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Linearization of differential equations (2)

Posted 9 years ago
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POSTED BY: B B
4 Replies

(fyi, the discussion title is a reference to http://community.wolfram.com/groups/-/m/t/449605)

POSTED BY: Bruce Miller
Posted 9 years ago

Here my code, to help for troubleshouting

Equation Psi

eq? = 
  (A*cos[?[t]]^2 - C*cos[?[t]]^2 + C)*[???][t] + 
    (-2*A*sin[?[t]]*cos[?[t]] + 2*C*sin[?[t]]*cos[?[t]])*
       Derivative[1][?][t]* Derivative[1][?][t] + 
    C*cos[?[t]]*Derivative[1][?][t]*? == 
  Subscript[m, 3]*g*L*sin[?[t]]*cos[?[t]]

Equation Theta

eq? = 
  A*(?^??)[t] + 
    cos[?[t]]*(A*sin[?[t]] - C*sin[?[t]])*(Derivative[1][?][t])^2 - 
    C*?*cos[?[t]]* Derivative[1][?][t] == 
  Subscript[m, 3]*g*L*cos[?[t]]*sin[?[t]];

Equations at equilibrium point

eq?E = eq? /. {? -> (?e &), ? -> (?e &)};

eq?E = eq? /. {? -> (?e &), ? -> (?e &)};

Equations with small variations around the equilibrium

eq?Ae = eq? /. {? -> (?e + ??[#] &), ? -> (?e + ??[#] &)};

eq?Ae = eq? /. {? -> (?e + ??[#] &), ? -> (?e + ??[#] &)};

Change of variables

?eq? = Inner[Subtract, eq?Ae, eq?E, Equal];

?eq? = Inner[Subtract, eq?Ae, eq?E, Equal];

repl =

 {??''[t] -> ??pp, ??'[t] -> ??p, ??[t] -> ??, 
   ??''[t] -> ??pp, ??'[t] -> ??p, ??[t] -> ??};

eq?Li = (?eq?) /. repl

eq?Li = (?eq? ) /. repl

Linearization

eq?Li = Normal @ 
  Series[eq?Li, 
    {??pp, 0, 1}, {??p, 0, 1}, {??, 0, 1}, {??pp, 0, 1}, {??p, 0, 1}, {??, 0, 1}];

eq?Li = 
  Normal @ 
    Series[eq?Li, 
      {??pp, 0, 1}, {??p, 0, 1}, {??, 0, 1}, {??pp, 0, 1}, {??p, 0, 1}, {??, 0, 1}];

Change of variables

eq?Li = eq?Li /. Map[Reverse, repl]

--> result :

???(t)(??(t)???(t)(?2Asin(?e)cos?(?e)?2Acos(?e)sin?(?e)+2Csin(?e)cos?(?e)+2Ccos(?e)sin?(?e))+???(t)(2Csin(?e)cos(?e)?2Asin(?e)cos(?e)))+????(t)(??(t)(2Acos(?e)cos?(?e)?2Ccos(?e)cos?(?e))+Acos(?e)2?Ccos(?e)2+C)+C???(t)cos?(?e)???(t)+C?cos(?e)???(t)=??(t)(gLm3cos(?e)sin?(?e)+gLm3??(t)cos?(?e)sin?(?e))+gLm3??(t)sin(?e)cos?(?e)
eq?Li = eq?Li /. Map[Reverse, repl]

--> Result :

A????(t)+???(t)(?C?cos(?e)?C???(t)cos?(?e))=??(t)(gLm3sin(?e)cos?(?e)+gLm3??(t)sin?(?e)cos?(?e))+gLm3??(t)cos(?e)sin?(?e)
POSTED BY: B B
Posted 9 years ago

Thank you for your information.

1) I tried to apply the function lin[] described in your link but I didn't manage to obtain the good results. I probably don't use it correctly. May you specify me which quantities you enter for the variables noms ? I believed that it was ye (equilibrium position) + deltay[t] (the small variation around the equilibrium). I attached in this post the code I have tried.

2) In the code that I have initially sent, do you see why some second order terms are not simplified ?

Thank you for your help.

Attachments:
POSTED BY: B B

This article in the Wolfram Library describes solving nonlinear boundary value problems by linearization.

http://library.wolfram.com/infocenter/MathSource/8484/

POSTED BY: Frank Kampas
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