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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
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 <> = 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) :
: ( 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 ) :
The comparison of two graphical solutions : non-linear and equivalent-linearization .
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Toán học thuần túy, theo cách của riêng nó, là thi ca của tư duy logic.
Pure mathematics is, in its way, the poetry of logical ideas.
Albert Einstein .
