96 TERMOTEHNICA Supliment 1/2013

THE RANDOM VIBRATIONS FOR THE NONLINEAR

OSCILLATORS WITH AN ELASTIC SINE-LIKE FEATURE

Petre STAN +, Marinic STAN ++

+ University of Pitesti, Romania ++University of Pitesti, Romania

Rezumat. In foarte multe aplicatii practice, vibratiile nu sunt liniare. Acestea au un caracter neliniar. De exemplu, vibratiile rotilor automobilelor pe caile de rulare sunt vibratii neliniare. Ele se produc datorita neregularitatilor cailor de rulare, care sunt descrise de functii aleatoare. Problema esentiala, este aici, in aflarea densitatii spectrale de putere a raspunsului. In teoria miscarilor seismice sau la diguri vibratiile sunt tot neliniare. Aceasta lucrare studiaza vibratiile neliniare cu un singur grad de libertate pentru oscilatorul Duffing cu caracteristica elastica de tip sinusoidal. Calculele facute sunt exemplificate cu grafice care reprezinta densitatea spectrala de putere a raspunsului in functie de frecventa.

Cuvinte cheie:Sistem neliniar, vibratie aleatoare, densitatea spectrala de putere.

Abstract. In many practical applications, the vibrations are not linear.They have a nonlinear character.For example, the cars wheel vibrations are random vibrations linear raceways. These occur due to irregularities of the runways who are described by random functions. The essential problem is here in finding the power spectral density of the response. In theory seismic or levees are all nonlinear vibrations.This paper studies the nonlinear vibrations of a single degree of freedom Duffing oscillator with elastic characteristics sinusoidal type. The computations are illustrated with graphs that represent the power spectral density of the frequency response function. The power spectral density of response functions so obtained show non-linear features such as multiple resonant peaks. Keywords: Non-linear system, random vibration, the power spectral density.

1 SYSTEM MODEL

We consider a nonlinear Duffing oscillator with nonlinear elastic component sinusoidal type. The reduced ordinary differential equation [1,2]of the motion can be written as:

.. .2

2

2 ( )1 sin ( )

x t p x t p x t

p x t f t (1)

where f(t) is a zero mean stationary Gaussian white noise excitation, with the power spectral density

'0 2 1FSS

m,

2cpm

, where c is the viscous

damping coefficient, F(t)=mf(t) is the external excitation signal with zero mean, is the nonlinear factor to control the type and degree of nonlinearity in the system, m is the mass and ( )x t is the displacement response of the system.

By linearization [2,4] of the above equation we get

.. .2( ) 2 ( ) ( ) ( ),e e ex t p x t p x t w t (2)

where ep is the undamped natural frequency and

e is the critical damping factor. For the linear system [2,4]

ee

pp

. (3)

The question that the error obtained by linearization of the system to a value as small white. We evaluate the error by the difference between the linear and nonlinear component.

The elastic component is sinusoidal, random excitation is applied to the system with the power

spectral density '0 2

FSSm

=1. It can be evaluated

by Taylor series expansion. The difference between the nonlinear stiffness

and linear stiffness terms [4] is 2 21[ ( ) sin ( ) ] ( )ee p x t x t p x t . (4)

TERMOTEHNICA Supliment 1/2013 97

Calculation of the equivalent linear system frequency ep uses the partial derivative with

respect to cancellation of this component 2{ }E e . The value of ep can be obtained by

minimizing the expectation of the square error 2

2

{ } 0e

dE edp

. (5)

We use the relationship [4]

2 2 2{ ( )} ( ) ( ( ))E x t x t P x t d (6)

This results [2,3] in 3 5

2 22

{ ( )( )}6 1201e

x

x xE x t xp p (7)

The probability density of the system is determined using the Fokker Planck Kolmogorov equation.

They are written in the form . .

2.

3 5 2'0 2.

[2 [ ( )

1 ( )] ] .6 120

P Px p x P p x tt x x

x x Px P Sx

(8)

Another form of this equation is 2

.

'3 50

'.0

. .

1{ [ ( ) (

)] }6 120 2

2 0.2

p x t xx

Sx x PPp x

S Pp x Px px x

(9)

The density function of the sistem is

2'0 0

3 5

2exp( [

1 ( )]) ,6 120

pP x C p zS

z zz dz

(10)

where C are normalisation constants. We obtain a solution for the stationary joint probability density function P as

3 52

'0 0

2 1[ 9 )]6 120( ) e

p z zp z z dzSP x C . (11)

The standard deviation [4] of ( )x t is

3 52

'0 0

2 2

2 1[ 9 )]6 1202 e

x

x

p z zp z z dzS

x P x dx

C x

(12)

The expression of ep can be obtained as 3 5

2 22

{ ( )}6 120(1 ).ex

x xE x xp p (13)

Using the Fourier transform of equation [1] we obtain the frequency response function [1,4] of the system is given by equation

2

1( )( 2 )

Hm p A p i

, (14)

where 3 5

2 22

{ ( )}6 120(1 )x

x xE x xA p . (15)

The power spectral density of response [1] is 2( ) ( ) ( )x FS H S (16)

or '0

2 2 2 2

2 2 2 2 2

( )( 4 )

.( 4 )

xe e

F

e e

SSm A p

Sm A p

(17)

2 THE NUMERICAL RESULTS

For 1,5 ,m kg 30 ,Nkm

4 ,Nscm

17 ,s with 21,5 ,FS N s we will find the statistical parameters of function.

We obtain 14, 47 ,p s (18)

0, 29 , (19)

3 52

'0 0

2 2

2 1[ 9 )]6 1202 e ,

x

x

p z zp z z dzS

x P x dx

C x

(20)

(for2

'0 31 mS

s).

98 TERMOTEHNICA Supliment 1/2013

Fig.1 . The power spectral density of the response

2( )[ ]xS m s for 1,7m kg , 35 Nkm

, 4 Nscm

, 13s .

Fig.2 . The power spectral density of the response

2( )[ ]xS m s for 1,5m kg , 30 Nkm

, 4 Nscm

, 17s .

The natural frequency is 17,19ep s . (21)

3 CONCLUSION In this paper we have discussed problem of

random vibration of nonlinear oscillators with sinusoidal type elastic component.

We have presented a stochastic linearization method with random parameters which allows us to calculate an approximation of the power spectral density functions of stationary responses.

The statistical linearization method is quite general in the sense that the power spectral density of the response can be calculated for multidimensional nonlinear systems.

Detailed numerical results are presented for of nonlinear oscillators under white noise excitation.

REFERENCES [1] N. Pandrea, S. Parlac – Mechanical vibrations, Pitesti

University, 2000 [2] J.B. Roberts, P.D. Spanos, Random Vibration and

Statistical Linearization, Wiley, Chichester, 1990. [3] P. Stan– The method of statistical linearization for non-

linear random oscillator, 3rd International Conference. Advanced Composite Materials Engineering, Brasov,

Romania, 2010. [4] M. Stan, . P. Stan,– Statistical linearization for non-linear

random oscillator, 4rd International Conference Computational Mechanics And Virtual

Engineering COMEC 2011, Brasov, Romania, 2011.