Pulse propagation in a linear and nonlinear diatomic periodic chain: effects of acoustic frequency band-gap

Abstract

One-dimensional nonlinear crystals have been assembled from periodic diatomic chains of stainless steel cylinders alternated with Polytetrafluoroethylene spheres. This system allows dramatic changes of behavior (from linear to strongly nonlinear) by the application of compressive forces practically without changes to the geometry of the system. The relevance of classical acoustic band-gap, characteristic for a chain with linear interaction forces and derived from the dispersion relation of the linearized system, on the transformation of single and multiple pulses in linear, nonlinear and strongly nonlinear regimes is investigated with numerical calculations and experiments. The limiting frequencies of the acoustic band-gap for the investigated system with a constant precompression force are within the audible frequency range (20-20,000 Hz) and can be tuned by varying the particle's material properties, mass and initial compression. In the linear elastic chain the presence of the acoustic band-gap was apparent through a fast transformation of incoming pulses within very short distances from the end of the chain. It is interesting that pulses with relatively large amplitude (nonlinear elastic chain) exhibit qualitatively similar behavior indicating the relevance of the acoustic band gap also for the transformation of nonlinear signals. The effects of an in situ band-gap created by a mean dynamic compression are observed in the strongly nonlinear wave regime.

Frequency filtering in disordered granular chains

“…This is equivalent to a linear chain where the mass and contact stiffness are both disordered. The non-dimensional contact stiffness is related to the sizes of the contacting particles as given in (15). Comparison of Fig.…”

“…Additionally, there is significant attention placed on the behavior of "designed" and ordered nonlinear chains, often motivated by energy modification and shock-protection applications. Studies have employed smoothly varying mass distributions [40], "decoration" (e.g., deliberate insertion of different sized masses) [10,13,14,27] tapering [5,32,47,49] and controlled variation of the particle material [3,15]. Combinations of both tapering and decoration have also been employed [6,13].…”

Granular acoustic switches and logic elements

“…This dynamically varying stiffness mechanism can be used to control acoustic waves when it is combined with the presence of acoustic frequency bandgaps. Similarly to what is seen in traditional phonon theory [25][26][27] , a chain of compressed spheres presents an acoustic band structure for weak travelling waves, with high frequencies decaying exponentially as they travel through the crystal 21 . The cutoff frequency f cutoff of the chain's pass band is determined to be…”

“…One-dimensional granular chains have been extensively studied for the past two decades, as they present a practically realizable example of a discrete system with a nonlinear contact relation due to the particles' contact geometry [19][20][21][22][23] . The forcedisplacement law between two solid spheres in contact is governed by a 3/2-power (Hertzian) interaction and has zero tensile force 24 .…”

Highly nonlinear solitary waves in chains of ellipsoidal particles

“…Changing the particle diameters, the particle material properties, and/or their periodicity, it is also possible to change the solitary waves' amplitude and traveling speed [9,13,15,16]. This ability to control the wave properties in such chains has been proposed for a variety of practical engineering applications; for example, it could be employed for shock absorbing materials [7,12,19,21], vibration damping [20], sound scramblers [9,10], and sound focusing [22].…”