MATLAB: Solve system of 2nd order differential equations with initial conditions

odeoscillations

I have system of two second-order ODE, and initail conditions

phi1 = 8 degree

phi2 = – 4 degree

phi1' = 0

phi2' = 0

And here is my func and conditions declaration

I'll try to solve this with ode45 but cant understand how i can use initial conditions and simplify this for solver

Best Answer

syms phi1(t) phi2(t) 
g = 9.8;
m = 1;
k = 1;
c = 1;
l = 1;
l0 = 0.75;
dphi1=diff(phi1,t);
dphi2=diff(phi2,t);
eq1 =  -(m * g * l * sind(phi1) + k * (l0^2) * cosd(phi1) * (sind(phi1) - sind(phi2)) + c * (l0^2) * cos(phi1) * (cos(phi1) * dphi1 - cosd(phi2) * dphi2))/(m * (l^2)) == diff(phi1, t ,2);
eq2 =  -(m * g * l * sind(phi2) - k * (l0^2) * cosd(phi2) * (sind(phi1) - sind(phi2)) - c * (l0^2) * cos(phi2) * (cos(phi1) * dphi1 - cosd(phi2) * dphi2))/(m * (l^2)) == diff(phi2, t ,2);
vars = [phi1(t) ; phi2(t)]
V = odeToVectorField([eq1,eq2])
M = matlabFunction(V,'vars', {'t','Y'})
interval = [0 1.5];  %time interval
y0 = [8 0 -4 0];      %initial conditions
ySol = ode45(M,interval,y0);
tValues = linspace(interval(1),interval(2),1000);
yValues = deval(ySol,tValues,1); %number 1 denotes first solution likewise you can mention 2 , 3 & 4 for the next three solutions
plot(tValues,yValues)

Related Solutions

MATLAB: How to solve system of second order nonlinear differential equations

Substitute these for what you currently have:

% Solve the System of First-Order ODEs
interval = linspace (0,20,250);
Init = [0 0 0 0 0 0]+1E-15; % theta_1(0) = 0, Dtheta_1(0) = 0, theta_2(0) = 0, Dtheta_2(0) = 0, theta_3(0) = 0, Dtheta_3(0) = 0,
[t,y] = ode45(M,interval,Init);
% tValues = linspace (0,20,100)
% yValues = deval(ySol,tValues,1)
sp = size(y,2);
figure
for k = 1:sp
    subplot(sp,1,k)
    plot(t, y(:,k))
    title(string(S(k)))
    grid
end
op = get(gcf, 'OuterPosition');
set(gcf, 'OuterPosition', op+[0 -500 0 500])

MATLAB: Velocity,Acceleration,Angle

This is a dynamics class. Dynamics involves derivatives. How can you be in a dynamics class without knowing calculus and how to take derivatives? All you have to do is know how to take the derivative of sqrt and trig functions and apply some chain rule stuff to get your answer. E.g., if you had the following for the x-coordinate of point B with respect to point A:

Bx = L1 * cos(phi1)

And you were asked to find the velocity of the x-coordinate of point B, then the derivative of Bx with respect to time would be

d(Bx)/dt = -L1 * sin(phi1) * d(phi1)/dt

And since they told you that phi1 is changing with constant angular rate w1, you get

d(Bx)/dt = -L1 * sin(phi1) * w1 <-- Velocity of Bx

If you were asked about the acceleration of Bx, then take another derivative:

d2(Bx)/dt^2 = -L1 * cos(phi1) * d(phi1)/dt * w1

And again knowing that d(phi1)/dt is the constant w1 you get

d2(Bx)/dt^2 = -L1 * cos(phi1) * w1^2 <-- Acceleration of Bx

This is simple calculus. You have to know this stuff to get by in a dynamics class. Just apply this method to the L3 equation I gave you above.

Best Answer

Related Solutions

MATLAB: How to solve system of second order nonlinear differential equations

MATLAB: Velocity,A​cceleratio​n,Angle

Related Question

MATLAB: Velocity,Acceleration,Angle