Using a De-Convolution Window for Operating Modal Analysis

Transcription

1 Using a De-Convolution Window for Operating Modal Analysis Brian Schwarz Vibrant Technology, Inc. Scotts Valley, CA Mark Richardson Vibrant Technology, Inc. Scotts Valley, CA Abstract Operating Modal Analysis (OMA) has been applied in a nuber of cases recently for finding the experiental odal paraeters fro structures when their excitation forces cannot be easured. Without force easureent, a classical Experiental Modal Analysis (EMA), which relies on the application of odal paraeter estiation, or curve fitting ethods to a set of Frequency Response Function (FRF) easureents, cannot be perfored. In this paper, we show that the application of a de-convolution window to cross power spectru data yields the effective recovery of the FRFs involving each vibration response signal. A set of recovered FRFs can then be curve fit using classical FRF-based curve fitting ethods to identify the experiental odal paraeters of the structure. Use of the de-convolution window is illustrated in an exaple that uses a FRF atrix odel to calculate structural responses to siulate a ultireference OMA. Introduction Experiental Modal Analysis (EMA), underwent a revolutionary change during the early 970 s with the ipleentation of the Fast Fourier Transfor (FFT) in coputer-based FFT analyzers []. Modal paraeter estiation is a key step in FFT-based EMA. This one step, also called curve fitting, has probably received ore attention than any other during the past 30 years. Nuerous ethods have been developed, and the technical literature contains 00 s of papers docuenting any different approaches. Two copletely different approaches to odal testing; one that relies on carefully controlled shaker excitation [3], and the other that strongly suggests that artificial excitation is not required at all [4], have been the topic of uch discussion recently. The first approach is coonly called an EMA, and the second an OMA. Regardless of whether artificial excitation is used or not, both approaches rely heavily on odal paraeter estiation. Both articles [3] & [4] devote the ajority of their discussion to this subject. Modal analysis is used to characterize resonant vibration in achinery and structures. A ode of vibration is defined by three paraeters; odal frequency, odal daping, and a ode shape. Modal paraeter estiation, or curve fitting is the process of deterining these paraeters fro experiental data. Furtherore, it can be shown that a set of odal paraeters copletely characterizes the dynaic properties of a structure. This set of paraeters is called a odal odel.

2 A odal odel will be used later in this paper to perfor a round trip exercise. The odal odel will be used to calculate the forced rando response of a structure, and its acceleration responses will be used to siulate a ulti-reference OMA. With application of the de-convolution window and FRF curve fitting to a set of response only Cross spectru easureents, odal paraeters will be recovered fro the response only data, and then copared to the paraeters of the original odal odel.. Which is Better, EMA or OMA? Although tie, budget, and physical constraints will ost certainly play a part, the odal testing ethod that you choose strongly depends on what you intend to do with the odal data. The two ost coon reasons for perforing a odal test are,. Trouble shooting a resonance related noise or vibration proble. 2. Verifying and updating a coputer generated finite eleent odel. Trouble shooting a noise or vibration proble only requires enough data to characterize the proble so that a solution can be found. Verifying and updating a finite eleent odel usually requires uch ore extensive and accurate odal testing. Finite eleent analysis (FEA) is coonly used today in the engineering developent of ost new achines, structures, and products of all kinds. Once a finite eleent odel is validated, it can be used for siulations, calculating stresses and strains, and for investigating the effects of structural odifications on a structure s acoustic or vibration behavior. Since EMA and FEA both yield a set of odes for a structure, odal paraeters are used for coparing results, and for updating the FEA odel to ore closely atch the experiental results. If no artificial excitation is required and excitation forces don t have to be easured, siply acquiring and processing operating, (response only, or output only) data appears to be an easier way to perfor a odal test. Siply acquiring the vibration response of a achine while it is operating or being excited in situ is easier than artificially exciting it and siultaneously easuring both the excitation forces and responses. However, the assuptions required for OMA are ore restrictive then when the excitation forces are easured. Therefore, controlling and easuring the excitation forces is the preferred way to do odal testing when possible. Nevertheless, when the excitation forces cannot be easured, then properly post-processed and curve fitting a set of response only easureents can still provide accurate odal paraeter estiates..2 What Is Operating Data? Operating data is what the nae iplies. It is data that is acquired while a achine or structure is undergoing vibratory otion during its operation or use. For odal paraeter estiation, the definition can be extended further. Operating Data is any vibration data that is acquired without siultaneously acquiring the excitation forces.

3 .3 Shape Data Whenever the vibration responses at two or ore points and directions (degrees-of-freedo or DOFs) on the surface of a structure are easured, a vibration shape is defined. That is, a shape defines the agnitude and phase of the otion of one DOF relative to another DOF. An Operating Deflection Shape (ODS) is the agnitudes and phases of two or ore DOFs of operating data acquired fro a achine or structure. Therefore, an ODS defines the relative otion between two or ore DOFs on a structure. An ODS can be defined for a specific frequency or for a oent in tie [2]..4 Broad Band Excitation All excitation forces can be classified as either narrow band like a single frequency sine wave, or broad band, or a cobination of both. The ost coon broad band signals are rando, swept sine or chirp, and transient or ipulsive. Variations of these signals include burst rando, burst chirp, and rando transient [5]. A sine wave is classified as narrow band because its spectru is very narrow, containing essentially a single non-zero frequency. All broad band signals have a non-zero frequency spectru over a broad range of frequencies..5 Curve Fitting Methods The two ost popular approaches to curve fitting either curve fit a paraetric odel of the FRF to experiental FRF data, or curve fit a paraetric odel of an Ipulse Response Function (IRF) to experiental IRF data. The Rational Fraction Polynoial (RFP) ethod is coonly used for curve fitting FRFs, and the Coplex Exponential (CE) for curve fitting IRFs. The FRF and its corresponding IRF for a Fourier transfor pair. That is, an IRF is obtained by applying the Inverse FFT to an FRF, and the FRF can be recovered by applying the Forward FFT to the IRF. Therefore, either FRFs or their equivalent IRFs can be curve fit by starting with either one and using the FFT to transfor to the other. Many variations of the RFP and CE ethods have been proposed and docuented [6], [7]. Both ethods of these ethods were used for curve fitting the operating data Figure. An IRF. presented in this paper. Other types of curve fitting based on state-space odels have also been used for curve fitting operating data [4].

4 .6 Ipulse Response Function Since the IRF is the Inverse Fourier transfor of the FRF, each eleent of an FRF atrix has and equivalent IRF in the tie doain. Modal paraeters are therefore estiated fro a set of IRFs in the siilar way as they are estiated fro a set of FRFs. 2 BACKROUND THEORY Recall that operating data is acquired in any situation where the excitation forces are not easured. It is possible to curve fit operating data using an FRF or IRF curve fitting odel, but an assuption regarding the spectru of the unknown excitation forces is required. 2. Roving & Reference Responses Assue that a certain nuber of the responses are used a Reference responses, i.e. they are easured at fixed DOFs through the OMA. The responses that are not fixed during an OMA are referred to as Roving responses. A ulti-input ulti-output (MIMO) FRF atrix odel for the Roving responses can be written as follows, { X( )} [A]{F( ω)} where: ω () { X(ω)} n-vector of Roving response Fourier transfors. [ A(ω)] (n by ) FRF atrix relating forces to Roving Responses. { F(ω)} -vector of (uneasured) force Fourier transfors. ω frequency variable. n nuber of Roving responses. nuber of (unknown) forces. Siilarly a MIMO odel for the Reference responses can be written as follows, { Y( ω )} [B]{F( ω)} (2) where: { Y(ω)} r- vector of Reference response Fourier transfors. [ B(ω)] (r by ) FRF atrix for Reference Responses. r nuber of Reference responses. Equations () & (2) both assue that forces are applied to the structure. The nuber of forces, and the DOFs where they are applied assued to be unknown. 2.2 Multi-Reference Cross Power Spectru Matrix A power spectru atrix can now be fored between the Roving and Reference response Fourier transfor vectors. This ulti-reference Cross Power spectru atrix is written as,

5 [ x ω)] where: T,y( ω )] [A][f,f ][B( (3) [ ω T x,y( ω )] {X}{Y( )} [ ω T f,f ( ω )] {F}{F( )} (n by r) Cross Power spectru atrix. ( by ) force Power spectru atrix. T denotes the transposed coplex conjugate. The force Power spectru atrix is syetric and real valued. The Roving & Reference FRF atrices are also assued to be syetric. Each eleent of the Cross Power spectru atrix (3) is the Cross Power spectru between a pair of Roving & Reference responses. Each colun of the Cross Power spectru atrix can be used for OMA after the De-Convolution window is applied to it. Measureent of eleents of the Cross Power spectru atrix (3) is a straightforward calculation in all current day Fourier analysis systes. A Cross spectru calculation has the following iportant advantages, Extraneous noise is reoved by spectru averaging. Non-linear responses due to rando excitation are reoved by spectru averaging. Triggering is not required. To see ore clearly how an FRF curve fitting odel can be applied to a colun of Cross Power spectru data, and hence OMA can be perfored, consider the case of one Reference. 2.3 Cross Power Spectru of One Reference Using equation (3), the colun of Cross Power spectra corresponding to Reference DOF k is written,,k 2,k M n,k where: i j i j i j A A A, j 2, j n, j C j,i C C B j,i j,i B B C j, i eleent(j,i) of the force Power Spectru atrix, [ f, f ( ω )]. - denotes the coplex conjugate. (4)

6 Equation (4) shows that each coponent of the k th colun of the Cross Power spectru atrix is a suation of ters, each ter taking the for, (Roving FRF) x (force spectru eleent) x (Reference FRF) 2.4 Flat Force Spectru Assuption If the force Spectru atrix is assued to be relatively flat in the frequency range of the resonances of interest, and the excitation forces are stationary, then the Force power spectru can be replaced with constants for the frequency range(s) of interest. There are practical cases when the excitation forces can be assued to have a relatively flat spectru. For instance, vehicle traffic on a bridge or wind blowing against a building are assued to be broad-band and rando in nature. If the excitation forces are ipulsive in nature, they too can be assued to have a relatively flat spectru. With the above assuptions, equation (4) can be re-written as a single suation of ters,,k 2,k M n,k where: Di i i i D A i D A i D A i,i 2,i B n,i B B constant due to the forces applied at the i th DOF. (5) 3 The De-Convolution Window Equation (5) akes it clear that each coponent of a colun of a ulti-reference Cross Power spectru atrix is a suation of ters involving the a Roving response FRF ultiplied by the coplex conjugate of a Reference response FRF. The inverse Fourier transfor of a Cross spectru is a Cross correlation function. Since ultiplication in one doain is equivalent to convolution in the other doain, the inverse Fourier transfor of equation (5) is a colun of Cross correlations, each colun being a suation of Roving response IRFs convolved with a Reference IRF. Figure (2) shows an exaple Cross correlation function. The unique property of these Cross correlations is that the Roving IRF doinates the first half of the signal while the Reference IRF (corresponding to the coplex conjugate of the Reference FRF), doinates the second half of the signal. To identify the odes of the structure, only the Roving IRF is needed. Hence applying a De- Convolution window to the signal in Figure 2 preserves the Roving IRF is while the ajority of the Reference IRF is reoved, as shown in Figure 3. Notice that even though the signal is non-zero in the iddle of the window, the De-convolution soothly transitions it to zero.

7 Figure 2. A Cross Correlation Function Figure 3. Cross Correlation After De- Convolution Windowing. When the windowed data in Figure 3 is transfored back to the frequency doain, an FRF curve fitting ethod can be applied to the Roving FRF to identify its odal paraeters. 4 Illustrative Exaple To deonstrate the use of the De-Convolution window, we will start with a odal odel of a structure and use equations () and (2) to synthesize its responses to a broad band rando excitation force. Then a set of Cross Power spectra calculated between ultiple Roving and Reference acceleration response will be post-processed using the De-Convolution window and FRF curve fitting. Finally, the resulting odal paraeters will be copared with the paraeters of the original odal odel to provide a round trip validation of this approach to OMA. The odal odel is a set of 0 ode shapes extracted fro a set of 99 FRF easureents taken in three directions at 33 points of the structure shown in Figure 4. An FRF atrix odel was synthesized using the odal odel of the bea, and the FRF odel was excited with a burst rando signal at the left front corner of the upper plate, at DOF 5Z. Several typical structural responses to the bust rando excitation are shown in Figure Burst Rando Excitation for Leakage Free Responses Of course, burst rando excitation of a real structure could never occur unless the Figure 4. Bea Structure. structure was artificially excited with shakers driven by burst rando signals. In ost practical cases where operating data is acquired, the responses would be purely rando, and therefore a Hanning window would be applied to the response signals in order to reduce leakage in the Cross spectru easureents. By using burst rando excitation, our MIMO siulation provided acceleration responses that would yield

8 leakage-free Cross Power spectru estiates, thus eliinating leakage as a source of error in this round trip experient. As a first siulated OMA, the acceleration response at the right front corner of the upper plate was used as a reference response, and Cross Power spectra were calculated between all 99 DOFs and the reference DOF 5Z. Soe typical Cross spectra are shown in Figure 6A before the De-convolution window was applied. These easureents are the results of 50 spectru averages, with 024 lines of resolution. The sae Cross spectra are shown in Figure 6B after the De-convolution window was applied. Notice that the agnitudes of the Cross spectra have less noise on the due to the effect of the De- Convolution window. Notice also that the phases now have the negative changing phase as frequency increases through a resonance peak. Figure 5. Several Acceleration Responses. Figure 6A. Cross Spectra before De-Convolution Windowing. Figure 6B. Cross Spectra after De-Convolution Windowing The windowed Cross spectra where curve fit using an FRF curve fitter, and the results are shown in Table. It indicates that all of the odal frequencies and daping were recovered with reasonable accuracy fro the Cross spectra, but it s clear fro its low Modal Assurance Criterion (MAC) value that the 460 Hz ode shape was not recovered. It can only be assued that the 460 Hz ode was not well defined in set of Cross Power spectra using Reference DOF 5Z, which iplicitly ultiplied together the Roving and Reference FRFs between the excitation DOF 5Z and the reference DOF 5Z.

9 Frequency (Hz) Daping (Hz) MAC (%) Mode Modal Model Curve Fit Modal Model Curve Fit Reference 5Z Table. Operating Modes fro Reference 5Z. 5 Multiple Reference OMA Since the Reference DOF 5Z was not enough to correctly identify all of the odes in the odal odel, a second siulated OMA was perfored using two Reference responses (3Z & 5Z). Table 2 below shows the results of the ultiple reference OMA. The results clearly indicate that the extra reference not only provided better estiates of odal frequency and daping, but ore accurate ode shape estiates fro both references as well. 6 CONCLUSIONS It was deonstrated the OMA can be perfored quite accurately using response only (or output only) acceleroeter data fro a structure, it the excitation forces have a relatively flat spectru over the frequency range of interest, and the data is processed with a De-Convolution window. It was shown in the background theory section that the De- Convolution window reoves the ajority of the Ipulse Response Function (IRF) of the Reference response fro the Cross correlation function, leaving the IRF of the Roving Figure 7. Typical Cross Spectru Curve Fitting response. Transforing the Roving IRF to the Result. frequency doain yields an FRF which can be processed using FRF curve fitting ethods to extract odal paraeters. Another advantage of this approach is that after applying the De-Convolution window, ore zero valued saples can be added to the IRFs to increase their tie length. Then, when the IRFs are transfored into FRFs, the frequency resolution of the FRFs in increased, thus providing ore saples of data surrounding each resonance peak, which iproves the curve fitting results.

10 This cobination of De-Convolution windowing and increased frequency resolution is a powerful cobination of techniques for perforing an OMA. Frequency (Hz) Daping (Hz) MAC MAC (%) (%) Mode Modal Model Curve Fit Modal Model Curve Fit Ref. 3Z Ref. 5Z References [] Schwarz, B. & Richardson, M., Modal Paraeter Estiation fro Operating Data, (Vibrant Tech. Paper No [2] Richardson, M., Is It A Mode Shape Or An Operating Deflection Shape?, Sound and Vibration Magazine, February 997. [3] Pickrel, C. R., Airplane round Vibration Testing Noinal Modal Model Correlation, Sound and Vibration Magazine, Noveber, [4] Batel, M. Operational Modal Analysis Another Way of Doing Modal Testing, Sound and Vibration Magazine, August [5] Richardson, M., Structural Dynaics Measureents, Structural 2000: Current Status and Future Directions, Research Studies Press, Ltd. Baldock, Hertfordshire, England Deceber, 2000, p-34. [6] Forenti, D. & Richardson, M. Paraeter Estiation Fro Frequency Response Measureents Using Rational Fraction Polynoials (Twenty Years Of Progress), Proceedings of International Modal Analysis Conference XX, February 4-7, 2002 Los Angeles, CA. [7] Brown, D. & Alleang, R. Multiple Input Experiental Modal Analysis, Fall Technical Meeting, Society of Experiental Stress Analysis, Salt lake City, UT, Noveber, 983. [8] Vold, H., Kundrat, J., Rocklin,. A Multi-Input Modal Estiation Algorith for Mini- Coputers, S.A.E. paper No , 982. [9] Ibrahi, S., Mikulcik, E. A Method for the Direct Identification of Vibration Paraeters fro the Free Response, Shock and Vibration Bulletin, Vol 47, Part 4, 977. [0] Iproveents in the Coplex Exponential Coputational Algorith, Office of NavalResearch, Washington, DC., prepared by Texas Instruents, Dallas, TX., March, 970.