ploting second derivative from second order differential equationn

Hi, Beloow is my code and i am solving second order differential equation numericaly and i am getting plot of first and zeroth derivation from ode45. is there any way to plot the second derivative as well by just calling ode45 ?

function fval = simulation( t,y )

x=y(1);
v=y(2);

   c=6850;
        m=450;
        k=0;
        f=2650;
        e=535;
        w=4410;

fval(1,1)=v;
 %fval(2,1)= -(c*v+k*x)/m
 fval(2,1)=(-c*v-f-e+w+k*x)/m;
end

%%%%main file%%%%%%
c=6850;
m=450;
 k=0;
f=2650;
e=535;
w=4410;
Y0= [0;3.86];
tspan= [0 1];
options= odeset ('Events', @event);
[tsol,ysol] = ode45(@(t,y) simulation(t,y), tspan,  Y0,options);
figure
plot(tsol,ysol(:,1), 'r-');
hold on;
figure
plot(tsol,ysol(:,2), 'g-');
hold on;
figure

Answer 1

[Jan]

The easiest solution is to let the function to be integrated accept vectors as input and call it with the time steps and positions replied by ODE45:

[tsol, ysol] = ode45(@(t,y) simulation(t,y), tspan,  Y0,options);

dy = simulation(tsol, ysol.').';
d2y = dy(:, 2);


function fval = simulation(t, y)
x = y(1, :);
v = y(2, :);
c = 6850;
m = 450;
k = 0;
f = 2650;
e = 535;
w = 4410;

fval(1, :) = v;
fval(2, :) = (-c * v - f - e + w + k * x) / m;
end

[Muhammad Sarmad Zahid]

Jan there is some syntax error in your code line 2

[Jan]

The 2nd line of my code is empty. So please post the error message, if you get one. Letting me guess which error appears anywhere in the code, is not efficient.

By the way: My code contains only the snippet you have to change to get the 2nd derivative. You have to insert this in your code.

[Muhammad Sarmad Zahid]

oh it was fine thank you for the help jan :)

[Muhammad Sarmad Zahid]

i have one more question i have this code. I am following you guidance to plot g against time. g is a contant value. How can i plot g ?

Y0= [-0.75 ;0];
tspan= [0 2];
options= odeset ('Events', @freefall);
[tsol,ysol] = ode45(@(t,y) pra(t,y), tspan,  Y0,options);
figure
plot(tsol,ysol(:,1), 'r-');
hold on;
figure
plot(tsol,ysol(:,2), 'g-');
hold on;
figure
%%%%function%%%
function fval = pra( t,y )
g=9.8;
y(1);
y(2);

% c=6850;
fval = [y(2); g)];
end

[Jan]

There is a not matching parenthesis in your code in:

fval = [y(2); g)];

You can plot the 2nd component of the output of pra() in exactly the same way, that I've showed you for simulation(): Convert pra() such that it accept vectors as input and use the output of ODE45 to get the wanted values. Afterwards use plot() as for the other components.

[Muhammad Sarmad Zahid]

i am tring to do that but its not returning me a contant value of g instead it is returning me big values starting from 2.7 to 55 m/s^2. can you please guideme by writing a code? it would be really appreaciat if you check that code as well.

you can chekc the values of f as well

[t,y] = ode45(@(t,y) pra(t,y), [0 1], [-0.75 ; 0 ]);
 f=simulation( t,y.').'
%%%%function%%%
function fval = pra( t,y )
g=9.8;
y(1);
y(2);
% c=6850;
fval = [y(2); g.*((t).^0)];
end

[Jan]

Please use the tools to format the code. This makes it easier to read you code. I've done this for you, but it is more efficient, if you do this by your own.

I've showed you how to convert the function simulation() such, that it accepts matrices as input. Then you can use the output of ODE45 to obtain the 2nd derivative also.

Now you have to modify your function pra() accordingly.

These lines are meaningless:

y(1);
y(2);

Instead of:

fval = [y(2); g.*((t).^0)];

you need:

fval = [y(2, :); repmat(g, size(t))];

Using g.*((t).^0) is smart, but the power operation is very expensive. repmat() is more efficient, or alternatively:

fval = [y(2, :); g * ones(size(t))];

Afterwards, do not call "f=simulation( t,y.').'", but:

df = pra(t, y.').';
plot(t, df(:, 2))

[Muhammad Sarmad Zahid]

Error using vertcat

Dimensions of arrays being concatenated are not consistent.

Error in pra (line 10)

fval = [y(2, :); g * ones(size(t))];

Error in practicemailfile (line 20)

df = pra(tsol, ysol.').';

[Jan]

Then call it like this:

df = pra(tsol.', ysol.').';
%            ^^

[Muhammad Sarmad Zahid]

it worked. Thank you so much jan

[Muhammad Sarmad Zahid]

jan kidly guide me on this question as well

https://www.mathworks.com/matlabcentral/answers/778502-ploting-second-derivative-from-second-order-differential-equation-using-for-loop

[Jan]

But this is the same question again?