Help with MATLAB code for 4a and 4b please! Thank you Use the MATLAB function od

Question

Help with MATLAB code for 4a and 4b please! Thank you

Use the MATLAB function ode45 to solve the following second order equation: (a minor modification to the code that we did on Wednesday) mx + cx + kx = f (t) (a) Use the following values and plot x (t) vs t for all the following conditions on the same plot: i. m = 1 kg, k = 900 N/m, c = 0 N s/m, x (0) = 1, x (0) = 0, f (t) = 0 ii. m = 1 kg, k = 900 N/m, c = 5 N s/m, x (0) = 1, x (0) = 0, f (t) = 0 iii. m = 1 kg, k = 900 N/m, c = 20 N s/m, x (0) = 1, x (0) = 0, f (t) = 0 iv. m = 1 kg, k = 900 N/m, c = 40 N s/m, x (0) = 1, x (0) = 0, f (t) = 0 v. m = 1 kg, k = 900 N/m, c = 60 N s/m, x (0) = 1, x (0) = 0, f (t) = 0 (b) Use the following values and plot x(t) vs t for all the following conditions on the same plot. Set m = 1 kg, k = 900 N/m, c = 2 N s/m, x(0) = 1, x = 0. Plot x(t) vs t on the same plot. i. f (t) = cos 10 t ii. f (t) = cos 25 t iii. f (t) = cos 30 t iv. f (t) = cos 45 t v. f (t) = cos 60t (c) Use the following values and plot x (t) vs t for the all the following conditions on the same plot. Set m = 1 kg, k = 90 N/m, c = 2 N s/m, f (t) = 0.

Explanation / Answer

The higher-order ordinary differential equations is first converted into systems of first-order differential equations, and then solve those systems.So lets convert given second order eqn into two first order eqn.

mx''+cx'+kx=f(t) // given 2nd order eqn

lets assume X1= x and X2= x'

differentiate above two eqns separately , we get X1' = x' = X2

and X2' = x'' = f(t) - (c/m)(X2) - (k/m)(X1)

therefore we got two first order equations(in bolded letters) , now lets use ordinary method to solve differential eduation in Matlab using ODE45 solver.

A. m-file:

function xdot=func(t,x)

% have global variables m,c,k,F , these are assumed to be co efficients in given eqn.

xdot(1) = x(2); // these two are the first order equations we obtained from the second order eqn

xdot(2) = F - (c/m)*x(2) - (k/m)*x(1);

xdot = xdot ' ;

command code:

(t,x) = ode45('func',[0 5],[1 0] );

m = 1; //value of m is given as 1kg in A(i)

c = 0; // value of c is given 0Ns/m in A(i)

k = 900; // value of k is given 900n/m in A(i)

F = 0; // value of f(t) is given 0 in a(i)

plot (t ,x(:,10)) ; hold on

m = 1;

c = 5;

k = 900;

F = 0;

plot (t ,x(:,10)) ; hold on

m = 1;

c = 20;

k = 900;

F = 0;

plot (t ,x(:,10)) ; hold on

m = 1;

c = 40;

k = 900;

F = 0;

plot (t ,x(:,10)) ; hold on

m = 1;

c = 60;

k = 900;

F = 0;

plot (t ,x(:,10)) ; hold on  

B. m-file: // set f alone as global variable since it is only the variable here

function xdot=func(t,x)

xdot(1) = x(2);

xdot(2) = F - 2*x(2) - 900*x(1);

xdot = xdot ' ;

command code:

(t,x) = ode45('func',[0 5],[1 0] );

F = cos10t;

plot (t ,x(:,10)) ; hold on

F = cos25t;

plot (t ,x(:,10)) ; hold on

F = cos30t;

plot (t ,x(:,10)) ; hold on

F = cos45t;

plot (t ,x(:,10)) ; hold on

F = cos60t;

plot (t ,x(:,10)) ; hold on

C. m-file: // no need global variable here since all values given constant

function xdot=func(t,x)

xdot(1) = x(2);

xdot(2) = 0 - 2*x(2) - 90*x(1);

xdot = xdot ' ;

command code:

(t,x) = ode45('func',[0 5],[1 0] );

plot (t ,x(:,10)) ; hold on

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