Uniqueness and stability of the simultaneous detection of the nonlinear restoring and excitation of a forced nonlinear oscillation

doi:10.1016/j.amc.2013.11.105

Applied Mathematics and Computation

Volume 228, 1 February 2014, Pages 234-239

https://doi.org/10.1016/j.amc.2013.11.105 Get rights and content

Highlights

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A new inverse problem for the simultaneous identification is presented.
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It identifies the restoring and excitations of a nonlinear physical system.
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We proved that the inverse problem has a unique solution and stability properties.

Abstract

This paper presents a new inverse problem for the simultaneous identification of the restoring characteristic and the excitations of a nonlinear physical system. An original mathematical formalism for the identification is developed based on the notion of a zero-crossing time and a new function, called the J-function. We analyze the developed formalism to prove that the inverse problem presented has a unique solution and the stability properties. That is, it is mathematically shown that the nonlinear restoring and excitation of a forced nonlinear oscillation can be detected in a unique and stable manner.

Introduction

In the area of mathematical science and engineering, the system identification of the mathematical modeling of nonlinear physical systems has received increasing attention over the past few decades. Especially, forced nonlinear oscillatory systems play an important role in many of applied sciences and engineering and thus the concerning identification has also been investigated for their practical purposes.

In forced nonlinear oscillations, there were much examples of nonlinear restoring identification, for example, Masri et al. [1] proposed a procedure for recovering nonlinear restoring based on neural networks. Chassiakos and Masri [2] and Liang et al. [3] extended the procedure to multi-degree-of-freedom nonlinear vibration systems further developed for a wider applicable range. Spina et al. [4] presented a new technique for the identification of nonlinearity, by using the Hilbert transforms. In addition, as examples of force identification, a wavelet-based deconvolution method was proposed by Doyle [5], who recovered impact forces of a system. Lu and Law [6] suggested a method of identification of system parameters and sinusoidal and impulsive forces from measurements. Chen and Li [7] also presented a time-domain scheme that iteratively identifies input excitations from measurements. Jang et al. [8] proposed a new method for measuring nonharmonic periodic excitation forces in nonlinear damped systems.

As mentioned above, a significant progress has been achieved on the identification of either restoring characteristic or excitations with regard to physical systems, however, little is known about their combination. That is, no studies have ever tried to identify simultaneously the restoring characteristic as well as the excitations. This is the primary motivation of the present study. In this paper, we newly present an inverse problem for the simultaneous identification of the restoring characteristic and the excitations of a system. Thereby, we develop an original mathematical formalism of the identification by introducing the notion of a zero-crossing time and a new function, called the J-function in this study. Finally, we prove not only the uniqueness of solution but also the stability properties of the inverse problem presented in this study.

Section snippets

Inverse problem for forced nonlinear oscillation

We consider an oscillatory dynamical system which consists of a nonlinear damping and a nonlinear restoring characteristic, subject to an external harmonic load $Γ \cdot \cos ω t$ , of an amplitude Γ > 0 and frequency ω > 0. The resulting motion, q(t), for t > 0, is usually governed by a second-order nonlinear ordinary differential equation: $m \ddot{q} + D (\dot{q}) + k_{1} q + k_{2} q^{3} = Γ \cdot \cos ω t, t > 0 .$ Here, the constant m > 0 denotes the mass of a particle of an oscillator and $D (\dot{q})$ a general nonlinear damping depending on the velocity $\dot{q}$ . The

Preliminary lemmas

The question raised in the previous section is interesting mathematically and practically. To answer it, we will begin with some preliminary lemmas to be used later.

Equations for nonlinear restoring

We will establish a relation for the nonlinear restoring force by using the preliminary lemmas of the previous section. Since the nonlinear restoring is parameterized by k₁ and k₂ as in Section 2, the relation established concerns the parameters of k₁ and k₂, which is to be illustrated as follows.

Theorem 1

Let us denote by a restoring coefficient vector $k \in R^{2}$ $k \equiv (\begin{matrix} k_{1} \\ k_{2} \end{matrix}) .$

Then, k satisfies a linear vector equation

Dk = b,

where D is a linear operator from

R^{2}

R^{2}

whose matrix form is expressed as

[D] \equiv [\begin{matrix} q [(t_{J}^{ze})] & q^{3} [(t_{J}^{} \end{matrix}]

Unique solvability

In Sections 3 Preliminary lemmas, 4 Equations for nonlinear restoring, we developed Eq. (2) for the amplitude Γ of the external harmonic load and Eq. (17) for the restoring coefficient vector k. The two equations Eqs. (2), (17) can be regarded as an inverse formalism for the present inverse problem. In this section, we will discuss whether the inverse formalism has a unique solution.

Theorem 2 uniqueness

The excitation and nonlinear restoring in Eq. (1) can be identified in a unique way whenever $\lim_{t_{0} \to \infty} \int_{t_{0}}^{T + t_{0}} (\cos ω t) \cdot$

Stability

While it is found that the present inverse problem is one-to-one, there remains the question of stability in the sense that the solution to the inverse problem depends continuously on the measured response data. We will examine whether the problem has the stability properties.

Theorem 3 excitation stability

The excitation in Eq. (1) depends on the velocity $\dot{q} (t)$ in a continuous manner.

Proof

The amplitude Γ of the external harmonic load in Eq. (1) may be considered as a functional of the velocity $\dot{q} (t)$ , that is, $Γ = Γ (\dot{q}) .$

Assume that

{q

Concluding remarks

The basic idea underlying this study is characterized by an inverse question: it is possible to identify both of nonlinear restoring force and the external forcing in a forced nonlinear system by measuring only system responses? To answer this question, we newly define the J-function and introduce the concept of zero-crossings with regard to the system responses. Thereby, we can obtain the uniqueness theorem as well as the two stability theorems concerning the question. Even though the present

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No.: 2011-0010090). In addition, it is a pleasure to express my gratitude to the student, Mr. Jinsoo Park in the Department of Naval Architecture and Ocean Engineering, Pusan National University for his help for the manuscript.

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