Vibration control using nonlinear damped coupling

The harmonic balance method is a very useful tool for characterizing and predicting the response of nonlinear dynamic systems undergoing periodic oscillations, either self-excited or due to harmonic excitation. The method and several of its variants were applied to nonlinear aeroelastic systems over the last two decades. This paper presents a detailed description of several harmonic balance methods and a continuation framework allowing the methods to follow the response of dynamic systems from the bifurcation point to any desired parameter value, while successfully negotiating further fold bifurcations. The continuation framework is described for systems undergoing subcritical and supercritical Hopf bifurcations as well as a particular type of explosive bifurcation. The methods investigated in this work are applied to a nonlinear aeroelastic model of a generic transport aircraft featuring polynomial or free-play stiffness nonlinearity in the control surface. It is shown that high-order harmonic balance solutions will accurately capture the complete bifurcation behavior of this system for both types of nonlinearity. Low-order solutions can become inaccurate in the presence of numerous folds in the limit-cycle oscillation branch but can still yield practical engineering information at a fraction of the cost of higher-order solutions. Time-domain harmonic balance schemes are shown to be more computationally expensive than the standard harmonic balance approach.

A dynamic physical system often undergoes phase transitions in response to fluctuations induced on system parameters. For example, hurricane activity is the climate system's response initiated by a liquid-vapor phase transition associated with non-linearly coupled fluctuations in the ocean and the atmosphere. Because our quantitative knowledge about highly non-linear dynamic systems is very meager, scientists often resort to linear regression techniques such as Least Absolute Deviation (LAD) to learn the non-linear system's response (e.g., hurricane activity) from observed or simulated system's parameters (e.g., temperature, precipitable water, pressure). While insightful, such models still offer limited predictability, and alternatives intended to capture non-linear behaviors such as Stepwise Regression are often controversial in nature. In this paper, we hypothesize that one of the primary reasons for lack of predictability is the treatment of an inherently multi-phase system as being phase less. To bridge this gap, we propose a hybrid approach that first predicts the phase the system is in, and then estimates the magnitude of the system's response using the regression model optimized for this phase. Our approach is designed for systems that could be characterized by multi-variate spatio-temporal data from observations, simulations, or both.

Among the irregular responses of nonlinear dynamic systems, chaotic responses of nonlinear systems are probably the most attractive phenomena along with the new observations in the last decades. A nonlinear deterministic system may behavior chaotically under regular such as periodic excitations. Regular motion of a system subjected to periodic exertions is usually periodic. In contrast with regular motions, final states of chaotic vibrations are extremely nonperiodic. This research is to analyzing the irregular behavior of dynamic systems with implementation of a newly developed criterion named Periodicity-Ratio. The development of a methodology for diagnosing the irregular motions from the regular motions of a dynamic system is presented. The Periodicity-Ratio describes the degree of periodicity of motion and can be conveniently used to distinguish a nonperiodic motion from a regular vibration or oscillation and to diagnose whether or not a motion is chaotic and the other irregular responses of the nonlinear dynamic systems, without plotting any figures. The analyses on the irregular behavior of nonlinear dynamic systems with the implementation of the Periodicity-Ratio will be demonstrated.

In this study, shaking table tests, theoretical research, and finite element modeling analysis were conducted on seismic responses of a transmission tower-line system under multi-component ground motions. The rocking component of ground motion was determined from the original seismic record by the wavelet analysis. The shaking table tests for scale models (single tower and tower-line system) that satisfy the similarity ratio of an actual tower-line system, were performed under multiple directions of motion. Dynamic equations under multi-component (horizontal, coupled horizontal, and tilting (CHT), and coupled vertical, horizontal, and tilting (CVHT)) ground motions were first derived in theoretical studies. The finite element model based on the actual transmission tower line system was built, and the response of the system under the identical working condition was calculated. It is found that the rotational component of the ground motion cannot be overlooked and significantly impacts the seismic response of the tower-line system, which is reflected especially as the vertical motion is coupled. The tilting ground motion, the additional second-order effect, and the foundation tilting jointly cause the increase in the response amplitude and significant asymmetric displacement effect. Since this effect of tilting ground motions, the weakening effect of the transmission line to reduce the responses for the displacement and acceleration of the tower-line system is also weakened. It is, therefore, essential to consider the adverse effects of multi-component ground motions on the designing of transmission tower systems in high-intensity areas.

Harmonic balancing method has more powerful superiority than other perturbation methods in the respect of solving steady-state response of strong nonlinear system. The amplitude-frequency equation of dry-friction vibration system considering cubic nonlinear displacement under harmonic excitation was derived by harmonic balancing method. Because the effect of even nonlinear term on vibration response of system was not evident, even nonlinear term of elastic restoring force was neglected. The first term coefficient and third term coefficient obtained by static experimental data for constitutive relationship of metallic rubber were substituted into frequency response equation. Amplitude-frequency characteristics curves of vibration response for cubic nonlinear displacement dry-friction system of metallic rubber were obtained. Natural frequencies obtained by computation were compared with natural frequencies by experiment, and the two Natural frequencies consisted with each other. It showed that the method that vibration response for cubic nonlinear displacement dry-friction system of metallic rubber was computed is reasonable.

Harmonic balance method (HBM) is one of the most popular and powerful methods, which is used to obtain response of nonlinear vibratory systems in frequency domain. The main idea of the method is to express the response of the system in Fourier series and converting the nonlinear differential equations of motion into a set of nonlinear algebraic equations. System response can be obtained by solving this nonlinear equation set in terms of the unknown Fourier coefficients. The accuracy of the solution is greatly affected by the number of harmonics included in the solution; hence, increasing the number of harmonics increases the accuracy of the solution at the expense of computational effort. Therefore, it is desirable to use an adaptive algorithm where the number of harmonics can be optimized in terms of both accuracy and computational effort. Until now, various adaptive harmonic balance methods have been formulated to perform this task. This paper presents an overview and a comparison of these adaptive harmonic balance methods in terms of their effectiveness.

This paper presents a study of a two-degree-of-freedom (2DOF) piezoelectric energy harvester (PEH) under concurrent aeroelastic and base excitation. The governing equations of the theoretical model under the combined excitation are developed and solved analytically using the harmonic balance method. Based on the electro-mechanical analogies, an equivalent circuit model is established. The energy harvesting performance of the 2DOF PEH under different wind speeds but the same base excitation is investigated. Voltage amplitudes of various response components with different frequencies are predicted by the analytical method and verified by the circuit simulation. The root-mean-square (RMS) voltage is used to measure the actual performance of the 2DOF PEH. Around the resonance state, the 2DOF PEH has been found to produce a larger voltage output than the conventional SDOF PEH. Moreover, several interesting phenomena, such as the quasi-periodic oscillation and the peak-to-valley transition, have been observed in the circuit simulation and explained by the analytical solution. The developed methodology in this paper can be easily adapted to analyze other similar types of multiple-degree-of-freedom (MDOF) PEHs under concurrent aeroelastic and base excitation.

Responsive systems have recently gained much interest in the scientific community in attempts to mimic dynamic functions in biological systems. One of the fascinating potential applications of responsive systems lies in catalysis. Inspired by nature, novel responsive catalytic systems have been built that show analogy with allosteric regulation of enzymes. The design of responsive catalytic systems allows control of catalytic activity and selectivity. In this Review, advances in the field over the last four decades are discussed and a comparison is made amongst the dynamic responsive systems based on the principles underlying their catalytic mechanisms. The catalyst systems are sorted according to the triggers used to achieve control of the catalytic activity and the distinct catalytic reactions illustrated.

Theoretical modeling and experimental analysis of nonlinear electromagnetic shunt damping

In the previous paper, the authors described the statistical studies on the response of nonlinear control systems subjected to an ergodic stationary Gaussian random input. This paper is devoted to the fundamental approaches to the non-stationary random time series which are considered to arise quite often for the automatic control systems in practice. Firstly, the response of nonlinear control systems subjected to a non-stationary random signal is evaluated. Secondly, for an example, a non-stationary Orenstein-Uhlenbeck process which is obtained by using the method of Fokker-Planck is treated. Detail illustrations are shown by several examples.

In recent years, the non-stationary random processes have attracted considerable attentions in the field of adaptive or self-optimizing control, but little seems to have been done for those cases. In this paper, a very simple but fundamental case of non-stationary random process of response of nonlinear control systems is treated. That is, the analytical study on the evaluation of the response of nonlinear control systems subjected to a suddenly applied stationary random signal is described The two methods are established here by introducing some extensive method of the equivalent linearization technique. A graphical method for evaluating the non-stationary random response is proposed and such procedures are explained by typical examples in detail.

Vibration characteristics analysis of fluid-conveying pipes concurrently subjected to base excitation and pulsation excitation

This paper investigates the dynamics of a spur gear system with time-varying and square nonlinearities, by both analytical method and numerical simulation. First, the equations of motion of a 2 degree-of-freedom system are established and the harmonic balance method is used to analyze the stability and the steady-state response of the system under the main resonant condition. Then the perturbation method is used to analyze the parameter instability under the main, subharmonic and nonresonance conditions. Finally, the interactions between the main and subharmonic resonant amplitudes and the nonlinearity and the contact ratio are analyzed. The results reveal that the system response contains various frequency components, such as meshing frequency and its higher harmonic terms due to the nonlinearity and the time-varying stiffness. The existence of the time-varying meshing stiffness can also result in the subharmonic resonance, and even chaos through period-doubling bifurcations as the input torque increases.

The characteristics of a passive nonlinear isolator are developed by combining a curved-mount-spring-roller mechanism as a negative stiffness corrector in parallel with a vertical linear spring. The static characteristics of the isolator are presented, and the configurative parameters are optimized to achieve a wider displacement range at the equilibrium position where the isolator has a low stiffness and the stiffness changes slightly. The restoring force of the isolator is approximated using a Taylor expansion to a cubic stiffness. Considering the overload and underload conditions, a dynamic equation is established as a Helmholtz-Duffing equation, and the resonance response of the nonlinear system is determined by employing the harmonic balance method (HBM). The frequency response curves (FRCs) are obtained for displacement excitations. The absolute displacement and acceleration transmissibility are defined and investigated to evaluate the performance of the nonlinear isolator, and they are compared with an equivalent linear isolator that can support the same mass with the same static deflection as the proposed isolator. The effects of the amplitude of the excitation, the offset displacement, and the damping ratio on the dynamic characteristics and the transmissibility performance are considered, and experiments are carried out to verify the above analysis. The results show that the overload and underload system can outperform the counterparts with the linear stiffness, softening stiffness, softening-hardening stiffness, and hardening stiffness with the magnitude of the excitation amplitude, and that its isolation performance is generally better than that of a linear system. The transmissibility, response, and resonance frequency of the system are affected by the excitation amplitude, offset displacement, cubic stiffness, and damping ratio. To obtain a better isolation performance, an appropriate mass, not-too-large amplitude, and larger damper are necessary for the proposed isolator.

To isolate low-frequency vibration, a novel single-degree-of-freedom vibration isolation system with quasi-zero stiffness (QZS) and nonlinear damping using geometric nonlinearity is proposed in this study. One of the remarkable features of this system is the use of scissor-like structures (SLSs) to achieve the nonlinear stiffness and damping. The length difference between the connecting rods in SLS is considered. First, both the stiffness and damping characteristics are derived and analyzed in detail. Then, the frequency response and force transmissibility are obtained using the harmonic balance method. Finally, the effects of structural parameters on the isolation performance are investigated. Theoretical results show that the proposed QZS vibration system can not only isolate low-frequency vibration but also suppress the high-amplitude vibration in the resonant region. Besides, increasing nonlinear damping has little influence on the isolation performance in high frequencies. The proposed QZS vibration system can outperform a classical counterpart.