Dynamics of Machines - Part II - IFS.pdf

1.6.4 Analytical and Numerical Solution of Equation of Motion using Matlab
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% DYNAMICS OF MACHINERY LECTURES (41035) %
% MEK - DEPARTMENT OF MECHANICAL ENGINEERING %
% DTU - TECHNICAL UNIVERSITY OF DENMARK %
% %
% Copenhagen, October 30th, 2003 %
% %
% IFS %
% %
% 1 D.O.F. SYSTEM - 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]
%Mass-Spring-Damping System Properties
L=0.610; %beam length
K= 3*E*Iz/L^3; %stiffness coefficient
M=m1+m2; %mass coefficient
xi=0.005; %damping factor [no-dimension]
D=2*xi*sqrt(K*M); %damping coefficient
wn=sqrt(K/M); %natural frequency [rad/s]
fn=wn/2/pi %natural frequency [Hz]
fnexp=0.875; %measured natural frequency [Hz]
dif=(fn-fnexp)/fnexp;%error between calculated
%and measured frequencies
%_____________________________________________________
%Initial Condition
y_ini= -0.000 % beam initial deflection [m]
v_ini= -0.000 % beam initial velocity [m/s]
freq_exc=0.95 % excitation frequency [Hz]
force=-0.100 % excitation force [N]
time_max=30.0; % integration time [s]
%_____________________________________________________
15
%_____________________________________________________
%EXACT SOLUTION % EQUATION (40)
n=600; % number of points for plotting
j=sqrt(-1); % complex number
w=2*pi*freq_exc; % excitation frequency [rad/s]
lambda1=-xi*wn+j*wn*sqrt(1-xi*xi);
lambda2=-xi*wn-j*wn*sqrt(1-xi*xi); AA=(force/M)/(wn*wn-w*w +
j*2*xi*wn*w); C1=(
lambda2*(y_ini-AA)-(v_ini-j*w*AA))/(lambda2-lambda1);
C2=(-lambda1*(y_ini-AA)+(v_ini-j*w*AA))/(lambda2-lambda1);
for i=1:n,
t(i)=(i-1)/n*time_max;
y_exact(i)=C1*exp(lambda1*t(i)) + ...
C2*exp(lambda2*t(i)) + ...
AA*exp(j*w*t(i));
end
%_____________________________________________________
%NUMERICAL SOLUTION % EQUATION (44)
% trying different time steps to observe convergence
% 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]
n_integ=time_max/deltaT; % number of points (integration)
% Initial Conditions
y_approx(1) = y_ini; % beam initial deflection [m]
yp_approx(1) = v_ini; % beam initial velocity [m/s]
for i=1:n_integ,
t_integ(i)=(i-1)*deltaT;
ypp_approx(i) =-(wn*wn)*y_approx(i) ...
-(2*xi*wn)*yp_approx(i) ...
+(force/M)*exp(j*w*t_integ(i));
yp_approx(i+1)= yp_approx(i) + ypp_approx(i)*deltaT;
y_approx(i+1) = y_approx(i) + yp_approx(i+1)*deltaT;
end
%_____________________________________________________
%Graphical Results
title(’Simulation of 1 D.O.F System in Time Domain’)
subplot(3,1,1), plot(t,real(y_exact),’b’)
title(’(a) Exact Solution - (b) Numerical Solution
(delta T = 0.01 s) - (c) Comparison’)
xlabel(’time [s]’)
ylabel(’(a) y(t) [m]’)
grid
subplot(3,1,2),
plot(t_integ(1:n_integ),real(y_approx(1:n_integ)),’r’)
xlabel(’time [s]’)
ylabel(’(b) y(t) [m]’)
grid
subplot(3,1,3), plot(t,real(y_exact),’b’,t_integ(1:n_integ),
real(y_approx(1:n_integ)),’r’)
xlabel(’time [s]’)
ylabel(’(c) y(t) [m]’)
grid