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Analytical and Experimental Study Using Output-Only Modal Testing for On-Orbit Satellite Appendages

Analytical and Experimental Study Using Output-Only Modal Testing for On-Orbit Satellite Appendages Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2009, Article ID 538731, 10 pages doi:10.1155/2009/538731 Research Article Analytical and Experimental Study Using Output-Only Modal Testing for On-Orbit Satellite Appendages 1 1 2 1 Mashiul Alam, Ramin Sedaghati, Yvan Soucy, and Rama B. Bhat Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Qc, Canada H3G 1M8 Canadian Space Agency, Space Technologies, John H. Chapman Space Center, Saint-Hubert, Qc, Canada J3Y 8Y9 Correspondence should be addressed to Ramin Sedaghati, sedagha@encs.concordia.ca Received 8 May 2008; Revised 20 January 2009; Accepted 25 February 2009 Recommended by Samir Gerges Output-only modal testing is an effective technique to identify the modal parameters of structural systems under ambient or operational conditions and has potential applications in civil, mechanical, and aerospace engineering. It may effectively be used for model validation, model updating, quality control, and health monitoring through the determination of modal characteristics of the structures. This approach to modal testing has great potential for ground and on-orbit modal testing of space hardware, especially for flexible structures such as membrane payloads where the operating and ambient excitations, such as firing of AC thrusters and ambient thermal shock, are difficult or impossible to measure. The main objective of this work is to conduct analytical and experimental study on output-only modal testing and to demonstrate its potential application to effectively extract modal parameters of an on-orbit satellite appendage. Copyright © 2009 Mashiul Alam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction density and, in all cases, damping estimation is uncertain or impossible. The concept of output-only modal testing is not new but The first time-domain technique that really became known for serious output-only identification was introduced the development of this technique has a brief history. For quite some time people working in modal analysis have by Ibrahim and Milkulcik [2–4] and is known as the been performing output-only modal identification using the Ibrahim Time Domain (ITD) method. Shortly after that the commonly accepted fact that if only one mode contributes to polyreference time-domain method was introduced by Vold a certain band of the cross-spectral matrix, then any row or et al. [5] and Vold and Rocklin [6]. The Eigen Realization column in that matrix can be used as a mode shape estimate. Algorithm (ERA) was developed by Juang and Pappa [7] By picking a peak in one of the spectral density functions and later employed in space application [8, 9]. The last one can get the mode shape from one of the columns or two techniques, that is, Polyreference and ERA, use multiple inputs but ITD can also be formulated as a multiple input rows in the cross-spectral matrix. This classical approach, also known as Basic Frequency Domain (BFD) technique, method as described by Fukuzono [10]. Recently, Zhang et al. is based on the simple signal processing technique using a [11] gave a common formulation for all these techniques. discrete Fourier transform and hinges on the fact that well- What they have in common is that they assume that a separated modes can be estimated directly from the matrix free response function can be obtained. These time domain of the cross-spectra [1]. techniques are all based on a function represented by In the case of closely spaced modes, it can be difficult to exponential decay. A more modern time-domain technique detect the modes and, even in the case where close modes called Stochastic Subspace Identification (SSI) has also been introduced. Three different implementations of Stochastic are detected, estimated frequencies and mode shapes become heavily biased. Furthermore, the estimated frequencies are Subspace Identification technique are: Unweighted Princi- limited by the frequency resolution of the estimated spectral pal Component (UPC), Principal Component (PC), and 2 Advances in Acoustics and Vibration Canonical Variate Analysis (CVA). These methods are time domain methods that directly work with time domain data, without any conversion to spectral functions. The algorithm Satellite body for this new stochastic subspace time domain technique was described by Van Overschee and De Moor [12–15]. Also, more detailed information on stochastic subspace method can be found in [16–19]. Two simulation examples including a spring-mass-system and a horizontal cantilever beam in transverse vibration were demonstrated by Lardies [20]to Thruster investigate the efficiency of this stochastic subspace method. The main advantage of the classical, that is, BFD Solar panel approach is that the structural properties can be easily found just by examining the density functions. The disadvantages associated with this approach are removed by another method called Frequency Domain Decomposition technique Figure 1: Simple model of a satellite. (FDD) that was extensively investigated by Rent and Zong [21] and Brincker et al. [22–24]. This method contains all the advantages of the classical technique and also provides spacecraft appendages will first be discussed. This will be clear indication of harmonic components in the response followed by the experimental and analytical investigation on signal. It has been described that the spectral matrix can be modal parameter identification of a simple plate structure decomposed with the Singular Value Decomposition (SVD) extracted by different output-only modal testing techniques. method into a set of auto-spectral density functions, each Finally a plate type structure representing the solar panel corresponding to a Single Degree of Freedom (SDoF) system. appendages of the Communication Technology Satellite The FDD technique can effectively handle close modes and (CTS) will be simulated to identify its modal parameters noise; however it cannot provide damping information. using different output-only modal testing techniques. The Enhanced Frequency Domain Decomposition (EFDD) is basically an extension of the FDD technique capable of providing damping information. In EFDD, the identified 2. Dynamics of Spacecraft Appendages frequency function around each resonant peak is transferred back to the time domain using Inverse Discrete Fourier The appendages of a spacecraft consist of lightweight, Transform, and damping can be obtained by the logarithmic flexible, deployable members in the form of solar panels, decrement of the corresponding SDoF normalized autocor- antennas, and booms. Satellites have to carry their own relation function. power source because they cannot receive power from earth Abdelghani et al. [25] presented the results of the per- and they must remain pointed in a specific direction, formance of output-only identification algorithm for modal or orientation, to accomplish their mission. Maintaining analysis of aircraft structures under white noise excitation. It proper orientation is typically accomplished by small rocket was found that the subspace-based algorithm performs well. engines, known as attitude thrusters. These thrusters/attitude The PolyMAX is a further evolution of the so-called controllers perform some functions such as pointing the Least-Square Complex Frequency (LSCF) domain method. antennas in a desired direction, pointing solar panels toward Originally, LSCF was introduced to find initial values of the sun, keeping sensors and sensitive equipment away from the iterative maximum likelihood method [26]. The method the sun’s light and heat, and pointing the control jets in the estimates a so-called common-denominator transfer func- desired direction to accomplish efficient maneuvers [31]. tion model [27]. The most important advantage of LSCF Attitude thrusters can make large changes to orientation estimator over the available and widely applied parameter quickly and can create significant vibration in the satellite estimator techniques [28] is that it provides very clear body. They are the largest source of torque on the spacecraft. stabilization diagrams. A thorough analysis of different If they are used to achieve accurate pointing, small amount variants of the common-denominator LSCF method can be of torque should be applied as consistent impulses for found in [29]. A complete background on frequency-domain minimum switch-on time of several milliseconds [31, 32]. system identification can be found in [30]. Figure 1 represents the simple model of satellite during The main aim of the project is to provide some insight attitude control. The satellite body rotates due to the torque into the application of the ouput-only modal testing for created by the forces exerted by the thrusters. on-orbit satellite appendages. In this work the performance Spacecraft thrusters may be categorized into cold-gas and of different out-put only modal testing techniques will hot-gas thrusters. Cold-gas thrusters produce small amount be examined both analytically and experimentally on a of thrust, typically 5 N or less and are useful for small simple plate type structure under different environmental spacecraft and for fine attitude control; whereas the hot-gas excitations. Then a solar panel appendage under represen- thrusters can produce thrusts from as low as 0.5 N up to tative operational conditions will be simulated to identify 9000 N, but are usually used for higher thrust applications its modal parameters using different output-only modal [33]. The excitation may exist for approximately 10–15 testing techniques. In the following sections dynamics of the seconds including steady-state thrust for 4 ∼ 5 seconds. Advances in Acoustics and Vibration 3 Table 1: Material and geometrical characteristics of the test plate article. Material type Aluminum 6061-T6 Modulus of Elasticity, E 68.7 × 10 N/m Length of the test article, a 1.22 m Shaker with test OOMT software Width of the test article, b 0.3048 m article UFF data Thickness of the test article, h 0.00635 m Data Density, ρ 2.71 × 10 kg/m Amplifier Poisson’s ratio, ν 0.3 LMS software Signal Multi - channel front end The analytical investigation of the structural dynamics of a flexible solar array had been done by Vigneron et al. Figure 2: Schematic diagram of the test set-up. [34]. The analysis focused on two representative array units of the Communication Technology Satellite (CTS) employed in ground-test program. Frequencies and mode shapes were identified by (a) Rayleigh-Ritz method on a simplified continuum model and by (b) NASTRAN finite-element program. Extrapolation of modal information to on-orbit Portable conditions has also been discussed. The formulation of com- exciter plex system dynamics due to interactions between librational motion, transverse vibration, and thermal deformations has Support been elaborated in [35]. frame 3. Analytical and Experimental Modal Parameter Identification of a Cantilever Aluminum Plate Using Output-Only Figure 3: Test-setup for point excitation. Modal Testing In this project the analytical and experimental study has been conducted on a cantilever aluminum plate type structure representing the solar panel. The dimensions of the plate is used to perform the driven base excitation; whereas a single are selected so that the fundamental frequency of the plate point portable exciter suspended on a frame and attached to to be in the practical range of 3-4 Hz. The material and the test article at the front centre mid-point using a force geometrical properties of the experiment test article are sensor, and a stinger is used to conduct point excitation presented in Table 1. experiment as shown in Figure 3. The accelerometers are Using Table 1, the plate is modeled in finite-element glued on the test article and connected to the front end of software PATRAN/NASTRAN, and its first four natural the modal system through cables. With the help of PolyMAX, frequencies are found to be: 3.51 Hz (1st bending), 22.00 Hz the data is processed to identify the modal parameters (2nd bending), 28.33 Hz (1st Torsion), and 61.90 Hz (3rd from the frequency response functions. For the case of base bending). As it can be seen, the fundamental frequency is in excitation, the response acceleration over input acceleration the practical range of interest. (transmissibility) is evaluated; whereas for the case of point Experiments have been carried out under both base and excitation response acceleration over input force (FRF) is single point excitations using facilities in Vibration lab of evaluated. Also, using cross-power densities of the processed the Canadian Space Agency (CSA) in Saint-Hubert, QC, data, the modal parameters are extracted with the help of Canada. The experimental data are acquired using LMS operational PolyMAX. Finally, time series data are exported Test Lab software and processed for modal information in universal file format to be processed in PULSE operational using LMS PolyMAX (for reference modal parameters), LMS modal analysis software to extract the modal parameters. Operational PolyMAX, and also B&K operational PULSE For the case of base excitations, 9 accelerometers are software. The results are presented and discussed hereafter. attached on the test article. Eight are attached on the front Vibration measurements are done on the test fixture at first face at points1,2,3,4,6,7,8,and 9, andone accelerometer to find out its natural frequencies, and to see whether the is attached at the center of the plate on the back (point 5b) as fixture remains rigid in the frequency range of interest in the shown in Figure 4. tests. For the case of point excitations, 9 accelerometers and a The schematic of the test setup is presented in Figure 2. force sensor are attached on the test article. Accelerometers The test article is placed vertically on the slip table of a shaker are located at the same locations as for the driven base using a fixture for both base and point excitation. The shaker excitation on the front face of the plate (except at point 5 4 Advances in Acoustics and Vibration 0 1 Test article 4 6 5b −7 0 0 50 130 (Hz) Cross PSD of point 1 w.r.t. point 1b Figure 6: Sample cross PSD for the driven base excitation. 8 9 Fixture complete cross-spectra matrix using all response locations as references. It is noted that the reference modal parameters are (a) (b) identified using PolyMAX from evaluation of transmissibility and FRF for the cases of driven base and portable exciter Figure 4: Accelerometer locations at (a) front face (b) back face of tests, respectively. Reference sensors have been located on the test article and fixture for base excitation. upper part of the fixture (point 10 in Figure 4)for driven base case and at point 5 on the front face of test article for case of point excitation. The parameters that are considered for both the driven base excitation and point excitation methods are presented in Table 2. It should be noted that for the driven base test, the experimental runs are open-loop, and hence the random input signals mentioned in Table 2 are uncontrolled and do not follow a predefined profile as for a typical vibration control setup. This is a driven base modal test, not a vibration test. −3 0 50 100 130 (Hz) 3.1. Fixture Survey. At the beginning of the experiment a transmissibility-based modal survey was carried out on the Transmissibility of point 1 w.r.t. point 10 fabricated fixture to ensure that the fixture is rigid enough Transmissibility of point 5b w.r.t. point 10 in the frequency range of interest. The fixture was rigidly Figure 5: Sample transmissibilities for driven base run. mounted to the slip table through 6 bolts. Ten accelerometers are mounted on the fixture, and a reference accelerometer is placed at the end side of the slip table. The first and second modes of the fixture are found to be 55 Hz and where it is on the back face) and also a force sensor and a 157 Hz. As it can be seen the fundamental frequency of the portable exciter are connected to point 5 on the front face. fixture (55 Hz) is well above the first 3 natural frequencies It should be noted that although an excitation location on of the test article which are about 3.48 Hz, 21.54 Hz, and either side of the plate (e.g., point 4) would have been a 28.15 Hz. In fact, this confirms that the fixture dynamics better choice for exciting the torsion modes, point 5 on the have negligible effects on the dynamics of the test article centerline of the plate was selected in order to have similar for at least the first three modes of the test article. For the capability (or lack of it) of exciting torsion modes as driven higher modes, since the reference modal parameters were base excitation. also obtained experimentally, any effects that the fixture It should be noted that for computation of the cross- might have in the modal parameters would be present for spectra required in Operational PolyMAX, the reference both the reference data and the data processed with the accelerometer sensor was located on top corner of the output-only techniques. plate (back of point 1) for both cases of driven base and portable exciter tests. As a general guideline for selection of the set of references for computing the cross-spectra, 3.2. Experimental Results-Base Excitation. The natural fre- one should select the response locations (and directions) quency and modal damping factors (damping ratios) of that correspond to maximum displacement for the key the test article are first identified using traditional modal modes of interest. Operational PULSE software computed a analysis approach by processing the transmissibility with log Tr (g/g) log PSD (g /Hz) Amplitude Advances in Acoustics and Vibration 5 1st mode 4th mode 2nd mode 6th mode −5 Figure 7: Mode shapes from output-only approaches for driven 0 50 100 130 base run. (Hz) FRF of point 5b w.r.t. point 5 PolyMAX. Thesemodal parameters aretobeusedas FRF of point 1 w.r.t. point 5 reference for comparison with the values extracted with Figure 8: Sample FRFs for point excitation run. operational modal techniques. Figure 5 shows two samples of trasnsmissibilities measured from the driven base excita- tion. −1 Theupper curvein Figure 5 represents the transmissibil- ity at the point 1 location on test article; whereas the lower curve represents the transmissibility at the point 5b at the back face of the test article as shown in Figure 4.Itisnoted that upper curve shows some torsion peaks at about 28 Hz and 87 Hz but the lower curve does not show the torsion modes as point 5 is located at the centerline along the length of the test article. To identify modal parameters both Operational Poly- −6 MAX and Pulse operational software are used. It is noted 0 50 100 130 that the cross power spectras are required in Operationl (Hz) PolyMAX to identify the modal parameters, while for Cross PSD of point 1 w.r.t. point 1b the PULSE operational methods (FDD, EFDD, SSI), time series output data should be extracted. An example of the Figure 9: Sample cross PSD for portable excitation run. cross power spectra for base excitation run is presented in Figure 6. Natural Frequencies and damping ratios for the first six not excite well the torsional modes of the test article, as may modes extracted using FRF PolyMAX (reference), Operation be expected. PolyMAX and PULSE are tabulated in Tables 3 and 4, respectively. 3.3. Experimental Results-Single Point Excitation. Sample It can be seen that very good agreement exists between FRFs for similar points (points 1 and 5b) discussed in operational modal results and reference modal parameters. driven base case are shown in Figure 8. As discussed before, As it can be seen from Tables 3 and 4,nomodal parameters these FRFs will be processed by PolyMAX to extract modal could be extracted for mode 3 (1st torsional mode) using parameters which will be used as reference values for both operational Polymax and Operational Pulse methods. comparison with those obtained from operational testing. Also operational Polymax was not able to detect the fifth As it can be seen from Figure 8, FRF at point 1 contains mode (2nd torsional mode) while most PULSE operational torsional mode peaks as expected. methods were able to identify the modal parameters for the fifth mode. The FDD method does not provide damping As mentioned before, for Operational PolyMAX, the information. It should be noted that driven base excitation cross power spectra are used to identify the modal param- mainly excites bending modes in symmetric structures and eters. A sample of cross-power spectra for point excitation thus it is not surprising that operational methods which run is shown in Figure 9. are based on only output responses have difficulty to detect The identified natural frequencies and damping factors torsional modes. using output-only modal testing approaches for this case are The mode shapes for driven base runs using output-only provided in Tables 5 and 6,respectively. modal testing technique are shown in Figure 7. Mode shapes It can be seen that there is close agreement in the values obtained from PolyMAX using transmissibility, Operational of natural frequencies identified using the different output- PolyMAX, and PULSE output-only modal testing are close to only modal testing approaches with those of reference FRF each other except for the 3rd and 5th modes in Operational except for the first mode. The reason for this discrepancy for PolyMAX and 3rd mode in PULSE, which are missing. It the identified fundamental mode can be attributed to the fact should be noted that the translational base excitation could that the fundamental mode found by operational techniques log PSD (g /Hz) log FRF (g/N) 6 Advances in Acoustics and Vibration Table 2: Parameters for both driven base excitation and point excitation. Method PolyMAX Operational PolyMAX PULSE Bandwidth 128 Hz 128 Hz 128 Hz No. of frequency lines 1025 1025 8192 No. of averages 30 15 (portable) and 10 (base excitation) 22 Signal type 75% burst random 100% random 100% random Table 3: Natural frequencies of test article under base excitation. PULSE methods Transmissibility Operational Mode PolyMAX SSI PolyMAX f (Hz) FDD f (Hz) EFDD f (Hz) (Ref.) f (Hz) UPC f (Hz) PC f (Hz) CVA f (Hz) 1 3.45 3.45 3.46 3.46 3.46 3.48 3.35 2 21.38 21.50 21.53 21.52 21.53 21.54 21.49 3 —— — — — 28.15 — 4 60.5 60.49 60.52 60.45 60.45 60.43 60.39 5 87.75 87.77 87.77 87.77 87.69 87.84 — 6 120.6 120.4 120.5 120.4 120.5 120.5 120.29 Table 4: Damping ratios of test article under base excitation. PULSE methods Transmissibility Operational Mode SSI PolyMax (Ref.) ξ (%) PolyMax ξ (%) FDD ξ (%) EFDD ξ (%) UPC ξ (%) PC ξ (%) CVA ξ (%) 1 — 0.59 0.78 0.59 0.84 0.40 1.43 2 — 0.39 0.23 0.21 0.23 0.14 0.28 3 —— — — — 0.02 — 4 — 0.19 0.27 0.33 0.34 0.13 0.2 5 — 0.24 0.24 0.33 0.34 0.20 — 6 — 0.35 0.23 0.54 0.36 0.14 0.11 Table 5: Natural frequencies of test article under point excitation. PULSE methods FRF PolyMax (Ref.) Operational Mode SSI f (Hz) PolyMax f (Hz) FDD f (Hz) EFDD f (Hz) UPC f (Hz) PC (Hz) CVA f (Hz) 1 4.85 4.60 4.60 4.75 4.84 3.47 4.98 2 21.0 21.11 21.21 21.33 21.41 21.50 21.39 3 28.25 28.31 28.28 28.28 28.28 28.26 28.25 4 59.50 59.38 59.49 59.43 59.45 60.39 59.28 5 87.75 87.78 87.81 87.81 87.81 87.84 87.91 6 120.1 120.1 120.5 120.0 120.1 119.8 120.1 Table 6: Damping ratios of test article under point excitation. PULSE methods FRF PolyMax (Ref.) Operational Mode SSI ξ (%) PolyMax ξ (%) FDD ξ (%) EFDD ξ (%) UPC ξ (%) PC ξ (%) CVA ξ (%) 1 0.95 1.86 — 0.76 1.85 1.36 2.0 2 0.21 0.44 — 0.26 1.87 2.51 2.23 3 0.08 0.02 — 0.59 0.20 0.19 0.19 4 0.34 0.74 — 1.96 1.60 1.96 1.95 5 0.27 0.02 — 0.29 0.23 0.21 0.22 6 0.22 0.11 — 0.37 0.26 0.36 0.29 Advances in Acoustics and Vibration 7 1st mode 2nd mode 3rd mode 4th mode 5th mode 6th mode Figure 10: Mode shapes from output-only approaches for point excitation. Table 7: Properties of solar panel and its dimensions. 21 22 23 24 25 16 17 18 19 20 Material type Isotropic material 11 12 13 14 15 2 67 8 910 Modulus of elasticity, E 207 × 10 N/m 1 23 4 5 Length of the panel, a 6.48 m Width of the panel, b 1.32 m Thickness of the panel, h 0.0228 m Figure 11: Finite-element model of the solar panel. Density, ρ 0.1548 × 10 kg/m Poisson’s ratio, ν 0.3 tional and one rotational degrees of freedom) modeled in PATRAN/NASTRAN is shown in Figure 11. Material properties and dimensions of the solar panel is basically related to the assembled structure and shaker due are given in Table 7. As mentioned before, the real panels to the interaction between the test article and shaker. of actual satellites include stiffened elements which are very It can also be observed that there are significant differ- difficult to model and here for the sake of simplicity the ences in the damping information due to the fact that the effect of stiffeners is smeared in the plate. Thus the density damping values are quite low, and it is more difficult to is adjusted so that the resulting natural frequency is between estimate them precisely. the 3-4 Hz which is similar to that of real panel. The mode shapes identified for single-point excitation Rotational transient excitation is applied to the base of method using FRF curve fit are shown in Figure 10.Mode the solar panel, and the acceleration responses at all the nodes shapes identified by Operational PolyMAX and PULSE are evaluated. The damping ratio considered in this analysis output-only modal testing are identical with the FRF modes. is 2% for all modes. The transverse acceleration response at More details of the work and results are presented in [36]. node 5 is evaluated. It is noted that the excitation time is 6 seconds and time step of 0.0025 seconds is considered for dynamic response. 4. Simulation of Solar Panel The nodal transverse acceleration responses are trans- formed into a universal file format. This universal file is then As shown in Section 3, the output-only modal testing processed in PULSE output-only modal testing software to techniques could accurately identify the modal parameters extract the modal parameters of the solar panel. Here three of a plate structure under translational random driven different cases are investigated. Case 1: acceleration responses base excitation. The Communication Technology Satellite for all nodes are transferred into a universal file for further (CTS) solar panel appendage is another example structure processing by operational PULSE modal testing software; studied due to the simplicity of its boundary conditions and Case 2: acceleration responses of just 4 nodes (node 3, node availability of some of its properties and characteristics [34]. 5, node 23, and node 25) are transferred; Case 3: acceleration The panel is 6.48 m long and 1.32 m wide. The materials used response of just node 5 is transferred. to manufacture these panels are glass-fibre with a Kapton layer on it. With the nominal thickness of such a panel, the fundamental frequency becomes very low. Normally, in the 4.1. Extracted Modal Parameters Based on Case 1. The real solar panel structure of actual satellites, supports and extracted natural frequencies of the solar panel using stiffeners are used to increase the fundamental frequency of PULSE output-only modal testing techniques for Case 1 are the appendages to about 3 ∼ 4 Hz. In the present study tabulated in Table 8. There is close agreement between the for design and analysis simplicity, the effects of stiffeners are identified natural frequencies and those obtained from finite- smeared over the panel in order to reach the fundamental element modal analysis. The identified damping ratios for frequency of 3 to 4 Hz. The panel is under base rotational Case 1 are also given in Table 9. excitation due to the torque created by the forces exerted by It should be noted that the damping ratios cannot the thrusters. be computed using FDD method. However EFDD and The finite-element model of the panel consisting of 16 UPC methods were able to capture the damping factors. plate elements and 25 nodes (each node has two transla- The reason that modal parameters could not be found by 8 Advances in Acoustics and Vibration Table 8: Identified natural frequencies for Case 1. Output-only modal testing methods Mode Finite-element modal analysis f (Hz) (Reference) SSI FDD f (Hz) EFDD f (Hz) UPC f (Hz) PC f (Hz) CVA f (Hz) 1 3.125 — 3.30 3.0 3.25 3.24 2 21.48 21.65 21.76 — 21.75 21.75 3 68.75 68.25 68.72 — 68.78 68.22 4 140.6 140.2 141.6 — 142.4 140.25 Table 9: Identified damping ratio for Case 1. Output-only PULSE methods Mode Finite-element modal analysis ξ (%) (Reference) SSI FDD ξ (%) EFDD ξ (%) UPC ξ (%) PC ξ (%) CVA ξ (%) 1 — — 2.1 2.4 3.3 2.0 2 — 1.98 2.0 — 2.3 2.0 3 — 1.99 2.0 — 1.96 2.0 4 — 1.88 2.0 — 2.4 2.0 Table 10: Identified natural frequencies for Case 2. Output-only modal testing methods Mode Finite-element modal analysis f (Hz) (Reference) SSI FDD f (Hz) EFDD f (Hz) UPC f (Hz) PC f (Hz) CVA f (Hz) 1 3.125 — 3.30 3.30 3.30 3.24 2 21.68 21.69 21.71 21.14 21.70 21.75 3 68.36 68.26 68.22 — 68.20 68.22 4 140.8 140.5 140.1 — 140.2 140.25 Table 11: Identified damping ratio for Case 2. Output-only PULSE methods Mode Finite-element modal analysis ξ (%) (Reference) SSI FDD ξ (%) EFDD ξ (%) UPC ξ (%) PC ξ (%) CVA ξ (%) 1 — — 2.0 2.0 2.7 2.0 2 — 1.85 2.0 1.9 2.0 2.0 3 — 1.80 2.0 — 2.0 2.0 4 — 1.60 2.0 — 2.0 2.0 Table 12: Identified natural frequencies for Case 3. Output-only modal testing methods Mode Finite-element modal analysis f (Hz) (Reference) SSI FDD f (Hz) EFDD f (Hz) UPC f (Hz) PC f (Hz) CVA f (Hz) 1 3.3 — 3.30 3.30 3.30 3.24 2 21.68 — 21.75 21.64 21.74 21.75 3 68.55 — 68.22 68.90 68.22 68.22 4 140.6 — 140.3 139.4 140.3 140.25 Table 13: Identified damping ratio for Case 3. Output-only PULSE methods Mode Finite-element modal analysis ξ (%) (Reference) SSI FDD ξ (%) EFDD ξ (%) UPC ξ (%) PC ξ (%) CVA ξ (%) 1 — — 2.0 2.0 2.0 2.0 2 — — 2.0 1.8 2.0 2.0 3 — — 2.0 1.6 2.0 2.0 4 — — 2.0 1.8 1.9 2.0 Advances in Acoustics and Vibration 9 some methods may be attributed to the fact that only few been conducted on a simple plate structure under different seconds, that is, 4 to 5 seconds of nodal responses have operating excitations to identify its modal parameters using been used to identify modal parameters. The real damping different output-only modal testing approaches. Finally the ratios can easily be identified by comparing the damping solar panel of the Communication Technology Satellite ratios identified using different methods. It should also be (CTS) has been simulated under representative operating noted that unlike UPC and CVA approaches, PC approach excitation provided by the spacecraft thrusters to find is not able to estimate modal parameters. Although these its modal parameters using different output-only modal three methods are similar, their algorithms have different testing techniques. The results are also compared with those weighting functions. CVA usually performs better for low- obtained using traditional modal analysis. It is aimed to order excited modes and may not provide information for demonstrate the potential application of output-only testing higher-order modes if they are not properly excited. in satellite appendages. It is concluded that output-only Mode shapes identified using output-only modal testing modal testing based on SSI techniques could capture the methods match very well with simulation results using modal parameters well using only a few nodal responses, classical FRF-based modal analysis approach. which is very important in satellite and space applications. 4.2. Extracted Modal Parameters Based on Case 2. The References extracted natural frequencies of the solar panel using PULSE [1] J. S. Bendat and A. G. Piersol, Engineering Applications of output-only modal testing techniques for Case 2 are given in Correlation and Spectral Analysis, John Wiley & Sons, New Table 10. The identified damping ratios for Case 2 are also York, NY, USA, 1993. provided in Table 11. [2] S. R. Ibrahim and E. C. Milkulcik, “A time domain modal Again the identified natural frequencies and damping vibration test technique,” Shock and Vibration Bulletin, vol. 43, factors are in close agreement with those obtained using no. 4, pp. 21–37, 1973. finite-element modal analysis. Similar to Case 1, in this [3] S. R. Ibrahim and E. C. Milkulcik, “The experimental case FDD method cannot find the damping factors of the determination of vibration parameters from time response,” structure, while EFDD and SSI methods were almost able to Shock and Vibration Bulletin, vol. 46, no. 5, pp. 187–196, 1976. identify damping factors for the first four modes. As it can [4] S. R. Ibrahim and E. C. Milkulcik, “A method for the direct be observed using acceleration nodal responses of only four identification of vibration parameters from the response,” nodes, it is possible to capture the modal parameters using Shock and Vibration Bulletin, vol. 47, no. 4, pp. 183–198, 1977. output-only modal testing techniques. [5] H. Vold, J. Kundrat, and G. T. Rocklin, “A multi-input modal estimation algorithm for mini-computers,” SAE Technical 4.3. Identified Modal Parameters Based on Case 3. The Paper 820194, International Congress and Exposition, Detroit, extracted natural frequencies and damping factors of the Mich, USA, 1982. solar panel based on nodal responses at node 5 are given in [6] H. Vold and G. T. Rocklin, “The numerical implementation of Tables 12 and 13,respectively. a multi-input modal estimation method for mini-computers,” in Proceedings of the 1st International Modal Analysis Confer- It is interesting to note that even using acceleration nodal ence & Exhibit (IMAC ’82), pp. 542–548, Orlando, Fla, USA, responses of one node (node 5), natural frequencies and November 1982. damping factors could be captured using output-only PULSE [7] J.-N. Juang and R. S. Pappa, “An eigensystem realization methods. Although FDD cannot identify the damping factor algorithm for modal parameter identification and model and EFDD was not able to find both natural frequencies and reduction,” Journal of Guidance, Control, and Dynamics, vol. damping factors, SSI method could accurately provide both 8, no. 5, pp. 620–627, 1985. natural frequencies and modal damping factors. [8] J. N. Juang and R. S. Pappa, “Galileo Spacecraft Modal It is found that the modal parameters identified for all identification using an eigensystem realization algorithm,” the three cases have good agreement with the finite-element NASA AIAA-1984-1070, Structures and Dynamics Division, results. Usually, in real cases, only a few nodal responses are NASA Langley Research Center, Hampton, Va, USA, 1984. recorded and processed as done in the present study. It is [9] J. E. Cooper and J. R. Wright, “Spacecraft in-orbit identi- shown that natural frequencies and damping ratios can be fication using eigensystem realization methods,” Journal of identified even by using nodal response at a single node. Guidance, Control, and Dynamics, vol. 15, no. 2, pp. 352–359, However, some optimization in the number and location 1992. of nodal responses is required to minimize the errors in [10] K. Fukuzono, Investigation of multiple reference Ibrahim time domain parameter estimation technique, M.S. thesis, Depart- identification. ment of Mechanical & Industrial Engineering, University of Cincinnati, Cincinnati, Ohio, USA, 1986. 5. Conclusions [11] L. Zhang, R. Brincker, and P. Andersen, “A unified approach for two-stage time domain modal identification,” in Proceed- In this work both analytical studies and output-only modal ings of the International Conference on Structural Dynamics testing are conducted on an on-orbit satellite appendage in Modeling, Test, Analysis, Correlation, and Validation,Madeira order to demonstrate the potential application of the output- Island, Portugal, June 2002. only modal testing method to effectively extract modal [12] P. Van Overschee and B. De Moor, Subspace Identification parameters. First an experimental and analytical study has for Linear Systems: Theory, Implementation and Applications, 10 Advances in Acoustics and Vibration Kluwer Academic Publishers, Dordrecht, The Netherlands, [28] W. Heylen, S. Lammens, and P. Sas, Modal Analysis Theory 1996. and Testing, Department Mechanical Engineering, Katholieke Universiteit Leuven, Leuven, Belgium, 1995. [13] P. Van Overschee and B. De Moor, “Subspace algorithms for the stochastic identification problem,” Automatica, vol. 29, no. [29] P. Verboven, Frequency domain system identification for modal 3, pp. 649–660, 1993. analysis, Ph.D. thesis, University of Brussels, Brussels, Bel- gium, 2002. [14] P. Van Overschee and B. De Moor, “N4SID: subspace algorithms for the identification of combined deterministic- [30] R. Pintelon and J. Schoukens, System Identification: A Fre- stochastic systems,” Automatica, vol. 30, no. 1, pp. 75–93, 1994. quency Domain Approach, IEEE Press, New York, NY, USA, [15] P. Van Overschee and B. De Moor, “A unifying theorem for 2001. three subspace system identification algorithms,” Automatica, [31] P. Fortescue and J. Stark, Spacecraft Systems Engineering,John vol. 31, no. 12, pp. 1853–1864, 1995. Wiley & Sons, New York, NY, USA, 1991. [16] S. M. Pandit, Modal and Spectrum Analysis: Data Dependent [32] J. R. Wertz and W. J. 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Andersen, “Modal identification of output-only systems using frequency domain decompo- sition,” in Proceedings of the European COST F3 Conference on System Identification and Structural Health Monitoring,pp. 273–282, Madrid, Spain, June 2000. [23] R. Brincker, C. E. Ventura, and P. Andersen, “Why output- only modal testing is a desirable tool for a wide range of practical applications,” in Proceedings of the 21st International Modal Analysis Conference on Structural Dynamics, pp. 1–8, Kissimmee, Fla, USA, February 2003. [24] R. Brincker and P. Andersen, “A way of getting scaled mode shapes in output-only modal testing,” in Proceedings of the 21st International Modal Analysis Conference on Structural Dynamics, pp. 141–145, Kissimmee, Fla, USA, February 2003. [25] M. Abdelghani, M. Goursat, T. Biolchini, L. 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Analytical and Experimental Study Using Output-Only Modal Testing for On-Orbit Satellite Appendages

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Copyright © 2009 Mashiul Alam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2009, Article ID 538731, 10 pages doi:10.1155/2009/538731 Research Article Analytical and Experimental Study Using Output-Only Modal Testing for On-Orbit Satellite Appendages 1 1 2 1 Mashiul Alam, Ramin Sedaghati, Yvan Soucy, and Rama B. Bhat Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Qc, Canada H3G 1M8 Canadian Space Agency, Space Technologies, John H. Chapman Space Center, Saint-Hubert, Qc, Canada J3Y 8Y9 Correspondence should be addressed to Ramin Sedaghati, sedagha@encs.concordia.ca Received 8 May 2008; Revised 20 January 2009; Accepted 25 February 2009 Recommended by Samir Gerges Output-only modal testing is an effective technique to identify the modal parameters of structural systems under ambient or operational conditions and has potential applications in civil, mechanical, and aerospace engineering. It may effectively be used for model validation, model updating, quality control, and health monitoring through the determination of modal characteristics of the structures. This approach to modal testing has great potential for ground and on-orbit modal testing of space hardware, especially for flexible structures such as membrane payloads where the operating and ambient excitations, such as firing of AC thrusters and ambient thermal shock, are difficult or impossible to measure. The main objective of this work is to conduct analytical and experimental study on output-only modal testing and to demonstrate its potential application to effectively extract modal parameters of an on-orbit satellite appendage. Copyright © 2009 Mashiul Alam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction density and, in all cases, damping estimation is uncertain or impossible. The concept of output-only modal testing is not new but The first time-domain technique that really became known for serious output-only identification was introduced the development of this technique has a brief history. For quite some time people working in modal analysis have by Ibrahim and Milkulcik [2–4] and is known as the been performing output-only modal identification using the Ibrahim Time Domain (ITD) method. Shortly after that the commonly accepted fact that if only one mode contributes to polyreference time-domain method was introduced by Vold a certain band of the cross-spectral matrix, then any row or et al. [5] and Vold and Rocklin [6]. The Eigen Realization column in that matrix can be used as a mode shape estimate. Algorithm (ERA) was developed by Juang and Pappa [7] By picking a peak in one of the spectral density functions and later employed in space application [8, 9]. The last one can get the mode shape from one of the columns or two techniques, that is, Polyreference and ERA, use multiple inputs but ITD can also be formulated as a multiple input rows in the cross-spectral matrix. This classical approach, also known as Basic Frequency Domain (BFD) technique, method as described by Fukuzono [10]. Recently, Zhang et al. is based on the simple signal processing technique using a [11] gave a common formulation for all these techniques. discrete Fourier transform and hinges on the fact that well- What they have in common is that they assume that a separated modes can be estimated directly from the matrix free response function can be obtained. These time domain of the cross-spectra [1]. techniques are all based on a function represented by In the case of closely spaced modes, it can be difficult to exponential decay. A more modern time-domain technique detect the modes and, even in the case where close modes called Stochastic Subspace Identification (SSI) has also been introduced. Three different implementations of Stochastic are detected, estimated frequencies and mode shapes become heavily biased. Furthermore, the estimated frequencies are Subspace Identification technique are: Unweighted Princi- limited by the frequency resolution of the estimated spectral pal Component (UPC), Principal Component (PC), and 2 Advances in Acoustics and Vibration Canonical Variate Analysis (CVA). These methods are time domain methods that directly work with time domain data, without any conversion to spectral functions. The algorithm Satellite body for this new stochastic subspace time domain technique was described by Van Overschee and De Moor [12–15]. Also, more detailed information on stochastic subspace method can be found in [16–19]. Two simulation examples including a spring-mass-system and a horizontal cantilever beam in transverse vibration were demonstrated by Lardies [20]to Thruster investigate the efficiency of this stochastic subspace method. The main advantage of the classical, that is, BFD Solar panel approach is that the structural properties can be easily found just by examining the density functions. The disadvantages associated with this approach are removed by another method called Frequency Domain Decomposition technique Figure 1: Simple model of a satellite. (FDD) that was extensively investigated by Rent and Zong [21] and Brincker et al. [22–24]. This method contains all the advantages of the classical technique and also provides spacecraft appendages will first be discussed. This will be clear indication of harmonic components in the response followed by the experimental and analytical investigation on signal. It has been described that the spectral matrix can be modal parameter identification of a simple plate structure decomposed with the Singular Value Decomposition (SVD) extracted by different output-only modal testing techniques. method into a set of auto-spectral density functions, each Finally a plate type structure representing the solar panel corresponding to a Single Degree of Freedom (SDoF) system. appendages of the Communication Technology Satellite The FDD technique can effectively handle close modes and (CTS) will be simulated to identify its modal parameters noise; however it cannot provide damping information. using different output-only modal testing techniques. The Enhanced Frequency Domain Decomposition (EFDD) is basically an extension of the FDD technique capable of providing damping information. In EFDD, the identified 2. Dynamics of Spacecraft Appendages frequency function around each resonant peak is transferred back to the time domain using Inverse Discrete Fourier The appendages of a spacecraft consist of lightweight, Transform, and damping can be obtained by the logarithmic flexible, deployable members in the form of solar panels, decrement of the corresponding SDoF normalized autocor- antennas, and booms. Satellites have to carry their own relation function. power source because they cannot receive power from earth Abdelghani et al. [25] presented the results of the per- and they must remain pointed in a specific direction, formance of output-only identification algorithm for modal or orientation, to accomplish their mission. Maintaining analysis of aircraft structures under white noise excitation. It proper orientation is typically accomplished by small rocket was found that the subspace-based algorithm performs well. engines, known as attitude thrusters. These thrusters/attitude The PolyMAX is a further evolution of the so-called controllers perform some functions such as pointing the Least-Square Complex Frequency (LSCF) domain method. antennas in a desired direction, pointing solar panels toward Originally, LSCF was introduced to find initial values of the sun, keeping sensors and sensitive equipment away from the iterative maximum likelihood method [26]. The method the sun’s light and heat, and pointing the control jets in the estimates a so-called common-denominator transfer func- desired direction to accomplish efficient maneuvers [31]. tion model [27]. The most important advantage of LSCF Attitude thrusters can make large changes to orientation estimator over the available and widely applied parameter quickly and can create significant vibration in the satellite estimator techniques [28] is that it provides very clear body. They are the largest source of torque on the spacecraft. stabilization diagrams. A thorough analysis of different If they are used to achieve accurate pointing, small amount variants of the common-denominator LSCF method can be of torque should be applied as consistent impulses for found in [29]. A complete background on frequency-domain minimum switch-on time of several milliseconds [31, 32]. system identification can be found in [30]. Figure 1 represents the simple model of satellite during The main aim of the project is to provide some insight attitude control. The satellite body rotates due to the torque into the application of the ouput-only modal testing for created by the forces exerted by the thrusters. on-orbit satellite appendages. In this work the performance Spacecraft thrusters may be categorized into cold-gas and of different out-put only modal testing techniques will hot-gas thrusters. Cold-gas thrusters produce small amount be examined both analytically and experimentally on a of thrust, typically 5 N or less and are useful for small simple plate type structure under different environmental spacecraft and for fine attitude control; whereas the hot-gas excitations. Then a solar panel appendage under represen- thrusters can produce thrusts from as low as 0.5 N up to tative operational conditions will be simulated to identify 9000 N, but are usually used for higher thrust applications its modal parameters using different output-only modal [33]. The excitation may exist for approximately 10–15 testing techniques. In the following sections dynamics of the seconds including steady-state thrust for 4 ∼ 5 seconds. Advances in Acoustics and Vibration 3 Table 1: Material and geometrical characteristics of the test plate article. Material type Aluminum 6061-T6 Modulus of Elasticity, E 68.7 × 10 N/m Length of the test article, a 1.22 m Shaker with test OOMT software Width of the test article, b 0.3048 m article UFF data Thickness of the test article, h 0.00635 m Data Density, ρ 2.71 × 10 kg/m Amplifier Poisson’s ratio, ν 0.3 LMS software Signal Multi - channel front end The analytical investigation of the structural dynamics of a flexible solar array had been done by Vigneron et al. Figure 2: Schematic diagram of the test set-up. [34]. The analysis focused on two representative array units of the Communication Technology Satellite (CTS) employed in ground-test program. Frequencies and mode shapes were identified by (a) Rayleigh-Ritz method on a simplified continuum model and by (b) NASTRAN finite-element program. Extrapolation of modal information to on-orbit Portable conditions has also been discussed. The formulation of com- exciter plex system dynamics due to interactions between librational motion, transverse vibration, and thermal deformations has Support been elaborated in [35]. frame 3. Analytical and Experimental Modal Parameter Identification of a Cantilever Aluminum Plate Using Output-Only Figure 3: Test-setup for point excitation. Modal Testing In this project the analytical and experimental study has been conducted on a cantilever aluminum plate type structure representing the solar panel. The dimensions of the plate is used to perform the driven base excitation; whereas a single are selected so that the fundamental frequency of the plate point portable exciter suspended on a frame and attached to to be in the practical range of 3-4 Hz. The material and the test article at the front centre mid-point using a force geometrical properties of the experiment test article are sensor, and a stinger is used to conduct point excitation presented in Table 1. experiment as shown in Figure 3. The accelerometers are Using Table 1, the plate is modeled in finite-element glued on the test article and connected to the front end of software PATRAN/NASTRAN, and its first four natural the modal system through cables. With the help of PolyMAX, frequencies are found to be: 3.51 Hz (1st bending), 22.00 Hz the data is processed to identify the modal parameters (2nd bending), 28.33 Hz (1st Torsion), and 61.90 Hz (3rd from the frequency response functions. For the case of base bending). As it can be seen, the fundamental frequency is in excitation, the response acceleration over input acceleration the practical range of interest. (transmissibility) is evaluated; whereas for the case of point Experiments have been carried out under both base and excitation response acceleration over input force (FRF) is single point excitations using facilities in Vibration lab of evaluated. Also, using cross-power densities of the processed the Canadian Space Agency (CSA) in Saint-Hubert, QC, data, the modal parameters are extracted with the help of Canada. The experimental data are acquired using LMS operational PolyMAX. Finally, time series data are exported Test Lab software and processed for modal information in universal file format to be processed in PULSE operational using LMS PolyMAX (for reference modal parameters), LMS modal analysis software to extract the modal parameters. Operational PolyMAX, and also B&K operational PULSE For the case of base excitations, 9 accelerometers are software. The results are presented and discussed hereafter. attached on the test article. Eight are attached on the front Vibration measurements are done on the test fixture at first face at points1,2,3,4,6,7,8,and 9, andone accelerometer to find out its natural frequencies, and to see whether the is attached at the center of the plate on the back (point 5b) as fixture remains rigid in the frequency range of interest in the shown in Figure 4. tests. For the case of point excitations, 9 accelerometers and a The schematic of the test setup is presented in Figure 2. force sensor are attached on the test article. Accelerometers The test article is placed vertically on the slip table of a shaker are located at the same locations as for the driven base using a fixture for both base and point excitation. The shaker excitation on the front face of the plate (except at point 5 4 Advances in Acoustics and Vibration 0 1 Test article 4 6 5b −7 0 0 50 130 (Hz) Cross PSD of point 1 w.r.t. point 1b Figure 6: Sample cross PSD for the driven base excitation. 8 9 Fixture complete cross-spectra matrix using all response locations as references. It is noted that the reference modal parameters are (a) (b) identified using PolyMAX from evaluation of transmissibility and FRF for the cases of driven base and portable exciter Figure 4: Accelerometer locations at (a) front face (b) back face of tests, respectively. Reference sensors have been located on the test article and fixture for base excitation. upper part of the fixture (point 10 in Figure 4)for driven base case and at point 5 on the front face of test article for case of point excitation. The parameters that are considered for both the driven base excitation and point excitation methods are presented in Table 2. It should be noted that for the driven base test, the experimental runs are open-loop, and hence the random input signals mentioned in Table 2 are uncontrolled and do not follow a predefined profile as for a typical vibration control setup. This is a driven base modal test, not a vibration test. −3 0 50 100 130 (Hz) 3.1. Fixture Survey. At the beginning of the experiment a transmissibility-based modal survey was carried out on the Transmissibility of point 1 w.r.t. point 10 fabricated fixture to ensure that the fixture is rigid enough Transmissibility of point 5b w.r.t. point 10 in the frequency range of interest. The fixture was rigidly Figure 5: Sample transmissibilities for driven base run. mounted to the slip table through 6 bolts. Ten accelerometers are mounted on the fixture, and a reference accelerometer is placed at the end side of the slip table. The first and second modes of the fixture are found to be 55 Hz and where it is on the back face) and also a force sensor and a 157 Hz. As it can be seen the fundamental frequency of the portable exciter are connected to point 5 on the front face. fixture (55 Hz) is well above the first 3 natural frequencies It should be noted that although an excitation location on of the test article which are about 3.48 Hz, 21.54 Hz, and either side of the plate (e.g., point 4) would have been a 28.15 Hz. In fact, this confirms that the fixture dynamics better choice for exciting the torsion modes, point 5 on the have negligible effects on the dynamics of the test article centerline of the plate was selected in order to have similar for at least the first three modes of the test article. For the capability (or lack of it) of exciting torsion modes as driven higher modes, since the reference modal parameters were base excitation. also obtained experimentally, any effects that the fixture It should be noted that for computation of the cross- might have in the modal parameters would be present for spectra required in Operational PolyMAX, the reference both the reference data and the data processed with the accelerometer sensor was located on top corner of the output-only techniques. plate (back of point 1) for both cases of driven base and portable exciter tests. As a general guideline for selection of the set of references for computing the cross-spectra, 3.2. Experimental Results-Base Excitation. The natural fre- one should select the response locations (and directions) quency and modal damping factors (damping ratios) of that correspond to maximum displacement for the key the test article are first identified using traditional modal modes of interest. Operational PULSE software computed a analysis approach by processing the transmissibility with log Tr (g/g) log PSD (g /Hz) Amplitude Advances in Acoustics and Vibration 5 1st mode 4th mode 2nd mode 6th mode −5 Figure 7: Mode shapes from output-only approaches for driven 0 50 100 130 base run. (Hz) FRF of point 5b w.r.t. point 5 PolyMAX. Thesemodal parameters aretobeusedas FRF of point 1 w.r.t. point 5 reference for comparison with the values extracted with Figure 8: Sample FRFs for point excitation run. operational modal techniques. Figure 5 shows two samples of trasnsmissibilities measured from the driven base excita- tion. −1 Theupper curvein Figure 5 represents the transmissibil- ity at the point 1 location on test article; whereas the lower curve represents the transmissibility at the point 5b at the back face of the test article as shown in Figure 4.Itisnoted that upper curve shows some torsion peaks at about 28 Hz and 87 Hz but the lower curve does not show the torsion modes as point 5 is located at the centerline along the length of the test article. To identify modal parameters both Operational Poly- −6 MAX and Pulse operational software are used. It is noted 0 50 100 130 that the cross power spectras are required in Operationl (Hz) PolyMAX to identify the modal parameters, while for Cross PSD of point 1 w.r.t. point 1b the PULSE operational methods (FDD, EFDD, SSI), time series output data should be extracted. An example of the Figure 9: Sample cross PSD for portable excitation run. cross power spectra for base excitation run is presented in Figure 6. Natural Frequencies and damping ratios for the first six not excite well the torsional modes of the test article, as may modes extracted using FRF PolyMAX (reference), Operation be expected. PolyMAX and PULSE are tabulated in Tables 3 and 4, respectively. 3.3. Experimental Results-Single Point Excitation. Sample It can be seen that very good agreement exists between FRFs for similar points (points 1 and 5b) discussed in operational modal results and reference modal parameters. driven base case are shown in Figure 8. As discussed before, As it can be seen from Tables 3 and 4,nomodal parameters these FRFs will be processed by PolyMAX to extract modal could be extracted for mode 3 (1st torsional mode) using parameters which will be used as reference values for both operational Polymax and Operational Pulse methods. comparison with those obtained from operational testing. Also operational Polymax was not able to detect the fifth As it can be seen from Figure 8, FRF at point 1 contains mode (2nd torsional mode) while most PULSE operational torsional mode peaks as expected. methods were able to identify the modal parameters for the fifth mode. The FDD method does not provide damping As mentioned before, for Operational PolyMAX, the information. It should be noted that driven base excitation cross power spectra are used to identify the modal param- mainly excites bending modes in symmetric structures and eters. A sample of cross-power spectra for point excitation thus it is not surprising that operational methods which run is shown in Figure 9. are based on only output responses have difficulty to detect The identified natural frequencies and damping factors torsional modes. using output-only modal testing approaches for this case are The mode shapes for driven base runs using output-only provided in Tables 5 and 6,respectively. modal testing technique are shown in Figure 7. Mode shapes It can be seen that there is close agreement in the values obtained from PolyMAX using transmissibility, Operational of natural frequencies identified using the different output- PolyMAX, and PULSE output-only modal testing are close to only modal testing approaches with those of reference FRF each other except for the 3rd and 5th modes in Operational except for the first mode. The reason for this discrepancy for PolyMAX and 3rd mode in PULSE, which are missing. It the identified fundamental mode can be attributed to the fact should be noted that the translational base excitation could that the fundamental mode found by operational techniques log PSD (g /Hz) log FRF (g/N) 6 Advances in Acoustics and Vibration Table 2: Parameters for both driven base excitation and point excitation. Method PolyMAX Operational PolyMAX PULSE Bandwidth 128 Hz 128 Hz 128 Hz No. of frequency lines 1025 1025 8192 No. of averages 30 15 (portable) and 10 (base excitation) 22 Signal type 75% burst random 100% random 100% random Table 3: Natural frequencies of test article under base excitation. PULSE methods Transmissibility Operational Mode PolyMAX SSI PolyMAX f (Hz) FDD f (Hz) EFDD f (Hz) (Ref.) f (Hz) UPC f (Hz) PC f (Hz) CVA f (Hz) 1 3.45 3.45 3.46 3.46 3.46 3.48 3.35 2 21.38 21.50 21.53 21.52 21.53 21.54 21.49 3 —— — — — 28.15 — 4 60.5 60.49 60.52 60.45 60.45 60.43 60.39 5 87.75 87.77 87.77 87.77 87.69 87.84 — 6 120.6 120.4 120.5 120.4 120.5 120.5 120.29 Table 4: Damping ratios of test article under base excitation. PULSE methods Transmissibility Operational Mode SSI PolyMax (Ref.) ξ (%) PolyMax ξ (%) FDD ξ (%) EFDD ξ (%) UPC ξ (%) PC ξ (%) CVA ξ (%) 1 — 0.59 0.78 0.59 0.84 0.40 1.43 2 — 0.39 0.23 0.21 0.23 0.14 0.28 3 —— — — — 0.02 — 4 — 0.19 0.27 0.33 0.34 0.13 0.2 5 — 0.24 0.24 0.33 0.34 0.20 — 6 — 0.35 0.23 0.54 0.36 0.14 0.11 Table 5: Natural frequencies of test article under point excitation. PULSE methods FRF PolyMax (Ref.) Operational Mode SSI f (Hz) PolyMax f (Hz) FDD f (Hz) EFDD f (Hz) UPC f (Hz) PC (Hz) CVA f (Hz) 1 4.85 4.60 4.60 4.75 4.84 3.47 4.98 2 21.0 21.11 21.21 21.33 21.41 21.50 21.39 3 28.25 28.31 28.28 28.28 28.28 28.26 28.25 4 59.50 59.38 59.49 59.43 59.45 60.39 59.28 5 87.75 87.78 87.81 87.81 87.81 87.84 87.91 6 120.1 120.1 120.5 120.0 120.1 119.8 120.1 Table 6: Damping ratios of test article under point excitation. PULSE methods FRF PolyMax (Ref.) Operational Mode SSI ξ (%) PolyMax ξ (%) FDD ξ (%) EFDD ξ (%) UPC ξ (%) PC ξ (%) CVA ξ (%) 1 0.95 1.86 — 0.76 1.85 1.36 2.0 2 0.21 0.44 — 0.26 1.87 2.51 2.23 3 0.08 0.02 — 0.59 0.20 0.19 0.19 4 0.34 0.74 — 1.96 1.60 1.96 1.95 5 0.27 0.02 — 0.29 0.23 0.21 0.22 6 0.22 0.11 — 0.37 0.26 0.36 0.29 Advances in Acoustics and Vibration 7 1st mode 2nd mode 3rd mode 4th mode 5th mode 6th mode Figure 10: Mode shapes from output-only approaches for point excitation. Table 7: Properties of solar panel and its dimensions. 21 22 23 24 25 16 17 18 19 20 Material type Isotropic material 11 12 13 14 15 2 67 8 910 Modulus of elasticity, E 207 × 10 N/m 1 23 4 5 Length of the panel, a 6.48 m Width of the panel, b 1.32 m Thickness of the panel, h 0.0228 m Figure 11: Finite-element model of the solar panel. Density, ρ 0.1548 × 10 kg/m Poisson’s ratio, ν 0.3 tional and one rotational degrees of freedom) modeled in PATRAN/NASTRAN is shown in Figure 11. Material properties and dimensions of the solar panel is basically related to the assembled structure and shaker due are given in Table 7. As mentioned before, the real panels to the interaction between the test article and shaker. of actual satellites include stiffened elements which are very It can also be observed that there are significant differ- difficult to model and here for the sake of simplicity the ences in the damping information due to the fact that the effect of stiffeners is smeared in the plate. Thus the density damping values are quite low, and it is more difficult to is adjusted so that the resulting natural frequency is between estimate them precisely. the 3-4 Hz which is similar to that of real panel. The mode shapes identified for single-point excitation Rotational transient excitation is applied to the base of method using FRF curve fit are shown in Figure 10.Mode the solar panel, and the acceleration responses at all the nodes shapes identified by Operational PolyMAX and PULSE are evaluated. The damping ratio considered in this analysis output-only modal testing are identical with the FRF modes. is 2% for all modes. The transverse acceleration response at More details of the work and results are presented in [36]. node 5 is evaluated. It is noted that the excitation time is 6 seconds and time step of 0.0025 seconds is considered for dynamic response. 4. Simulation of Solar Panel The nodal transverse acceleration responses are trans- formed into a universal file format. This universal file is then As shown in Section 3, the output-only modal testing processed in PULSE output-only modal testing software to techniques could accurately identify the modal parameters extract the modal parameters of the solar panel. Here three of a plate structure under translational random driven different cases are investigated. Case 1: acceleration responses base excitation. The Communication Technology Satellite for all nodes are transferred into a universal file for further (CTS) solar panel appendage is another example structure processing by operational PULSE modal testing software; studied due to the simplicity of its boundary conditions and Case 2: acceleration responses of just 4 nodes (node 3, node availability of some of its properties and characteristics [34]. 5, node 23, and node 25) are transferred; Case 3: acceleration The panel is 6.48 m long and 1.32 m wide. The materials used response of just node 5 is transferred. to manufacture these panels are glass-fibre with a Kapton layer on it. With the nominal thickness of such a panel, the fundamental frequency becomes very low. Normally, in the 4.1. Extracted Modal Parameters Based on Case 1. The real solar panel structure of actual satellites, supports and extracted natural frequencies of the solar panel using stiffeners are used to increase the fundamental frequency of PULSE output-only modal testing techniques for Case 1 are the appendages to about 3 ∼ 4 Hz. In the present study tabulated in Table 8. There is close agreement between the for design and analysis simplicity, the effects of stiffeners are identified natural frequencies and those obtained from finite- smeared over the panel in order to reach the fundamental element modal analysis. The identified damping ratios for frequency of 3 to 4 Hz. The panel is under base rotational Case 1 are also given in Table 9. excitation due to the torque created by the forces exerted by It should be noted that the damping ratios cannot the thrusters. be computed using FDD method. However EFDD and The finite-element model of the panel consisting of 16 UPC methods were able to capture the damping factors. plate elements and 25 nodes (each node has two transla- The reason that modal parameters could not be found by 8 Advances in Acoustics and Vibration Table 8: Identified natural frequencies for Case 1. Output-only modal testing methods Mode Finite-element modal analysis f (Hz) (Reference) SSI FDD f (Hz) EFDD f (Hz) UPC f (Hz) PC f (Hz) CVA f (Hz) 1 3.125 — 3.30 3.0 3.25 3.24 2 21.48 21.65 21.76 — 21.75 21.75 3 68.75 68.25 68.72 — 68.78 68.22 4 140.6 140.2 141.6 — 142.4 140.25 Table 9: Identified damping ratio for Case 1. Output-only PULSE methods Mode Finite-element modal analysis ξ (%) (Reference) SSI FDD ξ (%) EFDD ξ (%) UPC ξ (%) PC ξ (%) CVA ξ (%) 1 — — 2.1 2.4 3.3 2.0 2 — 1.98 2.0 — 2.3 2.0 3 — 1.99 2.0 — 1.96 2.0 4 — 1.88 2.0 — 2.4 2.0 Table 10: Identified natural frequencies for Case 2. Output-only modal testing methods Mode Finite-element modal analysis f (Hz) (Reference) SSI FDD f (Hz) EFDD f (Hz) UPC f (Hz) PC f (Hz) CVA f (Hz) 1 3.125 — 3.30 3.30 3.30 3.24 2 21.68 21.69 21.71 21.14 21.70 21.75 3 68.36 68.26 68.22 — 68.20 68.22 4 140.8 140.5 140.1 — 140.2 140.25 Table 11: Identified damping ratio for Case 2. Output-only PULSE methods Mode Finite-element modal analysis ξ (%) (Reference) SSI FDD ξ (%) EFDD ξ (%) UPC ξ (%) PC ξ (%) CVA ξ (%) 1 — — 2.0 2.0 2.7 2.0 2 — 1.85 2.0 1.9 2.0 2.0 3 — 1.80 2.0 — 2.0 2.0 4 — 1.60 2.0 — 2.0 2.0 Table 12: Identified natural frequencies for Case 3. Output-only modal testing methods Mode Finite-element modal analysis f (Hz) (Reference) SSI FDD f (Hz) EFDD f (Hz) UPC f (Hz) PC f (Hz) CVA f (Hz) 1 3.3 — 3.30 3.30 3.30 3.24 2 21.68 — 21.75 21.64 21.74 21.75 3 68.55 — 68.22 68.90 68.22 68.22 4 140.6 — 140.3 139.4 140.3 140.25 Table 13: Identified damping ratio for Case 3. Output-only PULSE methods Mode Finite-element modal analysis ξ (%) (Reference) SSI FDD ξ (%) EFDD ξ (%) UPC ξ (%) PC ξ (%) CVA ξ (%) 1 — — 2.0 2.0 2.0 2.0 2 — — 2.0 1.8 2.0 2.0 3 — — 2.0 1.6 2.0 2.0 4 — — 2.0 1.8 1.9 2.0 Advances in Acoustics and Vibration 9 some methods may be attributed to the fact that only few been conducted on a simple plate structure under different seconds, that is, 4 to 5 seconds of nodal responses have operating excitations to identify its modal parameters using been used to identify modal parameters. The real damping different output-only modal testing approaches. Finally the ratios can easily be identified by comparing the damping solar panel of the Communication Technology Satellite ratios identified using different methods. It should also be (CTS) has been simulated under representative operating noted that unlike UPC and CVA approaches, PC approach excitation provided by the spacecraft thrusters to find is not able to estimate modal parameters. Although these its modal parameters using different output-only modal three methods are similar, their algorithms have different testing techniques. The results are also compared with those weighting functions. CVA usually performs better for low- obtained using traditional modal analysis. It is aimed to order excited modes and may not provide information for demonstrate the potential application of output-only testing higher-order modes if they are not properly excited. in satellite appendages. It is concluded that output-only Mode shapes identified using output-only modal testing modal testing based on SSI techniques could capture the methods match very well with simulation results using modal parameters well using only a few nodal responses, classical FRF-based modal analysis approach. which is very important in satellite and space applications. 4.2. Extracted Modal Parameters Based on Case 2. The References extracted natural frequencies of the solar panel using PULSE [1] J. S. Bendat and A. G. Piersol, Engineering Applications of output-only modal testing techniques for Case 2 are given in Correlation and Spectral Analysis, John Wiley & Sons, New Table 10. The identified damping ratios for Case 2 are also York, NY, USA, 1993. provided in Table 11. [2] S. R. Ibrahim and E. C. 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