solving coupled system of second order differential equations

Hello everyone,

I want to solve a "second order coupled ordinary differential equation". I searched a lot but could not find the solution.

Please suggest me how can I solve this.

The structure of my equation is given below,

[M]{x''} + [K]{x} = {F}

where [M], [K] are the matrices, which contain time dependent terms.

{x} vector of unknown dependent variables.

{x''} is the second derivative of the vector {x} with respect to time.

Please note that [M], [K] contains time varying terms

Looking forward for your the response.

Thanks for your time..

[James Tursa]

Are you looking for a numeric solution with known initial values? E.g., using the ode45( ) function?

[aakash dewangan]

James, I am looking for Analytical solution..

[Paul]

Do you have a simple example for M, K, and F? Preferably one that you know what that solution should be?

[aakash dewangan]

M, K and F contains sinusoidal terms, which depend on time. Diagonal terms of M and K are in the form of a+sin(nwt), and non diagonal terms are like sin(nbt)*sin(nct). Where a,b,c,n are some constant parameters, and t is time.

Answer 1

[Sulaymon Eshkabilov]

Hi,

You can employ ode solvers (ode23, ode23tb, ode45, ode113, etc) as suggested or write scripts using function handles or anonymous functions by apply Euler or Runge-Kutta methods.

[aakash dewangan]

Thanks Sulaymon,

But I am looking for Analytical solution. Can you suggest me how I can solve this using Analytical approach?

[Sulaymon Eshkabilov]

Should you need to obtain an analytical solution, then dsolve() of Symbolic MATH toolbox needs to be employed. E..g.:

syms x(t) Dx(t) DDx(t)
Dx = diff(x, t);
DDx = diff(Dx, t);
M = [??];
K = [??];
EQN = DDx==inv(M)*(F-K*x);
SOL = dsolve(EQN, x(0)==??, Dx(0)==??)

Good luck