Question

Using Octave to solve (preferably with solving the differential equations and go through the process)

1. A harmonic oscillator obeys the equation dx dt dt which can be written as a set of coupled first order differential equations dx dt dt One procedure in Octave for coding these equations involves a global statement and the line solutionRC Isode(@dampedOscillator, [1, 0], timesR); Employ the help system to determine the properties of the Isode() function (or an equivalent solver such as ode23() in MATLAB, which has a different structure and is called with different arguments, however). Be careful to graph just the function r and not both x and the velocity v (recall that the iterator: can be employed as a matrix index) The three types of damped motion are termed underdamped, critically damped and overdamped. To illustrate the behavior in the three cases, set m=k=1 and run your program for values of equal to 0, 1, 2 and 4 with initial condition x(0)= 1, v(0)=0. The case of 7-2 is termed "critically damped", do you see why? This is the value employed in for example shock absorbers or door stops.

Answer 1

Given the equation

\frac{dx}{dt}=v

And

\frac{dv}{dt}=- \gamma v-\frac{k}{m}x........................(i)

Let

x_1=x,x_2= \frac{dx}{dt}

It follows x_2= \dot x_1

Therefore equation (i) can be written as

\dot x_2=- \gamma x_2-\frac{k}{m}x_1

For m=k=1 , the above equation reduces to

\dot x_2=- \gamma x_2-x_1

given the initial conditions x(0)=1, v(0)=0

Now we write a function file [ & save the file with the name 'odeshm' ]for the above equation, taking the value \gamma =1 . [ We can modify \gamma later for different values]

%%%%MATLAB CODE%%%%%%%

function f = odeshm(t,x)

gamm1=1;

f = zeros(2,1);
f(1) = x(2);
f(2) = -gamma*x(1)-x(2);

%%%%%% NOW IN SEPARATE SCRIPT FILE WRITE THE FOLLOWING CODE AND RUN IT%%%%%

clear; close all;
tspan = 0:0.05:10; % time span
x0 = [1;0]; % initial conditions
[t,x] = ode23('odeshm',tspan,x0);
plot(t,x(:,1));
xlabel('Time [s]')
ylabel('Displacement [m]')
title(' gamma =1');

gamma =1 1.2 0.8 0.6 0.4 0.2 -0.2 0 12345 6 78910 Time [s]

gammaO 1.8 1.4 드1.2 0.6 0.4 0.2 0 12345 6 78910 Time [s]

gamma 2 0.8 0.6 는 04 0.2 -0.2 0.4 0 12345 6 78910 Time [s]

gamma 4 0.5 0.5 0 12345 6 78910 Time [s]