[Paper Review] Theory of Truncation Resonances in Continuum Rod-based Phononic Crystals with Generally Asymmetric Unit Cells

Hasan B. Al Ba’ba’a, Carson L. Willey+3 more|arXiv (Cornell University)|Oct 11, 2022

Acoustic Wave Phenomena Research4 citations

TL;DR

This paper presents an analytical framework using the transfer matrix method to predict and tune truncation resonances in finite, continuum rod-based phononic crystals with asymmetric unit cells. It establishes that truncation resonances—localized vibrations within bandgaps—arise from boundary conditions and unit cell symmetry, with closed-form solutions showing that free-free and fixed-fixed systems exhibit at most one resonance per bandgap, while fixed-free/free-fixed systems may host up to two, all controllable via the symmetry parameter δ and material contrasts α and β.

ABSTRACT

Phononic crystals exhibit Bragg bandgaps, frequency regions within which wave propagation is forbidden. In solid continua, bandgaps are the outcome of destructive interferences resulting from periodically alternating layers. Under certain conditions, natural frequencies emerge within these bandgaps in the form of high-amplitude localized vibrations near a structural boundary, referred to as truncation resonances. In this paper, we investigate the vibrational spectrum of finite phononic crystals which take the form of a one-dimensional rod, and explain the factors that contribute to the origination of truncation resonances. By identifying a unit cell symmetry parameter, we define a family of finite phononic rods which share the same dispersion relation, yet distinct truncated forms. A transfer matrix method is utilized to derive closed-form expressions of the characteristic equations governing the natural frequencies of the finite system and decipher the truncation resonances emerging across different boundary conditions. The analysis establishes concrete connections between the localized vibrations associated with a truncation resonance, boundary conditions, and the overall configuration of the truncated chain as dictated by unit cell choice. The study provides tools to predict, tune, and selectively design truncation resonances, to meet the demands of various applications that require and uniquely benefit from such truncation resonances.

Motivation & Objective

• To understand the origins of truncation resonances in finite, continuum rod-based phononic crystals with asymmetric unit cells.
• To identify the structural and material parameters governing the emergence and localization of these resonances.
• To develop a predictive analytical model that extends beyond Bloch-wave theory for infinite systems.
• To establish connections between boundary conditions, unit cell symmetry, and the number and frequency of truncation resonances.

Proposed method

• Employing the transfer matrix method (TMM) to derive characteristic equations for natural frequencies in finite phononic rods.
• Defining a unit cell symmetry parameter δ to generate a family of truncated rods with identical dispersion relations but different boundary configurations.
• Deriving closed-form expressions for frequency response functions and characteristic equations that govern truncation resonances.
• Using the semi-infinite phononic crystal configuration to analytically prove coexistence of truncation resonances across different boundary conditions.
• Introducing frequency contrast α and impedance contrast β to parameterize unit cell properties and analyze their influence on resonance behavior.
• Solving transcendental equations (Eqs. 50 and 51) to predict resonance onset and intersections between free-free and fixed-fixed cases.

Experimental results

Research questions

• RQ1 What determines the existence and number of truncation resonances in finite phononic rods with asymmetric unit cells?
• RQ2 How do boundary conditions (free-free, fixed-fixed, fixed-free, free-fixed) influence the frequency and localization of truncation resonances?
• RQ3 What role does the unit cell symmetry parameter δ play in tuning truncation resonance frequencies and patterns?
• RQ4 How do the frequency contrast α and impedance contrast β affect the oscillatory behavior and amplitude of truncation resonances?
• RQ5 Can truncation resonances in different boundary conditions be analytically linked through a unified framework?

Key findings

• Truncation resonances in free-free and fixed-fixed configurations coincide exactly with the natural frequencies of a single unit cell under the same boundary conditions, enabling direct tuning via unit cell design.
• For asymmetric unit cells (δ ≠ 0), truncation resonances can occur in all standard boundary conditions, with up to one resonance per bandgap in free-free and fixed-fixed cases.
• In fixed-free and free-fixed configurations, up to two truncation resonances may emerge within each bandgap, depending on δ, α, and β.
• The oscillatory behavior of truncation resonance curves with respect to δ is governed by the first solution of Eq. (51), with frequency of oscillation decreasing as α becomes more negative.
• The amplitude of resonance fluctuations is bounded by the bandgap width and strongly dependent on the impedance contrast β, while the oscillation frequency is independent of β.
• The sign of β only interchanges the free-free and fixed-fixed resonance curves, without altering the overall resonance behavior, due to symmetry in the governing equations.