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Research Papers

Broadband Vibration Isolation for Rods and Beams Using Periodic Structure Theory

[+] Author and Article Information
Rajan Prasad

Department of Mechanical Engineering,
Indian Institute of Technology Madras,
Chennai 600036, India
e-mail: rajnitdgp2007@gmail.com

Abhijit Sarkar

Department of Mechanical Engineering,
Indian Institute of Technology Madras,
Chennai 600036, India
e-mail: asarkar@iitm.ac.in

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 7, 2018; final manuscript received November 9, 2018; published online December 3, 2018. Assoc. Editor: George Haller.

J. Appl. Mech 86(2), 021004 (Dec 03, 2018) (10 pages) Paper No: JAM-18-1515; doi: 10.1115/1.4042011 History: Received September 07, 2018; Revised November 09, 2018

The alternating stop-band characteristics of periodic structures have been widely used for narrow band vibration control applications. The objective of this work is to extend this idea for broadband excitations. Toward this end, we seek to synthesize a longitudinal and a flexural periodic structure having the largest fraction of the frequencies falling in the attenuation bands of the structure. Such a periodic structure when subjected to broadband excitation has minimal transmission of the response away from the source of excitation. The unit cell of such a periodic structure is constituted of two distinct regions having different inertial and stiffness properties. We derive guidelines for suitable selection of inertial and stiffness properties of the two regions in the unit cell such that the maximal frequency region corresponds to attenuation bands of the periodic structure. It is found that maximal dissimilarity between the neighboring regions of the unit cell leads to maximal attenuating frequencies. In the extreme case, it is found that more than 98% of the frequencies are blocked. For seismic excitations, it is shown that large, finite periodic structures corresponding to the optimal unit cell derived using the infinite periodic structure theory has significant vibration isolation benefits in comparison to a homogeneous structure or an arbitrarily chosen periodic structure.

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Figures

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Fig. 1

(a) Infinite one-dimensional periodic structure and (b) unit cell along the nondimensional coordinate ζ

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Fig. 2

Eigenvalues of the transfer matrix of the periodic rod (a) conjugate pair lying on the unit circle in the complex plane ((b) and (c)) lying on the real axis

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Fig. 3

(a) Eigenvalue map of a bimaterial unit rod with identical stiffness between adjacent regions (γ = 0) and (b) enlarged view of the same for mass ratio (ϵ/δ)=1. The gray and the white regions in the ϵδ parameter space represent stop-band and pass-band characteristics, respectively.

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Fig. 4

Eigenvalue map of a bimaterial periodic rod for mass ratio ξ → 2 and different values of stiffness ratio (γ). The gray and the white regions in the ϵδ parameter space represent stop-band and pass-band characteristics, respectively. (a) γ = 0.5, (b) γ = 1, (c) γ = 1.5, and (d) γ = 1.9.

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Fig. 5

Band gap evolution (shown as the gray region) of a bimaterial rod for different values of stiffness ratio (γ) with mass ratio ξ → 2

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Fig. 6

Finite bimaterial periodic rod comprising of ten unit cells

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Fig. 7

Tip response at P due to a harmonic base acceleration in the axial direction at: (a) optimal stiffness and mass ratio ξ, γ→2, (b) suboptimal ratio ξ, γ = 1, and (c) homogeneous material ξ, γ = 0

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Fig. 8

Eigenvalues of the transfer matrix of periodic beam: (a) all four eigenvalues lying on the real axis and (b) conjugate pair lying on the unit circle in complex plane and other pair lying on the real axis

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Fig. 9

Eigenvalue map of a bimaterial periodic beam for mass ratio ξ → 2 and different values of stiffness ratio (γ′). The gray and the white regions in the εδ parameter space represent stop-band and pass-band characteristics, respectively: (a) γ′=0.5, (b) γ′=1, (c) γ′=1.5, and (d) γ′=1.9

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Fig. 10

Band gap evolution (shown as the gray region) of a bimaterial beam for different values of stiffness ratio (γ′) with mass ratio ξ → 2

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Fig. 11

Tip response at P due to a harmonic base acceleration in the transverse direction at (a) optimal stiffness and mass ratio (ξ,γ′→2−), (b) suboptimal ratio (ξ,γ′=1), and (c) homogeneous material (ξ,γ′=0)

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