Numerical Study of the Vibrations of Beams with Variable Stiffness under Impulsive or Harmonic Loading

The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation method is a technique for numerical integration of partial differential equations involving both the space and time, with well-known initial conditions on time and boundary conditions on the space. This technique, although having been applied to beams with constant stiffness, is new for the case of beams with variable stiffness, and it aims to use a quadratic parabola (in time) to approximate the solutions of the differential equations of dynamics. The spatial part is studied using the successive approximation method of the partial differential equations obtained, in order to transform them into a system of time-dependent ordinary differential equations. Thus, the integration algorithm using this technique is established and applied to examples of beams with variable stiffness, under variable loading, and with the different cases of supports chosen in the literature. We have thus calculated the cases of beams with constant or variable rigidity with articulated or embedded supports, subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. In order to justify the robustness of the successive approximation method considered in this work, an example of an articulated beam with constant stiffness subjected to a distributed harmonic load was calculated analytically, and the results obtained compared to those found numerically for various steps (spatial h and temporal τ ¯ ) of calculus, and the difference between the values obtained by the two methods was small. For example for ( h=1/8 , τ ¯ =1/ 64 ), the difference between these values is 17%.

A PREDICT-CORRECT NUMERICAL INTEGRATION SCHEME FOR SOLVING NONLINEAR DYNAMIC EQUATIONS

A new numerical integration scheme incorporating a predict-correct algorithm for solving the nonlinear dynamic systems was proposed in this paper. A nonlinear dynamic system governed by the equation v = F（v, t） was transformed into the form as v = Hv ＋ f（v, t）. The nonlinear part f（v, t） was then expanded by Taylor series and only the first-order term retained in the polynomial. Utilizing the theory of linear differential equation and the precise time-integration method, an exact solution for linearizing equation was obtained. In order to find the solution of the original system, a third-order interpolation polynomial of v was used and an equivalent nonlinear ordinary differential equation was regenerated. With a predicted solution as an initial value and an iteration scheme, a corrected result was achieved. Since the error caused by linearization could be eliminated in the correction process, the accuracy of calculation was improved greatly. Three engineering scenarios were used to assess the accuracy and reliability of the proposed method and the results were satisfactory.

The efficiency of direct integration methods in elastic contact-impact problems

A comparison of direct integration methods is madeand their efficiency is investigated for impact problems.New-mark,Wilson-θ,Central Difference and Houbolt Methodsare used as direct integration methods.Impact analysisincludes that of elastic and large deformation based uponupdated Lagrangian including buckling check.The resultsshow that the direct integration methods give differentresults in different contact-impact cases.

Structural Reliability Analysis Based on Support Vector Machine and Dual Neural Network Direct Integration Method

Aiming at the reliability analysis of small sample data or implicit structural function,a novel structural reliability analysis model based on support vector machine(SVM)and neural network direct integration method(DNN)is proposed.Firstly,SVM with good small sample learning ability is used to train small sample data,fit structural performance functions and establish regular integration regions.Secondly,DNN is approximated the integral function to achieve multiple integration in the integration region.Finally,structural reliability was obtained by DNN.Numerical examples are investigated to demonstrate the effectiveness of the present method,which provides a feasible way for the structural reliability analysis.

Exact Solutions of the Harry-Dym Equation

The aim of this paper is to generate exact travelling wave solutions of the Harry-Dym equation through the methods of Adomian decomposition, He＇s variational iteration, direct integration, and power series. We show that the two later methods are more successful than the two former to obtain more solutions of the equation.

CONTINUOUS FRAME WAVELETS

Let π be a unitary representation of a locally compact topological group G on a separable Hilbert space H. A vector ψ∈ H is called a continuous frame wavelet if there exist A, B 〉 0 such that A｜｜Ф｜｜2≤fG｜〈π（G）ψ,Ф｜2dg≤B｜｜Ф｜｜2（Ф∈H）,in which dg is the left Haar measure of G. Similar to the study of wavelets, an essential problem in the study of continuous frame wavelets is how to characterize them under the given unitary representation. Moreover, we investigate a relation between admissible vectors of π and its components.

SPATIAL－TEMPORAL DISCRETE COORDINATION OF FEM AND DIRECT INTEGRAL METHOD FOR TRANSIENT DYNAMIC PROBLEMS

In this paper, the coordination of spatial-temporal discrets of FEM and direct integral method is investigates.By analyzing the numespatial-temporal discrete,the principle of balancing the principle of balancing the energy error induced by spatialdiscrete and the energy error induced by temporal discrete is presented, and the prioriprocess and adaptive method for the coordination of spatial discerand temporal discrete is obtained.

Research on Nonlinear Dynamic Characteristics ofStructures Supported on Slide-Limited Friction Base Isolation System

This paper presents a new type of base isolation system, i. e. , slide-limited friction (S-LF) base isolation system . Based on this system, the harmonic and subharmonic periodic response of S-LF subjected to harmonic motions is investigated by using Fourier-Galerkin-Newton (FGN) method with Flo-quet theory. The dynamic response of S-LF subjected to earthquake ground motions is calculated with a high order precision direct integration method, and the numerical results are presented in maximum acceleration response spectra of superstructure and maximum sliding displacement response spectrum form. The comparison of isolating effects of S-LF, pure-friction base isolation system (P-F) and resilient-friction base isolation system (R-FBI) shows that the isolating property of S-LF is superior to those of P-F and R-FBI. Finally, by analyzing an engineering example, it is observed that the distribution of the maximum shear between floors and absolute acceleration of S-LF to earthquake ground motion is very different from that of traditional structures.

A high order single step-βmethod for nonlinear structural dynamic analysis

A high order single step β algorithm, a new direct integration algorithm is proposed for solution of equations of motion. Whenβ=0.5, the accuracy of displacement, velocity and acceleration is of forth order (a truncation error of Δ t 5), and the algorithm is unconditionally stable and has no arithmetic damping and no overshooting. When >0.5, and an arithmetic damping is adopted, the algorithm is again unconditionally stable with a third order accuracy (a truncation error of Δ t 4). The analyses run with typical examples show that the algorithm proposed has higher speed, higher precision and better properties than other direct integration methods, such as Wilson θ method and Newmark β method in analysing linear elastic responses and nonlinear earthquake responses.

A Straightforward Direct Traction Boundary Integral Method for Two-Dimensional Crack Problems Simulation of Linear Elastic Materials

This paper presents a direct traction boundary integral equation method(DTBIEM)for two-dimensional crack problems of materials.The traction boundary integral equation was collocated on both the external boundary and either side of the crack surfaces.The displacements and tractions were used as unknowns on the external boundary,while the relative crack opening displacement(RCOD)was chosen as unknowns on either side of crack surfaces to keep the single-domain merit.Only one side of the crack surfaces was concerned and needed to be discretized,thus the proposed method resulted in a smaller system of algebraic equations compared with the dual boundary element method(DBEM).A new set of crack-tip shape functions was constructed to represent the strain field singularity exactly,and the SIFs were evaluated by the extrapolation of the RCOD.Numerical examples for both straight and curved cracks are given to validate the accuracy and efficiency of the presented method.

Anti-Windup Control Strategy of Drive System for Humanoid Robot Arm Joint

To address the problems of torque limit and controller saturation in the control of robot arm joint,an anti-windup control strategy is proposed for a humanoid robot arm,which is based on the integral state prediction under the direct torque control system of brushless DC motor. First,the arm joint of the humanoid robot is modelled. Then the speed controller model and the influence of the initial value of the integral element on the system are analyzed. On the basis of the traditional antiwindup controller,an integral state estimator is set up. Under the condition of different load torques and the given speed,the integral steady-state value is estimated. Therefore the accumulation of the speed error terminates when the integrator reaches saturation. Then the predicted integral steady-state value is used as the initial value of the regulator to enter the linear region to make the system achieve the purpose of anti-windup. The simulation results demonstrate that the control strategy for the humanoid robot arm joint based on integral state prediction can play the role of anti-windup and suppress the overshoot of the system effectively. The system has a good dynamic performance.

A modified highly precise direct integration method for a class of linear time-varying systems

It is always a bottleneck to design an effective algorithm for linear time-varying systems in engineering applications.For a class of systems,whose coefficients matrix is based on time-varying polynomial,a modified highly precise direct integration(VHPD-T method)was presented.Through introducing new variables and expanding dimensions,the system can be transformed into a timeinvariant system,in which the transfer matrix can be computed for once and used forever with a highly precise direct integration method.The method attains higher precision than the common methods(e.g.RK4 and power series)and high efficiency in computation.Some numerical examples demonstrate the validity and efficiency of the method proposed.

An efficient time-integration method for nonlinear dynamic analysis of solids and structures

This paper presents an efficient time-integration method for obtaining reliable solutions to the second-order nonlinear dynamic problems in structural engineering. This method employs both the backward-acceleration differentiation formula and the trapezoidal rule, resulting in a self-starting, single step, second-order accurate algorithm. With the same computational effort as the trapezoidal rule, the proposed method remains stable in large deformation and long time range solutions even when the trapezoidal rule fails. Meanwhile, the proposed method has the following characteristics: (1) it is applicable to linear as well as general nonlinear analyses; (2) it does not involve additional variables (e.g. Lagrange multipliers) and artificial parameters; (3) it is a single-solver algorithm at the discrete time points with symmetric effective stiffness matrix and effective load vectors; and (4) it is easy to implement in an existing computational software. Some numerical results indicate that the proposed method is a powerful tool with some notable features for practical nonlinear dynamic analyses.

Distortion correction for the elemental images of integral imaging by introducing the directional diffuser

A distortion correction method for the elemental images of integral imaging（Ⅱ） by utilizing the directional diffuser is demonstrated. In the traditional Ⅱ, the distortion originating from lens aberration wraps elemental images and degrades the image quality severely. According to the theoretical analysis and experiments, it can be proved that the farther the three-dimensional image is displayed from the lens array, the more serious the distortion is. To analyze the process of eliminating lens distortion, one lens and its corresponding elemental image are separated from the traditional Ⅱ. By introducing the directional diffuser, the aperture stop of the separated optical system changes from the eye＇s pupil to the lens. In terms of contrast experiments, the distortion of the improved display system is corrected effectively. In the experiment, when the distance between the reconstructed image and lens array is equal to 120 mm, the largest lens distortion is decreased from 46.6% to 3.3%.

Dynamic Response of Mold Oscillator Interacting with Steel Slabs

A mathematical model to show the dynamic response of the mold oscillator was suggested. The model con- sidered a frictional interaction between the mold oscillator and slab as several connected nodes. The governing equation considered the slab as a multi-degree-of-freedom （DoF） system, and included a hysteresis model to describe elastic-plastic behavior of the slab; the mold oscillator was given two DoF by utilizing pressure and displacement experiment data. Simulations indicate that the mold and slab execute various vibrations, and that mold oscillation marks are caused by a stick-and-slip phenomenon during intervals, in which the slab contacting the mold moves downward compared to the other slab （negative strip time）. The slab shows the formation of mold oscillation marks to previous formation criterion equally when the mold velocity is faster than the casting speed about downward. The oscillation mark will grow up over 2 Hz exciting frequency with constant 4 mm stroke in simulations. Finally, the negative strip time was compared to the frictional force, hysteresis variable, and plastic force to investigate formation mechanism of the oscillation marks.

112 Gbit/s transmitter optical subassembly based on hybrid integrated directly modulated lasers

Based on the hybrid integration technology, an ultra-compact and low cost transmitter optical subassembly module is proposed. Four directly modulated lasers are combined with a coarse wavelength division multiplexer operated at the O-band. The bandwidth for all channels is measured to be approximately 3 GHz. The 112 Gb∕s transmission is experimentally demonstrated for a 10 km standard single mode fiber（SSMF）, in which an optical isolator is used for avoiding the back-reflected and scattered light to improve the bit error rate（BER） performance. A low BER and clear eye opening are achieved for 10 km transmission.

Electrochemical performance of thermally-grown SiO2 as diffusion barrier layer for integrated lithium-ion batteries

Direct integration of lithium-ion battery （LIB） with electronic devices on the same Si substrate can significantly miniaturize autonomous micro systems. For achieving direct integration, a barrier layer is essential to be inserted between LIB and the substrate for blocking Li^＋ diffusion. In this paper, the feasibility of thermal SiOa film as the barrier layer is investigated by electrochemical characterization and X-ray photoelectron spectroscopy （XPS）. Due to the negligible side reactions of thermal SiO2 with electrolyte, the solid electrolyte interphase （SEI） layer formed on the surface of the barrier layer is thin and the SEI content mainly consists of hydrocarbon together with slight polyethylene oxide （PEO）, LixPOyFz, and Li2CO3. Although 8-nm thermal SiO2 effectively prevents the substrate from alloying with Li^＋, the whole film changes to Li silicate after electrochemical cycling due to the irreversible chemical reactions of SiO2 with electrolyte. This degrades the performance of the barrier layer against the electrolyte penetration, thus leading to the existence of Li^＋ （in the form of F-Si-Li） and solvent decompositions （with the products of hydrocarbon and PEO） near the barrier layer/substrate interface. Moreover, it is found that the reaction kinetics of thermal SiO2 with electrolyte decrease significantly with increasing the SiO2 thickness and no reactions are found in the bulk of the 30-nm SiO2 film. Therefore, thermal SiO2 with an appropriate thickness is a promising barrier layer for direct integration.