2 Dof coupled nonlinear ODEs

[ams31]

Can anyone please suggest whether I can use MATLAB ode45 for the numerical solution of the following equations?

mx ̈+ c_x x ̇ + k_x x= F_x0+ μ(v_r ) (K 〖VB〗^2 y ̇/v) sgn(v_r )
my ̈+ c_y y ̇+ k_y y= F_y0+ (K 〖VB〗^2 (y/v) ̇ )
Where,
m, c_x, k_x, c_y, k_y, F_x0, F_y0, K, v are known constants.
v_r = x ̇- v
x ̈= (d^2 x)/(dt^2 ), x ̇= dx/dt and so on.

If not, is there any other solver in MATLAB which I can use?

Thanks
ams

 

[JJacquelin]

What is the meaning of μ(v_r ) ? and of (K 〖VB〗^2 y ̇/v) ?

 

[ams31]

What is the meaning of μ(v_r ) ? and of (K 〖VB〗^2 y ̇/v) ?

μ(v_r ) is co-efficient of friction as a function of relative velocity.
K is a constant. 〖VB〗^2 is VB*VB and is a variable.
The whole thing can be expressed as: [K *VB*VB* (dy/dt)]/ v

 

[JJacquelin]

μ(v_r ) is co-efficient of friction as a function of relative velocity.
K is a constant. 〖VB〗^2 is VB*VB and is a variable.
The whole thing can be expressed as: [K *VB*VB* (dy/dt)]/ v

So, both μ(v_r ) and 〖VB〗^2 are functions of t. Altogether, these two functions and the two others x(t) and y(t), we have four fonctions of t. Since there are two equations only, the system in not solvable (as far as two relationships are missing).

 

[ams31]

Ohhhhh... I have to solve them anyway...what if I provide x(0), y(0), dx(0)/dt & dy(0)/dt?
Any other way I can make the system bit simple?

 

[JJacquelin]

what if I provide x(0), y(0), dx(0)/dt & dy(0)/dt?

What you propose in addition are boundary or initial conditions. They are not the missing relationships.
If the unknown functions are x(t) and y(t) and if there are only two equations, no other symbol, as μ and VB, must appear in the equations. Only known and explicite functions of x, y, and derivatives of x and y.
If μ appears in the two equations, a third equation or relationship has to be added. Not a symbolic as μ(v_r ), but an explicit formula.
If VB appears in the two equations, a fourth equation or relationship has to be added, which relates VB to x, y or known functions of x, y or derivatives.
If the added formulas or relationships are complicated, it would probably be impossible to anyliticaly solve the system of equations. Then, the only way would be numerical computation.

 

[AlephZero]

You need to convert each 2nd order equations into two first order equations.
That will give four first-order equations in four variables.

Call the variables as x, v_x, y, v_y. (v_x and v_y are the velocity components in the x and y directions).

Your second order equations then become first order equations;
m v_x' + c_x v_x + k_x x = whatever
m v_y' + c_y v_y + k_y y = whatever
And you have two more equations
x' = v_x
v' = v_y

If everything on the right hand side is a known function of x, v_x, y, v_y and t, that is all you need. If there is another indepedent variable you will need another equation. (I don't understand what you mean by "VB is a variable".)

 

[JJacquelin]

I agree with AlephZero's comments. That's clear.
On my opinion, the difficulty encountered at the present state by Ams31 isn't mathematical but rather physical. May be the modeling isn't fully carried out and a clear and unambiguous mathematical wording isn't derived yet.

 

[ams31]

Thanks a lot JJacquelin and AlephZero.
You are right, apart from lack of clear cut modeling, I am still new to nonlinear stuff.
Let me elaborate the problem & I hope that may help you in helping me.
In simplified form, I am trying to model the vibration of a mass on moving belt along with the effects of forces and wear. VB is wear due to friction. I am using the following model for the friction.

µ if vr > 0
µ(vr) Fcontact-y sgn (vr) = 0 if vr = 0
- µ if vr < 0

Also, force appears only during the contact period. So,


[K*VB^2 *(dy/dt)]/ v for t1 + nTp ≤ t ≤ t2 +nTp n=0,1,2,3...
[K *VB^2* (dy/dt)]/ v =
0 otherwise

Tp = 2π/ω (ω is frequency of mass vibration)
t1 = Tp/4
t2 = 3t1

 

[ams31]

I am attaching the figure of the simplified model here. Hope this will help.

 

[ams31]

Here is the better view of equations.

 

[JJacquelin]

I think that you would receive more help on the Physics Forums than on the Mathematics Forums.
Analytical solving appears too harduous. Numerical methods are probably preferable. So, I think that the Physics Forums are more approriate.

 

[AlephZero]

You shouldn't have any problems solving this with ODE45 in Matlab. Matlab calls your function and gives you some values of t, x, y, x' and y'. You just evaluate your formulas to calculate the corresponding values of dx/dy, dy/dt, dx'/dt and dy'/dt.

However in your PDF attachments, you say you have to "obtain the relationship between VB and time" but you don't have any equations to do that.

You need another equation that tells you how VB changes with respect to something (other than time) in the model. We can't help you include that in your Matlab model if we don't know what that equation is.