Proceedings of the IMAC-XXVIII February 1 4, 2010, Jacksonville, Florida USA 2010 Society for Experimental Mechanics Inc. Modal Parameter Estimation Using Acoustic Modal Analysis W. Elwali, H. Satakopan, and V. Shauche R. Allemang, and A. Phillips University of Cincinnati, Department of Mechanical Engineering, Cincinnati, Ohio, 45221, USA ABSTRACT Acoustic modal analysis (AMA) is of interest in cases where accelerometer measurements are limited by mounting techniques and where the mass of sensors affects the system dynamics. Major problems in performing AMA are time delay adjustment and the inability of obtaining true driving point measurements. For an impact test, the former problem causes difficulties because each measured acoustic frequency response function (AFRF) will have its own time delay as a function of the position of the reference microphone with respect to the structure. Thus, obtaining consistent modal parameters conventional multi-input, multi-output (MIMO) modal parameter estimation methods utilizing several microphones (MIMO AFRFs) becomes rather difficult. The latter problem complicates the computation of modal scaling which is frequently required in model validation. As an example, both microphone measurements and accelerometer measurements are utilized in an impact test for a heavy ringdisc structure. The results from each method are compared to study the effectiveness of estimating modal parameters from AFRFs compared to conventional FRFs. While some conventional modal parameter estimation tools such as the consistency diagram and the complex mode indicator function (CMIF) look slightly different, the frequencies, damping and mode shapes estimated using AFRFs are consistent with those of standard modal analysis. 1. INTRODUCTION Acoustic modal analysis is an interesting technique since it does not affect the system parameters such as mass, stiffness and damping. It is a useful technique especially when mounting accelerometers is difficult or in cases where accelerometers affect system parameters. Acoustic modal analysis is based on the assumption that emitted sound pressure levels vary linearly with vibration amplitudes at a certain frequency. Therefore for a linear structure, at a particular frequency, sound pressure level is directly proportional to the modal coefficient of the input point [1]. The proportionality is a function of the modal vibration pattern that is excited at each frequency in the structure and the coupled acoustic pattern that emanates from the structure. Based upon the impact point and the modal pattern, the input location may not result in a mode being excited (node at the input). Based upon the impact point and location, the coupled acoustic pattern may have a null at one of the reference microphones (node at the output). Finally, based upon the impact point and the modal pattern, if the modal pattern does not involve sufficient surface area to provide adequate vibro-acoustic coupling, the modal pattern cannot be observed (node at all outputs). There are mainly two problems with using acoustic modal analysis. The first one is the limitation of microphone to be placed in the near field since the distance affects the signal to noise ratio [1]. The other problem is the time delay problem. It is the time required for the sound wave to travel from impact point to microphone.
System dynamics Vibrations Fluid structure interaction Impact hammer Sound pressure Accelerometer Microphone Accelerometer measurement Microphone measurement Figure 1. Accelerometer and microphone measurements flow charts In this paper, a circular disc is experimentally studied in order to find its modal parameters such as natural frequencies, mode shapes and modal damping. This research study was initiated by a need to identify vibration characteristic of a hypoid gear so the disc is just a simpler test structure that can be more accurately and easily modeled. The main objective of this work is to estimate modal parameters using two techniques: microphone measurements and conventional accelerometer measurements. Microphone measurements are to be examined in terms of their ability to estimate modal parameters compared with respect to conventional measurement. Moreover, the difficulties and limitations accompanied with acoustic modal analysis including the time delay problem are also discussed. 2. FINITE ELEMENT CHECK Before conducting the experiment, natural frequencies were found using the finite element method. The purpose of using the finite element method (FEM) is to check whether the in-plane (radial) modes are in the frequency range of interest (0-9000Hz). FEM is also used to detect the presence of repeated roots and thus decide on the minimum number of references to be used. It is necessary to check the presence of in-plane modes in the frequency range of interest, since exciting both sets of modes, in-plane modes and transverse modes, requires more FRF measurements, even though the inplane modes may not be well identified using acoustic modal analysis. Using the finite element method, it was found that a significant number of modes lie in the range 0-9000Hz with all being out-of-plane (transverse) modes. In-plane modes are above 9000Hz and will not be accounted for. 3. EXPERIMENTAL PROCEDURE The experiment was performed by exciting the circular disc using an impact hammer at 17 points perpendicular to the plane of the disc. For accelerometer measurements, three uni-axial accelerometers were mounted on the disc. In this case frequency response functions [m/s 2 /N] are measured and then processed to estimate modal parameters. In the case of acoustic modal analysis, three microphones are mounted in the nearfield of the disc perpendicular to the plane of the disc. Acoustic frequency response functions [Pa/N] are then measured. These acoustic frequency response functions can be used to estimate modal parameters under the assumption that sound pressure level varies linearly with vibration amplitudes.
3.1. Experimental Setup There are a total of six fixed outputs (three accelerometers and three microphones). There are 17 input points on the disc. Impact hammer (Model E086C40) with metal tip is used for excitation since frequency range of interest is around 9 KHz. Since the response locations are fixed, the impact hammer is moved to various points for excitation. This is called a roving hammer method or a multiple reference impact test (MRIT). Uni-axial accelerometers (Model 352A56) are used to measure response (3 mounted in transverse direction). Three microphones (Model 130A10) are used for sound pressure measurements. The experimental setup is shown in Figure 2. Figure 2. The experimental setup In setting the digital signal processing (DSP) parameters in the MRIT VXI test setup, the span frequency is set at 12.58 KHz and the number of spectral lines is set to 1600 lines for better frequency resolution. Force-Exponential windows are used for this test, since an impact type of excitation is used, and the pre-trigger value is set to 10%. All three accelerometers are calibrated using a hand held calibrator. The calibration is done at 159 Hz with 1g RMS acceleration level. The nominal sensitivity values (taken from the PCB website) are used for the microphones and load cell. The basic assumptions of experimental modal analysis like linearity and reciprocity are checked. 3.2. FRF Measurement Auto-ranging of the sensors is done for every impact point to prevent overloading of sensors. Five spectral averages are done. Force-exponential windows are applied to the data. The accelerometers are mounted using adhesive mounting, as it is effective up to 9000 Hz. 3.3. Correction for Time Delay in Microphone Signals Since the microphones are kept at a distance of around 5.5 inch from the test component, there is some time delay in the microphone signals. This time delay leads to the phase wrap in the AFRF over the entire frequency range. The phase plot of AFRF with time delay is shown in Figure 3(a). By impacting at the center of the component, the time delay values were obtained for each microphone. The microphone data in each channel were corrected based on the time delay. The phase wrap of AFRF reduced after the time delay correction. The phase plot of AFRF after time delay correction is shown in Figure 3(b).
(a) (b) Figure 3. Phase plots of AFRF. (a) With time delay. (b) With time delay correction Once these values were fixed based on the impact at the center point, the same values were used when impacting at different points. Consistent poles were not obtained with this approach as shown in Figure 4. Figure 4. Consistency diagram for acoustic modal analysis with constant time delay values This led to the speculation that time delay at different microphones varies significantly with impact point. So the time delays for the three microphones were calculated for each of the 17 impact points. These delay values were used for correcting each microphone channel while impacting at different points and thus consistent poles were obtained.
If time delay is not corrected and the same time is used as that for the accelerometers, data will be lost out of the end of the data block contributing to more noise and poor coherence (not all data is observed).[2] The first correction mentioned in the above explanation reduces the coherence problem but is not sufficient to get optimum data. The second method in the above explanation is needed to be certain the best quality data is available, which is required by the multiple reference parameter estimation algorithms. 3.4. Creation of Wireframe Model and Trace Lines Using the Wireframe Editor option in X-Modal II software (developed by SDRL, University of Cincinnati), the wireframe model of the circular disc is created by entering the co-ordinates of all the measurement points and the creating trace lines connecting measurement points. This wireframe model is saved in universal file format (.ufb). These.ufb files are loaded into X-Modal II Data manager for parameter estimation. 4. PARAMETER ESTIMATION PROCEDURE There are several modal parameter estimation methods available to extract modal behavior of the system from the measured frequency response functions. These methods are implemented to estimate modal parameters from the measured data and classify the modes as consistent, non-realistic, etc. 4.1. Mode Indicator Function Mode indication functions (MIF) are normally real-valued, frequency domain functions that exhibit local minima or maxima at the natural frequencies of the modes [3]. Figures 5 and 6 show mode indicator functions from which the number of modes, including close or repeated roots, can be obtained. The number of peaks in the Complex Mode Indicator Function (CMIF) plot indicates the number of modes in the disc in the specified frequency range. Multiple peaks at the same frequency indicate repeated roots or close modes. Figure 5. CMIF plot using accelerometer measurement From Figure 5 showing the CMIF based on accelerometer measurements, a total number of 13 peaks can be found in the 0 to 9000 Hz frequency range and thus number of modes is 13. The CMIF plot based on the microphone measurements (without correction) shows the presence of additional peaks (apparent repeated roots) which confuses the analysis of the data. The CMIF plot for microphone measurement is shown in Figure 6.
Figure 6. CMIF plot for microphone measurement 4.2. MDOF Methods Time domain methods are usually limited to low damping case and may not perform accurately in the moderate or heavily damped cases. Knowing that the circular disc is a lightly damped structure, the Polyreference Time Domain (PTD) method, which is a higher order, matrix coefficient polynomial method, is used in the following analysis. 5. RESULTS AND DISCUSSION Modal order of 20 is used to get the 13 modes from the accelerometer measurements. The consistency diagram represents consistent poles and mode shapes; the poles with Mean Phase Correlation (MPC) values greater than 0.95 are selected. Figure 7 shows the consistency diagram for accelerometer measurements.
Figure 7. Consistency diagram for accelerometer measurements Modal order of 30 is used to get 13 modes from the microphone measurements. The consistency diagram is shown in Figure 8. Figure 8. Consistency diagram for microphone measurements
Due to the geometrical symmetry of the plate, it is expected to observe close or repeated roots. However, the presence of repeated roots does not mean that they must be due to symmetry. Two roots are considered to be repeated if they are very close in frequency and in damping. The comparison of frequencies and damping values obtained using both the methods are given in Table 1. Mode Table 1. Comparison of standard and acoustic modal analysis result Accelerometer measurements Frequency (Hz) Damping % Microphone Measurements Frequency (Hz) Damping % % Difference in frequency % Difference in damping 1 1288.701 0.502 1289.294 0.437 0.046 12.948 2 1298.466 0.505 1298.981 0.482 0.040 4.554 3 2103.88 0.38 2104.209 0.394 0.016 3.684 4 2987.763 0.591 2988.226 0.396 0.015 32.995 5 2991.433 0.283 2990.331 0.326 0.037 15.194 6 4637.023 0.205 4637.171 0.209 0.003 1.951 7 4660.377 0.204 4643.121 0.223 0.370 9.314 8 5163.566 0.173 5164.086 0.164 0.010 5.202 9 5184.923 0.157 5170.13 0.166 0.285 5.732 10 7745.799 0.37 7738.605 0.448 0.093 21.081 11 7780.38 0.191 7769.714 0.255 0.137 33.508 12 7796.185 0.145 7788.736 0.159 0.096 9.655 13 8380.989 0.154 8372.652 0.171 0.099 11.039 As seen from Table 1, the percentage difference in frequency values between the standard modal analysis and acoustic modal analysis is negligible. But there is considerable difference in some damping values, probably due to the vibro-acoustic nature of the test method involving the coupling of the vibration to the acoustic field and the characteristics of the acoustic pattern that develops. The close roots taken from standard modal analysis results are shown in Table 2. Table 2. Close roots from standard modal analysis Mode Frequency (Hz) Damping (%) 1, 2 1288.701, 1298.466 0.502, 0.505 4, 5 2987.763, 2991.433 0.591, 0.283 6, 7 4637.023, 4660.377 0.205, 0.204 8, 9 5163.566, 5184.923 0.173, 0.157 11,12 7780.380, 7796.185 0.191, 0.145 The modal vectors derived from the residues are generated using the poles selected from the consistency diagram. For the accelerometer measurement, the FRF correlation coefficient (comparing the FRF measurement and the synthesis of the FRF measurement based upon the model) is close to unity for all cases checked. In order to measure the linear dependence of mode shapes, the Modal Assurance Criterion (MAC) is used. MAC can take values between zero and one. If the modal assurance criterion has a value near zero, this is an indication that the modes are linearly independent. If the modal assurance criterion has a value near unity, this is an indication that the modes are linearly dependent. Figures 9 and 10 show MAC plots of standard modal analysis and acoustic modal analysis respectively.
Figure 9. MAC plot for Standard modal analysis Figure 10. MAC plot for Acoustic modal analysis The MAC plot of standard modal analysis (Figure 9) shows that most of the modes are linearly independent. Some small amount of linear dependence is observed in certain modes which is not unexpected. Note that there is good agreement (linear dependence) for the modal vectors and their complex conjugate pairs (lower diagonal of red squares). The MAC plot of acoustic modal analysis (Figure 10) shows that modal vectors associated with positive frequencies are again linearly independent of one another.
However, note that there is not good agreement for the modal vectors and their complex conjugate pairs (lower diagonal) as in the previous case. The modal vectors for the conjugate frequencies appear to be more linearly dependent. This may be because the acoustic frequency response function is a ratio of pressure to force where the modal vectors for the conjugate frequencies may not be necessarily the same. 5.1. Mode Shapes Table 3. Mode shapes description Mode number Description 1, 2 Bending mode (Saddle mode) 3 Bending mode (Umbrella mode) 4, 5 Bending mode 6, 7 Bending mode (Inner ring see-saw) 8, 9 Bending mode 10, 11 Bending mode (Inner saddle) 12 Bending mode 13 Bending mode (Reverse umbrella mode) The mode shapes obtained using microphone and accelerometer measurement are similar as shown in Figure 11. Accelerometer Microphone 1288.701 Hz 0.5025 % zeta 1289.2945 Hz 0.4369% zeta 8380.989 Hz 0.154 % zeta 8372.652 Hz 0.171 % zeta Figure 11. Comparison of mode shapes from standard and acoustic modal analysis 6. DISCUSSION OF ERRORS In conducting experiment, errors are usually generated by the following: Leakage error: The leakage error will exist anytime the digitized time domain data does not match the requirements of the FFT (periodic in observation time T or totally observed transient in observation time T). This was minimized using exponential windowing. The FRF referred to as the driving point FRF is approximate, because it was not possible to hit at the exact same location where the accelerometers are mounted. In taking averages, some errors may occur due to not hitting the same point and in the same direction, average to average. Force with same magnitude not applied during every measurement which should not be a problem if the system is linear.
The time delay correction factor for microphone measurements is measured to within the nearest time difference based on the measured pressure and impact at a particular point. The measured pressure signal is the superposition of sound radiated from various points and each point can have its own time delay. The time delay used for the correction is the minimum time delay. The PTD method is a time domain method that applies an inverse FFT to get the impulse response, which may have a small frequency domain truncation error. The selected poles from the consistency diagram may not be at the centroid of the pole cluster leading to variation in damping estimation. 7. SUMMARY AND CONCLUSIONS Accelerometer measurements and microphone measurements were conducted on a circular disc and modal analysis results were compared in terms of modal parameters. For accelerometer measurements, the setup consists of three accelerometers (outputs) and 17 impact points (inputs). At each point, impact excitation is done in the transverse direction. Time delay correction is done for each microphone channel during each measurement. In order to minimize leakage and noise errors, force-exponential windowing was applied to the measured data. For microphone measurements, it is critical to account for the time delay associated with the reference microphone s position relative to the part being tested. The PTD method was used to estimate modal parameters for both measurement techniques, from the measured FRFs and AFRFs independently. It was found that both techniques are able to find 13 modes in the frequency range 0 to 9000 Hz. In predicting natural frequencies, the acoustic modal analysis method is able to perform accurate estimation with an error of less than 0.096% relative to those found from accelerometer measurement. However, microphone and accelerometer measurements do not estimate the same value of damping with an average relative difference of 12.83% found. Some linear dependency was observed between modes 5 and 9 from the MAC plot of accelerometer measurements. More measurement points are required for better observability of these two modes. From the MAC plot of microphone measurements, it was observed that modal vectors associated with conjugate poles do not show a reasonable dependency. Modal scaling is not obtained from acoustic modal analysis since true driving point FRF measurements are not available. But, modal scaling can be obtained if a vibro-acoustic structure model with coupling is taken into consideration. It is probably easier to simply conduct a standard modal analysis test to get an estimate of the modal scaling even though the modal frequencies will be incorrectly identified. The two test cases can then be combined to give the complete modal model. REFERENCES [1] Allemang, R.J., Shapton, W.R., Using modal techniques to guide acoustic signature analysis, SAE, 780106, 1979. [2] Halvorsen, W.G., Brown, D.L., Impulse Technique for Structural Frequency Response Testing, Sound and Vibration Magazine,November,1977,pp. 8-21. [3] Allemang, R.J., Phillips, A.W., The Unified Matrix Polynomial Approach to Understanding Modal Parameter Estimation: An Update, ISMA 2004.