# Research of vibrations in induction machines during transient processes using a ferraris sensor/Indukciniu masinu vibraciju pereinamuju procesu metu tyrimas ferrario jutikliu.

1. Introduction

For root cause analysis it is necessary to measure the real input to a mechanical drive system. Induction machines are known for causing torsional vibration problems during start up, reversal or other transient phenomena . As to the fact that the electrical torque of an induction machine is not able to be measured by the electrical units of current and voltage, the only possibility is to measure the shaft torque and the angular acceleration [2, 3].

According to Newton's second law of motion "the acceleration of an object is proportional to the forces applied", the equation for the flywheel mass [[THETA].sub.1] (Fig. 1) is as follows:

[[THETA].sub.1][[??].sub.1](t)= [M.sub.e](t) - [M.sub.s](t), (1)

with [M.sub.e](t) the electrical torque of the induction machine caused by the electromagnetic field in the air gap. [M.sub.s](t) is the shaft torque, which is characterized by the spring- (c) and damping (k) factor of the shaft according to:

[M.sub.s] = c ([[alpha].sub.2] (t) - [[alpha].sub.1] (t)) + k d/dt ([[alpha].sub.2] (t) - [[alpha].sub.1] (t)). (2)

Under the precondition, that the damping is very low the damping torque:

[M.sub.d] = k d/dt ([[alpha].sub.2] (t) - [[alpha].sub.1] (t)), (3)

can be neglected and dynamic torque of the electrical machine can be measured by the difference of the angles of [[alpha].sub.2] (t) - [[alpha].sub.1] (t) and the acceleration [[??].sub.1] (t) of [[THETA].sub.1] [4, 5]:

[M.sub.e] = c([[alpha].sub.2] (t) - [[alpha].sub.1] (t)) + [[THETA].sub.1] [[??].sub.1] (t). (4)

2. Measurement principles for the angular acceleration

Acceleration sensors are well known for linear accelerations. The sensor market for angular acceleration sensors is very small. Two basic principles exist.

Fig. 2 shows the first principle of two linear acceleration sensors measuring the absolute angular acceleration.

The angular acceleration is given by Eq. (5):

[??](t) = r [a.sub.T1] (t) + r [a.sub.T2] (t). (5)

This configuration with two linear acceleration sensors causes different problems:

* at a constant speed the sensitivity of the sensor for lateral acceleration produces a constant signal caused by the centrifugal force at [??] (t) = 0;

* the power supply for the sensors and the measurement signal is difficult to handle at higher speeds;

* the adjustment of the sensors on the radius r and the mechanical differences of two sensors cannot by 100% compensated.

The second principle is shown in Fig. 3.

A constant magnetic field induces an electric field strength in a conductible cylinder rotating at constant speed according to the law of electromagnetic induction:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

This electric field causes a current I in the conductible cylinder with a constant magnetic flux [PHI] at constant speed [omega]. When the speed [omega](t) changes, the flux [PHI](t) changes accordingly and induces a voltage [U.sub.i](t) in the induction coil according equation:

[U.sub.i](t) = d/dt [PHI](t). (7)

As the flux [PHI](t) is proportional to the current I and I is proportional to [[??].sub.i] which is proportional to [omega](t) according Eq. 6, the induced voltage in the coil is proportional to the angular acceleration [??](t):

[U.sub.i](t) = d/dt [PHI](t) ~ d/dt [omega](t) = [??](t). (8)

This principle is named Ferraris principle according to the Italian Ingenieur Galileo Ferraris.

The cylinder is coupled to the end of the shaft and is the only rotating part of the sensor. So the signal for the angular acceleration can be picked up from the static coil. The sensitivity of the Ferraris sensor is typical 0.1 - 0.01 mV/rad/[s.sub.2].

To calibrate a Ferraris sensor a special test rig is necessary, see Fig. 4.

The aluminum cylinder of the Ferraris sensor is coupled to a highly dynamic DC motor with a disk-shaped rotor. The rotor is a disk which is a printed circuit board without any iron. So current of the motor is proportional to the torque without any distortion and can be used as angular acceleration reference.

With a frequency analyzer the frequency response (amplitude, phase) of the Ferraris sensor can be measured, see Fig. 5.

The sensitivity is 0.017 mV/rad/[s.sub.2] (0 dB) and the resolution is 1 rad/[s.sub.2]. The cut-off frequency is at 1.2 kHz (-3 dB). The phase in that frequency range is linear and the delay time:

[t.sub.d] = d/dt [empty set] = 0.12ms = const, (9)

is with 0.12 ms constant.

3. Torque measurement with resistance strain gauges

To determine the electrical torque according to Eq. (4) the displacement between the inertia of the rotor ([[THETA].sub.1]) and the flywheel ([[THETA].sub.2]) has to be determined. This has been done by four resistance strain gauges which have been glued under 45[degrees] on the shaft . The four strain gauges are electricaly coupled to a wheatstone bridge (Fig. 6).

A carrier frequency amplifier was used to supply the wheatstone bridge. The sensitivity of the torque sensor is 0.482 V/Nm. The cut-off frequency of the carrier frequency amplifier is at 2 kHz (-3 dB) and the delay time is 0.3 ms.

4. Measuring the electrical torque [M.sub.e]

To add signals in a measuring chain the delay time of each sensor has to be taken into account. As shown in Fig. 7 the lead time of a sinusidial signal is different, depending on the delay time of each measuring chain.

In the angular acceleration chain the phase shift after the DC amplifier is 46.8[degrees]. If the signals of the angular acceleration and the shaft torque would be added at that point, the result for the electrical torque would not be correct, because the shaft torque measurement via the elongation e has a phase shift of 108[degrees]. Therefore an electrical all pass module is necessary to shift the phase of the angular acceleration signal from 46.8[degrees] up to 108[degrees]. The amplitude of the signal is not modified by an all pass module .

After the calibration with the all pass module the two signals can be added by a normal operational amplifier.

The sensitivity for the electrical torque is 40 mV/Nm. The frequency range is determined by the Ferraris sensor with a cut-off frequency of 1.2 kHz. The delay time of 0.3 ms is determined by the carrier frequency amplifier.

5. Electrical torque measurements compensated by the angular acceleration

Fig. 8 shows a 3-phase 1.8 kW induction machine with a squirrel cage rotor that is coupled with a flywheel via a steel shaft.

This rig is very close to the representative model in Fig. 1. As there is no clutch with damping element the damping factor is with D = 0.007 very low and the precondition of Eq. 3 is fulfilled.

By variation of the diameter of the shaft and the flywheel mass it is possible to realize different resonance frequencies (fundamental mode of vibration), i.e. 33.5 Hz and 147 Hz. The flywheel mass of the clamping element is very small. It causes a resonance frequency of 1180 Hz (first harmonics) [8,9].

To show the compensation an impact pulse is given to the system (Fig. 9).

The electrical torque [M.sub.e] follows exactly the reference torque [M.sub.Ref] while the shaft torque [M.sub.w] and the angular acceleration signal are alternating with the resonance frequency of the mechanical system. The addition of both signals is zero what shows very clearly the compensation effect .

Fig. 10 shows the electrical torque during run-up of the induction machine (Fig. 8) with the compensation method.

After switch on the amplitudes of the electrical torque [M.sub.E] are with 36 Nm up to 6 times higher than the nominal torque [M.sub.Y] = 6 Nm of the induction machine. This forces the shaft torque to amplitudes up to 7 times of the nominal torque [M.sub.Y] (Fig. 11).

After about 1.8 seconds the resonance frequency of the system is excited by the induction machine. This phenomenon is to be explained by parametric excitation of the induction machine .

The Fast Fourier Transformation (FFT) of electrical torque signal (Fig. 12) shows higher amplitudes in the range of 45 Hz to 50 Hz and also around 33 Hz. The amplitudes of the torsional oscillator become up to 42 Nm so that the loop back into the electrical torque, which can be seen in Fig. 10.

Another very interesting experiment, the reversing, reveals additional alternating torques in the electrical torque of the induction machine [11, 12].

At the reversing two phase of the 3-phase power supply are swapped at full speed of 3000 rpm. When swapping two phases, the rotating electromagnetic field in the induction machine changes its direction and also does the electrical torque. This means, that the electrical torque works against the turning flywheel mass and brings it into the opposite direction with -3000 rpm (Fig. 13).

The electrical torque during reversing is shown in Fig. 14. The swapping of the electrical phases causes a peak in the electrical torque of about 25 times of the nominal torque (6 Nm). The time signal shows alternating torques with a changing frequency.

To analyze the electrical torque, the signal has been divided into 64 windows (Fig. 14). For each time window a FFT has been performed. All 64 spectra have been plotted as can be seen in Fig. 15.

The sweep of the alternating torque starts with about 1300 Hz, goes down to 0 Hz and increases up to

This sweep passes two times the resonance frequency ([F.sub.0] = 147 Hz) of the torsional vibration system and excites it two times as it can be clearly seen in Fig. 16.

6. Conclusions

To measure the input torque of a torsional vibration system it is necessary to take the vibrations of the rotor mass [[THETA].sub.1] into account. It is not sufficient only to measure the shaft torque. The shaft torque has to be compensated by the acceleration of the rotor mass. To get the exact electrical torque the delay time of the measuring chain has to be taken into account (Fig. 7).

The electrical torque shows several alternating vibrations in its torque signal, which causes high resonance excitations in low damped systems.

cross ref http:// dx.doi.org/10.5755/j01.mech.19.6.5985

Accepted October 10, 2013

References

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[11.] Hantel, P.; Bogdevicius, M.; Spruogis, B.; Turla, V.; Jakstas, A. Research of parametric vibrations of drive shafts in induction machine. In: Proceedings of VI Triennial International Conference HEAVY MACHINERY HM 2008, June 24-29, 2008, Kraljevo, Republic of Serbia, ISBN 978-86-82631-45-3, E.29E.34.

[12.] Spruogis, B.; Turla, V.; Jakstas, A.; Bogdevicius, M.; Hantel, P. 2009. Theory and Application of the Devices of Transmission and Stabilization of Rotary Motion: Monograph, Vilnius: Technika 479 p. (in Lithuanian).

P. Hantel, Hantel Consulting, Hornhang 13, Aachen, Germany, E-mail: peter.hantel@hantel-consulting.de

B. Spruogis, Vilnius Gediminas Technical University, Plytines 27, 10105 Vilnius-16, Lithuania, E-mail: bsp@vgtu.lt

V. Turla, Vilnius Gediminas Technical University, Basanaviciaus 28, 03224 Vilnius-6, Lithuania, E-mail: Vytautas.Turla@vgtu.lt

A. Jakstas, Vilnius Gediminas Technical University, Basanaviciaus 28, 03224 Vilnius- 6, Lithuania, E-mail: arunas.jakstas@vgtu.lt