Acoustic impact on the laminated plates placed between barriers

On the basis of previously derived equations, analytical solutions are established on the forced vibrations of two-layer and three-layers rectangular plates hinged in an opening of absolutely rigid walls during the transmission of monoharmonic sound waves. It is assumed that the partition wall is situated between two absolutely rigid barriers, one of them by harmonic oscillation with a given displacements amplitude on the plate forms the incident sound wave, and the other is stationary and has a coating of deformable energy absorbing material with high damping properties. The behavior of acoustic environments in the spaces between the deformable plate and the barriers described by classical wave equation based on the ideal compressible fluid model. To describe the process of dynamic deformation of the energy absorbing coating of fixed barrier, two-dimensional equations of motion based on the use of models transversely soft layer are derived with a linear approximation of the displacement field in the thickness direction of the coating and taking into account the damping properties of the material and the hysteresis model for it. The influence of the physical and mechanical properties of the concerned mechanical system and the frequency of the incident sound wave on the parameters of its insulation properties of the plate, as well as on the parameters of the stress-strain state of the plate has been analyzed.


Introduction
In the second half of the last century, a scientific direction in mechanics was founded on the study of stationary and non-stationary interactions of acoustic waves with barriers in the form of solid deformable bodies and thin-walled structural elements. This direction continues still to attract the attention of researchers by its actuality, complexity and diversity of the phenomena inherent in the process of interaction of bodies with different physical fields. Related to this direction, issues about the aero-hydroelasticity of thin-walled shell structures have been covered in a number of monographs and reviews ( [1][2][3][4], and et al.). However, they do not cover the issues about sound wave formation and the study on soundproofing problems as well as sound absorption by various deformable coatings, although in almost all publications devoted to the creation of various kinds of multi-layer constructions emphasizing that they have good soundproofing and sound absorption properties ( [5,6 ] et al.). Such problems of mechanics are problems acoustoelasticity, which is devoted to the rather extensive literature in the form of scientific articles ( [7][8][9][10], et al.), monographs ( [11][12][13], et al.) and review articles ( [14][15][16][17][18]

et al.).
Subject matter of this article -the problem of forced oscillations of a rectangular plate (two-layer and three-layer), hinged to an opening in absolutely rigid partitions, under the influence of monoharmonic external pressure. Without taking into account the interaction of the plate with the surrounding acoustic environments, as a rule, researches are conducted on forced vibrations of thinwalled structural elements ignoring the fact that the real structural elements are not in vacuum, but in acoustic environments. The following shows that the correct formulation of these problems requires consideration of external damping.

Problem statement
Let's consider multi-layer plate consist of We assume that the plate is placed in the opening of an absolutely rigid partition separating two adjacent spaces 1 V and 2 V . Coordinate plane x , respectively. One of the barriers makes z-axis direction in the harmonic oscillation with angular frequency and amplitude U  , and the second fixed coating has low-rigidity deformable layer with thickness c h . We assume that the boundary plane of coating We assume that p , applied to the first and N -th layers of the boundary planes of the plate, respectively, it is necessary to find solutions to the wave equation [ The process of dynamic deformation of the plate will be described by the equations of the theory of multilayer plates obtained in [20] with allowance for transverse compression and internal friction of the material of the plate on the model Thomson-Kelvin-Voigt. As the unknown components displacement components       12 ,, k k k u u w of the points on the facial planes of the first ( 1 k  ) and the last ( 1 kN  ) layers, as well as points of interlayer planes are taken. In the case of simply supported plate in the opening of an absolutely rigid partitions, for these functions the following representations are valid Hereafter, unless otherwise stated, and the summation is to be done over 1,3,... where When using the relations (4) - (7) for the case when the coating layer is subjected to the lateral load 2 p  , it is possible to get the system of three differential equations of motion in standard way for the coating layer  at all points of the boundary planes of the plate (the first pair of conditions (9)), and the first barrier coating.

Determination of aero-hydrodynamic loads acting on plate
In accordance with the assumptions made above for the functions   U   the representation of the form [21] must be taken i U U e    (10) by virtue of which, taking into account the first two terms of (9) and representation (3)  In turn, because of the last two terms of (9) and representation (10) -(11) for seeking solutions of equations (8) (14) and in accordance with the relations (2) (18) and subject the expression (15), the representation (18) and (12) (20) After introducing the expression (19) in (16) (21) and (22)

 
We note that stresses resulted in rubber layers x  , y  , xy  , at low frequencies are much less than the similar stresses resulted in the steel plate. In this regard, in the presented figures on the scale of the graph axes they look like zero. At high frequencies, stress shifts formed at the boundaries between the layers differing significantly by elastic characteristics are much more similar to stresses emerging in other areas.