Vibration control combining nonlinear isolation and nonlinear absorption

Nonlinear Dynamics - Quasi-zero stiffness system has been widely used to isolate structural vibrations. However, its small dynamic stiffness may result in drastic low-frequency vibration. This is a...     

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.     

Spectro-spatial analyses of a nonlinear metamaterial with multiple nonlinear local resonators

Nonlinear Dynamics - Recent focus has been given to spectro-spatial analysis of nonlinear metamaterials since they can predict interesting nonlinear phenomena not accessible by spectral analysis...     

Derivation of nonlinear damping from viscoelasticity in case of nonlinear vibrations

Nonlinear Dynamics - Experiments show a strong increase in damping with the vibration amplitude during nonlinear vibrations of beams, plates and shells. This is observed for large size structures...     

Literature survey on nonlinear components and chaotic nonlinear components of block ciphers

In the modern era of secure communication, it is important to create uncertainty in the original data in order to avoid unauthorized entities to extract or manipulate information. From simple methods such as permutations of original data to different mapping algorithms, the security of the ciphers rely on the substitution process. There are many types of components proposed in literature that are evolved by different methodologies and ideas. The prevailing ciphers use substitution boxes (S-boxes) to do this transformation process. In this work, we present a literature review of the design, construction, and analysis of the S-boxes used in block ciphers. The performance of S-boxes depends on the design and algebraic structure used for the construction and is contingent upon its ability to resist against cryptanalysis. We present the details of the S-box synthesis process and issues pertaining to creating resistance against various types of attacks, and highlight the consequences of a particular design methodology. In the infancy of the development of modern block ciphers, Shannon (Bell Syst. Tech. J. 28(4):656–715, 1949) presented the idea of encryption with the implementation of substitution-permutation network (SPN). In this process, the data is initially transformed by the substation process and then permuted that ends the first round supported by the secret key for this step. This substitution-permutation process is repeated several times to ensure reliability of encrypted data. The objective of using the substitution-permutation network is to create confusion between cipher text and secret key, and add diffusion in the plaintext.     

Weakly nonlinear acoustic instabilities

With the aim of eventually improving numerical solutions of small-scale phenomena, the Hunter-Keller theory of weakly nonlinear high-frequency waves is applied to the study of short wavelength instabilities in inviscid fluids driven by a heat or pressure source. A nonlinear damping effects is found which, for acoustic perturbations of a stationary, homogeneous state, reduces the growth rate to half the linear estimate. This is due primarily to the interactions of the expansion fan and the weak shock generated by the cumulative effect of the nonlinear convective term. For acoustic perturbations driven by an unbalanced heat source, the nonlinear damping actually stabilizes some modes which are unstable according to the linear theory. For the isentropic compression of a spherical shell of material obeying a γ-law equation of state, it is shown that the nonlinear damping again reduces the acoustic growth rate to the half the value predicted by conventional linear stability analyses.     

国际非均匀材料力学会议（ICHMM）简介

为了提高非线性有限元的教学和科研水平，经中国力学学会和北京市力学学会批准，清华大学工程力学系于2004年6月17日～19日在北京举办了全国非线性有限元讲习班，聘请了美国西北大学Ted Belytschko教授讲学3天，题目为非线性有限元与计算力学。     

Suppression of nonlinear vibration system described by nonlinear differential equations using passive controller

In this paper, the linear absorber is proposed to reduce the vibration of a nonlinear dynamical system at simultaneous primary resonance and the presence of 1:1 internal resonance. This leads to a two-degree-of-freedom system subjected to external excitation force. The method of multiple scales perturbation technique is applied throughout to determine the analytical solution up to first-order approximations. The stability of the system near the one of the worst resonance case is studied using the frequency response equations. The effects of the different system and absorber parameters on the behavior of the main system are studied numerically. For validity, the numerical solution is compared with the analytical solution and gets a good agreement. Effectiveness of the absorber ( \(E_{a})\) is about 800 for the nonlinear vibrating system. The simulation results are achieved using MATLAB programs. At the end of the work, the comparison with the available published work is reported.     

Discriminating linear from nonlinear elastic damage using a nonlinear time reversal DORT method

The DORT method is a selective detection and focusing technique originally developed to detect defects and damages which induce linear changes of the elastic moduli. It is based on the time reversal (TR) where a signal collected from an array of transducers is time reversed and then back-propagated into the medium to obtain focusing on selected targets. TR is based on the principle of spatial reciprocity. Attenuation, dispersion, multiple scattering, mode conversion, etc. do not break spatial reciprocity. The presence of defects or damage, may cause materials to show nonlinear elastic wave propagation behavior that will break spacial reciprocity. Therefore the DORT method will not allow focusing on nonlinear elastic scatterers. This paper presents a new method for the detection and identification of multiple linear and nonlinear scatterers by combining nonlinear elastic wave spectroscopy, time reversal and DORT method. In the presence of nonlinear hysteretic elastic scatterers, forcing the solid with a harmonic excitation, the time reversal operator can be obtained not only at the fundamental frequency of excitation, but also at the odd harmonics. At the fundamental harmonic, either inhomogeneities and linear damages can be individually selected but only at odd harmonics nonlinear hysteretic elastic damages exist. A procedure was developed where by decomposing the operator at the odd harmonics, it was possible to focus on nonlinear scatterers and to differentiate them from the linear inhomogeneities. A complete mathematical nonlinear DORT formulation for 1 and 2D structures is presented. To model the presence of nonlinear elastic hysteretic scatterers a Preisach–Mayergoyz (PM) material constitutive model was used. Results relative to 1 and 2 dimensional structures are reported showing the capability of the method to focus and discern selectively linear and nonlinear scatterers. Furthermore, an analysis was conducted to study the influence of the number of sources and their location on the imaging process showing that using a higher numbers of sensors does not automatically bring to a minor uncoupled behaviour between the nonlinear targets.     

A modified time domain subspace method for nonlinear identification based on nonlinear separation strategy

Nonlinear factors existing in engineering structures have drawn considerable attention, and nonlinear identification is a competent technique to understand the dynamic characteristics of nonlinear structures. Therefore, in this paper, a novel nonlinear separation subspace identification (NSSI) algorithm based on subspace algorithm and nonlinear separation strategy is proposed to conduct nonlinear parameter identification of nonlinear structures. For the proposed NSSI algorithm, the low-level excitation test is firstly conducted to obtain the transfer matrix in the linear response formula. Then, the obtained transfer matrix is used in the high-level excitation test to calculate the nonlinear response part by the proposed nonlinear separation strategy, and the subspace algorithm is utilized to identify the nonlinear parameter on the modified state-space model including only the nonlinear part. The proposed NSSI algorithm can reduce the coupling error caused by simultaneously processing both the large number part (corresponding to the linear part) and small number part (corresponding to the nonlinear part) in the traditional nonlinear subspace identification (NSI) algorithm. At last, two numerical experiments are given to validate the effectiveness of the developed novel nonlinear identification method. Furthermore, some influence factors are discussed to show the stability of the identification algorithm, and some comparisons between the proposed NSSI method and traditional NSI method are also conducted to demonstrate the advantages of the novel method.     

Isolated resonances and nonlinear damping

We analyze isolated resonance curves (IRCs) in single-degree-of-freedom systems possessing nonlinear damping. Through the combination of singularity theory and the averaging method, the onset and merging of IRCs, which coincide to isola and simple bifurcation singularities, respectively, can be analytically predicted. Numerical simulations confirm the accuracy of the analytical developments. Another important finding of this paper is that we unveil a geometrical connection between the topology of the damping force and IRCs. Specifically, we demonstrate that extremas and zeros of the damping force correspond to the appearance and merging of IRCs. Considering a damping force possessing several minima and maxima confirms the general validity of the analytical result. It also evidences a very complex scenario for which different IRCs are created, co-exist and then merge together to form a super IRC which eventually merges with the main resonance peak.     

A nonlinear composite shell theory

A general nonlinear theory for the dynamics of elastic anisotropic circular cylindrical shells undergoing small strains and moderate-rotation vibrations is presented. The theory fully accounts for extensionality and geometric nonlinearities by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. Moreover, the linear part of the theory contains, as special cases, most of the classical linear theories when appropriate stress resultants and couples are defined. Parabolic distributions of the transverse shear strains are accounted for by using a third-order theory and hence shear correction factors are not required. Five third-order nonlinear partial differential equations describing the extension, bending, and shear vibrations of shells are obtained using the principle of virtual work and an asymptotic analysis. These equations show that laminated shells display linear elastic and nonlinear geometric couplings among all motions.     

Orthogonal trajectories for nonlinear equations

Abdelkader developed a method for solving the van der Pol equation numerically. He observed that the orthogonal trajectories of the van der Pol equation satisfy a Riccati equation, the solution to which can be expressed in terms of Bessel functions. In the present paper, all nonlinear equations whose orthogonal trajectories satisfy Riccati or generalized Riccati equations are determined. Further on, another representation of the trajectories of the van der Pol equation is derived. This representation appears to be simpler than Abdelkader's.     

A nonlinear composite plate theory

A general nonlinear theory for the dynamics of elastic anisotropic plates undergoing moderate-rotation vibrations is presented. The theory fully accounts for geometric nonlinearities (moderate rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. The theory accounts for transverse shear deformations by using a third-order theory and for extensionality and changes in the configuration due to in-plane and transverse deformations. Five third-order nonlinear partial-differential equations of motion describing the extension-extension-bending-shear-shear vibrations of plates are obtained by an asymptotic analysis, which reveals that laminated plates display linear elastic and nonlinear geometric couplings among all motions.