Excitation of intrinsic localized modes in finite mass-spring chains driven sinusoidally at end/Mass-vedru tuupi lopliku ahela loomulike vonkemoodide sinusoidaalne ergastamine.
It is known that the intrinsic localized modes (ILMs) or the discrete breathers (DBs) are generic in spatially periodic, discrete, and nonlinear systems (see, for example [1-5]). It is also known that the mobile type of ILMs can be excited both in a spatially infinite system  and a semi-infinite system driven at one end sinusoidally at a frequency in a linear stopping band above the passing band [7-11]. However, most of the works are theoretical ones and, to the best our knowledge, only a few works are concerned experimentally with the ILMs (for example [12-14]). This paper considers motions of finite mass-spring chains driven sinusoidally at one end with the other fixed, and show existence of the ILMs numerically, aiming at experimental demonstration of them. Our model assumes that the restoring force of the spring is given by a piecewise linear function of a relative displacement of neighbouring masses, i.e. the spring constant changes at a threshold of the displacement.
Such a chain may be compared with the celebrated Fermi-Pasta-Ulam (FPU) chains as a paradigm of the ILMs where the restoring force is given by a cubic function of displacement called the FPU-[beta] model . Our model has similarity and also dissimilarity with the FPU-[beta] model. The piecewise linear spring behaves similarly to the cubic spring when the displacement is comparable with the threshold. However, when the displacement is far beyond the threshold, the piecewise linear spring is linear, not nonlinear, roughly speaking. On the other hand, the FPU-[beta] chains are infinite without ends and no external excitation is considered. Thus our model is different from the FPU-[beta] model, nevertheless, it is shown that the ILMs emerge commonly.
So far no experiments of the ILMs have ever been done on the FPU-[beta] model. One of the reasons behind the lack of experiments is speculated to be unavailability of appropriate cubic springs. This prompts us to consider the piecewise linear spring. In order to find suitable masses and springs to be used, simulations are done in the first place. The system of finite chains is solved numerically by exciting one end of the chains sinusoidally with the other fixed. The model includes a small, linear damping proportional to velocity, which is unavoidable in experiments. It is expected by the results of simulations that the ILMs can be excited experimentally as well.
2. NUMERICAL ANALYSIS
Chains' motions are described by the following set of equations for N([much greater than] 1) masses given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
with boundary conditions
[x.sub.0] (t) = A sin[[omega].sub.d]t and [x.sub.N+1](t)= 0, (2)
where [x.sub.j] (j = 1, ..., N) represent positions of the jth mass along the chains from the equilibrium point and [r.sub.j] = [x.sub.j+1] - [x.sub.j]; c is a damping coefficient, while A and [[omega].sub.d] are a driving amplitude and angular frequency, respectively. Let the function F(r) be composed of three linear functions with the inclination [k.sub.L] for small displacement, and [k.sub.push] and [k.sub.pull] for larger displacement r [less than or equal to] [r.sub.push] and r [greater than or equal to] [r.sub.pull], respectively, where [r.sub.push](< 0) and [r.sub.pull] (> 0) give the thresholds on r, where the spring constants change as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
If [k.sub.pull] is equal to [k.sub.push], F(r) is antisymmetric with respect to the equilibrium point. For reference, The FPU-[beta] model takes F(r) in the form of F(r) = [k.sub.L]r + [k.sub.C][r.sup.3], [k.sub.L] and [k.sub.C] being constant. A piecewise linear function is compared in Fig. 1 with a cubic function for appropriate values for parameters.
For the finite chains of 39 masses except for those at both ends, numerical values of the parameters are chosen as follows: m = 99.1 x [10.sup.-3] kg, c = 2.12 x [10.sup.-3] Ns/m, [k.sub.L] = 39.2 N/m, [k.sub.push] = 118 N/m, [k.sub.pull] = 99.1 N/m, and [r.sub.pull] = -[r.sub.push] = 20.0 x [10.sup.-3] m. The values [k.sub.L], [k.sub.push], [k.sub.pull], [r.sub.pull], and [r.sub.push] are determined by the measurements of springs to be used in experiments. Because [k.sub.pull] differs a little from [k.sub.push], the spring is not antisymmetric exactly. Details on the springs are not described here.
As is well known, there are N eigenfrequencies [[omega].sub.n] (n = 1, ..., N) in the linearized system given by
[[omega].sub.n] = 2 [square root of ([k.sub.L]/m)] sin [n[pi]/2(N + 1)], (4)
and they lie below the limit value [[omega].sub.[infinity]], = 2[square root of ([k.sub.L]/m)] . For the present chains, the cut-off angular frequency is calculated to be [[omega].sub.39] = 39.8 rad/s. The driving angular frequency [[omega].sub.d] is taken to be slightly higher than the cut-off one, [[omega].sub.d] = 41.5 rad/s.
Letting one end at [x.sub.0] be driven from a quiescent state sinusoidally at a frequency above the cut-off one, we solve Eq (1) with the boundary conditions (2) by the Runge-Kutta method. It is found numerically that while the driving amplitude is below the threshold, the oscillations are evanescent and confined near the end (Fig. 2). Small ripples are seen because the impulse at t = 0 excites various frequency modes. As the driving amplitude is increased beyond the threshold, the localized oscillations are excited intermittently at the driving end and propagated down the system at a constant speed (Fig. 3).
Figure 4 shows the temporal profile of [r.sub.0] for the displacement of spring adjacent to the driver at the driving amplitude A = 15.0 x [10.sup.-3] m. The localized oscillations are excited when [absolute value of [r.sub.0]] exceeds the threshold A = 20.0 x [10.sup.-3] m in Fig. 4. When the localized oscillations hit the other end, they are reflected and propagated back in the system, subject to nonlinear interactions between them, and with the driving end. The FFTs show that the highest peak of the oscillations is located in the linear stopping band (Fig. 5). In these respects, the oscillations may be regarded as the moving ILMs.
However, as the driving amplitude exceeds far beyond the threshold, the ILMs do not tend to be formed but to be evanescent. This is because the chains behave linearly so that the cut-off frequency is shifted to 2[square root of ([k.sub.push(pull)/m)], beyond which the evanescent mode revives.
3. CONCLUSIONS AND REMARKS
Excitation and propagation of the moving ILMs have been shown numerically in the finite mass-spring chains with the piecewise linear springs driven sinusoidally at one end with the other fixed. It has been revealed that the ILMs are generated when the relative displacement of the mass next to the driving end exceeds the threshold one at which the spring constant changes. The relation between the driving amplitude and frequency to excite ILMs is similar to the one for the ILMs in the case of the FPU-[beta] model unless the amplitude is too large. The present results are used to implement the experiments of the ILMs, and results of the experiments will be published in a forthcoming paper.
The authors thank Dr K. Yoshimura for useful comments. This work was supported by JSPS KAKENHI grant Nos 24654124 and 26400394.
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Yosuke Watanabe (a) *, Takunobu Nishida (a), and Nobumasa Sugimoto (a,b)
(a) Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
(b) Faculty of Engineering Science, Kansai University, Suita, Osaka 564-8680, Japan
Received 10 December 2014, accepted 25 June 2015, available online 28 August 2015
* Corresponding author, firstname.lastname@example.org
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|Title Annotation:||NONLINEAR WAVES|
|Author:||Watanabe, Yosuke; Nishida, Takunobu; Sugimoto, Nobumasa|
|Publication:||Proceedings of the Estonian Academy of Sciences|
|Date:||Sep 1, 2015|
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