Solving an ODE in matlab

Question

Kebels3 2020-3-11

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/510359-solving-an-ode-in-matlab

评论： Bjorn Gustavsson 2020-3-11

采纳的回答： Ameer Hamza

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Can you guys help me?

?̈(?) = -(0.1/50) ?̇(?)-2?(?)

How to numerically integrate to find a solution for ??

Speed of the mass in time-step ? + Δ? is:

?̇(? + Δ?) ≈ ?̇(?) + ?̈(?) ∗ Δ?

and the position in time-step ? + Δ? is:

?(? + Δ?) ≈ ?(?) + ?̇(?) ∗ Δ?

I want to put the previous two equations in a loop and loop from ? = 0 s to ? = 10 s(step Δ? = 0.001 s). And then plot those two graphs into in graph.

Assuming the mass is let go from initial displacement ?(0) = 0.5 m.

But I can’t get it to work in matlab.

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Answer 1

Ameer Hamza 2020-3-11

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编辑：Ameer Hamza 2020-3-11

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You can use ode45 to solve such ODEs. See this example: https://www.mathworks.com/help/matlab/ref/ode45.html#bu3uj8b to see how your differential equation is converted into two first order ODEs.

fun = @(t, x) [x(2); -0.1/50*x(2) - 2*x(1)];
time = 0:0.001:10;
initial_condition = [0.5 0];
[t, y] = ode45(fun, time, initial_condition);
plot(t,y)
legend({'position', 'velocity'});

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Ameer Hamza 2020-3-11

Bjorn, good that you mentioned it, I didn't know this. Can you point to an example of such a scenario?

Bjorn Gustavsson 2020-3-11

The simplest example (straigh off the cuff) is gyration of electrically charged particles in a static homogeouns magnetic field - i.e. particle accelerated by a force:

Only one of the odeNM functions were close to conserving the particle energy and and gyro-radius, the other ones had either rather bad growth or loss of kinetic energy.

This is rather understandable, the odeNM cannot know excplicityl what constants of motion there should be - we might implement the equations of motion as coupled first-order ODEs in rather arbitrary order of the components of position and velocity.

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Answer 2

Bjorn Gustavsson 2020-3-11

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First thing you should learn/revise/refresh is how to convert higher-order ODEs to sets of coupled first-order ODEs.

You have an equation of motion (well it might be something completely different, but...)

By noting that

and

, you can convert your second-order ODE to two coupled first-oder ODEs:

Now you have your equation in a format suitable for numerical integration with the odeNN-suite (starting with ode45 I think is the knee-jerk recommendation). To do that you write a function returning a vector with the left-hand-side of the two equations above as a function of time t and v and x. Something like this:

function dxdtdvdt = my_eq_o_motion(t,xv)
x = xv(1);
v = xv(2);
acceleration = % whatever you need for the acceleration as a function of x v and t
dxdtdvdt = [v;acceleration];
end

HTH