A Multi-Probe Setup for the Measurement of Angular Vibrations in a Rotating Shaft

T.Addabbo1, R.Biondi2, S.Cioncolini2, A.Fort1, M.Mugnaini1, S.Rocchi1, V.Vignoli1

(1) Information Engineering Dept. University of Siena (Italy)

(2) GE Oil & Gas Nuovo Pignone, Florence (Italy)

2012 IEEE Sensors Application Symposium February 7-9 2012 - Brescia - Italy

Overview: aims and results of this workWe aim at measuring the angular vibrations of a

rotating shaft

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assuming

^ ^ ^

Example application: monitoring the torsional vibrations in huge turbo-machines and compressors

Issue: the cogwheel shape can be very irregular, e.g., the “geared wheel” is represented by some bolts mounted on a joint in the shaft

Overview: aims and results of this workWe aim at measuring the angular vibrations of a

rotating shaft

➢ We propose a measurement setup using more than one probe to reject the effects of the irregular cogwheel shape

➢ The results were validated by means of numerical simulations and experimental measurements made at the GE Oil & Gas Facility in Florence, Italy

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we assume

^

OUR WORK:

MEASUREMENT PROBLEM: A THEORETICAL APPROACH

Analytical description of the probe electrical signal, as a function of the angular vibrations and the cogwheel shape

● Probe modeling● Analytical description of the cogwheel shape● Effects of the probe filtering (spatial and time domains)● Effects of the irregular cogwheel shape

Estimation of the vibrational signal● The zero crossing method● A method to reject the effects of the irregular cogwheel profile

Conclusions

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Presentation outline

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Electrical transfer function

Gaussian spatial reception lobe with 3dB half-angle θ

0

Probe modeling

Generic pass-band behavior

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Cogwheel profile: analytical description

The rotational motion of the cogwheel in front of the probe can be equivalently described

by “unrolling ” the periodic cogwheel profile and translating

the probe along a direction parallel to the y-axis

Equivalent relative probe motion

The “unrolled” periodic cogwheel profile can be described by means

of a Fourier series

The above series can be rearranged as an infinite sum of cosines

where and

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Effects of the probe filtering: spatial domain

The detected cogwheel profile is determined by a spatial convolution of the probe reception lobe with the “unrolled” cogwheel profile

after some calculations it can be proved that

where

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Effects of the probe filtering: time domainThe probe output signal can be

obtained relating the spatial variable y to the time, by means of

the cogwheel angular velocity

By properly substituting the above expression in the series of cosines, we have

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The probe voltage signal

Finally, for small vibrations and for probes with a

focused beam, the output probe voltage can be

obtained taking into account of the probe electrical transfer

function

The probe voltage signal is an infinite series of frequency modulated sinusoidal carriers with frequencies multiple of the fundamental shaft rotating

frequency ω0/2π, and amplitudes depending on the cogwheel profile.

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Effects of the cogwheel shape (example)

Spectrum of the probe voltage signal for a shaft rotating with fundamental frequency f0 = 60 Hz (no angular vibrations). The cogwheel has 36 irregular trapezoidal cogs.

(36 * 60Hz = 2160 Hz)

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Effects of the cogwheel shape (example)

Spectrum of the probe voltage signal for a shaft rotating with fundamental frequency f0 = 60 Hz (no angular vibrations). The cogwheel has 36 irregular trapezoidal teeth.

(36 * 60Hz = 2160 Hz)

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Estimation of the vibrational signal

● Extreme low frequencies if compared to typical radio-communication cases● Mutual interference between neighboring frequency modulated carriers● The probe signal is mainly affected by the cogwheel shape irregularities

ω(t n)≈φ

Δ t n≈

2πN Δ t n

Issues about the measurement problem

Demodulation method: the zero crossing

● Easy hardware implementation (with either digital or analog circuits)● It is possible to design a method to reject the effects of the cogwheel irregular

shape

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Irregular cogwheel profile rejectionThe estimation of the instantaneous angular velocity can be written as:

Constant term Angular vibration

Apparent angular vibrationdue to the irregular cogwheel shapeprofile

Z is periodic with frequency equal to the fundamental shaft rotation frequency f0 = 1/T0

Or, in other words, the sum of M multiple instances of the same signal Z, properly delayed, is a signal with period T0 divided by M

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Irregular cogwheel profile rejection

ω̂1(t)≈ω0+ω̃( t)+Z (t)

ω̂2(t )≈ω0+ω̃(t)+Z (t− 12f 0

)ω̂(t)≈2ω0+2 ω̃(t)+Z ' (t)

Periodic withperiod 1/2f0

Experimental results (example)Spectrum of the measurement system output.

Shaft rotating @ 60Hzwith two angular vibration tones @ 10Hz and 50Hz

(a) one probe

(b) two probes

(c) three probes

Conclusions

● We have proposed a theoretical description of the probe output signal, as a function of the angular vibrations and the cogwheel shape

● We have discussed a measurement method based on the zero-crossing FM demodulation method for estimating the angular vibrations of a rotating shaft

● The demodulation algorithm is applied directly to the probe output voltage, without any filtering

● On the basis of the theoretical results we have proposed a measurement method that can reject the effects of the irregular cogwheel shape. The method is based on the simple averaging of the measurement of two or more probes uniformly positioned around the rotating cogwheel.

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