I dont know how to integrate a function with variables depending on time

Question

Diego Martínez Caldas 2024-1-26

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评论： Diego Martínez Caldas 2024-1-26

采纳的回答： Sam Chak

在 MATLAB Online 中打开

I have a function

syms theta(t)

fmov(t) = (2943*sin(theta(t)))/200 - (27*sin(2*theta(t))*diff(theta(t), t)^2)/4 + (3*(9*cos(theta(t))^2 + 2)*diff(theta(t), t, t))/2

fmov(t) =

Which I want to integrate twice respect to time in order to obtain a function with just theta(t) and t as variables.

And I only have the min and max values of theta(t). It is an angle and range is from 60º until 0º.

I dont know what the final time will be and starting time is 0.

Can anyone help me?

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Torsten 2024-1-26

编辑：Torsten 2024-1-26

I have a function

fmov(t) =

(2943*sin(theta(t)))/200 - (27*sin(2*theta(t))*diff(theta(t), t)^2)/4 + (3*(9*cos(theta(t))^2 + 2)*diff(theta(t), t, t))/2

fmov(t) is a given function of t and it equals the expression

(2943*sin(theta(t)))/200 - (27*sin(2*theta(t))*diff(theta(t), t)^2)/4 + (3*(9*cos(theta(t))^2 + 2)*diff(theta(t), t, t))/2

, so e.g.

sin(t) = (2943*sin(theta(t)))/200 - (27*sin(2*theta(t))*diff(theta(t), t)^2)/4 + (3*(9*cos(theta(t))^2 + 2)*diff(theta(t), t, t))/2

for fmov(t) = sin(t) ?

?

John D'Errico 2024-1-26

This is a second order nonlinear differential equation, where theta(t) is the unknown function. You generally can't just integrate it twice. Instead, this is why entire courses and sequences of courses are taught about how to solve differential equations. Since the ODE is not at all in a standard form, I have a funny feeling there will be no analytical solution. And that means only a numerical solution is probably an option. Since it is second order, you will need two initial values or a pair of boundary values. Regardless, two pieces of information will need to be provided by you. Tools like ODE45 or a bvp solver will be necessary.

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Answer 1

Sam Chak 2024-1-26

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在 MATLAB Online 中打开

Hi @Diego Martínez Caldas

You can utilize the odeToVectorField() function to convert the 2nd-order differential equations to a system of 1st-order differential equations. Once you have the 1st-order form, you can then apply the ode45() solver to solve the problem.

syms theta(t)

myODE = (2943*sin(theta))/200 - (27*sin(2*theta)*diff(theta, t)^2)/4 + (3*(9*cos(theta)^2 + 2)*diff(theta, t, t))/2 == 0;

[V, S] = odeToVectorField(myODE)

V =

S =

M = matlabFunction(V, 'vars', {'t', 'Y'})

M = function_handle with value:

@(t,Y)[Y(2);((sin(Y(1)).*1.09e+2-sin(Y(1).*2.0).*Y(2).^2.*5.0e+1).*(-9.0./1.0e+2))./(cos(Y(1)).^2.*9.0+2.0)]

tspan = [0 30];

Y0 = [3*pi/4 0];

sol = ode45(M, tspan, Y0);

fplot(@(t) deval(sol, t, 1), tspan), grid on