4 Experimental Setup
4.3 Post-Processing of Operational Measurement Data
4.3.1 Deriving the RPM-Signal
Gaining an accurate signal of the rotational speed is essential and also provides the base for any further analysis. In the current project, two optical incremental encoders are used to record the rpm signal of the shaft and of the tumble sleeve. Tape with equidistantly-spaced black and white stripes – so called zebra tape – is wrapped around the outer piece of the chuck and around the tumble sleeve. During rotation, the tape acts as a coder: An optical probe detects the passing of the black and white stripes, transferring it into a square wave signal, thereby creating a pulse train, as in Figure 4.7. In a post-processing operation, the pulse train is transferred into a rpm signal by simply dividing the rotational angle that corresponds to one stripe by the time needed to pass the optical sensor. Since the encoder creates several pulses per revolution, this method generates a signal of sufficient accuracy.
Zebra tape is straightforward to install, but has one major disadvantage that needs to be taken care of. At the location where the two endings of the tape come together, it is inevitable to create a stripe that is either slightly smaller or larger than the other stripes. When this so called butt-joint passes the optical sensor, it will introduce a discontinuity in the pulse train and thereby also in the rpm signal. This results in two errors. The obvious thing that happens, is the fact that even at constant rotational speed, the butt-joint will have a
4.3 Post-Processing of Operational Measurement Data
Figure 4.7: [Meas]1Error in the rpm signal due to the butt-joint effect and its correction different passing time than the other stripes, which is misinterpreted as a fluctuation in the rpm signal. The second error results from the assumption that, in a tape with equidistantly spaced stripes, the peripheral angle corresponding to a single stripe can be calculated by simply dividing 360 degree (respectively 6.28 rad) by the total number of stripes. If one stripe is larger than the others, this assumption is violated because that actually means that all other stripes are in fact smaller than in the calculation just mentioned, leading to an error in which the rpm is calculated as too high. If the butt-joint is smaller than the rest of the stripes, it is the other way around, meaning that the rpm value is wrongfully calculated as too slow. In the underlying thesis, the rpm signal is corrected in a post-processing operation as described in [61]. The method allows one to eliminate both errors as can be seen in Figure 4.7.
The butt-joint issue is, however, not only a nuisance. The discontinuity created by the butt-joint provides information about the absolute position of the corresponding rotor. This way, the optical encoder measures not only the rotational speed, but also allows one to calculate the angular position by dead reckoning after the passing of the butt-joint. This information is essential for determining the imbalance of the drive shaft, as will be described in Chapter 7.2.1.
1In the further course of the thesis, captions of result plots will be marked by either [Meas] to indicate measurement data, or by [Sim] to indicate simulation results.
4 Experimental Setup
4.3.2 Order Analysis
When dealing with a vibrational problem, measurement data is usually analyzed in time and in frequency domain. In rotordynamics, significant dynamic events might repeat with each revolution and at a specific angle, like in the case of an imbalance excitation. Therefore, the time signal, and more importantly, the frequency content of the signal, depends on the rotational speed of the rotor, making it difficult to assign the dynamic response of the structure to a certain excitation mechanism. A helpful instrument in this case can be a so-called (vibrational) order analysis. In rotordynamics, orders are considered harmonics of the rotational speed, so the terminus first order refers to a sinusoidal vibration with the same frequency as the rotational frequency of the corresponding rotor. When analyzing the vibrational response of a structure, an order analysis allows one to determine the amount of vibration that is related to the different orders at varying rotational speeds, thereby creating a better understanding of the structural response to the different excitation mechanisms.
A vibrational order analysis usually requires one to record the signal at constant increments of shaft angles. Alternatively, software algorithms can be used to resample data that has been recorded at uniform time intervals ∆t to data at constant angular increments [9, 11].
However, both methods require a high-resolution rotary encoder which is not available in the current measurement setup. Therefore, a different approach is used: First, the data is recorded at uniform time intervals. Then, in a post-processing operation, a tracking algo-rithm determines the instants of time at which the rotor(s) run at a certain rotational speed.
Uniform rpm-increments (decrements during run-down) are used, i.e. every 10 rpm. After that, a time interval around the relevant instant of time is cut out of the measurement data and transformed into the frequency domain. Finally, the frequency content corresponding to the order of interest is evaluated.
The applicability of the afore mentioned approach depends very much on the slope of the rpm-curve and on the length of the time interval that is cut out at each rpm-increment to be transformed in the frequency domain. If the slope is very steep and/or the time interval is long, the time blocks will overlap each other and will thereby smooth out peaks in the order-signal. The length of the time interval depends on the desired frequency resolution, which in turn depends on the maximum order of interest. In the current case, the focus of interest lies on the first orders of the three rotors, the drive shaft, the tumble sleeve and the electric engine, as will be shown in Chapter 7.1. Therefore, the necessary length of the time interval can be kept rather short, making the approach described above suitable for the current problem.