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Research Papers

Experimental Application of Time-Domain Transmissibility Identification to Fault Detection and Localization in Acoustic Systems

[+] Author and Article Information
Khaled F. Aljanaideh

Department of Aeronautical Engineering,
Jordan University of Science and Technology,
Irbid 22110, Jordan
e-mail: kfaljanaideh@just.edu.jo

Dennis S. Bernstein

Professor
Department of Aerospace Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: dsbaero@umich.edu

1Corresponding author.

Contributed by the Noise Control and Acoustics Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received October 2, 2016; final manuscript received September 9, 2017; published online December 12, 2017. Assoc. Editor: Lei Zuo.

J. Vib. Acoust 140(2), 021017 (Dec 12, 2017) (11 pages) Paper No: VIB-16-1487; doi: 10.1115/1.4038436 History: Received October 02, 2016; Revised September 09, 2017

This paper considers a technique for fault detection and localization based on time-domain transmissibility identification. This technique takes the advantage of unknown external (ambient) excitation to identify a sensor-to-sensor model, which is independent of the excitation signal and the initial conditions of the underlying system. In the presence of unknown external excitation, the identified transmissibility operator is used to compute the sensor-to-sensor residual, which is the discrepancy between the predicted sensor output (based on the transmissibility operator) and the actual measurements. The sensor-to-sensor residuals are used to detect, diagnose, and localize faults in sensors and system dynamics. We consider an experimental setup consisting of an acoustic system with three speakers and six microphones. Each speaker is an actuator, and each microphone is a sensor that measures the acoustic response at its location. Measurements from the six microphones are used to construct transmissibility operators, which in turn are used to detect and localize changes in the dynamics of the acoustic system or the microphones by computing the resulting one-step residuals.

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References

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Figures

Grahic Jump Location
Fig. 4

For the acoustic system shown in Fig. 1 operating under healthy conditions, Spk1 and Spk2 are driven with realizations of band-limited white noise with a bandwidth of 500 Hz and Spk3 is not operating. This plot shows the entries of the estimated Markov parameters θ̂r,d,ℓ of T(q−1,θr,d) from each pseudo input y1 and y3 to the pseudo output y6 for two different runs of the experiment.

Grahic Jump Location
Fig. 3

For the acoustic system shown in Fig. 1 operating under healthy conditions, Spk1 is driven with a realization of bandlimited white noise with a bandwidth of 500 Hz. This plot shows the measurements of y6 and the computed one-step prediction ŷ6=T(q−1,θ̂r,d,ℓ)y1, where θ̂r,d,ℓ is obtained from run 1 shown in Fig. 2.

Grahic Jump Location
Fig. 2

For the acoustic system shown in Fig. 1 operating under healthy conditions, Spk1 is driven with a realization of bandlimited white noise with a bandwidth of 500 Hz and Spk2 and Spk3 are not operating. This plot shows the estimated Markov parameters θ̂r,d,ℓ of T(q−1,θr,d) from the pseudo input y1 to the pseudo output y6 for two different runs of the experiment.

Grahic Jump Location
Fig. 1

Experimental setup. The setup consists of an acoustic space with three speakers Spk1, Spk2, and Spk3 and six microphones Mic1,…,Mic6. Each speaker is an actuator, and each microphone is a sensor that measures the acoustic response at its location.

Grahic Jump Location
Fig. 8

For the acoustic system shown in Fig. 1 operating under healthy conditions, Spk1, Spk2, and Spk3 are driven with the harmonic signals (33)(35). This plot shows the measurements of y6 and the computed one-step prediction ŷ6=T(q−1,θ̂r,d,ℓ)[y1  y3  y5]T, where θ̂r,d,ℓ is obtained from run 1 shown in Fig. 6. This figure confirms that the identified transmissibility obtained under one type of excitation signal can be used under another type of excitation signal. This is advantageous for fault detection, where transmissibilities are identified under one set of conditions and used under a different set of conditions.

Grahic Jump Location
Fig. 9

For the acoustic system shown in Fig. 1 operating under healthy conditions, Spk1 is driven with a realization of bandlimited white noise with a bandwidth of 500 Hz and Spk2 and Spk3 are not operating. This plot shows the norm of the residual for the identified transmissibility operators with 1,…,5 pseudo inputs. Note that the benefits produced by using 2, 3, 4, or 5 pseudo inputs are not significant, which correctly suggests that the number of independent external excitation signals acting on the systems is 1.

Grahic Jump Location
Fig. 10

For the acoustic system shown in Fig. 1 operating under healthy conditions, Spk1 and Spk2 are driven with realizations of bandlimited white noise with a bandwidth of 500 Hz and Spk3 is not operating. This plot shows norm of the residual for the identified transmissibility operators with 1,…,5 pseudo inputs. Note that the benefits produced by using two pseudo inputs are significant, but the benefits produced by using 3, 4, or 5 pseudo inputs are not significant, which correctly suggests that the number of independent external excitation signals acting on the systems is 2.

Grahic Jump Location
Fig. 11

For the acoustic system shown in Fig. 1 operating under healthy conditions, Spk1, Spk2, and Spk3 are driven with realizations of band-limited white noise with a bandwidth of 500 Hz. This plot shows norm of the residual for the identified transmissibility operators with 1,…,5 pseudo inputs. Note that the benefits produced by using two or three pseudo inputs are significant, but the benefits produced by using four or five pseudo inputs are not significant, which correctly suggests that the number of independent external excitation signals acting on the systems is 3.

Grahic Jump Location
Fig. 12

For the acoustic system shown in Fig. 1 operating under healthy conditions, this plot shows the norm of the residuals for the identified transmissibility operators obtained using LS with 1,…,5 pseudo inputs for the case of 1, 2, or 3 speakers driven with realizations of bandlimited white noise with a bandwidth of 500 Hz. This plot shows that to reduce the residuals, the number of pseudo inputs has to be greater than or equal to the number of independent inputs acting on the system.

Grahic Jump Location
Fig. 13

For the acoustic system shown in Fig. 1, Spk1, Spk2, and Spk3 are driven with a band-limited white noise with a bandwidth of 500 Hz. At approximately t = 11 s, the blanket that covers the acoustic system is removed. This plot shows E(k|θ̂r,d,ℓ,w) for T(q−1,θ̂r,d,ℓ), where w = 500 steps and θ̂r,d,ℓ is the LS estimate obtained from run 1 shown in Fig. 6 using broadband excitation. Note the change in E(k|θ̂r,d,ℓ,w) at approximately t = 11 s due to the change in the dynamics of the acoustic system.

Grahic Jump Location
Fig. 14

For the acoustic system shown in Fig. 1, u1,u2,u3 are as shown in Eqs. (33)(35). At approximately t = 11 s, the blanket that covers the acoustic system is removed. This plot shows E(k|θ̂r,d,ℓ,w) for T(q−1,θ̂r,d,ℓ), where w = 500 steps and θ̂r,d,ℓ is the LS estimate obtained from run 1 shown in Fig. 6 using broadband excitation. Note the change in E(k|θ̂r,d,ℓ,w) at approximately t = 11 s due to the change in the dynamics of the acoustic system.

Grahic Jump Location
Fig. 5

For the acoustic system shown in Fig. 1 operating under healthy conditions, Spk1 and Spk2 are driven with realizations of band-limited white noise with a bandwidth of 500 Hz. This plot shows the measurements of y6 and the computed one-step prediction ŷ6=T(q−1,θ̂r,d,ℓ)[y1  y3]T, where θ̂r,d,ℓ is obtained from run 1 shown in Fig. 4.

Grahic Jump Location
Fig. 15

For the acoustic system shown in Fig. 1, Spk1, Spk2, and Spk3 are driven with a band-limited white noise with a bandwidth of 500 Hz. At approximately t = 10 s, a fault occurs in Mic2. This plot shows E(k|θ̂r,d,ℓ,w) for T(q−1,θ̂r,d,ℓ), where w = 500 steps. Note the change in E(k|θ̂r,d,ℓ,w) at approximately t = 10 s due to the faulty microphone.

Grahic Jump Location
Fig. 16

For the acoustic system shown in Fig. 1 with u1,…,u3 are as shown in Eqs. (33)(35). At approximately t = 10 s, a fault occurs in Mic3. This plot shows E(k|θ̂r,d,ℓ,w) for T(q−1,θ̂r,d,ℓ), where w = 500 steps. Note the change in E(k|θ̂r,d,ℓ,w) at approximately t = 10 s due to the faulty microphone.

Grahic Jump Location
Fig. 17

Fault localization algorithm. This algorithm is consistent with the transmissibility operators as defined in Table 1.

Grahic Jump Location
Fig. 18

For the acoustic system shown in Fig. 1, Spk1, Spk2, and Spk3 are realizations of bandlimited white noise with a bandwidth of 500 Hz. At approximately t = 10 s, the blanket that covers the acoustic system is removed. This plot shows E(k|θ̂r,d,ℓ,w) of the identified transmissibilities Ti(q−1,θ̂r,d,ℓ) of Ti, for i=1,…,6, where w = 500 steps. Note the change in E(k|θ̂r,d,ℓ,w) for all transmissibilities at approximately t = 10 s due to the change in the dynamics of the system. Using Fig. 17, we correctly conclude that the fault is in the system.

Grahic Jump Location
Fig. 19

For the acoustic system shown in Fig. 1, u1,u2,u3 are as shown in Eqs. (33)(35). At approximately t = 10 s, a fault occurs in Mic2. This plot shows E(k|θ̂r,d,ℓ,w) for the identified transmissibilities Ti(q−1,θ̂r,d,ℓ) of Ti, for i=1,…,6, where w = 500 steps. Note the change in E(k|θ̂r,d,ℓ,w) of all transmissibilities except T5 and T6 at approximately t = 10 s due to the faulty microphone. Using Fig. 17, we correctly conclude that Mic2 is faulty.

Grahic Jump Location
Fig. 20

For the acoustic system shown in Fig. 1, u1,u2,u3 are as shown in Eqs. (33)(35). At approximately t = 10 s, a fault occurs in Mic3. This plot shows E(k|θ̂r,d,ℓ,w) for the identified transmissibilities Ti(q−1,θ̂r,d,ℓ) of Ti, for i=1,…,6, where w = 500 steps. Note the change in E(k|θ̂r,d,ℓ,w) of all transmissibilities except T6 at approximately t = 10 s due to the faulty microphone. Using Fig. 17, we correctly conclude that Mic3 is faulty.

Grahic Jump Location
Fig. 21

For the acoustic system shown in Fig. 1, u1,u2,u3 are as shown in Eqs. (33)(35). At approximately t = 10 s, a fault occurs in Mic5. This plot shows E(k|θ̂r,d,ℓ,w) for the identified transmissibilities Ti(q−1,θ̂r,d,ℓ) of Ti, for i=1,…,6, where w = 500 steps. Note the change in E(k|θ̂r,d,ℓ,w) of all transmissibilities except T2 at approximately t = 10 s due to the faulty microphone. Using Fig. 17, we correctly conclude that Mic5 is faulty.

Grahic Jump Location
Fig. 6

Spk1, Spk2, and Spk3 are driven with realizations of bandlimited white noise with a bandwidth of 500 Hz. The plot shows the entries of the estimated Markov parameters θ̂r,d,ℓ of T(q−1,θr,d) from each pseudo input y1, y3, and y5 to the pseudo output y6 for two different runs of the experiment.

Grahic Jump Location
Fig. 7

For the acoustic system shown in Fig. 1 operating under healthy conditions, Spk1, Spk2, and Spk3 are driven with realizations of bandlimited white noise with a bandwidth of 500 Hz. This plot shows the measurements of y6 and the computed one-step prediction ŷ6=T(q−1,θ̂r,d,ℓ)[y1  y3  y5]T, where θ̂r,d,ℓ is obtained from run 1 shown in Fig. 6.

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