Nonlinear finite element model updating of an infilled frame based on identified time-varying modal parameters during an earthquake

A model updating methodology is proposed for calibration of nonlinear finite element (FE) models simulating the behavior of real-world complex civil structures subjected to seismic excitations. In the proposed methodology, parameters of hysteretic material models assigned to elements (or substructures) of a nonlinear FE model are updated by minimizing an objective function. The objective function used in this study is the misfit between the experimentally identified time-varying modal parameters of the structure and those of the FE model at selected time instances along the response time history. The time-varying modal parameters are estimated using the deterministic–stochastic subspace identification method which is an input–output system identification approach. The performance of the proposed updating method is evaluated through numerical and experimental applications on a large-scale three-story reinforced concrete frame with masonry infills. The test structure was subjected to seismic base excitations of increasing amplitude at a large outdoor shake-table. A nonlinear FE model of the test structure has been calibrated to match the time-varying modal parameters of the test structure identified from measured data during a seismic base excitation. The accuracy of the proposed nonlinear FE model updating procedure is quantified in numerical and experimental applications using different error metrics. The calibrated models predict the exact simulated response very accurately in the numerical application, while the updated models match the measured response reasonably well in the experimental application.     

Resonance response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random parametric excitation

The resonant response of a single-degree-of-freedom nonlinear vibro-impact oscillator with a one-sided barrier to a narrow-band random parametric excitation is investigated. The narrow-band random excitation used here is a bounded random noise. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, thereby permitting the applications of random averaging over "fast" variables. The averaged equations are solved exactly and an algebraic equation of the amplitude of the response is obtained for the case without random disorder. The methods of linearization and moment are used to obtain the formula of the mean-square amplitude approximately for the case with random disorder. The effects of damping, detuning, restitution factor, nonlinear intensity, frequency and magnitude of random excitations are analysed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak response amplitudes will reduce at large damping or large nonlinear intensity and will increase with large amplitude or frequency of the random excitations. The phenomenon of stochastic jump is observed, that is, the steady-state response of the system will jump from a trivial solution to a large non-trivial one when the amplitude of the random excitation exceeds some threshold value, or will jump from a large non-trivial solution to a trivial one when the intensity of the random disorder of the random excitation exceeds some threshold value.     

Non-trivial effect of fast vibration on the dynamics of a class of non-linearly damped mechanical systems

The effect of very high-frequency excitation on the slow dynamics of a class of non-linearly damped mechanical oscillators is considered. Two different models of damping namely, piecewise linear and pth power damping are considered. Fast excitation is modelled as triangular, sinusoidal and random base excitation. The effect of fast excitation is theoretically analyzed using the method of direct partition of motion (MDPM) and direct simulation. The method of numerical averaging is also used, where damping characteristics or excitations are not amenable to analytical techniques. Fast excitation has the non-trivial effect of increasing and decreasing the low-velocity damping of hard and soft dampers, respectively. The effect of fast excitation on the transient and steady state slow dynamics of the system is investigated by direct numerical integration of the equation of motion.     

Fundamental comparison of time-domain experimental modal analysis methods based on high- and first-order matrix models

All time-domain methods for experimental modal analysis (EMA) begin with a mathematical model. Based on either a high-order matrix polynomial model or a first-order state-space model, this paper emphasizes the comparison of numerical conditioning and stability, as well as the modal parameter estimation, among EMA methods. Numerical conditioning pertains to the perturbation behavior of a mathematical problem (model) itself and stability pertains to the perturbation behavior of an algorithm used to solve that problem on a computer. As various EMA methods are modeled differently with distinct solution algorithms, implementing these methods would have different conditioning and stability. In this paper, both deterministic and stochastic EMA methods are covered. Three different scenarios for the response signal are considered: (1) clean response from impulse loading, (2) noisy response from impulse loading, and (3) noisy response from ambient noise excitation. Comparing the numerical conditioning of various EMA methods, this paper theoretically illustrates that methods based on first-order state-space models are more likely to be well-conditioned (with a smaller conditioning number) than those based on high-order polynomial models. Furthermore, the numerical observation of a case study for a 6 degree-of-freedom system also suggests that first-order state-space model methods are more robust and accurate for the estimation of modal frequency and damping.     

Time and frequency response of structures with frequency dependent,non-proportional linear damping

A method to compute the non-stationary time and frequency response of structures with a frequency-dependent non-proportional linear damping, called the resonance modes method, is presented in this paper. It consists of two main steps. The first step aims at spotting the structure resonance modes, which are the solutions of the matrix nonlinear eigenvalue problem obtained using the finite element method in the complex plane. This step requires a complex eigensolver and an iterative scheme, a perturbation technique or a combination of both. The second step uses the computed resonance modes and an analytical expression of the inverse Laplace transform to deduce the time or frequency response of structures to general excitations. The response of an aluminum plate damped with an elastomer treatment to a point-force excitation, computed with the classical modal approach, the direct solution and the presented method shows its precision and efficiency. An acoustic power computation finally validates the implementation of a fast variant, based on the perturbation technique, for vibroacoustic applications.     

Green׳s function method for piezoelectric energy harvesting beams

The development of validated mathematical models for piezoelectric harvesters is important as it provides predictive capabilities of their performance and insight to their coupled electromechanical behavior. Advanced solutions to these models allows for more realistic parameters to be considered. In this paper, we present a Fourier Transform–Green?s Function (FTGF) solution approach to the distributed parameter coupled electromechanical equations for a piezoelectric beam excited by an arbitrary external transverse force. This method, as opposed to modal analysis, allows for frequency-dependent material properties and damping coefficients to be considered. The special case of a harmonic base excitation is considered and closed-form expressions for the frequency response functions of the voltage generated by piezoelectric layer, relative tip displacement and local bending strain are obtained. Finally, the FTGF solution to these frequency response functions is compared with the modal analysis solution along with experimental data for validation.     

基础激励下分数阶线性系统的响应特性分析

本文研究了基础激励下含分数阶阻尼的线性系统的响应特性. 当基础激励为简谐激励时, 通过待定系数方法求得系统的动力传递系数; 当基础激励为非简谐周期激励时, 首先将激励展开成傅里叶级数, 然后根据线性系统的叠加原理求得激励中各阶频率成分所引起的动力传递系数, 并根据展开的傅里叶级数解决了数值运算中的不可导问题. 用数值仿真的方法对解析结果进行了验证, 两者符合良好, 证明了解析分析的正确性. 研究表明, 基础激励引起的动力传递系数依赖于分数阶阻尼阶数的值, 通过调节阻尼阶数可以控制动力传递系数的大小. 对于基础激励为非简谐的周期激励情况, 当激励频率一定时, 激励中的高阶频率成分引起的动力传递系数可能大于激励中的低阶频率成分引起的动力传递系数. 因此, 激励中的高阶频率成分所起的作用是不可忽略的.     

On the application of blind source separation for damping estimation of bridges under traffic loading

The accurate and reliable estimation of modal damping from output-only vibration measurements of structural systems is a continuing challenge in the fields of operational modal analysis (OMA) and system identification. In this paper a modified version of the blind source separation (BSS)-based Second-Order Blind Identification (SOBI) method was used to perform modal damping identification on a model bridge structure under varying loading conditions. The bridge model was created with finite elements and consisted of a series of stringer beams supported by a larger girder. The excitation was separated into two categories: ambient noise and traffic loads with noise modeled with random forcing vectors and traffic simulated with moving loads for cars and partially distributed moving masses for trains. The acceleration responses were treated as the mixed output signals for the BSS algorithm. The modified SOBI method used a windowing technique to maximize the amount of information used for blind identification from the responses. The modified SOBI method successfully found the mode shapes for both types of excitation with strong accuracy, but power spectral densities (PSDs) of the recovered modal responses showed signs of distortion for the traffic simulations. The distortion had an adverse affect on the damping ratio estimates for some of the modes but no correlation could be found between the accuracy of the damping estimates and the accuracy of the recovered mode shapes. The responses and their PSDs were compared to real-world collected data and patterns similar to distortion were observed implying that this issue likely affects real-world estimates.     

An impact excitation system for repeatable,high-bandwidth modal testing of miniature structures

Miniature components and devices are increasingly seen in a myriad of applications. In general, the dynamic behavior of miniature devices is critical to their functionality and performance. However, modal testing of miniature structures poses many challenges. This paper presents a design and evaluation of an impact excitation system (IES) for repeatable, high-bandwidth, controlled-force modal testing of miniature structures. Furthermore, a dynamic model of the system is derived and experimentally validated to enable the identification of the system parameters that yield single-hit impacts with desired bandwidth and force magnitude. The system includes a small instrumented impact tip attached to a custom designed flexure-based body, an automated electromagnetic release mechanism, and various precision positioners. The excitation bandwidth and the impact force magnitude can be controlled by selecting the system parameters. The dynamic model of the system includes the structural dynamics of the flexure-based body, the electromagnetic force and the associated eddy-current damping, and the impact event. A validation study showed an excellent match between the model simulations and experiments in terms of impact force and bandwidth. The model is then used to create process maps that relate the system parameters to the number of hits (single vs. multiple), the impact force magnitudes and the excitation bandwidths. These process maps can be used to select system parameters or predict system response for a given set of parameters. A set of experiments is conducted to compare the performances of the IES and a (manual) miniature impact hammer. It is concluded that the IES significantly improves repeatability in terms of the impact bandwidth, location, and force magnitude, while providing a high excitation-bandwidth and excellent coherence values. The application of the IES is demonstrated through modal testing of a miniature contact-probe system.     

Response of a strongly non-linear oscillator to narrowband random excitations

The principal resonance of a van der Pol-Duffing oscillator subject to narrowband random excitations has been studied. By introducing a new expansion parameter the method of multiple scales is adapted for the strongly non-linear system. The behavior of steady state responses, together with their stability, and the effects of system damping and the detuning, and magnitude of the random excitation on steady state responses are analyzed in detail. Theoretical analyses are verified by some numerical results. It is found that when the random noise intensity increases, the steady state solution may change form a limit cycle to a diffused limit cycle, and the system may have two different stable steady state solutions for the same excitation under certain conditions. The results obtained for the strongly non-linear oscillator complement previous results in the literature for weakly non-linear systems.     

Effect of damping on the propensity of squeal instability: an experimental investigation

Friction induced vibrations in automotive brakes is recognized as a major problem in industry. Squeal is a difficult subject because of its unpredictability caused by a not completely understood sensitivity to variation of the system parameters. In the literature several analytical and numerical studies deal with the relationship between damping and system propensity to have instability. These studies highlight the existence of a nonintuitive effect of damping distribution on modal coupling that gives rise to the unstable vibrations. The complexity of commercial brakes and the difficulties to identify the values of modal damping in brake assemblies lead to the necessity to rely on experimental analysis using simplified test rigs. This paper presents an experimental investigation of the relationship between the distribution of modal damping and the propensity to develop squeal in a beam-on-disk setup, which reliably reproduces squeal events with easy control and measurement of the damping of the disk and the beam, respectively. The experiments highlight the key role played by the modal damping distribution on squeal: A nonuniform repartition of the modal damping causes an increase of the squeal propensity.     

Non-linear structural vibrations under combined parametric and external excitations

A set of second order equations with weak quadratic and cubic non-linearities is considered. Simultaneous parametric and external (forcing) excitations are included. The frequency of the parametric excitation is near a natural frequency of the system, and three cases are analyzed: (i) the external excitation is absent; (ii) the external excitation is present but is not involved in a resonance; and (iii) the external frequency is the same as the parametric frequency. Results are obtained by the method of multiple scales. Frequency-response curves are presented for various combinations of excitation amplitudes, damping coefficients, and phase shift between the excitations. It is found that stable multi-modal responses may exist in the first-order asymptotic solution, even though only one mode is involved in the resonance and no internal resonance condition is present.     

Excitation spectra of one-dimensional spin-1/2 Fermi gas with an attraction

Using an exact Bethe ansatz solution, we rigorously study excitation spectra of the spin-1/2 Fermi gas (called Yang–Gaudin model) with an attractive interaction. Elementary excitations of this model involve particle-hole excitation, hole excitation and adding particles in the Fermi seas of pairs and unpaired fermions. The gapped magnon excitations in the spin sector show a ferromagnetic coupling to the Fermi sea of the single fermions. By numerically and analytically solving the Bethe ansatz equations and the thermodynamic Bethe ansatz equations of this model, we obtain excitation energies for various polarizations in the phase of the Fulde–Ferrell–Larkin–Ovchinnikov-like state. For a small momentum (long-wavelength limit) and in the strong interaction regime, we analytically obtained their linear dispersions with curvature corrections, effective masses as well as velocities in particle-hole excitations of pairs and unpaired fermions. Such a type of particle-hole excitations display a novel separation of collective motions of bosonic modes within paired and unpaired fermions. Finally, we also discuss magnon excitations in the spin sector and the application of Bragg spectroscopy for testing such separated charge excitation modes of pairs and single fermions.     

Finite element analysis and experimental study on dynamic properties of a composite beam with viscoelastic damping

This paper investigates the frequency dependent viscoelastic dynamics of a multifunctional composite structure from finite element analysis and experimental validation. The frequency-dependent behavior of the stiffness and damping of a viscoelastic material directly affects the system's modal frequencies and damping, and results in complex vibration modes and differences in the relative phase of vibration. A second order three parameter Golla–Hughes–McTavish (GHM) method and a second order three fields Anelastic Displacement Fields (ADF) approach are used to implement the viscoelastic material model, enabling the straightforward development of time domain and frequency domain finite elements, and describing the frequency dependent viscoelastic behavior. Considering the parameter identification a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Agreement between the curve fits using both the GHM and ADF and experiment is within 0.001 percent error. Continuing efforts are addressing the material modulus comparison of the GHM and the ADF model. There may be a theoretical difference between viscoelastic degrees of freedom at nodes and elements, but their numerical results are very close to each other in the specific frequency range of interest. With identified model parameters, numerical simulation is carried out to predict the damping behavior in its first two vibration modes. The experimental testing on the layered composite beam validates the numerical predication. Experimental results also show that elastic modulus measured from dynamic response yields more accurate results than static measurement, such as tensile testing, especially for elastomers.     

Temperature dependence of quantal and thermal dampings of the hot giant dipole resonance

A systematic study of the damping of the giant dipole resonance (GDR) in ⁹⁰Zr, ¹²⁰Sn and ²⁰⁸Pb as a function of temperature T is performed. The double-time Green function technique is employed to determine the single-particle and GDR dampings. The single-particle energies, obtained in the Woods-Saxon potential for these nuclei, are used in the calculations. The results show that the coupling of collective vibration to the pp and hh excitations, which causes the thermal damping width, is responsible for the enlargement of the total width with increasing temperature up to T ≈ 3MeV and its saturation at higher temperatures. The quantal width, which arises from the coupling of the collective mode to the ph excitations decreases slowly with increasing temperature. The effect of single-particle damping on the GDR width is small. The results are found in an overall agreement with the experimental data for the GDR width, obtained in the inelastic α scattering and heavy-ion fusion reactions at excitation energies E^* ⩽ 450 MeV. At high excitation energies (E^* > 400 MeV) a behavior similar to the transition from zero to ordinary sounds is observed.     

Dynamic modelling of spur gear pair and application of empirical mode decomposition-based statistical analysis for early detection of localized tooth defect

Gears are one of the most common and important machine components in many advanced machines. An improved understanding of vibration signal is required for the early detection of incipient gear failure to achieve high reliability. This paper mainly consists of two parts: in the first part, a 6-degree-of-freedom gear dynamic model including localized tooth defect has been developed. The model consists of a spur gear pair, two shafts, two inertias representing load and prime mover and bearings. The model incorporates the effects of time-varying mesh stiffness and damping, backlash, excitation due to gear errors and profile modifications. The second part consists of signal processing of simulated and experimental signals. Empirical mode decomposition (EMD) is a method of breaking down a signal without leaving a time domain. The process is useful for analysing non-stationary and nonlinear signals. EMD decomposes a signal into some individual, nearly monocomponent signals, named as intrinsic mode function (IMF). Crest factor and kurtosis have been calculated of these IMFs. EMD pre-processed kurtosis and crest factor give early detection of pitting as compared to raw signal.     

Extension of the statistical modal energy distribution analysis for estimating energy density in coupled subsystems

The present article deals with an extension of the Statistical modal Energy distribution Analysis (SmEdA) method to estimate kinetic and potential energy density in coupled subsystems. The SmEdA method uses the modal bases of uncoupled subsystems and focuses on the modal energies rather than the global energies of subsystems such as SEA (Statistical Energy Analysis). This method permits extending SEA to subsystems with low modal overlap or to localized excitations as it does not assume the existence of modal energy equipartition. We demonstrate that by using the modal energies of subsystems computed by SmEdA, it is possible to estimate energy distribution in subsystems. This approach has the same advantages of standard SEA, as it uses very short calculations to analyze damping effects. The estimation of energy distribution from SmEdA is applied to an academic case and an industrial example.     

IN-SERVICE IDENTIFICATION OF NON-LINEAR DAMPING FROM MEASURED RANDOM VIBRATION

A non-linearly damped single-degree-of-freedom (s.d.o.f.) system under broadband random excitation is considered. A procedure for in-service identification of the damping characteristic from measured stationary response is described. The procedure is based on the stochastic averaging method. The explicit analytical solution is obtained for the integral equation, which relates the desired damping characteristics to the apparent force in the shortened equation for the slowly varying response amplitude, and thus to the measured probability density of the amplitude. The approach is of a non-parametric nature, which makes it convenient for testing hypotheses of damping mechanisms from measured random vibration data. Extensive results of numerical tests for the procedure are presented.     

Uninterrupted and accelerated vibrational fatigue testing with simultaneous monitoring of the natural frequency and damping

A mechanical system's modal parameters change when fatigue loading is applied to the system. In order to perform an accelerated vibration-based fatigue test these changes must be taken into account in order to maintain constant-stress loading. This paper presents an improved accelerated fatigue-testing methodology based on the dynamic response of the test specimen to the harmonic excitation in the near-resonant area with simultaneous monitoring of the modal parameters. The measurements of the phase angle and the stress amplitude in the fatigue zone are used for the real-time adjustment of the excitation signal according to the changes in the specimen's modal parameters. The presented methodology ensures a constant load level throughout the fatigue process until the final failure occurs. With the proposed testing methodology it is possible to obtain a S-N point of the Woehler curve relatively quickly and to simultaneously monitor the changes of the specimen's natural frequency and damping loss factor. The presented methodology with real-time control is verified on an aluminum Y-shaped specimen (10⁶ load cycles are achieved in 21 min) and is applicable to a specimen with arbitrary geometry. Besides the faster completion of the fatigue test the methodology can be adopted for the validation of the vibrational fatigue analysis.     

Spin 3/2 fermions with attractive interactions in a one-dimensional optical lattice: phase diagrams,entanglement entropy,and the effect of the trap

We study spin 3/2 fermionic cold atoms with attractive interactions confined in a one-dimensional optical lattice. Using numerical techniques, we determine the phase diagram for a generic density. For the chosen parameters, one-particle excitations are gapped and the phase diagram is separated into two regions: one where the two-particle excitation gap is zero, and one where it is finite. In the first region, the two-body pairing fluctuations (BCS) compete with the density ones. In the other one, a molecular superfluid (MS) phase, in which bound-states of four particles form, competes with the density fluctuations. The properties of the transition line between these two regions is studied through the behavior of the entanglement entropy. The physical features of the various phases, comprising leading correlations, Friedel oscillations, and excitation spectra, are presented. To make the connection with experiments, the effect of a harmonic trap is taken into account. In particular, we emphasize the conditions under which the appealing MS phase can be realized, and how the phases could be probed by using the density profiles and the associated structure factor. Lastly, the consequences on the flux quantization of the different nature of the pairing in the BCS and MS phases are studied in a situation where the condensate is in a ring geometry.