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• The compliant LLS exhibits various stiffness characteristics with different dimensions. • The topological dimensions were optimized to achieve the largest HSLDS region. • The compliant LLS exhibits a larger HSLDS region and a stronger stability than the rigid LLS. Bio-inspired isolators are a novel type of nonlinear isolators by mimicking the bionic structures and behaviors. Most of them are based on limb-like structures composed of rigid rods forming a geometric frame and linear springs storing elastic energy. A novel bio-inspired isolator with a compliant limb-like structure was proposed. Both the geometric constrained movement and the storage of the elastic energy are provided by the deformation of a compliant mechanism. A beam constraint model was adopted to characterize the nonlinear restoring force, and a harmonic balance method was used for the dynamic analysis. The minimum damping ratio ensuring the isolator operating within a working region was estimated, and the displacement transmissibility of the isolator was calculated. A systematic investigation was performed on the high-static-low-dynamic-stiffness characteristic and the vibration isolation performance of the proposed isolator. The investigation revealed that the compliant limb-like structure exhibits stiffness-softening, stiffness-softening-hardening, and negative-stiffness characteristics with different dimensions and deformation degrees. The dimensions can be optimized to achieve the largest high-static-low-dynamic-stiffness region, and the smallest dynamic/static stiffness ratio. The isolator exhibits a globally stable response and a weak nonlinearity if it is fully damped, while jump phenomena and unbounded response may occur with light damping. Compared with a three-spring isolator and a rigid limb-like structure, the compliant limb-like structure exhibits a strong stability, a strong load capacity, and a noticeable high-static-low-dynamic-stiffness characteristic at the same time. The combination of the bio-inspired isolation and the compliant mechanism may provide an innovative approach for the broadband vibration isolation. [ABSTRACT FROM AUTHOR]

Highlights • Vibration of fluid-conveying pipes is isolated by quasi-zero stiffness systems. • Nonlinear solution is obtained via Galerkin method and the finite difference. • The fluid flow can deteriorate the performance of vibration isolation. Abstract Fluid-conveying pipes are always subjected to various excitations to cause unwanted vibrations. A quasi-zero stiffness system consisting of three linear springs is adopted as the nonlinear isolator to attenuate the transverse vibrations of fluid-conveying pipes induced by foundation excitations. A dynamic model of nonlinear forced vibration of the fluid-conveying pipe coupled with two nonlinear isolators is established for the nonlinear continuous system and validated by using two methods, Galerkin method and the finite difference method. The influence of the quasi-zero stiffness isolators on the vibration characteristics and vibration transmission of the pipe is investigated by analyzing the natural frequency, vibration mode, and nonlinear vibration response. The effects of flow speed of the fluid and the system parameters of the isolator are studied to evaluate the isolation performance. It is found that the quasi-zero stiffness isolator and fluid flow can shift several natural frequencies of vibration of the pipeline to the low-frequency region. When the linear stiffness of the vibration isolation is zero in the vertical direction, the first two modes of the bending vibration of the fluid-conveying pipe tend to become rigid mode. While achieving high-efficiency vibration isolation in the high-frequency region, the vibration in the low-frequency region is complicated. The flow speed of the fluid can deteriorate the performance of vibration isolation. [ABSTRACT FROM AUTHOR]

Highlights • Resonance response interaction occurs without internal resonance. • Resonance response interaction enhances harvesting efficiency. • Bubble shaped response curve is uncovered in the power-frequency response. • Parametric effects on the bandwidth in the presence of resonance response interaction. Abstract Resonance response interaction in nonlinear multiple degrees of freedom vibration systems has always involved internal resonance. In this article, bubble shaped response curve is uncovered without pre-designated resonances relationship that is a necessary condition for internal resonance. The objective of this article is twofold: first to explore the connection between the resonance response interaction and bubble shaped response curve that may appear in the forced response of a nonlinear magnetoelectric coupled system; and, second, to exploit resonance response interaction to enhance the vibratory energy harvesting bandwidth. A nonlinear magnetoelectric oscillator coupled linearly to an additional oscillator is used to demonstrate the phenomena and the enhanced performance. The lateral springs could produce the geometrical nonlinear stiffness and its stiffness to adjust the two resonance responses such that the bubble shaped response curve appears in the power-frequency response plot. The method of harmonic balance is well established method for the analysis of such the power-frequency response. The results are also validated by some numerical work. The power-frequency response curves are generated for distinct harvesting mass, manifesting the resonance response interaction could increase the bandwidth and output power. At last, the resonance response interaction designed energy harvesting from Gaussian white noise excitation is numerically addressed to illustrate positive effects of geometrical nonlinearity. [ABSTRACT FROM AUTHOR]

Highlights • A modified harmonic balance approach is proposed for piezoelectric mechanical systems. • The proposed approach can handle non-polynomial nonlinear restoring forces, nonlinear damping, and nonlinear electric terms. • Automatic generation of harmonic balance equations and less analysis is required. • Numerical examples and experimental results well support the proposed method. • Both softening and hardening nonlinearities of a piezoelectric energy harvester are predicted and verified. Abstract A semi-analytical approach based on harmonic balance (HB) method is proposed to predict steady-state responses of nonlinear piezoelectric mechanical systems. In order to handle complicated non-polynomial forces of the systems, the alternating frequency/time domain (AFT) progress is applied to establish a set of implicit algebraic equations of HB method. The stabilities of periodic responses are analyzed via Floquet theory. The proposed approach is successfully implemented a nonlinear piezoelectric energy harvester with a magnetic oscillator and a Duffing energy harvester with strong nonlinear electric terms. The HB solutions are well supported by the simulation results of the Runge-Kutta method. Besides, an experiment piezoelectric device is developed by a circular laminated plate, and an experimental model is established by system identifications. The model includes both the nonlinear stiffness and nonlinear damping which are characterized by the combined functions of polynomials and hyperbolic tangents. The model is analyzed via the proposed approach. The numerical simulations under different excitations show good agreements with the harmonic solutions. The frequency sweeps experimentally pursue the system has both softening and hardening nonlinearities which are verified by HB results. The proposed approach requires less analysis during the solving processes, and shows high accuracy to the strongly nonlinear piezoelectric mechanical systems, even the system with complicated nonlinearities. [ABSTRACT FROM AUTHOR]

• An adjustable stiffness isolator implemented by a semicircular ring. • The quasi-zero-stiffness region shifted by adjusting the pre-deformed state. • Snap-through threshold depending on both pre-deformation and vibration conditions. • The proposed semi-active isolator applied in various working conditions. Nonlinear vibration isolators with quasi-zero-stiffness characteristics offer broadband vibration isolations, while passive quasi-zero-stiffness isolators are unable to adapt to variable working conditions. To address the issue, a nonlinear semi-active vibration isolator based on a semicircular ring structure is proposed. As an elastic component, the semicircular ring exhibits nonlinear stiffness under a vertical compression at the centre. With a semi-active control of the horizontal distance between the two ends of the semicircular ring, a pre-deformed state is induced, and the stiffness characteristics are adjusted accordingly. A chained beam-constraint model is adopted to characterize the pre-deformed process and the vertical compression process, respectively. The adjustable nonlinear restoring force is accurately calculated and adopted for vibration analysis. A harmonic balance method is used to analyze the frequency responses of the isolator under different working conditions. The semicircular ring exhibits softening stiffness under a vertical compression, and the quasi-zero-stiffness region can be effectively adjusted by the pre-deformed process. With the vertical compression reaching a threshold, negative stiffness is presented along with a snap-through phenomenon due to the bi-stability. With the semi-active pre-deformed, the isolator demonstrates different resonant frequencies and different risks in the occurrence of the snap-through during vibration. To avoid the snap-through-induced impact, the control threshold of the pre-deformed is determined, depending on the payload mass, the excitation amplitude, and the damping coefficient. Experiments are performed to validate the adjustable stiffness and the frequency responses of the isolator, and both the jump phenomenon and the snap-through phenomenon are observed. [ABSTRACT FROM AUTHOR]

[Display omitted] Due to high sensitivity to the changes in external excitation intensity, the implementation of nonlinear energy sinks (NESs) have been greatly limited in practical applications. This study presents a stable state adjustable NES (SA-NES) structure. By adjusting the structural geometric relationship, the SA-NES structure can easily switch its stable state among monostable, bistable, and tristable. The governing equation of the coupled system is obtained by Hamilton's principle. The approximate analytical solution is obtained by the harmonic balance method, and the slow invariant manifold is obtained complexification-averaging method, which are validated through numerical analysis. Through theoretical analysis and experimental validation, the dynamic response of the SA-NES and vibration reduction performance are studied under different levels of excitations. The obtained results show that the designed SA-NES can effectively suppress the vibration under different level excitation intensities. It is also shown that the SA-NES structure can be adjusted to the monostable state, and the vibration of the linear oscillator can be effectively suppressed by the strong modulation response under a certain excitation intensity. When the excitation intensity is weak or strong, the SA-NES can be adjusted to the bistable state and the tristable state, respectively, thus effectively suppressing the vibration through chaotic inter-well oscillation. This study reveals the vibration suppression mechanism of the SA-NES, and introduces a new NES configuration to suppress different levels of engineering excitations. [ABSTRACT FROM AUTHOR]

• Vertical random vibration is suppressed by a nonlinear energy sink. • The suppression performance increases with the NES cubic stiffness coefficient. • The account for the NES weight increases the peak values of probability density. • The account for the NES weight increases the suppression performance. A primary structure attached by a nonlinear energy sink (NES) moving vertically under a Gaussian white noise excitation is investigated. This paper demonstrates the stochastic responses and vibration suppression with emphasis on the effects of the weights. Four-dimensional state space equations which consider or ignore the weights in the vertical direction are given. Numerical probability density functions computed via the path integration method based on the Gauss-Legendre scheme are confirmed by Monte Carlo simulations. Probability density functions of the structure's responses are compared between two cases under various random excitation intensities. The root-mean-square (RMS) displacement of the primary structure or random vibration suppression is estimated through the path integration method. Results reveal that the NES is able to suppress broadband random vibration of the structure. As the stochastic excitation decreases, the NES weight has significant effects on both probability density functions of the structure's responses and RMS displacements of the primary structure. The NES parameters are discussed to explore random vibration suppression. For purpose of more accurate predictions, the effects of weights should be taken into consideration when the primary structure is excited by small random excitations or coupled with a large and reasonable NES mass. Simulations and discussions in this work provide theorical meanings to predict vibration suppression via a NES in stochastic environment. [ABSTRACT FROM AUTHOR]

In the evaluation of the vibration mitigation effect of a nonlinear energy sink (NES), the selection of performance measures is important for the optimal design. This paper aims to propose new reasonable performance measures and new design criteria of the NES in harmonic excitation. To broaden the application range of the slow invariant manifold, coordinate correction is carried out for the square root model of the bistable NES. Two existing performance measures (the displacement amplitude of the main system and the energy dissipation rate of the NES) and two new performance measures (the mechanical energy and the input energy of the main system) are compared. The optimal geometric configuration and response form of the NES are found in different excitation conditions. The results show that the energy dissipation rate of the NES is not competent to evaluate the performance of the NES in harmonic excitation, while the two new performance measures show competence. By adjusting the geometry parameter, the nonlinear vibration absorber with positive linear stiffness and the bistable NES show optimal vibration mitigation effect in small and large excitation amplitudes, respectively. Unusually, the optimal response form is the periodic response rather than the strongly modulated response. In summary, the proposed performance measures and design criteria will contribute to the design of NESs. [ABSTRACT FROM AUTHOR]

This study introduces a novel method to improve snap-through transitions in a parametrically excited cantilever-based energy harvester featuring repulsive magnetic coupling. The aim is to enhance the efficiency of a piezoelectric energy harvester while broadening its operational frequency range. The design incorporates an oscillating substructure that magnetically interacts with the beam, promoting easier transitions. Additionally, this setup capitalizes on the resonant energy of the substructure to optimize energy capture. This study addresses a previously unexplored limitation in the directly excited counterpart by optimizing the use of substructure's resonant energy to significantly improve the energy harvesting device's efficiency. We present a theoretical model based on the Hamilton principle, which is solved numerically. Comprehensive experimental studies are conducted to validate the theoretical results, showing a strong correlation between the experimental outcomes and theoretical predictions. The findings indicate that an oscillating substructure with a tailored natural frequency close to twice that of the parametrically excited piezoelectric beam can induce time-varying potential wells to facilitate snap-through transitions. Despite the damping coefficient of the piezoelectric beam being three times higher than that of its simple beam counterpart, periodic perturbations still effectively expand its instability region. This enlarged instability region, resulting from such perturbations, not only exhibits broadband characteristics but also facilitates efficient energy harvesting at lower excitation levels. [ABSTRACT FROM AUTHOR]

A NiTiNOL-steel wire rope is proposed as a nonlinear passive damper to suppress the vibration of a composite laminated truncated conical shell under harmonic and random excitations. The equation of motion of a composite laminated truncated conical shell hosting four NiTiNOL-steel wire ropes is derived from the generalized Hamilton principle and the first-order shear deformation theory under the assumption of small deformation. The restoring and damping force of a NiTiNOL-steel wire rope fitting the parameters identified via the extended Bouc-Wen model is integrated into the equation of motion to generate the overall governing equations. A generalized differential quadrature method is applied to discretize the governing equations into a set of algebraic equations obtained by choosing suitable weight coefficients and grid points coordinates. The discrete equations are solved to obtain the frequency distribution and the associated mode shapes validated via the finite element method. Under harmonic excitations, the resonant peak value and the amplitude reduction rate in the frequency response corroborate the vibration reduction performance of different NiTiNOL-steel wire ropes configurations and found the best type. The effects of different parameters on the vibration reduction performance of different NiTiNOL-steel wire rope configurations are also evaluated. Under Gaussian white noise or colored noise excitations, the limited bandwidth ergodic root means square reveals that the best type under random excitation has the highest probability of realizing the best vibration reduction throughout the entire random process. [ABSTRACT FROM AUTHOR]

This study validates the nonlinear broadband vibration reduction capability of Nitinol steel wire rope (NiTi-ST). Hysteresis tests on NiTi-ST provide hysteresis force-displacement data, and mechanical parameter calculations are performed by developing a fitting model. A dynamic model of a composite laminated cantilever plate embedded with NiTi-ST is developed. The modal analysis is performed using the Rayleigh-Ritz method, and accurate modal functions are obtained. The dynamical equations are solved using the Galerkin truncation and harmonic balance methods. Experimental results demonstrate NiTi-ST's global broadband vibration reduction at various temperatures, with a decrease in effectiveness as temperature rises. These experimental results are consistent with the theoretical analysis. This study firstly experimentally validates NiTi-ST's nonlinear behavior, and provides technical support for NiTi-ST's use as a nonlinear vibration reduction device. [ABSTRACT FROM AUTHOR]

Various nonlinear energy harvesters that harvest energy from rotational motion have been investigated, but their performance at low frequencies is always deficiency. To realize the efficient harvesting of low frequency rotational motion energy, rotational nonlinear energy harvesting via an orthogonally arranged dual-beam is proposed for the first time. Based on the energy method, the electromechanical coupling equation of orthogonally arranged dual-beam in rotating motion is derived in detail. The harmonic balance method together with arc length continuation is employed to approximate the frequency response functions of the output power. To deeply explore the nonlinear dynamic behavior of the system, numerical analysis is conducted to obtain the output voltage and phase diagram in the time domain. Finally, the theoretical analysis results are verified through experiments. The results demonstrate that the vertical arrangement of nonlinear dual-beam in rotational motion effectively improved the energy harvesting efficiency at low frequencies, with excellent energy harvesting performance. This study provides an effective technical approach to energy harvesting for low frequency rotating systems. [ABSTRACT FROM AUTHOR]

• An internal resonance piezoelectric energy harvester is proposed. • Experimental results depict double-jumps and bi-directional energy transfers. • The proposed device contributes to broadband and efficient utilization of piezoelectric effects by harnessing internal resonance. • Nearly commensurable conditions can lead to double-jumps. This study investigates the effects of nonlinear modal interactions on a piezoelectric energy harvester, specifically focusing on the absence of externally induced magnetic nonlinearities. While multi-mode vibration energy harvesters aim to utilize multiple resonance regions to broaden the effective frequency bandwidth, weak nonlinear dynamic behaviors can occur with small excitations, leading to the dominance of linear behavior in the system's response. To overcome this limitation, researchers have extensively explored nonlinear techniques like magnetic couplings. However, this study demonstrates that even without external magnetically induced nonlinearities, nonlinear modal interactions can significantly impact the performance of a piezoelectric energy harvester. The device has been designed, fabricated, and subjected to testing, both an approximate analytical method and a numerical method were employed to predict and qualitatively analyze the experimental results. The findings indicate that, when the natural frequency ratio of the first two modes of the system becomes nearly commensurable, the nonlinearities of the system become more prominent as the excitation levels increase. This leads to more noticeable interactions between the nonlinear modes, ultimately resulting in an internal resonance between the first two modes. The observed phenomena, such as double-jumps and resonance energy transfers between the coupled modes, contribute to broadening the overall bandwidth and enhancing the efficiency of utilizing piezoelectric effects. [ABSTRACT FROM AUTHOR]

Nonlinear energy sink (NES) has become a good passive vibration control technique with great potential. For a huge structure, it is unpractical to control the vibration by just one NES, let alone the structure has multi-modes. In this work, a technique who uses distributed nonlinear energy sink cells (NES-cells) to control the multi-modal resonance of the floating raft is proposed. The floating raft is simplified as a corner-point supported plate. Natural modes of it guide the distribution of NES-cells to make NESs locate on the position with strong vibration, which enhances the coupling ship between the plate platform and NES-cells. An example of installing NES-cells guiding by the first-two modes of the corner-point supported plate is presented in this paper. It is compared with the vibration reduction effect by a single NES. To make a fair comparison, total mass of NES-cells is the same with that of the single NES. The results show that NES-cells can deal with multi-modal vibration reduction better. The experimental results also fully illustrate this view. At the same time, the reaction of each NES-cell to the plate is effectively dispersed while comparing with the single NES, which decreases the stress at the attachment points and is more suitable for engineering applications. The number of NES-cells also has an effect on the vibration reduction efficiency. When the design parameters are fixed, there exists an optimal number to make the vibration reduction efficiency tend toward saturation. The multimodal vibration control of the plate platform is effectively solved by the proposed NES-cells technique. It also provides guidance for other practical engineering problems. [ABSTRACT FROM AUTHOR]

• Vibration control of interconnected composite beams is investigated. • Modal analysis is solved through the Rayleigh-Ritz approach and finite element method. • Investigation of frequency response of the structure is investigated. • A vibration experiment is established to explore NiTiNOL-steel damping ability. This paper presents a comprehensive theoretical investigation and experimental validation of vibration and nonlinear vibration control in interconnected composite beams with fixed ends operating within a thermal environment. The study employs the Rayleigh-Ritz method, utilizing a series of orthogonal polynomials as interpolation basis functions to derive the vibration mode functions of the composite structure under thermal conditions. Furthermore, the motion control equation for the system is derived based on Lagrange's equation. These equations of motion are then simplified using the Galerkin method, leading to ordinary differential equations. The response of the system is subsequently computed using the harmonic balance method and validated through the Runge-Kutta method. Finally, experimental results confirm that the NiTiNOL-steel wire rope effectively controls the vibration of interconnected composite beams in a thermal environment and is consistent with the theoretical results. [ABSTRACT FROM AUTHOR]

A conventional nonlinear energy sink (NES) proposed for spacecraft structure can achieve vibration reduction without changing the resonant frequencies. However, large additional mass limits its engineering applications, and the robustness needs to be improved. A novel nonlinear energy sink with active varying stiffness (NES-AVS) device is proposed for reducing the mass of the conventional NES and enhancing the vibration reduction effects. A two degrees-of-freedom equivalent model of the scaled spacecraft structure is proposed. A harmonic balance method is applied to analyze the effect of the vibration reduction of the equivalent model, and the results are verified numerical simulations. The results show that the NES-AVS can achieve better vibration reduction effects, especially in small damping and mass. A finite element model of the scaled spacecraft structure with the NES-AVS is developed. The NES-AVS device is experimentally embedded in the scaled spacecraft structure. The variable stiffness device is realized by a steel plate with the compression force adjusted by a piezoelectric actuator. An active controller is used to control the actuator by the dSPACE module. The experimental results demonstrate that the NES-AVS has a better effect of the vibration reduction of the spacecraft structure than that of the conventional NES in multiple modes and different excitations. Compared with the conventional NES, the mass of the NES-AVS is reduced by 90%. The experiment and the theory have the same trends that the vibration reduction performance of the NES-AVS can be improved by adjusting the active control parameters. This work devotes to providing a new design scheme of the NES. [ABSTRACT FROM AUTHOR]

To meet the requirements of vibration control for composite structures, the NiTiNOL steel wire ropes (NiTi-ST) is selected to control the vibration of the composite laminate. The dynamical modeling is built by integrating the composite laminate with the modified Bouc-Wen model. The dynamical model is discretized by Galerkin truncation method (GTM), and theoretical analysis is carried out using the harmonic balance method (HBM), Runge-Kutta method (RK) and finite element method (FEM). Based on the results of dynamical modeling and theoretical analysis, an experimental device is designed and manufactured. The results show that the NiTi-ST has excellent vibration control ability for composite laminate, and the impact on the natural frequency of the system is small. [ABSTRACT FROM AUTHOR]

Nonlinear localization is an important component of structural nonlinear identification. Its accuracy is of great significance for damage localization, health monitoring, and performance prediction of structural systems. In this paper, a two-step localization method is proposed for the nonlinear localization of multi-degree-of-freedom (MDOF) systems with multiple local nonlinearities. The approach achieves accurate nonlinear localization in two steps. In the first step, subset nonlinearity detection is performed to initially narrow down the nonlinear localization range. This effectively reduces the dimensionality of the candidate nonlinear basis functions in the second step. In the second step, candidate nonlinear basis functions are used to further confirm the doubtful connections. The accuracy and effectiveness of the method are verified by two numerical examples of a one-dimensional chain-type structure and a two-dimensional structure. In addition, an experimental study is conducted on a floor structure with two local nonlinearities to verify the effectiveness of the proposed method. Results show that the proposed method has good accuracy in a noisy environment and it requires only the input and output data of the system under a wide-band excitation. [ABSTRACT FROM AUTHOR]

[Display omitted] • A tristable nonlinear energy sink (TNES) is proposed for suppressing strong excitation vibration. • The TNES can eliminate the detached resonance curve and enlarge the effective range of the NES. • The TNES can perform chaotic inter-well oscillation and suppress strong excitation vibration. As well known, the vibration reduction efficiency of the nonlinear energy sink (NES) is poor under strong excitation. In this paper, a tristable NES (TNES) is proposed. The TNES can degenerate into bistable and mono-stable NES by adjusting the geometric parameters of the springs. The governing equations of a linear oscillator coupled with the TNES under harmonic excitation are derived. The approximate analytical solution of the coupled system is obtained by using the harmonic balance method and verified numerically. The vibration suppression efficiency of TNES and NES is compared. The dynamic behavior of TNES under strong excitation is demonstrated. The results show that the nonlinear restoring force is softened due to the wide distribution of the three stable points of TNES. Therefore, compared with NES, the proposed TNES can suppress stronger excitation vibration. In addition, the low side barrier depth is conducive to TNES to perform chaotic inter-well oscillation, which can effectively suppress the strong excitation vibration and obtain good vibration reduction performance. As a result, the proposed TNES can eliminate the detached resonance curve and enlarge the effective range of the NES. Under relatively weak excitation, the vibration suppression efficiency of TNES is slightly lower than that of NES, although the TNES is also relatively significant. Therefore, this paper reveals the vibration suppression mechanism of TNES, and provides a way to effectively solve the problem of low vibration reduction efficiency of NES under strong excitation. [ABSTRACT FROM AUTHOR]

Active acoustic insulation systems tend to suffer from narrow insulation frequency bandwidth and low-frequency deficiencies. In this paper, we employed a quasi-zero stiffness supported nonlinear moving-coil diaphragm to such a system to achieve the target energy transfer, and broaden the insulation frequency bandwidth. We established the electromechanical acoustic coupling model of the nonlinear active acoustic insulation system, and approximated the frequency response of the acoustic pressure transmissibility using the harmonic balance method. The analytical results were well supported by numerical methods. Further, we analyzed the effects of geometrical parameters and shunting circuit gains to explore the mechanical and electrical effects on insulation performance. An experimental design and a semi-physical simulation model were finally set up to validate theoretical analysis. It was revealed that a properly configured active acoustic insulation system could greatly reduce the sound levels and effectively expand the insulation frequency bandwidth. [ABSTRACT FROM AUTHOR]

Highlights • Nonlinear energy sink with an inerter is proposed for vibration systems. • NES-Inerter suppresses vibration more effectively in energy dissipation and amplitude frequency responses. • The use of NES-Inerter provides a novel NES application in practical engineering. Abstract A novel archetype of nonlinear energy sink (NES) enhanced by an inerter is proposed for vibration reduction of vibrating systems. The proposed NES-inerter suppressed vibration more effectively than a convenient NES measured by both energy dissipation and amplitude frequency responses. Complexification-averaging technique is applied to analyze a coupled system to find steady-state periodic solutions, and the analytical results are confirmed by numerical simulations. The analytical solutions are used to bifurcation analysis, and the results show that the saddle node and the Hopf bifurcations occur under a series of parameters in the steady-state responses. The effects of parameters, such as inertance, cubic stiffness, viscous damping, and amplitude of the external excitation, on the performance of the proposed NES are examined based on the analytical solutions. The investigation reveals that the optimal value of the inerter exists in a certain range. [ABSTRACT FROM AUTHOR]

Highlights • A generalized transmissibility is proposed for the vibration absorption system. • The root mean square is employed for redefined the transmissibility. • The quasi-periodic motion is an indication of the optimal transmissibility. • Two evaluation approaches based on the energetics and transmissibility are compared. • A base for the design and parameter optimization of the NES absorbers is provided. Abstract A generalized transmissibility is proposed to evaluate nonlinear vibration absorbers. Due to possible existence of higher order harmonics and aperiodic motion, the root mean square is employed to define the transmissibility. The transmissibility beyond harmonic motion is applied to measure vibration reduction of a structure coupled with a nonlinear energy sink. Periodic steady-state responses of the coupled system are analytically predicted by the method of harmonic balance, and the stabilities of the responses are determined. The effects of parameters of the nonlinear energy sink on the transmissibility are explored based on harmonic balance solutions and numerical simulations. Phase portraits, Poincaré map and frequency spectrum reveal that quasi-periodic motions occur when periodic responses are unstable. Finally, two evaluation approaches based on the energetics and the transmissibility are compared. The investigation demonstrates the quasi-periodic motion is an indication of the optimal transmissibility. The performance of nonlinear energy sink absorbers may be extremely sensitive to parameter variations. [ABSTRACT FROM AUTHOR]

• An axially loaded beam with parallel-coupled nonlinear isolation system is modelled. • Non- co -location of the excitation source and the isolation object is issued. • Double annular magnets and spiral springs aligned on the same axis is used to achieve high static and low dynamic stiffness. • The initial axial force load of the axially loaded beam can extend the isolation range. • An experiment is implemented to verify the accuracy of the analytical model. To improve low-frequency vibration isolation when the excitation source and the isolation object are not co-located, transverse vibration of axially loaded beam with parallel-coupled nonlinear isolators is explored both experimentally and analytically. The nonlinear isolators use double annular magnets and spiral springs aligned on the same axis to achieve high static and low dynamic stiffness, which is suitable for low-frequency vibration isolation. Using the Galerkin truncation, harmonic balance analysis, and arc-length continuation, an approach is developed to analyze the frequency response functions of power flow for the strongly nonlinear discrete and continuum coupled systems. Both analytical and numerical results demonstrate that frequency response function of power flow can be used to deal with transverse vibration which excitation source and isolation object are not co-located. Parallel-coupled nonlinear vibration isolator can decrease the energy transmission of high order modal vibration of continuum, compared with the discrete isolation system. The axially loaded beam with parallel-coupled nonlinear isolators achieves significant vibration suppression at low frequency. Parametric studies of strongly nonlinear discrete and continuum coupled systems demonstrate that increasing the length of the beam and increasing the initial axial force can reduce the resonant frequency, which broadens the vibration isolation frequency band. An experiment is implemented to verify the accuracy of the theoretical model. [ABSTRACT FROM AUTHOR]

[Display omitted] • A novel integral dynamic model of retaining clips on pipes is established. • The necessity of the integral pipe model is illustrated by comparing with the half pipe model. • Effects of the clip parameters on the natural vibration of the pipe are presented. • The clip width and thickness have a noticeable effect on the vibration of the pipe. As an indispensable accessory, retaining clips on pipes are usually modeled as simply supported or fixed end boundaries. In this paper, a dynamic model of a pipe without fluid, with fixed–fixed ends and restrained by a middle clip is established. Different from the previous modeling that ignores the width of a clip, the clip is modeled as a rigid body with a certain width, which is elastically connected to the base. The effects of clip width, thickness, material parameters, and connection stiffness on the natural vibration of the pipe are presented. Moreover, to illustrate the necessity of the proposed model, a dynamic model of a half pipe is extracted with a clip as one end boundary. According to the two models, the natural frequencies and modes of the pipe are calculated and compared. Some meaningful results are obtained. The clip parameters influence the natural frequencies of the pipe nonlinearly. When calculating the natural frequencies of the pipe, ignoring the clip width or modeling the clip as a boundary will cause a noticeable error. The symmetry of the modes and the numbers and locations of modal nodes change as the clip stiffness changes. Furthermore, modal shapes of the symmetric modes change noticeably as the clip thickness or width increases. The locations of the maximum modal displacements may change as the clip width varies. In summary, a novel and valuable dynamic model of the retaining clip is proposed in this study. The vibration of a pipe restrained by a clip can be more accurately described. Influence laws of clip parameters can be used to guide the design and vibration measurement of pipes with clips. [ABSTRACT FROM AUTHOR]

• Dynamics of a planar curved beam subjected to moving loads or masses. • Arbitrary Lagrangian-Eulerian formulation for a corotational curved beam element. • Treatment of geometric nonlinearity and viscoelasticity of the beam. • Driving force for a prescribed motion of a mass along the beam. This paper presents an arbitrary Lagrangian-Eulerian (ALE) formulation based on the consistent corotational method for the geometric nonlinear dynamic analysis of planar curved viscoelastic beams subjected to moving loads. In the ALE description, the beam nodes can be moved in arbitrarily specified ways to describe the moving loads' material positions accurately. The pure deformation and the deformation rate of the element are measured in a curvilinear coordinate system fixed on a curved reference configuration that follows the rotation and translation of a corotational frame. Based on Hamilton's principle, the global elastic force vector, the global internal damping force vector, the global inertia force vector, and the global external damping force vector are derived using the same shape functions to ensure the consistency and independence of the element. Then, a standard element can be embedded within the element-independent framework. An accurate two-node curved element and the Kelvin-Voigt model are introduced in this framework to consider the axial deformation, bending deformation, shear deformation, rotary inertia, and viscoelasticity of the beam. Three examples are given to verify the validity, computational efficiency, and versatility of the presented formulation. The effect of internal damping, external damping, the inertia force of a moving mass, and the moment of inertia of the mass on the dynamic response of the beam are investigated. Moreover, the driving force for a prescribed motion of a mass along the beam is investigated. [ABSTRACT FROM AUTHOR]

• Singularity analysis reveals all kinds of periodic responses of a NES system. • Essential dimensionless parameters are defined to determine the NES dynamics. • Fourteen different kinds of periodic responses exist in the NES system. • NES effective parameter regions are located by the singularity analysis. Nonlinear energy sink (NES) can be used to reduce the forced vibration of the primary system. Many different kinds of periodic responses of the linear system coupled with NES have been presented in the literature. In order to reveal all kinds of periodic responses of the NES system on given parameter planes, this study intends to achieve the classification of different kinds of periodic responses of linear oscillator coupled with a NES under harmonic excitation, which is modeled as a 2-degree-of-freedom system. The amplitudes of periodic responses of the nonlinear system are obtained by using the harmonic balance method. Then a general method based on the singularity theory is proposed to reveal all kinds of periodic responses under a given set of parameters. The periodic responses include 10 kinds known responses and 4 kinds of newly-revealed responses. The complete classification of the responses ensure the reliability and the efficiency of a NES. [ABSTRACT FROM AUTHOR]

• A nonlinear energy sink is proposed to restrict the vibration amplitude of the piecewise system. • In order to solve the piecewise system conveniently and accurately, a continuous and simple function is introduce to fit the discontinuous restoring force. • The vibration response of the piecewise system coupled with NES is studied theoretically and experimentally. The vibration reduction of a piecewise linear system coupled with a nonlinear energy sink (NES) is studied. Piecewise stiffness can restrain the large vibration while the NES enhances the control effect. An experimental device of the piecewise system coupled with the NES is deigned, and the governing equation is established with Newton's second law. Parameters of the system are obtained based on the experimental data. Especially, the nonlinear stiffness and the damping of the NES are identified by the restoring-force surface method. For a better accuracy during the analytical processing, a hyperbolic tangent function is introduced to fit the piecewise restoring force, instead of the Taylor series expansion. Considering the strong nonlinearity of the NES, the steady-state response of the coupled system is analyzed by the harmonic balance method (HBM), including the stability analysis. The numerical simulation and the experimental study verify that the result obtained with the HBM has a good accuracy. Compared with the response of the piecewise system without the NES, the NES has a significant damping performance for the piecewise structure. The NES even can restrain the vibration of the main system to a small region, in which the auxiliary spring in the piecewise system does not work. Based on the detailed investigations on the cubic nonlinear stiffness, the damping, and the mass of the NES, the control performance is optimized. A fork-shaped phenomenon exists at the top of the response curve of the main structure under optimized parameters. The discussion suggests that, a better control performance can be obtained with the NES which consists of a small mass, a strong nonlinear stiffness, and a well-designed damping. [ABSTRACT FROM AUTHOR]

Elastic wave propagation in solids composed of elastic metamaterials always exists as both transverse and longitudinal waves simultaneously. By using the Hamilton principle, Bloch theorem, and finite-element method, an approach is proposed to explore the band energy structures of the metamaterial, this is observed in an experiment on an ultralow frequency metamaterial periodically positioned nonlinear resonators and this means the broadband isolation of both longitudinal and transverse waves was verified. Comparison of the propagation of transverse and longitudinal waves in the ultralow frequency metamaterials, the Bragg band-gaps (BBGs) and local resonance band-gaps (LRBGs) differ with both the frequency of the bandwidth and the depth of the band-gap. [ABSTRACT FROM AUTHOR]

• The force-displacement relation is established for a compressed elliptical ring. • The elliptical ring isolator exhibits high-static-low-dynamic stiffness. • The isolator achieves the minimum resonance frequency with an optimal eccentricity. An elliptical ring is used to implement a novel type of high-static-low-dynamic vibration isolation. Under the compression deformation, the elliptical ring produces a nonlinear restoring force due to the coupling of the tension and the curvature caused by the stretching. For the elliptical ring under a uniaxial pressure, its force-displacement relation is derived from the Euler–Bernoulli beam theory. The force-displacement relation is experimentally validated via a digital force gauge. The dynamic equation is established for the elliptical ring isolator under a harmonic base excitation. It is revealed that the elliptical ring isolator is with high-static-low-dynamic stiffness. The harmonic balance method is applied to determine approximately steady-state responses of the vibration isolator, and thus the displacement transmissibility of the isolator can be analytically predicted. The analytical results show the effects of the eccentricity. The analytical outcomes are numerically confirmed by the direct integration of the dynamic equation. Under certain conditions, there is an optimal eccentricity for the fixed elliptical circumference, and the displacement transmissibility achieves a minimum value at the optimal eccentricity. The elliptical ring isolator is experimentally tested, and the experimental results validate the analytical outcomes. [ABSTRACT FROM AUTHOR]

• A novel vibration control and energy harvesting experimental device is proposed for the whole-spacecraft system. • The complexification-averaging (CX-A) method is devoted to analyze the whole-spacecraft system. • The experimental and theoretical analysis results show that the NES-GMM device has a positive effect on vibration reduction and energy harvesting. • The integration of NES-GMM device causes very little change in the resonance frequency of the whole-spacecraft system. Giant magnetostrictive material (GMM) is integrated into a nonlinear energy sink (NES) to construct a NES-GMM device for vibration control and energy harvesting. The NES-GMM device is experimentally embedded in a scaled model of a whole-spacecraft system. LMS Test.Lab software and an oscilloscope are used to collect vibration and energy signals of the whole-spacecraft system with and without the NES-GMM device under sine sweeping frequency excitations. The transmissibility amplitudes, resonant frequencies, and voltage values demonstrate the effeteness of vibration control and energy harvesting of the NES-GMM device under different various conditions. The experimental system is modeled as a two-degrees-of-freedom (2DOF) linear primary system with a nonlinear NES-GMM additional subsystem. The complexification-averaging (CX-A) method verified with numerical verifications is developed to investigate analytically the vibration reduction and energy harvesting of the NES-GMM device for the whole-spacecraft system. The experimental and theoretical results show that the NES-GMM device reduces the vibration and produces the electricity and that the device hardly changes the resonance frequencies of the original whole-spacecraft system. The saddle-node (SN) bifurcation is detected in the steady-state periodic response of the system. The performance of the NES-GMM device is examined for its varying parameters. [ABSTRACT FROM AUTHOR]

• A distributed model of a tri-stable energy harvester (TEH) is established. • The TEH with asymmetric potential wells extended the working bandwidth. • The asymmetric TEH realizes snap-through at weak excitations. • The superiority of an asymmetric TEH is experimentally verified. Scavenging energy from ambient vibration by tri-stable energy harvesters (TEHs) has been focused in recent years. Generally, classical TEHs adopt a design of symmetric potential wells. To improve the harvesting efficiency, a TEH with a staircase-shaped potential well (TEH-SSP) is proposed. This staircase-shaped potential well is established by adjusting the distances from the fixed magnets to the axis of symmetry. The distributed model of the proposed energy harvester is established, and the dynamical equations are derived by using the energy approach method. The nonlinear characteristics are explored theoretically and validated by experiments. Compared with the classical TEH with symmetric potential (TEH-SP), the proposed TEH-SSP could execute snap-through from a low frequency, and can obviously enlarge the frequency bandwidth of snap-through. Thus it can create a more dense high output voltages under weak random excitation. Both the theoretical analysis and experimental results prove the effectiveness of the proposed asymmetric design and vindicate that it is very beneficial for harvesting energy from the ambient vibration excitation. This work may serve as a novel insight into the design of more efficient harvesters with a broad bandwidth. [ABSTRACT FROM AUTHOR]

• A limited nonlinear energy sink is proposed with piecewise linear springs. • The vibration response of the limited NES is studied theoretically and experimentally. • The particle swarm optimization algorithm is used to optimize the design parameters. In the research of applying nonlinear energy sinks for vibration reduction, usually the vibration amplitude of nonlinear energy sinks is not limited. Since the linear stiffness of the nonlinear energy sink is zero, the vibration amplitude of the nonlinear energy sink may be very large. This is obviously not acceptable in many mechanical engineering fields. Based on theoretical and experimental investigations, this paper investigates a limited nonlinear energy sink by using a piecewise spring device. Thus, the vibration amplitude of the nonlinear energy sink can be restricted from being too large. For free vibration, the effects of the piecewise stiffness and the gap width on the vibration responses of the nonlinear energy sink and the primary system, i.e. the linear oscillator, are profoundly examined. The numerical results show that the vibration of the nonlinear energy sink is effectively suppressed. Nevertheless, the vibration damping effect of the nonlinear energy sink on the linear oscillator is weakened after the introduction of the piecewise spring. However, after improving parameters reasonably, the nonlinear energy sink can still achieve considerable damping effects. In addition, the experimental results are conducted to verify the theoretical results. Moreover, the particle swarm optimization algorithm is used to optimize the piecewise stiffness and the gap width so that the vibration of the linear oscillator is suppressed most efficiently. The optimized results are verified with the differential evolution algorithm to illustrate that the particle swarm optimization is effective and accurate. In conclusion, this work provides useful information for the design of the nonlinear energy sink and can promote the engineering application of the nonlinear energy sink. [ABSTRACT FROM AUTHOR]

• The proposed approach based on Chebyshev inclusion function and least square method. • An uncertain gear vibration experiment is firstly designed and performed. • Effects of different uncertain parameters on system amplitude-frequency responses. Gear transmission systems are widely used in mechanical equipment. Due to manufacturing and processing errors, material defects, wear and external environments, there are uncertain parameters such as mass, mesh stiffness and support stiffness. Parametric uncertainty plays an important role in response predictions of the gear transmission systems. In order to evaluate reasonably the dynamic characteristics of the gear transmission systems, a dynamic model of an uncertain single-stage gear system is established. The dynamic responses of the bounded uncertain gear transmission system are investigated by an interval analysis method based on the Chebyshev inclusion function. A matrix form of the least square method is applied to improve the calculation efficiency of the interval approach. The investigation reveals the effects of different uncertain parameters on amplitude-frequency responses of the gear system. It is the first time to design and to implement a test rig of an uncertain single-stage spur gear transmission system. The experimental results verify the accuracy and the effectiveness of the theoretical analysis results. The results demonstrate that the parametric uncertainty may propagate in the gear transmission system and the propagation results in significant variations of the dynamic responses. [ABSTRACT FROM AUTHOR]

• A bio-inspired multi-layer quasi-zero stiffness isolation system is proposed. • The approximate analytical results are verified by numerical method. • Vibration characteristics and the influence of system parameters are investigated. • The results show that the harmonic ultra-low frequency vibration is isolated. With the increasing of the accuracy requirement of data record for on-orbit instruments, it becomes a greater challenge and brings more opportunities in design of observing spacecraft to isolate micro-vibrations of low frequency and ultra-low frequency. By imitating both multi-vertebra structure of bird's neck and gazing stability of head, a multi-layer quasi-zero stiffness (QZS) nonlinear isolator is investigated in this paper. The mathematical model with high-order coupled nonlinear ordinary differential equations is established. Its isolation performance is analyzed by the harmonic balance method (HBM) combining with the pseudo arc length method. In comparison to the corresponding linear structure, the proposed multi-layer QZS nonlinear isolator can break the limitation of poor isolation performance in low frequency and ultra-low frequency vibration for traditional linear passive isolation systems. Also, the analysis results show that its High-Static-Low-Dynamic-Stiffness (HSLDS) property may solve the traditional contradiction between large isolation frequency band and loading capacity for reducing soft stiffness. The multi-layer QZS combined system as integration is quite capable of cumulatively expanding the effective range of dynamic displacement which is too narrow for a single QZS system. The relevant conclusions are instructive to develop preferable isolation equipment to isolate low frequency and ultra-low frequency vibration in practical engineering. [ABSTRACT FROM AUTHOR]

• A novel nonlinear vibration isolator in the shape of a circular ring is proposed. • Stretching-induced tension coupled with the curvature changes produce highly nonlinear stiffness. • The use of stiffness nonlinearity improves vibration isolation efficiency. • The results of transmissibility are validated through an experiment. A novel nonlinear vibration isolator in the shape of a circular ring is investigated in this study. When the ring is compressed along a diametral line, it exhibits highly nonlinear geometric stiffness due to the effects of stretching-induced tension coupled with the curvature changes. The use of stiffness nonlinearity improves vibration isolation efficiency. A mechanical model of the ring under compression is derived from beam theory. Methods of direct separation of motions and harmonic balance are employed to determine the frequency response and displacement transmissibility of the device under base excitations for different geometric configurations. The analytical results are numerically validated via direct time integration of the equation of motion and via experimental data. It is shown that an increase in the pre-deformation of the ring enhances the vibration isolation performance. [ABSTRACT FROM AUTHOR]

• Multimode resonances of composite plates are absorbed by inertial nonlinear absorbers. • Nonlinear solution is obtained approximately analytically and numerically. • A lightweight solution is proposed for wide-band suppressing vibration of plates. Bending vibration of the elastic structure has many resonance modes. Especially, the composite plate structure has many low frequency modes. Therefore, devices capable of achieving wide-band vibration suppression are required. Multimode resonances elimination of composite plates by an inertial nonlinear energy sink (NES) is proposed for the first time. Different with the traditional NES, the mass of the absorber in the present work is replaced by an inerter to reduce the weight of the attached device. The response of the plate is investigated by the Galerkin discretization together with the harmonic balance method (HBM). The gravity effect of the mass of the traditional NES is presented. It shows the benefits of the inertial NES without affecting the resonant frequency of the primary system. By comparing the resonance with and without the NES, the proposed device is certified that it has good efficiency in vibration eliminating for both low-order and high-order resonance. It finds that the mass of the inertial NES, which can achieve the effective suppression effect, is relatively tiny. The parameter effect of the NES is discussed fully to achieve the optimum design. Optimal parameters on the resonance response of different modes are not identical. This paper will provide a good reference for vibration elimination of multimode resonance of plates. [ABSTRACT FROM AUTHOR]