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 work We aim at measuring the angular vibrations of a rotating shaft assuming ^ ^ ^ Example application: monitoring the torsional vibrations in huge turbomachines 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 2
Overview: aims and results of this work We aim at measuring the angular vibrations of a rotating shaft we assume ^ OUR WORK: 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 3
Presentation outline 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 4
Probe modeling Electrical transfer function Generic pass-band behavior Gaussian spatial reception lobe with 3dB half-angle θ0 5
Cogwheel profile: analytical description Equivalent relative probe motion 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 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 6
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 7
Effects of the probe filtering: time domain The 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 8
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. 9
Effects of the cogwheel shape (example) (36 * 60Hz = 2160 Hz) 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. 10
Effects of the cogwheel shape (example) (36 * 60Hz = 2160 Hz) 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. 11
Estimation of the vibrational signal Issues about the measurement problem 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 Demodulation method: the zero crossing φ 2π ω(t n ) Δtn N Δtn 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 12
Irregular cogwheel profile rejection The estimation of the instantaneous angular velocity can be written as: Apparent angular vibration due to the irregular cogwheel shape profile Constant term Angular vibration 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 13
Irregular cogwheel profile rejection ω 1 (t ) ω0+ω(t )+Z (t ) Periodic with period 1/2f0 1 ω 2 (t ) ω0+ ω(t ) )+Z (t 2f 0 ω(t) 2 ω0+2 ω(t)+z ' (t) 14
Experimental results (example) Spectrum of the measurement system output. Shaft rotating @ 60Hz with 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. 16
References L. Zhen, Z. An, and Q. Li, ``Calculate engine crankshaft angular acceleration based on original flywheel data,'' in Mechanic Automation and Control Engineering (MACE), 2010, 2010, pp. 3141-3144. P. Wang, P. Davies, J. Starkey, and R. Routson, ``A torsional vibration measurement system,'' in Instrumentation and Measurement Technology Conference, 1992. IMTC '92., 9th IEEE, 1992. ------, ``A torsional vibration measurement system,'' IEEE Transactions on Instrumentation and Measurement, vol. 41, pp. 803 -- 807, 1992. M. Lebold, K. Maynard, K. Reichard, M. Trethewey, J. Hasker, C. Lissenden, and D. Dobbins, ``A non-intrusive technique for on-line shaft crack detection and tracking,'' in Aerospace Conference, 2005 IEEE, 2005. F. Yongqing and W. Kinsner, ``A marine engine torsion vibration measuring method and its implementation based on fpga,'' in Canadian Conference on Electrical and Computer Engineering, 2002. IEEE CCECE 2002., 2002. F. C. Gómez de León and P. Meroño Pérez, ``Discrete time interval measurement system: fundamentals, resolution and errors in the measurement of angular vibrations,'' Measurement Science and Technology, vol. 21, no. 7, July 2010. S. Haykin and M. Moher, Introduction to analog and digital communications. John Wiley & Sons, 2007. R. Wiley, ``Approximate {FM} demodulation using zero crossings,'' IEEE Trans. on Communications, vol. COM-29, no. 7, pp. 1061--1065, July 1981. H. Voelcker, ``Zero-crossing properties of angle-modulated signals,'' IEEETrans. on Communications, vol. COM-20, pp. 307--315, 1972. 17