Laser vibrometers are well-known interferometricinstruments1–3 for the measurement of minute vibra-tion of a remote target without physical contact andwith very good sensitivity. In recent years, a techniquecalled self-mixing interferometry (SMI) or optical feed-back interferometry1,3 has gained acceptance, thanksto its simple optical setup, requiring just a diode laser(DL) as the optical source and a focusing lens. In aself-mixing configuration, light from a DL is focused onthe remote target, and a fraction of the backscatteredlight is allowed to re-enter the SL cavity where it iscoherently mixed with the lasing field. This generatesa modulation of the emitted power in the form of aninterferometric signal carrying the phase informationrelated to the path length of light from the DL to thetarget and back. The interferometric signal can be de-tected by the monitor photodiode (PD) usually alreadyavailable in the DL package. SMI is a single-channelconfiguration, does not require a reference arm, and
has a minimum optical part count. It has been success-fully demonstrated in a variety of measurement appli-cations, such as displacement,3–5 absolute distance,6vibrations,7,8 and angles.9
SMI readily measures vibrations of the target withgood sensitivity, but, because of the inherent single-channel structure lacking a reference path, a differ-ential measurement of vibrations on two distinctpoints is not possible, at least in the basic form. Weshow how to circumvent this limitation of SMI. Be-cause SMI is single channel and so simple, we may aswell duplicate it and perform the difference of theoutputs of the two-channel structure, provided we areable to maintain the two channels stably and welltracked. This is ensured by the feedback loop action ofthe half-fringe stabilization loop. We show that thiselectrical subtraction is indeed viable and allows usto achieve sensitivity comparable to that of a conven-tional, referenced interferometric configuration.
The case of interest for our application was thecharacterization of the contact between two vibrat-ing bodies, requiring the measurement of small am-plitude differential displacement. The measurementhas to deal with the energy dissipated by the contactfriction forces and hence helps to understand the dy-namics of vibration damping.10 In particular, the re-lationship between tangential forces in the contactpoint and the relative displacement is linear whenthe amplitude of vibration is small, whereas it exhib-its hysteresis at large amplitude as the tangential
S. Donati ([email protected]), M. Norgia, and G. Giuliani([email protected]) were with the Dipartimento di Elettronica,Universitá di Pavia, Via Ferrata 1, Pavia I-27100, Italy. M. Norgiais now with the Dipartimento di Elettronica e Informazione, Po-litecnico di Milano, Via Ponzio 34-5, Milan I-20133, Italy.
Received 4 November 2005; revised 20 February 2006; accepted28 April 2006; posted 23 May 2006 (Doc. ID 65779).
force approaches a limiting value and the contactenters the gross-slip regime.
Friction damping is fundamental in quenching vi-bration in gas turbine blades, in order to improveendurance and aircraft safety. Usually, damping isprovided by underplatform dampers or by shroud-type contacts, which exploit friction at the contact todissipate the vibration energy.10,11 Modeling the con-tact regime requires the measurement of hysteresiscycles for different contact geometry and materialproperties. Also in favor of the optical approach, con-tact materials working in turbine blades becomeheated up to high temperatures, reaching 800 °C.
A test rig was developed to characterize frictionhysteresis.12 The aim of measurement is to plot inreal time and, simultaneously, the tangential contactforce (measured by a piezoelectric force transducer)and the differential displacement. The rig consists ofa beam carrying one specimen, to bring it in contactwith a second specimen held in position in a specialfixture. The two specimens are loaded against eachother in the normal contact direction by a constantload. When the beam is vibrated, the specimen at-tached to it rubs against the other, developing thecontact hysteresis.
The driving force is swept in frequency typicallyfrom a few hertz to hundreds of hertz, producingcommon-mode displacements up to 50–100 �m. Thedifferential vibration amplitude is up to 20–30 �m,and this quantity shall be measured with an accuracyof approximately 20 nm (for contact parameters ex-traction). The displacement point is very close to thecontact surface, thus ruling out measurement ap-proaches based on capacitive or eddy current.
2. Differential Self-Mixing Vibrometer
The optical head of each channel is shown in Fig. 1.Light from a single longitudinal mode Fabry–PerotDL is focused onto the target through an objectivelens. Backscattered light is focused back into the la-ser and modulates the cavity field. The monitor PD,on the rear laser mirror, collects a fraction of the SMIsignal, which is found1–4 as
P��� � P0�1 � mF����, (1)
where P0 is the power emitted by the unperturbed SL,F���, is a periodic function of phase � � 2 ks, wherek � 2���, and m is the modulation index (typically�10�6 to 10�3 in most cases). The exact shape of theinterferometric function F��� depends on feedbackparameter C, which, in turn, depends on parametersof the laser and also on the target distance s and thepower reflectivity of target Reff (see Refs. 3 and 4 fordetails).
It is important to note that for the weak feedback�C �� 1�, the function F��� is a cosine, just as theusual interferometric waveform, whereas for themoderate feedback (i.e., C � 1), the function F���becomes sawtoothlike (Fig. 2) and this allows for thediscrimination of the sign of target displacement, asexplained in Refs. 3–6. From this interferometric sig-nal, the vibration amplitude can be retrieved with a��2 accuracy just by counting the number of fringes.
To improve resolution further, a feedback-looptechnique has been developed so that the minimumdetectable vibration is ultimately limited by noiserather than by the ��2 quantization of fringe count-ing. The technique has been presented in Ref. 7 towhich the reader is directed for the details. Here, wejust discuss those features that are crucial for under-standing how precise the difference of vibrationsignals can be, i.e., to get a good common-mode sup-pression.
Basically, we construct a servo loop around theinterferometer (Fig. 3) by feeding back an error signalto the experiment, so as to keep the output phase read
Fig. 1. Schematic of the optical head: an 825 nm semiconductorlaser (DL) is used as the source, and its monitor photodiode (PD) ismounted internally in the package, behind the rear mirror, is thedetector of the SMI. An anamorphic objective lens is used to colli-mate the beam and direct it onto the diffusing target. A transim-pedance amplifier brings the photocurrent to a level large enoughfor subsequent signal processing.
Fig. 2. Driving the target with a sine wave, the SMI signalresembles the normal (Bessel-like) interferometric signal at lowinjection levels �C � 0.5�. At increased C factors, the signal isprogressively distorted and then becomes a sawtoothlike waveform(bottom, C � 3.5).
Fig. 3. Working at C � 1, the SMI can be stabilized at half-fringe(left) by a simple feedback loop (right) feeding back to the biascurrent of the DL from the amplified SMI signal.
by the SMI constant. In the experiment, the totalphase variation � �2ks� � 0 is nulled dynami-cally by changing the wavelength, so that 2ks �2sk � 0 or, being k�k � ����, with � � �s�s.The wavelength change is obtained by a bias-currentchange, as Ibias � �1�����, �� being a coefficientspecific of the DL. Additionally, phase produces asignal V, at the PD output, given by
V � RI0 sin � RI0, (2)
here, R is the transresistance of the preamplifierstage, and I0 is the peak SMI component of totalphotocurrent signal, or explicitly,
Iph � Iph0 � I0 cos . (3)
After a gain A in the main amplifier, a signal erroris developed that is fed to the current driver of thelaser as
Ibias � GmAV, (4)
Gm being the transconductance of the driver. Collect-ing the terms, we can see that the loop has a totalgain given by
Aloop � 2ks������I0GmAR, (5)
whereas from displacement signal s to the output Vof the main amplifier, we get a transfer factor given by
V � ���Gm��1s���s�. (6)
Now, the crucial point is that V depends on sthrough a factor ���Gm��1���s�, which does not con-tain the signal power nor its fluctuations, includingthose due to the speckle statistics introduced by thediffusing target. Indeed, as the zeroing effect sharesthe same channel as the displacement signal, anydependence from the transfer factors of the commonpath is canceled out.
Additionally, if we set the reference level in theamplifier at half-fringe amplitude, this level of thesignal will be kept constant by the feedback loop, anydisturbances being decreased by Aloop. In the sameway, the dynamic range will no more be limited to ��2as in a normal interferometer, rather it will be in-creased to Aloop���2� by virtue of the feedback effect,as already shown in Ref. 7 Also, the linearity errorcaused by the sin � dependence of the fringe signal isdecreased by the loop gain Aloop. All these effects areconsequences of the feedback loop action, as it is wellknown from feedback control theory.
Concerning speckle pattern statistics, certainlythere are fluctuations in the photodetected signal Iphas the spot on the target is moved from point to point.Yet the signal V � RI0 remains constant becauseof the feedback action. Stated in other terms, uponaiming the beam on the target we find bright anddark speckles with high and low values of Iph and I0.The corresponding Aloop gain varies accordingly, butnot the V. Of course, bright speckles are still pref-erable, because they provide us with the largest value
of Aloop for linearity, bandwidth, and dynamic rangeimprovement.
All the data presented below are for the favorablecondition of a fairly bright speckle (say �2–3 timesthe average intensity). This condition corresponds toachieving a loop gain of Aloop � 500. In the practicaloperation of the instrument, such a bright speckle isreadily found by slightly deflecting the laser spot onthe target while watching the amplitude Iph so as tofind the desired large value.
The block scheme of the SMI differential vibrom-eter is shown in Fig. 4. Each channel is just anoptical head with the LD, a monitor PD backed bythe transimpedance amplifier, a difference amplifier,and a current generator feeding the LD.
The output signals of the two channels are sub-tracted in a difference amplifier (Fig. 4), after theamplitude of one is trimmed to allow for a small(typically �10%) correction of relative amplitudes.The trimming operation is performed while the lightbeams of the two channels are aimed to the same po-sition on the target (see Fig. 5) and the target is excitedto vibrate to large amplitude (typically 100 �m). Theresidual amplitude of the difference signal after trim-ming is approximately 0.15–0.25 �m, or a factor of400–650 of suppression of the common-mode vibra-tion (or displacement) is achieved.
Fig. 4. Block scheme of the self-mixing differential vibrometerbased on the electronic subtraction of output signals of two nomi-nally identical channels.
Fig. 5. Top, geometric arrangement of the two optical heads andtargets, showing the laser beams in normal operation (solidlines) and during calibration (dashed lines). Bottom, arrange-ment of the forces applied to the rig assembly carrying the twobeadlike targets.
The initial matching of performance of the twochannels is good, despite the differences in the pa-rameters of the two DL samples, once again becauseof the feedback mechanism of the servo loop. In Fig.6 we plot, as a function of excitation frequency, themaximum signal amplitude handled by both chan-nels before exiting from the fringe and the minimummeasurable signal (at a signal-to-noise ratio of 1). Thedata are quite similar for the two channels and differby about �10% in most cases.
After subtraction of the channel output signals, thedifference signal has a maximum swing practicallyequal to the single channel (provided each channeldoes not exceed its maximum), or up to 100 �m in thefrequency range of 5–200 Hz. Here, the roll-off at highfrequency is due to the cutoff of the loop gain.
The minimum amplitude of the measurable sig-nal is approximately 200 pm��Hz in the range of10–2000 Hz, and thus in a B � 100 Hz bandwidtharound the excitation frequency, we can theoreticallyachieve a minimum detectable signal of 200 �100� 2 nm (at a signal-to-noise ratio of 1). However, asthe common-mode vibration is suppressed by about500, when the common mode is large, e.g., has an�100 �m amplitude filling the full dynamic range,the minimum measurable signal is limited by thedifference error to approximately 200 nm. The noiselimit prevails when the common-mode amplitudes aresmall, i.e., less than or equal to 2 nm 500 � 1 �m (ata signal-to-noise ratio of 1 and B � 100 Hz). All theabove data are for operation on a normal diffusingtarget, for example, plain white paper, on a medium-to-bright speckle.
About the trimming operation, it was found thatthe best way to minimize the difference is to tilt only
one beam, bringing it to be superposed on the otherbeam at the target location (Fig. 5). This procedureintroduces a negligible error in the ratio of voltage todisplacement � � V�s � ���Gm��1���s�, becausethe angle � (Fig. 5, top) is small. Indeed, it is����0 � 1 � cos � � 1 � cos�d�s� � 2.8 10�5, whenwe assume s � 0.4 m and d � 3 mm. On the contrary,the offset �s of the two target distances can producea sizable error, being ����0 � ��s�s � 10�3 fors � 0.4 m and �s � 400 �m. Thus we have to checkthe distance s and s � �s of the two targets, or weshall be prepared to tolerate a decrease of thecommon-mode rejection factor. (In practice, by re-trimming the gain as indicated in Fig. 4, we cancompensate for a �s�s error up to a few percent.)
3. Measurements
The instrument (whose prototype is shown in Fig. 7)has been tested in the field by looking at the differ-ential vibration generated in the mechanical exper-imental test rig set up at the Department ofMechanics, Politecnico di Torino. In this experiment,the two laser beams are aimed at the free surface oftwo steel samples that are put in contact (as shown inFig. 5, bottom). The master sample has a flat contactsurface and is driven by a sinusoidal force. The fol-lower sample has a spherical contact surface, and it issubject to a force perpendicular to the contact. Due tofriction, a tangential force is generated at the contactbetween the two samples, and this force, acting on thefollower sample, is measured by a separate forcetransducer (a quartz accelerometer). Our aim was tosimultaneously measure the tangential force and thedifferential vibration of the two samples. When theactuating force is small or the normal force is large,microslip of the contact occurs, and the differentialvibration is small, generally below 2–4 �m. As soon asthe actuating force is sufficiently large to overcomethe static friction, gross slip occurs, with much largerdifferential vibration.
Figure 8(a) shows an example of the acquiredwaveforms for the tangential force and the differen-tial vibration for the gross-slip regime. Because of thenonlinearity in the magnetic actuator used in theexperiment, the waveform of applied force deviatesfrom a pure sinusoid at the working frequency of
Fig. 6. Maximum measurable peak-to-peak vibration amplitudefor the two individual vibrometer channels as a function of fre-quency, and minimum detectable signal (at S�N � 1). Measure-ments are relative to a white paper surface, placed at a 40 cmdistance from the optical heads.
Fig. 7. Picture of the assembled differential vibrometer.
84 Hz, but this has no consequence in the develop-ment of the experiment. The measured vibration hasa differential amplitude of 20 �m peak to peak, andhas a distorted sine wave. The common-mode vibra-tion of the master sample had an approximately100 �m peak-to-peak amplitude. When the tangen-tial force is plotted versus the differential vibrationfor a single period of the vibration, the hysteresiscycle shown in Fig. 8(b) is obtained. The study of asequence of single-shot hysteresis cycles is impor-tant because trajectories can vary from cycle to cycleand can also depend on the wear of the contact and onits temperature.
Another important application is the measurementof the hysteresis cycle when the samples are kept athigh temperature 700–900 °C, that is, in a conditionclosely approaching that occurring in a turbine en-gine. The SMI vibrometer performed well for thehigh-temperature application, where other conven-
tional laser vibrometers may fail due to the largebackground noise caused by the blackbody emissionof the targets.
4. Conclusions
We have presented the development of a noncontactdifferential laser vibrometer for application to thecharacterization of vibrating contacts between twosamples in the presence of friction damping in themicroslip and gross-slip regimes. The approach basedon the electronic difference of signals from two dis-tinct half-fringe stabilized SMI has proved to be suc-cessful. Operation was at a substantial distance (400mm typically) on untreated diffusing surfaces for thetarget and achieved a submicrometer sensitivity forthe differential vibration amplitude superposed to an�100 �m common-mode vibration amplitude. Theprototype has been successfully tested on a purposelydeveloped test rig, and hysteresis cycles have beenmeasured for the microslip and gross-slip regimes.
The authors thank Muzio Gola and Sergio Filippi(Department of Mechanics, Politecnico di Torino) forkindly supplying the results of the SMI measure-ments on their mechanical apparatus and for theuseful discussions on the applicability of the opticaltechniques.
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Fig. 8. (a) Time-domain traces for the tangential force (as in Fig. 5)and the differential displacement for the case of gross slip of thecontact between the two vibrating samples. The master sample isdriven by at 84 Hz, and it vibrates harmonically with an amplitudeof �100 �m peak to peak. Bottom trace is the corresponding displace-ment waveform (amplitude �20 �m peak to peak). (b) Plot of thetangential force–differential displacement revealing the hysteresis.
