Unconditionally energy stable implicit time integration: application to multibody system analysis and design

This paper focuses on the development of an unconditionally stable time‐integration algorithm for multibody dynamics that does not artificially dissipate energy. Unconditional stability is sought to alleviate any stability restrictions on the integration step size, while energy conservation is important for the accuracy of long‐term simulations. In multibody system analysis, the time‐integration scheme is complemented by a choice of co‐ordinates that define the kinematics of the system. As such, the current approach uses a non‐dissipative implicit Newmark method to integrate the equations of motion defined in terms of the independent joint co‐ordinates of the system. In order to extend the unconditional stability of the implicit Newmark method to non‐linear dynamic systems, a discrete energy balance is enforced. This constraint, however, yields spurious oscillations in the computed accelerations and therefore, a new acceleration corrector is developed to eliminate these instabilities and hence retain unconditional stability in an energy sense. An additional benefit of employing the non‐linearly implicit time‐integration method is that it allows for an efficient design sensitivity analysis. In this paper, design sensitivities computed via the direct differentiation method are used for mechanism performance optimization. Copyright © 2000 John Wiley & Sons, Ltd.     

Studies on identification of an efficient and accurate integration scheme for piecewise linear dynamic problems

Engineering structures often have nonlinear characteristics. Many of these nonlinear systems could be modelled as piecewise linear. The dynamic analysis of such systems can be carried out by the direct integration of the mathematical model of the system. In this article, solution schemes are identified and qualified studying a single degree-of-freedom system and evaluated studying real-life engineering structures. As an alternative, a modified Newmark method with iteration for nonlinear forces and half-step error monitoring is presented. The modified scheme has been found to be efficient and accurate.     

一种多自由度阻尼体系动力问题的高阶分析方法

传统动力时程直接积分法多采用低阶数值格式,需要选择非常小的时间步距才能获得满足精度要求的动力分析结果.该文将结构动力时程分析的积分求微法推广至多自由度情形,发展了一种具有较高计算效率的多自由度阻尼体系的动力时程高阶分析方法.将相邻的ρ个时步组成一个待求解时段,基于多自由度体系动力响应积分解,以精细积分法结合秦九韶算法计...     

Sensitivity analysis for dynamic mechanical systems with finite rotations

This paper presents a sensitivity analysis for dynamic systems with large rotations using a semi‐analytical direct differentiation technique. The choice of a suitable time integration strategy for the rotation group appears to be critical for the development of an efficient sensitivity analysis. Three versions of the generalized‐α time integration scheme are considered: a residual form, a linear form, and a geometric form. Their consistency is discussed, and we show that the residual form, which is the most direct extension of the generalized‐α algorithm defined in structural dynamics, should not be used for problems with large rotations. The sensitivity analysis is performed and close connections are highlighted between the structure of the sensitivity equations and of the linearized dynamic equations. Hence, algorithms developed for the original problem can be directly reused for the sensitivities. Finally, a numerical example is analysed in detail. Copyright © 2007 John Wiley & Sons, Ltd.     

On the equivalence of dynamic relaxation and the Newton‐Raphson method

Dynamic relaxation is an iterative method to solve nonlinear systems of equations, which is frequently used for form finding and analysis of structures that undergo large displacements. It is based on the solution of a fictitious dynamic problem where the vibrations of the structure are traced by means of a time integration scheme until a static equilibrium is reached. Fictitious values are used for the mass and damping parameters. Heuristic rules exist to determine these values in such a way that the time integration procedure converges rapidly without becoming unstable. Central to these heuristic rules is the assumption that the highest convergence rate is achieved when the ratio of the highest and lowest eigenfrequency of the structure is minimal. This short communication shows that all eigenfrequencies become identical when a fictitious mass matrix proportional to the stiffness matrix is used. If, in addition, specific values are used for the fictitious damping parameters and the time integration step, the dynamic relaxation method becomes completely equivalent to the Newton‐Raphson method. The Newton‐Raphson method can therefore be regarded as a specific form of dynamic relaxation. This insight may help to interpret and improve nonlinear solvers based on dynamic relaxation and/or the Newton‐Raphson method.     

有阻尼动力方程显式积分方法的精度研究

在动力问题分析中,一种好的显式积分方法不仅要具有良好的稳定性,而且还要具有良好的计算精度。对几种有阻尼动力方程显式积分方法的精度问题进行了分析,研究了各自的局部截断误差、数值阻尼比和相对周期误差。这一研究将有助于更加全面地了解这些积分格式的性能。     

Computer-aided design of rolling mills

Computers have been used for design of rolling mills since 1960s. Easy access to high speed digital computers has facilitated use of more accurate rolling theories for design work and comprehensive computer simulations have been developed for all types of mills. These simulations include optimization of roll pass sequence, calculation of roll force, torque and temperature and detailed time studies for productivity calculations. These aspects of mill simulation are examined with specific examples. In the design of rolling mills equipment, computer-aided design has been widely used. The available software, which covers both steady state and dynamic analysis of items such as mill stands, drive systems, manipulators, cooling beds, roll cooling, and automatic gauge control systems, is reviewed with special reference to the nature of analysis and range of applications. Computer drafting is being increasingly used by rolling mills designers for both interactive and non-interactive applications. The integration of computer graphics with the design software holds promise for improving the productivity and creativity of designers and is an active area of development in rolling mills design organizations. A specific application where such integration has been achieved is discussed.     

A BEM formulation for inelastic transient dynamic analysis using domain decomposition and particular integrals

This study deals with the domain decomposition method and particular integrals for multi-region inelastic transient dynamic analysis. The particular integral formulation for single-region inelastic transient dynamic analysis is obtained by eliminating the acceleration volume integral and treating the initial stress term by volume cell. The Houbolt time integration scheme is used for the time- marching process. The Newton-Raphson algorithm for plastic multiplier is used to solve the system equation. In order to extend to multi-region problems, the domain decomposition method is examined. The domain of the original problem is subdivided into subregions. The interface boundary conditions are updated by using the iterative coupling employing Schwarz algorithm. Numerical results of two example problems are given to demonstrate the validity and accuracy of the present formulation.     

基于加速度泰勒展开的动力学方程显式积分方法

该文旨在提出兼顾适用性、可靠性与高效性的结构振动时域积分算法。基于加速度的泰勒展开式,引入截断系数考虑高阶项的影响,提出了具有4阶精度的加速度公式;通过积分并考虑典型时间步初始时刻系统动力平衡条件,建立了位移和速度的单步递推公式,运用终止时刻系统运动方程修正加速度。与多步积分法相比,单步积分法无需记录当前时间步以外时刻响应。稳定性分析表明,临界步长相比中心差分法增加40%。通过线性系统振动响应计算发现,当步长-系统固有周期（荷载周期）比达到0.2时,该文方法的振幅衰减率和周期延长率均小于5%;对于非线性系统,为降低算法阻尼和周期误差的影响,需控制步长周期比小于0.1。     

结构体系动力方程求解的显式积分格式的能耗特性

针对作者提出的结构体系动力方程求解的一种显式积分格式,探讨其数值计算能耗特性即算法阻尼特性,导出了其算法阻尼值随体系的物理阻尼和结构体系振动频率值变化的关系,并进一步讨论了该积分格式用于无限介质波动的数值模拟中抑制或消除透射边界引起的计算高频失稳问题,给出了该积分格式与中心差分格式的对比分析算例。     

Simulation of high speed impact into ceramic composite systems using cohesive-law fracture model

A numerical study for the analysis of oblique metal/ceramic/metal three layer composite systems against a long-rod has been performed. The study was done using a three-dimensional dynamic program NET3D, which uses the finite element Lagrangian method with explicit time integration. To model the discrete nature for fracture and damage of brittle materials, we implemented cohesive-law fracture model with a node separation algorithm for the tensile failure and Mohr–Coulomb model for the compressive loading. A tetrahedral element implemented in the code provides more potential fracture surfaces than a hexahedral element. As a verification of the scheme, an oblique impact into the composite system was conducted and the calculated penetration depth and propagating crack paths were found to be in good agreement with experiment. Next a series of three-dimensional numerical simulations have been conduced to examine the ballistic performance of three layer composite systems. The residual velocity and residual length of the rod were computed for different plate thickness ratios of equal areal density. The impact velocities considered are 1.5, 1.8 and 2.2 km/s. The oblique angle of the plate is 0° and 45°. The optimum thickness ratios of ceramic to metal are very similar to those obtained from the previous experiment.     

Efficient dynamic focus control for three-dimensional imaging using two-dimensional arrays

An efficient dynamic focus control scheme for a delay-and-sum-based beamformer is proposed. The scheme simplifies dynamic focus control by exploiting the range-dependent characteristics of the focusing delay. Specifically, the overall delay is divided into a range-independent steering term and a range-dependent focusing term. Because the focusing term is inversely proportional to range, approximation can be made to simplify dynamic focus control significantly at the price of minimal degradation in focusing quality at shallow depths. In addition, the aperture growth controlled by a constant f//sub number/ can also be utilized to devise a nonuniform quantization scheme for the focusing delay values. Efficacy of the proposed scheme is demonstrated using simulated beam plots of a fully sampled, two-dimensional array. Design procedures are also described in detail. One design example shows that, with the proposed dynamic focus control scheme, a 4096-element array only requires 227 independent controllers for the range-dependent focusing term. Moreover, only 28 non-uniform quantization levels are required to achieve the same focusing quality as that of a conventional scheme with 784 uniform quantization levels. The beam plots of a fully sampled array show that sidelobes are slightly increased below the -30 dB level for imaging depths less than 3 cm. At greater depths, there is no observable degradation.     

Manufacturing perspective of enterprise application integration: the state of the art review

Manufacturing enterprise today has become a matter of effective and efficient application of information technology and knowledge-based engineering. Several new manufacturing paradigms such as virtual enterprise (or extended enterprise) and mass customization have resulted in a highly distributed and autonomous manufacturing system. On the one hand, this will increase the competitiveness of a firm in terms of quickly meeting dynamic changes in the market; on the other hand, this will also increase the difficulty of integrating different information and knowledge systems residing in each member firm. This integration is also called enterprise application integration (EAI) (here the term application means information systems or software systems for supporting manufacturing or service activities). The methodology for EAI has been studied for at least a decade; but no satisfactory solution has been found from a practical viewpoint. EAI is becoming even more difficult due to the boom in various enterprise information and knowledge systems (and also to ever increasing competition in the technical software market). The study presented in this paper conducted a critical analysis of existing solutions to EAI. We consider EAI as having two generic issues: semantic integration and syntactic integration. The main problem of semantic integration reduces to the general problem of enterprise or business modelling. The main problem of syntactic integration reduces to the general problem of software architecture of enterprise applications which enables interoperability between any two EAs. While the first issue is studied in the manufacturing field, the second issue is studied in the computing engineering field. In this paper, both issues are put together under the context of EAI and studied. The result of the study, together with our experience with one Canadian manufacturing firm, has led to the identification of several issues to be addressed in the future. We also outline possible ways to approach these issues.     

Finite element implementation of virtual internal bond model for simulating crack behavior

The virtual internal bond (VIB) model has been recently proposed to describe material deformation and failure under both static and dynamic loading. The model is based on the incorporation of a cohesive type law in a hyperelastic framework, and is capable of fracture simulation as a part of the constitutive formulation. However, with an implicit integration scheme, difficulties are often encountered in the finite element implementation of the VIB model due to possible negative eigenvalues of the stiffness matrix. This paper describes the implementation of an explicit integration scheme of the VIB model. Issues pertaining to the implementation, such as mesh size and shape dependence, loading rate dependence, crack initiation and growth characteristics, and solution time are examined. Both quasi-static and dynamic loading cases have been studied. The experimental validation of the VIB model has been done by calibrating the model parameters using the experimental data of Andrews and Kim [Mech. Mater. 29 (1988) 161]. The simulations using the VIB model are shown to agree well with the experimental observations.     

Performance of time‐stepping schemes for discrete models in fracture dynamic analysis

In this work we present a critical evaluation of different time integration schemes within the setting of non‐linear dynamic analysis of brittle fracture problem represented by a discrete model. The discrete model of this kind consists of Voronoi cells representing the grains of a heterogeneous structure, which are interconnected by cohesive forces modelled by beam‐like links capable of taking properly into account both brittle dynamic fracture and large displacement of a still connected pack of grains that might split from the structure. The brittle behaviour of cohesive links requires that the dynamic analysis of such a model be treated with care, and the best possible integration scheme be selected. Four different schemes are explored and compared in application to a dynamic traction test, including Newmark explicit and implicit schemes, HHT‐α scheme and energy‐decaying scheme. Copyright © 2005 John Wiley & Sons, Ltd.     

Dynamic analysis of structures with unilateral constraints: Numerical integration and reduction of structural equations

Structural dynamic equations with unilateral constraints upon the displacements, velocities and accelerations are employed in order to represent vibrating elastic structures with normal, oblique impact and friction interaction points. For obtaining a numerical integration scheme the Lagrange multipliers and a minimum work approach are employed at each time step. The algorithm is presented as an extension of the generalized Newmark scheme. It seems to retain the asymptotic features of the original one. The reduction of the number of dynamic degrees of freedom of the unilaterally constrained structures is carried out by representing the equations of motion in modal co-ordinates of the unconstrained structure and truncating the dynamic contributions of higher modes. The presented techniques have been verified by investigating free longitudinal vibroimpact motion laws of an elastic vibroconverter and free longitudinal and bending vibration of a vibroconverter interacting with a moving rigid body by oblique impact and friction forces.     

结构动态载荷识1别的精细逐步积分法

对比例阻尼系统给出了基于精细逐步积分法的动态载荷识别方法.首先将系统进行模态坐标变换得到无耦合运动方程,然后应用精细逐步积分法构造一种高效精确的载荷识别公式,再由结构动态响应求出动态力的时间历程.数值算例验证了本方法的识别精度是好的.     

Buoyancy potential conserving technique for dynamic analysis of offshore pipes and risers

The paper describes the midpoint algorithm for the solution of dynamic equilibrium equations in the marine environment for the non‐linear dynamic analysis of flexible pipes and riser systems in 3D configuration. It has been shown that proposed method has an advantage over traditional time marching method by means of long‐term stability and larger time steps. The mid‐point algorithm used for the time integration is considered ‘approximate buoyancy‐energy conserving algorithm’. Emphasis is placed on elements that use the Reissner–Simo beam theory or a related co‐rotational approach. Copyright © 2005 John Wiley & Sons, Ltd.     

结构动态载荷识别的精细逐步积分法

对比例阻尼系统给出了基于精细逐步积分法的动态载荷识别方法．首先将系统进行模态坐标变换得到无耦合运动方程,然后应用精细逐步积分法构造一种高效精确的载荷识别公式,再由结构动态响应求出动态力的时间历程。数值算例验证了本方法的识别精度是好的。