Transient elastic wave propagation in a spherically symmetric bimaterial medium modeling the thorax

The transient response resulting from an impact wave on an elastic bimaterial, made out of a “hard” medium and a “soft” medium, welded at a spherical interface, have been investigated by using an integral transform technique. This technique permits isolation of the pressure and shear waves contributions to the wave field. The method of solution makes use of the generalized ray/Cagniard-de Hoop (GR/CdH) method associated with a “flattening approximation” (FA) technique, similar to the Earth flattening transformation used in geophysics. The GR/CdH method and the FA technique are briefly presented, together with their numerical implementations. The FA has proved to be useful in geophysical application, however, as far as the authors know, it has never been investigated for other applications. For the purpose of this paper, numerous tests of the method have been performed in order to check that the FA is appropriate to compute transient responses in the special case presented here. We could determine appropriate values for some parameters involved in the FA. This paper follows Grimal et al. [Int. J. Solid Struct. 39 (2002) 5345] in which we investigated the same bimaterial with a plane––instead of spherical––interface. Numerical examples are concerned with the propagation of an impact wave in the thorax modeled as a bimaterial (thoracic wall-lung). In addition to the effects of the weak coupling of the two media already observed in our previous study, we found that, for interface curvatures characteristic of those measured in the thorax, focalization of energy is manifest.     

A General Method for Estimating Dynamic Parameters of Spatial Mechanisms

Dynamic equations of motion require a large number of parameters for each element of the system. These can include for each part their mass, location of center of mass, moment of inertia, spring stiffnesses and damping coefficients. This paper presents a technique for estimating these parameters in spatial mechanisms using any joint type, based on measurements of displacements, velocities and accelerations and of external forces and torques, for the purpose of building accurate multibody models of mechanical systems. A form of the equations of spatial motion is derived, which is linear in the dynamic parameters and based on multibody simulation code methodologies. Singular value decomposition is used to find the essential parameter set, and minimum parameter set. It is shown that a simulation of a four-bar mechanism (with spherical, universal, and revolute joints) and based on the estimated parameters gives accurate response.     

A scanning PIV method for fine-scale turbulence measurements

A hybrid technique is presented that combines scanning PIV with tomographic reconstruction to make spatially and temporally resolved measurements of the fine-scale motions in turbulent flows. The technique uses one or two high-speed cameras to record particle images as a laser sheet is rapidly traversed across a measurement volume. This is combined with a fast method for tomographic reconstruction of the particle field for use in conjunction with PIV cross-correlation. The method was tested numerically using DNS data and with experiments in a large mixing tank that produces axisymmetric homogeneous turbulence at \(R_\lambda \simeq 219\) . A parametric investigation identifies the important parameters for a scanning PIV set-up and provides guidance to the interested experimentalist in achieving the best accuracy. Optimal sheet spacings and thicknesses are reported, and it was found that accurate results could be obtained at quite low scanning speeds. The two-camera method is the most robust to noise, permitting accurate measurements of the velocity gradients and direct determination of the dissipation rate.     

An analytic separation method for photoelasticity

A method for separating principal stresses in photoelasticity is presented. This method is based upon the series solution of Laplace's equation and the determination of the unknown coefficients arising in this series by a least-squares numerical technique. By selecting an adequate number of terms in the series, the representation of the boundary values of the first stress invariant can be established as accurately as the initial photoelastic data. This form of representation of the first stress invariant at interior points in the region is moe accurate than the boundary values employed.     

Scattering of plane SH waves by an isosceles trapezoidal hill

This paper presents a theoretical approach to study the surface motion of an isosceles trapezoidal hill impacted by incident SH waves. A rigorous solution has been derived by applying an accurate region-matching technique. The solution region is divided into three parts by an appropriate partitioning method. Based on complex function method and multipolar coordinates, a fractional factor is introduced to construct suitable wave functions which satisfy the governing equation and zero-stress condition on the free surface in each sub-region. According to the continuity condition at the auxiliary boundary, surface displacements are expressed in series of infinite algebraic equations, and the unknown coefficients of the series can be determined by Fourier series expansion technique in complex domain. Numerical results demonstrate the analytical results depend on the following parameters: The slope, the height and the width of the trapezoidal hill, the frequency content of the excitation and the incidence angle.     

Aerodynamic force and flow structures of two airfoils in flapping motions

Aerodynamic force and flow structures of two airfoils in a tandem configuration in flapping motions are studied, by solving the Navier-Stokes equations in moving overset grids. Three typical phase differences between the fore- and aftairfoil flapping cycles are considered. It is shown that: (1) in the case of no interaction (single airfoil), the time average of the vertical force coefficient over the downstroke is 2.74, which is about 3 times as large as the maximum steady-state lift coefficient of a dragonfly wing; the time average of the horizontal force coefficient is 1.97, which is also large. The reasons for the large force coefficients are the acceleration at the beginning of a stroke, the delayed stall and the “pitching-up” motion near the end of the stroke. (2) In the cases of two-airfoils, the time-variations of the force and moment coefficients on each airfoil are broadly similar to that of the single airfoil in that the vertical force is mainly produced in downstroke and the horizontal force in upstroke, but very large differences exist due to the interaction. (3) For in-phase stroking, the major differences caused by the interaction are that the vertical force on FA in downstroke is increased and the horizontal force on FA in upstroke decreased. As a result, the magnitude of the resultant force is almost unchanged but it inclines less forward. (4) For counter stroking, the major differences are that the vertical force on AA in downstroke and the horizontal force on FA in upstroke are decreased. As a result, the magnitude of the resultant force is decreased by about 20 percent but its direction is almost unchanged. (5) For 90°-phase-difference stroking, the major differences are that the vertical force on AA in downstroke and the horizontal force on FA in upstroke are decreased greatly and the horizontal force on AA in upstroke increased. As a result, the magnitude of the resultant force is decreased by about 28% and it inclines more forward. (6) Among the three cases of phase angles, inphase flapping produces the largest vertical force (also the largest resultant force); the 90°-phase-difference flapping results in the largest horizontal force, but the smallest resultant force.     

An iterative \mathit{GL}(n,\mathbb{R}) method for solving non-linear inverse vibration problems

For the inverse vibration problem, a differential-algebraic equation (DAE) method is proposed to simultaneously estimate the time-dependent damping and stiffness coefficients by using two sets of displacement and velocity as input data. We combine the equations of motion and the supplemental data into a set of DAEs. We develop an implicit $\mathit{GL}(n,\mathbb{R})$ scheme and a Newton iterative algorithm to stably solve the DAEs to find the unknown structural coefficients. The unknown force is also recovered by the present method. A linear oscillator and a non-linear Duffing oscillator are used as testing examples. The estimated results are rather accurate and robust against random noise; hence, the new method can be used in the solutions of non-linear inverse vibration problems.     

随机桁架结构几何非线性问题的混合摄动-伽辽金法求解

提出应用混合摄动$\!$-$\!$-$\!$伽辽金法求解随机桁架结构的几何非线性问题.将含位移项的随机割线弹性模量以及随机响应表示为幂多项式展开,利用高阶摄动方法确定随机结构几何非线性响应的幂多项式展开的各项系数.将随机响应的各阶摄动项假定为伽辽金试函数,运用伽辽金投影对试函数系数进行求解,从而得到随机桁架结构几何非线性响应的显式表达式.同已有的随机伽辽金法相比,本文所给的试函数由摄动解的线性组合而成,在求解非线性问题时,试函数的获取具有自适应性.数值算例结果表明,对于具有不同概率分布的多随机变量问题,本文方法无需对随机变量的概率分布形式进行转换,避免了转换误差,因而比同阶的广义正交多项式方法(generalizedpolynomial chaos, GPC)计算精度高.同时,在结果精度相当时,和GPC方法相比,本文方法得到的试函数系数的非线性方程维度不大,方程的求解工作量小且更易求解.当随机量涨落较大时,混合摄动$\!$-$\!$-$\!$伽辽金法计算所得的结构响应的各阶统计矩比高阶摄动法所得结果更逼近于蒙特卡洛模拟结果,显示了该方法对几何非线性随机问题求解的有效性.      

Rotary oscillations of axi‐symmetric bodies in an axi‐symmetric viscous flow with slip: Numerical solutions for sphere and spheroids

Rotary oscillations of several axi‐symmetric bodies in axi‐symmetric viscous flows with slip are investigated. A numerical method based on the Green's function technique is used wherein the relevant Helmholtz equation, as obtained from the unsteady Stokes equation, is converted into a surface integral equation. The technique is benchmarked against a known analytical solution, and accurate numerical results for local stress and torque on spheres and spheroids as function of the frequency parameter and the slip coefficients are obtained. It is found that in all cases, slip reduces stress and torque, and increasingly so with the increasing frequency parameter. The method discussed here can be potentially extended to the realistic case of an oscillating disk viscometer. Copyright © 2003 John Wiley & Sons, Ltd.     

Stochastic stability of non-linear SDOF systems

A semi-analytical procedure for obtaining stability conditions for strongly non-linear single degree of freedom system (SDOF) subjected to random excitations is presented using stochastic averaging technique. The method is useful for finding stability conditions for systems having highly irregular non-linear functions which cannot be integrated in closed form to yield analytical expressions for averaged drift and diffusion coefficients. In spite of numerical methods available for finding stability of SDOF system by determining Lyapunov exponent, the proposed technique may have to be adopted (i) when the excitation is non-white; and (ii) when numerical integration fails due to convergence problem. The method is developed in such a way that it lends itself to a numerical computational scheme using FFT for obtaining numerical values of drift and diffusion coefficients of Its differential equation and the corresponding FPK equation for the system. These values of averaged drift and diffusion coefficients are then fit into polynomial form using curve fitting technique so that polynomials can be used for stability analysis. Two example problems are solved as illustrations. The first one is the Van der Pol oscillator having non-linearities which can be treated purely analytically. The example is considered for the validation of the proposed method. The second one involves non-linearities in the form of signum function for which purely analytical solution is not possible. The results of the study show that the proposed method is useful and efficient for performing stability analysis of dynamic systems having any type of non-linearities.     

Renormalization calculations of immiscible flow

Oil reservoir properties can vary over a wide range of length scales. Reservoir simulation of the fluid flow uses numerical grid blocks have typical lengths of hundreds of metres. We need to specify meaningful values to put into reservoir engineering calculations given the large number of heterogeneities that they have to encompass. This process of rescaling data results in the calculation of effective or pseudo rock properties. That is a property for use on the large scale incorporating the many heterogeneities measured on smaller scales.For single phase flow, a variety of techniques have been tried in the past. These range from very simple statistical estimates to detailed numerical simulation. Unfortunately, the simple estimates tend to be inaccurate in real applications and the numerical simulation can be computationally expensive if not impossible for very fine grid representations of the reservoir. Likewise, pseudorelative permeabilities are time consuming to generate and often inaccurate.Real-space renormalization is an alternative technique which has been found to be computationally efficient and accurate when applied to single-phase flow. This approach solves the problem regionally rather than trying to solve the whole problem in one simulation. The effective properties of small regions are first calculated and then placed on a coarse grid. The grid is further coarsened and the process repeated until a single effective property has been calculated. This has enabled calculation of effective permeability of extremely large grids to be performed, up to 540 million grid blocks in one application.This paper extends the renormalization technique to two-phase fluid flow and shows that the method is at least 100 times faster than conventional pseudoization techniques. We compare the results with high resolution numerical simulation and conventional pseudoization methods for three different permeability models. We show that renormalization is as accurate as the conventional methods when used to predict oil recovery from heterogeneous systems.     

Application of hybrid and moment methods to the measurement of moisture diffusion coefficients of building materials

This work proposes two simple dynamic methods that provide an accurate method for measurement of diffusion coefficients in building materials. Experimental measurements of moisture diffusion coefficients covered three commonly used building materials and they were carried out for a range of the relevant parameters, as temperature and relative humidity. The diffusion coefficients obtained by the two dynamic methods show a deviation comparatively to the steady-sate cup method; however, this variance is in accordance with the results presented in literature.     

A variable angle method of calibration for X-probes applied to wall-bounded turbulent shear flow

A variable angle calibration technique for hot wire and hot film X-probes incorporating a new method of interpolation is described here along with measurements in a fully developed turbulent channel flow. Results based on the new method of calibration include the mean velocity profile, Reynolds stress, and probability density distributions for fluctuating velocity components u and and for the flow angle. Also skewness and flatness factors for u and are given. Measurement data were also evaluated using the conventional method. A comparison of both techniques shows that the new method does not yield appreciable differences in statistical flow analyses but is more accurate in measuring rare flow events associated with large flow angles. An extended version of the new method of calibration allowing three dimensional measurements in turbulent flow will be discussed.     

A hybrid finite-element–volume-of-fluid method for simulating free surface flows and interfaces

A numerical technique is developed for the simulation of free surface flows and interfaces. This technique combines the strength on the finite element method (FEM) in calculating the field variables for a deforming boundary and the versatility of the volume-of-fluid (VOF) technique in advection of the fluid interfaces. The advantage of the VOF technique is that it allows the simulation of interfaces with large deformations, including surface merging and breaking. However, its disadantage is that is solving the flow equations, it cannot resolve interfaces smaller than the cell size, since information on the subgrid scale is lost. Therefore the accuracy of the interface reconstruction and the treatment of the boundary conditions (i.e. viscous stresses and surface tension forces) become grid-size-dependent. On the other hand, the FEM with deforming interface mesh allows accurate implementation of the boundary conditions, but it cannot handle large surface deformations occurring in breaking and merging of liquid regions. Combining the two methods into a hybrid FEM-VOF method eliminates the major shortcomings of both. The outcome is a technique which can handle large surface deformations with accurate treatment of the boundary conditions. For illustration, two computational examples are presented, namely the instability and break-up of a capillary jet and the coalescence collision of two liquid drops.     

Bifurcation analysis for a modified Jeffcott rotor with bearing clearances

A HB (Harmonic Balance)/AFT (Alternating Frequency/Time) technique is developed to obtain synchronous and subsynchronous whirling motions of a horizontal Jeffcott rotor with bearing clearances. The method utilizes an explicit Jacobian form for the iterative process which guarantees convergence at all parameter values. The method is shown to constitute a robust and accurate numerical scheme for the analysis of two dimensional nonlinear rotor problems. The stability analysis of the steady-state motions is obtained using perturbed equations about the periodic motions. The Floquet multipliers of the associated Monodromy matrix are determined using a new discrete HB/AFT method. Flip bifurcation boundaries were obtained which facilitated detection of possible rotor chaotic (irregular) motion as parameters of the system are changed. Quasi-periodic motion is also shown to occur as a result of a secondary Hopf bifurcation due to increase of the destabilizing cross-coupling stiffness coefficients in the rotor model.     

非线性系统参数辩识的一种频域模型

本文基于对非线性系统的可分离性假设，将非线性弹性力和阻尼力分别分解为物理坐标下各点间相对位移和相对速度的幂级数函数，导出了一般多自由度非线性系统在恒幅激励下的广义频率响应函数与输入输出之间的迭代关系式，提出了非线性系统中基本线性部分的概念，进而了一种在实验条件下的系统物理参数辩识方法。     

A second-order approximation to natural convection for large Rayleigh numbers and small Prandtl numbers

The problem under investigation is that of fluid flow within an enclosed rectangular cavity. It is assumed that one wall is maintained at a constant temperature T₁ (hot wall) and the other wall is maintained at a constant temperature T₀ (cold wall). At the remaining walls, two separate cases are studied. In the first, an adiabatic boundary condition is assumed. That is, the normal derivative of the temperature function is assumed to be 0. In the second, it is assumed the temperature varies linearly from T₀ to T₁. The purpose of this paper is the application of a second order numerical technique to the problem of fluid flow within a heated closed cavity. The method is a modification of a method developed by Shay¹ and applied to the driven cavity problem. In order to test the viability of this technique, it was decided to extend the technique to the problem of natural convection in a square. Jones² proposed that this problem is suitable for testing techniques that may be applied to a wide range of practical problems such as reactor insulation, cooling of radioactive waste containers, solar energy collection and others.³ The technique makes use of second-order finite difference approximations to all derivatives in the governing equations. Furthermore, second-order approximations are also used to determine boundary vorticities and, when the adiabatic boundary condition is used, for the boundary temperatures as well. In some works, where second-order approximations are used at interior points, second-order boundary approximations have been sacrificed in favour of a more stable, but first-order boundary approximation. The current approximations are generated by writing the unknown value of a function at a given interior node as a linear combination of unknown function values at all of the neighbouring nodes. Then the function values at these neighbouring nodes are expanded in a Taylor series about the given node. Through appropriate regrouping of terms and the use of the equations to the solved, constraints are imposed on the coefficients of the linear combination to yield a second-order approximation. As it turns out, there are more unknowns than constraints and, as a result, we are left with some freedom in choosing coefficients. In this work this freedom was used to choose coefficients in such a way as to maximize stability of the resulting system of equations. In other words, the approximations to the governing partial differential equation are individually determined at each point dependent on the direction of flow in order to generate the best possible stability. This idea is analogous to that used in the derivation of the upwind method. However, the current method is second-order accurate where the upwind method is only first-order accurate. Thus, what is generated is an easily implemented second-order method that yields a system of equations that has proved easy to solve. The system of equations is solved via the method of successive overrelaxation. The stability of the method is shown in the convergence for a wide range of Rayleigh numbers, Prandtl numbers and mesh sizes. Level curves of the stream, vorticity and temperature functions are provided for Rayleigh numbers (Ra) as large as 100,000, Prandtl numbers (Pr) as small as 0.0001, and mesh sizes as small as 0.0125. Values of the Nusselt number have also been calculated through the use of Simpson's rule, and a second order approximation to the normal derivative of the temperature along the cold wall. Comparisons are made with other current works to aid in the verification of this methods' accuracy and also with the first-order upwind method to demonstrate superiority over the first-order method.     

An analytical approximate technique for a class of strongly non-linear oscillators

An analytical approximate technique for large amplitude oscillations of a class of conservative single degree-of-freedom systems with odd non-linearity is proposed. The method incorporates salient features of both Newton's method and the harmonic balance method. Unlike the classical harmonic balance method, accurate analytical approximate solutions are possible because linearization of the governing differential equation by Newton's method is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of non-linear algebraic equations without analytical solution. With carefully constructed iterations, only a few iterations can provide very accurate analytical approximate solutions for the whole range of oscillation amplitude beyond the domain of possible solution by the conventional perturbation methods or harmonic balance method. Three examples including cubic-quintic Duffing oscillators are presented to illustrate the usefulness and effectiveness of the proposed technique.     

Moisture transfer analysis during drying of slab woods

This article presents an analytical technique for determining the moisture diffusivities and moisture transfer coefficients for slab shaped woods subjects to drying process. The analysis of transient moisture diffusion is carried out on the basis of two important practical criteria: 0.1<Bi<100 and Bi>100. The drying coefficients and lag factors are defined for wood-drying applications and incorporated into the models. In order to verify the present models, the model results are compared with experimental measurements taken from the literature and good agreement was found. Results show that the technique presented here is capable of determining the moisture diffusivities and moisture transfer coefficients for slab woods in a simple and accurate manner for practical applications and will be beneficial to the relevant wood␣drying industries. This approach can be extended to␣different wood products of regular and irregular shapes. Received on 23 January 1998