Question

Consider the generic homogeneous spring-mass system

my″+γy′+ky=0my″+γy′+ky=0

Use either Euler’s Method or ode45 to investigate how the solution of the second order

equation

y

00

+ γy0 + 2y = 0

depends on the damping coefficient γ. Include nicely labeled plots y(t) vs. t for each

choice of γ in (a)-(d) below (either on the same or separate axes). Use the initial

conditions y(0) = 0.1, y0(0) = 0 and step size ∆t = 0.001. Write a few sentences

describing how the behavior of y(t) depends on γ and comment on which choice of γ

best matches the behavior we observed in the real live spring-mass system.

(a) γ = 0 (the case of an undamped or harmonic oscillator)

(b) γ = 0.1.

(c) γ = 1.

(d) γ = 3.

Answer 1

function dummy=ex_with_2eqs()
clc;
clear all;
t0 =0; tf = 4;

x0=0.1;
y0=0;

  
u0 = [x0;y0];% initial condtion
gama=0;
[t,y] = ode45(@f,[t0 tf],u0,[],gama);
u1 = y(:,1);
gama1=0.1;
[t,y1] = ode45(@f,[t0 tf],u0,[],gama1);
u2 = y1(:,1);
gama2=1;
[t,y2] = ode45(@f,[t0 tf],u0,[],gama2);
u3 = y2(:,1);
gama3=3;
[t,y3] = ode45(@f,[t0 tf],u0,[],gama3);
u4 = y3(:,1);

%%% Solution Plot
figure(1);
plot(t,u1,'b');
hold on
plot(t,u2,'r');
hold on
plot(t,u3,'m');
hold on
plot(t,u4,'k');
xlabel('t')
ylabel('y')
legend('\gamma=0','\gamma=0.1','\gamma=1','\gamma=3')

end
%----------------------------------------------------------------------
function dydt = f(t,y,gama)

k=1;
m=1;
  
%%%Here u is y(1), v is y(2),   
u = y(1); v = y(2);
dydt = [v ;-(k/m)*u-(gama/m)*v];
end