Qualitative analysis of a mechanical system of coupled nonlinear oscillators

. In this paper we investigate nonlinear systems of second order ODEs describing the dynamics of two coupled nonlinear oscillators of a mechanical system. We obtain, under certain assumptions, some stability results for the null solution. Also, we show that in the presence of a time-dependent external force, every solution starting from sufficiently small initial data and its derivative are bounded or go to zero as the time tends to + ∞ , provided that suitable conditions are satisfied. Our theoretical re-sults are illustrated with numerical simulations.


Introduction
Consider a mechanical system of coupled nonlinear oscillators, as shown in Figure 1.1.Specifically, the block of mass m 1 is anchored to a fixed horizontal wall and the block of mass m 2 by springs and dampers, and the block of mass m 2 is also attached to the wall by a pair of springs and dampers.Suppose that the stiffnesses and the dampings are represented by the functions k i : R + → R + and d i : R + → R + , i ∈ {1, 2, 3}, and g i : R + × R × R → R, i ∈ {1, 2}, denote external forces acting on the blocks.One may also consider an external force f (t) acting on the block of mass m 1 , but for the moment, we restrict our attention to the case f ≡ 0. We assume that when the two blocks are in their equilibrium positions, the springs and the dampers are also in their equilibrium positions.Let x(t) and y(t) be the vertical displacements of the blocks from their equilibrium positions.
The general case of a single 1-D damped nonlinear oscillator is described by the following equation which is well-known in the literature ẍ + 2 f * (t) ẋ + β * (t)x + g * (t, x) = 0, t ∈ R + .
(1.2) T. A. Burton and T. Furumochi [2] introduced a new method, based on the Schauder fixed point theorem, to study the stability of the null solution of Eq. (1.2) in the case β * (t) = 1.In [14] we reported new stability results for the same equation.Our approach was based on elementary arguments only, involving in particular some Bernoulli type differential inequalities.In [15] we considered Eq.(1.2) under more general assumptions, which required more sophisticated arguments.For other investigations regarding the asymptotic stability of the equilibrium of a single damped nonlinear oscillator, we refer the reader to [7,8,10,11,24], and the references therein.
In the present paper, in Section 2 we will study the stability of the null solution of system (1.1), by two approaches, based on classical differential inequalities and on Lyapunov's method.For other results regarding the asymptotic stability of the equilibria of coupled damped nonlinear oscillators, we refer the reader to [9,16,17,[20][21][22][23]25], and the references therein.For fundamental concepts and results in stability theory we refer the reader to [1,3,5,6,13,19].
In Section 3 we will consider that the block of mass m 1 is subject to the action of a time dependent external force f : R + → R. In this case, the system of ODEs describing the dynamics of the mechanical system is with the same functions as before, and f (t) := 1 m 1 f (t), and we will derive certain qualitative properties of the solutions of system (1.3) with initial data small enough.
The model in Figure 1.1 could be used, e.g., to describe the dynamics in vertical direction of vibration reduction systems for horizontal cranes with loadings suspended in two sides [12,28].For other models of coupled oscillators or for models from electric circuit theory, structural dynamics, described by systems of type (1.1) or (1.3), we refer the reader to the monographs [4,18,26].

A stability result for the system (1.1)
In this section we shall use the following hypotheses.
Remark 2.1.If (H1) and (H2) hold, then f i , ḟi are bounded, i ∈ {1, 2}.Indeed, by (H2) we see that This, combined with (H1), implies So, using again (H2), we obtain This concludes the proof, since, by (H1), Remark 2.2.Since we are going to discuss the stability of the null solution of system (1.1) and the large-time behavior of the solutions to (1.3) starting from small initial data, we can replace the inequalities (2.1) and (2.2) by possibly with M i r i (t) instead of r i (t), where M i > 0, and some functions g i instead of g i , ∀i ∈ {1, 2}.Indeed, from (2.1) there exist M 1 , a 1 > 0, such that If we define the function for all t ≥ 0, y ∈ R, then

A stability result via differential inequalities
We can state and prove the following stability result.Proof.By using the following transformation (inspired from [2]) where Using the boundedness of the functions f i , ḟi , f j , β, γ i , δ, r i , ∀i ∈ {1, 2}, ∀j ∈ {3, 4}, we easily deduce that our stability question of the null solution of the system (1.1) reduces to the stability of the null solution z(t) = 0 of the system (2.5) .
We have for all t ≥ t 0 .
From the relations (2.15)-(2.17)we get for all t ≥ t 0 .
In what follows we consider two cases.
Therefore, the null solution of (1.1) is uniformly stable.
Proof of b).If, in addition (H3) holds, then from (2.25) we can easily obtain that the null solution of (1.1) is asymptotically stable.

A stability result via Lyapunov's method
We are going to use the following additional assumptions.
Proof.Let us remark that using the classical change of variables x = x, u = ẋ, y = y, v = ẏ, the system (1.1) becomes ż = F(t, z), where and our stability question reduces to the stability of the null solution z(t) = 0 of the system (2.27).Let us remark that the global existence in the future of the solutions of (2.27) follows as in the proof of Theorem 2.3, this time the boundedness of the functions f 1 , f 2 being ensured by the hypothesis (H1*).
We are going to use again the norm ∥•∥ 0 defined by (2.14).Consider the function V : Obviously, ds , for all (t, z) ∈ R + × ∆.By using hypotheses (H1*), (H3*), (H4*), (H5), (H6), we deduce and so the function V is positive definite.The function V is also decrescent.Indeed, We prove that the time derivative of V along the solutions of the system (2.27) is less than or equal to 0. Indeed, for every (t, (2.28) From (2.28) and ( 2.3) for all (t, z) ∈ R + × ∆ we successively obtain Remark 2.7.Let us remark that by using the transformation (2.4) we obtained the uniform, the asymptotic, and the uniform asymptotic stability, while by using the classical transformation (x = x, u = ẋ, y = y, v = ẏ) and the Lyapunov's method we were only able to achieve the uniform stability of the null solution of (1.1).Hence the first method, based on the transformation (2.4), is more effective.Remark 2.8.Note that the null solution of the system (1.1) can be uniformly stable and not asymptotically stable.Indeed, this can be seen by considering the following functions These functions satisfy the hypotheses (H1*), (H3*), (H4*), (H5), (H6), with For small initial data, the solution of (1.1) and its derivative can be observed in Figure 2.5 on some time intervals.The plottings of the solution in the planes (x, ẋ), (y, ẏ) are given in Figure 2.6.t ∈ [0, 500] Figure 2.5: The solution of (1.1) and its derivative, with the initial data z 0 = [0.001,0.001, 0.001, 0.001] and the functions f 1 , f 2 , f 3 , f 4 , β, δ, γ 1 , γ 2 , g 1 , g 2 given in Remark 2.8. -

Analysis of the inhomogeneous system (1.3)
Suppose that the block of mass m 1 is subject to the action of a time-dependent external force f : R + → R. In this case, we obtain the inhomogeneous system (1.3).We are going to use the following hypotheses.

Qualitative properties of solutions via differential inequalities
Theorem 3.1.
Proof.This time we use the following transformation (of the same type as the one from [2]) and the system (1.3) becomes where and A(t) and B(t) are the same as in the proof of Theorem 2.3.
We distinguish two cases again.
and so every solution of (1.3) with initial data small enough is bounded.The boundedness of ż(t, t 0 , z 0 ) follows immediately.

Figure 1 . 1 :
Figure 1.1:A mechanical system of coupled nonlinear oscillators
and g 1 is locally Lipschitzian in x, y.Similar reasonings work for the functions g 2 and r 2 , possibly with another constant a 2 .