Comparison of Various Basic Wavelets for the ... - Science Direct

Journal of Fluids and Structures (1996) 10, 633 – 651

COMPARISON OF VARIOUS BASIC WAVELETS FOR THE ANALYSIS OF FLOW-INDUCED VIBRATION OF A CYLINDER IN CROSS FLOW M. N. HAMDAN, B. A. JUBRAN, N. H. SHABANEH AND M. ABU-SAMAK Department of Mechanical Engineering , Uniy ersity of Jordan , Amman , Jordan (Received 28 April 1995 and in revised form 14 September 1995) This paper is concerned with the application of wavelet transforms to the analysis of the self-excited flow-induced vibration of a single cylinder in a cross-flow. Published experimental data indicates that the vibration signals of the present system, and those of similar ones are, in general, nonstationary. However, these signals have thus far been analysed using classical methods, such as the FFT and correlation methods, wherein the signals are assumed to be stationary and they are analysed either in the time or frequency domain but not in both domains simultaneously. It is shown that wavelet transforms, which are applicable to stationary as well as to nonstationary processes and yield a joint time-frequency analysis and representation of the vibration signals with good localization, can reveal many important aspects of the dynamical process not shown by the classical methods. Various wavelets are used and recommendations regarding the most suitable ones, and the interpretation of the representations, are discussed. ÷1996 Academic Press Limited

1. INTRODUCTION FLOW-INDUCED VIBRATION ARISES in many practical engineering systems, such as heat exchangers, power transmission lines, bridges, and tall towers. For example, when an elastically mounted cylinder is placed in a cross-flow it is generally known that, as the flow separates from either side of cylinder, vortices are shed alternately resulting in an oscillating external force which causes the cylinder to oscillate. Numerous experimental and analytical investigations were carried out to study flow-induced vibration of a single elastically supported circular cylinder. Extensive reviews of these studies have been presented by Blevins (1977), Blake (1986) and Parkinson (1989). The fluid flow and the structure are interactive systems and their interaction is dynamic (Feng 1968). Many experimental investigations have been carried out to explore and understand the dynamical characteristics of this interaction. The characteristics of vortex shedding over bluff bodies were reported by Grffin et al. (1973) , Griffin & Ramberg (1974), Griffin (1989) and Griffin & Hall (1991). They observed both symmetric and asymmetric vortex patterns over a wide range of oscillation conditions. They also reported the occurrence of the vortex lock-on frequencies for different oscillation conditions and gave a detailed discussion on the spectral element computations of the perturbed flow over the cylinder. Shirakashi et al. (1985) investigated the spectral characteristics of the flow-induced vibration of a single cylinder and observed an unexpected subharmonic resonance at fy / fn 5 3 , in addition to the normal one at fy / fn 5 1 , where fy is the Karman vortex shedding-frequency and fn is the natural frequency of cylinder oscillation. Ota et al. (1987) investigated the dynamic response of an elliptic cylinder; they showed that at Reynolds number greater than a critical value, the flow around the cylinder becomes 0889 – 9746 / 96 / 060633 1 19 $18.00

÷ 1996 Academic Press Limited

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M. N. HAMDAN ET AL.

relatively stable with the peak frequency of the spectrum fluctuating from 760 to 890 Hz, and the power spectral density extends over a very wide range of frequency. Due to the complexity of the structural fluid dynamic interaction, an exact analytical model has not yet been developed. Most of the existing models reported in the literature are based on over-simplified assumptions of the actual physical process (Blevins 1977). The introduction of the absolute-convective theory of fluid dynamic stability has led to a promising new approach for understanding the physics of vortex formation and near wake flow development (Morkovin 1964; Iwan & Blevins 1974; Griffin & Ramberg 1976; Griffin 1978; Triantafyllou et al. 1987; Oertel 1990; Huerre & Monkewitz 1990). Recent stability calculations based upon computed and measured mean velocities in the wakes of stationary circular cylinders suggest that the vortex formation region is absolute unstable, while the fully formed vortex street is convectively unstable (Howell & Novek 1979; Rockwell 1990; Nakagawa 1986). It is to be noted that in the aforementioned investigations, as well as in similar ones, the spectral distribution using the Fourier transform (FT) was employed to study and analyse the data further. However, the FT, which is only a frequency-domain analysis method, does not provide information about the specific time at which the peak occurs or its duration (i.e., localization in time), and assumes that the analysed signal is infinite in extent and that the window used in discretizing the signal is representative throughout the signal. This method leads to acceptable results provided that the process is stationary. However, experimental investigations of flow-induced vibration showed that it is in general a nonstationary process, which may involve chaotic and transient behavior (Holmes & Marsden 1978; Sekar & Narayanan 1994; Li & Paidoussis 1994). Due to these inherent limitations of the stationary Fourier analysis, new methods for joint time-frequency analysis (JTFA) of nonstationary signals have been developed, and have become a hot research topic in many engineering fields, such as fluid mechanics, acoustics, and structural dynamics (Grossman et al. 1989; Arneodo & Grasseau 1988; David & Chapron 1989; Bacry et al . 1989; Everson et al . 1990; Miles 1991; Newland 1993, 1994a, b). More recently, Newland (1994a, b) introduced the harmonic wavelets, of complex coefficients, which are simple to generate and which can be used to analyse random vibration signals. In comparison with traditional waveform of frequency analysis techniques the JTFA has the advantage of revealing additional information about how the frequency content of the signal may change over time. The common methods used for JTFA include the short-time Fourier transform (STFT), the Gabor transform (GT), and the wavelet transform (WT). The traditional methods of signal analysis as well as new ones (JTFA methods), such as the wavelet transform, are outlined in Section 3. Discussion of the results and comparisons are presented in the remainder of this paper, whereas the experimental set-up and procedure are described in Section 2. 2. EXPERIMENTAL SET-UP AND PROCEDURE The experimental investigation was conducted in an open suction-type wind tunnel with a square cross-section of 30 cm 3 30 cm and of length equal to 200 cm, as shown in Figure 1. The freestream velocity was varied from 4?5 to 27 m / s; the freestream turbulence intensity level was 0?35%. The test cylinder was placed at 1?3 m from the inlet of the test-section were the flow was found to be fully developed. The range of

635

WAVELETS FOR FLOW-INDUCED VIBRATION

y

0·46 0·30

Signal to processor

Silencer Double butterfly valve Fan unit Plate spring Diffuser

0·30

0·54 0·48

0·50 0·67

0·02 0·50

0·90

0·90

Built-in end Test-section

1·30 2·00

0·90

Contraction Figure 1. Experimental set-up. All dimensions in meters.

Reynolds number was approximately 6?12 3 103 , Re , 3?78 3 104, based on the outside diameter of the vibrating cylinder. In this range, the Strouhal number for a circular cylinder is about 0?2. The test cylinder was made of aluminum tubing, of outer diameter D 5 21?5 mm, wall-thickness t 5 0?5 mm and length L 5 440 mm. This combination yields an aspect ratio L / D of 20?5 and mass per unit length m # of 0?1662 kg / m. The cylinder mountings in this experiment are similar to those used by Shirakashi et al. (1985). The cylinder was suspended by two similar clamped plates at its ends (Figure 1). The plates were placed outside the test-section of the wind tunnel to avoid interference with the flow. By adopting this sort of mounting, one can minimize the mode-coupling effects which may arise from the streamwise and the rotational motions. Motion in the horizontal direction for this mounting is difficult to excite since the plate axial rigidity is much higher than its rigidity in the vertical direction, i.e. the horizontal vibration natural frequency of the cylinder for this mounting is much higher than that of the vertical vibration. The cylinder passed through two slots of 28 mm 3 28 mm of the wind tunnel. Note that careful consideration was taken to ensure the two-dimensionality of the vortex wake. This was achieved by making the height of the slots at the two sides of the wind tunnel to be much less than nearly four times the diameter of the test cylinder, as was recommended by Graham (1969). This was confirmed by preliminary tests. In order to eliminate the influence of the flow through the slots, blocking plates were attached to the cylinder at both ends. This technique relies upon the isolation of interference effects that arise as a result of the interaction of the tunnel boundary layer with the cylinder (Argoul et al. 1989; Zabusky 1984). Fox (1992) reported that the rectangular plates should have an upstream location (distance of leading edge from the cylinder axis) sufficiently large to isolate the horseshoe vortex generated by the wall-cylinder interaction, but small enough to avoid substantial boundary layer growth on the plate itself, and tail dimensions adequate to prevent any wake interference. Using this concept, the end-plates were designed square with an area of 7D 3 7D , and with the hole to accommodate the cylinder at 2?5D from the leading edge of the plate; D is the cylinder diameter. An accelerometer (type B&K 4370) was placed on one end of the test cylinders. The

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accelerometer has a voltage sensitivity of 9?98 mV / (m / s2). The fundamental natural frequency of the accelerometer in the axial direction (direction of measurement) is about 25 kHz and its traverse fundamental natural frequency is about 4 kHz. The output signal from the accelerometer was simultaneously fed to a portable vibration meter (B&K 2511), which allowed direct read-out of vibration amplitude (acceleration, velocity or displacement) and to a conditional amplifier (B&K 2626). The output of the conditional amplifier was in turn fed to a computer via an interface card type (NI Lab PC1). The A / D converter has a resolution of 16 bits. The natural frequency, fn , and the logarithmic decrement, d , of the test cylinder were determined by an impulsive test, wherein the cylinder was set into free vibration by slightly tapping its center. The freestream velocity was measured using an inclined Pitot tube placed at 1D in the horizontal direction and 2D in the vertical direction from the static equilibrium position at the mid-point of the oscillating cylinder. Various time series were chosen for different freestream velocities. The digitization of the time series by the computer program was carried out at a sampling rate that obeys the Nyquist formula, given by fs $ 2fn ,

(1)

where fs is the sampling frequency. Noting that the natural frequency fn , of the test cylinder in the transverse direction is about 120 Hz, the sampling frequency, fs , was chosen to be 4000 Hz. This sampling rate was found to be satisfactory for the type of signals considered in this work. A time record of 2050 data points were taken in each experimental run. 3. SIGNAL ANALYSIS AND WAVELET TRANSFORM Random vibration signals are generally classified as stationary or nonstationary. In order to represent a stationary process adequately the time record should be longer than the basic cyclic time of the process. Many FFT analysing systems transform (210 5 1024) data samples into 512 positive frequency values; only the first 400 frequency lines are displayed so that the calculated frequency spectrum is for an effective time record of length T 5 400 / fs , where fs is the full-scale frequency; i.e., for a full-scale frequency fs 5 10 kHz the effective time record is 40 ms. If the length of the time record is comparable with the transform size of the analysing system T , then the full power spectrum of the data record is calculated in one pass. It should be noted that the FT decomposes the analysed signal canonically into sines and cosines assuming the signal to be periodic and stationary. Therefore, if the signal contains transient and nonperiodic events, these may be smeared across the time or ignored completely in the transformation process, as the FFT assumes the frequencies to be constant with time so that it does not provide temporal information about the frequency content; i.e., FFT does not provide information about time localization of high frequency transients which may exist in the process. This is so, since a sinusoidal function is highly localized in frequency domain and infinitely distributed in time domain. For a more accurate representation of a stationary process, data is usually collected over a time period as long as possible depending on the available memory and computational expenses, so that the length of the time record is much longer than the transform time size T of the analysing system. A scan analysis using the so-called short time Fourier transform (STFT) is used in this case to obtain an average spectrum, usually a moving window in overlapping steps along the time record. By plotting the spectra obtained for each segment versus time, i.e. at the middle of the moving window, a three-dimensional time-frequency amplitude plot may be obtained, from which one can study the


637

variation of various frequency components with time. Accordingly, the STFT is given by STFT(v , t ) 5

E

`

2`

g (t 2 b )e2jvtx (t ) dt ,

(2)

where g (t 2 b ) is a suitably shifted analysing window centered around b. It should be noted that the STFT is one of the most widely used traditional methods of joint time-frequency analysis (JTFA), which is also used for nonstationary signals where the time record is scanned in overlapping steps. In the case of nonstationary signals a heavy overlapping of the moving window is usually required in order to describe the various activities of the signal adequately. This task is usually time consuming and computationally expensive; therefore, one usually skips many of the data points in order to reduce the computational effort to an acceptable level. The STFT suffers from several pitfalls which include aliasing, the time-window effect and the picket-fence (Newland 1993). For example, for a large window, low frequencies can be calculated, while high frequency events are poorly localized. On the other hand, for a narrow window, high frequency events are localized while the frequency bandwidth is reduced; in other words STFT suffers from bandwidth-localization tradeoff. The STFT resolution in the time and frequency domains cannot be arbitrarily small simultaneously because their product has a lower bound (Gabor 1946) Time-Bandwidth Product 5 Dt Df $ 1 / 4π .

(3)

These window constraints reduce the effectiveness of STFT in the analysis of nonstationary and impulsive signals. In other words, traditional STFT analysis produces either good time-domain resolution or good frequency-domain resolution. Good joint time-frequency resolution is generally difficult to achieve because of the computational burden and the lack of variety for a suitable window. In order to overcome the many pitfalls of the STFT method, alternative JTFA methods have recently been developed, which include, among others, the wavelet transform. In analogy with the STFT, the wavelet transform correlates a selected analysing waveform (window) with the analysed time function, but with a few important differences as will briefly be discussed below. The wavelet transform (WT), firstly given by Grossman & Morlet (1985), assumes that any time signal can be decomposed canonically into a combination of time-shifted and dilated or compressed basic wavelets; accordingly, WT is given by WT(a , b ) 5

1 a

20?5

E

`

2`

x (t )h

St 2a bD dt,

(4)

where the wavelet function h (t ) is an appropriate window (called basic wavelet) like the modulated Gaussian. Hence, the WT can be defined as the convolution of a signal x (t ) with an analysis window h (t ) shifted in time by b and dilated by a scale parameter, a. The scale parameter, a , can be chosen such that it is inversely proportional to frequency. The factor ua u20?5 is used to ensure energy preservations (Daubechies 1988). The factor ua u20?5 is the normalizing factor which is obtained by correlating the signal with itself and equating the correlation to 1?0. Usually, in structural dynamics applications, the normalizing factor is calculated by correlating the signal with a sine

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Frequency

Frequency

Time

Time

Figure 2. Time-frequency resolution of (a) wavelet transform and (b) STFT.

wave and equating the correlation to 1?0. This leads to a normalization factor equal to 1 / a. The purpose of WT is to extract the localized conditions of the signal labeled by the two parameters a and b and then the signal can be computed by expanding it into a family of functions, i.e. into a set of frequency channels of equal bandwidth on a logarithmic scale (Daubechies 1988). In comparing equation (2) with equation (4) it can be seen that, whereas the STFT uses a window of constant width and envelope, the WT uses an analysing waveform modified by a given envelope which is scaled in time and magnitude to have a fixed number of oscillations inside the envelope. For high frequencies, the basic wavelet (the analysing window) is compressed in time, whereas for lower frequencies it is dilated in time. The width of the envelope scales along with the frequency, such that when the frequency is increased, the time envelope is narrowed, while the number of oscillations inside the envelope remains constant. This property results in good resolution in the frequency domain up to the theoretical Nyquist frequency, as well as good localization in the time domain. The time-frequency resolution of the STFT and WT are shown in Figure 2 for comparison (Rioul & Vetterli 1991; Khraisheh et al. 1995). The STFT transforms the signal in three steps; slicing the time data into segments, transforming these segments to the frequency domain, and then reconstructing these segments to form the joint time-frequency plot. On the other hand, WT directly transforms the time signal to the joint time-frequency domain. In general, the basic wavelet may be taken as any waveform; however, the best JFTA resolution is obtained if the basic wavelet is optimally localized in both the time and frequency domains. It should be pointed out that one reason that WT has not yet been used extensively is the fact that it is difficult to choose a basic wavelet function that will best transform a nonstationary signal. [The choice of the basic (generating) wavelet is critical, for example, the selected wavelet must be suitable for the problem at hand, which explains its lack of popularity, despite its powerfulness.] In order to compute the WT for discrete data, it is necessary to discretize equation (4). In general, good time-frequency localization properties can be obtained if discrete values for a and b are chosen such that, the parameter a is chosen to be equal to 2i where i is termed the octave of the transform (Mallat 1989a, b; Daubechies 1991; Shensa 1992), and the parameter b is taken to be a multiple of a , in particular b 5 n 2i. This leads to WT(a , b ) of the form WT(2i , 2in ) 5

1 42i

E hS2t 2 nDx(t) dt, `

2`

i

where n is the taken as the number of data samples.

(5)


639

Upon discretization, using the above set parameters a and b , equation (5) becomes WT(2i , 2in ) 5

1 42i

O hS2t 2 nDx(t). i

(6)

n

The above equation provides a multi-resolution analysis of the signal x (t ) for each i and n. Specifically, the coefficients WT(2i , 2in ) carry information about the signal x (t ) near the frequency 2i and near the time instant n 2i and the sum in equation (6) provides an approximation to x (t ) up to a scale of 2i , i.e., it provides a low-pass filtered version of x (t ) with less detail. This is one of the aspects which make WT decomposition a powerful tool in studying the scaling properties of the signal (Tewfik & Kim 1992; Tewfik et al. 1992; Rioul 1993). In the present investigation, the flow-induced vibration signal is analysed using a similar discretization procedure as above following Daubechies’ (1990) approach. Accordingly, the scale and shift parameters a and b are selected such that they m constitute discrete lattices of wavelets. Taking a 5 a m 0 and b 5 nb 0 a 0 , where m , n P Z , leads to m /2 hm ,n(t ) 5 a 2 h (a0 t 2m 2 nb 0) , 0

(7)

where a0 and b 0 are chosen to be 2?0 and 1?0, respectively. In this case, the family of the generated wavelets, hm ,n(t ) , constitutes an orthonormal basis for L2(R ) functions, where L2(R ) is the space of the squared integrable functions (Daubechies 1992). Typically, the Fourier transform, H (v ) , has a compact form. It should be emphasized at this point that the choice of the wavelet, h (t ) , plays an important role in identifying the properties of the analysed signal. The problem associated with the selection of a suitable wavelet is thoroughly discussed in references (Tewfik et al. 1992; Rioul 1993). In this work, for comparison purposes, four selected types of well-known wavelets are used to analyse the time series of the vibration signals, these wavelets are listed below. (a) Daubechies (1990, 1992) constructed the following tight frame wavelet, known to be adequate for the analysis of high frequency signals:

H (v ) 5 (log a 0)21/2

where

E

0,

F S F S

DG DG

π v 2l sin … , 2 l (a 0 2 l ) π v 2l cos … , 2 la0(a 0 2 l ) 0,

v ,l l # v # a0 l (8) a0 l # a 20l

v $ a 20l

l 5 2π / [b 0 a 20 2 1] k

(9)

`

and … is a C (or C ) function from R to R (Real to Real) that satisfies

… (x ) 5

H01

if x # 0 . if x $ 1

(10)

An example of a (C 1) function … is

… (x ) 5

5

0, x #0 2 sin (xπ / 2) 0 # x # 1 1, x $ 1.

(11)

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M. N. HAMDAN ET AL.

This construction leads to a family of tight wavelet frames with no restrictions on the choice of the parameters a0 and b 0 , other than a 0 . 1 and b 0 ? 0 . (b) The modulated Gaussian wavelet, used by Grossmann & Morlet (1985) in analysing seismic data, known to be adequate for analysing smooth signals, is given by h (t ) 5 π 21/4(e2jv 0t 2 e2v 0/2)e2t /2 , 2

2

(12)

H (v ) 5 π 21/4[e2(v 2v 0) /2 2 e2v 0/2e2v /2]. 2

2

2

The subtraction term in the definition of h and H ensures that H (0) 5 0 . Kadambe & Boudreaux-Bartels (1992a, b) used the modulated Gaussian wavelet with

v 0 5 π (2 / ln 2)1/2.

(13)

A fixed value v 0 will be used in the present work, which leads to a negligible value for the subtraction term in H (0). This also leads to a ratio between the highest and the second highest local real maxima of h equal to 1 / 2, approximately. 2 (c) The Mexican-hat wavelet, which is the second derivative of the Gaussian e2t /2 (Daubechies 1990), is known to be used for detecting discontinuities. If this wavelet is normalized such that its L2-norm is 1, one obtains h (t ) 5 H (v ) 5

2 21/4 2 π (1 2 t 2)e2t /2 , 43

(14)

2 21/4 2 2v 2/2 π v e . 43

When this function is plotted and rotated around its symmetrical axis, then one obtains a shape similar to that of a Mexican hat. (d) The eighth derivative of the Gaussian wavelet, which functions like the Mexican hat (Kadambe & Boudeaux-Bartels 1992b), is appropriate in applications of the wavelent transform to edge detection. It is given by

S215!7!D 2 7! H (v ) 5 S D 15! 15

h (t ) 5

15

1/2

π 21/4(t 8 2 28t 6 1 210t 4 2 450t 2 1 90)e2t /2 , 2

(15)

1/2

π 21/4v 8e2v /2. 2

These four different basic wavelets were used to compute the discrete wavelet transform (DWT) given by equation (4), for flow-induced vibration signals, which in the frequency domain takes the form WT(m , n ) 5

O a Ee m 0

jv nb 0am 0

H (a m 0 v )F (v ) dv ,

(16)

m ,n

where m , n P Z; F and H are the Fourier representation of the analysed signal and the chosen wavelet, respectively. A special computer program using NAG-library subroutines was written to calculate WT according to the previous discussion for every m and n. The program computes the WT for different basic wavelet types. Also, the program was tested by finding the WT of a known signal and then comparing the reconstructed signal with the original one. The correlation of the signal with the generating (basic) wavelet yields a twodimensional complex function defined in v and τ . The square modules of the complex functions are used to represent the energy density distribution which is presented in a three-dimensional plot. The x -axis represents the temporal position of the center of the correlated wavelet function in time, the y -axis represents the wavelet frequency, and the density is seen through the z -axis on the three dimensional plot.


641

Also in this work the following traditional functions are used: the autocorrelation function and the power spectral density function. The autocorrelation function provides information on the dependence of a random variable value at one time on the value at another time, and it is given by 1 T 5` T

Rx (τ ) 5 lim

E

T /2

2T /2

x (t )x (t 1 τ ) dt 5 E [x (t )x (t 1 τ )] ,

(17)

where E [x (t )] is the expectation operator. On the other hand, used for simplicity, power-spectrum and power-spectral density techniques arise as powerful classical methods to study the frequency content of the signal. When the signal transformed to the frequency domain, X (v ) , corresponds to a discrete signal X (n ) , then the power spectrum is defined as X (v ) 5

O X (n)e `

2jv n

.

(18)

2`

This transformation is not used for random signal analysis and instead the Fourier transform of the autocorrelation function, called the power spectral density, is often used. This function has two admirable properties; it represents the distribution of the power with respect to frequency and it measures the rate of the fluctuation of the random signal. The power spectral density is given by Sx (v ) 5

O R (n)e `

2`

x

2jv nt

.

(19)

A Labview program using the G-language was constructed to collect data and to calculate the autocorrelation and the power spectrum, with visual and output files capability. 4. RESULTS AND DISCUSSION The test cylinder was mounted as described in Section 2. The natural frequency, fn , and the damping ratio of that system were found to be 120 Hz and 0?02, respectively, through a simple impulse test. The time series of the vibration (acceleration) of the cylinder in the transverse direction in the steady-state region and the corresponding autocorrelation and power spectrum at freestream velocity of 19?0 m / s are shown in Figure 3. The 3-D wavelet plot for the above time series, obtained using the modulated gaussian as a mother wavelet, is shown in Figure 4. It should be noted that, for every collected data record, a WT plot was obtained for the full length of the record (i.e., for N 5 2050; t 5 512 ms). However, the displayed WT plots are shown, for convenience, for time t (t , 512 ms) after which the plot does not show any significant change in behavior with t. Also, the most clear view of the 3-D WT plot, obtained by rotating the plot about its vertical (energy distribution) axis, is presented. The autocorrelation function, Figure 3(b), in this case indicates that the process is narrow-banded (i.e., the autocorrelation decays slowly with shift time). The corresponding power spectrum, Figure 3(c), indicates that there are four dominant frequencies: subharmonic ( f 5 1–3 fn ) , fundamental ( f 5 fn 5 120 Hz), and superharmonics ( f 5 3fn and f < 5fn ). The third-order superharmonic ( f 5 3fn ) appears to have the dominant contribution, while the contributions form the fundamental and the fifth superharmonics are nearly equal; however, the subharmonic ( f 5 1–3 fn )

642 Acceleration (volts)

M. N. HAMDAN ET AL.

0·04

0·00

–0·04

0

500

1000

1500

2000

1500

2000

1500

2000

Data points, N

Autocorrelation

(a)

0·4 0·0

–0·4

0

500

1000

Power spectrum (dB)

Data points, N (b) –40 –60 –80 –100 –120

0

500

1000 Frequency (Hz) (c)

Figure 3. (a) The time series (1 volt 5 9?98 3 10 m / s ), (b) the corresponding autocorrelation, and (c) the corresponding power spectrum, at a freestream velocity of 19?00 m / s. 3

2

contribution is lower than that of the other three harmonics. This suggests that the process in this case is strongly nonlinear, probably of cubic type. It is to be noted that time-scale representation used for wavelet transforms replaces the time series or frequency distribution with a scale parameter related to time and frequency but not equal to them. The frequency scale obeys the Nyquist frequency, i.e. 1 , am 0 , fs , where fs is the sampling frequency, 2Tfs # 1 ; also the value of the frequency scale (m ) on the 3-D wavelet plot is related to the actual frequency ( f ) of the time series waveform by the relation am 0 5 4f0 / f , m 0

(20)

where a is the scale parameter as discussed in Section 3, and f0 is an arbitrary constant which depends on the type of wavelet used; i.e. f0 5 5?336 for modulated Gaussian, f0 5 1?42 for Mexican hat, f0 5 2?83 for eighth derivative of the Gaussian, and f0 5 4?188 for Daubechie’s tight frame. Thus, for the modulated Gaussian shown in Figure 4, for example, m 5 23?04 corresponds to an actual frequency f 5 360 Hz. On the other

643


Energy distribution

2·0

1·5

1·0

0·5

0·0

Fre –6 qu en

–4 105 cy 104 s cal –2 Fre e, 103 qu m en 100 cy (H 10 z) 5·3

200

0

50

250

150 100 e, N Tim

0

Figure 4. The 3-D wavelet plot using the modulated Gaussian as a mother wavelet, at a freestream velocity of 19?00 m / s. N is the number of data points, which is proportional to time; N 5 tfs , where fs is the sampling frequency and t is time.

hand, the time-scale for all the time series, the autocorrelation and the 3-D wavelet plots for all types of wavelets is related to the real time (t ) via t 5 (Time Scale) / fs ,

(21)

where fs is the sampling frequency, equal to 4 kHz. The wavelet transform, Figure 4, indicates that the energy of the vibration signal is concentrated at specific frequencies and times. It shows that the process is dominated by the above four harmonic components f 5 1–3 fn , f 5 fn , f 5 3fn and f 5 5fn ; however, the fifth harmonic has a lower contribution than the first, while as in the frequency spectrum results, the third harmonic has the dominant contribution, and the third subharmonic the least. The wavelet results, Figure 4, also indicate that while the third subharmonic and the fundamental persist (continuous with time), the third and the fifth harmonic show intermittent behavior with time, where for some time intervals only one of these two harmonics ( f 5 3fn , f 5 5fn ) is present in the response, with nearly equal duration. This may suggest the existence of transient disturbances in the flow or chaotic behavior, or some nonlinear modal coupling in the response of the cylinder, which requires further investigation. Figures 5 – 7 show the 3-D wavelet plots of the time series in Figure 3 obtained using as mother wavelets the Mexican hat, eighth derivative of the Gaussian and Daubechies tight frame, respectively. The Mexican hat wavelet, Figure 5, indicates that the process is dominated by the third superharmonic ( f 5 3fn ) of variable magnitude, and continuous with time, while the fundamental harmonic has a relatively small contribution and is discontinuous with time. The subharmonic ( f 5 1–3 fn ) and superharmonic ( f $ 3fn ) contributions shown in Figure 4 using the modulated Gaussian are not clearly identified in Figure 5, whereas another time-discontinuous contribution at ( f 5 12fn ) appears.

644

M. N. HAMDAN ET AL.

1·2

Energy distribution

1·0 0·8 0·6 0·4 0·2 0·0 –6 Fre 4 que –4 10 ncy 3 10 sca –2 Fre le, que m 100 ncy 0 10 (H z) 1·4

250 200 100 50

150 e, N Tim

0

Figure 5. The 3-D wavelet plot using the Mexican hat as a mother wavelet, at a freestream velocity of 19?00 m / s.

The eighth derivative of the Gaussian wavelet, Figure 6, shows that the process is totally dominated by a time continuous tenth subharmonic ( f 5 0?1fn ) of nearly constant magnitude and does not show any significant contribution from other harmonics except from a very high continuous superharmonic at ( f < 24fn ).

Energy distribution

0·20

0·15

0·10

0·05

0·00

100 75 Tim 50 e, N 25

0

–8

0 –2 m 2·8 –4 y scale, 10 c n –6 ue 100 Freq 103 4 (Hz) 10 ency 105 u q e Fr

Figure 6. The 3-D wavelet plot using the eighth derivative of the Gaussian as a mother wavelet, at a freestream velocity of 19?00 m / s.


645

Energy distribution

30

20

10

0

100 75 Tim 50 e, N 25

0

–8

0 –2 m 4·2 , e l –4 y sca 10 nc e –6 u 100 q Fre 103 (Hz) 104 5 ncy e 10 u q Fre

Figure 7. The 3-D wavelet plot using Daubechies’ tight frame as a mother wavelet, at a freestream velocity of 19?00 m / s.

The Daubechie’s tight frame wavelet, Figure 7, shows a time changing frequency content from low frequencies of about 4 Hz to higher ones of about 17 000 Hz of changing magnitudes with maximum peaks occurring at lower frequency f < 4 Hz. It is difficult in this case to make a precise judgment on the frequency content of the process. We expect the power spectrum in Figure 3 to give a good overall picture of the frequency distribution; hence, the wavelet which gives a distribution which most closely resembles it is used to give the time variation around the mean distribution. Based on the above results, in comparing Figure 4 with Figures 5 – 7 and Figure 3, it appears that the modulated Gaussian is the most adequate wavelet to be used for the timefrequency analysis of the process under consideration. Also, it is noted that similar comparisons between the results obtained using the various types of wavelets mentioned above with the frequency spectrum at different freestream velocities, showed the modulated Gaussian to be the most suitable and reliable mother wavelet to be used in this case. For this reason only the results using the modulated Gaussian are presented at different freestream velocities in the sequel. Figure 8 shows the time-series waveform, and the corresponding autocorrelation and the power spectrum for a freestream velocity of 4?98 m / s; Figure 9 shows the corresponding 3-D modulated Gaussian wavelet plot; and Figure 10 is another view of Figure 9, using a shorter time scale. In this case, the autocorrelation indicates that the process is broad-banded and the power spectrum shows the process to be dominated by the fundamental harmonic ( f 5 fn ) , a subharmonic ( f 5 1–3 fn ) and a superharmonic ( f 5 2fn ) , as well as the existence of other superharmonics up to f 5 10fn . The wavelet transform Figures 9 and 10 show the existence of a dominant time-continuous superharmonic ( f 5 3fn ) of variable magnitude, a time-continuous fundamental harmonic of smaller and nearly constant magnitude, intermittent superharmonics ( f . 3fn , up to f 5 10fn ) of various magnitudes, and no superharmonic ( f 5 2fn ). The fact that


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the process is broad-banded suggests that the process may be chaotic; to prove this, requires further investigation, using phase-plane portraits, Poincare´ maps and fractal dimension calculations. Figure 11 shows the time series, the corresponding autocorrelation and power spectrum, whereas Figure 12 shows the corresponding modulated Gaussian 3-D wavelet plot at a freestream velocity of 24?08 m / s. The autocorrelation and the power spectrum indicate that the process is broad-banded with the existence of a low subharmonic ( f 5 1–3 fn ) , fundamental ( f 5 fn ) and superharmonics up to f 5 8fn , all of appreciable magnitude. On the other hand, wavelet results indicate that the process is dominated by a time-continuous superharmonic ( f 5 3fn ) of variable magnitude, a smaller magnitude time-continuous fundamental harmonic, and very small intermittent sub- and superharmonic components. In this case, the power spectra do not correlate well with the wavelet transform results, as the process in this case is nonstationary and thus the power spectra alone cannot give enough information concerning the frequency content of the process.


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0 –2 m 5·3 , e –4 y scal 10 c n e –6 u 100 Freq 0 –8 103 z) 4 10 cy (H n e 105 u Freq Figure 9. The 3-D wavelet plot using the modulated Gaussian as a mother wavelet, at a freestream velocity of 4?98 m / s. 150 Tim 100 e, N 50

It is to be noted that the peak at f 5 1–3 fn observed in the power spectra at the above three freestream velocities (U` 5 4?98 , 19?0 and 24?08 m / s) does not appear in the wavelet transform at (U` 5 4?98 and U` 5 24?08 m / s), and appears as a component of very small magnitude in comparison with the other dominant harmonics for

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Figure 10. Another view of Figure 9, the 3-D wavelet plot using the modulated Gaussian as a mother wavelet, at a freestream velocity of 4?98 m / s.


M. N. HAMDAN ET AL.

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U` 5 19?0 m / s. Although, as indicated above, this peak appears in the power spectra for these flow velocities, it vanishes in the wavelet transforms; this agrees with assumption made by Shirkashi et al. (1985). Whereas the frequency spectrum indicates stable, with time, frequency content of the vibration signal, the wavelet transform indicates that some of the harmonic components of the signal are intermittent with time, which suggests that the process is associated with nonlinear dynamic and may lead the system to chaotic behavior; it may also suggest that some transient (intermittent) perturbations occur in the flow. It is to be noted that, in general, the frequency spectrum is difficult to read by visual observations, due to the presence of different sideband families around the harmonic of the system. These sidebands may for example be due to flow disturbances as a result of fan dynamics, background noise, etc. In the present work no prefiltering was carried out on the collected data to eliminate some of the modulated signals responsible for some of these sidebands as is usually done. An obvious advantage of the WT method is that


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10 9 Energy distribution

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0 –2 m 5·3 –4 y scale, 10 c n e –6 u 100 Freq 103 4 (Hz) 10 ency 105 u q e Fr

Figure 12. The 3-D wavelet plot using the modulated Gaussian as a mother wavelet, at a freestream velocity of 24?08 m / s.

the modulated frequencies are well isolated from the dominant harmonics. In order to verify these hypothesis, the time series should be further investigated using advanced dynamical analysis methods, such as the construction of phase-plane plots and Poincare´ -maps, fractal-dimension calculations and the determination of modulating frequencies. These are the subject of ongoing investigation and will be reported in future. 5. CONCLUSIONS When compared to other classical methods, such as the power spectral method which assumes the process to be stationary and provides time-averaged frequency information (and thus does not reveal all the aspects of a random process), the JFTA (joint time-frequency analysis) of flow-induced vibration signals using wavelet transform reveals more information about the frequency content of such a random signal. For example, at a freestream velocity of U` 5 24?08 m / s, whereas the power spectrum indicates that the response is dominated by the three peaks f 5 1–3 fn , f 5 fn and f 5 3fn , the wavelet transform indicates that the process is dominated by the third and fundamental harmonics ( f 5 3fn and f 5 fn ); the fundamental corresponds to translational mode, while f 5 3fn corresponds to the rotational mode in agreement with observations (Blevins 1977). Wavelet transform reveals more information regarding the nonlinear behavior of the dynamical process and the possibility of chaotic behavior of the system through precisely indicating the unsteady harmonics, magnitude vibrations, frequency bifurcations and transients of the system. The problem of selecting the most suitable type of wavelet, as well its parameters, in terms of computational effort and reliability still represents a challenge to the signal analyst. For the present process, which is basically a low-frequency one, the adaptation

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