Targeted energy transfer of a parallel nonlinear energy sink

A parallel nonlinear energy sink(NES) is proposed and analyzed. The parallel NES is composed of a vibro-impact(VI) NES and a cubic NES. The dynamical equation is given, and the essential analytical investigation is carried out to deal with the cubic nonlinearity and impact nonlinearity. Multiple time-scale expansion is introduced, and the zeroth order is derived to give a rough outline of the system. The underlying Hamilton dynamic equation is given, and then the optimal stiffness is expressed. The clearance is regarded as a critical factor for the VI. Based on the periodical impact treatment by analytical investigation, the relationships of the cubic stiffness, the clearance, and the zeroth-order attenuation amplitude of the linear primary oscillator(LPO) are obtained.A cubic NES under the optimal condition is compared with the parallel NES. Harmonic signals, harmonic signals with noises, and the excitation generated by a second-order?lter are considered as the potential excitation forces on the system. The targeted energy transfer(TET) in the designed parallel NES is shown to be more e?cient.     

Numerical and experimental investigations of a rotating nonlinear energy sink

In this work, passive nonlinear targeted energy transfer (TET) is addressed by numerically and experimentally investigating a lightweight rotating nonlinear energy sink (NES) which is coupled to a primary two-degree-of-freedom linear oscillator through an essentially nonlinear (i.e., non-linearizable) inertial nonlinearity. It is found that the rotating NES passively absorbs and rapidly dissipates a considerable portion of impulse energy initially induced in the primary oscillator. The parameters of the rotating NES are optimized numerically for optimal performance under intermediate and strong loads. The fundamental mechanism for effective TET to the NES is the excitation of its rotational nonlinear mode, since its oscillatory mode dissipates far less energy. This involves a highly energetic and intense resonance capture of the transient nonlinear dynamics at the lowest modal frequency of the primary system; this is studied in detail by constructing an appropriate frequency–energy plot. A series of experimental tests is then performed to validate the theoretical predictions. Based on the obtained numerical and experimental results, the performance of the rotating NES is found to be comparable to other current translational NES designs; however, the proposed rotating device is less complicated and more compact than current types of NESs.     

Identification method for backbone curve of cantilever beam using van der Pol-type self-excited oscillation

This study presents an experimental method for identification of the backbone curves of cantilevers using the nonlinear dynamics of a van der Pol oscillator. The backbone curve characterizes the nonlinear stiffness and nonlinear inertia of the resonator, so it is important to identify this curve experimentally to realize high-sensitivity and high-accuracy sensing resonators. Unlike the conventional method based on the frequency response under external excitation, the proposed method based on self-excited oscillation enables direct backbone curve identification, because the effect of the viscous environment is eliminated under the linear velocity feedback condition. In this research, the method proposed for discrete systems is extended to give an identification method for continuum systems such as cantilever beams. The actuation is given with respect to both the linear and nonlinear feedbacks so that the system behaves as a van der Pol oscillator with a stable steady-state amplitude. By varying the nonlinear feedback gain, we can produce the self-excited oscillation experimentally with various steady-state amplitudes. Then, using the relationship between these steady-state amplitudes and the corresponding experimentally measured response frequencies, we can detect the backbone curve while varying the nonlinear feedback gain. The efficiency of the proposed method is determined by identifying the backbone curves of a macrocantilever with a tip mass and a macrocantilever subjected to atomic forces, which are representative sources of hardening and softening cubic nonlinearities, respectively.

     

The X-structure/mechanism approach to beneficial nonlinear design in engineering

Nonlinearity can take an important and critical role in engineering systems, and thus cannot be simply ignored in structural design, dynamic response analysis, and parameter selection. A key issue is how to analyze and design potential nonlinearities introduced to or inherent in a system under study. This is a must-do task in many practical applications involving vibration control, energy harvesting, sensor systems, robotic technology, etc. This paper presents an up-to-date review on a cutting-edge method for nonlinearity manipulation and employment developed in recent several years, named as the X-structure/mechanism approach. The method is inspired from animal leg/limb skeletons, and can provide passive low-cost high-efficiency adjustable and beneficial nonlinear stiffness (high static & ultra-low dynamic), nonlinear damping (dependent on resonant frequency and/or relative vibration displacement), and nonlinear inertia (low static & high dynamic) individually or simultaneously. The X-structure/mechanism is a generic and basic structure/mechanism, representing a class of structures/mechanisms which can achieve beneficial geometric nonlinearity during structural deflection or mechanism motion, can be flexibly realized through commonly-used mechanical components, and have many different forms (with a basic unit taking a shape like X/K/Z/S/V, quadrilateral, diamond, polygon, etc.). Importantly, all variant structures/mechanisms may share similar geometric nonlinearities and thus exhibit similar nonlinear stiffness/damping properties in vibration. Moreover, they are generally flexible in design and easy to implement. This paper systematically reviews the research background, motivation, essential bio-inspired ideas, advantages of this novel method, the beneficial nonlinear properties in stiffness, damping, and inertia, and the potential applications, and ends with some remarks and conclusions.

     

Analysis of an electromechanical energy harvester system with geometric and ferroresonant nonlinearities

Enhancing the performance of vibrating energy harvesting systems has been the backbone of several research contributions for the last few years, and it is considered in this paper. Specifically, an electromechanical energy harvester is analyzed, and the effects of geometric and ferroresonant nonlinearities on the electric power are discussed. The geometric nonlinearity includes the small- and high-order terms in Euler internal force while the ferroresonant nonlinearity is included by assuming different levels of saturation in the circuit. Our results reveal regions in the parameter space where nonlinear stiffness is better than linear stiffness and vice versa. Similarly, increasing the saturation parameter can be used to enhance the electric power.     

Numerical optimisation of a classical stochastic system for targeted energy transfer

The paper studies stochastic dynamics of a two-degree-of-freedom system, where a primary linear system is connected to a nonlinear energy sink with cubic stiffness nonlinearity and viscous damping. While the primary mass is subjected to a zero-mean Gaussian white noise excitation, the main objective of this study is to maximise the efficiency of the targeted energy transfer in the system. A surrogate optimisation algorithm is proposed for this purpose and adopted for the stochastic framework. The optimisations are conducted separately for the nonlinear stiffness coefficient alone as well as for both the nonlinear stiffness and damping coefficients together. Three different optimisation cost functions, based on either energy of the system’s components or the dissipated energy, are considered. The results demonstrate some clear trends in values of the nonlinear energy sink coefficients and show the effect of different cost functions on the optimal values of the nonlinear system’s coefficients.     

Vibration Attenuation of a Nonlinear Oscillator by using the Bound on a system

This work examines dynamic optimization of an autonomous oscillator with nonlinearities in stiffness and damping. Lyapunov analysis is utilized to show boundedness of the solution. The ultimate bound on the system is found by using Lyapunov stability criterion. The optimal parameters are found by estimating the bound on the system. The proposed theory can predict the parameters of a nonlinear autonomous system to a relatively good precision and superior vibration attenuation can be predicted.     

滞回摩擦型调谐惯质阻尼器结构隔震研究

本文研究一种新型非线性阻尼器——滞回摩擦型调谐惯质阻尼器(HFTID)在工程结构抗震控制中的应用。HFTID由调谐惯质阻尼器(TID)和滞回弹簧摩擦元件并联组成。首先通过谐波平衡方法推导了HFTID单自由度系统力与位移的传递率。然后对HFTID进行了最佳调谐参数优化,得到HFTID最优参数的近似表达式,比较了HFTID和TID振动控制系统的减振效果。结果表明,HFTID相比TID可以进一步降低振动控制系统的传递率。最后,以一栋多层隔震结构为例,将HFTID与TID的隔震效果进行了对比,结果表明,HFTID相比TID在降低地震响应峰值和均方根值方面具有更大优势,验证了HFTID在降低地震响应方面的有效性和实用性。HFTID在建筑和桥梁结构抗震、车辆悬挂系统和其他机械隔震问题上具有潜在的应用前景。     

Vibration of fluid-conveying pipe with nonlinear supports at both ends

The axial fluid-induced vibration of pipes is very widespread in engineering applications. The nonlinear forced vibration of a viscoelastic fluid-conveying pipe with nonlinear supports at both ends is investigated. The multi-scale method combined with the modal revision method is formulated for the fluid-conveying pipe system with nonlinear boundary conditions. The governing equations and the nonlinear boundary conditions are rescaled simultaneously as linear inhomogeneous equations and linear inhomogeneous boundary conditions on different time-scales. The modal revision method is used to transform the linear inhomogeneous boundary problem into a linear homogeneous boundary problem. The differential quadrature element method (DQEM) is used to verify the approximate analytical results. The results show good agreement between these two methods. A detailed analysis of the boundary nonlinearity is also presented. The obtained results demonstrate that the boundary nonlinearities have a significant effect on the dynamic characteristics of the fluid-conveying pipe, and can lead to significant differences in the dynamic responses of the pipe system.

     

Adaptive Control of Nonlinear Dynamical Systems Using a Model Reference Approach

In this paper we consider using a model reference adaptive control approach to control nonlinear systems. We consider the controller design and stability analysis associated with these type of adaptive systems. Then we discuss the use of model reference adaptive control algorithms to control systems which exhibit nonlinear dynamical behaviour using the example of a Duffing oscillator being controlled to follow a linear reference model. For this system we show that if the nonlinearity is small then standard linear model reference control can be applied. A second example, which is often found in synchronization applications, is when the nonlinearities in the plant and reference model are identical. Again we show that linear model reference adaptive control is sufficient to control the system. Finally we consider controlling more general nonlinear systems using adaptive feedback linearization to control scalar nonlinear systems. As an example we use the Lorenz and Chua systems with parameter values such that they both have chaotic dynamics. The Lorenz system is used as a reference model and a single coordinate from the Chua system is controlled to follow one of the Lorenz system coordinates.     

Attractors of harmonically forced linear oscillator with attached nonlinear energy sink. II: Optimization of a nonlinear vibration absorber

This paper is the second one in the series of two papers devoted to detailed investigation of the response regimes of a linear oscillator with attached nonlinear energy sink (NES) under harmonic external forcing and assessment of possible application of the NES for vibration absorption and mitigation. In this paper, we study the performance of a strongly nonlinear, damped vibration absorber with relatively small mass attached to a periodically excited linear oscillator. We present a nonlinear absorber tuning procedure in the vicinity of (1:1) resonance which provides the best total system energy suppression, using analytical and numerical tools. A linear absorber is also tuned according to the same criterion of total system energy suppression as the nonlinear one. Both optimally tuned absorbers are compared under common parameters of damping, external forcing but different absorber stiffness characteristics; certain cases for which nonlinear absorber is preferable over the linear one are revealed and confirmed numerically.     

Nonlinear aeroservoelastic analysis of a controlled multiple-actuated-wing model with free-play

In this paper, the effects of structural nonlinearity due to free-play in both leading-edge and trailing-edge outboard control surfaces on the linear flutter control system are analyzed for an aeroelastic model of three-dimensional multiple-actuated-wing. The free-play nonlinearities in the control surfaces are modeled theoretically by using the fictitious mass approach. The nonlinear aeroelastic equations of the presented model can be divided into nine sub-linear modal-based aeroelastic equations according to the different combinations of deflections of the leading-edge and trailing-edge outboard control surfaces. The nonlinear aeroelastic responses can be computed based on these sub-linear aeroelastic systems. To demonstrate the effects of nonlinearity on the linear flutter control system, a single-input and single-output controller and a multi-input and multi-output controller are designed based on the unconstrained optimization techniques. The numerical results indicate that the free-play nonlinearity can lead to either limit cycle oscillations or divergent motions when the linear control system is implemented.     

A nonlinear subspace-prediction error method for identification of nonlinear vibrating structures

The industrial structural systems always contain various kinds of nonlinear factors. Recently, a number of new approaches have been proposed to identify those nonlinear structures. One of the promising methods is the nonlinear subspace identification method (NSIM). The NSIM is derived from the principals of the stochastic subspace identification method (SSIM) and the internal feedback formulation. First, the nonlinearities in the system are regarded as internal feedback forces to its underlying linear dynamic system. The linear and nonlinear components of the identified system can be decoupled. Second, the SSIM is employed to identify the nonlinear coefficients and the frequency response functions of the underlying linear system. A typical SSIM always consists of two steps. The first step makes a projection of certain subspaces generated from the data to identify the extended observability matrix. The second one is to estimate the system matrices from the identified observability matrix. Since the calculated process of the NSIM is non-iterative and this method poses no additional problems on the part of parameterization, the NSIM becomes a promising approach to identify nonlinear structural systems. However, the result generated by the NSIM has its deficiency. One of the drawbacks is that the identified results calculated by the NSIM are not the optimal solutions which reduce the identified accuracy. In this study, a new time-domain subspace method, namely the nonlinear subspace-prediction error method (NSPEM), is proposed to improve the identified accuracy of nonlinear systems. In the improved version of the NSIM, the prediction error method (PEM) is used to reestimate those estimated coefficient matrices of the state-space model after the application of NSIM. With the help of the PEM, the identified results obtained by the NSPEM can truly become the optimal solution in the least square sense. Two numerical examples with local nonlinearities are provided to illustrate the effectiveness and accuracy of the proposed algorithm, showing advantages with respect to the NSIM in a noise environment.     

Response of Systems Under Non-Gaussian Random Excitations

The approach of nonlinear filter is applied to model non-Gaussian stochastic processes defined in an infinite space, a semi-infinite space or a bounded space with one-peak or multiple peaks in their spectral densities. Exact statistical moments of any order are obtained for responses of linear systems jected to such non-Gaussian excitations. For nonlinear systems, an improved linearization procedure is proposed by using the exact statistical moments obtained for the responses of the equivalent linear systems, thus, avoiding the Gaussian assumption used in the conventional linearization. Numerical examples show that the proposed procedure has much higher accuracy than the conventional linearization in cases of strong system nonlinearity and/or high excitation non-Gaussianity. An erratum to this article is available at .     

Dynamic analysis and performance evaluation of nonlinear inerter-based vibration isolators

This paper investigates a nonlinear inertance mechanism (NIM) for vibration mitigation and evaluates the performance of nonlinear vibration isolators employing such mechanism. The NIM comprises a pair of oblique inerters with one common hinged terminal and the other terminals fixed. The addition of the NIM to a linear spring-damper isolator and to nonlinear quasi-zero-stiffness (QZS) isolators is considered. The harmonic balance method is used to derive the steady-state frequency response relationship and force transmissibility of the isolators subjected to harmonic force excitations. Different performance indices associated with the dynamic displacement response and force transmissibility are employed to evaluate the performance of the resulting isolators. It is found that the frequency response curve of the inerter-based nonlinear isolation system with the NIM and a linear stiffness bends towards the low-frequency range, similar to the characteristics of the Duffing oscillator with softening stiffness. It is shown that the addition of NIM to a QZS isolator enhances vibration isolation performance by providing a wider frequency band of low amplitude response and force transmissibility. These findings provide a better understanding of the functionality of the NIM and assist in better designs of nonlinear passive vibration mitigation systems with inerters.     

Parameter identification of systems with multiple disproportional local nonlinearities

Identification of nonlinear systems, especially with multiple local nonlinearities exhibiting disproportional ratios of the degree of nonlinearity and present at a single or multiple spatial locations, is a highly challenging inverse problem. Identification of such complex nonlinear systems cannot be handled easily by the existing conventional restoring force or describing function methods. Further, noise-corrupted measured time history responses make the parameter identification process much more difficult. Keeping this in view, we propose a new meta support vector machine (meta-SVM) model to precisely identify the type, spatial location(s) and also the nonlinear parameters present in disproportionate levels using the noisy measurements. Apart from the conventional SVM model, we also explore the effectiveness of the non-batch processing models like incremental learning for lesser computational cost and increased efficiency. Both incremental and conventional support vector regression models are explored to precisely identify the nonlinear parameters. A numerically simulated multi-degree of freedom spring-mass system with limited multiple local nonlinearities at a few selected spatial locations is considered to illustrate the proposed meta-SVM model for nonlinear parametric identification. However, the extension of the proposed meta-SVM model is rather straightforward to include all types of nonlinearities and cases with the simultaneous existence of multiple numbers of same or different nonlinearities (i.e. combined nonlinearities) at single or multiple locations. It is also clearly established from the numerical simulation studies that the proposed incremental meta-SVM model paves way for online real-time identification of nonlinear parameters which is not yet been addressed in the existing literature.

     

Effect of boundary condition nonlinearities on free large-amplitude vibrations of rectangular plates

The effect of boundary condition nonlinearities on free nonlinear vibrations of thin rectangular plates is analyzed. The method for analysis of the plate vibrations with geometrical nonlinearity and the boundary condition nonlinearity is suggested. The nonlinear boundary conditions for membrane forces are transformed into linear ones using the in-plane stress function. Additional boundary conditions for the in-plane displacements vanishing on the clamped edge of the plate are imposed on the stress function. Simply supported and cantilever plates are analyzed. The backbone curves obtained by satisfying linear and nonlinear boundary conditions are compared. It is shown that the results of the calculations with nonlinear boundary conditions differ essentially from the data obtained without these boundary conditions.     

Nonlinear damping in a micromechanical oscillator

Nonlinear elastic effects play an important role in the dynamics of microelectromechanical systems (MEMS). A Duffing oscillator is widely used as an archetypical model of mechanical resonators with nonlinear elastic behavior. In contrast, nonlinear dissipation effects in micromechanical oscillators are often overlooked. In this work, we consider a doubly clamped micromechanical beam oscillator, which exhibits nonlinearity in both elastic and dissipative properties. The dynamics of the oscillator is measured in both frequency and time domains and compared to theoretical predictions based on a Duffing-like model with nonlinear dissipation. We especially focus on the behavior of the system near bifurcation points. The results show that nonlinear dissipation can have a significant impact on the dynamics of micromechanical systems. To account for the results, we have developed a continuous model of a geometrically nonlinear beam-string with a linear Voigt–Kelvin viscoelastic constitutive law, which shows a relation between linear and nonlinear damping. However, the experimental results suggest that this model alone cannot fully account for all the experimentally observed nonlinear dissipation, and that additional nonlinear dissipative processes exist in our devices.     

Modified Parker-Sochacki method for solving nonlinear oscillators

A new approach is presented for solving nonlinear oscillatory systems. Parker-Sochacki method (PSM) is combined with Laplace-Padé resummation method to obtain approximate periodic solutions for three nonlinear oscillators. The first one is Duffing oscillator with quintic nonlinearity which has odd nonlinearity. The second one is Helmholtz oscillator which has even nonlinearity. The last one is a strongly nonlinear oscillator, namely; relativistic harmonic oscillator which has a fractional order nonlinearity. Solutions are also obtained using Runge-Kutta numerical method (RKM) and Lindstedt-Poincare method (LPM). However, the LPM could not be used to solve the relativistic harmonic oscillator since it is a strongly nonlinear oscillator. The comparison between these solutions shows that the convergence zone for the Parker-Sochacki with Laplace-Padé method (PSLPM) is remarkably increased compared to PSM method. It also shows that the PSLPM solutions are in excellent agreement with LPM solutions for Duffing oscillator and are superior to LPM solutions in case of Helmholtz oscillator. The PSLPM succeeded to give an accurate periodic solution for the relativistic harmonic oscillator. For a wide range of solution domain, comparing PSLPM with RKM prove the correctness of the PSLPM method. Hence, the PSLPM method can be used with satisfied confidence to solve a broad class of nonlinear oscillators.     

Pseudo-fault induction in engineering structures: Multi-input multi-output control

In a previous paper, it was demonstrated that any linear system could be made to respond to harmonic excitation as if a static nonlinearity of specified type and position were present, this response being obtained at a single predetermined point on the structure. The method requires the excitation of the linear structure by an additional or auxiliary input. In the present paper, the theory is extended to allow the possibility of producing a specified nonlinear response at more than one point on a linear structure. It is shown that N responses can be obtained by specifying N or less auxiliary inputs. The theory is also extended to provide for polynomial damping in addition to stiffness nonlinearity. The theory is validated using numerical simulation of MDOF lumped-parameter systems.