The use of transmissibility measurements in output-only modal analysis

An approach to identify modal parameters from output-only transmissibility measurements is introduced. In general, the poles that are identified from transmissibility measurements do not correspond with the system's poles. However, by combining transmissibility measurements under different loading conditions, it has been shown that modal parameters can be identified. In this paper the proposed technique is recapitulated and validated by means of an experimental test on a clamped beam.     

Direct calculation of modal parameters from matrix orthogonal polynomials

The object of this paper is to introduce a new technique to derive the global modal parameter (i.e. system poles) directly from estimated matrix orthogonal polynomials. This contribution generalized the results given in Rolain et al. (1994) [5] and Rolain et al. (1995) [6] for scalar orthogonal polynomials to multivariable (matrix) orthogonal polynomials for multiple input multiple output (MIMO) system.Using orthogonal polynomials improves the numerical properties of the estimation process. However, the derivation of the modal parameters from the orthogonal polynomials is in general ill-conditioned if not handled properly. The transformation of the coefficients from orthogonal polynomials basis to power polynomials basis is known to be an ill-conditioned transformation. In this paper a new approach is proposed to compute the system poles directly from the multivariable orthogonal polynomials. High order models can be used without any numerical problems.The proposed method will be compared with existing methods (Van Der Auweraer and Leuridan (1987) [4] Chen and Xu (2003) [7]). For this comparative study, simulated as well as experimental data will be used.     

Operational modal analysis by updating autoregressive model

This paper presents improvements of a multivariable autoregressive (AR) model for applications in operational modal analysis considering simultaneously the temporal response data of multi-channel measurements. The parameters are estimated by using the least squares method via the implementation of the QR factorization. A new noise rate-based factor called the Noise rate Order Factor (NOF) is introduced for use in the effective selection of model order and noise rate estimation. For the selection of structural modes, an orderwise criterion called the Order Modal Assurance Criterion (OMAC) is used, based on the correlation of mode shapes computed from two successive orders. Specifically, the algorithm is updated with respect to model order from a small value to produce a cost-effective computation. Furthermore, the confidence intervals of each natural frequency, damping ratio and mode shapes are also computed and evaluated with respect to model order and noise rate. This method is thus very effective for identifying the modal parameters in case of ambient vibrations dealing with modern output-only modal analysis. Simulations and discussions on a steel plate structure are presented, and the experimental results show good agreement with the finite element analysis.     

Output-only modal analysis of wind turbine tower based on vibration response under emergency stop

The identification technique of output-only modal parameters is proposed for the large wind turbine tower under emergency stop. Compared with the response of regular operating conditions, the immediate tower structural response under emergency stop much more resembles a state of free vibration, which is more appropriate for the modal identification of the wind turbine tower. The vibration response is measured in the nacelle, which is easy to perform in the field modal test. The variational mode decomposition (VMD) is applied to decompose the vibration response into several band-limited intrinsic mode functions. The free responses of decomposed functions are extracted by applying the random decrement technique (RDT). Finally, the modal damping ratio and natural frequency are identified from each free modal response by using the Hilbert transform method. Simulations and a 1.5 MW wind turbine field modal test results verify the effectiveness of the proposed identification method. The main modal parameters of wind turbine, including weak modes, are effectively extracted by using output-only vibration responses under emergency stop. The modal parameter identification method is provided for the large wind turbine structure under the engineering condition.     

ON-LINE MODAL MONITORING OF AIRCRAFT STRUCTURES UNDER UNKNOWN EXCITATION

In this paper, we present the application of recently proposed subspace-based damage detection and isolation algorithms in on-line structural monitoring of an airplane structure under unknown excitation. The first step of the approach is a subspace-based modal analysis of the safe structure, using output-only measurements. The monitoring step consists in the computation of a χ²-variable, which reflects possible deviations of newly collected output-only data w.r.t. the reference (safe) modal behaviour. No re-identification of the possibly damaged structure is required. An on-line version of the detection algorithm is proposed. Monitoring certain critical modes can also be achieved through sensitivity tests. Experimental results for the proposed monitoring algorithms are reported, in the case of the Paris MS760 airplane, on which mass as well as stiffness changes are introduced. Moreover, the results of the output-only subspace-based modal analysis are compared with the modal appropriation method.     

USING CROSS—CORRELATION THEORY TO EXTRACT MODAL PARAMETERS IN FREQUENCY—DOMAIN

Conventional modal parameter identifications are usually based on frequency response functions, which require measurements of both the input force and the resulting response. However, in many cases, only response data are available while the actual excitations (such as wind/wave load) are not measurable. Modal parameters estimation must base itself on response-only data. Over the past years, many time-domain modal parameter identification techniques from output-only are proposed. A poly-reference frequency-domain modal identification scheme on response-only is presented. It is based on coupling the cross-correlation theory with conventional frequency-domain modal parameter extraction. An experiment using an airplane model is performed to verify the proposed method.     

From operating deflection shapes towards mode shapes using transmissibility measurements

A transmissibility measurement is the traditional way of making a frequency response measurement when the excitation force cannot be measured. It is calculated in the same way as the admittance function, or transfer function. Whereas the admittance function is the ratio of response divided by force, transmissibility is the ratio of a response divided by a reference response. It is well known that operating deflection shapes can be displayed from a set of two or more transmissibilities. The use of transmissibility functions in the field of operational modal analysis has only recently been introduced by the authors of this paper. In general the “model” obtained from transmissibility functions is only valid for the concerned operating condition and is not related to any system parameter. In this paper a new post-processing method based on transmissibility measurements is introduced that allows the estimation of the modal parameters and in particular the unscaled mode shapes. The new method is introduced and illustrated with a numerical test.     

A framework for blind modal identification using joint approximate diagonalization

In this paper, a second-order statistical method employed in blind source separation (BSS) is adapted for use in modal parameter identification. Modal responses and mode shapes are estimated by the use of second-order blind identification (SOBI) on an expanded and pre-treated dataset. Frequency and damping can be obtained from the modal responses by simple single degree of freedom methods. Using this approach, a class of new non-parametric output-only modal identification algorithms is proposed and examples of its use are provided. It is demonstrated that the proposed methodology provides a novel and robust approach to modal identification. For the example shown, it is deduced that quality of the modal parameters produced by the method is competitive with the state of the art parametric methods.     

Modal parameter identification of stay cables from output-only measurements

Stay cables are one of the most critical structural components in modern cable-stayed bridges and the cable tension plays an important role in the construction, control and monitoring of cable-stayed bridges. We propose a time domain and a time-frequency domain approaches for modal parameter identification of stay cables using output-only measurements. The time domain approach uses the subspace algorithm which is improved with a new modal coherence indicator. The time-frequency approach uses the wavelet transform of signals which is improved with a new analyzing wavelet. The wavelet transform is applied to the free response of ambient vibration which is obtained using the random decrement technique. Two experiments of stay cables are presented. The first experiment concerns a stay cable in laboratory where the external load is applied through an impact hammer and the vibratory signals are acquired through four accelerometers. The second experiment concerns the Jinma cable-stayed bridge that connects Guangzhou and Zhaoqing in China. It is a single tower, double row cable-stayed bridge supported by 112 stay cables. Ambient vibration of each stay cable is carried out using accelerometers. From output-only measurements, the modal parameters of stay cables are extracted. Once the eigenfrequencies and the damping coefficients are obtained, the cable forces and the Scruton number are derived. In a continuous monitoring and modal analysis process, the tension forces and Scruton numbers could be used to assess the health of stay cables in cable-stayed bridges.     

User-assisting tools for a fast frequency-domain modal parameter estimation method

Recently, the least-squares complex frequency-domain (LSCF) estimator has been developed for modal analysis applications. This contribution elaborates in more detail the fast derivation of stabilisation charts and uncertainty bounds for the estimated poles. An alternative representation for the stabilisation chart as well as a robust cluster algorithm to identify clusters of poles using the chart information is presented. Based on the clusters, uncertainty bounds for the poles and an automation of the pole selection process are derived. The relation of these “variances” with the stochastic variances (or confidence bounds) introduced by the noise on the measurements is compared by means of Monte-Carlo simulations. The use of alternative representation for the stabilisation chart in combination with the robust cluster analysis as well as the availability of uncertainty bounds for the modal parameters, assist the user with the performance of an accurate modal parameter estimation.     

基于传递率函数的运行模态分析方法

传统的运行模态分析方法在推导过程中多假设激励为白噪声,造成在应用上有一定的局限性。提出了一种基于传递率函数的运行模态分析方法,无需采用白噪声假设,利用两种不同载荷情况下的传递率函数构造有理函数,通过有理分式Forsythe正交多项式法对其进行拟合,得到模态频率、阻尼和振型参数。最后,采用机翼模型仿真算例和悬臂梁实验,验证了在非白噪声激励情况下该方法的有效性与可靠性。     

Modal identification of linear non-proportionally damped systems by wavelet transform

A time-frequency identification technique based on wavelet transform is formulated and applied to free-decay responses of linear systems with non-proportional viscous damping. The Cauchy mother wavelet is used. Frequencies, modal damping ratios and complex mode shapes are identified from output-only free vibration signals. This identification technique has also shown to be effective when the (non-proportional) damping is significant.     

The inception of OMA in the development of modal testing technology for wind turbines

Wind turbines are immense, flexible structures with aerodynamic forces acting on the rotating blades at harmonics of the turbine rotational frequency. These harmonics are comparable to the modal frequencies of the structure. Predicting and experimentally measuring the modal frequencies of wind turbines have been important to their successful design and operation. Performing modal tests on wind turbine structures over 100 m tall is a substantial challenge, which has inspired innovative developments in modal test technology. For wind turbines, a further complication is that the modal frequencies are dependent on the turbine rotation speed. The history and development of a new technique for acquiring the modal parameters using output-only response data, called the Natural Excitation Technique (NExT), will be reviewed, showing historical tests and techniques. The initial attempts at output-only modal testing began in the late 1980s with the development of NExT in the 1990s. NExT was a predecessor to Operational Modal Analysis (OMA), developed to overcome these challenges of testing immense structures excited with natural environmental inputs. We will trace the difficulties and successes of wind turbine modal testing from 1982 to the present.     

Operational modal analysis in the presence of harmonic excitations by the use of transmissibility measurements

Operational modal analysis (OMA) is based on the assumption that the forces on the structure are the result of a stochastic process, so being white noise. In practice, however, structural vibrations observed in operation cannot always be considered as pure white-noise excitation. In many mechanical structures the loading forces are often more complex and even harmonic components can be present in the response. This is especially true, when measuring on mechanical structures containing rotating parts (e.g. cars, turbines, windmills), but also civil engineering structures may have responses superimposed by harmonic components. OMA procedures are, strictly speaking, not applicable in these situations. Current techniques may encounter difficulties to correctly identify the modal parameters, especially for modes with eigenfrequencies close to the harmonic frequencies. In this paper a recently proposed OMA technique based on transmissibility measurements will be applied. This method reduces the risk to wrongly identify the modal parameters due to the presence of harmonics. The unknown operational forces can be arbitrary (coloured noise, swept sine, impact, etc.) as long as they are persistently exciting in the frequency band of interest.     

SENSITIVITY-BASED OPERATIONAL MODE SHAPE NORMALISATION

In the past few years, in-operation modal analysis has become a valid alternative for structures where classic forced vibration tests would be difficult, if not impossible, to conduct. A disadvantage of this method is that the excitation forces are unknown. Therefore, not all modal parameters can still be estimated. As a result, operational mode shape estimates remain unscaled (dependent on the unknown level of excitation) what restricts the applicability of operational modal models. So far, no techniques are available for the correct re-scaling of operational mode shapes purely based on experimental output-only data. All known methods involve either a detailed knowledge of the material characteristics of the test structure (finite element model approach) or make very restrictive assumptions about the excitation signal. In this contribution, a sensitivity-based method is proposed for the scaling of operational mode shapes on a basis of operational modal models only.     

基于伪频响函数矩阵法的运行模态分析方法

基于传递率测量的伪频响函数法运行模态分析技术无需白噪声激励假设,具有强"谐波模态抑制"能力,但其模态识别结果依赖待分析伪频响函数的选取、稳定性欠佳。为解决该问题,建立了基于伪频响函数矩阵法的运行模态分析方法。该方法首先构造多参考测点、多参考工况的伪频响函数矩阵,然后基于伪频响函数矩阵采用最小二乘复频域法求解模态频率和阻尼比、基于传递率函数矩阵求解模态振型,最后通过建立相应的稳态图并结合以多参考测点平均的传递率函数矩阵第2阶奇异值的倒数为基础的模态指示函数来确定模态阶数与最终的模态结果。5自由度质量-弹簧-阻尼系统仿真结果及自由梁的运行模态试验结果均表明,提出方法既继承了基于伪频响函数法的运行模态分析方法的强"谐波模态抑制"能力,亦能有效解决伪频响函数法识别结果稳定性欠佳的问题、一定程度上提高模态识别精度,且更利于自动化。     

Cyclostationarity and the cepstrum for operational modal analysis of MIMO systems—Part II: Obtaining scaled mode shapes through finite element model updating

This paper presents a new technique for scaling mode shapes, obtained from cepstrum-based operational modal analysis (OMA) techniques, such as that described in the companion paper, using finite element model updating. This OMA technique estimated frequency response functions (FRFs) between a cyclostationary input and response measurements. If the input is frequentially white, the resulting FRFs can be obtained up to an overall scaling constant using the in-band poles and zeros identified in the OMA process and employing the response autospectrum as a reference to correct for the effect of out of band modes. In this way, the mode shapes would be scaled correctly relative to each other but would still have arbitrary overall magnitude. If the input is not white, then no reference is available to correct FRF regenerated from in-band poles and zeros, and so these FRFs will exhibit both an overall slope resulting from the effect of out-of-band poles and zeros, and an arbitrary magnitude. This overall slope will differ between measurement locations so even the relative scaling between the mode shapes will be lost. This paper describes a simple technique for recovering both the relative and overall scaling of the FRFs, and hence the mode shapes, based on finite element model updating.     

The role of anti-resonance frequencies from operational modal analysis in finite element model updating

Finite element model updating traditionally makes use of both resonance and modeshape information. The mode shape information can also be obtained from anti-resonance frequencies, as has been suggested by a number of researchers in recent years. Anti-resonance frequencies have the advantage over mode shapes that they can be much more accurately identified from measured frequency response functions. Moreover, anti-resonance frequencies can, in principle, be estimated from output-only measurements on operating machinery. The motivation behind this paper is to explore whether the availability of anti-resonances from such output-only techniques would add genuinely new information to the model updating process, which is not already available from using only resonance frequencies.This investigation employs two-degree-of-freedom models of a rigid beam supported on two springs. It includes an assessment of the contribution made to the overall anti-resonance sensitivity by the mode shape components, and also considers model updating through Monte Carlo simulations, experimental verification of the simulation results, and application to a practical mechanical system, in this case a petrol generator set.Analytical expressions are derived for the sensitivity of anti-resonance frequencies to updating parameters such as the ratio of spring stiffnesses, the position of the centre of gravity, and the beam's radius of gyration. These anti-resonance sensitivities are written in terms of natural frequency and mode shape sensitivities so their relative contributions can be assessed. It is found that the contribution made by the mode shape sensitivity varies considerably depending on the value of the parameters, contributing no new information for significant combinations of parameter values.The Monte Carlo simulations compare the performance of the update achieved when using information from: the resonances only; the resonances and either anti-resonance; and the resonances and both anti-resonances. It is found that the addition of anti-resonance information improves the updating performance for some combinations of parameter values, but does not improve the update in significant other regions.The simulated results are verified using resonance and anti-resonance frequencies measured on a steel beam test rig. The investigation is extended to include the updating of parameters of a petrol generator set. It is found that the contribution of the anti-resonances to the model update is heavily dependent on the geometry of the model and the choice of variables to be updated, suggesting that, for some models, the pursuit of anti-resonance information through expensive operational modal analysis may be inappropriate.     

Finite element model updating and structural damage identification using OMAX data

The main limitations in the finite element (FE) model updating technique lie in the ability of the FE model to represent the true behavior of the structure (modelling problem), and in the ability to identify enough modal parameters with sufficient accuracy, especially for large structures that are tested in operational conditions (identification problem). In this paper, the identification problem is solved with an OMAX approach, where an artificial force is used in operational conditions and a structural model is identified that takes both the forced and the ambient excitation into account. From an extensive case study on a real three-span bridge, it is observed that, while updating the FE model using the experimental output-only data yields a good fit, discrepancies show up when the more extensive set of OMAX data is used for validation, or even for updating. It can be concluded that an OMAX approach not only increases the well-posedness of the updating problem, it also allows to detect potential inaccuracies in the FE model.     

Output-only modal analysis using blind source separation techniques

The present study carries out output-only modal analysis using two blind source separation (BSS) techniques, namely independent component analysis and second-order blind identification. The concept of virtual source is exploited and renders the application of these BSS techniques possible. The proposed modal analysis method is illustrated using numerical and experimental examples.