enough to sufficiently excite the structure and ensure a reliable modal estimation.
All vertical lines in the time-frequency diagram represent highly excited harmonics of the fundamental rotational speed and they can all be related to rotational speed of the shafts of the different stages as well as to the gear meshing frequencies. As the response is dominated by narrow and closely spaced harmonic components, standard Operational Modal Analysis cannot be applied. As a consequence, to understand the response, only Operational Deflection Shapes can be analysed, but, as mentioned, it will be impossible to understand whether the high response at the receiver is due to the system or the source (Fig. 10.1). Of course one could compare the harmonic frequency with the natural frequencies identified by applying shaker excitation in stationary conditions. However, as the boundary conditions between standstill and operational conditions are significantly different, erroneous conclusion might be derived.
10.4.1 Operational Modal Analysis
A comparison of the two graphs reveals that some of the peaks in the spectrum (and in particular the sharpest ones) originate from order components suddenly stopping at the maximum rpms. Cursors were added to the pictures at frequencies that were identified as poles of the gearbox by classical OMA. Moreover, as the two gearboxes on the test rig were slightly different prototypes, they rotate at slightly different speeds, resulting on a doubling of these “end-of-order” related poles. This is the main weakness of this method: not only the real poles are identified, but also the so-called “end-of-order” related poles which are physically not present in the system and are purely processing artifact. The four identified frequencies correspond to some of the main order components ending at that frequency. The estimated modal model is consequently not correct because it considers them as poles of the system. While these poles could be ignored a-posteriori, in cases were many orders are present (such as this one) they can also affect the estimation of close modes, thus reducing the confidence in the identified model.
10.4.2 Order-Based Modal Analysis
The displayed results focus on the results obtained for the 100 % load cases using the three different discussed methods. Only few modes are found to be similar between OMA and OBMA and this is related to the high number of end-of-order poles identified. By comparing the results obtained using the two order tracking techniques, a discrete correlation is found for the modes in the mid-high frequency range. As expected from the results displayed also in Fig. 10.12, at lower frequencies the predicted orders differ significantly, thus a difference in the extracted modes is also expected.
Critical assessment of the different analyzed order tracking techniques
Order tracking technique
Advantages & drawbacks
Angle Domain Order Tracking
Suitable for real-time processing
Equidistant order lines
Huge number of parameters to be set
Great sensitivity of the resulting order on the settings
Very noisy phase
Modal parameter estimation is not reliable
Time Variant Discrete Fourier Transform
1 parameter to be set (number of rotation per order line)
Post-calculation to separate close and crossing orders
Non equidistant order lines
Low resolution at low frequency
Phase smoothness depends strongly on the number of rotation per order line
Difficult to fit higher frequency
Vold-Kalman Filter Order Tracking
2 parameters to be set (filter bandwidth and number of poles in the filter)
Very high order resolution (number of lines equal to the number of acquired samples)
Very good quality of the fit
Beat free extraction of close and crossing orders
Tracking capabilities are independent of the slew rate
Non equidistant order lines
the input parameters the user has to define.
the computational effort required for the estimation.
The performed comparison also includes standard Angle-domain resampling order tracking. Although the results are not discussed here, the method is one of those most employed in industry and implemented in commercial solutions.
On the other hand, damping values are more scattered, which is however expected with operational data. However, when comparing mode shapes using the Modal Assurance criterion, very poor correlation is observed although the shapes graphically compare. These results can be however be expected by taking into account the number of measurement points, the measurement and processing noise and uncertainties as well as the small inconsistencies between the different runs where the setup was changed.
Thus, at least when these conditions apply, it is envisaged to validate the analysis only qualitatively by comparing the mode shapes rather than performing a quantitative analysis based on the MAC.
When analysing the dynamic response of rotating machines in operating conditions, the techniques developed to perform operational modal analysis don’t hold. After clearly demonstrating that these methods don’t provide reliable results both during stationary and transient condition, a novel method is introduced, which is able to extract an operational modal model from run-up or run-down experiments.
Order-Based Modal Analysis (OBMA) relies on extracting high quality order from the throughput time data that can be then used for modal parameter identification using classical operational modal analysis algorithm. The advantages over classical Operational Deflection Shape (ODS) that also rely on order extraction are the possibility to clearly identify which part of the response can be associate to the system dynamics, which to the input forces and which to a combination of both. Moreover, damping information for each of the modes can be extracted and closely separated modes, as soon as the order resolution allows it, can be distinguished.
However, the use of an accurate phase reference signal is mandatory to obtain a reliable modal model and because of this an accurate tacho measurement is even more critical than in standard ODS analysis. Also, by using more advanced order tracking techniques, although sometimes computationally demanding, can lead to significant improvement of the results.
OBMA has demonstrated to be suitable for operational processing and to give very good results. The method was significantly improved and the two new methods (TVDFT and VK) showed much improved results compared to older implementations. The extra effort compared to calculate the ODS are limited, although the Vold-Kalman order tracking is computationally more demanding and the quality of the tacho signal should be high enough to allow a reliable processing of the data. The results of the proposed approach were used in Vanhollebeke et al. (2015) to validate the operational response of the analysed gearbox predicted using a flexible multibody model.
The authors would link to kindly acknowledge Frederik Vanhollebeke, Sonja Goris and Joris Peeters of ZF Wind Power for the possibility of using the data acquired on the gearbox test rig to carry out this research. Additionally, the authors kindly acknowledge the Institute for Promotion of Innovation through Science and Technology in Flanders, Belgium (IWT Vlandereen) for the O&O grant ALARM in which the aforementioned experimental campaign was performed. The ALARM project was furthermore supported by an Eureka label in the framework of international co-operations.
- Blough J (1998) Improving the analysis of operating data on rotating automotive components. Dissertation, University of CincinnatiGoogle Scholar
- Blough J, Brown DL, Vold H (1997) The time variant discrete fourier transform as an order tracking method. In: Publications: Technical Paper Number 972006. Society of Automotive Engineers. Available via SAE. http://papers.sae.org/972006/. Accessed 08 Apr 2016
- Goris S, Vanhollebeke F, Ribbentrop A et al (2013) A validated virtual prototyping approach for avoiding wind turbine tonalities. Paper presented at the 5th international conference on wind turbine noise, INCE-Europe, Denver, 28–30 August 2013Google Scholar
- Heylen W, Lammens S, Sas P (2013) Modal analysis theory and testing. KU Leuven, BelgiumGoogle Scholar
- Janssens K, Kollar Z, Peeters B et al (2006) Order-based resonance identification using operational Polymax. In: Abstracts of the IMAC-XXIV: conference and exposition on structural dynamics – looking forwards: technologies for IMAC, Society for Experimental Mechanics, St. Louis, 30 January–02 February 2006Google Scholar
- Manzato S, White JR, LeBlanc B et al (2013) Advanced identification techniques for operational wind turbine data. In: Allemang R, De Clerck J, Niezrecki C et al (eds) Topics in modal analysis volume 7: proceedings of the 31st IMAC, a conference on structural dynamics, 2013. Conference Proceedings of the Society for Experimental Mechanics Series, Springer, New YorkGoogle Scholar
- Manzato S, Di Lorenzo E, Medici A et al (2015) Order-based modal analysis versus standard techniques to extract modal parameters of operational wind turbine gearboxes. In: Mains M (ed) Topics in modal analysis volume 10: proceedings of the 33rd IMAC, a conference and exposition on structural dynamics, 2015. Conference Proceedings of the Society for Experimental Mechanics Series, Springer, New YorkGoogle Scholar
- Vold H, Leuridan J (1993) High resolution order tracking at extreme slew rates, using Kalman tracking filters. In: Publications: Technical Paper Number 931288. Society of Automotive Engineers. Available via SAE. http://papers.sae.org/931288/. Accessed 08 Apr 2016
Open Access This chapter is distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the work’s Creative Commons license, unless indicated otherwise in the credit line; if such material is not included in the work’s Creative Commons license and the respective action is not permitted by statutory regulation, users will need to obtain permission from the license holder to duplicate, adapt or reproduce the material.