Measurement 44 (2011) Contents lists available at SciVerse ScienceDirect. Measurement

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1 Measurement 44 (2011) Contents lists available at SciVerse ScienceDirect Measurement journal homepage: Enhancement of coherence functions using time signals in Modal Analysis Abdul Ghaffar Abdul Rahman a, Zhi Chao Ong a,, Zubaidah Ismail b a Mechanical Engineering Department, Faculty of Engineering, University of Malaya, Malaysia b Civil Engineering Department, Faculty of Engineering, University of Malaya, Malaysia article info abstract Article history: Received 25 February 2011 Received in revised form 13 June 2011 Accepted 16 August 2011 Available online 27 August 2011 Keywords: Modal Analysis Dynamic characteristics Spectral Averaging Impact-Synchronous Time Averaging Transfer functions Coherence functions Experimental Modal Analysis (EMA) and Operational Modal Analysis (OMA) are two widely used techniques in the identification of modal parameters. EMA is synonymous with a laboratory environment requiring complete system shutdown while OMA is implemented in a real environment where the ambient forces cannot be isolated. A new method, namely Impact-Synchronous Modal Analysis (ISMA) utilising the modal extraction techniques commonly used in EMA but performed in the presence of the ambient forces, is proposed. Transfer functions, from where the modal parameters are extracted, are obtained from Fourier transform of cross and auto correlation functions. These functions are estimated quantities and their outcomes are dependable on the averaging techniques used. The coherence functions are commonly used to measure the acceptability of the estimations. Impact-Synchronous Time Averaging is compared against Spectral Averaging while performing Modal Analysis in a situation containing ambient and operating forces. Results showed that while the transfer functions obtained from both the averaging techniques were of similar quality, the Impact-Synchronous Time Averaging indicated better coherence than the Spectral Averaging. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Corresponding author. Address: Department of Mechanical Engineering, Engineering Faculty, University of Malaya, Kuala Lumpur, Malaysia. Tel.: ; fax: addresses: agar@um.edu.my (A.G.A. Rahman), zhichao83@um.edu.my (Z.C. Ong), zu_ismail@um.edu.my (Z. Ismail). Three parameters, namely modal frequencies, modal shapes and modal damping, comprehensively define the dynamic characteristics of a structure. Prior to developing a mathematical model for the dynamic behaviour of the structure, these modal parameters need to be identified. Modal identification started from nonparametric determination of modal parameters based on different representation of measured frequency response functions (FRF) [1]. Parametric modal identification advanced significantly in the 1970s with the development of Complex Exponential algorithm based on Prony s method [2]. This method can be classified as the first important SISO or Single Input Single Output parametric modal identification method [3]. In 1977, a well-known Ibrahim Time Domain (ITD) method was proposed [4]. ITD was the modal identification algorithm formulated in Single-Input/Multi-Output (SIMO) version. During the 1980s, two milestones were created. Firstly, modal identification was performed in the frequency domain. Secondly, modal identification developed from SIMO to MIMO (Multi-Input/Multi-Output). Rational Fraction Polynomial (RFP) developed in 1982 based on orthogonal polynomials was the first major frequency domain modal identification technique [5,6]. It has been observed that modal identification accuracy can be improved by applying a correlation filter or data correlation to noisy time response data. A number of modal identification algorithms have been developed individually by different researchers with different formulations and matured in the late 1980s and early 1990s [7 13]. Much work has been done in the last decade to further develop the modal identification techniques. First is the development from traditional Experimental Modal /$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi: /j.measurement

2 A.G.A. Rahman et al. / Measurement 44 (2011) Analysis (EMA) using I/O (input output) measurement data to Operational Modal Analysis (OMA) using O/O (output-only) data. Second is the modal parameter estimation from a two-stage approach to a one-stage approach. This is continued with the development modal from deterministic to statistical framework. This has increased the estimation accuracy by reducing the influence of the measurement noise and system distortion. It provides not only the modal parameters but also their confidence intervals. Various researchers have been conducting OMA in order to perform modal analysis while the structures and systems are in operation. It is used for accurate modal identification under actual operating conditions and in situations where it is difficult or impossible to control an artificial excitation of the structure. OMA holds certain advantages over EMA in terms of its practicality and is easiness to carry out the procedure. Also, it performs the analysis while the system is in operation and the measured responses are representative of the real operating conditions of the structure. However, the lack of knowledge of the input forces does affects the parameters extracted. For example, mode shapes obtained from OMA cannot be normalised accurately, subsequently affecting the mathematical models. Over the years, in the effort to improve the estimation accuracy in OMA and EMA, the focus has been mainly in the development of modal identification algorithms. Relatively less effort is made on the digital signal processing aspects, especially upstream of the collected data. In this paper, a method, named Impact-Synchronous Modal Analysis or ISMA, that utilises Impact-Synchronous Time Averaging [14] is proposed. ISMA has the advantages of the OMA and EMA combined. It carries out the analysis while the system is in operation and at the same time is able to provide the actual input forces in the transfer functions, hence, allowing for better modal extractions. 2. Overview of modal analysis The fundamental difference in the two techniques (OMA and EMA) lies in the manner by which the system is excited and subsequently the modal identification algorithms Operational Modal Analysis (OMA) In practical situations where the system cannot have a complete shutdown or the structure is too huge to respond to artificial excitation, OMA is sought. Here, the operating forces coming from the structure cyclic loads or ambient forces are used as the exciters. As these quantities cannot be measured, OMA utilised only information from the measurable responses and some patented algorithms are used to extract the three parameters [15 18]. Researchers have combined OMA with correction technique of spectrum analysis (CTSA) [19,20], harmonic wavelet filtering (HWF), random decrement technique (RDT) and Hilbert transform (HT) method to obtain modal parameters [21]. In OMA, structural modal parameters can be computed without knowing the input excitation to the system. It is therefore a valuable tool to analyse structures subjected to excitation generated by their own operation. Presently, operational modal analysis procedures are limited to the case when excitation to the system is white stationary noise [22]. The main advantage of OMA is that the measured responses are used for modal identification of structures under real operation without the excitation from a hammer and shaker. Finally, the measured response is representative of the real operating conditions of the structure EMA during operation If EMA is carried out while the machine is in operation, the response of the system measured, X(x), will be the linear superimposition of all the forces induced as shown by Eq. (1). This includes the artificial excitation from the measured impact force input, F 1, plus all other unaccounted operating forces and ambient forces, F 2, F 3, and so forth. In other words, the transfer function registered on the analyser is the total response of the system cross spectrum only by the one force induced by the impact hammer. Hence, there will be errors in the transfer function, H(x), and coherence will be low. XðxÞ ¼H 1 ðxþf 1 þ H 2 ðxþf 2 þ H 3 ðxþf 3 þ... ð1þ Transfer function H 1 (x) is from the measured force input and transfer function H 2 (x), H 3 (x), etc. are due to other unaccounted forces Impact Synchronous Modal Analysis (ISMA) 2.1. Experimental Modal Analysis (EMA) EMA requires the system to be in a complete shutdown situation. In other words, there should be no unaccounted excitation force induced into the system. Measurable impact or random forces are used to excite the system. The responses of the system are auto-correlated and cross-correlated with the measured inputs. The correlation functions are transformed to frequency domain to obtain the transfer functions. This procedure is repeated at a discrete set of geometrical positions sufficient enough to describe the structure. Various curve-fitting algorithms are then used to extract the three parameters namely natural frequencies, mode shapes and modal damping. In ISMA, performed while machine is in running condition, all the responses contributed by the unaccounted forces in Eq. (1) are filtered out in the time domain, leaving only the response due to the impact hammer. This is achieved by utilising the Impact-Synchronous Time Averaging [14] prior to performing the Fast Fourier Transformation (FFT) operation. Apart from this, the process of modal parameters extraction follows the EMA procedures. The limitation of ISMA is perceived to be in it is application on large structures where the impact response may be too small as compared to the operating cyclic loads. Excessive impacts may result in non-linearity. It is worthwhile to note that responses from unaccounted forces that contain even the same frequency as that

3 2114 A.G.A. Rahman et al. / Measurement 44 (2011) contained in the impulse response is diminished if the phase is not consistent with respect to the impact signature. 3. Signal processing and averaging techniques Digital signal processing (DSP) is concerned with the representation of signals by a sequence of numbers or symbols and subsequent processing of these signals. It converts the signal from an analogue to a digital form by sampling it using an analogue-to-digital converter, turning the analogue signal into a stream of numbers. The digital signals are studied in time, spatial, frequency and autocorrelation domain. The domain chosen depends on its ability to represent the essential characteristics of the signal. A sequence of samples from a measuring device produces a time or spatial domain representation, whereas a discrete Fourier transform produces the frequency domain information, which is the frequency spectrum. Autocorrelation, on the other hand, is defined as the cross-correlation of the signal with itself over varying intervals of time or space [23]. Time domain is a term used to describe the analysis of mathematical functions, or physical signals, with respect to time. In the time domain, the signal or function s value is known for all real numbers, for the case of continuous time, or at various separate instants in the case of discrete time. Frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time. A given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called a transform. An example is the Fourier transform, which decomposes a function into the sum of potentially infinite number of sine wave frequency components. Spectrum of frequency components is the frequency domain representation of the signal. The inverse Fourier transform converts the frequency domain function back to a time function [24] Averaging techniques Averaging can be described as a data reduction process, reducing undesired quantities such as noise and randomness from the raw data. The amount of raw data collected is reduced to an optimum and useful quantity. However, such process does diminish or eliminate other irrecoverable information contained therein. Different kinds of averaging generate different averaged outputs. The choice of averaging techniques depends on what is to be extracted or removed Spectral Averaging In industrial application of EMA, Spectral Averaging is normally used. Block averaging is performed in the frequency domain. In this case, the real and imaginary components of the transfer function are averaged separately as shown by the following equation: YðwÞ ¼ 1 N X N r¼1 XðwÞ ð2þ where Y(w) is averaged vibration signal in frequency domain, N is number of running averages, X(w) is vibration signal in frequency domain (real and imaginary components) and r is number of running average Impact-Synchronous Time Averaging In Impact-Synchronous Time Averaging, block averaging is performed on the time block of both the force and response. Each time block is initiated by the impulse generated from the force trace of the impact hammer. Assuming sufficient number of averages is taken, this has the effect of filtering out most of the signatures non-synchronous in frequency and phase to the impact time block. The periodic responses of cyclic loads and ambient excitations are no more in the same phase position for every time block acquired. Eq. (3) shows that averaging process of repetitive impact will slowly diminish these components hence leaving behind only the structure impulse responses which are synchronous to the repetitive impact force. yðtþ ¼ 1 N X N 1 r¼0 xðt þ rt o Þ ð3þ where y(t) is the averaged vibration signal in time domain, N is the number of rotations, x(t) is the vibration signal in time domain, and T o is the time interval. The averaged time block of impulse responses is auto and cross spectrum with the averaged time block of impact signatures to generate the transfer function [14]. 4. Mathematical background In EMA, FFT is performed on the input and output signals and averaging is performed on the power spectrums. On the other hand, for ISMA, averaging is carried out on the time signals for both input and output before FFT is applied to them. After this averaging is completed, transfer and coherence functions are calculated. In addition, Modal Assurance Criterior or MAC is utilised to identify the similarity between two modal vectors Correlation function and power spectrum The correlation function examines whether there is any correlation between signals at two points in time. If a signal is a sine wave with period s, then there is an excellent correlation, since the response at these two times will be identical. A purely random signal should be uncorrelated for any time s besides zero since the signal changes in a completely unpredictable way. It has been realised that identification accuracy can be improved by applying correlation functions instead of noisy time response data [25]. The auto-correlation of discrete input signal is given by Eq. (4) R qq ðsþ ¼E½qðtÞqðt þ sþš ð4þ by definition is the mean of the product of q(t) and q(t + s), and, where q(t + s) is the value of input function q measured at s s after t. Similarly, the autocorrelation of the output is defined as given by the following equation: R xx ðsþ ¼E½xðtÞxðt þ sþš ð5þ

4 A.G.A. Rahman et al. / Measurement 44 (2011) Seeking correlation between two different signals, the cross-correlation of input x(t) and output y(t) can be stated as given by the following equation: R xq ðsþ ¼E½xðtÞqðt þ sþš Cross-correlation would be useful in trying to determine whether a vibration at one point of a structure is being influenced by vibrations at some other point. If the cross correlation is high for some value of s, then it could be deduced that there is a relationship between the two vibration and that the transit time for the motions to be transmitted from one point to the other is equal to fraction or multiples of s. The Fourier Transform of the auto-correlation function is called Auto Power Spectrum S qq (x), and for the input is given by the following equation: S qq ðxþ ¼ Z 1 1 R qq ðsþe ixs ds and for the output is given by the following equation: S xx ðxþ ¼ Z 1 1 R xx ðsþe ixs ds The Fourier Transform of the cross-correlation function is called Cross Power Spectrum S xq (x), given by the following equation: S xq ðxþ ¼ Z 1 1 R xq ðsþe ixs ds ð6þ ð7þ ð8þ ð9þ In time domain, the link between the Auto and Cross Power Spectrum with the Frequency Response Function H(x) is given by the following equation: XðtÞ ¼HðxÞqðtÞ ð10þ Multiply both sides of Eq. (10) by x(t + s) and finding the expected values, will produce Eq. (12). E½xðtÞxðt þ sþš ¼ HðxÞE½qðtÞxðt þ sþš R xx ðsþ ¼HðxÞR xq ðsþ ð11þ ð12þ The transfer function can be derived by dividing the Auto Power Spectrum of output by the Cross Power Spectrum of the input and output. Performing Fourier Transform on both sides of Eq. (12) results in Eq. (13), which after rearrangement gives Eq. (14). S xx ðxþ ¼HðxÞS xq ðxþ ð13þ Similarly the transfer function in the form of Eq. (15) can be derived by dividing the Cross Power Spectrum of the input and output by the Auto Power Spectrum of the input. H 2 ðxþ ¼ S xqðxþ S qq ðxþ ð15þ Both Eqs. (14) and (15) can be used in a multi-channel FFT analyser to determine the frequency response function for FRF modal analysis Transfer functions As schematically represented in Fig. 1, the input and output is related to the transfer function, H(x), of the system in frequency domain by HðxÞ ¼ F xðxþ F q ðxþ Given F x ðxþ F q ðxþ ¼ F x ðxþ F q ðxþ Comparing Eqs. (16) and (17), HðxÞ ¼ F x ðxþ F q ðxþ ð16þ ð17þ ð18þ In digital signal analysis, the auto and cross correlation are normally performed in the frequency domain in terms of auto and cross spectrum. The Fourier transforms of input, output and their conjugates ( ) are multiplied to estimate the Transfer Functions in Eqs. (19) and (20). Multiplying numerator and denominator on the right hand side of Eq. (18) by F x (x) gives H 1 ðxþ ¼ F xðxþ F x ðxþ F x ðxþ F q ðxþ ¼ S xxðxþ S xq ðxþ ð19þ and multiplying numerator and denominator on the right hand side of Eq. (16) by F qðxþ gives H 2 ðxþ ¼ F xðxþ F q ðxþ F q ðxþ F q ðxþ ¼ S xqðxþ S qq ðxþ ð20þ H 1 ðxþ ¼ S xxðxþ S xq ðxþ ð14þ Fig. 1. Input output relationship. Fig. 2. Noise in input output relationship. Fig. 3. Fault simulation rig.

5 2116 A.G.A. Rahman et al. / Measurement 44 (2011) Fig. 4. Structural model of the fault simulation rig Coherence function Noise in the measurements enters into the system in different ways, and they can be used together in order to find the best overall transfer function estimate. Normally, due to experimental problems, the estimates are not exactly the same. The coherence function, c 2, is the ratio of the two estimates shown in Eqs. (19) and (20) as given in Eq. (21). c 2 ¼ S xxðxþ S nn ðxþ S xx ðxþ Note that S xx ðxþ ¼F x ðxþf x ðxþ ð21þ ð22þ and S qq ðxþ ¼F q ðxþf q ðxþ ð23þ Thus, from Eqs. (16) and (17) the transfer function can also be estimated as jhðxþj 2 ¼ S xxðxþ S qq ðxþ ð24þ From Fig. 2 and Eq. (21), noise in the measurement is defined as S nn ðxþ ¼S xx ðxþ S qq ðxþjhðxþj 2 and can be rewritten as ð25þ Fig. 5. Frequency response function using Spectral Averaging.

6 A.G.A. Rahman et al. / Measurement 44 (2011) Fig. 6. Coherence function using Spectral Averaging. S xq ðxþ S nn ðxþ ¼S xx ðxþ S qq ðxþ S qq ðxþ and reduced to the following form S nn ðxþ ¼S xx ðxþ js xqðxþj 2 S qq ðxþ Thus, substituting Eq. (27) into Eq. (21), c 2 is defined as c 2 js xq ðxþj 2 ¼ S qq ðxþs xx ðxþ 2 ð26þ ð27þ ð28þ Rearranging Eq. (28) from Eqs. (19) and (20) will produce the coherence function, Eq. (29). c 2 ¼ H 2ðxÞ H 1 ðxþ ð29þ This frequency-dependent function is widely used to measure believability of measurements. It can be shown that c 2 lies between 0 and 1. Coherence near to 1 indicates that both H 1 (x) and H 2 (x) are in very close agreement, which implies that the measurements are believable and that noise and non-linear effects are not significant. If the coherence is near zero, it means that the measurements should be viewed with great suspicion. It is therefore advisable to display the coherence function while making measurements of transfer function [25]. Fig. 7. Frequency response function using Impact-Synchronous Time Averaging.

7 2118 A.G.A. Rahman et al. / Measurement 44 (2011) Fig. 8. Coherence function using Impact-Synchronous Time Averaging. Running Speed: 30 Hz Spectral Averaging Overlaid Frequency Response Function Spectral Impact- Synchronous Time Averaging Fig. 9. Comparison of overlaid frequency response functions between Spectral Averaging and Impact-Synchronous Time Averaging.

8 A.G.A. Rahman et al. / Measurement 44 (2011) Running Speed: 30 Hz Spectral Averaging 1 st Mode Natural Frequency (Hz) 19.3 Damping ratio (%) 4.16 Impact-Synchronous Time Averaging Natural Frequency (Hz) 19.3 Damping ratio (%) 4.16 Fig. 10. Comparison of dynamic characteristics of 1st mode between Spectral Averaging and Impact-Synchronous Time Averaging Modal Assurance Criterion (MAC) The original development of the Modal Assurance Criterion (MAC) was modelled after the development of the ordinary coherence calculation associated with computation of the frequency response function. It is important to recognise that this least squares based form of linear regression analysis yields an indicator that is most sensitive to the largest difference between comparative values. Meanwhile, it results in a modal assurance criterion which is insensitive to small changes or small magnitudes. This was considered an advantage since small modal coefficient values are often seriously biased by frequency response function (FRF) measurements or modal parameter estimation errors. The function of the MAC is to provide a measure of consistency (degree of linearity) between estimates of a modal vector. This provides an additional confidence factor in the evaluation of a modal vector from different excitation (reference) locations or different modal parameter estimation algorithms [26]. Modal Assurance Criterion is a method for quantitatively comparing a pair of shapes, and it is computed between two complex shape vectors {X}, {Y} using the following formula,

9 2120 A.G.A. Rahman et al. / Measurement 44 (2011) fxg T fyg 2 MACðfXg; fygþ ¼ ð30þ fxg T fxg fyg T fyg MAC values range between 0 and 1, and should be interpreted as follows MAC = 1.0 means the two mode shapes are identical. MAC > 0.9 means the two mode shapes are similar. MAC < 0.9 means the two mode shapes are different. 5. Measurement procedures and instrumentation A test rig consisted of a motor coupled to rotor shaft system as displayed in Fig. 3. It was used in the laboratory to compare both the frequency response functions and coherence functions generated from EMA and ISMA in operating condition. The instrumentation and procedures used in ISMA was the same as in EMA. The only difference was that the averaging techniques allowed the procedure to be carried while the machine was in operation. Rahman et al. [14] explained the complete experimental procedures of ISMA. Data were Running Speed: 30 Hz Spectral Averaging 2 nd Mode Natural Frequency (Hz) 30.4 Damping ratio (%) 1.68 Impact-Synchronous Time Averaging Natural Frequency (Hz) 30.4 Damping ratio (%) 1.7 Fig. 11. Comparison of dynamic characteristics of 2nd mode between Spectral Averaging and Impact-Synchronous Time Averaging.

10 A.G.A. Rahman et al. / Measurement 44 (2011) obtained by using a data acquisition system together with an impact hammer and tri-axial accelerometer. One hundred averages or impacts were made during rotating condition for both EMA and ISMA. The signals were processed by the virtual instruments using two different averaging techniques to generate the frequency response functions and coherence functions. Spectral Averaging was applied in EMA while Impact-Synchronous Time Averaging was used in ISMA. The modal extraction techniques applied to EMA could also be applied in ISMA. Fig. 4 shows a threedimensional structural model of the test rig which was drawn in coordinate points and connected by straight lines using a modal analysis software called ME scope. This model was used to display the mode shapes of the rig from the acquired data. In addition, the software performed post-processing of the acquired data for modal parameters extractions. 6. Results and discussions Transfer functions and coherence functions obtained from ISMA and EMA are now compared. Running Speed: 30 Hz Spectral Averaging 3 rd Mode Natural Frequency (Hz) 34.7 Damping ratio (%) 5.12 Impact-Synchronous Time Averaging Natural Frequency (Hz) 34.6 Damping ratio (%) 5.3 Fig. 12. Comparison of dynamic characteristics of 3rd mode between Spectral Averaging and Impact-Synchronous Time Averaging.

11 2122 A.G.A. Rahman et al. / Measurement 44 (2011) Frequency response function and coherence function determination using Spectral Averaging in operating condition Fig. 5 shows that Spectral Averaging has the same effect of filtering out the frequency components that were nonsynchronous to the impact. These non-synchronous frequencies seemed to diminish with the increase in the number of averages. However, Fig. 6 shows that the coherence function obtained using this averaging technique gave poor coherence at these frequencies. In regions where the modal frequencies were close to the unaccounted excitation frequencies, the coherence results deteriorated. This poor coherence indicated low acceptability of the estimated transfer function Frequency response function and coherence function determination using impact-synchronous time averaging in operating condition Impact-Synchronous Time Averaging incorporates the time synchronous averaging technique to rapidly improve the signal to noise ratio of the response obtained while the machine is in an operating condition. The application of Impact-Synchronous Time Averaging is effective in filtering out the non-synchronous running speed frequency components, its harmonics and noise as displayed in Fig. 7. This averaging technique also gives a good coherence function (Fig. 8) which indicates that the estimated transfer function is acceptable Comparison between using Spectral Averaging and Impact-Synchronous Time Averaging in Modal Testing during operating condition Fig. 9 shows that the overlaid Frequency Response Function spectral obtained from modal analysis using Spectral Averaging and Impact-Synchronous Time Averaging are very similar. The modal parameters obtained for the first three modes show that both averaging techniques give quite identical results in natural frequencies, dampings and mode shapes. The results are as displayed in Figs Modal Assurance Criterion (MAC) values between these two averaging techniques are compared as shown in Figs. 13 and 14. The MAC values of almost 1 indicate good correlations of dynamic characteristics for the first three modes obtained between using Spectral Averaging and Impact-Synchronous Time Averaging. Although poor coherence shows low reliability of transfer function obtained using Spectral Averaging, the dynamic characteristics determined by these two averaging techniques show good correlation in terms of natural frequency, damping ratio, mode shape and MAC values of the first three modes. In summary, Impact-Synchronous Time Averaging produced better coherence than Spectral Averaging in spite of both generating almost identical transfer functions. Eqs. (15) and (28) were scrutinised to explain this observation. Spectral Averaging performs averages of real and imaginary components separately after the Fourier transformation. The amplitudes and phases are then obtained from the averaged real and imaginary components. As phases are involved, averaging of the cross spectrum S xq has the same effect as averaging the data synchronised to the impact input, that is filtering out the responses that are excited by all unaccounted forces. S qq is obtained from the impact hammer only and hence do not contain other unac- Fig. 13. Modal Assurance Criterion (MAC) between using Spectral Averaging and Impact-Synchronous Time Averaging in Determination of Modal Parameters (3D view). Fig. 14. Modal Assurance Criterion (MAC) between using Spectral Averaging and Impact-Synchronous Time Averaging in Determination of Modal Parameters (Top view).

12 A.G.A. Rahman et al. / Measurement 44 (2011) counted forces. Therefore, H 2 gives a transfer function as though the unaccounted forces do not exist. Thus Spectral Averaging of transfer function is equivalent to the Impact- Synchronous Time Averaging as shown in Figs. 5 and 7. In the coherence function c 2, S xx exists in the denominator. Auto power spectrum S xx are all real and contained no phase and averaging it would continually take into account the responses generated from the impact hammer as well as from all the unaccounted forces. Meanwhile S xq as mentioned above, filters out the responses that are excited by all unaccounted force as number of averages increase, thus, reducing the numerator of c 2, subsequently reducing the coherence at the frequency positions of these unaccounted forces. Evidently, this could have caused the poor coherence as shown in Fig. 6. On the other hand, Impact-Synchronous Time Averaging filters out most of the responses that are non-synchronous to the impact prior to Fourier transformation. Both S xx and S xq in Impact-Synchronous Time Averaging always contain the same amount of unaccounted forces and are simultaneously and proportionately reduced with the number of averages. Hence, the coherence c 2 will not result in much variation with respect to the number of averages used. A better coherence even at the frequency positions of the unaccounted forces could be obtained as shown in Fig Conclusions Transfer functions or so-called Frequency Response Functions are estimated quantities. A coherence function is commonly used to measure the acceptability of estimations. In this paper, Impact-Synchronous Time Averaging was compared against Spectral Averaging when performing EMA during machine operating condition. Results showed that Impact-Synchronous Time Averaging produced the same quality of transfer function as compared to Spectral Averaging. However, a better coherence function was established using Impact-Synchronous Time Averaging. The degeneration of coherence functions by Spectral Averaging is due to the auto spectrum of the output, S xx being averaged in frequency domain. Consequently, EMA using Impact-Synchronous Time Averaging is named Impact-Synchronous Modal Analysis or ISMA. Acknowledgement The authors wish to acknowledge the financial support and advice given by Fundamental Research Grant Scheme (FP A) and Advanced Shock and Vibration Research (ASVR) Group of University of Malaya, and other project collaborators. References [1] W.D. Pilkey, R. Cohen (Ed.), System Identification of Vibrating Structures, ASME. [2] N.M.M. Maia, J.M.M. Silva, Theoretical and Experimental Modal Analysis, Research Studies Press, Taunton, Somerset, UK, [3] D.L. Brown, R.J. Allemang, R. Zimmerman, M. 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