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We excite the 81-bead diatomic crystal by driving the actuator with a higher-amplitude (relative to the linear-spectrum experiments) 90 ms sine voltage with frequency close to the lower optical cutoff frequency f₂exp. We place force sensors in particles 2, 6, 10, 14, 18, 22, and 26. The experimental results in figure3.4 show the MI onset and subsequent DB formation. figure 3.4(a) shows the force versus time at particles 2 (near the actuator) and 14 (close to the DB pinning site), and figure3.4(b) shows the corresponding PSDs. The peak force amplitude near the actuator is 8.6 N '0.43F0 (where F0 = 20 N). figure 3.4(c) shows the normalized power versus lattice site at both the driving and DB frequencies, before and after the formation of the DB. The normalized power is the PSD at a given frequency divided by the spectral power— i.e., the integral of the PSD over all frequencies. The force at particle 14 shows an exponential increase (at t ' 20 ms), which is indicative of the onset of MI. This is followed by the DB formation at t ' 55 ms. Both figure 3.4(b) and (c) show the

5.5 7 8.5 0 0.5 1 1.5 f b (kHz) max(F i )/F 0 5.5 7 8.5 1 1.1 1.2 1.3 1.4 f b (kHz) max(| λ j |) −1 0 1 −1 −0.5 0 0.5 1 f b=8.63kHz Re(λ j) Im( λ j ) 8.5 8.7 0 f b dE/df b 20 40 60 −0.5 0 0.5 1 lattice site F i /F 0 20 40 60 real instability oscillatory instabilities (a) _(b) f b=8.35kHz fb=8.75kHz

Figure 3.3: Bifurcation diagram of the continuation of the DB solutions. (a) Maximal dynamic force of the wave versus frequencyfb. The insets show spatial profiles at two values offb. (b) Maximal deviation of Floquet multipliers from the unit circle, which indicates the instability growth strength. The right inset shows a typical multiplier picture, and the left inset shows the connection between the strong (real multiplier) instability and the change in sign of dE/dfb.

appearance of a frequency component f_bexp '8.28 kHz in the gap and localization of the energy over approximately 15 beads around site 14. Before the DB generation, for t ≤35 ms, the lattice mostly vibrates at the driving frequency, and the power is uniformly distributed over the lattice [see figure 3.4(c1)]. After the DB formation, fort ≥55 ms, part of the energy is pumped from the driving to the DB frequency, as shown in figure 3.4(c2). The decay of the vibrations after the actuator is turned off, which does not occur in the numerical simulations, arises from dissipation [116,133]. However, analysis of the PSD after the actuator is turned off indicates that the power at DB frequency is longer-lived than at the driving frequency.

Figure 3.4: Experimental observations of MI and DB at f_bexp ' 8.28 kHz, with

f₁exp < f_bexp < f₂exp, while driving the chain at 8.90 kHz ' f₂exp (see Table 3.1) for 90 ms. (a1, a2) Forces versus time and (b1, b2) PSDs at particles 2 and 14. Normalized power versus lattice site at the driving (open symbols) and the DB (filled symbols) frequencies, before (c1) and after (c2) DB formation. Vertical lines in (b) mark the driving frequency and the DB frequency. Blue (red) curves in (a, b, c) refer to time regions of 30 ms before (after) the DB formation, while the black curves refer to the entire signal.

3.8

Conclusions

We have characterized the dynamics of compressed 1D diatomic granular crystals using theory, numerical simulations, and experiments. We found good agreement for the linearized spectrum, explored the mechanism leading to the formation of DBs via MI, and provided clear experimental proof of their existence. Our results provide a first step toward achieving a deeper understanding and classifying ILMs in 1D granular crystals and pave the way for their manifestation in 2D and 3D lattices, which might eventually lead to their exploitation in energy-harvesting applications.

3.9

Author Contributions

This chapter is based on [5]. G.T., P.G.K., M.A.P., and C.D. proposed the study. N.B. and S.J. led the experimental work. G.T. led the theoretical and numerical anal- ysis. C.D., P.G.K., and M.A.P provided guidance and contributed to the design and analysis throughout the project. All authors contributed to the writing and editing of the manuscript.

Chapter 4

Existence and Stability of Discrete

Breather Families in Diatomic

Granular Crystals

We present a systematic study of the existence and stability of discrete breathers that are spatially localized in the bulk of a one-dimensional chain of compressed elastic beads that interact via Hertzian contact. The chain is diatomic, consisting of a periodic arrangement of heavy and light spherical particles. We examine two families of discrete gap breathers: (1) an unstable discrete gap breather that is centered on a heavy particle and characterized by a symmetric spatial energy profile and (2) a potentially stable discrete gap breather that is centered on a light particle and is characterized by an asymmetric spatial energy profile. We investigate their existence, structure, and stability throughout the band gap of the linear spectrum and classify them into four regimes: a regime near the lower optical band edge of the linear spectrum, a moderately discrete regime, a strongly discrete regime that lies deep within the band gap of the linearized version of the system, and a regime near the upper acoustic band edge. We contrast discrete breathers in anharmonic FPU-type diatomic chains with those in diatomic granular crystals, which have a tensionless interaction potential between adjacent particles, and highlight in that the asymmetric nature of the latter interaction potential may