In this chapter the identifiability of the torsional natural frequency of a VFD system using motor’s torque, speed and current signals were tested experimentally. The goal was to find out if the signals that are obtainable from a commercial frequency converter could be used to detect the first torsional natural frequency without additional sensors. The signals were analysed in frequency domain with Welch’s method for obtaining the PSD or frequency response estimates between different signals. The plant was excited with a known PRBS signal. First, the location of the actual reso-nance frequency was identified from the estimated frequency response be-tween the torque reference of the VFD and the measured rotational speed.
The signals were analyzed first with the encoder feedback to find out if the resonance frequency could be identified with the studied methods. The speed feedback was then changed to the drive’s speed estimator to test the applicability to the sensorless identification. From the studied signals, the resonance frequency could be detected from the torque and the speed. The current signal would have needed more advanced signal processing and more thorough analysis. Due to limited resources the current signal
analy-4.4 Discussion 41
Table 4.2: Estimated resonances from the torque and speed signals with and without the encoder for two couplings with different hardness.
Encoder Estimator
Load Coupling Reference resonance Torque Speed Torque Speed
0 % 42 Hz 41 Hz 41 Hz 40 Hz 41 Hz
25 % 98 Shore A 69 Hz 68 Hz 67 Hz 51 Hz 66 Hz
50 % 110 Hz 106 Hz 111 Hz 92 Hz 110 Hz
0 % 35 Hz 33 Hz 33 Hz 32 Hz 36 Hz
25 % 92 Shore A 64 Hz 65 Hz 63 Hz 57 Hz 64 Hz
50 % 88 Hz 91 Hz 88 Hz 82 Hz 91 Hz
sis were carried out only to some extent. In addition, the experiments show that the resonance frequency could be identified from the speed signal with quite minimal signal processing so the more demanding analysis was con-sidered unjustified.
A summary of the estimated resonance frequencies are shown in Table 4.2.
As can be seen the drive’s speed estimator shows the most promising re-sults. With the sensorless identification from the speed signal, the error of the estimated resonance frequencies are < 5 % for the tested cases. An important remark is that the reference resonance is also just an estimation.
However, the identified frequencies with the encoder are showing similar results so it is considered to be reasonably accurate.
As mentioned before, the characteristics of the couplings are usually dy-namic and non-linear. This can be seen from the results where the reso-nance rises as the load gets higher. In this thesis, the non-linear effects are emphasized with the smaller loads since the PRBS amplitude is higher than the static load torque. Due to this the torque of the motor is transitively neg-ative which increases the effects of the backlash. With this in mind, when considering the application for the proposed methods, it should be noted that the identified resonance for a no-load case might not be applicable at the actual operating point of the drive. However, in some cases it’s useful to identify the resonance characteristics for varying loads. Another remark is that the proposed methods are carried out as a one time ID-experiment.
Therefore the applicability for a continuous monitoring purposes should be studied in more detail. It is clear that the amplitude of the excitation
signal in such case should be significantly lower than the 20 % of the rated torque used in this thesis. Likewise, for the ID-test application the ampli-tude should be minimized.
43
5 Direct on line application
In this chapter a simulation model is presented to simulate a direct con-nected induction motor. The purpose of the simulations is to find out if the resonance frequency of the mechanical system can be observed from the stator current of the motor in the direct on line (DOL) applications. The simulated cases are also experimentally tested in LUT laboratory.
5.1 Simulation model and results
The simulation model is build using MATLAB Simulink®. The induction motor is direct connected to a 400 V 50 Hz grid. The simulation model is shown in Fig. 5.1. The mechanical model of the two-mass system used in the simulation is shown in Fig. 5.2. The rotor friction coefficients,bm and bl, are included in the model as a viscous damping for both the motor and the load, respectively. The electrical parameters for a same size motor as in the laboratory are used for the induction machine model. The electrical parameters of the induction motor and the parameters of the mechanical model are given in Table 5.1.
Table 5.1: The electrical parameters of a 7.5 kW 400 V 50 Hz 1440 rpm induction motor and the initial mechanical parameters that were used in the simulation.
Electrical Mechanical
Parameter Value Parameter Value
Stator resistanceRs 0.7384Ω Motor inertiaJm 0.034 kgm2 Stator leakage inductanceLsσ 3.045 mH Motor frictionbm 0.005 Nm·s/rad
Rotor resistanceR0r 0.7402Ω Load inertiaJl 0.103 kgm2 Rotor leakage inductanceL0sσ 3.045 mH Load frictionbl 0.005 Nm·s/rad Magnetizing inductanceLmag 124.1 mH Torsional stiffnesskt 7.8·103Nm/rad
Pole pairsp 2 Torsional dampingct 3 Nm·s/rad
As shown in Fig. 5.1, the PRBS is now superposed to the load torque. This due to experimental system where the load side drive is used to generate excitation to the system in order to identify the flexible behavior by mea-suring the grid side machine currents. The frequency response of the me-chanical model can be estimated from the load side with the Welch method
N Stator current𝐼s,a
Stator current𝐼s,a
Load torque
Stator current𝐼s,b
Stator current𝐼s,b
Stator current𝐼s,c Stator current𝐼s,c Grid
Figure 5.1: A representation of the Simulink®model. The mechanical model is shown in Fig. 5.2 in more detail.
1
Figure 5.2: The mechanical model of the two-inertia system, where theτ is the electro-magnetic torque,b is the friction coefficient,ωis the angular velocity. The subscripts m and l are representing the motor and the load, respectively.
5.1 Simulation model and results 45
Resonance = 90 Hz
Offset = 50 Hz Frequency 50 Hz
Figure 5.3: The frequency response of the mechanical model estimated with the Welch method (left y-axis) and the Welch PSD of the phase A current of the induction motor (right y-axis) from the simulation of varying torsional damping. The used PRBS ampli-tude is 20 % of the rated torque.
fromτltoωl. The load torque is set to 50 % of the rated torque. The PRBS signal used in the simulation is the same as the one used in Chapter 4 with amplitude of 20 % of the rated torque. The simulation is carried on for three different torsional damping values. The frequency response of the mechanical system estimated with the Welch method and the Welch PSD of the Phase A current are shown in Fig. 5.3. The resonance of 90 Hz is observed from frequency response where the ct = 1 and is shown with the dashed line. The frequencies of the frequency response and the resonance are offset by 50 Hz, which is the supply frequency of the motor. The reso-nance can be observed from the current PSD atfin+fres. It can be seen that with lower damping the resonance can be seen more clearly from both, the current PSD and the frequency response. Fig. 5.4 shows that increasing the PRBS amplitude does not improve the observability of the resonance from the current PSD. However, the distortion of the frequency response decreases. One important remark should be emphasized; the studied sim-ulation is ideal as there is no noise sources or non-idealities included that have an influence to the results.
101 102 103
(Frequency response) Magnitude [dB]
-100
(Phase A current PSD) Magnitude [dB]
PRBS amplitude A = 20 % PRBS amplitude A = 50 % PRBS amplitude A = 80 %
Resonance = 90 Hz
Offset = 50 Hz Frequency 50 Hz
Figure 5.4: The frequency response of the mechanical model estimated with the Welch method (left y-axis) and the Welch PSD of the phase A current of the induction motor (right y-axis) from the simulation of varying PRBS amplitude. The used PRBS amplitudes are 20 %, 50 % and 80 % of the rated torque and the torsional dampingctis 3 Nm·s/rad.
5.2 Experimental results
The experimental setup presented in Chapter 4 is modified so that the mo-tor is supplied directly from the 400 V 50 Hz grid. As in the simulations, the PRBS is now superposed to the load torque. The load torque and the PRBS amplitude are set to 50 % and 20 % of the rated torque, respec-tively. The Welch PSD of the measured phase A current and the frequency response estimated with the Welch method are shown in Fig. 5.5. The measurements for the frequency response were carried on with the setup presented in Chapter 4 where the load was 50 % of the rated torque and the used coupling was the KTR ROTEX® 92 Shore A spider. The reso-nance of 88 Hz is observed from the frequency response and is shown with dashed line in Fig. 5.5. The frequency response and the resonance are offset by the supply frequency of 50 Hz as was in the simulations. It can be seen that the shape of the current PSD is similar to the simulated case.
However, the damping seems to be high enough that the resonance can’t be observed from the current in this case even thought the excitation am-plitude is selected according to typical guidelines for similar mechanical systems.
5.2 Experimental results 47
101 102 103
Frequency [Hz]
-60 -40 -20 0 20 40 60
Magnitude [dB]
Phase A current PSD Frequency response estimate Resonance
Frequency 50 Hz
Offset = 50 Hz
Resonance = 88 Hz
Figure 5.5: The Welch PSD of the phase A current signal from the DOL mechanical ID-experiment and the frequency response estimate obtained with the setup presented in Chapter 4.
6 Conclusions and summary
Vibration analysis is an important phase of a mechanical design. The main outcome of the vibration analysis is to secure safe operation on the op-eration range of the motor. However, smaller machines, and also larger machines to some extent, continue to produce problems related to torsional vibrations since they are hard to detect without external measurement tools.
Typically, the torsional analysis after the delivery requires expensive and time consuming measurements. During commissioning of a VFD appli-cation a ”mechanical identifiappli-cation” -feature could be beneficial to detect critical speeds that should be avoided.
In this thesis, the identifiability of the first torsional natural frequency from selected signals of a VFD system is studied. The research is carried out with simulations and laboratory experiments. The mechanical system used in the simulations and experiments is simplified to a two-mass system rep-resenting a motor and a load that are connected with a coupling. The stud-ied signals were torque, rotational speed and stator current of the motor from which the first two are obtained from the VFD and the latter is mea-sured with an external power analyzer in the experiments. A PRBS signal is used as an excitation signal in all the experiments and simulations. The signals are analyzed in frequency domain using Welch’s power spectrum density estimate. The frequency responses from the system are estimated with Welch method. The experiment was first carried out as a closed-loop system. In addition, the stator current was also studied in DOL applica-tion where the motor is supplied directly from the 400 V, 50 Hz grid. The experiment is repeated for three different loads and for two different cou-plings.
As the used couplings allow backlash the torsional stiffness becomes un-defined as the applied torque approaches zero. The effects of the backlash and the non-linearity of the coupling are emphasized with smaller loads due to the ratio of the PRBS amplitude and the static load torque which causes the motor torque to be transitively negative. The system’s mechan-ical frequency response was first estimated with the Welch method from the closed-loop system using the speed feedback from the encoder. The
49
resonance frequency obtained from the frequency response is used as a reference for the other identification methods. The experiments show pos-itive results for sensorless identification of the resonance. From the stud-ied signals, the resonance frequency was clearly visible in the PSD of the VFD’s estimated rotational speed. From the frequency response from the excitation signal to the torque, the resonance was also observable to some extent. The resonance could not be identified from the stator current in the closed-loop case. The simulations of the DOL case show that with higher torsional damping the resonance becomes harder to observe from the cur-rent signal. The DOL experiments show similar results to the simulations.
However, the damping of the coupling seems to be high enough that the resonance was not clearly visible.
As conclusion, the proposed method to identify the torsional natural fre-quency from the speed estimate could be developed more towards the in-tegrated feature of a VFD. As the test lasts for 10 seconds, the method provides a solution for a fast estimation of the mechanical resonance. The applicability for condition monitoring should be investigated more since a constant excitation would most likely disturb the actual process’ perfor-mance.
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