Assessment of the Frequency Domain Decomposition Method: Comparison of Operational and Classical Modal Analysis Results

Transcription

1 Assessment of the Frequency Domain Decomposition Method: Comparison of Operational and Classical Modal Analysis Results Ales KUYUMCUOGLU Arceli A. S., Research & Development Center, Istanbul, Turey Prof. Dr. Kenan Yuce SANLITURK Istanbul Technical University, Mechanical Engineering Department, Istanbul, Turey ABSTRACT: Operational Modal Analysis (OMA) is originally used for extracting modal parameters (natural frequency, mode shape and damping) of mainly civil engineering structures. However, in recent years it is also becoming popular for modal analysis of mechanical structures. The main advantage of the method is that neither artificial excitation needs to be applied to the structure nor force signal(s) is/are to be measured. The parameter estimation is based upon the response signals only, thereby minimizing the cost for modal testing. One of the techniques that is often used for OMA is called Frequency Domain Decomposition (FDD) method which is a non-parametric technique. This study addresses identifying modal parameters of some relatively simple mechanical systems under operating conditions using FDD method. The FDD procedure requires auto- and cross-spectrums between measured response data. The success of the FDD technique is examined using both numerical simulations as well as experimental studies. Various types of excitations have been applied in order to assess the advantages and limitations of the FDD method. The results of OMA are compared to those of classical modal testing. It is shown that OMA can yield satisfactory results if the underlying assumptions behind this approach, the most significant one being that the excitation is stationary white noise, are satisfied. However, it is also seen that the accuracy of the method starts to decline when a structure is subjected to complicated excitations as in rotating machinery. 1 INTRODUCTION Classical modal analysis methods can be applied to those cases where the input forces can be measured. However, if it is not possible or practical to measure the input forces, these methods lose their applicability. In some cases, it is impossible or very difficult to excite structures using shaers or impact hammers. These conditions are usually faced when engineers deal with very large structures such as bridges, buildings and large mechanical systems. Also, in some other cases, the level of excitation forces might be so large that they cannot be measured by available sensors. The measured responses from linear systems contains: i) Responses due to the input forces ii) Environmental noises iii) Measurement Noises. In classical modal analysis, measured responses are supposed to depend only on the input forces which are generated and applied in a controlled manner. Therefore, to minimize the other effects, input forces must be sufficiently strong. This requirement in classical modal analysis brings additional cost, especially in terms of hardware. However, Operational Modal Analysis (OMA) techniques do not require any artificial excitation to be applied to the structure in order to determine the dynamic properties of systems under actual woring conditions. As a result, if OMA approach is applicable, the whole modal analysis process can become relatively easier and more economical. Large civil engineering structures are often excited by natural loads under operating conditions that cannot easily be controlled, for instance wave loads (offshore structures), wind loads (buildings) or traffic loads (bridges). A similar argument can also be made for most

2 machines under operating conditions. They are also excited by natural sources such as noise from bearings or vibrations from other machines nearby or internally generated random forces. For example, a typical vehicle on the road in operating conditions is excited by the road, wind and the engine in very complicated manner. Another advantage of OMA is that the dynamic properties of systems are estimated under actual operating conditions. These dynamic properties are the system s natural frequencies, damping and mode shapes. In practice, many forces have some randomness and contain some characteristics of noise. It is nown that this type of forces can excite all the modes of structures. The most important assumption in most Operational Modal Analysis techniques is that the excitation to the system is stationary white noise. This white noise is able to excite the structure in all frequencies and has a flat spectrum. Therefore, the spectrum of white noise can be used as an appropriate excitation for any system. Based on this fundamental assumption, Operational Modal Analysis methods are able to determine the dynamic properties of systems by using the measured vibration levels under operating conditions without the need for measuring the excitation forces. For the past 15 years or so, different OMA methods have been proposed by researchers to determine the dynamic characteristics of systems and these methods are applied to many different situations. In 1995, G.H. James [1] used Natural Excitation Techniques (NExT) to determine the physical parameters of the systems under woring conditions. R.Brincer [2] introduced Frequency Domain Decomposition Technique and later Enhanced Frequency Domain Decomposition (EFDD) Technique. At the same time, different time domain methods [3] were also developed and published in the literature, for example, Stochastic Subspace Identification (SSI) method [4]. Among all the proposed OMA techniques, FDD and SSI are the most well-nown and used methods. FDD uses Complex Mode Indicator Function [5] approach for processing the measured responses and then determines the natural frequencies and mode shapes of the structure with simple Pea Picing technique. Therefore, the main advantage of this method is that it is relatively easy to implement and use as it requires relatively fewer complex calculations. On the other hand, as stated, SSI method is a time-domain method and it is more complex and time consuming when compared to FDD. For detail information interested reader may refer to [3], [4], [6], [7], [8]. The FDD method is used in this paper. This method is based on the formulation of the relationship between the input and the output power spectral density functions, and the modal parameters are estimated via Singular Value Decomposition (SVD) of the cross-spectrum matrix. Singular values contain frequency and damping information and singular vectors contain mode shape information. The aim of this study is to determine the advantages and the limitations of FDD technique and to investigate the applicability of this method in practice. The main reason for choosing this method is its simplicity, requiring less computation and practical applicability being easier than other methods. This paper is organized as follows: First, the theory behind the FDD method is described briefly. Then, some numerical simulations are made in order to assess the performance of the FDD method. In these simulations, dynamic forces are applied to the test structure and structural responses are calculated at all selected coordinates. However, the excitation forces in this paper are considered unnown for OMA purposes. In numerical simulations, finite element model of a simple plate structure is developed and used. The modal parameters obtained from simulated OMA are compared with those obtained via traditional approach. After that, experimental studies are presented. In the experimental studies, modal shaers or modal hammers are used to excite the plate structure but excitation forces are not utilized during OMA. Piezoelectric accelerometers are used to measure the system responses. Then, the measured data are processed using a MATLAB code developed for FDD analysis so as to determine the system modal parameters. These modal parameters are then compared with those obtained via classical modal testing approach that required excitation forces to be measured. After the plate structure, a practical applicability of FDD method is assessed by applying this approach to a rotating machinery under operating conditions. The results are again compared with those obtained via traditional approach. Finally, some concluding remars are given about the accuracy, reliability and applicability of FDD method.

3 2 THEORY Frequency Domain Decomposition technique determines the modes by applying the method of Singular Values Decomposition to the spectral density matrices. This decomposition corresponds to a single degree of freedom identification of the system for each singular value. The relationship between the input x(t) and the output y(t) of a linear system can be written as: [9] T [ G ( )] [ H ( ω) ] [ G ( ω) ] [ H ( ω) ] ω = (1) mxm Where [ (ω)] mxr xx rxr G xx is the input power spectrum matrix that is constant in case of stationary zero mean white noise input and this is expressed in terms of constant C. Also r indicates the number of inputs. [ G (ω)] is the output power spectrum matrix. m indicates the number of the responses. [ H (ω)] is the Frequency Response Function (FRF) matrix. and T superscripts refers to complex conjugate and transpose, respectively. As seen in Equation (1), [ G (ω)] is very sensitive to the input constant C. The next equations and single degree of freedom system identification are based on the assumption that the input power spectrum is represented by a constant value. As shown in Eq. (2) the FRF matrix can be expressed in the form of poles and residues as in classical modal analysis. [ H ( )] [ R ] [ R ] m + = 1 λ mxr ω = λ (2) λ = σ + (3) d Here, m is the total number of modes, λ is the pole of the mode, σ modal damping (decay constant), ωd is the damped natural frequency of the mode which can be expressed as Eq. (4). ω d = ω ς (4) σ ς = (5) ωo In Eq. (5), ς is the damping ratio for the mode and ω o is the undamped natural frequency for the mode. [R] in Eq.(2) is given as: = ψ γ (6) In Eq. (6), ψ is the mode shape of the mode and γ is the modal participation vector of the mode. If the input to the system is considered as the white noise, power spectral density matrix can be taen as a constant matrix ( G xx ( jw) = C ) and Eq. (1) taes the following form.

4 m m [ ] [ ] [ ] T R R [ Rs ] [ Rs ] G ( ) = + C + H ω = 1 s= 1 λ λ λs λ (7) s Where superscript H represents the complex conjugate transpose. With using the expression [ G ] in Eq. (1) and with some mathematical operations, output power spectrum matrix can be written as: [2] m [ ] [ A ] [ A ] [ B ] [ B ] G ( ω) = Where, [ ] = 1 λ λ λ A is the residue matrix of [ ] and can be expressed as: λ (8) G matrix. This matrix is a Hermitian matrix [ A ] = [ R ] C H [ R ] [ R ] m T s s + s= 1 λ λ λs λ s (9) The contribution of the residue for the mode has the following expression: [ A ] [ R ] C[ R ] H = (10) 2σ Here σ is the negative of the real part of the pole as λ = σ +. If lightly damped model is considered, modal contribution matrix becomes proportional to the mode shape vector and can be written as: T T T [ A ] = [ R ] C[ R ] = { ψ }{ γ } C{ γ }{ ψ } = d { ψ }{ ψ } T lim (11) damping light Where d is a scalar constant. At a certain frequency (ω ) only a limited number of modes will contribute significantly, typically one or two modes. These modes are indicated by Sub (ω) As a result, for the lightly damped structures, output spectral density matrix is expressed as the final form in Eq. (12) [2]. [ G ( )] = T H d ψ ψ d ψ ψ ω + (12) λ λ Sub( w) This final form of the matrix is decomposed to singular values and singular vectors using the singular value decomposition technique. This decomposition is performed to determine modal parameters of the system.

5 3 NUMERICAL SIMULATIONS In order to assess the performance and the applicability of FDD technique in practice, some numerical simulations are performed first. In these simulations, the system response is calculated and these results are used as input to the FDD method for operational modal analysis. In other words, measured dataa are simulated in a numerical environment. Finite element method is used here to create numerical models of the test structure which has dimensions of 450 mm x 200 mm and a thicness of 3 mm. Material is steel. The finite element model of the plate is created and shown in Figure 1. Then, the natural frequencies and mode shapes are calculated using classical modal analysis. After that, the structure is excited using a specified force and the forced response of the system is calculated. The resulting accelerations over the surface of the plate at many locations are recorded for further operational modal analysiss using FDD. The results of the OMA are then compared to those obtained via traditional modal analysis. Figure 1: Finite element model of the plate structure In this simulation, a swept sine force is applied to a point in the numerical model of the plate shown in Figure 1. For OMA purposes, this excitation force is considered unnown and forced responses of the system are recorded at 50 co-ordinates. A typical response spectrum in terms of acceleration is plotted in Figure 2. Figure 2: Frequency dependent acceleration spectrum

6 As can be seen from Figure 2, this plate is analyzed up to 400 Hz. In this simulation, all the accelerations data on the structure are taen simultaneously. This simulatess the measurement situation where the response levels are recorded at desired co-ordinates simultaneously using as many sensors as required. In such cases, there is no need to use a reference point. A matrix with a dimension 50 x 50, containing power and cross spectrums is obtained using the acceleration data taen from 50 locations of the structure. Natural frequencies of the structure are obtained using singular value decomposition of the spectral density matrix. A plot of singular values is presented in Figure 3 as function of frequency. In this graph, there are seen 50 curves as accelerations are measured at 50 co-ordinates on the structure. Natural frequencies are identified from this figure considering the frequencies with highest singular values. It should be noted in Figure 3 that the first singular values are much larger than the other singular values, indicating that there are no double modes. As seen in Figure 3, this plate has 4 natural frequencies within 0 to 400 Hz. Figure 3: Singular Value Plot for plate structure The corresponding mode shape is obtained via singular value decomposition of the spectral density matrix at a natural frequency. In figure 4a and 4b there is shown the comparison of first two natural frequencies and mode shapes that are obtained via classical and operational modal analysis. a)

7 b) Figure 4a 4b: First two modes obtained via Classical Modal Analysis and OMA It is also aimed to assess the performance of the FDD method in terms of damping estimation. For this purpose, a nown level of (i.e. 1%) structural damping is specified in the FE model. Then, the response levels in terms of acceleration are recorded and the response spectrums are analyzed using the half-power bandwidth method [10]. The actual and the estimated damping levels are listed in Table 1. It is seen that the estimated damping level in this OMA analysis deviates from the actual levels by about 5%. Although this level of deviation from the actual damping level may be considered acceptable, it is still significant since the OMA is simulated here under ideal conditions. Table 1: Estimated damping levels In this section, both classical and operational modal analyses are applied to a plate structure and the corresponding natural frequencies, mode shapes and damping ratios are obtained. It is seen that the modal parameters obtained from FDD technique are very close to those obtained from classical modal analysis. However, it should be restated here that the OMA simulation here represents ideal conditions. In the next section, the performance of the FDD is examined in real situation using measured data. 4 EXPERIMENTAL STUDY After validating FDD method using numerical simulations, this section aims to assess the performance and the applicability of this OMA method in practice using experimental data. Two experimental test cases are performed here. In the first case, a plate structure is used. The

8 second case, however, is using an actual rotating machinery in order to assess the capabilities and limitations of FDD method in practice. First of all, classical modal analyses are performed on the experimental structures. For this purpose, Frequency Responsee Functions (FRFs) are measured using either hammer or shaer testing. Then measured FRFs are analyzed using classical modal analysis approach so as to have reference modal parameters to compare with OMA results. Shaer or modal hammer is also used for operational modal analyses in this study. However, the forces applied to the structure are not measured for OMA purposes. Instead, only the resulting vibrations are recorded. The first experimental study is carried out on a steel plate structure which has dimensions of 450 mm x 200 mm x 3 mm. As stated, FRFs are measured using modal hammer as shown in Figure 5a. 3 accelerometers are used simultaneously and a total of 50 FRFs are measured for the purpose of classical modal analysis. A typical measured FRF is presented in Figure 6. Then, a random white noise type of excitation is applied to this plate using a shaer as shown in Figure 5b. However, this force is not measured or recorded for the purpose of OMA. Instead, only the resulting vibrations are recorded. Again 3 accelerometers are used simultaneously during data collection for OMA and vibration data are collected at 50 co-ordinates again. However, the location of one of the accelerometers is ept fixed during these measurements, the response measured by this accelerometer being used as a reference for phase determination. a) b) Figure 5a 5b: Plate structure excited using a modal hammer and a shaer Figure 6: A Measured FRF on a plate structure

9 Power and cross spectrums are calculated using 4 x 4 matrices. The magnitude and phase plots of a typical cross spectrum are given in Figure 7a and 7b, respectively. This cross spectrum is obtained by using the response measurements at two points on the plate. Then, the natural frequencies of the system are identified via singular value decomposition of the spectral density matrix as shown in Figure 8. Each pea in this graph identifies a natural frequency and the corresponding mode shape is obtained via singular value decomposition of the spectral density matrix at a natural frequency. a) b) Figure 7a 7b: Plate structure: Measured Cross Spectrum Magnitude & Phase

10 Figure 8: Plate structure: Singular Value Graph In this study the plate structure is analyzed up to 800 Hz with a 0.5Hz frequency resolution. 9 modes are identified up to 800 Hz. The natural frequencies determined via classical and operational modal analyses are listed and compared in Table 2. Some of the corresponding mode shapes are compared in Figure 9. It is seen that the natural frequencies identified using two methods are fairly close to each other. Also, the mode shapes are found to be quite compatible. The damping level for the bare steel plate is very low, hence it was not appropriate to compare the damping levels. Table 2: Plate Natural Frequencies

11 a) b) Figure 9a 9b: First two mode shapes obtained via Classical Modal Analysis and OMA FDD method s practical applicability is also tested here. The dynamic properties of a rotating machine under actual operating conditions are determined via OMA and the results are compared to those obtained using traditional modal analysis method. In this case, the vibration data are acquired from one of the side panels of the rotating machine. Figure 10 shows the result of singular value decomposition of the spectral density matrix.

12 Figure 10: Singular Value Plot for a Rotating Machine In this study, the particularr rotating machine is running at 690 rpm which corresponds to 11.5 Hz, so in frequency spectrums peas will appear at rotation frequency 11.5 Hz and its harmonics. This situation is one of the major disadvantages of operational modal analysis, as it leads to the need for distinguishing between the real physical modes of the structure and the virtual modes. If the rotational frequency or its harmonics are very close to the natural frequencies of the structure, it will be very difficult to determine modal parameters at those frequencies. In this study, the rotational frequency of the machine under test was measured. Therefore, it was possible to identify the physical modes and the corresponding natural frequencies via OMA. The identified natural frequencies are listed and compared to those estimated from classical approach in Table 3. The corresponding mode shapes, which are quite compatible, are shown in Figure 11. Table 3: Natural Frequencies of the side panel of a rotating machine.

13 a) b) Figure 11a 11b: 4 th and 7 th modes obtained using Classical Modal Analysis and OMA 5 CONCLUDING REMARKS The accuracy and the applicability of the FDD method is examined using both numerical and experimental case studies. The results show that OMA can yield satisfactory answers if the underlying assumptions behind this approach, the most significant one being that the excitation is stationary white noise, are satisfied. Mode shapes obtained from operational modal analysis are not scaled as the excitation forces are not nown. It is found that the accuracy of the method starts to decline when a structure is subjected to complicated excitations as in rotating machinery. In operating conditions, if the main excitation frequency and its harmonics are quite close to the one or more natural frequencies of the structure, it appears that it will be difficult to determine modal parameters at those frequencies. REFERENCES [1] G.H.James III, T.G.Carne and J.P.Laufer, The Natural Excitation Technique (NExT) for modal parameter extraction from operating structures, The International Journal of Analytical and Experimental Modal Analysiss 10(4), pp

14 [2] R.Brincer, L.Zhang and P.Andersen, "Modal Identification from Ambient Responses using Frequency Domain Decomposition", Proceedings Of the 18th International Modal Analysis Conference. San Antonio, Texas. [3] P.Andersen, R.Brincer, B.Peeters, G.De Roec, L.Hermans, Comparison of System Identification Methods Using Ambient Bridge Test Data, Proceedings of 17th International Modal Analysis Conference (IMAC), Kissimee, Florida. [4] B.Peeters and G.De Roec, Reference-based stochastic subspace identification for output-only modal analysis, Mechanical Systems and Signal Processing, Volume 13, Issue 6, pp [5] A. W. Phillips, R. J. Allemang, and W. A. Fladung, "The Complex Mode Indicator Function (CMIF) as a parameter estimation method.",international Modal Analysis Conference. [6] H.Herlufsen, N.Møller, "Operational Modal Analysis of a Wind Turbine Wing using Acoustical Excitation", Brüel&Kjær Application Note, Denmar. [7] R.Brincer, P.Andersen and N.Møller, "Ambient Response Analysis of the Heritage Court Tower Building Structure", Proceedings of the 18 th International Modal Analysis Conference, San Antonio, Texas [8] B.Peeters, G.De Roec, Stochastic system identification for operational modal analysis: a review, Journal of Dynamic Systems Measurement and Control (123), pp [9] Bendat, Julius S and Allan G. Piersol, Random Data, Analysis and Measurement Procedures, New Yor, USA [10] Ewins, D. J., Modal Testing: Theory, Practice and Application. Research Studies Press Ltd., Hertfordshire, England