科学计算科学计算 (37).pdf

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1、6.36.3 Application Application Examples Examples of of System System of of Linear EquationsLinear EquationsProblems of the force analysis of plane truss structuresProblems of the calculating asteroid orbits1 1Problems of the Problems of the F Force orce A Analysis of nalysis of P Plane lane T Truss

2、russ S StructurestructuresTruss is a common structure in engineering.A truss in which members are in the same plane is called a plane truss.A plane truss structure is composed of 13 member bars connected to 8 nodes.A specified load is applied to node B,E and G,and find the axial force on each member

3、 of the truss.For a statically balanced truss,its nodes must also be balanced,which means the sum of forces in the horizontal or vertical directions must be zero at any node.Therefore,we can list two independent equilibrium equations for each node,thus get the axial force of the members.For 8 nodes,

4、16 equations can be listed,and the number of equations is more than the number of unknown numbers,13.In order to make the truss statically determinate,which means to make the question have only one answer,we assume that nodeAis rigidly fixed in the horizontal and vertical directions,while node H is

5、rigidly fixed only in the vertical direction.Take node E as an example,its force is shown in the figure.Write the equilibrium equation:horizontal direction:f5cos45+f6=f9cos45+f10vertical direction:f5sin45+f7+f9sin45=15Define the argment ,then the equilibrium equation at point E can be obtained.=2/2N

6、ode B:f2=f6f3=10Node C:f1=f4+f5f1+f3+f5=0Node D:f4=f8f7=0Node E:f5+f6=f9+f10f5+f7+f9=15Node F:f10=f13f11=20Node G:f8+f9=f12f9+f11+f12=0Node H:f13+f12=0Node H is rigidly fixed only in the vertical direction,and only has equilibrium equation in horizontaldirection.This is a system of linear equations

7、containing 13unknowns.Vector f,the axial force of the member,can be easily calculated by MATLAB.alpha=sqrt(2)/2;A=0,1,0,0,0,-1,0,0,0,0,0,0,0;0,0,1,0,0,0,0,0,0,0,0,0,0;alpha,0,0,-1,-alpha,0,0,0,0,0,0,0,0;alpha,0,1,0,alpha,0,0,0,0,0,0,0,0;0,0,0,1,0,0,0,-1,0,0,0,0,0;0,0,0,0,0,0,1,0,0,0,0,0,0;0,0,0,0,al

8、pha,1,0,0,-alpha,-1,0,0,0;0,0,0,0,alpha,0,1,0,alpha,0,0,0,0;0,0,0,0,0,0,0,0,0,1,0,0,-1;0,0,0,0,0,0,0,0,0,0,1,0,0;0,0,0,0,0,0,0,1,alpha,0,0,-alpha,0;0,0,0,0,0,0,0,0,alpha,0,1,alpha,0;0,0,0,0,0,0,0,0,0,0,0,alpha,1;b=0;10;0;0;0;0;0;15;0;20;0;0;0;f=Ab;disp(f)-28.2843 20.0000 10.0000 -30.0000 14.1421 20.

9、0000 0 -30.0000 7.0711 25.0000 20.0000 -35.3553 25.0000Truss members mainly bear axial tension or pressure.When the member is pulled,the axial force is tension,which points away from the section.When the member is pressed,the axial force is pressure,which points towards the section.In the two answer

10、s,the positive number indicates tension,while the negative number indicates pressure.2 2Problems of the Problems of the C Calculating alculating A Asteroid steroid OOrbitsrbitsNow,five observations have been made on an asteroid at five different times,and the coordinate data of five points on the or

11、bit are listed in the following table.Its unit is astronomical measurement unit.Try to determine the orbit equation of the asteroid.i i1 12 23 34 45 5x xi i1.021.020.870.870.670.670.440.440.160.16y yi i0.390.390.270.270.180.180.130.130.130.13(1)Model analysisAccording to Keplers first law,the orbits

12、 of asteroids are ellipses:a1x2+2a2xy+a3y2+2a4x+2a5y+1=0The coefficients to be determined are ai(i=1,2,3,4,5).According to the known data,we might as well set the measured coordinates of the 5 points as(xi,yi)(i=1,2,3,4,5).Determining the coefficient ai equals to solving the following system of line

13、ar equations.11221131241511222223224252132233332435314224434244541522553524555a x2a x ya y2a x2a y10a x2a x ya y2a x2a y10a x2a x ya y2a x2a y10a x2a x ya y2a x2a y10a x2a x ya y2a x2a y10+=+=+=+=+=aaaaa121112112222222232333233424442445255525512345x2x yy2x2yx2x yy2x2yx2x yy2x2yx2x yy2x2yx2x yy2x2y11

14、111=Rewrite the equations like this:Substituting the data(xi,yi)(i=1,2,3,4,5)in the table into the coefficient matrix,then we can find ai(i=1,2,3,4,5),thus get the orbit equation of the asteroid.(2)Model solution Ax2x yy2x2yx2x yy2x2yx2x yy2x2yx2x yy2x2yx2x yy2x2y121112112222222232333233424442445255

15、5255=xi=1.02,0.87,0.67,0.44,0.16;yi=0.39,0.27,0.18,0.13,0.13;A=zeros(length(xi);for i=1:length(xi)A(i,:)=xi(i)*xi(i),2*xi(i)*yi(i),yi(i)*yi(i),2*xi(i),2*yi(i);endb=-ones(length(xi),1);ai=Ab ai=2.4645-0.44236.4917-0.6819-3.6008Draw the asteroids orbit:f=(x,y)2.4645*x.2-0.8846*x.*y+6.4917*y.2-1.3638*x-7.2016*y+1;h=ezplot(f,-0.5,1.2,0,1.2);Therefore,the orbital equation of the asteroid is:2.4645x2-0.8846xy+6.4917y2-1.3638x-7.2016y+1=0