paramath

* B7

Image 
Investigation of the Power Spectral Density of Duffing's Equation By Equivalent Linearization Method 

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   :    

Image    (*)

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 :  

                                                 Image                  Image( 1.0 )

                              ( fig . 1)
 

2. The equivalent linearization method 

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

The linear operator   :      Image           ( 2 2 )  

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

The impulse response :     Image   ( 2.4 )
 

The power spectral density  :     
                                                Image          ( 2. 5 )


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

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

The function  h(z) : 
 
 

> restart;
 

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

1/2/((rho^2*z^2+(Gamma-z^2)^2)*Pi) 
And the equation :  (2.8) 

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

rho^2*z^2+(Gamma-z^2)^2 = 0 
Roots of  (2.8)  : 

> cdiem:=solve(eqn,z);
 

1/2*(-2*rho^2+4*Gamma+2*(rho^4-4*rho^2*Gamma)^(1/2))^(1/2), -1/2*(-2*rho^2+4*Gamma+2*(rho^4-4*rho^2*Gamma)^(1/2))^(1/2), 1/2*(-2*rho^2+4*Gamma-2*(rho^4-4*rho^2*Gamma)^(1/2))^(1/2), -1/2*(-2*rho^2+4*Ga...
1/2*(-2*rho^2+4*Gamma+2*(rho^4-4*rho^2*Gamma)^(1/2))^(1/2), -1/2*(-2*rho^2+4*Gamma+2*(rho^4-4*rho^2*Gamma)^(1/2))^(1/2), 1/2*(-2*rho^2+4*Gamma-2*(rho^4-4*rho^2*Gamma)^(1/2))^(1/2), -1/2*(-2*rho^2+4*Ga...
 
(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);
 

-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));
 

1/2*I/(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)) 
The  formula  of   Image  :   

> 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);
 

-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;
 

-3/2*mu*beta*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:=subs(rho=2*omega[0]*psi,delta);
 

-3/2*mu*beta*S[0]/(Pi*(8*omega[0]^2*psi^2-4*Gamma-2*(-4*omega[0]^2*psi^2*(-4*omega[0]^2*psi^2+4*Gamma))^(1/2))^(1/2)*(-4*omega[0]^2*psi^2*(-4*omega[0]^2*psi^2+4*Gamma))^(1/2)) 

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

-3/2*mu*beta*S[0]/(Pi*(8*omega[0]^2*psi^2-4*omega[0]^2-4*Delta-2*(-4*omega[0]^2*psi^2*(-4*omega[0]^2*psi^2+4*omega[0]^2+4*Delta))^(1/2))^(1/2)*(-4*omega[0]^2*psi^2*(-4*omega[0]^2*psi^2+4*omega[0]^2+4*... 

> eqndelta:=Delta=deta;
 

Delta = -3/2*mu*beta*S[0]/(Pi*(8*omega[0]^2*psi^2-4*omega[0]^2-4*Delta-2*(-4*omega[0]^2*psi^2*(-4*omega[0]^2*psi^2+4*omega[0]^2+4*Delta))^(1/2))^(1/2)*(-4*omega[0]^2*psi^2*(-4*omega[0]^2*psi^2+4*omega...
Delta = -3/2*mu*beta*S[0]/(Pi*(8*omega[0]^2*psi^2-4*omega[0]^2-4*Delta-2*(-4*omega[0]^2*psi^2*(-4*omega[0]^2*psi^2+4*omega[0]^2+4*Delta))^(1/2))^(1/2)*(-4*omega[0]^2*psi^2*(-4*omega[0]^2*psi^2+4*omega...
 
     Image        ( 2.10)   and   Image  (2.11) 
                 

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

Int(1/2*mu*beta*2^(1/2)*x^4*exp(-1/2*x^2/sigma^2)/(sigma*Pi^(1/2)), 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);
 

piecewise(csgn(1/sigma^2) = 1, 3*mu*beta*sigma^4*csgn(1/sigma), infinity) 
(2.13) 
The coefficient of equivalent linearization  :    Image       ( 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);
 

Delta = -0.3000000000e-1/(Pi*(4*omega[0]^2-4*Delta-2*(-16*omega[0]^2*Delta)^(1/2))^(1/2)*(-16*omega[0]^2*Delta)^(1/2)) 
Delta = -0.3000000000e-1/(Pi*(1.00-4*Delta-2*(-4.00*Delta)^(1/2))^(1/2)*(-4.00*Delta)^(1/2)) 

> nodelta:=solve(eq,Delta);
 

-.2286403831, -0.3981894531e-1, -.2675483392 

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

                       Image   ( 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):
 

((`@@`(D, 2))(x))(t)+2*psi*omega*(D(x))(t)+omega^2*x(t)+mu*beta*x(t)^3 = x^3 
1 
.5 
.1 
.2 

> 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`);
 

Plot 
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;
 

((`@@`(D, 2))(x))(t)+1.0*(D(x))(t)+(.25-0.3000000000e-1*S[0]/(Pi*(8*.5[0]^2-4*Gamma-2*(-4*.5[0]^2*(-4*.5[0]^2+4*Gamma))^(1/2))^(1/2)*(-4*.5[0]^2*(-4*.5[0]^2+4*Gamma))^(1/2)))*x(t) = sin(.5*t)
((`@@`(D, 2))(x))(t)+1.0*(D(x))(t)+(.25-0.3000000000e-1*S[0]/(Pi*(8*.5[0]^2-4*Gamma-2*(-4*.5[0]^2*(-4*.5[0]^2+4*Gamma))^(1/2))^(1/2)*(-4*.5[0]^2*(-4*.5[0]^2+4*Gamma))^(1/2)))*x(t) = sin(.5*t)
 
1 
.5 
-0.3981894531e-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`);
 

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

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B7.COHONGTRAN-EQUIVALENT-LINEARIZATION
 
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