The force transmissibility of MDOF structures with a non-linear viscous damping device

In the present study, the concept of the Output Frequency Response Function (OFRF), recently proposed by the authors, is applied to theoretically investigate the force transmissibility of MDOF structures with a cubic non-linear viscous damping device. The results analytically show that the introduction of cubic non-linear damping can significantly reduce the transmissibility over all resonance regions for a Multiple Degree of Freedom (MDOF) structure and at the same time leave the transmissibility over the isolation region virtually unaffected. The analysis also indicates that a strong linear damping may shift the system resonances and compromise the beneficial effects of cubic non-linear viscous damping on the force transmissibility of MDOF structures. This suggests that a less significant linear damping together with a strong cubic non-linear damping can be used in MDOF structures to achieve a desired vibration isolation performance. This research work has a significant implication for the design of viscously damped MDOF structures for a wide range of practical applications.     

The effects of Coulomb friction on the performance of centrifugal pendulum vibration absorbers

We investigate analytically and experimentally the effects of Coulomb friction on the performance of centrifugal pendulum vibration absorbers (CPVAs), which are used to reduce torsional vibrations in rotating machinery. The analysis is based on perturbation methods applied to the nonlinear equations of motion for a rotor subjected to an engine order applied torque and equipped with a circular path CPVA with viscous and Coulomb damping. The experimental work is based on quantifying parameters for the damping model using free vibration measurements with a viscous and Coulomb damping identification scheme that is enhanced to better handle measurement noise, and running tests for steady-state operation under a range of loading conditions. The level of Coulomb damping is varied by adjusting the friction of the absorber connection bearing. Good agreement is found between the analytical predictions and the experimental data. It is shown that the absorber sticks up to a level of excitation that allows it to release, after which the Coulomb damping acts in the expected manner, resulting in lowered response amplitudes. The results obtained are of general use in assessing absorber performance when dry friction is present in absorber suspensions.     

A robust online parametric identification method for non-deteriorating and deteriorating distributed element models with viscous damping

The online parametric identification of deteriorating and non-deteriorating distributed element models (DEMs) with viscous damping is studied using a generalization of Masing model to provide the proper framework for identification. The approach renders the hysteretic response of the DEM into a time-independent single-valued mapping from equivalent displacement values into equivalent force values, while considering the effect of damping as a parallel element. This approach allows for parametric identification of this non-linear rate-dependent hysteretic behavior to be performed using non-linear optimization techniques. A changing objective function, defined as a norm of force estimation error over a shifting window of recent data, is employed so that classic non-linear optimization techniques can be used for the online identification problem. A variation of the steepest descent method is used with significant modifications. Special measures are taken to guarantee robustness of the results in presence of noise. The results show that the proposed identification method exhibits a very good performance in identifying the correct values of the parameters in real time, and is robust in dealing with noise. The proposed method can be applied to many other types of hysteretic behavior as well.     

Identifying Coulomb and Viscous Friction from Free-Vibration Decrements

This study focuses on an algorithm for the simultaneous identification of Coulomb and viscous damping effects from free-vibration decrements in a damped linear single degree-of-freedom (DOF) mass-spring system. Analysis shows that both damping effects can indeed be separated. Numerical study of a combined-damping system demonstrates a perfect match between the simulation parameters and the estimated values. Experimental study includes two types of real systems. The method is applied to an experimental industrial bearing. Experimental results are compared with numerical simulations to illustrate the reliability of this method. An analysis provides conservative bounds on error estimates. An example of the effect of quantization error on the estimations is included.     

Detection and description of non-linear phenomena in experimental modal analysis via linearity plots

Ground vibration tests (GVTs) on aircraft prototypes are mainly performed to experimentally identify the structural dynamic behaviour in terms of a modal model. This assumes a linear dynamic behaviour of the structure. However, in the practice of ground vibration testing it is often observed that structures do not behave in a perfectly linear manner. Non-linearities can be determined, for example, by free play in junctions, hydraulic systems in control surfaces, or friction. This paper compiles measured, typical, non-linear phenomena from various GVTs on large aircraft. The standard procedure in GVTs nowadays is the application of the Harmonic Balance method which linearizes the dynamic behaviour on the level of excitation. The procedure requires a harmonic excitation of the structure which is usually performed during phase resonance testing. The non-linear behaviour is investigated in terms of linearity plots in which the resonance frequency of a mode is plotted as a function of the excitation level. The experimental data is then compatible with all post-processing procedures for the measured results, e.g. updating of the finite element model or flutter calculations. This paper shows measured linearity plots for some typical non-linear phenomena. In the second part of the paper analytical linearity plots for different non-linear stiffness and damping models are considered in order to investigate whether the type of non-linearity can be identified from measured linearity plots. The analytical linearity plots are discussed with respect to their application limits. The analytical linearity plots are used to interpret the experimental linearity plots stemming from various GVTs on different aircraft prototypes. Finally, the observability of non-linear stiffness and non-linear damping characteristics via linearity plots is assessed.     

Damping in a parametric pendulum with a view on energy harvesting

The present article addresses the quantification of damping in a parametric pendulum, with a view on further applications in the design of energy harvesting devices. Detailed new experimental data is obtained for such purpose, and a novel mathematical model is presented. Linear and quadratic viscous damping and also dry friction are taken into account. To introduce the dry friction component, the pendulum axis is mounted on ball bearings. This is considered as a very realistic situation of a harvester. Damping parameters are determined by minimizing the difference between numerical and experimental time histories. It is shown that the damping model here presented is more adequate to replicate experiments than commonly used linear models, which consider only a linear viscous damping term characterized by means of free decay tests. It is also pointed that linear models are not adequate for refined studies, since they can lead to erroneous predictions of rotation zones, and consequently to wrong considerations in the design of pendulum harvesters.     

A comprehensive study on the behavior of a rigid block on an oscillating ground with friction,elastic and viscous forces

This paper investigates the behavior of a non-linear mechanical model where a block is driven by an oscillating ground through Coulomb friction, a linear viscous damper and a linear spring. The governing equation is solved analytically for different partial configurations: friction only, friction with viscous damping, friction with a linear restoring force, and for the complete model. Using dimensionless groups, the analysis of the block motion provides a comprehensive set of information on the motion regime (stick, stick-slip or permanent sliding), on the dominant energies or forces, on the resonance and on the amplification of the ground oscillation by the system. The limit between the stick-slip regime and the permanent slipping regime is found either analytically or numerically. It is also shown that there exists a set of parameters for which the friction force, the viscous dissipative force and the elastic restoring force are equal.     

Reducing force transmissibility in multiple degrees of freedom structures through anti-symmetric nonlinear viscous damping

In the present study, the Volterra series theory is adopted to theoretically investigate the force transmissibility of multiple degrees of freedom (MDOF) structures, in which an isolator with nonlinear anti-symmetric viscous damping is assembled. The results reveal that the anti-symmetric nonlinear viscous damping can significantly reduce the force transmissibility over all resonance regions for MDOF structures with little effect on the transmissibility over non-resonant and isolation regions. The results indicate that the vibration isolators with an anti-symmetric damping characteristic have great potential to solve the dilemma occurring in the design of linear viscously damped vibration isolators where an increase of the damping level reduces the force transmissibility over resonant frequencies but increases the transmissibility over non-resonant frequency regions. This work is an extension of a previous study in which MDOF structures installed on the mount through an isolator with cubic nonlinear damping are considered. The theoretical analysis results are also verified by simulation studies.     

Transient response of a ship to an abrupt flooding accounting for the momentum flux

Numerical non-linear time domain simulation method for damaged ship motions is presented. Floodwater motion modelling is based on the lumped mass method with a moving free surface. The ship and floodwater motions are fully coupled. The variation of the floodwater mass is accounted for. A model to account for the flooding ingress transporting the momentum is presented. The experiments of abrupt flooding have shown that the ship may experience the first large roll towards the undamaged side, especially when a large undivided compartment is flooded. The presented time domain model is validated against the experimental data on the roll damping of the flooded ship and transient flooding. Two different initial stability conditions and two different compartment layouts are studied. Viscous dissipation of the floodwater motions is modelled with an equivalent friction coefficient. The impact of the viscous damping is studied. Transient flooding tests show that the inflow momentum has to be accounted for when the undivided compartment is flooded. The simulation model is capable of capturing the impact of the inflooding jet and the first roll on the opposite side of the damage is reproduced.     

Damping for large-amplitude vibrations of plates and curved panels,Part 1: Modeling and experiments

Theoretical and experimental non-linear vibrations of thin rectangular plates and curved panels subjected to out-of-plane harmonic excitation are investigated. Experiments have been performed on isotropic and laminated sandwich plates and panels with supported and free boundary conditions. A sophisticated measuring technique has been developed to characterize the non-linear behavior experimentally by using a Laser Doppler Vibrometer and a stepped-sine testing procedure. The theoretical approach is based on Donnell's non-linear shell theory (since the tested plates are very thin) but retaining in-plane inertia, taking into account the effect of geometric imperfections. A unified energy approach has been utilized to obtain the discretized non-linear equations of motion by using the linear natural modes of vibration. Moreover, a pseudo arc-length continuation and collocation scheme has been used to obtain the periodic solutions and perform bifurcation analysis. Comparisons between numerical simulations and the experiments show good qualitative and quantitative agreement. It is found that, in order to simulate large-amplitude vibrations, a damping value much larger than the linear modal damping should be considered. This indicates a very large and non-linear increase of damping with the increase of the excitation and vibration amplitude for plates and curved panels with different shape, boundary conditions and materials.     

Experiments and Material Parameter Identification Using Finite Elements. Uniaxial Tests and Validation Using Instrumented Indentation Tests

In this article, we focus our attention on the relation between instrumented indentation tests and the prediction by means of finite element calculations. To this end, a finite strain viscoplasticity model of Perzyna-type with non-linear isotropic and kinematic hardening is calibrated at experimental data of steel S690QL. A particular concept for conducting uniaxial tensile and compression tests is taken up in order to represent the basic rate-dependent material behavior. In this respect, an algorithmic framework of material parameter identification using finite elements is proposed leading to a two-stage procedure in the case of the underlying rate-dependent constitutive model. On the basis of the termination points of relaxation the rate-independent equilibrium stress state can be identified and all viscous parts of the model are obtained using rate-dependent loading paths. Finally, use is made of finite elements for predicting indentation experiments, which results in a critical view on modeling and parameter identification on the basis of experimental results occurring in instrumented indentation tests.     

基于等效阻尼理论的金属橡胶弹性迟滞力学模型及实验研究

针对弹性多孔金属橡胶非线性迟滞特性力学行为,将迟滞恢复力-位移曲线分解为非线性单值曲线和椭圆,并将等效阻尼理论用于动态力学性能参数识别,从而建立了一种新型的适用于黏弹性阻尼材料的宏观唯象力学模型。采用不同相对密度的环形金属橡胶进行动态实验测试,以验证理论模型的准确性,结果表明该模型可将具有非线性特性的金属橡胶系统进行降阶处理,提高金属橡胶力学模型的预测效率,并能很好地描述金属橡胶的迟滞力学行为。另外,研究了在不同激励频率条件下金属橡胶的阻尼耗能特性。实验结果表明:在高频加载的条件下,黏性阻尼系数对动态加载频率不敏感,阻尼耗能与加载幅值之间呈线性正相关。基于等效阻尼理论的弹性迟滞力学模型具有一定的普适性,可进一步推广应用于类似弹性多孔材料的力学性能表征,为其工程应用提供理论基础。     

Identification of Material Damage Model Parameters: an Inverse Approach Using Digital Image Processing

A reliable prediction of ductile failure in metals is still a wide-open matter of research. Several models are available in the literature, ranging from empirical criteria, porosity-based models and continuum damage mechanics (CDM). One major issue is the accurate identification of parameters which describe material behavior. For some damage models, parameter identification is more or less straightforward, being possible to perform experiments for their evaluation. For the others, direct calibration from laboratory tests is not possible, so that the approach of inverse methods is required for a proper identification. In material model calibration, the inverse approach consists in a non-linear iterative fitting of a parameter-dependent load–displacement curve (coming from a FEM simulation) on the experimental specimen response. The test is usually a tensile test on a round-notched cylindrical bar. The present paper shows a novel inverse procedure aimed to estimate the material parameters of the Gurson–Tvergaard–Needleman (GTN) porosity-based plastic damage model by means of experimental data collected using image analysis. The use of digital image processing allows to substitute the load–displacement curve with other global quantities resulting from the measuring of specimen profile during loading. The advantage of this analysis is that more data are available for calibration thus allowing a greater level of confidence and accuracy in model parameter evaluation.     

On estimating site damping with soil non-linearity from earthquake recordings

This study proposes an approach for estimating the damping factor of non-linear soil and its role in seismic wave responses at soil sites from earthquake recordings. In particular, the paper first reviews the Fourier-based approach to characterizing the dynamic features of soil sites in seismic ground motion and shows the deficiencies for assessing non-linear site amplification and damping. It then offers an alternative approach, i.e., the Hilbert-Huang transform (HHT) for non-linear and non-stationary data process, in addressing the non-linearity issues. This study focuses on discerning the signature of non-linear site damping from the general non-linear features in the motion that are typically shown in the frequency-dependent site amplification. As a result, the study proposes an HHT-based approach to estimate site damping with non-linear soil from earthquake recordings and to measure the influences in the motion at soil sites. With the use of recordings from the mainshock and aftershock of the 2001 Nisqually earthquakes, this study shows that the proposed approach is able to assess appropriately the site damping that can be used to estimate the influences of soil damping in non-linear site responses.     

Effects of anti-symmetric nonlinear viscous damping on the force transmissibility of multi-degree of freedom structures

In the present study, the concept of the output frequency response function is applied to theoretically investigate the force transmissibility of multi-degree of freedom (MDOF) structures with a nonlinear anti-symmetric viscous damping. The results reveal that an anti-symmetric nonlinear viscous damping can significantly reduce the transmissibility over all resonance regions for MDOF structures while it has almost no effect on the transmissibility over non-resonant and isolation regions. The results indicate that the vibration isolators with an anti-symmetric damping characteristic have great potential to overcome the dilemma in the design of linear viscously damped vibration isolators where an increase of the damping level reduces the force transmissibility over resonant region but increases the transmissibility over non-resonant regions.     

Research on linear/nonlinear viscous damping and hysteretic damping in nonlinear vibration isolation systems

A nonlinear vibration isolation system is promising to provide a high-efficient broadband isolation performance. In this paper, a generalized vibration isolation system is established with nonlinear stiffness, nonlinear viscous damping, and Bouc-Wen(BW)hysteretic damping. An approximate analytical analysis is performed based on a harmonic balance method(HBM) and an alternating frequency/time(AFT) domain technique.To evaluate the damping effect, a generalized equivalent damping ratio is defined with the stiffness-varying characteristics. A comprehensive comparison of different kinds of damping is made through numerical simulations. It is found that the damping ratio of the linear damping is related to the stiffness-varying characteristics while the damping ratios of two kinds of nonlinear damping are related to the responding amplitudes. The linear damping, hysteretic damping, and nonlinear viscous damping are suitable for the small-amplitude, medium-amplitude, and large-amplitude conditions, respectively. The hysteretic damping has an extra advantage of broadband isolation.     

Effect of the geometry on the non-linear vibration of circular cylindrical shells

The non-linear vibration of simply supported, circular cylindrical shells is analysed. Geometric non-linearities due to finite-amplitude shell motion are considered by using Donnell's non-linear shallow-shell theory; the effect of viscous structural damping is taken into account. A discretization method based on a series expansion of an unlimited number of linear modes, including axisymmetric and asymmetric modes, following the Galerkin procedure, is developed. Both driven and companion modes are included, allowing for travelling-wave response of the shell. Axisymmetric modes are included because they are essential in simulating the inward mean deflection of the oscillation with respect to the equilibrium position. The fundamental role of the axisymmetric modes is confirmed and the role of higher order asymmetric modes is clarified in order to obtain the correct character of the circular cylindrical shell non-linearity. The effect of the geometric shell characteristics, i.e., radius, length and thickness, on the non-linear behaviour is analysed: very short or thick shells display a hardening non-linearity; conversely, a softening type non-linearity is found in a wide range of shell geometries.     

Variation of Small Amplitude Vibration Dynamic Properties with Displacement in Reinforced Concrete Structures

Correlating damage level and changes in dynamic characteristics of a structure forms the basis for damage detection techniques in structural health monitoring. In reinforced concrete building structures such correlation is not well established. A damage detection technique capable of identifying the structural condition of the system based on its small amplitude vibration response is desirable because such response is easier to obtain. It is a common practice in engineering applications to estimate dynamic parameters from small-amplitude vibrations assuming a linear behavior of the structure. This simplification causes inaccurate estimation of the dynamic properties in reinforced concrete structures due to the presence of nonlinear elastic behavior. In this study no such assumption is made and a linear model is only used for sets of data corresponding to the same displacement amplitude of a nonlinear elastic system. The trends found between small-amplitude vibration dynamic properties and past levels of maximum displacement in various reinforced concrete structures are reported. In addition to analytical and numerical studies, results from a series of laboratory tests are reported to demonstrate the use of the approach. One full-scale three-story reinforced concrete flat-plate building and six small-scale reinforced concrete beams were examined. In this study, small displacements are defined as displacements below an overall drift ratio of 0.03%. The displacement dependence of the dynamic properties is considered explicitly. It was found that while fundamental frequencies of the examined reinforced concrete specimens were found to decrease uniformly as past peak displacement level increased, the equivalent viscous damping ratio was found to increase until the past peak displacement reached the neighborhood of nominal yield displacement and then observed to decrease when the specimens are pushed beyond the nominal yield displacement level, which has not been reported in literature before. Recommendations are provided as to how small amplitude vibration tests should be set up to avoid misleading observations due to nonlinear response at small amplitude response, observations that could lead to erroneous conclusions regarding the damage state of a structure.     

Non-linear dynamics of asymmetric structures under 2:2:1 resonance

The non-linear dynamic behavior of a novel model of a single-story asymmetric structure under earthquake and harmonic excitations and near two-to-one internal resonance is investigated. The non-linearities of the proposed model, ignored in conventional linear models, are caused by non-linear inertial coupling between translational and torsional degrees of freedom defined in the directions of a non-inertial rotational system of reference, attached to the center of mass of the floor. The multiple scales method is used to achieve approximately linear solutions for the originally non-linear equations near a two-to-one ratio of external and internal resonant conditions. The suitability of the proposed model is justified by the similarity between the simulated response of the non-linear model and the experimental results. The numerical results of time history and frequency domain analyses illustrate the difference between the non-linear and linear models. Energy transfer from a lower natural frequency excited mode to a higher one due to non-linear interaction in the novel model is shown. The effects of amplitude, frequency detuning parameters, uncoupled lateral and torsional frequencies, and damping ratio on the responses are inspected and some non-linear phenomena such as hysteresis, jumping, hardening, and softening are observed.     

液体边界层对平板结构振动阻尼效应分析

本文分析了液体边界层对平板振动的阻尼效应，基于液体的Stokes边界层理论，建立了考虑液体边界层阻尼效应的平板振动单自由度系统运动方程，利用Laplace变换方法求解了运动方程，得到了平板结构的位移响应解答，制作了一个单自由度质量-弹簧系统来测量水体粘滞阻尼效应，并将理论与试验结果进行了对比分析，表明Stokes边界层理论能较好模拟水体的粘滞阻尼特性，边界层阻尼明显减小了结构的动力响应，本文最后讨论了所提的分析方法的适用范围．