Matlab 3-Higher order ODE (1)
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Name: MECH 2220 Matlab Lab 3, Fall 2020
Matlab Lab 3: Numerical Solutions to Higher Order ODE’s
Assigned: 11/02/ 2020
Due: 2 weeks from the lab attended
Problem Statement #1:
1.
Write an uwf that will solve a system of first order ordinary differential numerically using
Euler’s method. Code for problems:
%Matlab Lab 3: Higher Order
%Enoch Crittenden
%Number 1 Code
function out = EulerODE(slope,x_initial,dx,x_final,y_initial)
x(1) = x_initial;
G(1,:) = y_initial;
n = 1;
while x(n) < x_final
G(n+1,:) = G(n,:) + slope(x(n),G(n,:))*dx;
x(n+1) = x(n) + dx;
n=n+1;
end
out = [x',G];
Name: MECH 2220 Matlab Lab 3, Fall 2020
2.
Find the analytic solution to the following third order ode. Transform the ode into a system
of three first order equations. Solve the system with the uwf developed in task one. Solve the
system with the built in function ode45. Compare the three solutions. '''
cos( ); ''(0)
1; '(0)
0; (0)
1
y
x
y
y
y
Code for problems:
%Matlab Lab 3: Higher Order
%Enoch Crittenden
clc
clear all
slope = @(x,G)[cos(x),G(1),G(2)]; %Slope form
x_initial = 0; %initial x value
x_final = pi; %final x value
dx = (pi/20); %derivative of value
IV = [1,0,-1];
fprintf(
'\t\tX\t\tY''''\t\tY''\t\tY\n'
)
Matrix = EulerODE(slope,x_initial,dx,x_final,IV) %Matrix derivation
x = Matrix(:,1);y = Matrix(:,4);
plot(x,y) %Plot
hold on
y_analytical = -sin(x)+(x.^2)/2 + x-1;
plot(x,y_analytical, 'r+'
) %Plot
%Now the ode45 method
slope2 = @(x,G) [cos(x),G(1),G(2)]';
[x,GV] = ode45(slope2,[x_initial:dx:x_final],IV);
y_ode=GV(:,3);
plot(x, y_ode,
'k.'
) %plot for graph
xlabel(
'x'
)
ylabel(
'y'
)
legend(
'Euler Method'
,
'Analytical'
,
'ode45'
)
Name: MECH 2220 Matlab Lab 3, Fall 2020
Solution for problems:
3.
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Name: MECH 2220 Matlab Lab 3, Fall 2020
4.
Do problem 22.15 in your text using the function developed in #1 above. Repeat using
ode45. The motion of a damped spring-mass system (Fig. P22.15) is described by the following
ordinary differential equation:
m
d
2
x
dt
2
+
c
dx
dt
+
kx
=
0
where x
= displacement from equilibrium position (m), t
=time (s), m
= 20-kg mass, and c
=
the damping coefficient (N · s/m). The damping coefficient c
takes on three values of 5
(underdamped), 40 (critically damped), and 200 (over damped). The spring constant k
= 20
N/m. The initial velocity is zero, and the initial displacement x
= 1 m. Solve this equation
using a numerical method over the time period 0 ≤ t
≤ 15 s. Plot the displacement versus time
for each of the three values of the damping coefficient on the same plot.
Code for problems:
%Matlab Lab 3: Higher Order
%Enoch Crittenden
%Number 3 Code
clc
clear all
%Initial values for formula
%time in seconds
t_initial = 0;
t_final = 15;
dt = .05;
c = 200; %Damped value (change for each value)
k = 20; %N/m
m = 20; %kg(s)
%Slope forumula 1
slope = @(t,G)[(-c*G(1) - k*G(2))/m, G(1)];
IV = [0,1];
Name: MECH 2220 Matlab Lab 3, Fall 2020
Matrix = EulerODE(slope,t_initial,dt,t_final,IV);
t = Matrix(:,1);
x = Matrix(:,3);
plot(t,x)
hold on
%Slope formula 2
slope_2 = @(x,G)[(-c*G(1) - k*G(2))/m, G(1)]';
[t2,G] = ode45(slope_2,[t_initial:dt:t_final],IV);
ode = [t2,G];
t1 = ode(:,1);x2 = ode(:,3);
%Plotting the line
plot(t1,x2,
'r.'
);
xlabel(
'Time (seconds)'
);
ylabel(
'Displacement (meters)'
);
legend(
'Euler'
,
'ode45'
);
Solution for problems:
Name: MECH 2220 Matlab Lab 3, Fall 2020
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Name: MECH 2220 Matlab Lab 3, Fall 2020
Solution for problems:
5.
Dragster: An application of ODE solvers
After reaching a top speed of 250 mph a 3000 lbm dragster deploys the semispherical
parachute and engages wheel brakes to stop the car. The diameter of the parachute is 10 feet,
and the wheel brakes can result in a constant -2g acceleration (the wheel brake force is 2
times the weight of the car).
The force exerted by the parachute may be approximated by
Name: MECH 2220 Matlab Lab 3, Fall 2020
The drag coefficient for the parachute has been experimentally determined to be 1.4.
The governing differential equation is then developed with Newton’s Second Law.
Write this second order equation as a system of first order equations assuming the following
units:
Use MATLAB to determine the following questions.
A.
How long and how many feet does it take to stop the car?
B.
Create formatted output that tabulates time, position, velocity, and parachute force.
C.
Compute the total work performed by the parachute on the car.
Code for problems:
%Matlab Lab 3: Higher Order
%Enoch Crittenden
%Number 4 Code
clc
Name: MECH 2220 Matlab Lab 3, Fall 2020
clear all
%Given values and formula
t_initial = 0; %seconds
t_final = 3.5; %seconds
dt = 0.01; %derivative (seconds)
slopeode = @(t,G)[-((4.123*(G(1)^2))/3000) - 64.4, G(1)]';
IV = [366.67,0];
%Time Matrix
[time,Matrix] = ode45(slopeode,[t_initial:dt:t_final],IV);
velocity = Matrix(:,1);
distance = Matrix(:,2);
p_force = 4.123*velocity.^2;
fprintf(
'\ttime\t\tV\t\t\tD\t\tF'
)
T = [time,Matrix,p_force]
%Graph Formation
plot(distance,velocity,
'r.'
) %Answer for A
xlabel(
'distance(ft)'
);
ylabel(
'velocity'
);
fprintf(
'%6.2f,T'
); %Answer for B
total_work = trapz(distance,p_force) %Part C Answer
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Name: MECH 2220 Matlab Lab 3, Fall 2020
Solution for problems:
Name: MECH 2220 Matlab Lab 3, Fall 2020
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