Application of a modified Lindstedt–Poincaré method in coupled TDOF systems with quadratic nonlinearity and a constant external excitation

An analytical approach is developed for the nonlinear oscillation of a conservative, two-degree-of-freedom (TDOF) mass-spring system with serial combined linear–nonlinear stiffness excited by a constant external force. The main idea of the proposed approach lies in two categories, the first one is the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation. Another is the treatment a quadratic nonlinear oscillator (QNO) by the modified Lindstedt–Poincaré (L-P) method presented recently by the authors. The first-order and second-order analytical approximations for the modified L-P method are established for the QNOs with satisfactory results. After solving the nonlinear differential equation, the displacements of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, the modified L-P method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and classical harmonic balance methods. Two examples of nonlinear TDOF mass-spring systems excited by a constant external force are selected and the approximate solutions are verified with the exact solutions derived from the Jacobi elliptic function and also the numerical fourth-order Runge–Kutta solutions.     

Organometallic complexes for nonlinear optics: Part 21. Syntheses and quadratic hyperpolarizabilities of alkynyl complexes containing optically active 1,2-bis(methylphenylphosphino)benzene ligands

The preparation of the chloro complex trans-[FeCl₂{(R,R)-diph}₂] (1) and the alkynyl complexes trans-[M(4-CCC₆H₄R)Cl{(R,R)-diph}₂] [M=Fe, R=NO₂ (2); M=Ru, R=H (4), NO₂ (5), (E)-CH=CH-4-C₆H₄NO₂ (6); M=Os, R=NO₂ (7)], incorporating the optically active diphosphine 1,2-bis(methylphenylphosphino)benzene (diph), are described. Oxidation potentials, as determined by cyclic voltammetry, increase as 2<7<5. Molecular quadratic nonlinearities by hyper-Rayleigh scattering at 1064 nm increase upon introduction of an acceptor group (4<5), chain-lengthening of bridging group (5<6), and proceeding from 3d to 4d and 5d metal (2≤5≤7). Two-level-corrected nonlinearities reproduce the first two trends, but metal variation follows the sequence 2<7<5. The experimental and two-level-corrected nonlinearities for 6 (2795×10⁻³⁰ and 406×10⁻³⁰ esu, respectively), are amongst the largest observed thus far for organometallic complexes. Crystals of complexes 2 and 7 exhibit second-harmonic generation (assessed using the Kurtz powder technique), with an efficiency for the former of twice that of urea.     

Divisible Designs Admitting GL(3, q) as an Automorphism Group

Starting from a 3-dimensional projective space we construct divisible designs admitting GL(3,q) as an automorphism group.     

Error detection capability of shortened hamming codes and their dual codes

1. IntroductionIn ant omat ic--rep eat-- request (ARQ ) error- cont rol syst em 3 t he u-ndet ect ed error probability (UEP) of an error-detecting code is one of the most importallt performance characteristics. There are a number of papers dedicated to examining the error detection capabilityfor some well known classes of linear codes, suCh as Reed-Solomon codes, BCH codes andReed-Muller codes. For a general introduction to the theory of error detecting codes, werefer the readers to [if …     

关于信源的Tunstall编码方法

在信源编码理论中,Ｔｕｎｓｔａｌｌ码是渐近最优的Ｖ－Ｂ码（Ｖ代表变长消息,Ｂ代表定长码字）。本文进一步研究了Ｔｕｎｓｔａｌｌ码的性质,给出了Ｔｕｎｓｔａｌｌ码的码率的新的上界,刻划了Ｔｕｎｓｔａｌｌ树和扩展次数之间的一些较深刻的内在联系,并且给出了一个寻找ε－最优的Ｔｕｎｓｔａｌｌ码的扩展次数的算法。     

Quasi-cyclic codes as cyclic codes over a family of local rings

We give an algebraic structure for a large family of binary quasi-cyclic codes. We construct a family of commutative rings and a canonical Gray map such that cyclic codes over this family of rings produce quasi-cyclic codes of arbitrary index in the Hamming space via the Gray map. We use the Gray map to produce optimal linear codes that are quasi-cyclic.     

Time series prediction of nonlinear ship structural responses in irregular seaways using a third-order Volterra model

To predict the nonlinear structural responses of a ship traveling through irregular waves, a third-order Volterra model was applied based on the given irregular data. A nonlinear wave–body interaction system was identified using the nonlinear autoregressive with exogenous input (NARX) technique, which is one of the most commonly used nonlinear system identification schemes. The harmonic probing method was applied to extract the first-, second- and third-order frequency response functions of the system. To achieve this, a given set of time history data of both the irregular wave excitation and the corresponding midship vertical bending moment for a certain sea state was fed into the three-layer perceptron neural network. The network parameters are determined based on the supervised training. Next, the harmonic probing method was applied to the identified system to extract the frequency response function of each order. While applying the harmonic probing method, the nonlinear activation function (i.e., the hyperbolic tangent function) was expanded into a Taylor series for harmonic component matching. After the frequency response functions were obtained, the structural responses of the ship under an arbitrary random wave excitation were easily calculated with rapidity using a third-order Volterra series. Additionally, the methodology was validated through the in-depth analysis of a nonlinear oscillator model for a weak quadratic and cubic stiffness term, whose analytic solutions are known. It was confirmed that the current method effectively predicts the nonlinear structural response of a large container carrier under arbitrary random wave excitation.