Question

Use matlab for the following:

Frequency Response of a mass-spring-dashpot system Consider a mass-spring-dashpot system driven by a unit amplitude harmonic input mdx/dt+ cdx/dt + kx- Sin (wt) Use Matlab to simulate time response for ten well-chosen values of w for 3 different values of dimensionless damping factor : 0, between 0 and 1, larger than 1. Record and plot the steady state values of amplitude.

Answer 1

MATLAB Script:

%%
clear
clc

%% Parameters
global m c k omega

m = 1;
k = 1;
zeta = [0 0.5 1.5];

omega_n = sqrt(k/m);

omega_vals = [0.1 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25].*omega_n;

%% Simulating all cases

for i=1:length(zeta)
c = zeta(i)*2*sqrt(m*k);
for j=1:length(omega_vals)
omega = omega_vals(j);
  
[t,X] = ode45('eom',[0 100],[0; 0]);
  
SS(i,j) = X(end,1);
  
end
end

%% Plot of Steady State Value vs Omega
plot(omega_vals,abs(SS(1,:)))
hold on
plot(omega_vals,abs(SS(2,:)))
hold on
plot(omega_vals,abs(SS(3,:)))

xlabel 'Omega (rad/s)'
ylabel 'Magnitude of Steady State Value'

legend 'zeta = 0' 'zeta = 0.5' 'zeta = 1.5'
grid on

Function 'eom' (save as new script)

function X_dot = eom (t,x0)

global m c k omega
  
x1 = x0(1);
x2 = x0(2);
  
y = sin(omega*t);
  
x1_d = x2;
x2_d = -(k/m)*x1 -(c/m)*x2 + y;
  
X_dot = [x1_d;x2_d];
  
end

Graph of Steady State value vs Omega

45 -zeta = 0 40 zeta = 0.5 zeta = 1.5 6 S 35 30 25 S 10 0 0.5 1.5 2.5 3 2 0 9 Omega (rad/s)