A method of non-linear modal identification from frequency response tests (original) (raw)
Related papers
2008
Dynamic structural modification matrix {f} Generalized external forcing vector {F} Amplitude vector of harmonic external forcing i Unit imaginary number [H] Linear structural damping matrix [I ] Identity matrix [K] Linear stiffness matrix [M] Linear mass matrix {N} Internal non-linear forcing vector n Number of degrees of freedom v Describing function {x} Displacement vector {X} Complex amplitude vector of steady state harmonic displacements [α] Receptance matrix [Δ] Non-linearity matrix [ΔC] Viscous damping matrix of modifying system [ΔK] Stiffness matrix of modifying system [ΔM] Mass matrix of modifying system [γ] Receptance matrix of modified system ω Excitation frequency ABSTRACT One of the major problems in structural dynamics is to identify nonlinearity, which is usually local in large structural systems, and to conduct dynamic analysis of the non-linear system. In this work, a new approach is suggested for modal identification of a non-linear system. Modal parameters obtained through modal identification are used in harmonic response prediction at different forcing levels. The response at only the fundamental harmonic is considered. The model can also be used to predict the response of the non-linear system coupled with a linear system and/or subjected to structural modification. An iterative solution method is used in structural dynamic analyses. The identification method proposed is for systems where nonlinearity is between a single coordinate and the ground. Response dependent modal parameters of the non-linear system are obtained via modal testing at different response levels. The method presented is verified through case studies. In the case studies presented, a subsystem with cubic stiffness type non-linearity is considered and the simulated FRFs of the subsystem generated for various response levels are used as pseudo experimental values.
NON-LINEAR MODAL ANALYSIS METHODS FOR ENGINEERING STRUCTURES
This thesis presents two novel nonlinear modal analysis methods, aimed at the identification of representative engineering structures. The overall objective is to detect, localize, identify and quantify the nonlinearities in large systems, based on nonlinear frequency response functions (FRFs) as input data. The methods are first introduced in a direct-path, by analyzing a general theoretical system. Then, the concepts are extended to tackle a nonlinear identification via the reverse-path of the same methodologies.
Modal identification of non-classically damped structures
2015
This thesis focuses on modal identification of structural systems. System Identification can be used to create mathematical models of structures which can then be used in simulation and design. Modal identification provides a valuable means of calibrating, validating, and updating finite element models of structures. This thesis reviews the standard techniques of system identification. Basic theoretical background of the different methods of practical use is presented. Demonstrations of the described methods are provided by using response simulated from known systems which are then used in system identification. Starting with single degrees of freedom systems, basic theory in relation to dynamics and modal analysis of multi degree of freedom systems are presented with examples used in the identification procedure, along with methods for non-classically damped systems. As a case study, system identification of a base-isolated bridge is undertaken. Modal properties of the bridge are computed from a finite element model. A simulated response is used in system identification to check whether the identified modal properties match that of the finite element model. One of the issues investigated is the effect of damping provided by the rubber bearings of the base isolation system. This damping makes the system nonclassically damped. The results indicate that system identification for a structure like this is feasible and reliable estimates of vibration periods and damping ratios are obtained. It was also observed that increase in damping ratio in the rubber bearings results in modal damping ratios that are different than the commonly used Rayleigh damping model.
A new method for harmonic response of non-proportionally damped structures using undamped modal data
Journal of Sound and Vibration, 1987
A method of calculating the receptances of a non-proportionally damped structure from the undamped modal data and the damping matrix of the system is presented. The method developed is an exact method. It gives exact results when exact undamped receptances are employed in the computation. Inaccuracies are due to the truncations made in the calculation of undamped receptances. Numerical examples, demonstrating the accuracy and speed of the method when truncated receptance series are used are also presented. Advantages of the method over classical methods are discussed, and it is concluded that the method is most advantageous when used for a structure with frequency and/or temperature dependent damping properties, or when the non-proportional part of the damping is local. The technique suggested can easily be applied to structural modification problems if there is no additional degree-of-freedom due to the modifying structure.
Mechanical Systems and Signal Processing, 2011
Engineering structures seldom behave linearly and, as a result, linearity checks are common practice in the testing of critical structures exposed to dynamic loading to define the boundary of validity of the linear regime. However, in large scale industrial applications, there is no general methodology for dynamicists to extract nonlinear parameters from measured vibration data so that these can be then included in the associated numerical models. In this paper, a simple method based on the information contained in the frequency response function (FRF) properties of a structure is studied. This technique falls within the category of single-degree-of-freedom (SDOF) modal analysis methods. The principle upon which it is based is effectively a linearisation whereby it is assumed that at given amplitude of displacement response the system responds at the same frequency as the excitation and that stiffness and damping are constants. In so doing, by extracting this information at different amplitudes of vibration response, it is possible to estimate the amplitude-dependent 'natural' frequency and modal loss factor. Because of its mathematical simplicity and practical implementation during standard vibration testing, this method is particularly suitable for practical applications. In this paper, the method is illustrated and new analyses are carried out to validate its performance on numerical simulations before applying it to data measured on a complex aerospace test structure as well as a full-scale helicopter.
MODAL ANALYSIS OF NONLINEAR SYSTEMS WITH NONCLASSICAL DAMPING
This paper presents a mode-superposition procedure for the analysis of nonlinear problems in structural dynamics where damping cannot be assumed proportional. The procedure consists of treating the nonlinearity as a pseudo force and using a complex eigenvalue solution to decouple the equations of motion. The response time history of a twenty-degree-of-freedom system with nonproportional damping to a base excitation is obtained using the proposed procedure and compared with that from a direct integration of the equations of motion. The comparison indicates excellent agreement. Few studies have used the mode-superposition procedure to solve nonlinear problems in structural dynamics (Molnar e t al. ; Riead ' ; Shah e t al. ; Stricklin and Haisler '). Such procedures have been limited to classically damped systems where proportional damping is assumed to uncouple the equa--tions of motion.
Noise and Vibration: Emerging Methods, NOVEM 2012. In: INTER-NOISE and NOISE-CON Congress and Conference Proceedings, Sorrento, Italy, 2012
The paper presents an output-only technique for the identification of the mass distribution of a vibrating system. The methodology exploits the orthogonality of the natural modes of vibration and is applied to a spring-mass system with n masses. A system of n algebraic equations is built using: (a) a number of n − 1 inner products between the natural modes; (b) the total mass (known) as the sum of the lumped masses. The solution of the system provides the mass distribution in terms of lumped masses. The evaluation of the natural modes is performed through time domain decomposition (TDD). TDD consists in the proper orthogonal decomposition (POD, also known as Karhunen-Loève decomposition, KLD) of the time-dependent displacement vector, suitably filtered to have one frequency (at once); this is equivalent to frequency domain decomposition (FDD). TDD (or FDD) provides a set of eigenvectors and eigenvalues. The former correspond to the natural modes; the latter equal the signal-energy of the corresponding modes. Modes with higher energy are considered for the algebraic equations system (a). Numerical results show a good agreement between identified mass distribution and (known) input values. The method can be extended to continuous structures in the discrete approach. given by the standard KLD coincide with the natural modes of vibration of the system [7]. In its standard form, the KLD or POD is limited to uniform-mass systems. The standard formulation for the Karhunen-Loève decomposition has been extended to the modal identification of non-uniformmass structures and addressed by Feeny and Kappagantu [7] for lumped parameters systems, and by Feeny [10] for continuos structures. Iemma et al. shed light on some relevant mathematical aspect of the method, by embedding the overall formulation for the KLD in a different Hilbert space. This extension is referred herein as the generalized Karhunen-Loève decomposition. The techniques proposed in Refs. for the extension of the KLD or POD to non-uniform-mass systems require the knowledge of the mass distribution, which, in certain cases, may be difficult to evaluate and provide. In 2002, Han and Feeny [12] suggest a technique that allow for the modal identification of non-uniform-mass systems in the case that the mass distribution is not available. This is based on the frequency-filtering of the measured signals, and on the consequent spectral analysis of the time-averaged autocorrelation tensor. After signal filtering, the analysis is conducted in the time domain and the technique is referred as the time domain decomposition (TDD, see Kim et al. [13]). Moreover, Mariani [14] and Mariani and Dessi [15] broaden the method and successfully applied the technique (referred in their work as band-pass POD, BP-POD) to the modal identification of a hydroelastic system. Diez and Leotardi applied this frequencyfiltered version of the KLD to the modal analysis of a wing with a tip-mounted engine in a uniform flow. The TDD is closely related to the frequency domain decomposition (FDD). This was first introduced by Brincker et al. , and applied to the modal identification of buildings structures (see also Brincker et al. [18]). The technique is based on the evaluation of the power spectral density (PSD) function matrix, related to a single natural 1 frequency. The consequent decomposition of the matrix through singular value decomposition (SVD) gives a set of functions, corresponding to the natural modes of vibration of the system. As the TDD, the method is independent of the knowledge of the mass distribution. The formulation in Brincker et al. [18] is based on the frequency response function (FRF) and yields a response spectral density matrix, defined in terms of poles and residues. The similarity of FDD and TDD has been studied in, e.g., Mariani [14].
Study of Non-Linear Behavior of Vibrating System
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. In the case of the real structures a linear model will be insufficient to describe the dynamic behavior correctly. It thus appears natural to introduce non-linear models of structures which are able to predict the dynamic behavior of the real structures. This seminar includes study of non-linear vibrations, it different types and various applications. Here the vibratory behavior of an absorber of vibration related to a system subjected to a harmonic load, in the presence of uncertainties on the design parameters is optimised. The total system is modeled by two degrees of freedom (2 dof) with a shock absorber and a generalized non-linear stiffness. one proposes to optimize the vibratory behavior of an absorber of vibration related to a system subjected to a harmonic load, in the presence of uncertainties on the design parameters. The total system is modelled by two degrees of freedom (2 dof) with a shock absorber and a generalized non-linear stiffness. The resolution is carried out in the temporal field according to a traditional diagram. It is a question of seeking the optimal responses envelopes of the deterministic and stochastic case and this for the non-linear displacements, phases and forces.
Use of reciprocal modal vectors for nonlinearity detection
Archive of Applied Mechanics (Ingenieur Archiv), 2000
This paper seeks to exploit the reciprocal modal vector orthogonality between experimentally-derived mode shapes and the corresponding measured frequency response functions in order to derive a criterion for the detection of structural non-linearities. Being based on the use of measured data and experimentally-derived quantities only, the method is directly applicable to practical engineering cases for which there are usually no spatial descriptions. A brief outline of the reciprocal modal vector theory is given ®rst, followed by a short description of a frequency-domain nonlinear response simulation technique using a harmonic balance approach. A detailed study of a 4-DOF system with cubic stiffness and friction damping nonlinearities is presented next. It is shown that the proposed nonlinearity detection criterion is relatively insensitive to measurement noise. It is concluded that the method is effective for detecting nonlinear behaviour in a consistent and quantitative fashion.