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In recent decades, nonlinear damping identification (NDI) in structural dynamics has attracted wide research interests and intensive studies. Different NDI strategies, from conventional to more advanced, have been developed for a variety of structural types. With apparent advantages over classical linear methods, these strategies are able to quantify the nonlinear damping characteristics, providing powerful tools for the analysis and design of complex engineering structures. Since the current trend in many applications tends to more advanced and sophisticated applications, it is of great necessity to work on developing these methods to keep pace with this progress. Moreover, NDI can provide an effective and promising tool for structural damage detection purposes, where the changes in the dynamic features of structures can be correlated with damage levels. This review paper provides an overview of NDI methods by explaining the fundamental challenges and potentials of these methods based on the available literature. Furthermore, this research offers a comprehensive survey of different applications and future research trends of NDI. For potential development and application work for nonlinear damping methods, the anticipated results and recommendations of the current paper can assist researchers and developers worldwide to find out the gaps and unsolved issues in the field of NDI.

A geometric nonlinear damping is proposed and applied to a quasi-zero stiffness (QZS) vibration isolator with the purpose of improving the performance of low-frequency vibration isolation. The force, stiffness and damping characteristics of the system are presented first. The steady-state solutions of the QZS system are obtained based on the averaging method for both force and base excitations and further verified by numerical simulation. The force and displacement transmissibility of the QZS vibration isolator are then analysed. The results indicate that increasing the nonlinear damping can effectively suppress the force transmissibility in resonant region with the isolation performance in higher frequencies unaffected. In addition, the application of the nonlinear damping in the QZS vibration isolator can essentially eliminate the unbounded response for the base excitation. Finally, the equivalent damping ratio is defined and discussed from the viewpoint of vibration control.

A nonlinear vibration isolator with quasi-zero-stiffness (QZS) is designed inspired by the bi-stable Miura-origami tube. The isolator comprises a negative stiffness element with four rods and one horizontal spring, while a vertical linear spring is used to provide positive stiffness. The static characteristics of the negative stiffness element are studied and then the design procedure for the QZS isolation system with a lumped mass is provided. The static displacement is considered as an input parameter to illustrate the high-static-low-dynamicstiffness (HSLDS) property of the isolator. To further display the isolation ability of the present isolator, the dynamic model of the system under base excitation is established with the linear air damping and nonlinear damping caused by rotational joints. The nonlinear terms are then approximated by Taylor series and the averaging method is used to predict the dynamic performance. The present theory is verified by comparing the system’s free and forced responses under different frequencies of base excitation with those calculated by numerical simulation. The displacement transmissibility of the present nonlinear vibration isolator demonstrates the much improved vibration isolation performance as compared to its linear counterpart.

In recent years, the increasing demand for vibration control has driven the development of viscous damping isolators, and there have been many attempts to apply the shear thickening fluid (STF) system to vibration control to achieve customized damping property according to different application requirements. Previous studies on the performance of STF-based isolators have observed a peculiar \"collapse\" phenomenon, but researchers have no clear explanation for the mechanism causing this unfavorable phenomenon. The main focus of this work is to explore the nonlinear characteristics and mechanism of STF-based isolators, especially for introducing a novel carbon fiber powder STF system (CFP-STF). The damping mechanism of the CFP-STF isolator is theoretically derived based on Poiseuille's law, the governing equations of arbitrary rheology are solved and demonstrated in details, and the theoretical model is established for multi-scale analysis under multi-frequency loading. To explore the influence of each parameter on the damping coefficient/force–displacement-velocity relationship, attention is paid to the initial viscosity and peak viscosity of the CFP-STF system, and as a result the concept of damping coefficient decrease rate is proposed in this paper. The results show that the the new CFP-STF isolator avoids the \"collapse\" phenomenon of the force–displacement hysteresis curve at high frequencies, where the damping force does not decrease and the mechanical dissipation tends to increase linearly after the threshold is exceeded, since the viscosity decreases more slowly than the velocity increases with increasing frequency.

The mechanism of energy dissipation in mechanical systems is often nonlinear. Even though there may be other forms of nonlinearity in the dynamics, nonlinear damping is the dominant source of nonlinearity in a number of practical systems. The analysis of such systems is simplified by the fact that they show no jump or bifurcation behaviour, and indeed can often be well represented by an equivalent linear system, whose damping parameters depend on the form and amplitude of the excitation, in a 'quasi-linear' model. The diverse sources of nonlinear damping are first reviewed in this paper, before some example systems are analysed, initially for sinusoidal and then for random excitation. For simplicity, it is assumed that the system is stable and that the nonlinear damping force depends on the nth power of the velocity. For sinusoidal excitation, it is shown that the response is often also almost sinusoidal, and methods for calculating the amplitude are described based on the harmonic balance method, which is closely related to the describing function method used in control engineering. For random excitation, several methods of analysis are shown to be equivalent. In general, iterative methods need to be used to calculate the equivalent linear damper, since its value depends on the system's response, which itself depends on the value of the equivalent linear damper. The power dissipation of the equivalent linear damper, for both sinusoidal and random cases, matches that dissipated by the nonlinear damper, providing both a firm theoretical basis for this modelling approach and clear physical insight. Finally, practical examples of nonlinear damping are discussed: in microspeakers, vibration isolation, energy harvesting and the mechanical response of the cochlea.

Passive vibration isolation is always preferable in most engineering practices. To this aim, a novel, compact and passive vibration isolation mount is studied in this paper, which is designed by using the X-shaped structure with a much smaller size. The novel mount is adjustable to different payloads due to a special oblique and tunable spring mechanism, and of high vibration isolation performance with a wider quasi-zero-stiffness range due to the deliberate employment of negative stiffness of the X-shaped structure. The X-shaped structure has been well studied recently due to its excellent nonlinear stiffness and damping properties. In this study, it is for the first time to explore the utilization of the negative stiffness property within the X-shaped structure such that the resulting design, the X-structured mount (X-mount), can have an obviously larger vibration displacement range which maintains the quasi-zero-stiffness property. A special oblique spring is thus introduced such that the overall equivalent stiffness of the X-mount can be much easily adjusted according to different payloads. Systematic parametric study is conducted to reveal the critical design parameters and their relationship with vibration isolation performance. A prototype and experimental validations are implemented to validate the theoretical results. It is believed that the X-mount would provide an innovative technical upgrade to many existing vibration isolation mounts in various engineering practices and it could also be the first prototyped mount which can offer adjustable quasi-zero stiffness and adjustable loading capacity conveniently.

Experimental data clearly show a strong and nonlinear dependence of damping from the maximum vibration amplitude reached in a cycle for macro- and microstructural elements. This dependence takes a completely different level with respect to the frequency shift of resonances due to nonlinearity, which is commonly of 10–25% at most for shells, plates and beams. The experiments show that a damping value over six times larger than the linear one must be expected for vibration of thin plates when the vibration amplitude is about twice the thickness. This is a huge change! The present study derives accurately, for the first time, the nonlinear damping from a fractional viscoelastic standard solid model by introducing geometric nonlinearity in it. The damping model obtained is nonlinear, and its frequency dependence can be tuned by the fractional derivative to match the material behaviour. The solution is obtained for a nonlinear single-degree-of-freedom system by harmonic balance. Numerical results are compared to experimental forced vibration responses measured for large-amplitude vibrations of a rectangular plate (hardening system), a circular cylindrical panel (softening system) and a clamped rod made of zirconium alloy (weak hardening system). Sets of experiments have been obtained at different harmonic excitation forces. Experimental results present a very large damping increase with the peak vibration amplitude, and the model is capable of reproducing them with very good accuracy.

This study is to present a novel way to achieve superior passive vibration isolation by employing a specially designed and compact linkage mechanism. The proposed anti-vibration system has beneficial nonlinear inertia, inspired by swinging motion of human arms, and is constructed with an adjustable nonlinear stiffness system inspired by animal or human leg skeleton. It is shown with comprehensive theoretical analysis and consequently validated by a series of well-designed experiments that the nonlinear stiffness, nonlinear damping and nonlinear inertia of the proposed system are very helpful for significantly reducing resonant frequency and enhancing damping effect in a beneficial nonlinear way. This results in excellent vibration isolation performance with lower resonant frequency and resonant peak of faster decay rate. This study provides an innovative solution to a cost-efficient vibration control demanded in various engineering systems.

An improved incremental harmonic balance method (IHBM) is proposed by Wang (J Sound Vib 441:111–125, 2019) to solve the periodic responses of the continuous nonlinear stiffness systems. However, the nonlinear damping systems remain unsolved. This paper aims to investigate the nonlinear damping parts by the proposed IHBM method, which is based on the principle that any continuous curve can be approximated by a piecewise-linear curve with discrete nodes. The piecewise-linear function can be considered a unified benchmark function that can convert the complex IHBM Galerkin process of arbitrary nonlinear damping systems to that of unified piecewise-linear damping systems. The general process of the proposed method for this piecewise-linear system is derived considering the stability of the solutions. Then, a polynomial nonlinear damping system is investigated to validate the accuracy of the method. Furthermore, five typical cases of single-degree-of-freedom (SDOF) nonlinear damping systems are carried out, and this method is also extended to multi-degree-of-freedom (MDOF) systems where each nonlinear force in the systems is expressed by the function of only one independent DOF. The results illustrate that the proposed method shows convenience and accuracy in obtaining the dynamics of nonlinear systems.

The nonlinear modes of a non-conservative nonlinear system are sometimes referred to as damped nonlinear normal modes (dNNMs). Because of the non-conservative characteristics, the dNNMs are no longer periodic. To compute non-periodic dNNMs using classic methods for periodic problems, two concepts have been developed in the last two decades: complex nonlinear mode (CNM) and extended periodic motion concept (EPMC). A critical assessment of these two concepts applied to different types of non-conservative nonlinearities and industrial full-scale structures has not been thoroughly investigated yet. Furthermore, there exist two emerging techniques which aim at predicting the resonant solutions of a nonlinear forced response using the dNNMs: extended energy balance method (E-EBM) and nonlinear modal synthesis (NMS). A detailed assessment between these two techniques has been rarely attempted in the literature. Therefore, in this work, a comprehensive comparison between CNM and EPMC is provided through two illustrative systems and one engineering application. The EPMC with an alternative damping assumption is also derived and compared with the original EPMC and CNM. The advantages and limitations of the CNM and EPMC are critically discussed. In addition, the resonant solutions are predicted based on the dNNMs using both E-EBM and NMS. The accuracies of the predicted resonances are also discussed in detail.