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A novel lever-type stiffness-based grounded damping dynamic vibration absorber with grounded stiffness is presented in this paper, and the analytical design parameters are derived in detail. At the first, the equations of motion are established and the analytical solution of the primary structure displacement is obtained. It is found that with the introduction of grounded stiffness, the coupled system could be unstable and the stability condition is established. Then, the optimum stiffness ratio, the optimum damping ratio and the optimum grounded stiffness ratio are expressed as the function of mass ratio and lever ratio by minimizing the mean squared displacement response of the primary structure previously established. From the results analysis, the system stability is verified, and it is found that with the change in the lever ratio when the mass ratio is selected, there are three cases for the optimum grounded stiffness ratio, i.e., negative, zero and positive. Thus, for the vibration reduction of primary structure, the proposed dynamic vibration absorber (DVA) with positive grounded stiffness has the best control performance among the three cases. Compared with some typical designed DVAs under harmonic and random excitation, the results show that with the proposed optimum DVA the resonance amplitude and the frequency band of vibration reduction can greatly reduce and broadened, respectively, and the random vibration mitigation can be greatly increased. According to the existing literature, the proposed lever-type stiffness mechanism is justified, which means that the proposed DVA is practical and can be used in many engineering applications. [ABSTRACT FROM AUTHOR]

Improving the control performance of dynamic vibration absorbers has recently been effective by introducing a grounded negative stiffness device. However, the negative stiffness structure is unstable and difficult to achieve in engineering practice , and its major drawback is that it amplif ies the vibration response of the primary system at low frequency region. Meanwhile, some mechanical devices can be combined to make the DVA work even better with a grounded positive stiffness. For this purpose, this paper combines for the first time the control effect of the inerter device and grounded positive stiffness into a three-element DVA model in order to better improve vibration reduction of an undamped primary system under excitation. First, the dynamic equation of motion of the system is written according to Newton 's second law. Then, the steady-state displacement response of the primary system under harmonic excitation is calculated. In order to minimize the resonant response of the primary system around its natural frequency, the extended fixed point theory is applied. Thus, the optimized parameters such as the tuning frequency ratio, the stiffness ratio , and the approximate damping ratio are determined as a function of mass ratio and inerter – mass ratio. From the results analysis, it was found that the inerter – mass ratio has a better working range to guarantee the stability of the coupled system. Then , study on the effect of inerter – mass ratio on the primary system response is carried out. It can be seen that increasing the inerter – mass ratio in the optimal working range can reduce the response of the primary system beyond its uncontrolled static response. However, it is necessary to avoid the situation where the inerter – mass ratio is very large because it can lead to unrealistic optimal parameters. Finally, comparison with other DVA models is show n under harmonic and random excitation of the primary system. It is found that the proposed DVA model in this paper has high control performance and can be used in many engineering practice s. [ABSTRACT FROM AUTHOR]

The (2 + 1) -dimensional chiral nonlinear Schrödinger equation (CNLSE), which specifies the edge states of the Hall effect, is presented in this study. A complicated transformation is performed, and the bifurcation conditions are determined using the theory of planar dynamical systems. The phase pictures of the system are then produced using quantitative analysis in order to predict the family of solutions which can be found for the equation studied. It is important to note that this prediction is usually not made. The qualities of phase pictures are then used to obtain the exact solutions. As a result, this model produces some bright solitons, dark solitons and periodic wave solitons. Some of the solutions are graphically depicted in three dimensions (3D) using Matlab software to aid comprehension, and they play an important part in the creation of realistic Quantum Hall effects when Chiral excitations are known to occur. The method applied in this paper is simple and does not need an ansatz to predict the solutions as in the literature. [ABSTRACT FROM AUTHOR]

This article presents the results of the study of a novel inerter-based vibration absorber with an amplified inertance mechanism and grounded stiffness, to control excessive vibrational movements of an excited primary structure. The inerter vibration absorber used in this study acts as a passive tuned inerter damper. An undamped primary structure model with a single degree of freedom controlled by the proposed inerter vibration absorber is developed and used to derive the equations of motion of the coupled system. The optimum frequency ratio and the optimum damping ratio of inerter vibration absorber are found using the fixed point theory for harmonic force-excited primary structures. Then, the optimum grounded stiffness ratio is deduced. Based on the inclusion of an amplified inertance mechanism, it is found that for given inertance mass ratio, the change in the amplification ratio results in three cases for the optimum grounded stiffness ratio, that is, negative, zero, and positive. From these three cases of grounded stiffness, the inerter vibration absorber with positive grounded stiffness has demonstrated the best control performance. Under optimum parameters, the results indicate that the inerter vibration absorber in this article outperforms some existing inerter vibration absorbers under the harmonic excitation, in terms of decreases in the peak vibration response of the primary system and widens the suppression bandwidth. Finally, the further comparison among the inerter vibration absorber under random (white noise) excitation also shows that the model in this article is superior to other inerter vibration absorbers in terms of smallest mean square response and smallest variance of the time history of the primary system. [ABSTRACT FROM AUTHOR]

This study proposes a series–parallel inerter system with negative stiffness for the passive vibration control of an undamped single–degree–of–freedom system under base excitation. The necessary and sufficient conditions for stability of series-parallel inerter system with negative stiffness are established by Routh–Hurwitz criterion, and the stability boundary is obtained. The tuning parameters of the series-parallel inerter system with negative stiffness are determined through fixed point theory, and a comparison between the vibration mitigation performance of series-parallel inerter system with negative stiffness, series–parallel inerter system (without negative stiffness), and tuned mass damper is presented considering both harmonic excitation, transient excitation, and random (white noise) excitation. The results of this study demonstrate that under base harmonic excitation, series-parallel inerter system with negative stiffness outperforms the series–parallel inerter system and tuned mass damper in terms of suppression bandwidth and reducing the peak vibration amplitude of the primary mass. In the case of base acceleration–excited primary structure, more than 49.84% and 67.53% improvement can be obtained from series-parallel inerter system with negative stiffness as compared with tuned mass damper in terms of suppression bandwidth and reducing the peak vibration amplitude, respectively. Whereas in the case of base displacement–excited primary structure, more than 78% and 80% improvement can be obtained from series-parallel inerter system with negative stiffness, respectively, following the same criteria. A slightly lower improvement has been obtained from series-parallel inerter system with negative stiffness as compared with series–parallel inerter system, which justified the superiority of series–parallel inerter system compared to tuned mass damper. The transient response investigation showed that series-parallel inerter system with negative stiffness outperforms the series–parallel inerter system and tuned mass damper in terms of much shorter stabilization times and lower peak amplitude of the primary mass. Finally, the further comparison among these devices (series-parallel inerter system with negative stiffness, series–parallel inerter system, and tuned mass damper) under white noise excitation also shows that series-parallel inerter system with negative stiffness is superior to series–parallel inerter system and tuned mass damper for a small inertance mass ratio. This result could provide a theoretical basis for the design of inerter-based isolators with negative stiffness. [ABSTRACT FROM AUTHOR]

This study is concerned with the problem of analysis and optimization of inerter-based systems. A main inerter system is generally composed of an inerter, a spring, and viscous damper. Series – parallel inerter system s and series inerter system s are two commonly used configurations of inerter-based system s. First , in this study , the H ∞ optimum parameters of inerter-based isolators are derived to minimize the compliance and mobility transfer function of a single-degree -of-freedom system under a harmonic ground acceleration excitation. Under the optimum tuning condition, it is shown that the proposed inerter-based isolators when compared with the traditional dynamic vibration absorber provide larger suppression of the peak value of the magnitude of compliance and mobility transfer function s of the primary system. For the studied cases, more than 40% and 45% improvement can be attained in terms of minimizing the compliance and mobility transfer function s , respectively, as compared with the traditional dynamic vibration absorber for the series – parallel inerter system and 15% and 11% improvement can be attained respectively , for the series inerter system. Finally, further comparison between the inerter-based isolators and traditional dynamic vibration absorber under white noise excitation also shows that the series – parallel inerter system and series inerter system s are superior to the traditional dynamic vibration absorber. The results of the studied systems show that m ore than 23% and 16% improvement are attained in terms of minimizing the compliance and mobility transfer function s respectively , as compared with the traditional dynamic vibration absorber for the series – parallel inerter system and 26% and 13% improvement can be attained respectively , for the series inerter system. The optimal parameters for different cases are obtained. It is shown that the optimal parameters obtained using the minimized mobility transfer function are smaller than those using the compliance transfer function at all mass ratios or inertance-to-mass ratio. The results of this study can provide theoretical basis for design of the optimal inerter-based isolators in engineering practice. [ABSTRACT FROM AUTHOR]