On the forced vibration of a damped linear system

For a damped linear system in forced vibration, a bound on the norm of the response vector is derived in terms of the smallest eigenvalue of the stiffness matrix and of the damping matrix, and the largest eigenvalue of the mass matrix.     

On a critique of a numerical scheme for the calculation of fractionally damped dynamical systems

Recently Yuan and Agrawal [L. Yuan and O.P. Agrawal, A numerical scheme for dynamic systems containing fractional derivatives, Journal of Vibration and Acoustics 124 (2002) 321–324] presented a new numerical scheme to calculate the dynamic response of mechanical systems, the damping forces of which are described by fractional derivatives. When solving the resulting equation of motion by time integration, it is necessary to store the entire displacement history of the system due to the non-local character of the fractional derivatives. The cited scheme appears to overcome this drawback by transforming the equation of motion with the fractional term into a set of ordinary differential equations. It can be shown that this scheme is equivalent to a classical spring-dashpot representation and thus does not imply the benefits derived from fractional-derivative models. In addition, it is less flexible and incorrectly predicts the asymptotic behavior.     

Analysis of damped guided waves using the method of multiple scales

We analytically investigate the influence of damping on Lamb waves, which are a specific type of guided wave in two-dimensional plates. Considering material attenuation, we suppose that Lamé constants are complex numbers. This leads to the associated wavenumbers being complex, with the imaginary part of the wavenumber being associated with effect of attenuation of the guided wave. In this paper, we show how dispersion curves and attenuation coefficients can be obtained using the self-adjointness and the method of multiple scales (MMS), which is a type of perturbation method. Using the self-adjointness and the MMS, we can calculate the frequency- and wavenumber-dependent attenuation coefficients from the integral values and boundary values of a corresponding eigenfunction with respect to each propagation mode. This analytical method can yield not only dispersion curves but also mode-by-mode attenuation coefficients regardless of the numerical initial values, unlike numerical approaches using the Newton method. Thus, the proposed method can more easily calculate the attenuation coefficients with respect to a particular mode than conventional methods. Furthermore, the results obtained by proposed method were in good agreement with those obtained by the semi-analytical finite element (SAFE) method, which validates the proposed method.     

On the multiple-attractor coexsting system with parameter uncertainties using generalized cell mapping method

In this paper the generalized cell mapping (GCM) method is used to study multiple-attractor coexisting system with parameter uncertainties. The effects that the uncertain parameters has on the global properties of the system are presented. And It is obtained that the attractor with much smaller value of protect thickness, will disappear firstly with the degree of the uncertainty of parameter increasing. Project supported by the National Natural Science Foundation of China (19672046)     

基于微分求积法求解非粘滞阻尼系统的时程响应

研究一种含有指数型非粘滞阻尼线性多自由度振动系统的时程分析问题。该非粘滞阻尼模型假设阻尼力与质点速度的时间历程相关,数学表述为质点速度与核函数的卷积。由于阻尼模型的改变,常用的数值积分方法（如Newmark-β法、Wilson-θ法）不能直接应用于这种非粘滞阻尼系统。基于一种无条件稳定的微分求积方法,给出了这种非粘滞阻...     

On applications of the microlocal parameter method in modelling of temperature distributions in composite cylinders

Summary The problems of transient heat conduction in a periodically stratified medium consisting of a large number of alternating concentric cylinders of two homogeneous isotropic rigid materials and in a rotationally periodic cylinder consisting of a large number of circular homogeneous isotropic rigid sectors are considered. The equations of the homogenized models with microlocal parameters are derived by using the homogenization procedure given in [17]. The obtained models take into account certain microlocal effects connected with the microperiodic structure of the considered composites. Some examples of the application of the presented models to the problems of temperature distributions in composite cylinders are detailed. Received 10 March 1997; accepted for publication 23 October 1997     

Analytical solution of fractionally damped beam by Adomian decomposition method

The analytical solution of a viscoelastic continuous beam whose damping characteristics are described in terms of a fractional derivative of arbitrary order was derived by means of the Adomian decomposition method.The solution contains arbitrary initial conditions and zero input.For specific analysis,the initial conditions were assumed homogeneous,and the input force was treated as a special process with a particular beam. Two simple cases,step and impulse function responses,were considered respectively. Subsequently,some figures were plotted to show the displacement of the beam under different sets of parameters including different orders of the fractional derivatives.     

Numerical method for the system of reaction-diffusion equations with a small parameter

This paper deals with the numerical method for the system of reaction-diffusionequations with a small parameter.It is difficult to solve the problems of this kindnumerically because of the boundary layer efect.Besed on singular perturbed theory andGreens’function,we have established the difference scheme that is suited for the solution tothe problems.It e introduce an idea of feasilbe equidistant degree a here.And this provesthat if a≥2.the scheme converges in l,m norm with speed O.h \t)uniformly.     

Time integration method to determine the characteristic parameter of a light extinctiometer

The linear relation between the particle concentration and the output voltage of a light extinctiometer used to measure particle concentrations has been derived and calibrated. A time integration method was introduced to determine the characteristic parameter of such a measuring system. This characteristic parameter which is proportional to the extinction coefficient is found to be in good agreement with its theoretical value. As an example, in a vertical tube the increasing particle concentration was measured behind a developing flat interface between a gas and a gas-particle mixture as a function of time.On leave from East China Institute of Technology, Nanjing 210014, The People's Republic of China     

Flow of a relaxed gas in the vicinity of a solid surface

Behavior of a polyatomic relaxed gas in the vicinity of a solid surface was studied. The case in which the size of the relaxation zone exceeds considerably the mean free path between the elastic collisions (suppressed exchange of translational and internal energies) was considered. A smooth validity of the asymptotic expansion for the distribution function with defined assumptions was indicated. A solution for the zero-approximation equation and boundary conditions for the surface flow, based on this solution and generalized model for a diffusive gas reflection from the surface were derived. The latter problem was partially studied in [1] by the Grad method and in [2] in terms of an analysis of the temperature jump.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 33–39, July–August, 1973.     

非比例阻尼线性体系平稳随机地震响应计算的虚拟激励法

应用复振型分解方法,将非比例阻尼线性体系在地震作用下的动力方程求解问题转化为若干个广义复振子的求解与叠加问题。通过假定地震地面运动为一零均值的平稳随机激励,应用虚拟激励法原理,推导得到了广义复振子动力坐标的解析计算公式,进而得到了以复振型为基础的非比例阻尼线性体系随机地震响应计算的一般实数解析解答。算例证实了这种方法的可靠性及高效率。     

Explicit computational method of dynamic response for non-viscously damped structure systems

Non-viscous damping models in which the damping forces depend on the past history of velocities via convolution integrals over some kernel functions have been raised in many engineering fields. This paper describes an explicit computational method of dynamic response for the non-viscously damped structure systems. The explicit formula is derived using the differential property of convolution and the central difference formula of acceleration. The explicit computational procedure of dynamic response is given in detail. Finally, the dynamic responses of MDOF structure system with double exponential model dampers and SDOF structure system with Gaussian model damper are computed using the proposed explicit method. The accuracy and efficiency are discussed by comparison with other two developed methods.     

On damped non-linear dynamic systems with many degrees of freedom

Non-linear mass-spring-damper systems with many degrees of freedom are studied; all springs and/or all dampers may be strongly non-linear. It is shown that the ultimate state of completely damped systems is always rest, that of incompletely damped systems may be either rest or a periodic normal mode motion. Necessary and sufficient conditions are given for the existence of classical normal mode motion in completely or incompletely damped systems. When the system is linear, these reduce to known results found by Rayleigh and generalized by Caughey and O'Kelly.     

Random impacts of a complex damped system

We investigated the random impacts of a complex damped system. Firstly the interested deterministic complex damped system was revisited and the unstable periodic attractors could be found by means of Poincaré map, time evolution and phase plot since the top Lyapunov exponent could not be applied to decide the unstable states of the proposed system. Secondly the stochastic complex damped system was examined and random impacts would be discovered, namely, the initial deterministic system will be stabilized using the stochastic force properly. The top Lyapunov exponent versus the noise intensity will be observed and one can find the change of dynamical behaviors from instability to stability. Also we implemented Poincaré map analysis, time history and phase plot to confirm the obtained results of top Lyapunov exponent, and we can find excellent agreement between these results. Therefore random noise can be applied to control the dynamical behaviors.     

采用两步优化器的深度配点法与深度能量法求解薄板弯曲问题

随着计算机技术的进步以及机器学习算法的进一步发展,深度学习方法逐渐被广泛引用于各行各业中。本文发展并比较了适应于工程计算的深度配点法与深度能量法并应用于求解薄板弯曲问题。深度配点法采用物理驱动的深度神经网络来,并将物理信息（偏微分方程强形式）引入到损失函数中,最终将求解薄板弯曲问题简化为优化问题。深度能量法则是采用系统总势能驱动的神经网络。根据最小势能原理,在所有的可能位移场中,真实位移场的总势能取最小值,因此我们可以使用总势能构造损失函数,从而求解薄板弯曲问题。对于边界条件,通过罚函数法将有约束最优化问题转化为求解无约束最优化问题。深度配点法与深度能量法的适用性基于神经网络的通用近似定理。由于物理信息跟总势能的引入,增加了神经网络训练的困难,为了解决这个问题,我们发展了两步优化器方法。数值结果表明,深度配点法与深度能量法很适合求解薄板弯曲问题,并且程序实现简单,实现了真正意义上的“无网格法”。     

Damage detection using a new regularization method with variable parameter

In this research, a sensitivity approach to finite element model updating is used to determine stiffness reduction factors from measured structural response. The used method causes a set of nonlinear ill-conditioned equations that need to be linearized and regularized in order to find the solution. A new approach to solve the problem is presented using variable regularization parameter. Utilization of variable regularization parameter eliminates dependency on the number of iterations and prevents the loss of regularization effect due to iterations. A new stopping criteria is used which is based on the difference between mean and variance of last iterations. Furthermore the results show that using wavelet transform to update the model yields better results than modal parameters. Expedient performance of the proposed method is shown through a numerical simulation.     

On the analytical approximation of damped oscillations of autonomous single degree of freedom oscillators

In this paper, we discuss the generalization of a recently developed perturbation method for conservative single degree of freedom systems to the cases of damping forces. We show that by explicitly considering the solution as a function of the amplitude parameter, frequency and phase of oscillation, one can merge the classical Krylov–Bogoliubov–Mitropolsky method and a modified Lindstedt–Poincaré method to derive better accurate slow-flow for damped oscillation of single degree of freedom systems.