Research Papers

An Analytical Model for the Interaction of Mass Inclusions in Heterogeneous (HG) Blankets

[+] Author and Article Information
A. Wagner

TU Darmstadt, SzM, Dynamics and Vibrations Group, Bartningstr. 53, 64289 Darmstadt, Germanywagner@dyn.tu-darmstadt.de

M. E. Johnson

Vibration and Acoustics Labs, 136 Durham Hall, Virginia Tech,Blacksburg, VA 24061martyj@vt.edu

K. Idrisi

Vibration and Acoustics Labs, 136 Durham Hall, Virginia Tech,Blacksburg, VA 24061kamal.idrisi@daimler.com

D. P. Bartylla

Vibration and Acoustics Labs, 136 Durham Hall, Virginia Tech,Blacksburg, VA 24061bartylla@vt.edu

J. Vib. Acoust 133(6), 061014 (Nov 28, 2011) (10 pages) doi:10.1115/1.4005017 History: Received April 27, 2010; Revised July 21, 2011; Accepted July 25, 2011; Published November 28, 2011; Online November 28, 2011

The heterogeneous (HG) blanket is a passive treatment used to reduce the low frequency transmission of sound through partitions. HG blankets, glued onto a structure, consist of an elastic medium with embedded mass inhomogeneities that mechanically replicate a mass-spring-damper system to reduce efficient radiating structural modes at low frequencies. The elastic layer typically used has sound absorption properties to create a noise control device with a wide bandwidth of performance. The natural frequency of an embedded dynamic vibration absorber is determined by the mass of the inhomogeneity as well as by its effective stiffness due to the interaction of the mass inclusion with the elastic layer. A novel analytical approach has been developed to describe in detail the interaction of the mass inclusions with the elastic layer and the interaction between the masses by evaluating special elastomechanical concepts. The effective stiffness is predicted by the analytical approach based on the shape of the mass inclusions as well as on the thickness and material properties of the layer. The experimental validation is included and a simplified direct equation to calculate the effective stiffness of a HG blanket is proposed. Furthermore, the stress field inside the elastic material will be evaluated with focus on the stresses at the base to assess the modeling of one or more masses placed on top of the elastic layer as dynamic vibration absorbers. Finally, the interaction between two (or more) masses placed onto the same layer is studied with special focus on the coupling of the masses at low distances between them.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 2

Schematic of the “real” HG blanket with a mass glued on top of an elastic layer Fig 2a. Approximation with a force F distributed over the area of the mass leading to the pressure distribution p Fig. 2b.

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Figure 3

The coordinate system for the “3D Halfspace” of the Boussinesq solution. The force F is applied at a point whose coordinates are described with the vector xf, while the actual position in the material is marked by x. The vector r=x-xf represents the difference of both vectors and has the magnitude R.

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Figure 4

The Image method leads to the introduction of a limited layer thickness d when |F1|=|F2|

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Figure 9

Change of the effective stiffness with the area of the mass shape. A comparison between measurements (

), predictions of the constant pressure (), and constant displacement () model is included for all mass shapes. Part (a) shows a square mass shape, part (b) a rectangular shape with a side length ratio of 1:3, and part (c) a circular shape.

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Figure 10

Prediction of the effective stiffness based on the thickness of the elastic layer and the area of a square-shaped mass. The evaluation was conducted with the constant displacement strategy.

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Figure 11

Contour plots of the compressive stress in 3-direction [Pa] for two applied masses on top of an elastic layer of 0.04 m thickness. Both masses are represented by a constant pressure p = 104  Pa applied on an area of 0.01 m * 0.01 m and their middle points are 0.08 m away from each other and symbolized in the plots as black quadrangles. Part (a) shows the x2 -x3 plane at x1  = 0 (“side view” of the layer) while part (b) shows the stress distribution in the x1 -x2 -plane at x3  = 0.04 m (“top view” onto the baseplate).

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Figure 12

Part (a) Dependence of effective stiffness terms kgand kcon the distance between two masses of area 0.01 m * 0.01 m. kg(

) and kc () computed by inverting the effective 2 by 2 compliance matrix Veff. kg() and kc () are computed by Eq. 33. Part (b) Dependence of the resulting natural frequencies f1 and f2 of the effective 2 by 2 mass spring system on the distance between two identical masses with a mass of 0.006 kg. f1 () and f2 () computed using the inverse of the 2 by 2 compliance matrix Veff. f1 () and f2 () computed using kg and kc computed by Eq. 33.

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Figure 1

Function of a HG blanket. Masses distributed inside or onto the elastic layer of the HG blanket act as dynamic vibration absorbers, whose stiffness and damping is provided by the elastic layer.

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Figure 5

Simulation of the displacement in 3-direction at x3 = 0 when a force of 1 N is distributed over an area of 0.01 m x 0.01 m with its center at x1  = 0 and x2  = 0

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Figure 6

Simulation of the pressure distribution necessary to create a constant displacement of 0.0003 m with a mass of an area of 0.01 m x 0.01 m. The relative deviation between the applied pressure p and the necessary average pressure of approximately 15 kPa is shown. The mass center is at x1  = 0 and x2  = 0.

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Figure 7

The “effective model” of two identical masses placed onto the same layer a distance t apart. kg is the effective stiffness connecting the masses with the ground and kc is the effective coupling stiffness describing the interaction of the two masses. F1 and F2 are the forces applied to the masses, while u1 and u2 are the two degrees of freedom of the “effective model.”

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Figure 8

Part (a) Relationship between effective stiffness of a square mass and its area for a fixed layer thickness. Part (b) Dependence of the effective stiffness on the side length ratio a/b of a rectangular mass shape with cross-sectional area of 0.0001 m2 . Part (c) Dependence of the effective stiffness on the thickness of the elastic layer of the HG blanket. The cross-sectional area of the square mass again was 0.0001 m2 . The evaluations were done with the constant displacement strategy.




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