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1、,CHAPTER 7. Numerical Evaluation of Dynamic Responses,7.1 Time Stepping Methods,CHAPTER 7. Numerical Evaluation of Dynamic Responses,CHAPTER 7. Numerical Evaluation of Dynamic Responses,7.2 Methods Based on Interpolation of Excitation,CHAPTER 7. Numerical Evaluation of Dynamic Responses,CHAPTER 7. N
2、umerical Evaluation of Dynamic Responses,CHAPTER 7. Numerical Evaluation of Dynamic Responses,CHAPTER 7. Numerical Evaluation of Dynamic Responses,CHAPTER 7. Numerical Evaluation of Dynamic Responses,7.3 Central Deference Method,CHAPTER 7. Numerical Evaluation of Dynamic Responses,Start the computat
3、ion:,CHAPTER 7. Numerical Evaluation of Dynamic Responses,CHAPTER 7. Numerical Evaluation of Dynamic Responses,CHAPTER 7. Numerical Evaluation of Dynamic Responses,7.4 Newmarks Method,CHAPTER 7. Numerical Evaluation of Dynamic Responses,Special Cases,CHAPTER 7. Numerical Evaluation of Dynamic Respon
4、ses,Non-iterative Formulation,Incremental quantities:,CHAPTER 7. Numerical Evaluation of Dynamic Responses,CHAPTER 7. Numerical Evaluation of Dynamic Responses,This equation is needed to obtain to start the computation,In Newmarks method, equilibrium condition may also be satisfied at step of (i+1).
5、 Such methods are called implicit methods.,CHAPTER 7. Numerical Evaluation of Dynamic Responses,CHAPTER 7. Numerical Evaluation of Dynamic Responses,CHAPTER 7. Numerical Evaluation of Dynamic Responses,7.5 Stability and Computational Error,CHAPTER 7. Numerical Evaluation of Dynamic Responses,CHAPTER
6、 7. Numerical Evaluation of Dynamic Responses,CHAPTER 7. Numerical Evaluation of Dynamic Responses,7.6 Nonlinear Response Analysis : Central Difference Method,CHAPTER 7. Numerical Evaluation of Dynamic Responses,7.7 Nonlinear Response Analysis : Newmarks Method,Newmarks method for linear system may
7、be extended to nonlinear systems,This is a incremental linear equation, non-iterative formulation for linear systems may also be used for nonlinear systems. But it is preferable to obtain by the follow equation,CHAPTER 7. Numerical Evaluation of Dynamic Responses,The above procedure with constant ti
8、me step can lead to unacceptably inaccurate results. Significant errors arose for two reasons: the tangent stiffness was used instead of the secant stiffness, and use of a constant time step delays detection of the transitions in the force-deformation relationship.,CHAPTER 7. Numerical Evaluation of
9、 Dynamic Responses,The above procedure with constant time step can lead to unacceptably inaccurate results. Significant errors arose for two reasons: the tangent stiffness was used instead of the secant stiffness, and use of a constant time step delays detection of the transitions in the force-defor
10、mation relationship.,First , The second source of errors,The error can be minimized using an iterative procedure.,CHAPTER 7. Numerical Evaluation of Dynamic Responses,Iterative Procedure,CHAPTER 7. Numerical Evaluation of Dynamic Responses,This additional displacement is used to find a new value of
11、the residual force, and the process is continued until convergence is achieved. The iterative process for step i to i+1 summarized in the following table is known as Newton Raphson method.,CHAPTER 7. Numerical Evaluation of Dynamic Responses,CHAPTER 7. Numerical Evaluation of Dynamic Responses,Newmarks Method: Nonlinear Systems,CHAPTER 7. Numerical Evaluation of Dynamic Responses,CHAPTER 7. Numerical Evaluation of Dynamic Responses,CHAPTER 7. Numerical Evaluation of Dynamic Responses,CHAPTER 7. Numerical Evaluation of Dynamic Responses,CHAPTER 7. Numerical Evaluation of Dynamic Responses,