I want to plot n vs theta1 from the function tan(n*theta1) + tan(n*theta2) + tan(n*thetas)= 0 in MATLAB; where n, theta1 and theta2 are variables and thetas is given. Can anyone give any pointers how to do so ?
I have tried using ezplot. It didn't work.
Next, I used this bit of code,
c = 1;
thetas = pi/4;
for n = 1.01:0.01:3.50
for theta1 = pi/2 : 0.01 : 5*pi/6
for theta2 = 0: 0.01: pi/2
if(tan(n*theta1) + tan(n*theta2) + tan(n*thetas) >= -0.0001 && tan(n*theta1) + tan(n*theta2) + tan(n*thetas) <= 0.0001 && (n*theta1) ~= pi/2 ...
&& (n*theta2) ~= pi/2 && (n*thetas) ~= pi/2)
ns(c) = n;
theta1s(c) = theta1;
theta2s(c) = theta2;
c = c + 1;
end
end
end
end
And I tried to plot the result using the plot command. Didn't work out either. Mostly the plot should come as three different curves for every given value of thetas.
Can anyone help ?
Something like this might work? -
%%// Data (Random numbers)
thetas_array = [2 4 6]; %// Put different thetas here
theta2 = 2:2:20;
n = 1:10;
figure,
for k = 1:numel(thetas_array)
thetas = thetas_array(k);
%%// Since tan(n*theta1) + tan(n*theta2) + tan(n*thetas)= 0;
theta1 = (1./n).*atan(- tan(n.*theta2) - tan(n.*thetas));
subplot(3,1,k), plot(n,theta1), xlabel('n'), ylabel(strcat('theta1 [thetas = ',num2str(thetas),']'));
end
Output
Related
I have a code that I would like to add an additional plot inside. I want to plot the relative error of two of my functions. I am unsure where in the loop to place this so that it doesn't give me an error message and plots all three cases as single graphs. Also, my relative error code may have some mistakes, which is what may be causing the problem.
This is the relative error code:
rel_error = (y_exact1 - Y(:,2)')./y_exact; %relative error
figure()
plot(T,rel_error,'r')
This is the function I need to add it into
function ivp1()
clear;clc;close all;
t=linspace(0,2.5);
K=[.02 .1 1.5];
for i=1:3
k =K(i);
[T,Y] = ode45(#prblm1_fun,t,0); %Solving for the approximate solution to the IVP
figure()
plot(T,Y)
hold on
y_exact1 = 1/(k^2+pi^2)*(pi*exp(k*t)-pi*cos(pi*t)-k*sin(pi*t));
y_exact2 = 1/(2*k)*(exp(k*(t-1))-1) + pi/(k^2+pi^2)*(exp(k*t) + exp(k* (t-1)));
y_exact3 = 1/2/k*(exp(k*(t-1))-exp(k*(t-2))) + pi/(k^2+pi^2)*(exp(k*t) + exp(k*(t-1))) + 1/2/(k-1)*(exp(k*(t-2)) - exp(t-2));
for i=1:length(t)
if t(i)<1
plot(t(i),y_exact1(i),'mo')
hold on
elseif t(i)<2
plot(t(i),y_exact2(i),'mo')
hold on
else
plot (t(i),y_exact3(i),'mo')
hold on
end
end
end
function dy= prblm1_fun(t,y) %This is the function of the IVP for varying values of t
dy = zeros(1,1);
if t < 1
dy(1)= y(1)*k + sin(pi*t);
elseif t < 2
dy(1)= y(1)*k + 0.5;
else
dy(1)= y(1)*k + exp(t-2)/2;
end
end
end
This is the desired result for one of the k values:
Two erroneous places:
1) Y is a 100x1 matrix from the result of ode45. Trying to access Y(:,2) leads into a out-of-boundary error.
2) y_exact is not defined. You only have y_exact1, y_exact2 and y_exact3.
Sample code:
function ivp1()
clear;clc;close all;
t=linspace(0,2.5);
K=[.02 .1 1.5];
for i=1:3
k =K(i);
[T,Y] = ode45(#prblm1_fun,t,0); %Solving for the approximate solution to the IVP
figure()
plot(T,Y)
hold on
y_exact1 = 1/(k^2+pi^2)*(pi*exp(k*t)-pi*cos(pi*t)-k*sin(pi*t));
y_exact2 = 1/(2*k)*(exp(k*(t-1))-1) + pi/(k^2+pi^2)*(exp(k*t) + exp(k* (t-1)));
y_exact3 = 1/2/k*(exp(k*(t-1))-exp(k*(t-2))) + pi/(k^2+pi^2)*(exp(k*t) + exp(k*(t-1))) + 1/2/(k-1)*(exp(k*(t-2)) - exp(t-2));
for j=1:length(t)
if t(j)<1
y_exact = y_exact1(j);
elseif t(j)<2
y_exact = y_exact2(j);
else
y_exact = y_exact3(j);
end
plot(t(j),y_exact,'mo')
end
rel_error = (y_exact1 - Y(:,1)')./y_exact; %relative error
plot(T,rel_error,'r')
end
function dy= prblm1_fun(t,y) %This is the function of the IVP for varying values of t
dy = zeros(1,1);
if t < 1
dy(1)= y(1)*k + sin(pi*t);
elseif t < 2
dy(1)= y(1)*k + 0.5;
else
dy(1)= y(1)*k + exp(t-2)/2;
end
end
end
Sample output:
Introduction
I am using Matlab to simulate some dynamic systems through numerically solving systems of Second Order Ordinary Differential Equations using ODE45. I found a great tutorial from Mathworks (link for tutorial at end) on how to do this.
In the tutorial the system of equations is explicit in x and y as shown below:
x''=-D(y) * x' * sqrt(x'^2 + y'^2)
y''=-D(y) * y' * sqrt(x'^2 + y'^2) + g(y)
Both equations above have form y'' = f(x, x', y, y')
Question
However, I am coming across systems of equations where the variables can not be solved for explicitly as shown in the example. For example one of the systems has the following set of 3 second order ordinary differential equations:
y double prime equation
y'' - .5*L*(x''*sin(x) + x'^2*cos(x) + (k/m)*y - g = 0
x double prime equation
.33*L^2*x'' - .5*L*y''sin(x) - .33*L^2*C*cos(x) + .5*g*L*sin(x) = 0
A single prime is first derivative
A double prime is second derivative
L, g, m, k, and C are given parameters.
How can Matlab be used to numerically solve a set of second order ordinary differential equations where second order can not be explicitly solved for?
Thanks!
Your second system has the form
a11*x'' + a12*y'' = f1(x,y,x',y')
a21*x'' + a22*y'' = f2(x,y,x',y')
which you can solve as a linear system
[x'', y''] = A\f
or in this case explicitly using Cramer's rule
x'' = ( a22*f1 - a12*f2 ) / (a11*a22 - a12*a21)
y'' accordingly.
I would strongly recommend leaving the intermediate variables in the code to reduce chances for typing errors and avoid multiple computation of the same expressions.
Code could look like this (untested)
function dz = odefunc(t,z)
x=z(1); dx=z(2); y=z(3); dy=z(4);
A = [ [-.5*L*sin(x), 1] ; [.33*L^2, -0.5*L*sin(x)] ]
b = [ [dx^2*cos(x) + (k/m)*y-g]; [-.33*L^2*C*cos(x) + .5*g*L*sin(x)] ]
d2 = A\b
dz = [ dx, d2(1), dy, d2(2) ]
end
Yes your method is correct!
I post the following code below:
%Rotating Pendulum Sym Main
clc
clear all;
%Define parameters
global M K L g C;
M = 1;
K = 25.6;
L = 1;
C = 1;
g = 9.8;
% define initial values for theta, thetad, del, deld
e_0 = 1;
ed_0 = 0;
theta_0 = 0;
thetad_0 = .5;
initialValues = [e_0, ed_0, theta_0, thetad_0];
% Set a timespan
t_initial = 0;
t_final = 36;
dt = .01;
N = (t_final - t_initial)/dt;
timeSpan = linspace(t_final, t_initial, N);
% Run ode45 to get z (theta, thetad, del, deld)
[t, z] = ode45(#RotSpngHndl, timeSpan, initialValues);
%initialize variables
e = zeros(N,1);
ed = zeros(N,1);
theta = zeros(N,1);
thetad = zeros(N,1);
T = zeros(N,1);
V = zeros(N,1);
x = zeros(N,1);
y = zeros(N,1);
for i = 1:N
e(i) = z(i, 1);
ed(i) = z(i, 2);
theta(i) = z(i, 3);
thetad(i) = z(i, 4);
T(i) = .5*M*(ed(i)^2 + (1/3)*L^2*C*sin(theta(i)) + (1/3)*L^2*thetad(i)^2 - L*ed(i)*thetad(i)*sin(theta(i)));
V(i) = -M*g*(e(i) + .5*L*cos(theta(i)));
E(i) = T(i) + V(i);
end
figure(1)
plot(t, T,'r');
hold on;
plot(t, V,'b');
plot(t,E,'y');
title('Energy');
xlabel('time(sec)');
legend('Kinetic Energy', 'Potential Energy', 'Total Energy');
Here is function handle file for ode45:
function dz = RotSpngHndl(~, z)
% Define Global Parameters
global M K L g C
A = [1, -.5*L*sin(z(3));
-.5*L*sin(z(3)), (1/3)*L^2];
b = [.5*L*z(4)^2*cos(z(3)) - (K/M)*z(1) + g;
(1/3)*L^2*C*cos(z(3)) + .5*g*L*sin(z(3))];
X = A\b;
% return column vector [ed; edd; ed; edd]
dz = [z(2);
X(1);
z(4);
X(2)];
I only have about two weeks of experience with programming/Matlab, so I'm just a beginner. In my code I would like to plot mu as a function of alpha. When I display mu it shows the 10 values of mu for each value of alpha. However, when I plot the graph it gives the values of mu as seperate points. But I want the points to be connected with just one line. How can I solve this problem?
n=40; %number of months
p=0.23; %probability of success
num_of_simulations=100;
s=rng; x = rand(1,n)<p;
rng(s);
hold on;
for alpha=0.01:0.01:0.1;
for i=1:num_of_simulations
x = rand(1,n)<p;
S0=5000; %initial value
Y(1)=S0*alpha; %deposit
for k=1
if x(1,1)==1;
S(1, i)=S0+2*Y(1);
else
S(1, i)=S0-Y(1);
end
end
for k=2:n
Y(k)=S(k-1, i)*alpha;
if x(1,k)==1;
S(k, i)=S(k-1, i)+2*Y(k);
else
S(k, i)=S(k-1, i)-Y(k);
end
end
Sn(i)=S(n,i); %end value for each simulations
end
mu=mean(Sn);
disp(mu);
plot(alpha,mu);
end
The reason your points aren't connected is because you plot each point separately. If we take a different approach and take alpha = 0.01:0.01:0.1; out of the for loop definition and then change the for loop definition to for j=1:numel(alpha) we can still loop over every element of alpha. Now we need to change each use of alpha in the loop to alpha(j) so that we are referring to the current element of alpha and not every element. Following on from this we need to change mu to mu(j). What this means is that when the entire loop has finished we have all of the values of alpha and mu stored and 1 call to plot(alpha, mu) will plot the data with the points connected as in
This also enables us to remove hold on; too as we only plot once.
I've included the complete edited code here for you to see. The changes are minuscule and should make sense.
clear all
close all
n = 40; %number of months
p = 0.23; %probability of success
num_of_simulations = 100;
s = rng;
x = rand(1, n) < p;
rng(s);
alpha = 0.01:0.01:0.1;
for j = 1:numel(alpha)
for i = 1:num_of_simulations
x = rand(1, n) < p;
S0 = 5000; %initial value
Y(1) = S0*alpha(j); %deposit
for k = 1
if x(1, 1) == 1;
S(1, i) = S0 + 2*Y(1);
else
S(1, i) = S0 - Y(1);
end
end
for k = 2:n
Y(k) = S(k-1, i)*alpha(j);
if x(1, k) == 1;
S(k, i) = S(k-1, i) + 2*Y(k);
else
S(k, i) = S(k-1, i) - Y(k);
end
end
Sn(i) = S(n, i); %end value for each simulations
end
mu(j) = mean(Sn);
disp(mu(j));
end
plot(alpha, mu);
I'm writing a script for an aerodynamics class and I'm getting the following error:
Undefined function or variable 'dCt_dx'.
Error in Project2_Iteration (line 81)
Ct = trapz(x,dCt_dx)
I'm not sure what the cause is. It's something to do with my if statement. My script is below:
clear all
clc
global dr a n Vinf Vr w rho k x c cl dr B R beta t
%Environmental Parameters
n = 2400; %rpm
Vinf = 154; %KTAS
rho = 0.07647 * (.7429/.9450); %from mattingly for 8kft
a = 1084; %speed of sound, ft/s, 8000 ft
n = n/60; %convert to rps
w = 2*pi*n;
Vinf = (Vinf*6076.12)/3600; %convert from KTAS to ft/s
k = length(c);
dr = R/k; %length of each blade element
for i = 1:k
r(i) = i*dr - (.5*dr); %radius at center of blade element
dA = 2*pi*r*dr; %Planform area of blade element
x(i) = r(i)/R;
if x(i) > .15 && x(i-1) < .15
i_15 = i;
end
if x(i) > .75 && x(i-1) < .75
i_75h = i;
i_75l = i-1;
end
Vr(i) = w*r(i) + Vinf;
%Aerodynamic Parameters
M = Vr(i)/a;
if M > 0.9
M = 0.9;
end
m0 = 0.9*(2*pi/(1-M^2)^0.5); %lift-curve slope (2pi/rad)
%1: Calculate phi
phi = atan(Vinf/(2*pi*n*r(i)));
%2: Choose Vo
Vo = .00175*Vinf;
%3: Calculate Theta
theta = atan((Vinf + Vo)/(2*pi*n*r(i)))-phi;
%4:
if option == 1
%calculate cl(i) from c(i)
sigma = (B*c(i))/(pi*R);
if sigma > 0
cl(i) = (8*x(i)*theta*cos(phi)*tan(phi+theta))/sigma;
else
cl(i) = 0;
end
else %option == 2
%calculate c(i) from cl(i)
if cl(i) ~= 0
sigma = (8*x(i)*theta*cos(phi)*tan(phi+theta))/cl(i);
else
sigma = 0;
end
c(i) = (sigma*pi*R)/B;
if c(i) < 0
c(i) = 0;
end
end
%5: Calculate cd
cd(i) = 0.0090 + 0.0055*(cl(i)-0.1)^2;
%6: calculate alpha
alpha = cl(i)/m0;
%7: calculate beta
beta(i) = phi + alpha + theta;
%8: calculate dCt/dx and dCq/dx
phi0 = phi+theta;
lambda_t = (1/(cos(phi)^2))*(cl(i)*cos(phi0) - cd(i)*sin(phi0));
lambda_q = (1/(cos(phi)^2))*(cl(i)*sin(phi0) + cd(i)*cos(phi0));
if x(i) >= 0.15
dCt_dx(i) = ((pi^3)*(x(i)^2)*sigma*lambda_t)/8; %Roskam eq. 7.47, pg. 280
dCq_dx(i) = ((pi^3)*(x(i)^3)*sigma*lambda_q)/16; %Roskam eq. 7.48, pg 280
else
dCt_dx(i) = 0;
dCq_dx(i) = 0;
end
%calculate Mdd
t(i) = (0.04/(x(i)^1.2))*c(i);
Mdd(i) = 0.94 - (t(i)/c(i)) - cl(i)/10;
end
%9: calculate Ct, Cq, Cd
Ct = trapz(x,dCt_dx)
Cq = trapz(x,dCq_dx)
D = 2*R;
Q=(rho*(n^2)*(D^5)*Cq)
T=(rho*(n^2)*(D^4)*Ct)
When I step through your script, I see that the the entire for i = 1:k loop is skipped because k=0. You set k = length(c), but c was never initialized to a value, so it has length zero.
Because of this, dCt_dx is never given a value--and more importantly the majority of your script is never run.
If you're going to be using MATLAB in the future, I really suggest learning how to do this. It makes it a lot easier to find bugs. Try looking at this video.
Got one more problem with matrix multiplication in Matlab. I have to plot Taylor polynomials for the given function. This question is similar to my previous one (but this time, the function is f: R^2 -> R^3) and I can't figure out how to make the matrices in order to make it work...
function example
clf;
M = 40;
N = 20;
% domain of f(x)
x1 = linspace(0,2*pi,M).'*ones(1,N);
x2 = ones(M,1)*linspace(0,2*pi,N);
[y1,y2,y3] = F(x1,x2);
mesh(y1,y2,y3,...
'facecolor','w',...
'edgecolor','k');
axis equal;
axis vis3d;
axis manual;
hold on
% point for our Taylor polynom
xx1 = 3;
xx2 = 0.5;
[yy1,yy2,yy3] = F(xx1,xx2);
% plots one discrete point
plot3(yy1,yy2,yy3,'ro');
[y1,y2,y3] = T1(xx1,xx2,x1,x2);
mesh(y1,y2,y3,...
'facecolor','w',...
'edgecolor','g');
% given function
function [y1,y2,y3] = F(x1,x2)
% constants
R=2; r=1;
y1 = (R+r*cos(x2)).*cos(x1);
y2 = (R+r*cos(x2)).*sin(x1);
y3 = r*sin(x2);
function [y1,y2,y3] = T1(xx1,xx2,x1,x2)
dy = [
-(R + r*cos(xx2))*sin(xx1) -r*cos(xx1)*sin(xx2)
(R + r*cos(xx2))*cos(xx1) -r*sin(xx1)*sin(xx2)
0 r*cos(xx2) ];
y = F(xx1, xx2) + dy.*[x1-xx1; x2-xx2];
function [y1,y2,y3] = T2(xx1,xx2,x1,x2)
% ?
I know that my code is full of mistakes (I just need to fix my T1 function). dy represents Jacobian matrix (total derivation of f(x) - I hope I got it right...). I am not sure how would the Hessian matrix in T2 look, by I hope I will figure it out, I'm just lost in Matlab...
edit: I tried to improve my formatting - here's my Jacobian matrix
[-(R + r*cos(xx2))*sin(xx1), -r*cos(xx1)*sin(xx2)...
(R + r*cos(xx2))*cos(xx1), -r*sin(xx1)*sin(xx2)...
0, r*cos(xx2)];
function [y1,y2,y3]=T1(xx1,xx2,x1,x2)
R=2; r=1;
%derivatives
y1dx1 = -(R + r * cos(xx2)) * sin(xx1);
y1dx2 = -r * cos(xx1) * sin(xx2);
y2dx1 = (R + r * cos(xx2)) * cos(xx1);
y2dx2 = -r * sin(xx1) * sin(xx2);
y3dx1 = 0;
y3dx2 = r * cos(xx2);
%T1
[f1, f2, f3] = F(xx1, xx2);
y1 = f1 + y1dx1*(x1-xx1) + y1dx2*(x2-xx2);
y2 = f2 + y2dx1*(x1-xx1) + y2dx2*(x2-xx2);
y3 = f3 + y3dx1*(x1-xx1) + y3dx2*(x2-xx2);