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The thermomechanically tunable properties of shape memory alloys (SMAs) have long been exploited for the development of adaptive hybrid composite materials. Most of the literature focuses on composites integrated either with wires or thin homogeneous laminae. The possibility of tuning the behavior of the laminae and the resulting energy dissipation characteristics (i.e. the effective damping) by exploiting the geometric patterning as well as the pre-strain of individual SMA laminae remains unexplored. This work investigates, by means of numerical simulations, the nonlinear damping characteristics of hybrid composite plates (HCPs) assembled by stacking fiber composites and SMA layers (either monolithic or patterned). The model of the HCP developed for the numerical study accounts for both geometric and material nonlinearities, and for the possibility to apply pre-strain on individual SMA layers. The damping properties of the HCP are assessed via free decay analysis, where the effects of the position, the pre-strain level, and the pattern imprinted on the SMA layer are investigated under different operating regimes. The effective damping indicates that the arrangement and the total removed volume ratio associated with the patterning can significantly affect the damping capacity of the HCP under both uni-axial and bi-axial pre-strain conditions. The damping capacity of the HCP is also estimated as a function of the SMA total transformed volume fraction in order to identify the types of patterns and the pre-strain profiles capable of improving the overall damping capacity of the HCP. [ABSTRACT FROM AUTHOR]

This article addresses the measurement of the nonlinear modes of highly flexible structures vibrating at extreme amplitude, using a Phase-Locked Loop experimental continuation technique. By separating the motion into its conservative and dissipative parts, it is theoretically proven for the first time that phase resonance testing organically allows for measurement of the conservative nonlinear modes of a structure, whatever be its damping law, linear or nonlinear. This result is experimentally validated by measuring the first three nonlinear modes of a cantilever beam. Extreme amplitudes of motion (of the order of 120° of cross section rotation for the first mode) are reached for the first time, in air with atmospheric pressure condition, responsible for a strong nonlinear damping due to aeroelastic drag. The experimental backbone curves are validated through comparison to the conservative backbone curves obtained by numerical computations, with an excellent agreement. The classical trends of cantilever beams are recovered: a hardening effect on the first nonlinear mode and softening on the other modes. The nonlinear mode shapes are also measured and compared to their theoretical counterparts using camera capture. Finally, it is shown that the damping law can be estimated as a by-product of the phase resonance measurement of the conservative nonlinear modes. As an original result, the damping law is observed to be highly nonlinear, with quadratic and cubic evolutions as a function of the structure's amplitude. [Display omitted] • Phase resonance enables measurement of conservative nonlinear modes for arbitrary damping. • The experimental damping law around a resonance is obtained as a by-product. • Phase-Locked Loop testing experimental continuation is used. • The nonlinear modes of a cantilever beam are measured at extreme amplitude. • Quadratic nonlinear aeroelastic drag damping is estimated. [ABSTRACT FROM AUTHOR]

In this paper, we incorporate the effect of nonlinear damping with the concept of locally resonant metamaterials to enable vibration attenuation beyond the conventional bandgap range. The proposed design combines a linear host cantilever beam and periodically distributed inertia amplifiers as nonlinear local resonators. The geometric nonlinearity induced by the inertia amplifiers causes an amplitude-dependent nonlinear damping effect. Through the implementation of both modal superposition and numerical harmonic methods with Alternating Frequency Time and numerical continuation techniques, the finite nonlinear metamaterial is accurately modeled. The resulting nonlinear frequency response reveals the bandgap is both amplitude-dependent and broadened. Furthermore, the nonlinear interaction between the local resonators and the mode shapes of the host beam is discussed, which leads to efficient modal frequency dissipation ability. The theoretical results are validated experimentally. By embedding the nonlinear damping effect into locally resonant metamaterials, wideband and shock wave attenuation of the proposed metamaterial is achieved, which opens new possibilities for versatile metamaterials beyond the conventional bandgap ranges of their linear counterparts. • A nonlinear metamaterial attenuates vibration beyond the bandgap range of the linear counterparts. • Nonlinear dispersion and frequency response by homogenization and harmonic balance methods. • Nonlinear damping effect enables efficient modal frequency dissipation. • Experiments validate the broadening of bandgap and modal frequency dissipation. [ABSTRACT FROM AUTHOR]

Highlights • The nonlinear damping of the limb-like (LLS) or X-shaped structure is studied. • The damping is a nonlinear function of vibration displacement and varies at different frequencies. • The damping is high at resonance (large-displacement) but low at other frequencies. • The structural parameters can be tuned to achieve a desired passive and nonlinear damping. • The study demonstrates an innovative and passive solution for desired nonlinear damping design in engineering. Abstract Natural biological systems can always demonstrate amazing properties or performance subject to external excitation. The nonlinear damping characteristics of a passive bio-inspired limb-like structure (LLS) or X-shaped structure is studied in this paper with theoretical modelling and experimental validation. It is revealed that, (1) the linear horizontally-placed damper and joint friction can both produce equivalent nonlinear damping in the vertical direction; (2) the equivalent damping characteristic is shown to be a nonlinear function of the vibration displacement and varies at different frequency; (3) the equivalent damping can be beneficially varying with vibration displacement and frequencies, i.e., automatically high at around resonance frequency (for large-displacement vibration) but low at other frequencies (for low-displacement vibration), showing an ideal passive and nonlinear damping property; (4) the structural rod length, asymmetric rod length ratio, layer number, and assembly angle can all affect the equivalent nonlinear damping and thus can be used to tune the damping characteristic to the ideal case conveniently in practice. The results of this study demonstrate an innovative and passive solution for designing desired nonlinear damping characteristics in various engineering practices by employing the LLS. [ABSTRACT FROM AUTHOR]

Highlights • A unique human body inspired anti-vibration system is presented. • The muscle function of human body is explored as special nonlinear stiffness and damping units. • The rotation motion of upper body or arm swinging are employed in a simple but effective passive anti-vibration design. • Some nonlinear features are discovered for the first time leading to very beneficial vibration isolation. Abstract This paper presents a unique human body inspired passive vibration isolation system with a special coupled nonlinear inertia design. This human body inspired anti-vibration structure with nonlinear inertia consists of a compact X-shaped structure with a horizontally-installed spring system to simulate the functions of legs and muscles, and a compact rotational unit to mimic the arms swinging (or upper body movement) and/or muscle extension-flexion motion during human walking. It is particularly focused on the beneficial nonlinear effect incurred by the rotational unit, including a nonlinear equivalent mass and a very special nonlinear inertia incurred conservative force. It is shown that, the nonlinear properties (in equivalent stiffness, damping and mass simultaneously) can obviously improve the vibration isolation at low frequencies and/or in a broadband frequency range, and increase the stability of the mass center of the overall isolation system during vibration, irrespective of the loading and excitation conditions. The muscle function and swinging arms of human body are for the first time employed with an innovative and simple mechanical design for vibration control and successfully validated by experimental prototypes for their excellent beneficial nonlinear features. This paper presents a unique simple passive and adjustable anti-vibration solution of great potential in extensive engineering applications. [ABSTRACT FROM AUTHOR]

Highlights • Behaviour is hardening. • Deviation between elastic and viscoelastic results is larger for larger excitations. • MCS theory predicts primary resonance at larger excitation frequencies. Abstract A viscoelastic model for a shear-deformable microplate is developed in this paper while accounting for geometric nonlinearities. Nonlinear numerical solutions are conducted to examine the resonant oscillations of the microsystem. The geometrically nonlinear theoretical model is developed utilising the Kelvin-Voigt viscoelastic model, to account for nonlinear dissipation, the modified version of the couple-stress theory, to account for small-scale characteristics, and the third-order deformation theory, to account for shear stress. The constitutive relations for both the classical and higher-order stress tensors are constructed and are divided into elastic and viscous components. The elastic components are used to develop the potential strain energy and the viscous components are employed to model the virtual work of damping (energy dissipation). Additionally, the microplate motion energy is developed while accounting for all in-plane, out-of-plane, and rotational motions. A distributed harmonic load, as the representative an external force, is applied to the microsystem and the corresponding virtual work is obtained. The generalised Hamilton's principle is applied to the virtual works and variations of the energy terms, resulting in the nonlinear equations of motion of the microsystem. Being of partial differential type, the equations of motion are discretised into a set of nonlinearly coupled ordinary differential equations consisting of nonlinear geometric and nonlinear damping terms. A solution procedure for the forced oscillation analysis of the microsystem is developed using a continuation method. Different diagrams are constructed to examine the nonlinear resonant characteristics of the viscoelastic shear deformable microplate and to highlight the nonlinear dependency of the Kelvin-Voigt viscoelastic damping mechanism on the oscillation amplitude, for a geometrically nonlinear model. [ABSTRACT FROM AUTHOR]

Highlights • The proposed quasi-static method computes effective natural frequency and damping. • The method enables updating of Iwan element parameters in a bolted joint model. • Modal coupling is addressed by updating to measurements where one mode is dominant. • The updated model is verified to accurately predict modal coupling in measurements. Abstract Structures with bolted joints are known to exhibit amplitude-dependent shifts in the natural frequency and damping of its modes that are challenging to model accurately. These two properties could be derived from dynamic simulations (for example, by computing the response of the structure to an impulsive force), but such simulations are expensive. Any updating strategy based on dynamic simulation therefore becomes cumbersome. A solution was provided by Festjens et al. (2013) whereby the joint is treated as a static subcomponent in an otherwise dynamic global model. This work proposes a few modifications to their theory, resulting in a highly-efficient quasi-static algorithm that can be used to compute the amplitude-dependent frequency and damping by loading the finite element model in the shape of one of its modes. This new quasi-static method was then utilized to update a set of Iwan joint parameters so that the shift in the frequencies and damping seen in the finite element model matched those from measurements on an experimental beam containing three bolted joints. The updated model was then verified to be capable of capturing with good accuracy the effects of modal coupling seen in the impact response of the beam. [ABSTRACT FROM AUTHOR]

Highlights • A novel passive human body inspired anti-vibration structure (HBIAVS) is proposed. • The HBIAVS is to mimic human legs' structure and rotational motion of human arms and upper body. • The HBIAVS demonstrates advantageous nonlinear stiffness, nonlinear damping and nonlinear inertia simultaneously. • Ultra-low resonant frequency, beneficial anti-resonance, and broadband low transmissibility etc can all be achieved. • This study presents an innovative technology to many passive vibration control. Abstract This paper investigates, for the first time, a human body inspired anti-vibration structure (HBIAVS) for exploring its vibration isolation potential. The anti-vibration structure HBIAVS consists of an X-shaped supporting structure to simulate legs of human body and a rotational unit with mass to mimic rotational motion of arms and upper body during human walking. The nonlinear property of the HBIAVS can obviously improve the vibration isolation at low frequencies and/or in a broadband frequency range. With mathematical modeling, the influence incurred by different structural parameters on system isolation performance is systematically studied. It is revealed for the first time that, the HBIAVS has passive and very beneficial nonlinear stiffness, nonlinear damping and nonlinear inertia simultaneously and it can achieve a tunable ultra-low resonant frequency and an advantageous anti-resonance characteristic. All these beneficial properties are adjustable with respect to structural parameters, compared with other benchmark vibration isolation systems and validated by experimental results. [ABSTRACT FROM AUTHOR]

The input shaping technique alters the human-operator commands to reduce the payload oscillation. A single radial spring-damper can simultaneously produce three dampings to suppress the tridirectional vibration of a crane payload. Combination of input shaping and radial spring-damper can be a sensorless approach to reduce the vibration induced by both operator commands and external disturbances. A numerical simulation of a boom crane is carried out to clarify the effectiveness of each component in the combination. An experiment of a laboratory boom crane is presented to show the effect of radial spring-damper. [ABSTRACT FROM AUTHOR]

Three different dissipation models have been used to identify the increase of damping with the vibration amplitude for a rubber rectangular plate. For this purpose, a square rubber plate made of silicone with fixed edges has been tested and its linear and nonlinear responses have been measured by laser Doppler vibrometers. First, a reduced-order model, using energy based approach and global discretization, has been constructed, taking into account geometric imperfections; the linear viscous damping at each excitation level in the nonlinear regime has been identified from the experimental data. This numerical model with linear viscous damping has been widely validated and constitutes the basis for comparison with subsequent damping identifications. Then, three different single degree of freedom models have been fitted to the same experimental data; each model has a different damping description. Specifically, the models are based on a modified Duffing oscillators with linear, quadratic and cubic stiffness and: (i) a linear viscous damping; (ii) a nonlinear viscoelastic dissipation described by the loss factor; (iii) a standard linear solid viscoelastic model with nonlinear springs. The dissipation identified by the different models is discussed and confirms the major nonlinear nature of damping as a function of the vibration amplitude. [ABSTRACT FROM AUTHOR]

Inspired by the smooth motions of a running kangaroo, a bio-inspired quadrilateral shape (BIQS) structure is proposed to suppress the vibrations of a free-floating spacecraft subject to periodic or impulsive forces, which may be encountered during on-orbit servicing missions. In particular, the BIQS structure is installed between the satellite platform and the capture mechanism. The dynamical model of the BIQS isolation system, i.e. a BIQS structure connecting the platform and the capture mechanism at each side, is established by Lagrange’s equations to simulate the post-capture dynamical responses. The BIQS system suffering an impulsive force is dealt with by means of a modified version of Lagrange’s equations. Furthermore, the classical harmonic balance method is used to solve the nonlinear dynamical system subject to periodic forces, while for the case under impulsive forces the numerical integration method is adopted. Due to the weightless environment in space, the present BIQS system is essentially an under-constrained dynamical system with one of its natural frequencies being identical to zero. The effects of system parameters, such as the number of layers in BIQS, stiffness, assembly angle, rod length, damping coefficient, masses of satellite platform and capture mechanism, on the isolation performance of the present system are thoroughly investigated. In addition, comparisons between the isolation performances of the presently proposed BIQS isolator and the conventional spring-mass-damper (SMD) isolator are conducted to demonstrate the advantages of the present isolator. Numerical simulations show that the BIQS system has a much better performance than the SMD system under either periodic or impulsive forces. Overall, the present BIQS isolator offers a highly efficient passive way for vibration suppressions of free-floating spacecraft. [ABSTRACT FROM AUTHOR]

[Display omitted] • Realising a linear elastic oscillator using an inherently nonlinear stiffness configuration. • Achieving a strong geometrical nonlinear damping characteristic. • Prototyping a test-rig device for validation and damping insight. • Proposing an alternative design solution to classical linear oscillators. This paper proposes a paradigm shift in the perspective of designing nonlinear oscillators, i.e., the exploitation of nonlinearity to achieve a linear behaviour to good engineering purposes. An elastic suspension with four inclined springs is studied, which has an inherently strong geometric nonlinear stiffness characteristic. Such a configuration has attracted remarkable research efforts in last couple of years, because, compared to other classical nonlinear spring configurations, it has more design parameters, which can be wisely selected to attain a tailored force–displacement characteristic. A particular relationship among these parameters is found so that the overall characteristic becomes exactly linear. Compared to the use of classical linear springs mounted along the direction of motion, the proposed configuration with inclined springs has the potential to allow more freedom in the dimensioning of an engineering device. Also, while the equivalent spring obtained is linear, the equivalent damping is not, and this has the potential advantage of practically realising a linear elastic behaviour with the benefit of geometrical nonlinear damping. Experiments are performed for validation on a prototype device, and results confirm the linear behaviour predicted by the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this study model identification of the nonlinear dynamics of a micro-speaker is carried out by purely electrical measurements, avoiding any explicit vibration measurements. It is shown that a dynamic model of the micro-speaker, which takes into account the nonlinear damping characteristic of the device, can be identified by measuring the response between the voltage input and the current flowing into the coil. An analytical formulation of the quasi-linear model of the micro-speaker is first derived and an optimisation method is then used to identify a polynomial function which describes the mechanical damping behaviour of the micro-speaker. The analytical results of the quasi-linear model are compared with numerical results. This study potentially opens up the possibility of efficiently implementing nonlinear echo cancellers. [ABSTRACT FROM AUTHOR]

• Augmenting nonlinear system identification with model reduction and virtual sensing. • Releasing the need of measuring data in the location of nonlinear elements for identification. • Automating the discovery of stiffness and damping nonlinear models from measured data. • Physics-informed identification through an efficient reduced-order model. • Experimental identification of nonlinear assembly with multiple bolted joints. The identification of nonlinearities that have a significant impact on dynamic behaviour of complex mechanical structures is necessary for ensuring structural efficiency and safety. A new methodology for structural identification of nonlinear assemblies is proposed in this paper that enables the discovery of stiffness and damping nonlinear models especially when it is not possible to directly measure the degrees of freedom where non-trivial nonlinearities are located. Input-output time-domain data collected at accessible locations on the structure are used to learn nonlinear models in the unmeasured locations. This is accomplished by making use of virtual sensing and model reduction schemes along with a physics-informed identification method recently developed by the authors (Safari and Londoño 2021). The methodology is suited for weakly nonlinear systems with localised nonlinearities for which their location is assumed to be known. It also takes into account dominant modal couplings within the identification process. The proposed methodology is demonstrated on a case study of a nonlinear structure with a frictional bolted joint, in numerical and experimental settings. It is shown that the model selection and parameter estimation for weakly nonlinear elements can be carried out successfully based on a reduced-order model which includes only a modal equation along with relevant modal contributions. Using the identified localised nonlinear models, both the reduced and full-order models can be updated to simulate the dynamical responses of the structure. Results suggests that the identified nonlinear model, albeit simple, generalises well in terms of being able to estimate the structural responses around modes which were not used during the identification process. The identified model is also interpretable in the sense that it is physically meaningful since the model is discovered from a predefined library featuring different nonlinear characteristics. [ABSTRACT FROM AUTHOR]

Control of vibrations in fluid transmitting pipes has emerged as a major engineering challenge. For higher fluid flow rate, support vibration, and excitation load, the vibrations in pipes that convey fluid can be complex and, at times, catastrophic. Recent works have demonstrated that Nonlinear Energy Sink (NES) can be effectively employed for passive vibration control of fluid-conveying pipes. The present work explores the possibility of using NES with nonlinear damping to overcome the limitations of conventional NES-based vibration control in fluid-transmitting pipes. Geometric nonlinear damping (velocity-displacement dependent damping), whose physical realization is possible by a suitable geometric configuration of the linear viscous damper, is considered in this study. Euler–Bernoulli beam theory models the fluid conveying pipe under an external harmonic load. The reduced order model obtained through the Galerkin procedure is investigated analytically using the complex averaging method. The occurrence of strongly modulated and weakly modulated responses are demonstrated, and their effect on vibration mitigation is shown. The frequency range exhibiting strongly modulated response in the system is significantly increased with the addition of nonlinear damping to NES. This assists in mitigating pipe vibrations by initiating targeted energy transfer from pipe to NES. The Saddle–Node and Hopf bifurcation boundaries in the model are identified using the slow-flow dynamics obtained by complex averaging. The effect of nonlinear damping, external excitation, the location of NES, and the flow rate on the bifurcation boundaries are investigated in detail. It is shown that the external excitation needed to trigger Saddle–Node bifurcation is largely reduced for NES having nonlinear damping. Furthermore, a study of Hopf bifurcation in the frequency domain shows that using NES with nonlinear damping significantly decreases the frequency range in which high-amplitude isolated solutions occur. The method of multiple scales is used to derive the analytical expression for the slow invariant manifold of the pipe-NES system. The vibration suppression mechanism and the occurrence of strongly modulated responses are explained with the help of flow on slow invariant manifolds. A comparison of the topology of the slow invariant manifolds shows that using nonlinear damping with NES substantially reduces the vibration amplitude of the pipe structure. The percentage of energy transferred to the NES is quantified to show the effectiveness of the proposed absorber over normal NES for different flow speeds. [ABSTRACT FROM AUTHOR]

• Combined linear and non-linear cubic damping is more efficient. • Combined damping can significantly suppress maximum amplitude. • Combined damping eliminates jump effects. • Combined damping significantly reduces the amplitude of the vibration frequency. • Cubic nonlinear damping significantly narrows the unstable region. The article examines the effect of linear damping and combined linear and nonlinear cubic damping of an elastic support on the dynamics of a gyroscopic rigid rotor with a non-ideal energy source, taking into account cubic nonlinear stiffness of the support material. Analysis of the research results shows that both linear damping and combined linear and nonlinear cubic damping can significantly suppress the resonance peak of the fundamental harmonic, reduce the amplitude of vibration frequency variation and stabilize the shaft rotation speed, but the effect of combined damping is more significant. In non-resonant regions, where the speed is higher than the natural frequency of the rotor system, both types of damping shorten the distance between jumps in nonlinear resonance curves and eliminate them. If linear damping mainly affects the boundaries of the instability region close to the resonant frequency, then nonlinear cubic damping significantly narrows the width of the instability region throughout the entire range beyond the resonant rotation speed. These results can be successfully used for the development of passive vibration isolators used to damp vibrations generated by rotary machines [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a study of nonlinear vibrating mechanical structures whose resonances have nonlinearities due to stiffness and/or damping. A method is presented to detect the presence of nonlinear distortions qualitatively and quantitatively. Basically, a device under test is configured as a second-order, single-input–single-output closed-loop feedback system where static nonlinearities are confined to the feedback path, and the dynamic linear part is modeled as the forward gain. The system is excited by a random phase multisine, which allows for the modeling of the linear part by its frequency response function, thus facilitating the characterization of the nonlinear part. The identification is formulated in a linear-in-parameters framework, using input and output regressors. A good agreement between the estimated data and the measurements indicates the presence of nonlinearities due to stiffness and damping with distinguishable levels. [Copyright &y& Elsevier]

A procedure is presented to identify the nonlinear damping and stiffness parameters of harmonically forced continuous systems showing nonlinear vibrations with a one-to-one internal resonance. The identification is obtained by using a two-degree-of-freedom (2DOF) model. Cubic nonlinear damping is introduced in the 2DOF model in addition to the classical viscous one. Different damping coefficients are considered within or outside the internal resonance frequency range. The parameter estimation relies on the harmonic balance method. It is shown that a three-harmonic expansion is needed to obtain a converged solution, although often only the first harmonic, without mean component, is experimentally measured with accuracy. The proposed novel cascade procedure consists in (i) a single-term harmonic balance parameter estimation used as initiation of (ii) a minimization of the distance between data and a three-term harmonic balance model by means of a genetic algorithm. These procedures are validated by parameter identification on synthetic (numerically generated) and experimentally measured frequency–responses. The nonlinear damping model is validated by comparison with level-adjusted linear damping model estimated at each level of harmonic force, demonstrating its ability to account for the evolution of damping with the vibration amplitude for different branches of solutions. [ABSTRACT FROM AUTHOR]

• An analytical SDoF model with nonlinear stiffness and damping terms is introduced. • The model is applied to the large oscillations of a prototype wing. • The model parameters are calibrated in the steady state frequency domain. • The model calibration accounts for different levels of input energy. • Different structural conditions (with additional masses) are considered. In this paper, a single-degree-of-freedom dynamic model is described, with displacement- and velocity-dependent nonlinearities represented by power laws. The model is intended to support the dynamic identification of structural components subjected to harmonic excitation. In comparison to other analytical expressions, the data-driven estimation of the nonlinear exponents provides a large versatility, making the generalised model adaptable for a wide number of different nonlinearities in both stiffness and damping. For instance, the proposed damping formulation can naturally accommodate air drag (quadratic) damping as well as dry friction. Differently to purely data-driven methods (e.g. black boxes), the obtained model is fully inspectable. The proposed formulation is here applied to the large oscillations of a prototype highly flexible wing and fitted on its steady state response in the frequency domain. These large-amplitude flap-wise bending oscillations are known to be affected by nonlinearities in both the stiffness (nonlinear hardening) and the velocity-dependent damping terms. The model is validated against experiments for different structural configurations and input amplitudes, as both these nonlinearities are energy-dependent. [ABSTRACT FROM AUTHOR]

• Identification tool of geometric nonlinear stiffness and nonlinear damping. • Genetic algorithm for parameter identification. • Optimized algorithms for softening and hardening systems. • Tool verified on different experimental data. A procedure is presented to identify the nonlinear damping and stiffness parameters of a single-degree-of-freedom (SDOF) model from large-amplitude vibrations of harmonically forced continuous systems, in absence of internal resonances. Cubic nonlinear damping is introduced in the SDOF model in addition to the classical viscous one. The parameter estimation relies on the harmonic balance method. It is shown that the mean value, the first and the second harmonics are needed for a softening response, although often only the first harmonic is experimentally measured with accuracy. The identification methodology is thus split between (i) purely hardening and (ii) softening behaviour. The absence of coupling enables for an independent estimation of the nonlinear stiffness from the backbone curve, followed by an evaluation of damping at resonance. For the purely hardening case, the classical least-squares method is applied by using experimentally measured first harmonic. For the softening case, a novel cascade procedure consists in (i) a single-term harmonic balance parameter estimation used as initiation of (ii) a minimization of the distance between data and a two-term harmonic balance model by means of a genetic algorithm. These procedures are validated by parameter identification on synthetic (numerically generated) and experimental frequency-responses. In particular, the identified nonlinear damping model is compared to level-adjusted linear viscous damping with a different coefficient identified at any level of harmonic force, demonstrating its ability to account for the evolution of damping with the vibration amplitude. The identification of one-to-one internal resonances is treated in a companion paper. [ABSTRACT FROM AUTHOR]

• A direct damping identification method for nonlinearly damped system is proposed. • The generally valid nonlinear damping model is identified under displacement amplitude controlled stepped sine test. • The identification method is experimentally validated on a MDOF system with a nonlinear magnetic damper. • The identified coefficients are in good agreement with the linearised reference values. In the identification of mechanical systems through inverse receptance methods, a linear model is usually assumed, at least on a first approximation. However, most of the systems have a degree of nonlinearity in their elastic or dissipative properties. Usually, stiffness nonlinearities are identified, while damping nonlinearities are neglected or even not detected. Indeed, an equivalent linear damping model, correctly representing the nonlinear dissipation of the system under testing condition, can always be found, but it is test-specific and not generally valid. This paper addresses the identification of dissipative models for multi degrees of freedom mechanical systems with a single damping nonlinearity. Based on the direct damping matrix identification through inverse receptance methods, this research proposes an extension of the Stabilised Layers Method, valid for linear systems, to damping nonlinearities, resulting in the identification of the test-independent linear and nonlinear damping matrices of the system. The proposed method is theoretically derived for the identification of nonlinear damping forces depending on powers of displacement and velocity. The sinusoidal input describing function approximation of the nonlinear damping is exploited to identify the coefficients of the nonlinear damping force from the system receptances, measured under sinusoidal sweeps at constant amplitudes of oscillation across the nonlinearity. The presented method is applied to identify the linear and nonlinear damping matrices of a multi degrees of freedom system with a localised nonlinear magnetic damper. The coefficients of the nonlinear magnetic damping force are identified for two configurations of the magnetic damper. The proposed approach is a parametric method to identify the nonlinear damping models of mechanical systems using standard experimental techniques usually adopted for linear systems. The identified model is valid under general excitation forces, predicting the system behaviour in a broader range of operation than the test-specific linear equivalent model. [ABSTRACT FROM AUTHOR]

• Nonlinear electromagnetic shunt damping (N-EMSD) is proposed and modeled. • We derive the theoretical model of a nonlinear vibration isolator with N-EMSD. • N-EMSD can achieve nonlinear damping to improve vibration isolation performance. • Negative inductance can reduce peak frequency to broaden the isolation bandwidth. • The optimal resistance and inductance for vibration isolation are studied. This paper investigates the nonlinear mass and damping effects of nonlinear electromagnetic shunt damping (N-EMSD) for vibration isolation performance enhancement of a permanent magnets (PMs) based nonlinear vibration isolator (NVI). An electromagnetic structure composing of two coils and two PMs is designed to realize equivalent nonlinear damping and mass. The nonlinear mechanic-electromagnetic coupling coefficient is analyzed and modeled. The voltage frequency of the circuit is twice the displacement frequency that is totally different with that of the linear electromagnetic shunt damping (L-EMSD). The amplitude frequency response function of the NVI with N-EMSD is theoretically derived via the harmonic balance method (HBM) and the stability is judged with Jacobian matrix. Then the equivalent nonlinear damping and nonlinear mass of the N-EMSD is derived. The numerical predictions agree with the experimental results, which demonstrate that the use of inductance in shunt circuit can change the equivalent nonlinear mass and the "jump" frequency of the NVI. Large value of inductance deteriorates the vibration isolation performance and the optimal inductance is slightly smaller than zero. Furthermore, N-EMSD can realize nonlinear damping to achieve wide band vibration isolation of the NVI. The optimal negative resistance is discussed. [ABSTRACT FROM AUTHOR]

• Partially-covered piezoelectric cantilever with self-sensing SSDI circuit is analyzed. • Governing equations considering nonlinear piezoelectricity and damping are derived. • Two-way coupled simulation based on governing equation and SSDI circuit is developed. • Simulation replicates experimentally observed nonlinear frequency response. We examined a partially covered piezoelectric cantilever connected to a self-sensing synchronized switch damping on an inductor (SSDI) circuit. Nonlinear behavior, which cannot be explained by the single-degree-of-freedom model based on linear piezoelectricity, is observed in the frequency response as the excitation level increases. Here, we propose a new method that allows us to predict the attenuation performance in such a smart damping structure. We first derived the governing equation for a practical cantilever structure in which the piezoelectric elements are attached with glue and partially cover the cantilever. We include the nonlinear piezoelectricity and damping in the model. Then, we used a two-way coupled simulation technique to investigate the complex electromechanical dynamics for the piezoelectric cantilever connected to the self-sensing SSDI circuit. Our model replicates the nonlinear behavior observed experimentally. Intriguingly, the simulation revealed that, when connected to the SSDI circuit, the lower shift of the peak frequency observed in the open circuit vanished and the peak frequency shifted to higher frequency because of the increased piezoelectric coupling force. We emphasize the importance of nonlinear piezoelectricity and damping in the model and show that ignoring the peak frequency shift and the decrease in the displacement may lead to the wrong estimation in the attenuation performance, especially as the excitation level increases. [ABSTRACT FROM AUTHOR]