INVERSE PROBLEMS IN MECHANISMS FOR PACKAGING MACHINES G. Dalpiaz and A. Rivola University of Bologna - DIEM Viale Risorgimento 2, I-40136 Bologna, Italy. E-mail:
[email protected]
Abstract An inverse-filtering technique for the in-service recovery of mechanism motion from casing response measurements is presented. Different processing options are discussed based on experimental results. In particular, the capability of some FRF estimators and cepstral smoothing is pointed out. The results are satisfactory.
1. INTRODUCTION Vibration analysis is widely used in the field of condition monitoring and fault diagnostics of machinery, with applications in predictive maintenance, quality control and automation. This paper deals with diagnostics in automatic packaging machines of high performances. These machines are composed of sub-assemblies, each embodying one or more mechanisms for reciprocal or intermittent motion, designed to achieve the proper motion of the output link, acting on the product. Unfortunately, due to mechanism vibrations, the actual motion may be quite different from the theoretical one. These vibrations are excited by variable forces, such as inertia forces, and internal shocks which occur in the kinematic pairs with backlash when contact forces reverse [1-3]. Joint wear causes backlashes to increase and more intense shocks to take place. These effects may produce stronger deviations of the actual motion from the theoretical one and, consequently, unacceptable product quality and poor production; in addition, higher accelerations and dynamic overloads may arise. In this kind of mechanism, mechanical breakage is very infrequent, so the predominant aim of condition monitoring is preventing the mechanism from the above improper conditions. In this context, the best strategy would be to measure the output motion, which can be directly related to the mechanism functional limits and to the actual inertia forces. Moreover, mathematical models of the mechanism can be easily used as a diagnostic tool [3]. However, in automatic machines, as well as in many other engineering applications, it is practically impossible to mount
transducers on the mechanism links during machine operation; as a matter of fact, monitoring and diagnostics are generally performed by analysing the machine casing vibrations [4, 5]. It is noteworthy that casing vibrations are excited by the forces transmitted by the mechanisms, whose dynamic behaviour is the result of numerous factors, including the kinematic and dynamic properties of the mechanism itself, external forces, parametric excitations, friction, backlash effects in joints and faults. However, the casing response is generally distorted with respect to mechanism forces because machine structures have complicated transfer functions, due to dispersion, multi-path transmission and reverberation phenomena. The effect of the transfer function (TF) between mechanism components and casing can be so important that information about faults might be partially or completely hidden in the casing vibration [6, 7], i.e. low sensitivity to faults and poor diagnostic capability might be the result. Besides, to correlate the casing vibration parameters to machine condition may be difficult, because of the result scattering between nominally identical machines in analogous health conditions [8, 9]. For the mentioned reasons, it is desirable to recover the actual quantities of interest - i.e. mechanism forces or motion - from the casing response. This can be carried out by means of an inversefiltering technique [10], that consists of the experimental estimation of the TF between these quantities and the casing vibration. The TF estimate can be used in production to recover the desired functional quantities from the measurement of the casing vibration. Different approaches and implementations can be found in literature, including applications to reciprocating machinery [7, 9, 10], gear systems [11-13] and cam mechanisms [3-5, 14]. In the specific case of automatic machines, previous works [3-5] have shown that the actual motion of one of the mechanisms embodied in a sub-assembly can be satisfactorily recovered, but in some cases poor results may be produced by errors affecting the TF estimation. In particular, difficulties were found for sub-assemblies containing two mechanisms [14]. This work is a contribution to overcome the above mentioned problems and to assess an effective inverse-filtering technique. To this end, some estimators of the inverse filter have been considered and developed for the application to the specific case. The results are compared on the basis of experimental tests carried out on a subassembly containing two mechanisms, both in sound and faulted conditions. The capability of the TF estimators is firstly compared; the application of cepstral-smoothing [7, 9, 10] is then discussed for assessing the effectiveness of these techniques in order to increase the robustness of the inverse-filtering process.
2. INVERSE-FILTERING PROCESS 2.1. General outline. The various sources of excitation in the machinery produce both output link motion and casing vibrations. With reference to a specific sub-assembly embodying only one cam mechanism, the actual motion of the output link can be considered as the response of the mechanism to the input motion imposed by the curve manufactured on the cam working surface. On the other hand, the casing vibration can be considered as the response of the machine structure to the same excitation, though the path is now more complicated. Therefore, two transfer functions exist between the cam curve and, respectively, the output motion and the casing vibration: H’(f) = Ym(f) / X0(f), and H”(f) = Yc(f) / X0(f), where X0(f), Ym(f) and Yc(f) are the Fourier transforms of the cam curve, the output link acceleration and the casing acceleration at a specific location, respectively. Consequently, the TF between the acceleration of the output link and the casing, H(f), can be defined as the ratio between H”(f) and H’(f): H(f) = Yc(f) / Ym(f) = H”(f) / H’(f) .
(1)
The complex inverse of this TF, Hinv(f)=1/H(f), is the filter function (or inverse filter) [10] needed in order to recover the output acceleration by filtering the casing acceleration: Ym(f) = Hinv(f) Yc(f). Since the exact filter function Hinv(f) is not known and its estimation is required, the following recovering technique was proposed [3, 14]. Preliminarily, the inverse filter is estimated for each individual machine, for example at the installation stage [Fig. 1(a)]: both the output link acceleration, ym(t), and the casing acceleration at a properly selected location, yc(t), are experimentally achieved; as an estimate of H(f), the Frequency Response Function (FRF), H ( f ) , is then evaluated considering the output link acceleration as the input; finally, the filter function estimate, H inv ( f ) , is obtained by complex inversion of the FRF. During machine working life this filter function is used for recovering the output acceleration [Fig. 1(b)]: only the acceleration of the selected point on the casing is periodically or continually picked up; the output link acceleration spectrum, Ym ( f ) , is then computed based on the casing acceleration spectrum,Yc ( f ) , and the inverse filter estimate; finally, the time pattern of the output acceleration, y m ( t ) , can be recovered and analysed for monitoring and diagnostic purposes. This recovering operation is possible in any health condition on the reasonable hypothesis that the filter function is practically independent
of the faults that may take place. ym(t) yc(t)
Fourier transform Fourier transform
Ym(f) Yc (f)
FRF ESTIMATION
^ H(f)
Complex Inversion
^ (f) H inv
(a)
^ H inv(f) yc (t)
Fourier transform
Yc (f)
^ Ym(f)
Inverse Fourier transform
y^ (t) m
(b)
Fig. 1 - Inverse-filtering process. (a) Inverse filter estimation in the preliminary stage; (b) use of the inverse filter for the mechanism motion estimation in production.
It was noted in [3] that the above described technique is unable to properly restore the first cam revolution harmonics due to the fact that the casing response is generally very low in that frequency range and consequently the FRF estimation is poor. However, it was found that the first harmonics are almost insensitive to mechanism conditions; in fact, they essentially describe the theoretical pattern of output acceleration, while the superposed vibrations are due to higherfrequency components. Thus, the output acceleration spectrum can be recovered by copying lower components from an output acceleration spectrum measured and stored in the preliminary stage, and by recovering only higher components by the above described process of Fig. 1(b), see [3]. In the case considered in this paper, this copying operation concerns the range 0 to 40 Hz, containing the first four revolution harmonics. 2.2. Transfer function estimation. FRF measurements are the base of several techniques for the frequency domain identification of the dynamic characteristics of structural systems. They are employed to such technologies as experimental modal analysis, structural systems identification, and force determination to name a few. In the strict sense of the term, the FRF can be defined only for a linear, time invariant system, so it cannot be defined for the system under consideration. However, since the purpose is not the accurate system identification but the approximate signal recovery, the use of the FRF can be admissible. An additional cause of errors is that the casing response is generally affected by vibrations due to other mechanisms,
as occurs in particular when two or more mechanisms are embodied in the same sub-assembly [14]. Also, measurement and computational errors are present. For a generic linear time invariant system, in the case of single input and single output (SISO), the true FRF is defined as H0(f)=Y(f)/X(f), where X(f) and Y(f) are respectively the Fourier transforms of the input and output signals [15]. Several FRF estimation techniques have been proposed in literature in order to minimize the error effects: estimates obtained with Least Square techniques (i.e. H1 and H2), with Total Least Square Techniques (i.e. HV) and by means of Average Methods (i.e. H3 and H4) [16-18]. The most popular estimation methods for uncorrelated input and output noise are [16] H1 ( f ) =
Sxy ( f ) Sxx ( f )
,
H2 ( f ) =
Syy ( f ) Syx ( f )
,
(2)
where Sxx(f) and Syy(f) are the autopower spectra of the input and output, respectively, Sxy(f)=E[X*(f)Y(f)] is the crosspower spectrum (the symbol E[⋅] denotes the expectation operator) and Syx(f) the complex conjugate of Sxy(f). The estimator H1 is the least square solution for the true FRF which minimises the bias effects of output noise; conversely, the estimator H2 minimises the effects of input noise [16]. It should be noted that the estimator H2 is more accurate at the resonances because the output signal to noise ratio (SNR) is high, so that the noise to be minimised is that which affects the excitation. On the contrary, at the anti-resonances the better estimator is H1 and H2 overestimates the true FRF substantially. As the assumption that errors are confined to the input or the output signal is usually unrealistic, other techniques have been proposed. Among them, the HV technique [17] aims to minimize the total squared error on both input and output and proved to be effective in many circumstances. Besides, according to the observation that H1 and H2 are respectively a lower and upper bound estimators of H0, new estimators defined as their arithmetic mean, H3, and geometric mean, H4, were proposed [16, 18]: H3 ( f ) = ( H1 ( f ) + H2 ( f )) / 2 ,
H 4 ( f ) = H1 ( f ) ⋅ H2 ( f ) .
(3)
It is worth noting that the phase of estimators H1 and H2 is the same and, consequently, this phase is also the one of estimators H3 and H4. Since the HV estimator reduces to H4 for SISO systems [17], and the application of other estimators proposed in literature does not
seem suitable in the present case, only the results obtained by using estimators H1, H2, H3 and H4 were compared in this work. 2.3. Cepstral smoothing. The cepstral smoothing [9, 10] of a signal is performed by applying a window to the signal complex cepstrum, c(τ) - i.e. the inverse Fourier transform of the logarithm of the signal spectrum - in order to extract from the signal only the low-quefrency information. In details, the smoothed complex cepstrum, csm(τ), is obtained as: csm(τ)=c(τ)⋅w(τ), where c(τ) is the complex cepstrum computed on the signal spectrum X(f), after phase unwrapping, and w(τ) is the cepstral window which eliminates the high-quefrency components. By coming back to the frequency domain, a smoothed version of the spectrum, Xsm(f), is obtained, as windowing the cepstrum is equivalent to a smoothing operation in the frequency domain. The use of a window function other than the rectangular one is needed in order to avoid discontinuity in the cepstrum and related spectrum errors at high frequencies [9]. If only smoothed magnitude is required, the original phase must be used. Cepstral smoothing was applied in literature both to the TF and the casing response [7, 9, 10, 13] in order to improve the robustness of the inverse filter. In the case of automatic machines, it was found [14] that the complex inversion of the FRF may produce poor results at those frequencies where the amplitude is very low; in this case, cepstral smoothing could make it possible to overcome the problem. 3. EXPERIMENTAL RESULTS AND DISCUSSION Tests were carried out on a sub-assembly mounted on a complete machine, set-up for normal operation. This sub-assembly contains two cam mechanisms acting on the same output link, in order to give it two motions: a reciprocal translating motion and a reciprocal rotational motion around the link axis. Thus the motions of both mechanisms contribute to the case vibration, possibly causing difficulties in recovering one of the motions, as indicated by the results presented in [14]. This paper only concerns the recovery of the translating motion. Such a motion is obtained by means of the mechanism of Fig. 2, made up of a cylindrical cam and a follower system composed of a lever with two rollers which engage the cam rib, a small connecting rod and the output link acting on the product. The results deal with two health conditions: sound condition and artificially increased backlash between the cam and rollers of the mechanism for the translating motion. The backlash in the latter condition - simulating the typical fault of this type of machines, i.e.
wear in joints - was about twelve times greater than in the first: it was judged to be a limit condition for production, so that maintenance is required as soon as possible.
Fig. 2 - Schematic of the mechanism for reciprocal translating motion.
Acceleration [m/s^2]
600 (a)
400 200 0 -200 -400 -600
0
0.02 0.04 0.06 0.08 Time [s]
0.1
0.12
15 Amplitude [m/s^2]
(b) 10
5
0
0
1000
2000 3000 4000 Frequency [Hz]
5000
Fig. 3 - Measured output link acceleration in sound conditions. (a) Time pattern; (b) amplitude spectrum.
The output link acceleration was measured using an accelerometer, mounted at the extremity of the output link with its
sensitivity axis coincident with the link axis, while the casing acceleration was picked up next to the lever bearings by means of a second accelerometer; moreover a one-per-cam revolution tachometer signal was collected by an inductive proximity probe.
Acceleration [m/s^2]
600 400
(a)
(b)
(c)
(d)
200 0 -200 -400
Acceleration [m/s^2]
-600 600 400 200 0 -200 -400 -600
0
0.02 0.04 0.06 0.08 Time [s]
0.1
0.12 0
0.02 0.04 0.06 0.08 Time [s]
0.1
0.12
Fig. 4 - Output link acceleration in sound conditions recovered by means of inverse filters estimated through various algorithms: (a) H1, (b) H2, (c) H3, (d) H4.
Figure 3(a) shows the time pattern of the measured output link acceleration, corresponding exactly to one machine cycle (one cam revolution) and beginning at the bottom of the output link stroke. It is worth noting the presence of oscillations superimposed to the theoretical acceleration, produced by the internal shocks occurring when the roller hits the cam surface; this happens at the beginning of the motion and, during the motion, when the inertia force direction reverses (at about 50 and 95 ms), the backlash is traversed and the contact side between rollers and cam rib changes. Figure 3(b) shows the amplitude spectrum of the same signal, computed over a time pattern corresponding exactly to five cam revolutions with rectangular window (as well as the other spectra presented in this paper). The lowfrequency components exceed the diagram amplitude range, as the scale is set with the aim of showing the high-frequency components more clearly.
The recovery procedure described in Sect. 2.1 was applied. The FRF between the output link acceleration and the casing acceleration was estimated for the mechanism in sound condition by means of the autopower and crosspower spectra evaluated over 20 averages, in the range 0-5 kHz, using Hanning window and frequency resolution of 1.5 Hz (blocksize of 8192 points). Since only data acquired at the maximum cam speed of 500 rpm were employed, the inverse-filtering process may give bad results at different speeds. However, properly recovering the output link motion only at one particular speed, taken as reference, may be enough for diagnostics. In other tests [3], on the contrary, these spectra were evaluated by averaging data relative to several cam speeds, in order to obtain an inverse filter useful for different speeds, with the drawback of longer test time.
Amplitude [m/s^2]
15 (a)
(b)
(c)
(d)
10
5
Amplitude [m/s^2]
0 15
10
5
0
0
1000
2000 3000 4000 Frequency [Hz]
5000 0
1000
2000 3000 4000 Frequency [Hz]
5000
Fig. 5 - Amplitude spectrum of the output link acceleration in sound conditions, recovered by means of inverse filters estimated through various algorithms: (a) H1, (b) H2, (c) H3, (d) H4.
The capability of the FRF estimators mentioned in Sect. 2.2 is assessed by comparing the output link acceleration recovered by means of the related inverse filters (Figs. 4 and 5) to the acceleration directly measured in sound conditions (Fig. 3). Estimator H1 produces completely unsatisfactory recovery [Fig. 4(a)] because the FRF is underestimated and consequently causes the inverse filter and the
recovered spectrum to assume incorrectly high amplitude over wide frequency ranges [Fig. 5(a)]. The other estimators give better results. The recovered acceleration waveforms [Figs. 4(b)-(d)] are quite good; deviations from the measured waveform are observable mainly at 35 to 50 ms, where the recovered waveforms exhibit oscillations of higher amplitude. Moreover, the amplitude relationships between the oscillations at 0 to 25 ms and 95 to 120 ms are altered by using H2, while a satisfactory recovery is obtained through H4, in spite of a slight disagreements in the oscillation amplitude. In addition, the highfrequency vibrations superimposed to the time pattern are rather well recovered only by means of H4; in fact, the amplitude spectra recovered by H2 and H3 are globally underestimated, while the spectrum obtained through H4 is rather good [Figs. 5(b)-(d)]. As general results, the application of H4 gives the most satisfactory recovery in sound conditions, while the use of H3 produces results of intermediate quality between H2 and H4.
Acceleration [m/s^2]
600 400
(a)
(b)
(c)
(d)
200 0 -200 -400
Acceleration [m/s^2]
-600 600 400 200 0 -200 -400 -600
0
0.02 0.04 0.06 0.08 Time [s]
0.1
0.12 0
0.02 0.04 0.06 0.08 Time [s]
0.1
0.12
Fig. 6 - Output link acceleration in the case of increased backlash. (a) Measured pattern. Recovered patterns using algorithm H4: (b) without smoothing; (c) with 1/2-blocksize cepstral smoothing; (d) with 1/4-blocksize cepstral smoothing.
Similar results about the applications of the estimators were found in the case of increased backlash; however the recovery is not so
good as in sound conditions, as shown by Figs. 6(a)-(b), comparing the measured output member acceleration and the recovered one by means of H4. In particular, the important oscillations that take place at about 50 ms and between 90 to 115 ms are recovered with lower amplitude; in addition, the amplitude of the oscillations at about 18 ms, between 30 to 50 ms and 80 to 90 ms is overestimated in the recovered pattern. These deviations are produced by inverse filter errors which mainly affect the frequency range 150 to 270 Hz, that appears to be a resonance band of the mechanism [Figs. 7(a)-(b)].
Amplitude [m/s^2]
50 (a)
(b)
(c)
(d)
40 30 20 10
Amplitude [m/s^2]
0 50 40 30 20 10 0
0
100 200 Frequency [Hz]
300 0
100 200 Frequency [Hz]
300
Fig. 7 - Amplitude spectrum of the output link acceleration in the case of increased backlash. (a) Measured pattern. Recovered patterns using algorithm H4: (b) without smoothing; (c) with 1/2-blocksize cepstral smoothing; (d) with 1/4-blocksize cepstral smoothing.
The cepstral smoothing was applied in [9] with the aim of evaluating a robust inverse filter, able to recover waveforms of rather simple shapes, and scarcely affected by the TF variability among nominally identical machines, as well as errors in transducer locations. In that work, not only the TF is smoothed, but also the casing response spectrum employed for the source waveform recovery; in addition, a quite narrow cepstral window seems to be used. On the other hand, the application considered in the present paper has different needs, as the objective is to accurately recover a link motion pattern of
complicated shape; so the evaluation of the inverse filter for each individual machine is required and the use of the measured casing response - without smoothing - is suitable. Thus, according to the procedure described in Sect. 2.3, the cepstral smoothing of the magnitude of the FRF estimated in sound condition through algorithm H4 has been carried out, in order to eliminate only the narrow peaks and dips, which may be affected by localized but important estimation errors. 0 Amplitude [dB]
-10 -20 -30 -40 -50 -60 -70
(a) 0
100 200 Frequency [Hz]
300 0
(b) 100 200 Frequency [Hz]
300
Fig. 8 - FRF estimated by means of algorithm H4 in sound conditions. (a) Without smoothing (thin line) and using 1/2-blocksize cepstral smoothing (thick line); (b) without smoothing (thin line) and using 1/4-blocksize cepstral smoothing (thick line).
The length of the cepstral window must be properly set, taking into account that too narrow windows would cause the loss of meaningful information and, consequently, unacceptable alterations of the recovered waveform. Figure 8 compares the original FRF with the smoothed versions obtained applying Hanning cepstral windows of length equal to 1/2 and 1/4 of the blocksize (4096 and 2048 points, respectively). The two smoothed FRF’s exhibit meaningful differences only in the low frequency range 0-160 Hz, where quite wide peaks and dips are present; the use of 1/2-blocksize window appears to be suitable, while the 1/4-blocksize window seems to introduce excessive simplifications in the low-frequency range. This is verified by considering the effects of the cepstral smoothing on the recovered signals. In the case of increased backlash, the 1/2-blocksize smoothing produces significant improvement in the time waveform [Fig. 6(c)], reducing the overestimation of the oscillation amplitude at about 18 ms and in the zones 30 to 50 ms and 80 to 90 ms; this is the consequence of the improvement of the spectrum recovery [Fig. 7(c)] chiefly in the range 150 to 270 Hz. On the other hand, the 1/4-blocksize smoothing does not introduce further improvements, but it conversely gives
errors in the 40 to 80 Hz range of the recovered spectrum [Fig. 7(d)], where the relationships between component amplitude are strongly altered with respect to the measured spectrum; the reason is that the FRF is excessively smoothed in that range. In sound conditions it has been shown that the recovery without smoothing is satisfactory; the cepstral smoothing does not introduce meaningful changes, but only a slight improvement at 25 to 50 ms. Besides, the phase smoothing does not give significant advantages in the considered tests. 4. CONCLUSIONS This paper presents an inverse-filtering technique for the inservice recovery of the motion of the output link in mechanisms for reciprocal or intermittent motion, used in high performance packaging automatic machines. In this case, waveforms of complicated shape have to be accurately recovered - taking into account TF differences between nominally identical machines - in order to directly assess health condition and obtain diagnostic information. On the basis of experimental results concerning a sub-assembly embodying two cam mechanisms, the effectiveness of different processing options is discussed with the aim of improving the robustness of inverse filter and reducing estimation errors. In particular, the capability of some FRF estimators is firstly compared and the application of cepstral smoothing is then discussed. Satisfactory recovery both in sound and in faulted conditions is obtained by means of algorithm H4, while the cepstral smoothing of the FRF magnitude produces significant improvement, particularly in faulted conditions, on condition that the length of the cepstral window is properly set. In fact, too narrow windows may cause the loss of meaningful information and consequent alterations of the recovered waveform. As a final remark, the proper selection of the FRF estimation technique appears to be essential for correct waveform recovery, while cepstral smoothing may give significant improvement, but it may not compensate relevant errors due to FRF estimation. The application of the proposed technique to other mechanisms presenting faults of different severity is needed in order to establish its general effectiveness.
REFERENCES 1.
Dresner, T.L. and Norton, R.L., Cam Dynamics and Modeling. In Modern Kinematics: Developments in the Last Forty Years, ed. A. G. Erdman, John Wiley, New York, 1993, 305-312.
2. 3. 4.
5.
6.
7.
8.
9.
10. 11.
12. 13.
14.
15. 16.
17.
18.
Koster, M.P., Vibrations of Cam Mechanisms, MacMillan, London, 1974. Dalpiaz, G. and Maggiore, A., Monitoring Automatic Machines. Mechanical Systems and Signal Processing, 1992, 6, 517-534. Dalpiaz, G. and Rivola, A., Fault Detection and Diagnostics in Cam Mechanisms. Proceedings of the 2nd International Symposium on Acoustical and Vibratory Surveillance Methods and Diagnostic Techniques, Senlis, France, 1995, 327-338. Dalpiaz, G. and Rivola, A., Condition Monitoring and Diagnostics in Automatic Machines: Comparison of Vibration Analysis Techniques. Mechanical Systems and Signal Processing, 1997, 11, 53-73. McFadden, P.D. and Smith, J.D., Effect of Transmission Path on Measured Gear Vibration. Journal of Vibration, Acoustics, Stress, and Reliability in Design, 1986, 108, 377-378. McCarthy, D.J. and Lyon, R.H., Recovery of Impact Signatures in Machine Structures. Mechanical Systems and Signal Processing, 1995, 9, 465483. Dalpiaz, G. , Maggiore, A. and Meneghetti, U., Monitoring Cam Mechanisms by Vibration Analysis. Atti XI Congresso Nazionale AIMETA, Meccanica delle macchine, Trento, Italy, 1992, 133-138. (In Italian). Kim, J.T. and Lyon, R.H., Cepstral analysis as a tool for robust processing, deverberation and detection of transients. Mechanical Systems and Signal Processing, 1992, 6, 1-15. Lyon, R.H., Machinery Noise and Diagnostics, Butterworths, Boston, 1987. Umezawa, K., Houjoh, H. and Maki, H., Estimation of the Vibration of In-Service Gears by Monitoring the Exterior Vibration. JSME Int. Journal, 1988, 31, 588-592. Dalpiaz, G. and Meneghetti, U., Monitoring Fatigue Cracks in Gears. NDT&E International, 1991, 24, 303-306. Chai, J. and Lyon, R.H., Determination of Component Motions in a Motor-Operated Valve Using Housing Vibration. Mechanical Systems and Signal Processing, 1996, 10, 113-123. Dalpiaz, G., Rivola, A. and Meneghetti. U., Use of Filtering Function for Machine Health Monitoring. Proceedings of an International Conference on Condition Monitoring, Swansea, UK, 1994, 217-228. Oppenheim, A.V. and Schafer, R.W., Discrete-time Signal Processing, Pretice-Hall Int., 1989. Mitchell, L.D., Improved Methods for the Fast Fourier Transform (FFT) Calculation of the Frequency Response Function. Journal of Mechanical Design, 1982, 104, 277-279. Rocklin, G.T., Crowley, J. and Vold, H., A Comparison of H1, H2, and Hv Frequency Response Functions. Proceedings of the 3rd International Modal Analysis Conference, Orlando, FL, 1985, 272-278. Zhang, L. and Yao, Y., Advances of Frequency Resonse Function Estimation in Modal Analysis. Proceedings of the 6th International Modal Analysis Conference, Kissimmee, FL, 1988, 107-112.
