实时子结构实验Chang算法的稳定性和精度

与慢速拟动力子结构实验相比,实时子结构实验的优点在于它能真实地反映速度相关型试件的特性。实时子结构实验的逐步积分算法通常借用拟动力算法,但是目前液压伺服作动器很难实现速度反馈控制,因而试件速度不能实现原算法的假定值,这样一来算法的稳定性和计算精度将发生改变。台湾学者S.Y.Chang提出一种无条件稳定的显式拟动力算法,本文分析了这种方法应用于实时子结构实验时的稳定性和计算精度。研究发现在实时子结构实验中该方法由无条件稳定变成了有条件稳定的,精度也发生了改变。     

等效力控制方法及其在混合试验中的应用

对于大型复杂结构的实时(拟动力)子结构试验,更适宜用无条件稳定的逐步积分方法。隐式逐步积分方法通常是无条件稳定的,然而需要复杂耗时的迭代求解非线性方法。为了避免迭代过程,等效力控制方法用反馈控制求解非线性方程,使隐式逐步积分方法在实时子结构试验中的应用成为可能。本文首先以平均加速度法为例介绍等效力控制方法的原理、关键参数的选取;然后介绍基于等效力控制的能量守恒子结构试验方法和隐式中点法;最后介绍这些方法在以防屈曲支撑阻尼器为试件的单自由度简化结构、以磁流变阻尼器为试件的海洋平台结构的实时子结构试验,以及装配式钢筋混凝土剪力墙结构和框支配筋砌块短肢剪力墙结构拟动力试验中的应用。研究结果表明:这三种等效力控制方法都具有很好的精度,等效力控制方法相对于中心差分法具有更好的稳定性。     

一族基于模型的积分算法在负刚度时的数值稳定性

在正刚度条件下，基于模型的积分算法兼具无条件稳定和显式的特性。为了研究基于模型的积分算法在负刚度条件下的数值稳定性，对一族新近提出来的GCR(generalized Chen-Ricles, GCR)算法进行分析。首先，推导出GCR族算法在负刚度条件下的稳定性准则，得到满足无条件稳定的参数取值范围。其次，通过分析积分参数、阻尼比及时间步长对GCR族算法数值稳定性的影响规律，验证推导的稳定性准则。再次，对多自由度体系负刚度条件下隐式算法和GCR族算法的数值稳定性进行分析和对比，并提出适用于正负刚度条件下GCR族算法的分析流程和策略。最后，通过2个典型算例验证了所提策略，表明GCR算法在正负刚度条件下可以同时满足稳定。     

高阶单步实时动力子结构试验技术研究

结构联机试验可分为两类：拟静力及拟动力试验技术,它们都需要建立一套显式的逐步积分算法。国内外学者在这方面已经进行了许多的研究,取得了很好的成果。随着振动控制技术在结构工程上的应用,一些速度相关型的装置开始用于被控系统,它给原有的实时子结构试验带来了新问题。如何建立更好的高精度、无条件稳定的实时动力子结构试验算法日趋重要。本文在前人早期高阶单步逐步积分算法研究成果的基础上,提出了一种新的高阶单步实时动力子结构试验算法。数值模拟分析表明,新算法不仅是显式的,而且具有高精度、无算法阻尼、无超越现象等算法特点,均比目前所见到的已有算法优越。如果能实现实时子结构试验,就能同时控制位移和速度,则应用本文算法必将取得更好的试验结果。     

结合动量方程及显式γ函数法的拟动力试验方法

研究了一种适合实时或快速拟动力试验的数值积分方法--动量方程方法,阐述了动量方程方法的原理,结合显式γ函数法求解隐式方程,得到了拟动力试验实用的显式位移表达式.选用合适的参数,对一根悬臂钢柱进行了拟动力试验.试验结果与中心差分法得到的试验结果吻合较好,从而验证了积分方法的可行性和有效性,可以作为速度相关型结构或构件拟动力试验的数值积分方法.     

基于螺旋线上谱因式分解的地震波场隐式辛算法

均匀介质、复杂各向同性介质和各向异性介质中的地震波传播过程,可用统一形式的标量声波方程描述.考虑到在无损耗条件下,地震波方程描述了地震波场这一个无穷维的哈密顿体系随时间的演化过程,该过程为一个单参数连续辛变换,因而可以在其哈密顿形式表述下导出其辛格式.与显式辛算法相比,隐式辛格式对应的隐式辛几何算法具有无条件稳定的特点,可以允许较大的计算步长.但是由于隐式算法不可避免地面临高阶矩阵的求逆,其每一步的计算速度较慢.为实现矩阵快速求逆,文中采用了螺旋边界条件下谱因式分解的方法.在螺旋边界条件下,需要求逆的矩阵化为带状矩阵,而且其各列非零元素的位置和大小具有非常好的相似性,因而可以采用谱因式分解的方法实现快速LU分解.文中采用二阶精度的隐式蛙跳辛格式和谱因式分解方法,计算了常速度、层状介质和Marmousi模型中的波场.计算表明,隐式辛算法不失为波场计算的一种好方法.     

一种基于离散控制理论的无条件稳定显式算法

基于离散控制理论,结合CR法和RST法提出一种无条件稳定的动力学显式新算法。以算法精度和稳定性为条件,通过离散传递函数推导参数表达式和极点,使得新算法可满足零振幅衰减率和零周期延长率。算法参数α和γ作为传递格式选择参数,当α和γ分别取1时,新算法对应CR法和RST法的位移速度表达式。对新算法的精度和稳定性理论分析表明:新算法可满足无振幅衰减和周期延长,且对于线性系统和非线性刚度软化系统为无条件稳定,对非线性刚度硬化系统为条件稳定,并给出了非线性刚度硬化系统的稳定性范围。算例分析验证了新算法的精度和稳定性,证明提出的新算法是可靠有效的。     

隐式时间积分方法的拟动力实验

本文介绍了采用隐式时间积分方法实现的拟动力实验。目前拟动力实验中所用的时间积分方法是显式条件稳定的，所以时间步Δｔ的选择受到试件刚度和自由度的限制，对于刚度大自由度很多的试件需采用很小的Δｔ，而Δｔ太小将造成实验累积误差增大，实验结果失真。隐式时间积分方法是无条条件稳定的Δｔ的选择不受试件特性的限制，可以比显式算法的稳定极限大很多，从而拓宽了拟动力的应用范围。     

使用Chang算法的剪切型子结构振动台试验方法及数值模拟

提出了使用无条件稳定显式Chang积分算法的剪切型子结构振动台试验方法,包括子结构定义及试验流程。进行某12层剪切型结构的整体结构及子结构方法的时程分析,整体结构分析结果作为子结构方法分析结果的参照。对于无时滞的情形,采用两种误差指标来量化子结构方法结果的准确性,研究了积分时间步长及时程分析类型对结果的影响。最后,考虑时滞对子结构方法结果的影响。结果表明:当系统不存在时滞时,即便在较大的积分时间步长的情况下,子结构方法的分析结果依然可以较好地吻合整体结构的分析结果;当系统存在时滞时,时滞对子结构方法结果的准确性产生不利的影响。     

基于零阶和一阶优化算法的建筑结构抗震优化设计

针对第三水准“大震不倒”的抗震设防目标,提出一种建筑结构的抗震优化设计方法,依据“用相同的投资获最好的设计”的设计理念,建立了在强烈地震波的作用下,以建筑结构最大的层间相对位移最小化作为优化目标,同时满足体积约束的优化数学模型,并采用动力有限元分析模型和高效的显式动力分析方法进行结构分析获得最大的层间相对位移,采用零阶和一阶优化算法分别求解优化数学模型,在显式动力分析软件ANSYS／LS—DYNA的基础上进行二次开发,实现了一个三维框架结构的抗震优化设计。数值结果表明该方法能获得较高质量的解,经优化设计后建筑结构的抗震性能得到了很大的改善。     

A two-step unconditionally stable explicit method with controllable numerical dissipations

A family of unconditionally stable direct integration algorithm with controllable numerical dissipations is proposed. The numerical properties of the new algorithms are controlled by three parameters α, β and γ. By the consistent and stability analysis, the proposed algorithms achieve the second-order accuracy and are unconditionally stable under the condition that α≥-0.5, β≤ 0.5 and γ≥-(1+α)/2. Compared with other unconditionally stable algorithms, such as Chang's algorithms and CR algorithm, the proposed algorithms are found to be superior in terms of the controllable numerical damping ratios. The unconditional stability and numerical damping ratios of the proposed algorithms are examined by three numerical examples. The results demonstrate that the proposed algorithms have a superior performance and can be used expediently in solving linear elastic dynamics problems.     

Operator‐splitting method for real‐time substructure testing

It has been shown that the operator‐splitting method (OSM) provides explicit and unconditionally stable solutions for quasi‐static pseudo‐dynamic substructure testing. However, the OSM provides only an explicit target displacement but not an explicit target velocity, so that it is essentially an implicit method for real‐time substructure testing (RST) when the velocity‐dependent restoring force is considered. This paper proposes a target velocity formulation based on the forward difference of the predicted displacements so as to render the OSM explicit for RST. The stability and accuracy of the resulting OSM‐RST algorithm are investigated. It is shown that the OSM‐RST is unconditionally stable so long as the non‐linear stiffness and damping are of the softening type (i.e. the tangent stiffness and damping never exceed the initial values). The stability of the OSM‐RST for structures with infinite tangent damping coefficient or stiffness is also proved, and the stability of the method for MDOF structures with a non‐classical damping matrix is demonstrated by an energy criterion. The effects of actuator delay and compensation are analysed based on the bilinear approximation of the actuator step response. Experiments on damped SDOF and MDOF structures verify that the stability of the OSM‐RST is preserved when the experimental substructure generates velocity‐dependent reaction forces, whereas the stability of real‐time substructure tests based on the central difference method is worsened by the damping of the specimen. Copyright © 2005 John Wiley & Sons, Ltd.     

Stability of central difference method for dynamic real‐time substructure testing

This paper studies the stability of the central difference method (CDM) for real‐time substructure test considering specimen mass. Because the standard CDM is implicit in terms of acceleration, to avoid iteration, an explicit acceleration formulation is assumed for its implementation in real‐time dynamic substructure testing. The analytical work shows that the stability of the algorithm decreases with increasing specimen mass if the experimental substructure is a pure inertia specimen. The algorithm becomes unstable however small the time integration interval is, when the mass of specimen equal or greater than that of its numerical counterpart. For the case of dynamic specimen, the algorithm is unstable when there is no damping in the whole test structure; a damping will make the algorithm stable conditionally. Part of the analytical results is validated through an actual test. Copyright © 2009 John Wiley & Sons, Ltd.     

Energy-consistent integration method and its application to hybrid testing

The time-integration algorithm is an indispensable element to determine response of the boundary of the numerical as well as physical parts in a hybrid test. Instability of the time-integration algorithm may directly lead to failure of the test, so stability of an integration algorithm is particularly important for hybrid testing. The explicit algorithms are very popular in hybrid testing, because iteration is not needed. Many unconditionally stable explicit-algorithms have been proposed for hybrid testing. However, the stability analysis approaches used in all these methods are valid only for linear systems. In this paper, a uniform formulation for energy-consistent time integrations, which are unconditionally stable, is proposed for nonlinear systems. The solvability and accuracy are analyzed for typical energy-consistent algorithms. Some numerical examples and the results of a hybrid test are provided to validate the effectiveness of energy-consistent algorithms.     

逐步积分法求解复阻尼结构运动方程的稳定性问题

平均常加速度法是无条件稳定的，但用它来解复阻尼振动方程时，现了不稳定现象，我们通过理论分析和数值计算结果的比较，探明了这种现象不是由于逐步积分算法本身有不稳定的问题，而是由于复阻尼振动方程解集本身含有不稳定子集的缘故。     

Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation

The implicit dissipative generalized‐ α method is analyzed using discrete control theory. Based on this analysis, a one‐parameter family of explicit direct integration algorithms with controllable numerical energy dissipation, referred to as the explicit KR‐α method, is developed for linear and nonlinear structural dynamic numerical analysis applications. Stability, numerical dispersion, and energy dissipation characteristics of the proposed algorithms are studied. It is shown that the algorithms are unconditionally stable for linear elastic and stiffness softening‐type nonlinear systems, where the latter indicates a reduction in post yield stiffness in the force–deformation response. The amount of numerical damping is controlled by a single parameter, which provides a measure of the numerical energy dissipation at higher frequencies. Thus, for a specific value of this parameter, the resulting algorithm is shown to produce no numerical energy dissipation. Furthermore, it is shown that the influence of the numerical damping on the lower mode response is negligible. It is further shown that the numerical dispersion and energy dissipation characteristics of the proposed explicit algorithms are the same as that of the implicit generalized‐ α method. A numerical example is presented to demonstrate the potential of the proposed algorithms in reducing participation of undesired higher modes by using numerical energy dissipation to damp out these modes. Copyright © 2014 John Wiley & Sons, Ltd.     

An unconditionally stable time-stepping procedure with algorithmic damping: a weighted integral approach using two general weight functions

An accurate algorithm for the integration of the equations of motion arising in structural dynamics is presented. The algorithm is an unconditionally stable single-step implicit algorithm incorporating algorithmic damping. The displacement for a Single-Degree-of-Freedom system is approximated within a time step by a function which is cubic in time. The four coefficients of the cubic are chosen to satisfy the two initial conditions and two weighted integral equations. By considering general weight functions, eight additional coefficients arise. These coefficients are selected to (i) minimize the difference between exact and approximate solutions for small time steps, (ii) incorporate specified algorithmic damping for large time steps, (iii) ensure unconditional stability and (iv) minimize numerical operations in forming the amplification matrix. The accuracy of the procedure is discussed, and the solution time is compared with a widely used algorithm. Copyright © 1999 John Wiley & Sons, Ltd.     

Stability of average acceleration method for structures with nonlinear damping

The energy approach is used to theoretically verify that the average acceleration method （AAM）, which is unconditionally stable for linear dynamic systems, is also unconditionally stable for structures with typical nonlinear damping, including the special case of velocity power type damping with a bilinear restoring force model. Based on the energy approach, the stability of the AAM is proven for SDOF structures using the mathematical features of the velocity power function and for MDOF structures by applying the virtual displacement theorem. Finally, numerical examples are given to demonstrate the accuracy of the theoretical analysis.     

Analysis of implicit HHT‐α integration algorithm for real‐time hybrid simulation

Real‐time hybrid simulation is a viable experiment technique to evaluate the performance of structures equipped with rate‐dependent seismic devices when subject to dynamic loading. The integration algorithm used to solve the equations of motion has to be stable and accurate to achieve a successful real‐time hybrid simulation. The implicit HHT α‐algorithm is a popular integration algorithm for conducting structural dynamic time history analysis because of its desirable properties of unconditional stability for linear elastic structures and controllable numerical damping for high frequencies. The implicit form of the algorithm, however, requires iterations for nonlinear structures, which is undesirable for real‐time hybrid simulation. Consequently, the HHT α‐algorithm has been implemented for real‐time hybrid simulation using a fixed number of substep iterations. The resulting HHT α‐algorithm with a fixed number of substep iterations is believed to be unconditionally stable for linear elastic structures, but research on its stability and accuracy for nonlinear structures is quite limited. In this paper, a discrete transfer function approach is utilized to analyze the HHT α‐algorithm with a fixed number of substep iterations. The algorithm is shown to be unconditionally stable for linear elastic structures, but only conditionally stable for nonlinear softening or hardening structures. The equivalent damping of the algorithm is shown to be almost the same as that of the original HHT α‐algorithm, while the period elongation varies depending on the structural nonlinearity and the size of the integration time‐step. A modified form of the algorithm is proposed to improve its stability for use in nonlinear structures. The stability of the modified algorithm is demonstrated to be enhanced and have an accuracy that is comparable to that of the existing HHT α‐algorithm with a fixed number of substep iterations. Both numerical and real‐time hybrid simulations are conducted to verify the modified algorithm. The experimental results demonstrate the effectiveness of the modified algorithm for real‐time testing. Copyright © 2011 John Wiley & Sons, Ltd.     

Implementation and application of the unconditionally stable explicit parametrically dissipative KR‐α method for real‐time hybrid simulation

In real‐time hybrid simulations (RTHS) that utilize explicit integration algorithms, the inherent damping in the analytical substructure is generally defined using mass and initial stiffness proportional damping. This type of damping model is known to produce inaccurate results when the structure undergoes significant inelastic deformations. To alleviate the problem, a form of a nonproportional damping model often used in numerical simulations involving implicit integration algorithms can be considered. This type of damping model, however, when used with explicit integration algorithms can require a small time step to achieve the desired accuracy in an RTHS involving a structure with a large number of degrees of freedom. Restrictions on the minimum time step exist in an RTHS that are associated with the computational demand. Integrating the equations of motion for an RTHS with too large of a time step can result in spurious high‐frequency oscillations in the member forces for elements of the structural model that undergo inelastic deformations. The problem is circumvented by introducing the parametrically controllable numerical energy dissipation available in the recently developed unconditionally stable explicit KR‐α method. This paper reviews the formulation of the KR‐α method and presents an efficient implementation for RTHS. Using the method, RTHS of a three‐story 0.6‐scale prototype steel building with nonlinear elastomeric dampers are conducted with a ground motion scaled to the design basis and maximum considered earthquake hazard levels. The results show that controllable numerical energy dissipation can significantly eliminate spurious participation of higher modes and produce exceptional RTHS results. Copyright © 2014 John Wiley & Sons, Ltd.