Dynamics of Machines - Part II - IFS.pdf

1.7.5 Analytical and Numerical Solutions of Equation of Motion using Matlab
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% DYNAMICS OF MACHINERY LECTURES (72213) %
% MEK - DEPARTMENT OF MECHANICAL ENGINEERING %
% DTU - TECHNICAL UNIVERSITY OF DENMARK %
% %
% Copenhagen, February 11th, 2000 %
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% IFS %
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% 2 D.O.F - EXACT AND NUMERICAL SOLUTION %
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clear all;
close all;
%Concentred Masses
m1= 0.191; %[Kg]
m2= 0.191; %[Kg]
m3= 0.191; %[Kg]
m4= 0.191; %[Kg]
m5= 0.191; %[Kg]
m6= 0.191; %[Kg]
%Elastic Properties of the Beam of 600 [mm]
E = 2e11; %elasticity modulus [N/m^2]
b = 0.030 ; %width [m]
h = 0.0012 ; %thickness [m]
Iz= (b*h^3)/12; %area moment of inertia [m^4]
% (1.CASE) Data for the mass-spring system
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M1=m1+m2; %concentrated mass [Kg] |
M2=m3+m4; %concentrated mass [Kg] |
L1= 0.310; %length for positioning M1 [m] |
L2= 0.610; %length for positioning M2 [m] |
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% Coefficients of the Stiffness Matrix
LL=(L1-4*L2)*(L1-L2)^2;
K11= -12*(E*Iz/LL)*L2^3/L1^3; % Stiffness coeff.[N/m]
K12= -6*(E*Iz/LL)*(L1-3*L2)/L1; % Stiffness coeff.[N/m]
K21= -6*(E*Iz/LL)*(L1-3*L2)/L1; % Stiffness coeff.[N/m]
K22= -12*(E*Iz/LL); % Stiffness coeff.[N/m]
% Coefficients of the Damping Matrix
% (damping factor xi=0.005)
D11= 2*0.005*sqrt(M1/K11); % Damping coeff.[Ns/m]
D12= 0; % Damping coeff.[Ns/m]
D21= 0; % Damping coeff.[Ns/m]
D22= 2*0.005*sqrt(M2/K22); % Damping coeff.[Ns/m]
%Mass Matrix
M= [M1 0; 0 M2];
%Damping Matrix
D=[D11 D12; D21 D22];
%Stiffness Matrix
K= [K11 K12; K21 K22];
%State Matrices A & B % EQUATION (52)
A= [ M D ;
zeros(size(M)) M ] ;
B= [ zeros(size(M)) K ;
-M zeros(size(M))];
%Dynamical Properties of the Mass-Spring System
[u,w]=eig(-B,A); %eigenvectors u
%eigenvalues w
w1=abs(imag(w(3,3)))/2/pi; %first natural freq.[Hz]
w2=abs(imag(w(1,1)))/2/pi; %second natural freq.[Hz]
wexp1=0.78125; %measured natural freq.[Hz]
wexp2=5.563; %measured natural freq.[Hz]
dif1=(w1-wexp1)/wexp1; %error calculated and measured freq.
dif2=(w2-wexp2)/wexp2; %error calculated and measured freq.
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%Initial Condition
y1_ini = -0.000 % beam initial deflection [m]
y2_ini = -0.000 % beam initial deflection [m]
v1_ini = -0.001 % beam initial velocity [m/s]
v2_ini = -0.000 % beam initial velocity [m/s]
freq_exc = 0.000 % excitation frequency [Hz]
force1 = -0.000 % excitation force 1 [N]
force2 = -0.000 % excitation force 2 [N]
time_max = 30.0; % integration time [s]
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40
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%EXACT SOLUTION % EQUATION (68)
n=2000; % number of points for plotting
j=sqrt(-1); % complex number
w_exc=2*pi*freq_exc; % excitation frequency [rad/s]
z_ini = [v1_ini v2_ini y1_ini y2_ini]’;
force_exc = [force1 force2 0 0 ]’;
vec_aux = z_ini - inv((j*w_exc*A + B))*force_exc;
lambda1=w(1,1);
lambda2=w(2,2);
lambda3=w(3,3);
lambda4=w(4,4);
u1=u(1:4,1);
u2=u(1:4,2);
u3=u(1:4,3);
u4=u(1:4,4);
C=inv(u)*(vec_aux);
c1=C(1);
c2=C(2);
c3=C(3);
c4=C(4);
for i=1:n,
t(i)=(i-1)/n*time_max;
y_exact=c1*u1*exp(lambda1*t(i)) + ...
c2*u2*exp(lambda2*t(i)) + ...
c3*u3*exp(lambda3*t(i)) + ...
c4*u4*exp(lambda4*t(i)) + ...
inv((j*w_exc*A + B))*force_exc*exp(j*w_exc*t(i));
end
y1_exact(i) = y_exact(3);
y2_exact(i) = y_exact(4);
figure(1)
title(’Simulation of 2 D.O.F System in Time Domain’)
subplot(2,1,1), plot(t,real(y1_exact),’b’)
title(’Exact Solution ’)
xlabel(’time [s]’)
ylabel(’ y1_{exact}(t) [m]’)
grid
subplot(2,1,2), plot(t,real(y2_exact),’b’)
xlabel(’time [s]’)
ylabel(’y2_{exact}(t) [m]’)
grid
pause;
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%NUMERICAL SOLUTION % EQUATION (73)
% deltaT=0.3605; % time step [s]
% deltaT=0.3; % time step [s]
% deltaT=0.1; % time step [s]
% deltaT=0.05; % time step [s]
% deltaT=0.01; % time step [s]
deltaT=0.005; % time step [s]
n_integ=time_max/deltaT; % number of points (integration)
% Initial Conditions
y1_approx(1) = y1_ini; % beam initial deflection [m]
y2_approx(1) = y2_ini; % beam initial deflection [m]
yp1_approx(1) = v1_ini; % beam initial velocity [m/s]
yp2_approx(1) = v2_ini; % beam initial velocity [m/s]
for i=1:n_integ,
t_integ(i)=(i-1)*deltaT;
ypp1_approx(i)=-1/M1*(K11*y1_approx(i)+K12*y2_approx(i) ...
+ D11*yp1_approx(i)+D12*yp2_approx(i) ...
-(force1)*exp(j*w_exc*t_integ(i)));
ypp2_approx(i)=-1/M2*(K21*y1_approx(i)+K22*y2_approx(i) ...
+D21*yp1_approx(i)+D22*yp2_approx(i) ...
-(force2)*exp(j*w_exc*t_integ(i)));
yp1_approx(i+1)=yp1_approx(i) + ypp1_approx(i)*deltaT;
yp2_approx(i+1)=yp2_approx(i) + ypp2_approx(i)*deltaT;
y1_approx(i+1)=y1_approx(i)+yp1_approx(i+1)*deltaT;
y2_approx(i+1)=y2_approx(i)+yp2_approx(i+1)*deltaT;
end
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