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A second order oscillator with nonlinear restoring force and nonlinear damping is considered: it is subject to both external and internal (parametric) excitations of Gaussian white noise type. The nonlinearities are chosen in such a way that the associated Fokker-Planck-Kolmogorov equation is solvable in the steady state. Different choices of some system parameters give rise to different and interesting shapes of the joint probability density function of the response, which in some cases appears to be multimodal. The problem of the determination of the power spectral density of the response is also addressed by using the true statistical linearization method.

From the twentieth century in almost all fields of the sciences the deterministic visions have been abandoned for a probabilistic point of view. Thus, the study of the response of the dynamical system subjected to random excitations is of paramount importance. If the excitations are Gaussian white noise stochastic processes, the response of the system is a diffusive Markov vector process, whose transition probability density function (PDF) is governed by a partial differential equation that is named Fokker-Planck-Kolmogorov (FPK) equation [

Unfortunately, for second and higher order dynamical systems analytical solutions of the FPK equation are known only in the final state of equilibrium or, in other words, in the stationary state of motion of the system. Moreover, if the system exhibits nonlinear damping, and if the excitations appear in the coefficients of the unknown, the so-called internal or parametric excitation, the situation is much more cumbersome. Much work has been done in this direction: see Chapter 5 of [

This paper is aimed at studying the statistical characteristics of the response of a class of second order nonlinear systems, and highlighting the shape of response PDF by varying some system coefficients. In order to avoid using approximate methods or numerical methods, the dynamic system is chosen in a class for which the associated FPK equation admits an analytical solution [

The problem of the determination of the power spectral density (PSD) of the system displacement is also addressed. While the PSD of linear systems is immediate, for nonlinear systems to writer’s knowledge only an exact solution exists [

Even in this paper the PSD of the displacement has the form of the PSD of a linear oscillator. Its parameters are found by means of the statistical linearization. As the response PDF is known in this case, it is a “true” linearization [

The present study has been conducted at the Technical University of Milan (Politecnico Milano), Italy.

Consider the following nonlinear stochastic second order oscillator:

X ¨ ( t ) + g 0 ( X ) + g 1 ( X ) X ˙ ( t ) + g 2 ( X ) X ˙ ( t ) 2 + g 3 ( X ) X ˙ ( t ) 3 = k 1 W 1 ( t ) + k 2 X ( t ) W 2 ( t ) (1)

where the dots mean derivative with respect to time. The functions g 0 , ⋯ , g 3 are deterministic ones and do not depend explicitly on time, while k 1 and k 2 are real constants. W_{1} and W_{2} are stationary Gaussian white noises with autocorrelation functions of the form E [ W j ( t ) W j ( t + τ ) ] = 2 w j δ ( τ ) ( j = 1 , 2 ) ; without loss of generality they are assumed to be uncorrelated.

The Fokker-Planck-Kolmogorov (FPK) equation associated with the dynamic system (1) is

− ∂ ∂ x [ y p ( x , y , t ) ] + ∂ ∂ y { [ g 0 ( x ) + g 1 ( x ) y ( t ) + g 2 ( x ) y ( t ) 2 + g 3 ( x ) y ( t ) 3 ] p ( x , y , t ) } + ( k 1 2 w 1 + k 2 2 x 2 w 2 ) ∂ 2 p ( x , y , t ) ∂ y 2 = ∂ p ( x , y , t ) ∂ t ( y = x ˙ ) (2)

where p(x, y, t) is the joint transition probability density function of the Markov vector { X , X ˙ } . If the dynamic system is stable (to have stability, first it is sufficient that the damping is globally positive), asymptotically it tends to the equilibrium: the right-hand-side of Equation (2) becomes zero, and the reduced equation is satisfied by the equilibrium PDF. As no confusion is possible, it will equally denoted by p.

A clarification is necessary: the stochastic differential Equation (1) may interpreted in different ways (strictly speaking infinite) according to the rule of integration that is adopted, but Itô’s interpretation and Stratonovich’s one are the most important and popular. The necessity of specifying the interpretation that one adopts arises when a parametric excitation is present. However, in the present case the parametric term affects the restoring force so that Itô and Stratonovich coincide (in other words the Wong-Zakai-Stratonovich corrective terms are zero in this case [

In general, Equation (2) does not admit an analytical solution even in state of equilibrium, that is with the right-hand-side null. Wang and Yasuda [_{i} have the following forms:

g 1 ( x ) = ϕ ( x ) [ c 1 − 2 c 2 G ( x ) ] , g 2 ( x ) = 0 , g 3 ( x ) = c 2 ϕ ( x ) (3)

where c 1 and c 2 are real constants; G(x) is a potential function, that is G ( x ) = ∫ g 0 ( x ) d x , and ϕ ( x ) = k 1 2 w 1 + k 2 2 x 2 w 2 . The above relationships impose strict conditions on the damping function, which too is made dependent on the strengths of the white noises. If the functions g_{i} obey to the relations in (3), the equilibrium PDF is

p ( x , y ) = C exp { − [ g 0 ( x ) ( c 1 + 2 c 2 G ( x ) ) ] − [ c 2 y 4 4 + ( c 1 2 + c 2 G ( x ) ) ] y 2 } (4)

where C is a normalization constant.

If the parametric excitation is absent, say k 2 = 0 , the PDF simplifies into

p ( x , y ) = C exp [ − 1 k 1 2 w 1 ( c 1 Λ + c 2 Λ 2 ) ] (5)

where Λ = y 2 / 2 + G ( x ) is the mechanical energy of the oscillator. It is noted that Equation (5) is also the PDF of the energy, if this is considered an independent variable.

As advanced in the Introduction, approximately it is assumed that the response power spectral density (PSD) has the same form as that of a linear oscillator, that is

S X X ( ω ) = k 1 2 w 1 ( ω e 2 − ω 2 ) 2 − β e 2 ω 2 (6)

In Equation (6) the parameters of the PSD ω e , β e will be computed by means of the statistical linearization method [

The nonlinear system (1) is replaced by the following linear system equivalent to (1) with k 2 = 0 in some statistical sense:

X ¨ ( t ) + β e X ˙ ( t ) + ω e 2 X ( t ) = k 1 W 1 ( t ) . (7)

Using Equation (7) instead of (1) causes the following error:

E = g 0 ( X ) + 2 k 1 2 c 2 w 1 G ( X ) X ˙ + k 1 2 c 2 w 1 + β X ˙ − β e X ˙ − ω e 2 X (8)

where β = k 1 2 c 1 w 1 . Minimizing the error E in mean square, one obtains

β e = β + E [ z X ˙ ] E [ X ˙ 2 ] , ω e 2 = E [ z X ] E [ X 2 ] (9)

where z = g 0 ( X ) + 2 k 1 2 c 2 w 1 G 0 ( X ) X ˙ + k 1 2 c 2 w 1 . The question arises as the statistical averages in (9) are to be evaluated. In [

The function g 0 ( X ) represents the restoring force of the dynamic system: it has been chosen

g 0 ( x ) = d G ( x ) d x = a x + b x 3 , G ( x ) = 1 2 a x 2 + 1 4 b x 4 (10)

The constant b must be positive, otherwise the system diverges, while a my be negative or positive. If it is positive, the potential function G has only a minimum in the origin. When it is negative, G has an unstable maximum in the origin and two minima in ± a / b (

The function F d = ( k 1 2 w 1 + k 2 2 w 2 x 2 ) ( c 1 + 2 c 2 G ( x ) ) x ˙ + c 2 ( k 1 2 w 1 + k 2 2 w 2 x 2 ) x ˙ 3 is the damping function of the system. If c 2 < − 1 , there is an interval in which the damping is negative (

All the computations and the plots have been performed by using the software MAPLE ruled on a work-station. The computational charges have been low in every case: at the most few seconds.

In

Things are very different when the parameter a in Equation (10) becomes negative: a = −1 has been chosen (the other parameters are unchanged). The potential function is now double-well (see

in x-direction are always bimodal, while the sections in orthogonal direction are always unimodal Gaussian like. It is not new that the PDF of the displacement is bimodal in the case of double-well potential: as there are many and many papers that report this fact, we renounce to cite them. However, it is emphasized that in the present case in x-direction the PDF is rather flat and the saddle is not deep: see

As advanced at the beginning of this section, the damping function is negative over a certain interval when c 2 < − 1 : it has been chosen c 2 = − 2 in the next two examples, while a may be −1 or +1 with the other parameters unchanged.

The negative damping even in a short interval changes the things dramatically. The overall aspect of the PDF is crater-like, but a cone is inner as it is revealed by the section x ˙ = 0 , which has three modes (

x = 1 is not shown). Another characteristic that has not been plotted is that the level curves (p = constant) tend to be angular nearly rectangular.

In next application of this section the parameters are a = 1 , c 2 = − 2 . The three-dimensional plot of the PDF and its sections are in

In the case of external excitation only the constant k 2 is zero: for clarity’s sake we report the motion equation of the system, which is

X ¨ ( t ) + g 0 ( X ) + k 1 2 w 1 [ c 1 + 2 c 2 G ( X ) ] X ˙ ( t ) + k 1 2 w 1 c 2 X ˙ ( t ) 3 = k 1 W 1 ( t ) (11)

The joint PDF of X and X ˙ is given by Equation (5). The response PSD is computed by using the method of Section 2.2. In the analyses a is worth −1, while c 2 takes the values 1, −2. The other parameters are: b = 0.5 , c 1 = 1 , k 1 = 1 , w 1 = 1 .

In

displacement X(t) is defined as

S X X ( ω ) = 1 2 π ∫ − ω + ω R X X ( τ ) e − i ω τ d τ (12)

where R X X ( τ ) is the autocorrelation function of X(t), and i = − 1 (using the method of Sec. 2.2, it is not necessary to know R X X ( τ ) . Thus, S X X ( τ ) extends on the whole real axis, and it is symmetric with respect to the vertical axis in the origin. It is expressed by Equation (6), in which the parameters are computed by means of the formulae in Equation (9): ω e = 0.9442 , β e = 3.6674 . Being the linearized frequency ω e less than one, the PSD has a maximum only, which is in the origin.

^{1}It is recalled that the nonlinear system method allows replacing a dynamic system for which the FPK equation is no solvable by means of one for which the FPK is analytically solvable. However, the computations are generally cumbersome.

PDF as in

In this paper, the statistical characteristics of strongly nonlinear second order oscillators are studied by examining the joint probability density function of the response X , X ˙ . It has been chosen a class of oscillators for which the Fokker-Planck equation is exactly solvable in the equilibrium regime. In the general case, the excitation is formed by an external stationary Gaussian white noise and by an internal (parametric) Gaussian white noise proportional to the displacement. But, in order to compute the power spectral density function of the displacement the internal excitation is absent as the method of analysis, statistical linearization, is defective in the presence of internal excitations.

Both the damping function and the restoring force are of polynomial form, being the polynomials defined by some real constants. For the Fokker-Planck equation having an analytical solution, strict relationships must be satisfied, which link the strengths of the white noises with the system parameters. This fact was already known [^{1}. The rationale of choosing such a type of system is to avoid using approximate methods or numerical ones. In fact, the paper is aimed to study as the response probability density function varies when the system parameters are varied. The applications show that the response statistics vary substantially by varying the constants that define the damping function and the restoring force of the system. Crater-like multimodal PDFs are detected that ceil complicated dynamics. To writer’s knowledge, such a type of studies are lacking in literature. Other systematic analyses are surely necessary.

The dynamic characteristics of a system are difficult to be estimated, and in practice they are adapted to a linear model, whose response might be very far from the actual response, if in reality the oscillator is nonlinear. The lesson that can be drawn is that one must be very cautious in choosing a linear model.

The determination of the power spectral density of a nonlinear system is still an open problem, for which the studies are not numerous [

The author declares no conflicts of interest regarding the publication of this paper.

Floris, C. (2019) Random Response of a Strongly Nonlinear Oscillator with Internal and External Excitations. Journal of Applied Mathematics and Physics, 7, 331-342. https://doi.org/10.4236/jamp.2019.72025