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第十一讲 理论和模型导论之后通常是理论和模型部分，主要用于介绍所采用或开发的理论模型，或技术的理论基础。两大类：数学分析；模型介绍1数学分析数学分析类的理论部分通常包括以下资料：1）介绍题目及目地2）叙述相关的假设、条件或定义3）说明主要问题或基本方程式4）进行分析并得出结果5）说明或讨论上述结果，或根据结果作出推论Example one（介绍题目）For the purpose of discussion, a single story damped structure with a dynamic absorber having adaptable stiffness and damping is used, as shown in Figure 1, to illustrate the control law. m is the mass of the structure; k and c are the stiffness and the damping coefficients of the structure, respectively. ks(u) and cs(u) are adaptable stiffness and damping coefficients of the dynamic absorber which depend on the control parameter u. The control parameter u is different for different control materials.In an electro-rheological type material, the control parameter u denotes voltage. ks(u) and cs(u) increase monotonically with u. When the voltage is off, both ks(u) and cs(u) take their minimum values. When the voltage is on, ks(u) and cs(u) take their maximum values. But, it must be noted that very complicated nonlinear models are proposed for the electro-rheological materials in the literature (Gavin et al. 1996).（叙述相关的假设及说明基本方程式）In this paper, for the sake of simplicity, it is assumed that the dynamic absorber has adaptable stiffness ks(u) and adaptable damping cs(u) and these can be changed independently of each other. They can only provide two different values of stiffness, (ks)min and (ks)max, and damping, (cs)min and (cs)max. The equation of motion of a base-excited single story damped structure given in Figure 1 can be presented as (1)where is the earthquake acceleration and fu is the control force which occurs in the dynamic absorber and is given by (2)For the system given by equation (1), the mechanical energy of the main structure can be expressed as: (3)where Er and Es are the relative kinetic energy and the elastic strain energy of the main structure, respectively, and are given by the equations (4) (5)It must be noted here that if the absolute kinetic energy is used instead of the relative kinetic energy this will result in a closed-open loop control algorithm.（进行分析并得出结果）From equation (3), the energy rate of the system is obtained as (6)Using equation (1), the rate of change of energy can be rewritten as (7)Upon substituting the control force fu given by equation (2) into equation (7), equation (7) becomes (8)where and To make the time derivative of system energy given by equation (8) as negative as possible, the last term in parentheses must be forced to be as positive as possible by changing ks and cs. Control policy for changing the active stiffness ks and cs is obtained as follows: (9) (10)（说明或讨论上述结果）Since the term b is always positive, the adaptive damping coefficient cs must be set to its maximum value during the control process. Therefore the variable damping part of the dynamic absorber serves as a passive damping device. This will result in a variable stiffness control. For the application of this algorithm, it is assumed that the quantities x and can be measured in real time.2模型介绍在介绍理论或数学模型时常出现的资料包括：1）模型的背景或理论基础，必要时引述其它学者的成果2）模型所采用的方法的基本假设或根据3）基本方程及分析4）模型的详细描述5）说明或讨论模型的重要特点6）说明在特定情况下如何应用此模型7）对实例的解答进行讨论Example two（基本方程）In structural dynamics, the behaviour of a structure is defined from the dynamic equilibrium equation (1)where K, C and M are the stiffness, damping and mass matrices respectively, while the right-hand side is the external force vector.The procedure described in this work refers to a joint contribution between a numerical model involving equation (1) and an experimental step, which leads to the calculation of a force vector called the internal structure force or structural restoring force vector. This force vector corresponds to the effective physical (or mechanical) behaviour of the structure at the level of the selected degrees of freedom. This analysis may deal with a linear or non-linear structural behaviour.（模型的背景或理论基础）The dynamic behaviour of the structure is numerically approached by direct time integration of equation (1). A necessary condition to deal with pseudo-dynamic techniques demands that an implicit time integration algorithm must be used, which in this case is the Newmark method. In its usual version, as described for example by Bathe 24 or Adina Research and Development Incorporated 25, the primary variable to be iterated is the structure global displacement vector, which allows further determination of the velocity and acceleration vectors by numerical derivation. However, a Newmark version presenting slight differences was used in this analysis. Here, an updated structure state identified by the displacement vector referred to a time step t+Dt depends on the evaluation of the structure internal restoring force vector, experimentally measured in a previous time step t. The analysis may deal with linear or non-linear behaviour, as mentioned. （模型所采用方法的根据）In order to make it possible to couple the variables included in the numerical iteration algorithm with experimentally measured parameters, as described, an efficient version proposed by Liu 26 was used. The version considers at time step t the experimental evaluation of the structure internal restoring force vector at the level of degrees of freedom. This vector is given by the product KUt, where K is the global structure stiffness matrix and Ut is the displacement vector. The procedure holds for linear elastic behaviour or not, assuming that a tangent stiffness matrix could be defined between two consecutive time steps.（模型的详细描述）The iterative method used here deals with the acceleration as the iteration variable. Considering that viscous damping may be neglected in the majority of dynamic problems, as in the application under discussion here, the iteration scheme is set up as follows: (2)At each time step t+Dt it is possible to define the structure kinematic state from the previous iteration at instant t: (3a) (3b)Equation (2) can be written as the compact expression (4)where (5a)and (5b)where Rst is the internal restoring force vector (experimentally measured).（说明模型的重要特点）The iterative procedure starts once the initial conditions for displacement and velocity are defined. The acceleration at t = 0 may be calculated from . Therefore, the starting conditions for a pseudo-dynamic test need numerical definitions for the matrices and vectors included in equations (4) and (5). Once equation (5) is solved for the acceleration vector at time step t+Dt, the velocity and displacement vectors are calculated from equations (3a) and (3b). Also, it is necessary to solve KUt only for the first iteration of the method, in order to start running the process, defining the velocity and acceleration vectors for the next time step.（说明如何应用此模型）Once the degrees of freedom from the structure discretization are defined, the displacement a