paramath

* B7

 

Co. H. Tran.
Faculty of Mathematics, University of Natural Sciences - VNU-HCM 

Abstract 

We  consider the non-linear random vibration model  demonstrated  by the  Duffing's  differential equation   :    

   (*)

The stationary random process  is  f( t)  which is  satisfied    < f(t) >  = 0          
with  the spectral density  function  Sf ( w  ) . To find  the solution  Sx ( w  ) of  (*)  we use  the equivalent linearization method . 

 

1. Model  Definition 

The  non-linear random vibration model  includes  the mass   (m) - dashpot   (c) -spring   (k)
 ( fig.1 ) .  This model moves on the rough surface  which is  described by the random variable  y(s)  with  the  constant velocity   v  . If  we have the  relation  s =  vt   and  the mass  m  is also  influenced  under  the non-linear stimulating  force    , then the  vibration differential equation of the mass    m  can be rewritten as :  

                                                                   ( 1.0 )

                              ( fig . 1)  

2. The equivalent linearization method 

The conditions of the stationary  solution and equivalent approximation :         ( 2.1 )

The linear operator   :                 ( 2 2 )  

Substitute   D  =  iy1    into  (2.2)  we obtain  the frequency response :  
       ( 2.3 )

The impulse response :        ( 2.4 )
 

The power spectral density  :     
                                                          ( 2. 5 )


Assuming  S f ( y1  )  =  So  : const  ( white-noise)  then we have  :
            ( 2. 6 )

By altering    :        and    choosing  S f ( y1  )  =  So  =  1  ( to simplify the next  algorithm )  , we take into account the integral expression  :
                ( 2.7 ) 

The function  h(z) : 

 

 

>

restart;

 

>	h(z):=(1/((rho^2*z^2+(Gamma-z^2)^2))/(2*Pi));

 

 

And the equation :  (2.8) 

>	eqn:=((rho^2*z^2+(Gamma-z^2)^2))=0;

 

 

Roots of  (2.8)  : 

>	cdiem:=solve(eqn,z);

 

 

(2.9) 

We choose the main value  of  (2.9)  

>	z1:=-1/2*I*(2*rho^2-4*Gamma-2*(rho^4-4*rho^2*Gamma)^(1/2))^(1/2);

 

 

Use ( 2.9 ) to  find  the residue of h(z)  : 

>	simplify(residue(h(z),z=z1));

 

 

The  formula  of     :   

>	Ex2 := -1/2*S[0]/Pi/(2*rho^2-4*Gamma-2*(-rho^2*(-rho^2+4*Gamma))^(1/2))^(1/2)/(-rho^2*(-rho^2+4*Gamma))^(1/2);

 

 

>	delta:=3*mu*beta*Ex2;

 

 

>	delta:=subs(rho=2*omega[0]*psi,delta);

 

 

>	deta:=subs(Gamma=omega[0]^2+Delta,delta);

 

 

>	eqndelta:=Delta=deta;

 

 

             ( 2.10)   and     (2.11) 

                 

>	Int((mu*beta/(sigma*sqrt(2*Pi)))*x^4*exp(-x^2/(2*sigma^2)),x=-infinity..infinity);

 

 

(2.12) 

>	Exg(x):=int((mu*beta/(sigma*sqrt(2*Pi)))*x^4*exp(-x^2/(2*sigma^2)),x=-infinity..infinity);

 

 

(2.13) 

The coefficient of equivalent linearization  :           ( 2.14 ) 

Calculation in details  :  

>	eq:=subs(psi=1,mu=0.1,beta=0.2,S[0]=1,Gamma=omega[0]^2+Delta,eqndelta);eq:=subs(omega[0]=0.5,eq);

 

 

 

>	nodelta:=solve(eq,Delta);

 

 

The Duffing's  equation can be approximated in the linear  form  with the values  of  nodelta :

                          ( 2.15 )  
The investigation on components of  the Duffing's differential equation will be calculated by other methods of linear random vibration , and we can obtain the  corresponding approximate values in the meaning of minimum variance .  

 

3. Parameters - Solution of the equivalent differential equation 

The graph of Duffing's differential equation  ( non-linear random )  : >
 

>	D(D(x))(t)+2*psi*omega*D(x)(t)+(omega^2)*x(t)+mu*beta*(x(t)^3)=x^3;psi:=1;omega:=0.5;mu:=0.1;beta:=0.2;with(DEtools):with(plottools):with(plots):

 

 

 

 

 

 

>	DEplot({D(D(x))(t)+2*psi*omega*D(x)(t)+(omega^2)*x(t)+mu*beta*(x(t)^3)=sin(omega*t)},{x(t)},t=0..30,[[x(0)=1,D(x)(0)=1]],stepsize=0.5,title=`Nghiem  cua pt Duffing phi tuyen_non-linear random`);

 

 

The graph of Duffing's differential equation  ( equivalent -linearization  random )  :>  

>	D(D(x))(t)+2*psi*omega*D(x)(t)+((omega^2)+delta)*(x(t))=sin(omega*t);psi:=1;omega:=0.5;delta:=-.3981894531e-1;

 

 

 

 

 

>	DEplot({D(D(x))(t)+2*psi*omega*D(x)(t)+((omega^2)+delta)*(x(t))=sin(omega*t)},{x(t)},t=0..30,[[x(0)=1,D(x)(0)=1]],stepsize=0.05,title=`Nghiem  cua pt Duffing tuyen tinh hoa tuong duong_equivalent-linearization`);

 

 

The comparison of two graphical solutions : non-linear and  equivalent-linearization .  

 

 

 

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B7.COHONGTRAN-EQUIVALENT-LINEARIZATION