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THE LASER TORSIONAL VIBROMETER: A STEP FORWARD IN ROTATING MACHINERY DIAGNOSTICS
Journal of Sound and Vibration (1996) 190(3), 399–418
THE LASER TORSIONAL VIBROMETER: A STEP FORWARD IN ROTATING MACHINERY DIAGNOSTICS N. A. H Department of Mechanical Engineering, Loughborough University of Technology, Loughborough, LE11 3TU, England (Received 1 November 1995) This paper describes the development of the laser torsional vibrometer since its invention at the Institute of Sound and Vibration Research, University of Southampton in 1983. The theory of laser vibrometry measurements is first introduced together with a description of early attempts to harness the technology for torsional vibration measurement through use of a cross-beam laser Doppler velocimeter. This is followed by the theory for the laser torsional vibrometer itself. A measurements section then describes results taken by the first prototype built at ISVR and compares these with the cross-beam velocimeter in a controlled experiment. Further tests with the commercial version of the instrument, produced by Bru¨el and Kjaer Ltd as the Type 2523 Laser Torsional Vibration Meter, are described, involving checks with the well known Hooke’s joint. Results from an investigation into torsional vibration damper performance are also presented. The sensitivity of the instrument is dealt with in detail before practical considerations are described. The laser torsional vibrometer has solved a particularly difficult metrology problem and is now used on a worldwide basis. It is rapidly gaining acceptance as the standard means of measuring torsional vibration. 7 1996 Academic Press Limited
PREFACE
Elfyn Richards returned to the I.S.V.R. from his position as Vice-Chancellor of Loughborough University in 1975. He immediately began to form a research group working in the area of machinery noise. After a three-year period as research fellow in the Aeronautics Department, working with laser technology, I joined the staff of the I.S.V.R. in 1977. I will never forget Elfyn walking into my office and advising me to ‘‘stop playing with lasers’’ and to join his research group as his right-hand man. After much thought, and persuasion by the then Director, Professor Large, I joined Elfyn’s team. There then began a most lively and interesting period in my career. Long car journeys, when the two of us travelled to the Midlands and back to promote the group’s work, were particularly enjoyable. He would put the world to rights in terms of acoustic and vibration research, and I learned a tremendous amount from him. He was a man of great vision, and I was privileged to work so closely with him. The machinery noise work spawned many interesting problems, and I returned to playing with lasers in 1979. 1. INTRODUCTION
Since the advent of the laser in the early 1960s, the field of optical metrology has provided accurate experimental data in situations in which, hitherto, it would have been considered unobtainable. The technique of laser Doppler velocimetry (LDV) [1] is now well 399 0022–460X/96/080399+20 $12.00/0
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established and was initially applied to obtain non-intrusive measurements in fluid flows. Although the use of LDV for solid surface velocity measurement was recognized at an early stage, its development in this area received little attention compared with the effort in fluid mechanics. Accordingly, the measurements of vibration was still extensively achieved with accelerometers or other forms of transducer which rely on contact with the measurement surface for successful operation. There are, however, many cases of engineering interest where this approach is either impractical or impossible. Typical examples are the measurement of very hot or light surfaces, such as exhaust pipes or loudspeakers, and measurement on rotating surfaces which preclude their use. In the latter area the measurement of torsional vibration of rotating components presented a particularly difficult measurement problem. When designing rotating machinery components, an engineer must be careful to suppress torsional oscillations, since incorrect or insufficient control may lead to fatigue failure, rapid bearing wear, gear hammer, fan belt slippage and can produce associated excessive noise problems. Torsional oscillations are a particular problem in engine crankshaft design where torsional dampers are commonly used to maintain oscillations at an acceptable level over the working speed range of the engine. Torsional transducers have formerly included optical, seismic and mechanical torsiographs, strain gauges and slotted discs. The latter system has found common use in the automotive industry and consists of a slotted disc fixed to the end of the crankshaft. A proximity transducer monitors the slot passing frequency, which is then demodulated to provide a voltage analogue of the crankshaft speed and hence torsional oscillations, but within a limited frequency range. Strain gauges and associated telemetry or slip ring systems are notoriously difficult to fix, calibrate and use successfully. In summary, the measurement of torsional oscillations presented difficult problems for contacting transducer technology and, of course, necessitated machinery ‘‘downtime’’ and special arrangements being made for fitting, calibration, etc. Very often, the cost of this machinery stoppage would preclude a measurement being attempted, even though the vibration engineer had concluded that it was vital if a design improvement is to be made. There was therefore a real need for a torsional vibration transducer which was user friendly and could provide data immediately in on-site situations. It was not until the advent of laser technology that a solution was found and, in what follows, the laser torsional vibrometer is described, which allows the engineer to point low powered laser beams at a rotating target component and obtain torsional vibration information.
2. LASER VIBROMETRY
2.1. Laser Doppler velocimetry (LDV) or laser doppler anemometry (LDA) is now a well-established technique, which is used primarily for non-intrusive measurement in fluid flows [1]. Fundamentally, LDV measures the velocity of a light-scattering particle which is seeded into the flow of interest, and it is assumed that the particle follows the flow faithfully. Under normal operating conditions the velocity of a group of seeding particles in the measurement region is detected. Clearly, the number density of scattering particles is important since they essentially ‘‘sample’’ the flow, and much of the early work in LDV was concerned with the validity or otherwise of statistical conclusions drawn from a particular measurement. An obvious practical problem to be overcome was the
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intermittence of seeding particles. An analysis system which claimed time-resolved measurement had to be able to cope with periods of signal drop-out in which, instantaneously, either no particles were present in the measurement area or the vector addition of the light signals from particles combined to produce a very low signal amplitude. For solid surface velocity measurement, light-scattering particles can be considered to be replaced by light-scattering surface elements. Then, theoretically, there is never any loss of signal due to absence of scatterers, and it was probably this single reason which meant that this form of measurement was considered to be relatively straightforward compared with its fluid counterpart. However, when coherent light is scattered by a diffusely reflecting surface, a laser speckle pattern is formed in space in front of the target [2], and the signal is produced by detecting the intensity of one or more speckles. It is this fact which distinguishes a solid surface velocity measurement and, indeed, it is the speckle pattern dynamics which limit the performance of the LDV system (see section 4.1). The physical principle of all LDVs relies upon the detection of the Doppler frequency shift in coherent light which occurs when it is scattered from a moving object. In Figure 1 is shown a schematic diagram of the effect when a particle moving with velocity U scatters light of wavelength l in a direction K2 from a laser beam of essentially single frequency travelling in a direction K1 (K1 and K2 are unit vectors). The change in light frequency Df produced by the moving particle is given by Df=U · (K2−K1 )/l=U · K/l,
(1)
where K=K2−K1 . A formal proof of this equation can be found in the text by Watrasiewicz [3]. It has been assumed that the refractive index of the medium in which scattering occurs is unity. In Figure 2 is shown an optical geometry which is appropriate for a solid surface measurement where the laser beam is directed normal to the surface. In this geometry K1 and K2 are parallel, i.e., K1=−K2 , so that the Doppler frequency shift fD measured corresponds to the surface velocity component in the direction of the incident beam and is given by fD=2U/l,
(2)
where U==U=. In this way, by measuring and tracking the change in fD , which is of the order of MHz, a time resolved measurement of the solid surface velocity U can be made.
Figure 1. The Doppler effect. Df=U · (K2−K1 )/l.
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Figure 2. The optical geometry for solid surface velocity measurement.
The scattered light shown in Figures 1 and 2 has a frequency which is of the order of 1015 Hz, and a frequency modulation of the order of MHz cannot be demodulated directly. The Doppler shift fD is measured electronically by mixing the scattered light with a reference beam derived from the same coherent source, onto the surface of a photodetector. The latter responds to the intensity from the total light collected and this non-linear detection produces a heterodyne or ‘‘beat’’ in the current output, the frequency of which is equal to the difference in frequency between the two collected beams. This is shown schematically in Figure 3, where a beam splitter has been used to mix the two beams. Unfortunately, the system as depicted in Figure 3 is ambiguous in the measurement of the direction of motion. In Figure 4 is shown the frequency spectrum of the photodetector output. The Doppler signal assumes a zero frequency value twice per vibration period when the solid surface velocity is zero. When the Doppler signal has a non-zero frequency value it is not possible to distinguish whether the target surface is moving away from or towards the detector. Vibration engineers require amplitude and phase of the target surface motion and this is usually provided by frequency pre-shifting the reference beam. In Figure 5 is shown an equivalent optical geometry in which the reference beam has been frequency pre-shifted by an amount fR . When considering the frequency spectrum of the photodetector output, it is now clear that the frequency pre-shift provides a carrier frequency which the target surface velocity can frequency modulate. The frequency shifted signal is often referred to as the Doppler signal, and when this is demodulated it is possible
Figure 3. Reference beam heterodyning.
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Figure 4. Doppler signal ambiguity.
to produce a time-resolved analogue measurement of the solid surface velocity in both amplitude and phase, as shown schematically in the figure. A complete description of the various means of demodulating a Doppler signal is beyond the scope of this paper, and readers are again referred to the text by Durst et al. [1]. The choice of method is dictated by the characteristics of the Doppler signal itself, which are directly related to the particular measurement problem. All demodulation techniques produce a time-resolved voltage analogue of the Doppler frequency with an associated frequency response limit.
Figure 5. Frequency shifting and time-resolved measurement.
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In the case of solid surface vibration measurement Doppler signals are continuous, but they can be subject to periods of low amplitude due to speckle dynamic effects. Periods of low signal amplitude are analogous to ‘‘drop-outs’’ in the fluid flow case. Where it can be reasonably assured that a target surface is moving parallel to the incident laser beam, the Doppler signal takes the form of a frequency modulated carrier of essentially constant amplitude. This form of signal can be demodulated with extreme accuracy and sensitivity and measurement of vibration velocity as low as nanometres per second are possible. If the target motion induces spatial or temporal changes in the speckle pattern formed on the detector, then performance is degraded. This is the case when torsional vibration measurements are attempted on rotating targets, and this issue is addressed in section 4.1. Laser Doppler velocimeters for solid surface target use are often referred to as laser Doppler vibrometers, or simply laser vibrometers. They all work on the physical principle described in this section and differ only in the choice of optical geometry and the type of frequency shifting device used. Just as frequency shifting is paramount for solid surface vibration measurement, it is also included as a standard item in all commercially available LDV systems which are used for flow measurement. It is obviously necessary for measurements in highly oscillatory flows, and in practice it is extremely useful to have a carrier frequency corresponding to zero motion for alignment and calibration purposes. For a discussion of frequency shifting devices, readers are referred to the text by Durst et al. [1]. Fortunately, for the purpose of torsional vibration measurements, the unidirectional motion of a surface scattering element on a rotating target means that use of frequency shifting is not necessary. 2.2. - The cross-beam, or dual beam, laser Doppler velocimeter [1] provided a means of measuring tangential surface velocities and hence torsional vibration velocity without surface contact, and this is shown schematically in Figure 6. With this optical geometry
Figure 6. The cross-beam velocimeter instrument.
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the backscattered light produces a Doppler beat frequency fD in the photodetector output which is given by fD=(2U/l) sin (u/2),
(3)
where U is the tangential surface velocity at the point of intersection of the beams, l is the wavelength of the laser light and u is the included angle between the incident laser beams. Demodulating the Doppler signal produces a time-resolved voltage analogue of U, the fluctuating part of which is proportional to the torsional vibration velocity. This design was used to measure torsional oscillations of the crankshaft of a six cylinder in-line diesel engine [4]. A comparison of the results obtained with those given by using a slotted disc system mentioned in the Introduction is shown in Figure 7. A further laser measurement taken with the engine operating under no load conditions confirms the expected overall reduction in vibration level. Broad agreement with the slotted disc system is achieved except at very low levels, which can be attributed to the limited sensitivity of the disc system used. With reference to Figure 6, the cross-beam laser velocimeter suffers two major disadvantages in practice. Firstly, the intersection region of the laser beams (where the target surface must remain at all times) is typically less than 1 mm in length and consequently the target must have a circular cross-section and the instrument must be tripod-mounted at a fixed distance. Clearly, gross solid body movement of the target or instrument will prevent a measurement being taken. Secondly, because the target has a solid body oscillation in practice, the component of oscillation in the direction of the tangential surface velocity contaminates the data, so that with this cross-beam geometry torsional oscillations cannot be distinguished from solid body movement. A final practical point is that the mean Doppler frequency is dictated by the angle of the beam intersection u, which is fixed. If the instrument is to be used over a wide speed range (100–10 000 rpm), then the associated electronic processing must have a very wide bandwidth which adds expense and complexity to the final design.
Figure 7. A comparison of cross-beam velocimeter and slotted disc torsiograph results; eighth order of rotation. w, Slotted disc, full load; q, laser, full load; W, laser, no load.
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These points prevent the cross-beam geometry providing a reliable and user friendly instrument, and laser technology for torsional vibration measurement did not develop for over a decade, until the invention of a new geometry which will be described in the next section. 2.3. The optical geometry for the laser torsional vibrometer is shown schematically in Figure 8. With reference to this figure, the theory will be developed for the system operating on the side of a shaft of arbitrary cross-sectional area which is rotating about an axis defined by the unit vector zˆ which is assumed to be perpendicular to the plane of the latter. The shaft itself is allowed to oscillate as a rigid body with an instantaneous velocity vector V. The laser beam is divided into two equal intensity parallel beams of separation d, which impinge on the shaft surface at points A and B in a direction defined by the unit vector i . The instantaneous shaft surface velocities, with respect to the axis of rotation, at these points are V1 and V2 , respectively. Under normal circumstances the laser used would be a low powered (22 mW) helium–neon which produces red light at a wavelength l=6328×10−10 m. The essentially single frequency light from this laser undergoes a Doppler shift fD when scattered by the moving surface and light collected in direct backscatter is shifted by an amount given by equation (2) of fD=2U:l,
(4)
where U is the instantaneous velocity in the direction of the incident laser beam and l is the laser wavelength. Accordingly light backscattered from the points A and B undergoes Doppler shifts fA and fB given by fA=(2/l)i · (V+V1 ),
fB=(2/l)i · (V+V2 ),
(5, 6)
respectively. When this backscattered light is mixed onto the surface of a photodetector, heterodyning takes place and the output current from the detector is modulated at the difference or ‘‘beat’’ frequency, fD , given by fD=fA−fB=(2/l)i · (V1−V2 ), where now the sensitivity to solid body motion V has been removed.
Figure 8. The laser torsional vibrometer optical geometry.
(7)
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Considering the vector V1−V2 , one can write V1=2pN(R1×zˆ),
V2=2pN(R2×zˆ),
(8, 9)
where N is the shaft revolutions per second. Thus V1−V2=2pN(R1−R2 )×zˆ=2pNBA×zˆ, where BA represents the vector R1−R2 . Equivalently, fD=(4pN/l)zˆ · (i ×BA).
(10)
Examining the vector product term (i ×BA)==i = =BA(sin d)t ,
(11)
where t is a unit vector perpendicular to i and BA, and d is their included angle, suggests a simplification, in that =BA=(sin d) is equal to the beam separation, d, and hence fD=(4pN/l) d zˆ · t .
(12)
The scalar product term zˆ · t can be expanded to give the final general result, fD=(4pd/l)(cos u)N,
(13)
where u is the angle between the normal to the plane of the incident beams and the rotational axis of the shaft. If the instrument is held so that the plane of the incident laser beams is parallel to the shaft cross-section, then u=0 and hence fD=(4pd/l)N.
(14)
Doppler frequency demodulation of the photodetector output now provides a time-resolved voltage analgoue of the speed of rotation of the target component, the fluctuating part of which is the torsional vibration. The frequency response of the instrument is dictated by that of the demodulation system used and the usual bandwidth of practical interest is up to 1 kHz. The optical geometry used makes the instrument insensitive to solid body oscillation of the target or operator. The incident laser beams are parallel and therefore any solid body motion produces an identical Doppler frequency shift in the light scattered from the points A and B. The subsequent heterodyning of this light at the photodetector, which produces a current output proportional to any frequency difference, is therefore insensitive to this form of motion. With reference to Figure 8, the frequency shift in the backscattered light from A or B is not dependent on their radial distance from the rotational axis and is therefore independent of the cross-sectional shape of the component. This fact allows the instrument to work successfully on targets of arbitrary cross-section, such as gear wheels, and there is no restriction to components of circular cross-section as with the cross-beam geometry discussed in the last section. When the light is heterodyned, the ‘‘beat’’ frequency is dependent only upon the beam separation d. With reference to equation (14), if this position is considered as a reference a lateral tilt or rotation of the laser beam plane will result in a value of u other than zero. For end of shaft use, it is necessary to note that careful choice of u is needed which, together with suitable adjustment of the beam separation, d, can provide immediate control of the mean value of the beat frequency. This aids electronic design for the frequency to voltage converter required to demodulate the output signal from the photodetector. Although the laser torsional vibrometer is insensitive to solid body movement of the
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operator or shaft, hand holding the instrument can produce a tilt of the laser beam plane. A tilting motion of the operator or shaft rotational axis will modulate the Doppler frequency fD as a result of changes in u as defined by equation (13). For a discussion of these effects, readers are referred to the paper by Halliwell and Eastwood [5] and the recent work of Miles et al. [6]. Tilt effects have been shown to be negligible in practice if the instrument is used such that the plane of the incident beams is parallel with the shaft cross-section (u=0°). They must be considered, however, if use is envisaged at a non-zero value of u (end of shaft use). 3. LASER TORSIONAL VIBROMETER MEASUREMENTS
3.1. A photograph of the first prototype laser torsional vibrometer built at ISVR is shown in Figure 9. This instrument was used to measure the rotational speed variations of a disc driven by a brushless d.c. motor, the drive voltage of which was modulated by a sinusoidal voltage. For comparison, the speed variations were also measured with a cross-beam torsional vibrometer as described in section 2.2. In Figure 10 are shown the angular displacements measured by the two instruments against variations in the frequency of the voltage supply. Agreement to within 0·5 dB is demonstrated over the range 30–130 Hz achieved in this test. In Figure 11 two torsional vibration spectra with a fundamental frequency of 100 Hz which were obtained in this way are compared and excellent agreement is observed. A photograph of the commercial version of the laser torsional vibrometer produced by Bru¨el and Kjaer Ltd of Denmark is shown in Figure 12. This is the Type 2523 Torsional Displacement Meter. This instrument was used to measure the torsional vibrations induced in a rotating shaft by a Hooke’s joint [7] which provides a means of checking the levels of torsional vibration velocity. With reference to Figure 13, speed fluctuations are generated in the driven shaft due to the time-variant torque transmission characteristics of the Hooke’s joint. The amplitude of the fluctuations depends upon the angle f shown in the figure and the amplitude of the second order fluctuation is readily calculated [7]. When the driven shaft is measured by the Type 2523 comparison between experiment and theory is straightforward. The results of this comparison are shown in Figure 14 where
Figure 9. The prototype laser torsional vibrometer.
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Figure 10. A comparison of results from the cross-beam velocimeter (W) and laser torsional vibrometer (w).
the abscissae represents negative and positive inclination angles f and the ordinate represents angular displacement in degrees peak. Experimental results are shown for measured values of the second order of torsional vibration at three different speeds of rotation. These results demonstrate the accuracy and consistency of the Torsional Vibration Meter Type 2523. The theoretical curve is independent of the rotational speed and consequently the arbitrarily chosen rotational speeds all provide the same measurement results to close agreement. This demonstrates the capability of the Torsional Vibration Meter Type 2523 precisely to measure the torsional vibration experienced by any rotating component. 3.2. : The majority of diesel engines require some form of damping or detuning device to be fitted in order to prevent the build-up of large vibration amplitudes and stresses at torsional resonance. Two basic designs are in common use, these being the viscous shear damper and the elastomeric (rubber) equivalent. The former is the usual choice for higher power output engines because of its superior heat dissipation capability, while the latter is a comparatively cheap unit and is very popular for automotive type engines. The laser torsional vibrometer has provided a means of assessing the performance of a torsional vibration damper in situ, and readers are referred to the papers by Halliwell and Eastwood [8, 9] for a full description of this application. With reference to Figure 15, elastomeric dampers consist of a heavy seismic mass (the inertia ring) and a relatively lighter hub which is attached to the free end of the engine crankshaft. The two members are coupled by an elastomer element which provides stiffness and damping. A vibration damper of this design, therefore, adds another mass and elasticity to the basic equivalent dynamic system of the engine and introduces an additional torsional natural frequency. It therefore has a ‘‘tuning’’ capability. Laser torsional vibrometry was used to assess the performance of an elastomeric (bonded rubber) damper which was fixed to the end of the crankshaft of a six-cylinder turbocharged diesel. A simple, two-mass model [8] of the crankshaft system predicted two torsional resonances excited by the ninth order harmonic at engine speeds of 1050 rpm and
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Figure 11. A comparison of torsional vibration spectra. (a) Laser torsional vibrometer; (b) cross-beam velocimeter.
1850 rpm respectively. This theoretical result is shown in Figure 16. For comparison, in Figure 17 is shown an experimentally determined ninth order plot obtained by using a laser torsional vibrometer, which verifies the use of the model for resonance prediction to within 5% error. Loss of elastomeric damper performance can be attributed to changes in the material properties of the elastomer element over time. In particular, changes in the stiffness provided by the damper under optimal conditions will rapidly produce a decline in performance through de-tuning. It is well known that stiffness of the element changes with temperature and, consequently, monitoring the performance of a damper from ‘‘cold’’ start of the engine to normal operating conditions would represent that which might be anticipated from a damper progressively degrading with age and/or wear. It should be noted, however, that in the former case the behaviourial trend will occur in reverse and at a greatly accelerated rate. Two laser torsional vibrometers simultaneously measured the response of the hub (u) and the inertia ring (ud ). Data was recorded immediately following engine start-up and subsequently after intervals of ten minutes. The results are presented in Table 1. For the engine under test, the sixth order was monitored at an engine speed of 1895 rpm.
Figure 12. The Bru¨el & Kjaer type 2523 Laser Torsional Displacement Meter.
Figure 13. The Hooke’s joint checking method.
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Figure 14. A measurement comparison with Hooke’s joint predictions. ——, Theoretical; ×– – –, 700 rpm; W · · ·, 100 rpm; q — – —, 1300 rpm.
Measurements obtained after a period of 30 minutes are omitted since after this time the ratio (ud /u) remained constant. For the two-mass model [8] these results show that a change in stiffness of the damper element of approximately 62% has occurred between ‘‘cold’’ and stable operating conditions. Such a difference is not uncommon; manufacturers report differences of over 100% in some circumstances.
4. PRACTICAL CONSIDERATIONS 4.1.
It is the dynamics and characteristics of the laser speckle pattern [2] collected by the photodetector which dictate the sensitivity of the laser torsional vibrometer. This is best understood by considering the geometry of a standard laser vibrometer as shown in Figure 5 and analyzing the sensitivity of this device. When engineering target surfaces which are rough on the scale of the optical wavelength scatter coherent light, a speckle pattern forms in space in front of the target. This can be
Figure 15. The elastomeric damper.
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Figure 16. The theoretical prediction of ninth order torsional vibration [8, 9].
observed on a screen as a continuous random array of dark and bright spots which have a grainy speckly appearance and hence the name laser speckle pattern. The phenomenon occurs because each scattering surface element within the laser spot acts like a point source of coherent light. At a point in space, individual scattered wavelets from these sources interefere constructively or destructively to produce the bright or dark speckle, respectively, which is observed on the screen. The speckle pattern is a continuous random distribution of light amplitude and phase although in practice we simply observe the corresponding intensity distribution. An average speckle size can be calculated and attributed to the distribution observed on the screen [10]. Within this correlation region the amplitude and phase of the light may be assumed to be constant. In laser vibrometry, the photodetector active area in the instrument is sampling a speckle pattern. One or more speckles may be sampled, depending upon the optical geometry used in a given measurement situation and also the collecting optics used (if any) for the scattered light. If the detector area collects several speckles, the current output is proportional to the instantaneous sum of the intensity distribution which has been sampled.
Figure 17. As Figure 16, but experimental measurement.
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T 1 The simulation of elastomeric torsional damper failure
Elapsed time from ‘‘cold start’’ (mins) 0 10 20 30
increasing stiffness 3
Amplitude of sixth order torsional harmonic (degrees) ZXXXXXCXXXXXV Damper Inertia hub, u ring, ud 0·497 0·251 0·226 0·179
1·179 0·923 0·899 0·774
Amplitude ratio, ud /u 2·37 3·68 3·98 4·33
When a frequency-shifted reference beam is superposed onto the photodetector, a heterodyne beat occurs as described in section 2.1. For simplicity, consider what happens when the reference beam has a constant spatial phase distribution across the dector area and the target is stationary. It is useful to think of each speckle region as an interferometer which has a distinct value of intensity and phase. In practice, the detector output is the resultant vector addition of the outputs from all these separate interferometers corresponding to each speckle collected. In this way, for a stationary target the output from the detector is at the beat frequency with a constant amplitude and phase. When the target moves normal to surface and parallel to the incident laser beam, the Doppler effect produces a uniform rate of change of phase across the detector area and a pure frequency modulation of the detector output occurs with negligible amplitude modulation. A word of caution is necessary here. Speckles form small, continuous ‘‘cigar-shaped’’ volumes in space and the detector plane effectively moves along the major
Figure 18. The torsional vibration spectrum: speckle pattern periodicity.
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Figure 19. The torsional vibration spectrum with the speckle pattern periodicity removed.
axis of each volume when the target vibrates. Movement along speckle volumes would therefore cause amplitude modulation, but this effect is usually negligible. If, however, the target tilts or moves other than normal to surface, the characteristics of the speckle pattern sampled by the detector will change spatially and/or temporally. This effect produces a phase modulation of the detector output, which is demodulated and is indistinguishable from the change in Doppler frequency associated with normal-to-surface movement. This is a major source of noise in laser vibrometers and has been referred to as pseudo-vibration [11]. When the scattering elements in the laser spot rotate locally due to surface tilt, the associated speckle pattern on the detector moves in sympathy, and this is usually the dominant noise source. However, if a significant change in population of scattering elements occurs (perhaps due to in-plane target motion) then the speckle pattern will decorrelate in sympathy with this change and produce noise. In practice, both these mechanisms contribute noise in the laser vibrometer output. The light intensity incident at a point on the photodetector surface Ip (t) can be written as [2]
$ g
%
Ip (t)=IR+IT+2zIR IT cos vR t− vT dt+fR−fT ,
(15)
where IR and fR are the reference beam intensity and initial phase, respectively, IT and fT are the target scattered light intensity and resultant phase, respectively, vR is the constant reference beam angular frequency pre-shift and vT is the ‘‘instantaneous’’ Doppler angular frequency shift produced by target movement. This equation is derived after neglecting polarization effects. The very high frequency modulation of Ip caused by the summed frequency term in the heterodyne process is beyond the frequency response of the photodetector.
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In practice, the photodetector takes the sum of Ip (t) over its active area A, and one must therefore consider the spatial distributions of IR and IT and their respective phases fR and fT . For convenience, one can assume that reference beam intensity and phase is uniform in space and one need consider only the speckle pattern distribution. Accordingly, one can write the detector output current i(t) as i(t)=
g
A
$ g
%
[IR+IT (a)+2zIR IT (a) cos vR t− vT dt+fR−fT (a) dA,
(16)
where A is the photodetector area, IT (a) is the speckle intensity distribution and fT (a) is the speckle phase distribution. Examination of equation (16) confirms the earlier discussion that if the spatial distribution of the speckle pattern changes during a measurement period (i.e., if IT (a) and fT (a) are functions of time) then spurious information will be contributed when i(t) is frequency demodulated. Unfortunately, in practice, this noise is usually linked to the vibration frequency of interest, introducing uncertainty into the measurement. When a laser vibrometer is pointed at a rotating target, the spatial and temporal characteristics of the speckle pattern sampled by the photodetector change with time. With reference to equation (16), the terms IT (a) and fT (a) describing the speckle pattern intensity and phase distribution across the detector surface become periodic at the rotation frequency and are demodulated as pseudo-vibration. Consequently, when the target surface is not vibrating but simply rotating, the laser vibrometer will produce a noise floor which has a spectrum typical of a pseudo-random signal. This consists of a fundamental at the rotation frequency and higher order harmonics of similar magnitude. It is produced because the changing speckle pattern produces random phase and amplitude noise which repeats exactly in sympathy with the rotating target. For a further discussion of this effect, the reader is referred to the papers by Rothberg et al. [11–13]. Unfortunately, in practice, the vibration frequencies of interest are usually direct integer multiples of the rotation frequency and consequently the distribution of this speckle noise is a worst possible case. It is at this stage that a degree of engineering judgement is required before deciding whether the vibration levels measured in a given bandwidth are not those associated with the pseudo-vibration of speckle noise. Users need to be wary of the very low noise floors and sensitivities quoted by instrument manufacturers in the literature. These have usually been measured on a stationary target and, in practice, as discussed in the last section, the true noise floor is peculiar to the speckle pattern behaviour caused by target surface dynamics during the actual measurement. For the laser torsional vibrometer, and with reference to Figure 8, the photodetector collects two speckle patterns from points A and B which then heterodyne on its active area. If the target component has no torsional vibration whatsoever, the frequency spectrum of the instrument output due to the steady rotation will take the form of a periodogram at the fundamental rotation frequency. An example of this is shown in Figure 18, where the instrument was used to measure the speed fluctuation of a d.c. motor from a tape recorder drive. The displacement spectrum shown clearly exhibits the periodic nature of the speckle noise which completely masks the intended measurement. Fortunately, the insensitivity of the instrument to solid body oscillation means that the speckle periodicity can be destroyed by moving the laser beam plane during the course of the measurement. Hand holding the instrument and moving the beams from side to side means that the speckle pattern incident on the photodetector does not exactly repeat. In Figure 19 is shown the equivalent measurement taken on the d.c. motor when the incident laser beam plane was moved from side to side at very low frequency of about 1 Hz. The figure shows the effect of removing the speckle pattern periodic noise, in this way, and
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the ‘‘wow’’ of the motor can now be distinguished. For most mechanical engineering applications, a level of −60 dB re 1° peak means that torsional vibrations have ceased to be a problem and therefore speckle periodicity is not troublesome. The Type 2523 Laser Torsional Vibration Meter has a ‘‘noise smearing’’ facility. This is intended for use at very low levels of torsional vibration and allows the speckle pattern periodicity to be destroyed by modulating the spatial position of the incident laser beams from side to side in a random manner. It is not necessary to use this when measuring levels greater than −40 dB re 1° peak displacement, which is normally the case in practice. 4.2. In practice, the highest levels of torsional vibration will be found in the crankshafts of large diesel engines of reciprocating compressors. These may be as high as a few degrees of peak displacement, but in frequency terms these levels still represent a relatively small modulation of the mean beat frequency resulting from the target speed. Consequently, measuring high levels does not present a problem, and it is simply necessary to ensure that they are within the frequency response range of the Doppler signal processor. It is the range of possible rotational speeds which represents much more of a practical problem for signal processors. At a beam separation of 1 cm for beams perpendicular to the rotational axis (u=0), speed variation of 500 rpm to 15 000 rpm produces a beat frequency range of 1·5 MHz to 45 MHz! The theory for the instrument is developed on the assumption that the incident laser beams are infinitely thin and therefore, by an order of magnitude argument, a minimum beam separation of one cm is required. The lower speed range can therefore be extended by widening the beam separation in order to match the input bandwidth of the processor. At the higher rotational speeds, using the instrument so that the incident laser beam plane is at an angle to the rotational axis will reduce the mean beat frequency. In this situation, care should be taken to account for the effects of operator or component tilt [5, 6]. When a low powered helium–neon laser is used as a light source so that the outgoing beams have about 1 mW power, the target distance for operation is up to several metres, but retroreflective tape or paint must be used. Calibration of the laser torsional vibrometer should be carried out in situ. It is not necessary to attempt to measure the beam separation d or angle of inclination of the normal to the laser beam plane u with the rotational axis. In most circumstances it is possible to rotate the target at several known speeds so that an in situ calibration of mean d.c. volts per rpm is possible. If it is not possible to rotate the target at known speeds, then a small disc can be used, which is driven at known speeds and interposed in the laser beams in place of the target. It is only necessary to monitor the d.c. voltage from the instrument at three known speeds to produce a calibration factor giving negligible error. It is usual to output the instrument signal straight to a real time spectrum analyzer which will integrate the output directly to produce vibration spectra in terms of torsional vibration displacement. The alternative, of course, is to tape record the analogue output on site for later analysis, but it must be remembered that the d.c. level is important if an in situ calibration has taken place. 5. CONCLUSIONS
The problems of torsional vibration measurement have been solved by the invention of the laser torsional vibrometer. The use of optical, seismic and mechanical torsiographs, together with strain gauges, slip rings and slotted discs, has now been overtaken by the use of laser technology. The laser torsional vibrometer offers in situ measurement, thus
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avoiding machinery downtime and can function on rotating components of arbitrary shape whilst maintaining an immunity to solid body motion of the operator or component. The instrument is robust, user friendly and can be calibrated in situ. Invented at ISVR in 1983 and subsequently produced commercially by Bru¨el and Kjaer Ltd of Denmark as the Type 2523 Torsional Displacement Meter, the vibrometer represents a significant step forward in rotating machinery diagnostics. It is now used on a worldwide basis and is rapidly gaining acceptance as the standard means of measuring torsional vibration. ACKNOWLEDGMENTS
The author wishes to acknowledge the provision of Hooke’s joint results by T. J. Miles. REFERENCES 1. F. D, A. M and J. H. W 1981 Principles and Practice of Laser Doppler Anemometry (second edition). London: Academic Press. 2. J. C. D (editor) Laser Speckle and Related Phenomena. New York: Springer-Verlag. 3. B. M. W and M. J. R 1976 Laser Doppler Measurements. London: Butterworth. 4. N. A. H, H. L. P and J. B 1983 Proceedings of the Society of Automotive Engineers, International Off-Highway Meeting and Exposition, Milwaukee, Wisconsin, USA, September, paper 831324, 986–994. Diesel engine health: laser diagnostics. 5. N. A. H and P. G. E 1985 Journal of Sound and Vibration 101, 446–449. The laser torsional vibrometer. 6. T. J. M, M. L and S. J. R 1995 Proceedings of the ASME 15th Biennial Conference on Mechanical Vibration and Noise Boston, U.S.A., September 3(C), 1451–1460. The laser torsional vibrometer: successful operation during lateral vibrations. 7. E. J. N 1958 A Handbook on Torsional Vibration. Cambridge: Cambridge University Press. 8. N. A. H and P. G. E 1990 Proceedings of the I. Mech. E. Conference on Automotive Diagnostics, 115–122; London, I. Mech. E. Torsional vibration damper performance: diagnosis using laser technology. 9. N. A. H and P. G. E 1992 Optics and Lasers in Engineering 16, 337–350. Marine vehicles and offshore installations: laser diagnostics of machinery health. 10. M. F 1979 Laser Speckle and Applications in Optics. London: Academic Press. 11. S. J. R, J. R. B and N. A. H 1989 Journal of Sound and Vibration 135, 516–522. Laser vibrometry: pseudo vibrations. 12. S. J. R, J. R. B and N. A. H 1990 Journal of Laser Applications 2(1), 29–36. On laser vibrometry of rotating targets: effect of in-plane and torsional vibration. 13. S. J. R and N. A. H 1994 Transactions of the American Society of Mechanical Engineers, Journal of Vibration and Acosutics 116(3), 326–331. Vibration measurements on rotating machinery using laser Doppler velocimetry.
THE LASER TORSIONAL VIBROMETER: A STEP FORWARD IN ROTATING MACHINERY DIAGNOSTICS N. A. H Department of Mechanical Engineering, Loughborough University of Technology, Loughborough, LE11 3TU, England (Received 1 November 1995) This paper describes the development of the laser torsional vibrometer since its invention at the Institute of Sound and Vibration Research, University of Southampton in 1983. The theory of laser vibrometry measurements is first introduced together with a description of early attempts to harness the technology for torsional vibration measurement through use of a cross-beam laser Doppler velocimeter. This is followed by the theory for the laser torsional vibrometer itself. A measurements section then describes results taken by the first prototype built at ISVR and compares these with the cross-beam velocimeter in a controlled experiment. Further tests with the commercial version of the instrument, produced by Bru¨el and Kjaer Ltd as the Type 2523 Laser Torsional Vibration Meter, are described, involving checks with the well known Hooke’s joint. Results from an investigation into torsional vibration damper performance are also presented. The sensitivity of the instrument is dealt with in detail before practical considerations are described. The laser torsional vibrometer has solved a particularly difficult metrology problem and is now used on a worldwide basis. It is rapidly gaining acceptance as the standard means of measuring torsional vibration. 7 1996 Academic Press Limited
PREFACE
Elfyn Richards returned to the I.S.V.R. from his position as Vice-Chancellor of Loughborough University in 1975. He immediately began to form a research group working in the area of machinery noise. After a three-year period as research fellow in the Aeronautics Department, working with laser technology, I joined the staff of the I.S.V.R. in 1977. I will never forget Elfyn walking into my office and advising me to ‘‘stop playing with lasers’’ and to join his research group as his right-hand man. After much thought, and persuasion by the then Director, Professor Large, I joined Elfyn’s team. There then began a most lively and interesting period in my career. Long car journeys, when the two of us travelled to the Midlands and back to promote the group’s work, were particularly enjoyable. He would put the world to rights in terms of acoustic and vibration research, and I learned a tremendous amount from him. He was a man of great vision, and I was privileged to work so closely with him. The machinery noise work spawned many interesting problems, and I returned to playing with lasers in 1979. 1. INTRODUCTION
Since the advent of the laser in the early 1960s, the field of optical metrology has provided accurate experimental data in situations in which, hitherto, it would have been considered unobtainable. The technique of laser Doppler velocimetry (LDV) [1] is now well 399 0022–460X/96/080399+20 $12.00/0
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established and was initially applied to obtain non-intrusive measurements in fluid flows. Although the use of LDV for solid surface velocity measurement was recognized at an early stage, its development in this area received little attention compared with the effort in fluid mechanics. Accordingly, the measurements of vibration was still extensively achieved with accelerometers or other forms of transducer which rely on contact with the measurement surface for successful operation. There are, however, many cases of engineering interest where this approach is either impractical or impossible. Typical examples are the measurement of very hot or light surfaces, such as exhaust pipes or loudspeakers, and measurement on rotating surfaces which preclude their use. In the latter area the measurement of torsional vibration of rotating components presented a particularly difficult measurement problem. When designing rotating machinery components, an engineer must be careful to suppress torsional oscillations, since incorrect or insufficient control may lead to fatigue failure, rapid bearing wear, gear hammer, fan belt slippage and can produce associated excessive noise problems. Torsional oscillations are a particular problem in engine crankshaft design where torsional dampers are commonly used to maintain oscillations at an acceptable level over the working speed range of the engine. Torsional transducers have formerly included optical, seismic and mechanical torsiographs, strain gauges and slotted discs. The latter system has found common use in the automotive industry and consists of a slotted disc fixed to the end of the crankshaft. A proximity transducer monitors the slot passing frequency, which is then demodulated to provide a voltage analogue of the crankshaft speed and hence torsional oscillations, but within a limited frequency range. Strain gauges and associated telemetry or slip ring systems are notoriously difficult to fix, calibrate and use successfully. In summary, the measurement of torsional oscillations presented difficult problems for contacting transducer technology and, of course, necessitated machinery ‘‘downtime’’ and special arrangements being made for fitting, calibration, etc. Very often, the cost of this machinery stoppage would preclude a measurement being attempted, even though the vibration engineer had concluded that it was vital if a design improvement is to be made. There was therefore a real need for a torsional vibration transducer which was user friendly and could provide data immediately in on-site situations. It was not until the advent of laser technology that a solution was found and, in what follows, the laser torsional vibrometer is described, which allows the engineer to point low powered laser beams at a rotating target component and obtain torsional vibration information.
2. LASER VIBROMETRY
2.1. Laser Doppler velocimetry (LDV) or laser doppler anemometry (LDA) is now a well-established technique, which is used primarily for non-intrusive measurement in fluid flows [1]. Fundamentally, LDV measures the velocity of a light-scattering particle which is seeded into the flow of interest, and it is assumed that the particle follows the flow faithfully. Under normal operating conditions the velocity of a group of seeding particles in the measurement region is detected. Clearly, the number density of scattering particles is important since they essentially ‘‘sample’’ the flow, and much of the early work in LDV was concerned with the validity or otherwise of statistical conclusions drawn from a particular measurement. An obvious practical problem to be overcome was the
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intermittence of seeding particles. An analysis system which claimed time-resolved measurement had to be able to cope with periods of signal drop-out in which, instantaneously, either no particles were present in the measurement area or the vector addition of the light signals from particles combined to produce a very low signal amplitude. For solid surface velocity measurement, light-scattering particles can be considered to be replaced by light-scattering surface elements. Then, theoretically, there is never any loss of signal due to absence of scatterers, and it was probably this single reason which meant that this form of measurement was considered to be relatively straightforward compared with its fluid counterpart. However, when coherent light is scattered by a diffusely reflecting surface, a laser speckle pattern is formed in space in front of the target [2], and the signal is produced by detecting the intensity of one or more speckles. It is this fact which distinguishes a solid surface velocity measurement and, indeed, it is the speckle pattern dynamics which limit the performance of the LDV system (see section 4.1). The physical principle of all LDVs relies upon the detection of the Doppler frequency shift in coherent light which occurs when it is scattered from a moving object. In Figure 1 is shown a schematic diagram of the effect when a particle moving with velocity U scatters light of wavelength l in a direction K2 from a laser beam of essentially single frequency travelling in a direction K1 (K1 and K2 are unit vectors). The change in light frequency Df produced by the moving particle is given by Df=U · (K2−K1 )/l=U · K/l,
(1)
where K=K2−K1 . A formal proof of this equation can be found in the text by Watrasiewicz [3]. It has been assumed that the refractive index of the medium in which scattering occurs is unity. In Figure 2 is shown an optical geometry which is appropriate for a solid surface measurement where the laser beam is directed normal to the surface. In this geometry K1 and K2 are parallel, i.e., K1=−K2 , so that the Doppler frequency shift fD measured corresponds to the surface velocity component in the direction of the incident beam and is given by fD=2U/l,
(2)
where U==U=. In this way, by measuring and tracking the change in fD , which is of the order of MHz, a time resolved measurement of the solid surface velocity U can be made.
Figure 1. The Doppler effect. Df=U · (K2−K1 )/l.
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Figure 2. The optical geometry for solid surface velocity measurement.
The scattered light shown in Figures 1 and 2 has a frequency which is of the order of 1015 Hz, and a frequency modulation of the order of MHz cannot be demodulated directly. The Doppler shift fD is measured electronically by mixing the scattered light with a reference beam derived from the same coherent source, onto the surface of a photodetector. The latter responds to the intensity from the total light collected and this non-linear detection produces a heterodyne or ‘‘beat’’ in the current output, the frequency of which is equal to the difference in frequency between the two collected beams. This is shown schematically in Figure 3, where a beam splitter has been used to mix the two beams. Unfortunately, the system as depicted in Figure 3 is ambiguous in the measurement of the direction of motion. In Figure 4 is shown the frequency spectrum of the photodetector output. The Doppler signal assumes a zero frequency value twice per vibration period when the solid surface velocity is zero. When the Doppler signal has a non-zero frequency value it is not possible to distinguish whether the target surface is moving away from or towards the detector. Vibration engineers require amplitude and phase of the target surface motion and this is usually provided by frequency pre-shifting the reference beam. In Figure 5 is shown an equivalent optical geometry in which the reference beam has been frequency pre-shifted by an amount fR . When considering the frequency spectrum of the photodetector output, it is now clear that the frequency pre-shift provides a carrier frequency which the target surface velocity can frequency modulate. The frequency shifted signal is often referred to as the Doppler signal, and when this is demodulated it is possible
Figure 3. Reference beam heterodyning.
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Figure 4. Doppler signal ambiguity.
to produce a time-resolved analogue measurement of the solid surface velocity in both amplitude and phase, as shown schematically in the figure. A complete description of the various means of demodulating a Doppler signal is beyond the scope of this paper, and readers are again referred to the text by Durst et al. [1]. The choice of method is dictated by the characteristics of the Doppler signal itself, which are directly related to the particular measurement problem. All demodulation techniques produce a time-resolved voltage analogue of the Doppler frequency with an associated frequency response limit.
Figure 5. Frequency shifting and time-resolved measurement.
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In the case of solid surface vibration measurement Doppler signals are continuous, but they can be subject to periods of low amplitude due to speckle dynamic effects. Periods of low signal amplitude are analogous to ‘‘drop-outs’’ in the fluid flow case. Where it can be reasonably assured that a target surface is moving parallel to the incident laser beam, the Doppler signal takes the form of a frequency modulated carrier of essentially constant amplitude. This form of signal can be demodulated with extreme accuracy and sensitivity and measurement of vibration velocity as low as nanometres per second are possible. If the target motion induces spatial or temporal changes in the speckle pattern formed on the detector, then performance is degraded. This is the case when torsional vibration measurements are attempted on rotating targets, and this issue is addressed in section 4.1. Laser Doppler velocimeters for solid surface target use are often referred to as laser Doppler vibrometers, or simply laser vibrometers. They all work on the physical principle described in this section and differ only in the choice of optical geometry and the type of frequency shifting device used. Just as frequency shifting is paramount for solid surface vibration measurement, it is also included as a standard item in all commercially available LDV systems which are used for flow measurement. It is obviously necessary for measurements in highly oscillatory flows, and in practice it is extremely useful to have a carrier frequency corresponding to zero motion for alignment and calibration purposes. For a discussion of frequency shifting devices, readers are referred to the text by Durst et al. [1]. Fortunately, for the purpose of torsional vibration measurements, the unidirectional motion of a surface scattering element on a rotating target means that use of frequency shifting is not necessary. 2.2. - The cross-beam, or dual beam, laser Doppler velocimeter [1] provided a means of measuring tangential surface velocities and hence torsional vibration velocity without surface contact, and this is shown schematically in Figure 6. With this optical geometry
Figure 6. The cross-beam velocimeter instrument.
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the backscattered light produces a Doppler beat frequency fD in the photodetector output which is given by fD=(2U/l) sin (u/2),
(3)
where U is the tangential surface velocity at the point of intersection of the beams, l is the wavelength of the laser light and u is the included angle between the incident laser beams. Demodulating the Doppler signal produces a time-resolved voltage analogue of U, the fluctuating part of which is proportional to the torsional vibration velocity. This design was used to measure torsional oscillations of the crankshaft of a six cylinder in-line diesel engine [4]. A comparison of the results obtained with those given by using a slotted disc system mentioned in the Introduction is shown in Figure 7. A further laser measurement taken with the engine operating under no load conditions confirms the expected overall reduction in vibration level. Broad agreement with the slotted disc system is achieved except at very low levels, which can be attributed to the limited sensitivity of the disc system used. With reference to Figure 6, the cross-beam laser velocimeter suffers two major disadvantages in practice. Firstly, the intersection region of the laser beams (where the target surface must remain at all times) is typically less than 1 mm in length and consequently the target must have a circular cross-section and the instrument must be tripod-mounted at a fixed distance. Clearly, gross solid body movement of the target or instrument will prevent a measurement being taken. Secondly, because the target has a solid body oscillation in practice, the component of oscillation in the direction of the tangential surface velocity contaminates the data, so that with this cross-beam geometry torsional oscillations cannot be distinguished from solid body movement. A final practical point is that the mean Doppler frequency is dictated by the angle of the beam intersection u, which is fixed. If the instrument is to be used over a wide speed range (100–10 000 rpm), then the associated electronic processing must have a very wide bandwidth which adds expense and complexity to the final design.
Figure 7. A comparison of cross-beam velocimeter and slotted disc torsiograph results; eighth order of rotation. w, Slotted disc, full load; q, laser, full load; W, laser, no load.
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These points prevent the cross-beam geometry providing a reliable and user friendly instrument, and laser technology for torsional vibration measurement did not develop for over a decade, until the invention of a new geometry which will be described in the next section. 2.3. The optical geometry for the laser torsional vibrometer is shown schematically in Figure 8. With reference to this figure, the theory will be developed for the system operating on the side of a shaft of arbitrary cross-sectional area which is rotating about an axis defined by the unit vector zˆ which is assumed to be perpendicular to the plane of the latter. The shaft itself is allowed to oscillate as a rigid body with an instantaneous velocity vector V. The laser beam is divided into two equal intensity parallel beams of separation d, which impinge on the shaft surface at points A and B in a direction defined by the unit vector i . The instantaneous shaft surface velocities, with respect to the axis of rotation, at these points are V1 and V2 , respectively. Under normal circumstances the laser used would be a low powered (22 mW) helium–neon which produces red light at a wavelength l=6328×10−10 m. The essentially single frequency light from this laser undergoes a Doppler shift fD when scattered by the moving surface and light collected in direct backscatter is shifted by an amount given by equation (2) of fD=2U:l,
(4)
where U is the instantaneous velocity in the direction of the incident laser beam and l is the laser wavelength. Accordingly light backscattered from the points A and B undergoes Doppler shifts fA and fB given by fA=(2/l)i · (V+V1 ),
fB=(2/l)i · (V+V2 ),
(5, 6)
respectively. When this backscattered light is mixed onto the surface of a photodetector, heterodyning takes place and the output current from the detector is modulated at the difference or ‘‘beat’’ frequency, fD , given by fD=fA−fB=(2/l)i · (V1−V2 ), where now the sensitivity to solid body motion V has been removed.
Figure 8. The laser torsional vibrometer optical geometry.
(7)
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Considering the vector V1−V2 , one can write V1=2pN(R1×zˆ),
V2=2pN(R2×zˆ),
(8, 9)
where N is the shaft revolutions per second. Thus V1−V2=2pN(R1−R2 )×zˆ=2pNBA×zˆ, where BA represents the vector R1−R2 . Equivalently, fD=(4pN/l)zˆ · (i ×BA).
(10)
Examining the vector product term (i ×BA)==i = =BA(sin d)t ,
(11)
where t is a unit vector perpendicular to i and BA, and d is their included angle, suggests a simplification, in that =BA=(sin d) is equal to the beam separation, d, and hence fD=(4pN/l) d zˆ · t .
(12)
The scalar product term zˆ · t can be expanded to give the final general result, fD=(4pd/l)(cos u)N,
(13)
where u is the angle between the normal to the plane of the incident beams and the rotational axis of the shaft. If the instrument is held so that the plane of the incident laser beams is parallel to the shaft cross-section, then u=0 and hence fD=(4pd/l)N.
(14)
Doppler frequency demodulation of the photodetector output now provides a time-resolved voltage analgoue of the speed of rotation of the target component, the fluctuating part of which is the torsional vibration. The frequency response of the instrument is dictated by that of the demodulation system used and the usual bandwidth of practical interest is up to 1 kHz. The optical geometry used makes the instrument insensitive to solid body oscillation of the target or operator. The incident laser beams are parallel and therefore any solid body motion produces an identical Doppler frequency shift in the light scattered from the points A and B. The subsequent heterodyning of this light at the photodetector, which produces a current output proportional to any frequency difference, is therefore insensitive to this form of motion. With reference to Figure 8, the frequency shift in the backscattered light from A or B is not dependent on their radial distance from the rotational axis and is therefore independent of the cross-sectional shape of the component. This fact allows the instrument to work successfully on targets of arbitrary cross-section, such as gear wheels, and there is no restriction to components of circular cross-section as with the cross-beam geometry discussed in the last section. When the light is heterodyned, the ‘‘beat’’ frequency is dependent only upon the beam separation d. With reference to equation (14), if this position is considered as a reference a lateral tilt or rotation of the laser beam plane will result in a value of u other than zero. For end of shaft use, it is necessary to note that careful choice of u is needed which, together with suitable adjustment of the beam separation, d, can provide immediate control of the mean value of the beat frequency. This aids electronic design for the frequency to voltage converter required to demodulate the output signal from the photodetector. Although the laser torsional vibrometer is insensitive to solid body movement of the
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operator or shaft, hand holding the instrument can produce a tilt of the laser beam plane. A tilting motion of the operator or shaft rotational axis will modulate the Doppler frequency fD as a result of changes in u as defined by equation (13). For a discussion of these effects, readers are referred to the paper by Halliwell and Eastwood [5] and the recent work of Miles et al. [6]. Tilt effects have been shown to be negligible in practice if the instrument is used such that the plane of the incident beams is parallel with the shaft cross-section (u=0°). They must be considered, however, if use is envisaged at a non-zero value of u (end of shaft use). 3. LASER TORSIONAL VIBROMETER MEASUREMENTS
3.1. A photograph of the first prototype laser torsional vibrometer built at ISVR is shown in Figure 9. This instrument was used to measure the rotational speed variations of a disc driven by a brushless d.c. motor, the drive voltage of which was modulated by a sinusoidal voltage. For comparison, the speed variations were also measured with a cross-beam torsional vibrometer as described in section 2.2. In Figure 10 are shown the angular displacements measured by the two instruments against variations in the frequency of the voltage supply. Agreement to within 0·5 dB is demonstrated over the range 30–130 Hz achieved in this test. In Figure 11 two torsional vibration spectra with a fundamental frequency of 100 Hz which were obtained in this way are compared and excellent agreement is observed. A photograph of the commercial version of the laser torsional vibrometer produced by Bru¨el and Kjaer Ltd of Denmark is shown in Figure 12. This is the Type 2523 Torsional Displacement Meter. This instrument was used to measure the torsional vibrations induced in a rotating shaft by a Hooke’s joint [7] which provides a means of checking the levels of torsional vibration velocity. With reference to Figure 13, speed fluctuations are generated in the driven shaft due to the time-variant torque transmission characteristics of the Hooke’s joint. The amplitude of the fluctuations depends upon the angle f shown in the figure and the amplitude of the second order fluctuation is readily calculated [7]. When the driven shaft is measured by the Type 2523 comparison between experiment and theory is straightforward. The results of this comparison are shown in Figure 14 where
Figure 9. The prototype laser torsional vibrometer.
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Figure 10. A comparison of results from the cross-beam velocimeter (W) and laser torsional vibrometer (w).
the abscissae represents negative and positive inclination angles f and the ordinate represents angular displacement in degrees peak. Experimental results are shown for measured values of the second order of torsional vibration at three different speeds of rotation. These results demonstrate the accuracy and consistency of the Torsional Vibration Meter Type 2523. The theoretical curve is independent of the rotational speed and consequently the arbitrarily chosen rotational speeds all provide the same measurement results to close agreement. This demonstrates the capability of the Torsional Vibration Meter Type 2523 precisely to measure the torsional vibration experienced by any rotating component. 3.2. : The majority of diesel engines require some form of damping or detuning device to be fitted in order to prevent the build-up of large vibration amplitudes and stresses at torsional resonance. Two basic designs are in common use, these being the viscous shear damper and the elastomeric (rubber) equivalent. The former is the usual choice for higher power output engines because of its superior heat dissipation capability, while the latter is a comparatively cheap unit and is very popular for automotive type engines. The laser torsional vibrometer has provided a means of assessing the performance of a torsional vibration damper in situ, and readers are referred to the papers by Halliwell and Eastwood [8, 9] for a full description of this application. With reference to Figure 15, elastomeric dampers consist of a heavy seismic mass (the inertia ring) and a relatively lighter hub which is attached to the free end of the engine crankshaft. The two members are coupled by an elastomer element which provides stiffness and damping. A vibration damper of this design, therefore, adds another mass and elasticity to the basic equivalent dynamic system of the engine and introduces an additional torsional natural frequency. It therefore has a ‘‘tuning’’ capability. Laser torsional vibrometry was used to assess the performance of an elastomeric (bonded rubber) damper which was fixed to the end of the crankshaft of a six-cylinder turbocharged diesel. A simple, two-mass model [8] of the crankshaft system predicted two torsional resonances excited by the ninth order harmonic at engine speeds of 1050 rpm and
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Figure 11. A comparison of torsional vibration spectra. (a) Laser torsional vibrometer; (b) cross-beam velocimeter.
1850 rpm respectively. This theoretical result is shown in Figure 16. For comparison, in Figure 17 is shown an experimentally determined ninth order plot obtained by using a laser torsional vibrometer, which verifies the use of the model for resonance prediction to within 5% error. Loss of elastomeric damper performance can be attributed to changes in the material properties of the elastomer element over time. In particular, changes in the stiffness provided by the damper under optimal conditions will rapidly produce a decline in performance through de-tuning. It is well known that stiffness of the element changes with temperature and, consequently, monitoring the performance of a damper from ‘‘cold’’ start of the engine to normal operating conditions would represent that which might be anticipated from a damper progressively degrading with age and/or wear. It should be noted, however, that in the former case the behaviourial trend will occur in reverse and at a greatly accelerated rate. Two laser torsional vibrometers simultaneously measured the response of the hub (u) and the inertia ring (ud ). Data was recorded immediately following engine start-up and subsequently after intervals of ten minutes. The results are presented in Table 1. For the engine under test, the sixth order was monitored at an engine speed of 1895 rpm.
Figure 12. The Bru¨el & Kjaer type 2523 Laser Torsional Displacement Meter.
Figure 13. The Hooke’s joint checking method.
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Figure 14. A measurement comparison with Hooke’s joint predictions. ——, Theoretical; ×– – –, 700 rpm; W · · ·, 100 rpm; q — – —, 1300 rpm.
Measurements obtained after a period of 30 minutes are omitted since after this time the ratio (ud /u) remained constant. For the two-mass model [8] these results show that a change in stiffness of the damper element of approximately 62% has occurred between ‘‘cold’’ and stable operating conditions. Such a difference is not uncommon; manufacturers report differences of over 100% in some circumstances.
4. PRACTICAL CONSIDERATIONS 4.1.
It is the dynamics and characteristics of the laser speckle pattern [2] collected by the photodetector which dictate the sensitivity of the laser torsional vibrometer. This is best understood by considering the geometry of a standard laser vibrometer as shown in Figure 5 and analyzing the sensitivity of this device. When engineering target surfaces which are rough on the scale of the optical wavelength scatter coherent light, a speckle pattern forms in space in front of the target. This can be
Figure 15. The elastomeric damper.
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Figure 16. The theoretical prediction of ninth order torsional vibration [8, 9].
observed on a screen as a continuous random array of dark and bright spots which have a grainy speckly appearance and hence the name laser speckle pattern. The phenomenon occurs because each scattering surface element within the laser spot acts like a point source of coherent light. At a point in space, individual scattered wavelets from these sources interefere constructively or destructively to produce the bright or dark speckle, respectively, which is observed on the screen. The speckle pattern is a continuous random distribution of light amplitude and phase although in practice we simply observe the corresponding intensity distribution. An average speckle size can be calculated and attributed to the distribution observed on the screen [10]. Within this correlation region the amplitude and phase of the light may be assumed to be constant. In laser vibrometry, the photodetector active area in the instrument is sampling a speckle pattern. One or more speckles may be sampled, depending upon the optical geometry used in a given measurement situation and also the collecting optics used (if any) for the scattered light. If the detector area collects several speckles, the current output is proportional to the instantaneous sum of the intensity distribution which has been sampled.
Figure 17. As Figure 16, but experimental measurement.
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T 1 The simulation of elastomeric torsional damper failure
Elapsed time from ‘‘cold start’’ (mins) 0 10 20 30
increasing stiffness 3
Amplitude of sixth order torsional harmonic (degrees) ZXXXXXCXXXXXV Damper Inertia hub, u ring, ud 0·497 0·251 0·226 0·179
1·179 0·923 0·899 0·774
Amplitude ratio, ud /u 2·37 3·68 3·98 4·33
When a frequency-shifted reference beam is superposed onto the photodetector, a heterodyne beat occurs as described in section 2.1. For simplicity, consider what happens when the reference beam has a constant spatial phase distribution across the dector area and the target is stationary. It is useful to think of each speckle region as an interferometer which has a distinct value of intensity and phase. In practice, the detector output is the resultant vector addition of the outputs from all these separate interferometers corresponding to each speckle collected. In this way, for a stationary target the output from the detector is at the beat frequency with a constant amplitude and phase. When the target moves normal to surface and parallel to the incident laser beam, the Doppler effect produces a uniform rate of change of phase across the detector area and a pure frequency modulation of the detector output occurs with negligible amplitude modulation. A word of caution is necessary here. Speckles form small, continuous ‘‘cigar-shaped’’ volumes in space and the detector plane effectively moves along the major
Figure 18. The torsional vibration spectrum: speckle pattern periodicity.
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Figure 19. The torsional vibration spectrum with the speckle pattern periodicity removed.
axis of each volume when the target vibrates. Movement along speckle volumes would therefore cause amplitude modulation, but this effect is usually negligible. If, however, the target tilts or moves other than normal to surface, the characteristics of the speckle pattern sampled by the detector will change spatially and/or temporally. This effect produces a phase modulation of the detector output, which is demodulated and is indistinguishable from the change in Doppler frequency associated with normal-to-surface movement. This is a major source of noise in laser vibrometers and has been referred to as pseudo-vibration [11]. When the scattering elements in the laser spot rotate locally due to surface tilt, the associated speckle pattern on the detector moves in sympathy, and this is usually the dominant noise source. However, if a significant change in population of scattering elements occurs (perhaps due to in-plane target motion) then the speckle pattern will decorrelate in sympathy with this change and produce noise. In practice, both these mechanisms contribute noise in the laser vibrometer output. The light intensity incident at a point on the photodetector surface Ip (t) can be written as [2]
$ g
%
Ip (t)=IR+IT+2zIR IT cos vR t− vT dt+fR−fT ,
(15)
where IR and fR are the reference beam intensity and initial phase, respectively, IT and fT are the target scattered light intensity and resultant phase, respectively, vR is the constant reference beam angular frequency pre-shift and vT is the ‘‘instantaneous’’ Doppler angular frequency shift produced by target movement. This equation is derived after neglecting polarization effects. The very high frequency modulation of Ip caused by the summed frequency term in the heterodyne process is beyond the frequency response of the photodetector.
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In practice, the photodetector takes the sum of Ip (t) over its active area A, and one must therefore consider the spatial distributions of IR and IT and their respective phases fR and fT . For convenience, one can assume that reference beam intensity and phase is uniform in space and one need consider only the speckle pattern distribution. Accordingly, one can write the detector output current i(t) as i(t)=
g
A
$ g
%
[IR+IT (a)+2zIR IT (a) cos vR t− vT dt+fR−fT (a) dA,
(16)
where A is the photodetector area, IT (a) is the speckle intensity distribution and fT (a) is the speckle phase distribution. Examination of equation (16) confirms the earlier discussion that if the spatial distribution of the speckle pattern changes during a measurement period (i.e., if IT (a) and fT (a) are functions of time) then spurious information will be contributed when i(t) is frequency demodulated. Unfortunately, in practice, this noise is usually linked to the vibration frequency of interest, introducing uncertainty into the measurement. When a laser vibrometer is pointed at a rotating target, the spatial and temporal characteristics of the speckle pattern sampled by the photodetector change with time. With reference to equation (16), the terms IT (a) and fT (a) describing the speckle pattern intensity and phase distribution across the detector surface become periodic at the rotation frequency and are demodulated as pseudo-vibration. Consequently, when the target surface is not vibrating but simply rotating, the laser vibrometer will produce a noise floor which has a spectrum typical of a pseudo-random signal. This consists of a fundamental at the rotation frequency and higher order harmonics of similar magnitude. It is produced because the changing speckle pattern produces random phase and amplitude noise which repeats exactly in sympathy with the rotating target. For a further discussion of this effect, the reader is referred to the papers by Rothberg et al. [11–13]. Unfortunately, in practice, the vibration frequencies of interest are usually direct integer multiples of the rotation frequency and consequently the distribution of this speckle noise is a worst possible case. It is at this stage that a degree of engineering judgement is required before deciding whether the vibration levels measured in a given bandwidth are not those associated with the pseudo-vibration of speckle noise. Users need to be wary of the very low noise floors and sensitivities quoted by instrument manufacturers in the literature. These have usually been measured on a stationary target and, in practice, as discussed in the last section, the true noise floor is peculiar to the speckle pattern behaviour caused by target surface dynamics during the actual measurement. For the laser torsional vibrometer, and with reference to Figure 8, the photodetector collects two speckle patterns from points A and B which then heterodyne on its active area. If the target component has no torsional vibration whatsoever, the frequency spectrum of the instrument output due to the steady rotation will take the form of a periodogram at the fundamental rotation frequency. An example of this is shown in Figure 18, where the instrument was used to measure the speed fluctuation of a d.c. motor from a tape recorder drive. The displacement spectrum shown clearly exhibits the periodic nature of the speckle noise which completely masks the intended measurement. Fortunately, the insensitivity of the instrument to solid body oscillation means that the speckle periodicity can be destroyed by moving the laser beam plane during the course of the measurement. Hand holding the instrument and moving the beams from side to side means that the speckle pattern incident on the photodetector does not exactly repeat. In Figure 19 is shown the equivalent measurement taken on the d.c. motor when the incident laser beam plane was moved from side to side at very low frequency of about 1 Hz. The figure shows the effect of removing the speckle pattern periodic noise, in this way, and
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the ‘‘wow’’ of the motor can now be distinguished. For most mechanical engineering applications, a level of −60 dB re 1° peak means that torsional vibrations have ceased to be a problem and therefore speckle periodicity is not troublesome. The Type 2523 Laser Torsional Vibration Meter has a ‘‘noise smearing’’ facility. This is intended for use at very low levels of torsional vibration and allows the speckle pattern periodicity to be destroyed by modulating the spatial position of the incident laser beams from side to side in a random manner. It is not necessary to use this when measuring levels greater than −40 dB re 1° peak displacement, which is normally the case in practice. 4.2. In practice, the highest levels of torsional vibration will be found in the crankshafts of large diesel engines of reciprocating compressors. These may be as high as a few degrees of peak displacement, but in frequency terms these levels still represent a relatively small modulation of the mean beat frequency resulting from the target speed. Consequently, measuring high levels does not present a problem, and it is simply necessary to ensure that they are within the frequency response range of the Doppler signal processor. It is the range of possible rotational speeds which represents much more of a practical problem for signal processors. At a beam separation of 1 cm for beams perpendicular to the rotational axis (u=0), speed variation of 500 rpm to 15 000 rpm produces a beat frequency range of 1·5 MHz to 45 MHz! The theory for the instrument is developed on the assumption that the incident laser beams are infinitely thin and therefore, by an order of magnitude argument, a minimum beam separation of one cm is required. The lower speed range can therefore be extended by widening the beam separation in order to match the input bandwidth of the processor. At the higher rotational speeds, using the instrument so that the incident laser beam plane is at an angle to the rotational axis will reduce the mean beat frequency. In this situation, care should be taken to account for the effects of operator or component tilt [5, 6]. When a low powered helium–neon laser is used as a light source so that the outgoing beams have about 1 mW power, the target distance for operation is up to several metres, but retroreflective tape or paint must be used. Calibration of the laser torsional vibrometer should be carried out in situ. It is not necessary to attempt to measure the beam separation d or angle of inclination of the normal to the laser beam plane u with the rotational axis. In most circumstances it is possible to rotate the target at several known speeds so that an in situ calibration of mean d.c. volts per rpm is possible. If it is not possible to rotate the target at known speeds, then a small disc can be used, which is driven at known speeds and interposed in the laser beams in place of the target. It is only necessary to monitor the d.c. voltage from the instrument at three known speeds to produce a calibration factor giving negligible error. It is usual to output the instrument signal straight to a real time spectrum analyzer which will integrate the output directly to produce vibration spectra in terms of torsional vibration displacement. The alternative, of course, is to tape record the analogue output on site for later analysis, but it must be remembered that the d.c. level is important if an in situ calibration has taken place. 5. CONCLUSIONS
The problems of torsional vibration measurement have been solved by the invention of the laser torsional vibrometer. The use of optical, seismic and mechanical torsiographs, together with strain gauges, slip rings and slotted discs, has now been overtaken by the use of laser technology. The laser torsional vibrometer offers in situ measurement, thus
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avoiding machinery downtime and can function on rotating components of arbitrary shape whilst maintaining an immunity to solid body motion of the operator or component. The instrument is robust, user friendly and can be calibrated in situ. Invented at ISVR in 1983 and subsequently produced commercially by Bru¨el and Kjaer Ltd of Denmark as the Type 2523 Torsional Displacement Meter, the vibrometer represents a significant step forward in rotating machinery diagnostics. It is now used on a worldwide basis and is rapidly gaining acceptance as the standard means of measuring torsional vibration. ACKNOWLEDGMENTS
The author wishes to acknowledge the provision of Hooke’s joint results by T. J. Miles. REFERENCES 1. F. D, A. M and J. H. W 1981 Principles and Practice of Laser Doppler Anemometry (second edition). London: Academic Press. 2. J. C. D (editor) Laser Speckle and Related Phenomena. New York: Springer-Verlag. 3. B. M. W and M. J. R 1976 Laser Doppler Measurements. London: Butterworth. 4. N. A. H, H. L. P and J. B 1983 Proceedings of the Society of Automotive Engineers, International Off-Highway Meeting and Exposition, Milwaukee, Wisconsin, USA, September, paper 831324, 986–994. Diesel engine health: laser diagnostics. 5. N. A. H and P. G. E 1985 Journal of Sound and Vibration 101, 446–449. The laser torsional vibrometer. 6. T. J. M, M. L and S. J. R 1995 Proceedings of the ASME 15th Biennial Conference on Mechanical Vibration and Noise Boston, U.S.A., September 3(C), 1451–1460. The laser torsional vibrometer: successful operation during lateral vibrations. 7. E. J. N 1958 A Handbook on Torsional Vibration. Cambridge: Cambridge University Press. 8. N. A. H and P. G. E 1990 Proceedings of the I. Mech. E. Conference on Automotive Diagnostics, 115–122; London, I. Mech. E. Torsional vibration damper performance: diagnosis using laser technology. 9. N. A. H and P. G. E 1992 Optics and Lasers in Engineering 16, 337–350. Marine vehicles and offshore installations: laser diagnostics of machinery health. 10. M. F 1979 Laser Speckle and Applications in Optics. London: Academic Press. 11. S. J. R, J. R. B and N. A. H 1989 Journal of Sound and Vibration 135, 516–522. Laser vibrometry: pseudo vibrations. 12. S. J. R, J. R. B and N. A. H 1990 Journal of Laser Applications 2(1), 29–36. On laser vibrometry of rotating targets: effect of in-plane and torsional vibration. 13. S. J. R and N. A. H 1994 Transactions of the American Society of Mechanical Engineers, Journal of Vibration and Acosutics 116(3), 326–331. Vibration measurements on rotating machinery using laser Doppler velocimetry.