B
Ingenieurmathematik Prüfung 2
9.Dez.2010
Zeit 90 Minuten, Reihenfolge beliebig, 8 Punkte pro Hauptaufgabe,
40 Pt. = N.6.
en
und dkrit
an für die Hesse'sche Normalform der zur x-z-Ebene
parallelen Ebene y = 2
.
en = [0 1 0], dkrit = 2
z = exp(i*18*pi/2) = exp(i*9*pi) = exp(i*pi) = -1
u = [q ; 4 ; -2]
,
und v = [q ; 1 ; 4]
zueinander orthogonal sind.
q*q + 4 - 8 = 0 ; q = 2
A = [-4 -4 0]' B = [4 -4 0]' S = [0 0 7]' SA = A - S SB = B - S w = acosd(SA'*SB/(norm(SA)*norm(SB))) %SA = -4 -4 -7 %SB = 4 -4 -7 %w = 52.7756
xc = 5, yc = 0, r = 5 w0 = pi/2 gh = 4/2 n = 2 t = 0:pi/200:n*2*pi; x = r*cos(t+w0) + xc; y = r*sin(t+w0) + yc; z = gh*t/(2*pi); plot3(x,y,z) hold on axis equal PSE = [5 5 5; 5 5 5; 0 2 4] plot3(PSE(1,:),PSE(2,:), PSE(3,:),'ko') PM = [5 5 ; -5 -5 ; 1 3] plot3(PM(1,:),PM(2,:), PM(3,:),'ko') P1 = [0 0 0.5]' plot3(P1(1),P1(2), P1(3),'ro') P2 = [0 0 2.5]' plot3(P2(1),P2(2), P2(3),'ro') plot3([0 0],[0 0],[ 0 4],'r') box on view(22,25) hold off
A = [0 0 0]' , B = [-4 0 0]' C = [-4 3 0]' , D = [0 3 0]' E = [0 0 3.2]' , F = [-4 0 3.2]' G = [-4 3 3.2]' , H = [0 3 3.2]' u = G - B v = D - B N = cross(u,v) lN = norm(N) en1 = N/lN dkrit1 = en1'*B % Distanztests A H F dA = en1'*A-dkrit1 dF = en1'*F-dkrit1 dH = en1'*H-dkrit1 %u = 0 -3.0000 3.2000 %v = -4 -3 0 %N = -9.6000 12.8000 -12.0000 %lN = 20 %en1 = -0.4800 0.6400 -0.6000 %dkrit1 = 1.9200 %dA = -1.9200 %dF = -1.9200 %dH = -1.9200 en2 = en1 dkrit2 = en2'*A % = 0
Koi = [4 6 6 4 4; 0 0 6 6 0; 1 1 1 1 1] Trax = [1 0 -5; 0 1 0 ; 0 0 1] Mx = [-1 0 0; 0 1 0; 0 0 1] Trbx = [1 0 5; 0 1 0 ; 0 0 1] Tray = [1 0 0; 0 1 -3 ; 0 0 1] My = [1 0 0; 0 -1 0; 0 0 1] Trby = [1 0 0; 0 1 3 ; 0 0 1] Ttot = Trby * My * Tray * Trbx * Mx * Trax Kof = Ttot*Koi