chap2-微分方程模型-基本理论+机械系统建模.ppt
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1、Differential Equation Models东华大学信息学院自动化系 陈亮 office:信息学院楼:信息学院楼522室室2014-12By L. Chen2建模概述建模概述 机理建模机理建模 Analytical reasoning: the ability to write down mathematical expressions that represent a system; Computational skills: the ability to use a computer to either evaluate or predict characteristics o
2、f a model that are too challenging to solve analytically; (Physical) knowledge: crucial ability to think critically about analysis and computations in the context of real (, physical) knowledge about the world. 实验建模实验建模 System Identification Data ScienceBy L. Chen3Everything is the Same Modeling Eng
3、ineered SystemsSource: www.coursera.orgBy L. Chen4By L. Chen5By L. Chen6Spring-damper-mass mechanical system Idea component: Whenever the spring is in extension, its position xs is defined to be positive; Whenever a damper is in extension, its velocity vd is defined to be positive. The acceleration
4、of a mass is defined to be positive to the right; the external forces are always positive if they are acting to the right. The only allowed motion is horizontal motion. By L. Chen7Case1: without mass Ordinary differential equation (ODE) Linear, constant coefficient, first ordersdxv( )( )dxxf xdt0ext
5、for F0extfor F减震器、弹力绳E.g.By L. Chen8Numerical Solution - Euler Integration( )xf x Euler integration is the easiest method for numericaly approximating a solution to an ODE, but there are certainly other options; Euler integration doesnt do a very good job, but if you make dt small enough, you will a
6、lways end up approximating the limit well enough.By L. Chen9Example:By L. Chen10By L. Chen11Analytic Solution - Exponential solutionsFor linear, constant coefficient, first order ODEs:( )xf xh = ?E.g.Plot By L. Chen12By L. Chen13 If I extend the spring-damper system by 0.5 units, then I will see it
7、move forever, which of course a real system will never do. This is largely because damping tends to be nonlinear at very low velocity - suggesting we can never fully believe our analytical models or computational models - but the solution will capture the behavior of many spring-damper systems.物理解释:
8、By L. Chen14欧拉积分欧拉积分-评注评注 Euler integration allows us to approximate solutions to ODEs whether they are linear or not so long as dt is small enough. Euler integration requires having an initial condition to get started.By L. Chen15Superposition The principle of superposition: for the linear system.
9、Superposition: if I give you two solutions regardless of how I obtained them you can get a whole bunch of potential solutions from those two a really powerful idea. By L. Chen16叠加原理叠加原理-解释解释 Example: The principle of superposition: if x1(t) and x2(t) are both solutions to the same linear ,constant c
10、oefficient, first order ODE, then: is a solution, where and are constant numbers, even can be imaginary numbers.By L. Chen17叠加原理叠加原理-验证验证E.g.E.g.By L. Chen18 Why would you need to do this? You might have already calculated the solution for one initial condition, and then someone tells you that the o
11、riginal initial condition was wrong and you need to do it again for a new initial condition. For a simple system, you would probably just calculate the whole solution again, but as systems get more complex - if there were thousands or millions of spring-damper components all connected together - rec
12、omputing the solution might take a very long time.By L. Chen19Case 2: with mass Second-order ODE F are the forces acting on a point, defined to be positive if they are acting to the right; The acceleration a is the first time derivative of vm of the point; vm of the mass is measured relative to the
13、left wall.Shocks on a car or bicycle符号说明:符号说明:By L. Chen20To use Euler integration and the analytic exponential solutions, we need to convert this system to a first-order ODE.E.g.?x By L. Chen21Case 3: with several masses 12xv22va44va( )xxfBy L. Chen22 The first-order ODE with four states. The numbe
14、r of first-order equations you get will go up with the number of masses. Typically, you should expect to have two equations for every mass. The total distance between walls, L, is playing the role of an external force in the equation, and this is quite common any time there is a constraint between v
15、ariables.Same k and same m =1几点说明:几点说明:By L. Chen23About oscillation When you have mass you should expect the possibility of oscillation.1123( )cos2tx tcet( )xf xSolution e.g.Same to sin()By L. Chen24How to get oscillation Exponential solutions for linear, constant-coefficient, first-order ODEs How
16、do we get an exponential function to give us cos() and sin() ? 123( )cos2tx tcet0( )ktbx texBy L. Chen25Imaginary numbers and Eulers formula Taylor series expansion: cos()sin()htettEulers formula:By L. Chen26 Eulers formula : Example: the spring-mass system( )htx tce( )kjthtmx tcece( )cossinkkx tctj
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