Identification of modal parameters of unmeasured modes using multiple FRF modal analysis method

As all experimental procedures, Experimental Modal Analysis (EMA) is subject to a wide range of potential testing and processing errors. The modal identification methods are sensitive to these errors, yielding modal results which are uncertain up to certain error bounds. The question hence is what these error bounds on test data and modal parameters are. In this paper, the studied source of uncertainty is related to the variance (noise) on the Frequency Response Function (FRF) measurements. Under the H1 assumptions and in single-input cases, the FRF variances can be computed from the coherences and the FRFs. In multiple-input cases, some more measurement functions are required. Advanced system identification methods like the Maximum Likelihood Estimator (MLE) and PolyMAX Plus have the possibility to take the uncertainty on the measurement data into account and to propagate the data uncertainty to (modal) parameter uncertainty. This paper will review FRF variance estimation techniques, including some pragmatic approaches. The basic concepts of Maximum Likelihood Estimation and the calculation of confidence bounds will be discussed. Some typical structural testing and modal analysis cases will be used as illustration of the discussed concepts.

Viscoelastic materials are used in vibration damping applications as they have the ability to dissipate energy, they are used in machine basis, bridges and many other applications. In this paper four plates with the same thickness but different material lamination are investigated for their modal properties. Namely,natural frequencies, mode shapes and damping loss factors. Experimental modal analysis is carried out using impact hammer for excitation and accelerometer for measurement. Data is processed using computer package to obtain frequency response function (FRF) for each plate and modal parameters are extracted from the plot. Results are compared to finite element model that was created using ANSYS package. It showed very well agreement between experimental and finite element. Damping effect on the shift in natural frequencies was obvious due to the viscoelastic material thickness in the laminated plates. It is concluded that the finite element analysis using ANSYS software is a powerful tool in predicting natural frequencies and mode shapes even in laminated plates with viscoelastic core, it is also confirmed the effect of damping increase due to the use of viscelastic cores on the plate natural modes and vibration behaviour.

Performing forced vibration tests on full-scale structures is the most reliable way of determining the relevant modal parameters in structural dynamics, such as modal frequencies, mode shapes, modal damping, and modal masses. This study describes the modal identification of a double-level curved cable-stayed bridge with separate deck systems for pedestrians and vehicles via forced vibration tests. The steady-state structural responses to sinusoidal excitations produced by an electrodynamic shaker are recorded under varying excitation frequencies, and the frequency response functions are established. The measured frequency response functions are curve fitted to estimate the modal parameters. The numerical simulation of frequency response function–based modal parameter identification of an elastically multi-supported continuous beam structure is carried out, and the emphasis has been placed on the evaluation of the effect of an additional shaker mass, excitation frequency step and range, multi-mode vibration, and noise on identification results. Finally, the modal parameters for the first lateral mode of a double-level curved cable-stayed bridge are identified by forced vibration experiments, and the results are compared with those from ambient vibration tests and free vibration tests. The effect of the unmeasured wind excitation on identification is discussed. It is shown that the effect of ambient vibration is minor for wind velocity of 3–5 m/s. The damping ratios identified by forced and free vibration tests are comparable, while those from ambient vibration are subject to large variations. The modal mass obtained from forced vibration tests is in good agreement with finite element prediction, which provides design basis for mass-type dampers.

Measurement of Frequency Response Functions (FRFs) on the Millennium Bridge in London was a key part in the validation of the retrofit solution for excessive sway vibration caused by walking pedestrians during the bridge opening in June 2000. The measured FRFs were employed to verify the analytical response analysis and to estimate and confirm analytical predictions of natural frequencies, modal damping ratios, mode shapes and modal masses corresponding to various configurations of prototype viscous and tuned mass dampers. These results were used to experimentally assess the effectiveness of the proposed retrofit solution. Although FRF measurement technology is fairly standard in many engineering disciplines, its application on the Millennium Bridge was quite problematic and required some unique solutions which are described in this paper. Modal testing of the Millennium Bridge clearly demonstrated that it was possible to measure good quality FRFs around a frequency as low as 0·5 Hz, which was the natural frequency of the first lateral mode of vibration. This was achieved by designing, constructing and installing a bespoke hydraulic shaker featuring a translatory moving mass of 1000 kg. As the shaker was not able to produce perfectly sinusoidal excitation for stepped-sine testing, FRFs were successfully estimated using the so-called Resolver method in conjunction with an Hc estimator. The standard H1 FRF estimator, which is used in the vast majority of modern digital spectrum analysers nowadays, was demonstrably unable to produce meaningful FRF estimates when using the Resolver approach. The FRF estimation and curve-fitting performed in situ proved to be crucial for the success of the modal testing of the Millennium Bridge. By evaluating the estimates of the experimental FRFs and the corresponding modal properties during the testing programme on site, a number of test errors were spotted. This, in turn, enabled immediate repetition of measurements and/or evaluation of the structural behaviour which would have been impossible if the FRF data analysis had been performed after the testing had been completed.

In modal testing, the measured Frequency Response Functions (FRFs) are often affected by transducer mass effects. Especially in Multi-Input-Multi-Output (MIMO) modal test, multiple sensors are employed and the effects of additional masses are more significant. This paper deal with removing masses effects of multiple sensors from the measured FRFs. The proposed method offers some advantages over the available techniques in that extra FRFs measurements with different configurations are not required during the correction process. First, a general correction formulation is derived theoretically. Then, validations of the proposed methods are demonstrated using simulated data. It demonstrates a good performance in numerical simulation. To investigate the performance of the proposed method in practical application, further simulations are presented by employing noise-polluted data. And it is shown that the accuracy of correction results will be affected to some extent by the noise. It is thus suggested that the measured FRFs be preprocessed using the curve-fitting procedure or noise reduction processing before applying the proposed method. Since the FRFs measurements required for corrections refer to different excitation points, it requires moving the exciter to different locations. This is usually easy to achieve in hammer impact testing. However, moving the exciter is usually inconvenient in shaker modal testing. Further work is necessary to extend this proposed method so as to make it also applicable for shaker modal test case.

The identification of the rigid body properties of a structure is an important matter in various structural dynamic applications, namely in structural modification (coupling/uncoupling), optimization and vibration control. In most situations, the experimental route is the only via to obtain the desired dynamic properties. This goal is achieved by an inverse process which starts from the measurement of Frequency Response Functions (FRFs) and ends on matrices describing the mass distribution of a rigid body.The objective of the present article is the evaluation and comparison of the performance of three different kinds of identification methods that share a common characteristic – they all use FRFs as inputs to the inverse process. One of them, the Inertia Restraint Method (IRM) makes use of the accelerance FRF (information condensed at zero frequency), while the other two methods make use of receptance FRFs as inputs, although in different manners; in one of them the properties are evaluated directly from the measurement data – Direct Physical Parameter Identification Method (DPPIM) – whereas the other type requires a modal identification procedure as an intermediate step – Modal Method (MM). An actual experimental structure forms the case study to illustrate the performance of the three different techniques.

Modal analysis has developed into a major technology for the study of structural dynamics in the past several decades. Through it, complex structural dynamics phenomena can be represented in terms of structural invariants, i.e., the modal parameters: natural frequencies, damping ratios and mode shapes. Operational Modal Analysis (OMA) deals with the estimation of modal parameters on vibration data measured for operational conditions, when the excitation on the structure is not measured. In this work, OMA is performed on a wind turbine blade undergoing wind tunnel testing. These tests included different wind speeds, pitch angles and also different health conditions of the blade, where masses of different magnitude were fixed to the blade on different locations to emulate damage conditions. In order to monitor the modal parameters across multiple days of varied tests in the wind tunnel, the Polymax modal parameter estimator was implemented, coupled with an Automated Modal Analysis methodology. This methodology included an automatic modal parameter selection technique, using a Machine Learning (ML) clustering algorithm, coupled with a modal tracking procedure which applied statistical thresholds on the modal parameters’ values. The tracking procedure searches for modes similar to the ones calculated for healthy conditions. The results show how the modal parameters of the wind turbine blade vary with the different measured conditions in the wind tunnel. Moreover, a damage detection methodology is implemented to differentiate between the healthy and damaged conditions on the blade, by leveraging an anomaly detection algorithm using the Multivariate Gaussian Distribution (MGD). This algorithm takes as input the modal parameters calculated by the previous Automated Modal Analysis methodology and detects statistical deviations among them which could indicate the presence of damage. All steps of this work contribute to developing an automatic framework able to detect damages on a wind turbine blade, and therefore perform Structural Health Monitoring (SHM) for different operational conditions.

Stator end winding of turbine generator experiences high vibrations at resonance frequency in operation which result into basket loosening, insulation failure and ultimately end with no power from machine. Modal analysis is a powerful tool of vibration test to visualize vibration characteristics viz. natural frequencies, amplitudes and mode shapes of dynamic structures. This article elucidates the vibration behavior of stator end winding structure in a 150 MVA, 16 kV, 50 Hz turbine generator of decennial period. Natural frequency test (NFT) was conducted on it in off-line condition. An impact hammer and an accelerometer were jointly used for excitation force and respond vibration respectively. Modal parameters were extracted from the frequency response function through FFT analyzer. Natural frequencies and vibration magnitudes had been obtained. Overhang was reinforced based on the test results. Efficacy of the rectification work was endorsed by post modal analysis. Vibration amplitudes had been brought down to low values in the resonance band width of 95 to 110 Hz.

In modal testing, the measured frequency response functions (FRFs) are often affected by transducer mass effects. Especially when multiple sensors are employed, transducers mass effects are more significant. This paper deals with removing sensors mass effects from the measured FRFs in hammer impact testing. The proposed method offers some advantages over the available techniques in that extra FRFs measurements with different configurations are not required during the correction process. Firstly, correction method for single accelerometer case is discussed. Further, general correction formulation for multiple accelerometers case is derived. Then, validations of the proposed method are demonstrated using simulated and experimental data. It shows good effectiveness in experimental validation for single accelerometer case. Numerical simulation also demonstrates a good performance for multiple accelerometers case. However, further simulations with noisy-polluted data show that the accuracy of correction results will be affected to some extent by the noise. It is suggested that the measured FRFs be preprocessed using the curve-fitting procedure or noise reduction processing before applying the proposed method.

This paper presents the system identification technique by applying PZT actuators and PVDF sensors. An array of finite-length PVDF films is equally spaced and distributed over a cantilever beans acting as sensors. Two finite-length PZT patches are bonded symmetrically and excited 180°out-of-phase for pure bending excitation. The theoretical base of modal analysis for system identification of piezoceramic beams is presented to derive the frequency response functions (FRFs) in conventional modal format. The physical interpretation of P11 and PVDF mode shape functions are characterized respectively. The FRFs between the PZT actuators and PVDF sensors are experimentally measured. By the operation of a series of FRFs and the application of modal parameter extraction method, the modal parameters. including system natural frequencies, mode shapes and modal damping ratios, can be obtained. Both theoretical and experimental approaches agree very well. The system information can be used for control applications and also be applicable to structural failure diagnosis.

High-speed digital image correlation is a camera-based displacement measurement technique that has experienced increasing popularity for modal analysis. It is due to a high spatial density of measurement that provides an outstanding interpretation of mode shapes or operational deflection shapes. However, its sensitivity is typically lower than other techniques and transducers such as accelerometers or laser vibrometry. With these measurements, the estimation of any transfer function in the frequency domain is noisier in valley regions of the response, including anti-resonances. Therefore, an accurate analytical model of the response should be used for modal identification. Modal analysis algorithms perform an identification of the modal parameters and a reconstruction of the response based on the transfer function between force excitation and displacement response. This is called the frequency response function. However, force measurements are not available in many experiments, where the motion excitation is recorded instead. This is the case when the vibration is applied to the specimen through a motion in its base. This transfer function, also known as the transmissibility function, relates the motion of excitation and response, whose shape differs from the frequency response function. This difference in the response function is relevant in a relatively noisy measurement like using digital image correlation. Hence, in this work, a transformation of the transmissibility function into equivalent frequency response function, based on theoretical models, is performed prior to modal identification in base motion tests. More suitable experimental response functions are obtained, increasing the accurateness of the modal parameter estimation.KeywordsModal analysisTransmissibility functionsFull-field measurement

This paper provides data-driven tools to detect and quantify approximately the influence of the time variation of a system under test in classical frequency response function (FRF) measurements. To achieve this, the best linear time-invariant approximation of a linear time-varying system is defined and is estimated using existing FRF estimators. An analysis of the residuals of the latter estimation reveals the frequency band in which the contributions from the time variation dominates the disturbing measurement noise and, thus, is significant. All concepts are illustrated on a simulation and real measurement examples.

In modal testing, the force transducer and accelerometer mounted on the test structure will introduce extra mass effects to the system and then affect the measured Frequency Response Functions (FRFs). This paper proposed a method for assessing sensor mass effects on the measured FRFs in shaker modal testing. The assessment method offers some distinct advantages where very few FRFs measurements are required, and more importantly it does not require calculations involving several measured FRFs, hence avoiding further contaminations of the measured data. In view of two different ways of response measurements, two cases have been discussed: (1) shaker + laser Doppler vibrometer and (2) shaker + accelerometer. In case (1), only force transducer mass need be assessed. In case 2, however, both force transducer and accelerometer masses should be considered. Especially in Case 2, the overall as well as the individual mass effects of the two transducers were assessed. It is found that the assessment method is quite effective in the experimental validation for Case 1. A simple numerical example for Case 2 presented also illustrates the good theoretical performance of the method. The same example is extended to incorporate simulated noise, simulating an experimental situation, and it is shown that the accuracy of assessment results will be affected to some extent by the noise. It is suggested that the measured FRFs be preprocessed using the curve-fitting procedure before applying the proposed method.

This paper presents an application of the Experimental Acoustic Modal Analysis (AMA) on a vintage 10 M Conn tenor saxophone. This technique originates from the traditional Modal Analysis which is commonly adopted to determine the dynamic behaviour of solid structures. The methodology is based on the measurement of Frequency Response Functions (FRFs) as a ratio of the acoustic pressure, measured by means of a set of microphones positioned along the saxophone body, and the volume velocity produced by an acoustic source and measured at the instrument input. The microphones are housed on the stripped saxophone keys by means of tailored three-dimensional printed adapters. The AMA makes use of the FRFs to extract modal parameters, such as resonance frequencies, loss factors, and mode shapes. These parameters pertain to the cavity modes of the saxophone passive resonator. The analysis has been performed for three different notes (B ♭, B, and C) and for two registers of the instrument in order to show the potential of the technique. Moreover, the influence of the mouthpiece volume on the cavity modes has been shown. The information obtained by AMA can find useful application in the validation of analytical or numerical models of this kind of musical instruments.

Vibration testing based on impulse response excited by pulsed-laser ablation: Measurement of frequency response function with detection-free input