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Abstract

In this research, we have proposed a methodology for experimental identification of modal parameters based on measurement of the frequency responses of structures with complex geometries and performed an overall investigation of structural behavior on a funnel-shaped inlet of magnetic resonance tomograph. Several identification methods are implemented and compared: complex exponential, least-squares complex exponential, and polyreference least-squares complex exponential. We have implemented the modal parameter identification methodology within our own graphical user interface supported by MATLAB to create an independent tool for modal analysis. The estimation methods are compared and the comparison results are summarized showing based on tabular representation and stabilization diagrams significant advantage of the proposed methodology for determining eigenfrequencies, damping coefficients, mode shapes, and residues for complex structures investigated in broad band of frequencies. Runtime for the execution of algorithms vary depending on the applied method, assumed order of the model used for estimation, and the number of measurements, that is, inputs and outputs.

Keywords

Experimental modal analysis identification of modal parameters stabilization diagrams frequency response functions magnetic resonance tomograph funnel

Introduction

Despite powerful simulation tools, modal analysis still remains an indispensable method for reliable investigation of the structural behavior. Numerical structural models like the finite element (FE)-based ones can, in many cases, capture the structural behavior to a satisfactory extent of reliability, nevertheless some information might still be missing or cannot be accurately represented in such models. This is especially valid for material damping properties, which can still not be captured accurately enough merely by a numeric simulation, where they can be implemented only as assumed values. Modal analysis can contribute to FE model improvement in such cases and it brings a number of other advantages such as machine diagnosis, trouble-shooting problems, health monitoring, or other fields.

Development of tools which are used in estimation of modal parameters based on experimental measurements dates as early as development of some numerical algorithms, which were originally developed for solving some other problems.¹ Yet, a first major breakthrough in modal estimators appears with development of the maximum likelihood estimator in combination with the least-squares complex frequency domain estimator.^2,3 An algorithm based on the polyreference least-squares estimator in the frequency domain has been proposed by Guillaume et al.⁴ and Peeters et al.⁵ Recently, Khader⁶ employed the unified matrix polynomial approach for modal parameter estimation. Previous works mainly consider just a single approach to identification of modal parameters. Regarding the implementation of the proposed algorithms, they are mainly applied on geometrically simple structures, such as flexible beams.^6–8 In this article, we have proposed a procedure for modal parameter estimation which involves several modal estimation algorithms. In addition, we have tested not only our approach with geometrically simple structures (performed test example on the plate structure is here omitted, due to space limitation) but also the procedure with complex flexible three-dimensional (3D) shell structures, like the funnel-shaped inlet of a magnetic resonance tomograph (MRT).

In implementation of the techniques for estimation of modal parameters based on measurement data, one is often confronted with requirement of using expensive commercial tools for modal analysis. Another arising problem is that usually standard techniques based on mere implementation of fast Fourier transform (FFT) are not satisfactory in determining modal parameters if even slight deviation from assumed linear properties of structures under consideration is present or if structures are not lightly damped. In such cases, it is very difficult, almost impossible to clearly distinguish the picks from the frequency response. To overcome these problems, we have contributed in this article a methodology for reliable estimation of modal parameters, which is based on conducting several steps, thorough out implementation of required algorithms in order to determine modal parameters. We have developed our own tool, which is independent of any commercial platform for experimental modal analysis, and therefore represents a reliable, but inexpensive, way for experimental modal analysis of demanding structures. Our further contribution is development of a tool based on a MATLAB graphical user interface (GUI), which enables interactive manipulating of measurement data and implementation of parameter identification algorithms. The tool is advantageous for wide implementation in academia, research, and so on. Our modal parameter estimation methodology involves several estimation steps. The procedure begins with estimation of the mode indicator functions (MIFs). Subsequently, we have implemented, tested, and compared several modal parameter estimation algorithms. Due to limited space, we have presented here only selected algorithms and investigation results, although different others are also included in our tool (e.g. peak-picking method for estimation of the eigenfrequencies). Decision about relevant (resonant) frequencies is made based on stabilization diagrams, which are automatically generated as the output of our tool. Furthermore, detailed algorithms for complex exponential (CE), least-squares complex exponential (LSCE), polyreference least-squares complex exponential (PRCE), and PRCE in frequency domain are presented. The feasibility of the proposed methodology for the identification of the modal parameters of structures with complex geometries is documented on an example of a funnel-shaped structure, the inlet of the MRT. Obtained results include estimated eigenfrequencies, damping coefficients, and residues for different implemented algorithms. They are systematically represented by comparison tables and stabilization diagrams. In addition, the visualization of characteristic mode shapes of the funnel is presented.

The funnel-shaped MRT inlet investigated in this article is characterized by complex geometry, and therefore, the characterization of the modal parameters requires a careful investigation under implementation of the identification methods in order to produce reliable statements about the eigenfrequencies, mode shapes, and damping coefficients. For that purpose, a detailed investigation was conducted in this article: several identification methods presented in Appendix 1 were implemented and their results were evaluated and compared. Experimental modal analysis is based on estimation of the frequency response functions (FRF) - the transfer functions between measured outputs and inputs of a structure, that is, on identification of modal parameters from the FRF. Complexity of the investigated structure can significantly influence the procedure for identification of its modal parameters. Due to lack or inaccessibility of some functionalities in available commercial tools for experimental modal analysis, the authors of this article have developed their own tool for the identification of modal parameters based on the experimental measurement of the excitations and responses of structures and implemented it within a MATLAB-based GUI. Raw measurement data are read into GUI in universal file format. Required analyses are performed within GUI according to the execution procedure shown in the chart of Figure 1. Corresponding results can be called and graphically represented in time and frequency domain, as shown by examples in Figures 2 and 3. In subsequent sections, the results of the tool implementation for the modal analysis of the MRT funnel inlet are presented.

Figure 1.

Flow chart of the frequency response and modal parameter identification implemented in GUI.

Figure 2.

GUI for experimental modal analysis: representation of raw measurement data in time domain.

Figure 3.

GUI for experimental modal analysis: FRF estimation.

Experimental determination of the MRT funnel frequency responses

The structure under investigation is the funnel-shaped inlet of an MRT represented in Figure 4. Due to rapidly changing magnetic field, this diagnostic device is characterized by high noise emission, where the acoustic air pressure ranges from 60–100 dB and can therefore be very unpleasant, and if longer and frequently exposed, even harmful for the patients undergoing diagnostic treatment. If the frequency of a periodic excitation is close to one of the eigenfrequencies of the funnel, the vibration amplitudes will drastically increase, which in turn causes increased noise level. The MRT funnel is made of acrylonitrile–butadiene–styrene (ABS) plastics and it weighs about 15 kg. To avoid strong resonant vibration and noise effects, active vibration control^9–12 can be applied using piezoelectric actuator–sensor patches which are glued to the surface of the funnel and connected by cables via Bayonet Neill–Concelman (BNC) connectors with appropriate AD and DA converters. In addition, the funnel structure is reinforced by aluminum profiles attached to its back surface.

Figure 4.

MRT with funnel-shaped inlet (source: Siemens AG).

For efficient model-based active control, reliable modeling in early development phases before the controller design plays an important role. Modeling can be performed using numerical FE analysis.^10,13,14 Yet, it is often difficult and sometimes even not possible to precisely model by the FE approach all important influences like material properties of the funnel (especially damping properties), aluminum reinforcements, piezo-patches, and the cables, which have significant influence to modal parameters. Experimental modal analysis has therefore a great importance. FE models of the funnel including piezoelectric patches^10,12–14 are primarily used to determine from numerical modal analysis the critical eigenfrequencies, which are required in the controller design. In order to provide boundary conditions comparable with the FE analysis, for the purpose of the experimental modal analysis, the funnel was hanged using four elastic springs, as shown in Figure 5. The cables with connectors—interface between the piezoelectric patches and AD–DA converters for the control purposes—are also hanged in such a way that they exert minimal or possibly no impact on the funnel.

Figure 5.

Experimental setup with the funnel hanged on four steel springs (positions 1 and 2 denote transducer positions).

Linearity test

In the linearity test, the funnel is excited at point 1 (Figure 5) using the shaker Brüel & Kjær (B&K) 4809. The force produced by the shaker is measured at the same point by the force transducer B&K 8230 placed at the top of the stinger connected with the shaker. The response of the funnel is measured by the one-dimensional (1D) accelerometer B&K 4507B at two different positions (1 and 2 in Figure 5), according to Table 1. Measurements 1–3 are performed with accelerometer applied at drive point 1 (collocated with the shaker). Direction of the acceleration measurement at point 1 coincides with the excitation direction, which are in this case both vertical. Excitation is a pseudo random signal obtained through averaging of 10 blocks. Obtained frequency responses are shown in Figure 6 (also see Figure 7).

Table 1.

Preview of measurements in the linearity test with transducer positions 1 and 2 (red in Figure 5).

No.	Pos. force	Pos. acceleration	Excitation (V RMS)
1	1	1	1
2	1	1	0.8
3	1	1	0.3
4	1	2	0.6
5	1	2	1

Figure 6.

Linearity test for the funnel: measurements 1–3, sensing at drive point. Measurements: 1 (black) 1.0 V RMS, 2 (red) 0.8 V RMS, and 3 (blue) 0.3 V RMS.

Figure 7.

Zoomed portion of Figure 6.

Measurements 4 and 5 are performed with accelerometer placed at position 2, further from the excitation point 1. Both the excitation and the response (acceleration measurement) directions are vertical. Obtained frequency responses are represented in Figure 8. Good agreement can be observed. It should be noted that the signal level of the accelerometer for measurement 4 indicates a measurement under range. This can be avoided by increasing the excitation signal level to 1.0 V RMS, which improves the signal-to-noise ratio (Figure 9).

Figure 8.

Linearity test for the funnel: measurements: 4 (black) and 5 (magenta); sensing far from excitation point.

Figure 9.

Zoomed portion of Figure 8.

Single-input-multiple-output versus multiple-input-multiple-output measurement and effect of the accelerometer mass

In order to obtain complete overview of the structural behavior of the funnel, measurements of the structural response are performed in three perpendicular directions using a 3D accelerometer. Single-input-multiple-output (SIMO) and multiple-input-multiple-output (MIMO) measurement cases were investigated in detail. Frequency responses for MIMO $H_{1}$ measurements were performed according to Maia and Silva.¹⁵ Implementation of the MIMO measurement with two shakers with perpendicular directions of excitation could contribute on one hand to a uniformer energy distribution, supposedly excitation of all modes and lower signal levels, but on the other hand, it has resulted in lower partial coherence (correlation coefficient, which describes possible causal relation between one output and all inputs¹²) about resonant frequencies. Therefore, for further investigations, a SIMO structural response measurement is performed using a rowing 3D accelerometer B&K 4524B in combination with the single shaker excitation. Due to adopted SIMO measurement procedure, only the investigation of the sensor mass effect could have relevance. The accelerometer used for the measurement has the mass which is approximately only 0.03% of the funnel mass, and therefore, it can be expected that it has a negligible influence to frequency response measurements. Comparison of the frequency responses for measurements 1 and 5 from Table 1 confirms this. For the first pronounced resonances, Figure 10 also confirms this, since only a negligible discrepancy between the resonant frequencies is present. Also, a classical test for investigation of the sensor mass influence, similarly as proposed in Baharin and Abdul Rahman,¹⁶ has shown that the accelerometer mass can be neglected. Two frequency responses were measured, both with collocated input and output, but in the measurement of the second frequency response, additional sensor was applied at the neighboring degree of freedom, in this case ca. 10 cm from the drive point. Comparison of the frequency responses in Figure 11 shows that they are almost identical, which confirms that the influence of the sensor mass can be neglected.

Figure 10.

Overlaying of two frequency responses with different sensor positions on the funnel shows just a negligible shift of the resonant frequencies.

Figure 11.

SISO measurement test for investigation of the sensor mass influence. Black: measurement without neighboring additional sensor, magenta: measurement with additional sensor; below: zoomed portion.

Determination of the frequency responses

As described above, for the SIMO measurements, the force transducer B&K 8230 and the 3D accelerometer B&K 4524B are used. In the frequency range up to 200 Hz, 800 spectral lines cover quite a large number of eigenfrequencies. The measurements are performed using B&K PULSE system, at predefined 107 points of the mesh represented in Figure 12, which define the positions of the 3D accelerometer. Since the transducer measures acceleration in three perpendicular directions, totally 321 measurement sets are obtained. Based on exported measurements, a diagram of the overlaid frequency responses with corresponding coherence (Figures 13 and 14) is created in MATLAB.

Figure 12.

PULSE model of the funnel with shaker and accelerometer positions; arrows represent measurement directions: x (red), y (blue), and z (black).

Figure 13.

Overlaid frequency responses of 321 measurement results for 107 points on the funnel with positions in Figure 12.

Figure 14.

Overlaid coherence diagrams of 321 measurement results for 107 points on the funnel with positions in Figure 12.

From Figures 13 and 14, it can be seen that first clearly separated eigenfrequencies appear in the frequency range up to 60 Hz. The range with bad coherence (under 10 Hz) is present due to shaker specification. Still it can be observed that bad coherence appears also for some frequencies above 10 Hz, but it occurs mainly at anti-resonant frequencies of lower importance. At some frequencies below 5 Hz, double modes could be observed. They could represent either rigid body modes or elastic modes, but due to strong corruption by the measurement noise, the coherence corresponding to those modes is very bad and it does not allow a clear statement about them.

Identification of the MRT funnel modal parameters

In order to identify the modal parameters of the funnel, a detailed analysis and comparison of several estimation methods presented in Appendix 1 has been performed. Main results of these investigations are presented subsequently. Due to space limitation, only selected results are represented by appropriate diagrams or tables, others are explained and commented. Selected results of the implementation of methods for modal parameter identification are presented in terms of the stabilization diagrams.

Stabilization diagrams

Presented time-domain methods CE, LSCE, PRCE, and the polyreference least-squares complex frequency domain (PLSFD) method are based on the assumed order n of underlying models. The order is related to calculated number of eigenfrequencies. Due to noisy measurement data, weak highly damped modes, or inexact model assumptions, the number of eigenfrequenceis usually cannot be reliably determined based on measured frequency responses. If the number of modes is assumed to be higher than the actual number of modes, computational or fictitious modes will appear. Usual method which is used to distinguish between the real and the computational modes is based on repeated computation of the modes with one iteration over the order n. If the estimated poles are plotted versus iterations, it can be noted that the real poles remain stable with regard to eigenfrequency, damping, and residues, whereas the computational modes are randomly scattered. In this way, the real poles can be selected by the user and kept for further processing. For models with high orders and high degrees of freedom, the runtime can be large, depending on the algorithm.

Table 2 represents four types of stable ( $Im (s_{r}) < 0$ ) underdamped poles. Unstable overdamped poles ( $ξ_{r} > 1$ ) are not represented. Stabilization diagrams for different estimation methods are represented for the magnetic resonance imaging (MRI) funnel inlet. The fourth type of poles from Table 2 (black dots) are not represented in diagrams, since they do not show stabilization and in addition for high orders of the model n, the stabilization diagram would not be clearly represented.

Table 2.

Criteria for poles in stabilization diagrams.

Symbol	Stabilization
	Frequency	Damping
	$\| \frac{ω_{r} - ω_{(r - 1)}}{ω_{r}} \|$	$\| \frac{ξ_{r} - ξ_{(r - 1)}}{ξ_{r}} \|$
	<1.5%	<2%
	<1.5%	<5%
	<1.5%	≥5%
	≥1.5%	≥5%

Methods for identification of modal parameters

MIF₁ and CMIF mode indicator functions

MIF₁ and CMIF (complex mode indicator function) mode indicator functions gave similar identification results. Due to negligible sensor mass effect, the methods allow SIMO measurement, but on the other hand, multiple eigenfrequencies could not be identified. Significant lower eigenfrequencies can be clearly distinguished. Results for identified eigenfrequencies based on MIF₁ are presented in a comparative tabular overview in section “Comparison of identified parameters.” Diagrams in Figure 15 represent the MIF₁ and CMIF mode indicator functions.

Figure 15.

Mode indicator functions for the funnel.

CE

CE method can identify most of the eigenfrequencies obtained from the mode indicator functions. Since some measurement points pertain to vibration nodes, not all eigenfrequencies in all frequency responses can be identified. The method performs fitting of a frequency response within entire range. In order to achieve appropriate pole stabilization, higher order for fitting has to be selected, which in turn may excite many unstable fictitious computational modes. Figure 16 represents stabilization diagram for the frequency response 139.

Figure 16.

Frequency response 139: measured (blue) and identified using complex exponential (magenta).

LSCE

LSCE requires calculation of a pseudo inverse matrix, and it is therefore ineffective for large number of outputs. According to equation (13) in Appendix 1, for order n and number of outputs $n_{o}$ , the pseudo inverse of a $(2 n n_{o}) \times (2 n)$ matrix should be calculated. In the case of the funnel, for $n_{o} = 321$ and $n = 50$ , this matrix has dimension $(32, 100) \times (100)$ . Calculation of the inverse matrix takes about 70 s. In some cases, limited computer memory can influence the limitation of the selected order. Similarly, as with the method of CE, also here, the entire frequency range is used which requires increased orders to capture all frequencies, but in turn can invoke fictitious modes. The presence of fictitious modes makes the selection of the poles more difficult. Here, the LSCE has been implemented with order $n = 40, \dots, 50$ . Mode indicator function MIF₁ supports selection of poles in stabilization diagram. One representative result is shown in Figure 17.

Figure 17.

Frequency response 139: measured (blue) and estimated using least-squares complex exponential (magenta).

PRCE

Implementation of this method takes ca. 65 s for one run with $n = 2, \dots, 70$ . Stabilization diagram in Figure 18 shows worse stabilization of the poles than for the CE and LSCE methods due to data inconsistency and due to the fact that the method was developed for MIMO measurements. Determining the eigenfrequencies from the stabilization diagram is, therefore, in this case, not adequate.

Figure 18.

Frequency response 139: measured (blue) and estimated using PRCE (magenta).

Polyreference least-squares frequency domain

In order to show the advantages of the method without influence of data inconsistency, the properties of the method are first tested for a single-input-single-output (SISO) case. As a result, the stabilization diagram in Figure 19 is obtained. Frequencies lower than 10 Hz are excluded due to specification of the shaker which is responsible for low coherence in this range.

Figure 19.

Frequency response 139: measured (blue) and estimated using SISO-PLSFD (magenta).

Due to properties of the method, it is also possible to set the upper limit for the investigated frequency range. In this way, the frequency responses can be well fitted even at higher frequencies with strongly coupled modes, without having to increase additionally the order of the method. This is shown exemplarily in Figure 20. Using this method, the frequency responses with strongly coupled and weak modes can be fitted well. Yet, implementation of the method to predefined narrow frequency ranges of SISO frequency responses in stabilization diagrams enables fitting of almost arbitrarily weak peaks in the frequency response, which could result, for example, from measurement noise. In that way, almost any frequency response could be fitted. For example, the frequency close to 110 Hz in Figure 20 identified by PLSFD could not be determined using the mode indicator functions and it could originate from the measurement noise.

Figure 20.

Frequency response 139: measured (blue) and estimated using SISO-PLSFD (magenta) in the frequency range 90–120 Hz.

For the frequency range 10–120 Hz of the first 15 eigenfrequencies of interest, the stabilization is further improved, as Figure 21 for the SISO case shows.

Figure 21.

Frequency response 139: measured (blue) and estimated using SISO-PLSFD (magenta) in the frequency range 10–120 Hz.

Applied to SIMO data, this method significantly improves identification of global resonant frequencies (Figure 22), since in this case, information from all outputs and for each frequency are used. The stabilization is especially good for the first eigenmodes with good consistency and small effect of the sensor mass, and clearly better than for the time-domain methods. Even by obvious increase in the method order n, less fictitious modes will be induced. In comparison with time-domain methods, PLSFD seems to also have a better robustness with respect to data inconsistency. Weak resonance at approximately 110 Hz in Figure 20 showing the SISO-PLSFD case appears also in stabilization diagram for the SIMO case (Figure 22), but no significant stabilization can be observed.

Figure 22.

Frequency response 139: measured (blue) and estimated using SIMO-PLSFD (magenta) in the frequency range 10–120 Hz.

Computation of the PLSFD algorithm requires efficient programming in order to achieve fast runtime. Matrices $[X_{o}]$ and $[Y_{o}]$ in equation (32), Appendix 1, should be calculated only once for high-model orders n, since their order grows high with increasing number of inputs and spectral lines.⁵ This is possible, since the matrices $[X_{o}]$ and $[Y_{o}]$ represent for small orders n only submatrices of their larger versions. Time-consuming calculation of the left-hand side of the Kronecker product in both matrices of equation (32) should be performed only once. This submatrix can be saved and used twice, each time in $[X_{o}]$ and $[Y_{o}]$ . Further runtime savings could be achieved by calculation of this submatrix using FFT,⁴ but in this work, we have not implemented this approach. Nevertheless, the computation is faster than with time-domain methods CE, LSCE, and PRCE.

Comparison of identified parameters

Estimated modal parameters of the funnel are represented in Tables 3 and 4. Due to limited space, a representative frequency response 139 obtained by implementation of mentioned identification methods is shown on diagrams in the previous subsection. In addition, for another frequency response, 225, the results are summarized in the tables for comparison purposes. Stabilization diagrams show, as expected, that SISO methods are not reliable to capture all eigenfrequenceis. However, SIMO methods— LSCE and PLSFD—are capable of a more reliable identification of all expected eigenmodes owing to larger data sets which capture information from all outputs. Eigenfrequencies are characterized by faster stabilization than the corresponding dampings.

Table 3.

Eigenfrequencies (Hz) based on MIF₁ and presented identification methods.

No.	MIF₁	CE (139)	CE (225)	LSCE	SISO	SISO	SIMO
No.	MIF₁	CE (139)	CE (225)	LSCE	PLSFD (139)	PLSFD (225)	PLSFD
1	13	13.05	13.02	13.03	13.03	13.03	13.03
2	16	15.85	15.87	15.93	15.83	15.89	15.92
3	26	25.98	25.97	25.99	25.95	25.98	26.02
4	28	28.87	28.09	28.05	27.81	28.15	28.12
5	32.25	32.24	32.30	32.24	32.25	32.31	32.26
6	35	34.94	34.94	34.90	34.94	34.93	34.90
7	42.25	42.23	42.26	42.15	42.29	42.26	42.14
8	45.5	45.72	45.53	45.89	45.75	45.93	45.67
9	47.5	47.72	47.60	47.90	47.71	47.58	47.54
10	50.25	50.29	−	−	50.32	50.13	50.34
11	53.25	53.39	53.24	52.53	52.54	53.02	53.06
12	64.25	63.46	63.16	64.28	63.35	66.71	64.98
13	69.25	69.17	69.27	69.04	69.18	69.16	69.07
14	80.75	80.23	81.20	80.71	80.07	80.42	80.99
	85.25	85.11	85.34	85.44	86.92	85.18	87.00

MIF: mode indicator function; CE: complex exponential; LSCE: least-squares complex exponential; SISO: single-input-single-output; SIMO: single-input-multiple-output; PLSFD: polyreference least-squares complex frequency domain.

Table 4.

Damping ratio $ξ$ (in %) for corresponding eigenmodes of the funnel: comparison of different methods.

No.	CE (139)	CE (225)	LSCE	SISO	SISO	SIMO
No.	CE (139)	CE (225)	LSCE	PLSFD (139)	PLSFD (225)	PLSFD
1	1.29	1.30	1.33	1.47	1.31	1.27
2	1.79	1.77	1.92	1.53	1.38	1.61
3	1.40	1.30	1.43	1.48	1.31	1.48
4	0.59	1.00	1.08	0.80	0.69	0.90
5	1.19	1.05	1.28	1.05	1.05	1.07
6	1.16	1.17	1.18	1.17	1.18	1.16
7	0.85	0.82	0.91	0.88	0.82	0.86
8	2.30	2.54	2.97	2.44	1.97	2.10
9	0.90	1.22	2.16	0.82	0.94	0.98
10	0.27	−	−	0.66	0.31	0.29
11	1.15	1.19	2.04	2.71	1.18	1.11
12	4.19	3.14	6.82	2.00	3.78	5.58
13	2.09	2.19	2.22	2.10	1.53	2.02
14	3.24	1.09	3.29	2.50	1.44	1.07
15	1.77	2.36	2.29	1.16	2.04	1.48

Approximate runtime for the execution of the identification algorithms is represented in Table 5. Among the implemented algorithms, SIMO-PLSFD shows best stabilization, runtime, and agreement of frequency responses. LSCE method is also characterized by acceptable stabilization, but it requires much more runtime.

Table 5.

Approximate runtime for the algorithm execution on a Core2Duo E8400 4GB RAM computer.

Method (order)	No. of FRFs	Runtime (s)
Method (order)	No. of FRFs	CE (2–80)	LSCE (40–50)	PRCE (2–70)	PLSFD (10–150)
SISO	1 × 1	11	−	−	7.6
SIMO	321 × 1	321 × 11	198	75	33

CE: complex exponential; LSCE: least-squares complex exponential; PRCE: polyreference least-squares complex exponential; PLSFD: polyreference least-squares complex frequency domain.

Additional check of the results can be performed by animation of the vibration modes of the structure using appropriate software such as Labshop or PULSE. Multiple or rigid body modes can be distinguished through animation of the modes using imaginary parts of frequency responses as shown in Figure 23. For complex modes which exist in case of non-proportional damping, it is possible to perform the animation of modes using PULSE REFLEX software.

Figure 23.

Screen shots of the funnel eigenmodes obtained from the PULSE REFLEX animation.

One drawback of the experimental modal analysis with used shaker B&K 4809 is caused by its specification of the measured frequency range, which results in bad coherence under 10 Hz and uncertain modal parameter estimation in this range. In order to investigate this low-frequency region, another modal test has been performed, using impact hammer for excitation of the funnel. Frequency spectrum up to 100 Hz with resolution $Δ f = 0, 25 Hz$ has been investigated.

For excitation by the hammer, eight points on the funnel were predefined according to the mesh shown in Figure 24(a) (black: positions of the hammer). For output measurements, an accelerometer was glued by wax to the surface of the funnel. Red arrow in Figure 24(a) shows the position and measurement direction of the accelerometer. One frequency response with corresponding coherence is shown in Figure 25. Frequency responses show a good coherence. In this way, both rigid body modes presented in Figure 24(b) and (c) could be distinguished. With their corresponding eigenfrequencies of 1.75 and 2.5 Hz, respectively, these two rigid body modes lay under 20% of the first elastic mode (Figure 24(d)) and have therefore a minimal effect on elastic modes. Furthermore, this investigation has confirmed the effectiveness of the funnel support by hanging it on springs, which simulates free body motion, similarly as assumed in the FE analysis. The rigid body modes are present due to supports and spring stiffness.

Figure 24.

(a) Mesh of the impact hammer test model. (b)–(d) Modes of the funnel obtained from PULSE: rigid body modes: first (b), second (c), and elastic mode (d).

Figure 25.

FRF and coherence from the hammer test of the funnel.

Conclusion

This article presents the identification of the modal parameters (eigenfrequencies, damping coefficients, residues, and mode shapes) based on measured FRF. Several time and frequency domain estimation algorithms are implemented within our MATLAB-based tool for the modal parameter estimation. An overall analysis of the structural behavior of the funnel-shaped inlet of MRT is performed based on the implemented estimation algorithms. The estimation methods are compared and the comparison results are summarized showing based on tabular representation and stabilization diagrams significant advantage of the proposed methodology for determining modal parameters in a broad band of frequencies.

Footnotes

Academic Editor: Francesco Massi

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: Article Processing Charge was funded partly by the German Research Foundation (Deutsche Forschungsgemeinschaft - DFG) and the Open Access Publication Fund of Ruhr-Universität Bochum.

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Identification of modal parameters for complex structures by experimental modal analysis approach

Abstract

Keywords

Introduction

Experimental determination of the MRT funnel frequency responses

Linearity test

Single-input-multiple-output versus multiple-input-multiple-output measurement and effect of the accelerometer mass

Determination of the frequency responses

Identification of the MRT funnel modal parameters

Stabilization diagrams

Methods for identification of modal parameters

MIF1 and CMIF mode indicator functions

CE

LSCE

PRCE

Polyreference least-squares frequency domain

Comparison of identified parameters

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References

MIF₁ and CMIF mode indicator functions