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Modelling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties
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Modelling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties夽 N.V. Larin, S.A. Skobel’tsyn, L.A. Tolokonnikov Tula State University, Tula, Russia
a r t i c l e
i n f o
Article history: Received 29 December 2015 Available online xxx
a b s t r a c t The problem of determining the laws governing the inhomogeneity of the coating of a plane elastic plate that ensure the least reflection with a given angle of incidence of a plane sound wave is considered. On the basis of the direct problem solution, a functional representing the intensity of reflection is constructed, and an algorithm for its minimisation is proposed. Analytical expressions describing the mechanical parameters of the inhomogeneous coating are obtained. © 2016 Elsevier Ltd. All rights reserved.
The sound-reflecting characteristics of a body can be changed by means of a coating in the form of a continuous inhomogeneous elastic layer. The scattering of sound waves by flat, cylindrical, and spherical elastic bodies with through-thickness inhomogeneous coatings has been investigated.1–5 A continuous inhomogeneous coating can be modelled by a system of thin homogeneous elastic layers with dissimilar mechanical parameters (density and elastic constants).6 The present work is devoted to finding the laws governing the inhomogeneity of the coating of an elastic plate that would ensure the required sound reflection. The reflection and passage of sound through a plane homogeneous isotropic elastic plate have been investigated in many studies (see, for example, Brekhovskikh7 ). The problem of reflection and refraction of a plane sound wave by an inhomogeneous plane layer has been solved for an isotropic layer,8 for a transverse isotropic layer,9 for a layer with anisotropy of the general type,10 and for a thermoelastic layer.11 The recovery of elastic layer characteristics arbitrarily changing over the depth of the layer has been examined.12 Problems of the recovery of the properties of isotropic and orthotropic through-thickness inhomogeneous layers for a known displacement field at the layer boundary have been solved by analysing steady-state vibrations.13–15 The linear laws governing the inhomogeneity of a plane elastic layer having least reflection with a given angle of incidence of a plane sound wave have been determined.16 1. Statement of the problem Let us examine a homogeneous isotropic elastic plate of thickness H, the material of which is characterised by a density 0 and elastic constants 0 and 0 . The plate has a coating in the form of a through-thickness inhomogeneous isotropic elastic layer of thickness h (Fig. 1). We assume that the density (z) of the material of the inhomogeneous layer is described by a continuous function, and that the elastic moduli (z) and (z) are described by differentiable functions of the z coordinate. Here, a system of rectangular coordinates x, y, z is selected such that the x axis lies in the plane separating the plate and the coating, and the z axis is directed downwards along a normal to the plate surface. The external surfaces of the coating and plate are contiguous to ideal homogeneous liquids that have densities 1 and 2 and velocities of sound c1 and c2 respectively. Suppose that, from a half-space z < −h, onto a plate with a coating, there falls, at an arbitrary angle, a plane monochromatic sound wave of amplitude A0 , the velocity potential of which
夽 Prikl. Mat. Mekh. Vol. 80, No. 4, pp. 480–488, 2016. E-mail address: [email protected] (L.A. Tolokonnikov). http://dx.doi.org/10.1016/j.jappmathmech.2016.09.009 0021-8928/© 2016 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Larin NV, et al. Modelling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.009
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Fig. 1.
where k1x and k1z are projections of the wave vector k1 onto the x and z axes, k1 = /c1 is the wave number in half-space z < −h, is the angular velocity, and 0 is the angle of incidence of the plane wave, formed by the normal to the wave front with the z axis. Without limiting the generality, we assume that the wave vector k1 lies in the xz plane. The time factor exp(−it) is omitted below. The velocity potentials of a wave reflected from a plate with a coating 1 and of a wave passing through it 2 are written in the form1
where k2x and k2z are the projections of the wave vector k2 onto the x and z axes, and k2 = /c2 is the wave number in half-space z > H. The vector of displacement of particles of the elastic homogeneous plate is presented in the form1
where
Here, and are the scalar and vector potentials of dispacement, ey is the unit vector of the y axis, kl = /cl and k = /c are the wave
(0 + 20 )/0 and c = 0 /0 are the velocities of the longitudinal numbers of the longitudinal and transverse elastic waves, cl = and transverse waves, and klx , klz , and kx , kz are the projections of the wave vectors of the longitudinal kl and transverse k waves onto 2 = k2 − k2 , k2 = k2 − k2 . According to Snell’s law,7 k = k = k = k . The expressions for the coefficients A , B , C the coordinate axes; klz x 2x 1x j j j lx x l lx z (j = 1, 2) were given earlier.1 The projections of the displacement vector u onto the coordinate axes in the inhomogeneous coating are written in the form1
The functions U1 (z) and U3 (z) are the solution of the boundary-value problem for a system of linear, homogeneous, ordinary second-order differential equations: (1.1) (1.2) )T ,
where U = (U1 , U3 and the primes denote differentiation with respect to z. The expressions for the second-order matrices A, B, C, E1 , and E2 and the column vector D were given earlier.1 The linear boundary-value problem (1.1), (1.2) was solved by Godunov’s orthogonal sweep method.17 We will introduce the dimensionless quantities
˜ and where , ˜ , ˜ are characteristic quantities of the mechanical properties of the coating. Then, the dimensionless coefficient of reflection is defined by the expression
Please cite this article in press as: Larin NV, et al. Modelling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.009
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Using the solution of the direct problem, we will determine the laws of inhomogeneity of the coating material for which we will have the least reflection of sound with a given angle of incidence of a plane wave 0 . Below, the asterisk at all the dimensionless quantities will be omitted. 2. Determination of the laws governing the inhomogeneity of the coating We will assume that the functions (z), (z), and (z) are approximated by second-degree polynomials relative to variable z, i.e., we will consider the following parabolic laws of inhomogeneity of the coating material:
(2.1) We will construct the functional
determined for the class of parablic functions (z), (z) and (z), and expressing the coefficient of reflection with respect to energy. We will find the values of the coefficients (k) , (k) and (k) (k = 0, 1, 2) of functions (2.1) with which the functional F reaches its minimum value. For the functions (z), (z) and (z), determined on the segment [−1,0], we will introduce the constraints
(2.2) where ␣j , j and ␥j (j = 0, 1) are certain positive constants. Geometrically, each of inequalities (2.2) of the form
assigns in a rectangular system of coordinates with abscissa axis z and ordinate axis f an infinite set of curves lying in the rectangular region
shown in the left-hand part of Fig. 2. Here, the triads (a(0) , a(1) , a(2) ) stand for each of the triads ((0) , (1) , (2) ), ((0) , (1) , (2) ) and ((0) , (1) , (2) ), and b0 and b1 for the corresponding boundaries. In region , each parabola f(z) is determined in a unique way by three points
Substituting the coordinates of the points G0 , G1 and G2 into the expression for the function f(z) and solving the obtained system of equations relative to a(0) , a(1) and a(2) , we find (2.3)
Fig. 2.
Please cite this article in press as: Larin NV, et al. Modelling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.009
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Selecting from the segment [b0 , b1 ] the values for the ordinates f0 , f1 and f2 , and calculating by means of relations (2.3) the values of the coefficients a(0) , a(1) and a(2) , we obtain the parabolic (linear when a(2) = 0) laws of inhomogeneity of the coating material f(z). Here, not all the parabolic laws are to be considered. If the condition
(2.4) is fulfilled, this means that the abscissa of the apex of the parabola belongs to the segment [−1, 0]. In this case, the parabola should be considered only when the ordinate of its apex belongs to the segment [b0 , b1 ], i.e., when the condition
(2.5) is fulfilled. We will find the values of the unknown quantities (k) , (k) , (k) (k = 0, 1, 2) satisfying the condition (2.2) and minimising the function of nine variables (2.6) using the following algorithm. We will divide the segments MN, PQ, and RS in the region respectively into n0 , n1 , and n2 equal parts, i.e., on the segment [b0 , b1 ] we will construct three uniform grids with nodes (2.7) where lq = 0, 1, . . ., nq is the number of the node, and hq = (b1 −b0 )/nq is the step of the qth grid. Using relations (2.3) and conditions (2.4) and (2.5), we calculate the triad of coefficients a(0) , a(1) and a(2) , which determine the laws of inhomogeneity of the coating material
We will introduce grids (2.7) for each of the three regions (2.8) On these grids, we obtain the sets of values of the coefficients
The optimum set of parameters (k) , (k) and (k) (k = 0, 1, 2) is found using a procedure for seeking the minimum of the function of many variables (2.6). The calculation procedure was based on a combination of the methods of random search and coordinate descent18 and includes two stages. Here, the nine sought coordinates were not the parameters (k) , (k) and (k) themselves but the sets of quantities f0 , f1 and f2 present in expressions (2.3) that corresponded to them. At the first stage, the initial point – the set of nine values
is randomly selected from the set of permissible discrete combinations on the introduced grid. Here, the additional subscript indicates that the quantities f0 , f1 and f2 belong to the corresponding parameter. At the second stage, in a random order, one of the coordinates is selected, and a search is carried out for the minimum of function F with change in the values of this coordinate at all possible nodes with numbers lq . Here, the values of the other eight coordinates remain unchanged. This is repeated until all the coordinates have been selected. At the end of the second stage, we obtain the value of the local minimum of function F and the corresponding set of coordinates f, by which, using formulae (2.3), the sought parameters (k) , (k) and (k) are calculated. As, in the general case, the function F is not unimodal, the local minimum of F and the set of material parameters corresponding to it will also depend on the choice of the initial point and on the order of exhaustive search for the coordinates in coordinate descent. Therefore, the procedure for seeking the local minimum is repeated N times. As the final solution, the set of parameters (k) , (k) , (k) is selected that ensures the lowest value of F among the local solutions. The optimum solution obtained in this way is approximate, its accuracy depending on the choice of the step of the grid hq and the number N. 3. Results of calculations The parameters of the laws governing the inhomogeneity of the plate coating that ensure the least reflection at the given wave size of the plate and angle of incidence of the plane wave were calculated. In implementing the proposed algorithm, a homogeneous plate of thickness H = 0.05 m with a density 0 = 2700 kg/m3 and elastic moduli 0 = 5.3 × 1010 N/m2 and 0 = 2.6 × 1010 N/m2 (aluminium) was examined. It was assumed that the inhomogeneous coating has a thickness h = 0.005 m, a characteristic density ˜ = 1070 kg/m3 , and charac˜ = 3.9 × 109 N/m2 and teristic elastic moduli ˜ = 9.8 × 108 N/m2 (polyvinyl butyral). The coated plate lies in water (1 = 2 = 1000 kg/m3 , c1 = c2 = 1485 m/s). A plane sound wave of unit amplitude falls at an angle 0 = 25◦ . Please cite this article in press as: Larin NV, et al. Modelling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.009
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Table 1 (0)
(1)
(2)
(0)
(1)
(2)
(0)
(1)
(2)
F
1 0.5
2 −0.5
2 0
1.5 1.5
1 0
0 0
0.5 1
0 2
0 2
2.33 × 10−3 1.86 × 10−4
In the calculations it was assumed that
Such values of the boundaries of inequalities (2.2) ensure a fairly wide range of variation in the functions (z), (z) and (z), when the maximum possible values of the functions are 3 times greater than the minimum permissible values. Table 1 presents the results of calculating the sought parameters for a plate with a wave size k1 H = 10. The first row gives values of the optimum parameters (k) , (k) and (k) (k = 0, 1, 2) obtained when into each of the three regions (2.8) are introduced grids (2.7) with identical steps h0 = h1 = h2 = 0.5. The triads calculated on these grids determine in each of the regions all the permissible laws of inhomogeneity of the coating material, shown in the right-hand part of Fig. 2. From these laws, in accordance with the algorithm set out above, those laws are selected to which the minimum value of F corresponds. To the minimum value of F found with N = 65 and equal to 2.33 × 10−3 there corresponds a coating with the following properties: (3.1) The results given in the second row of Table 1 were obtained on condensed grids (h0 = h1 = h2 = 0.25). In this case, all permissible laws of inhomogeneity of the coating material are depicted in Fig. 3. Reduction in the size of the grids leads to a considerable increase in the number of permissible solutions. The least reflection with a value F = 1.86 × 10−4 obtained with N = 6800 is achieved with the following laws of inhomogeneity: (3.2) Thus, condensing the grids made it possible to find the laws governing the inhomogeneity of the coating (3.2) that led to a considerable reduction in sound reflection by comparison with the laws (3.1).
2
To assess the influence of an inhomogeneous coating on the sound reflection of a plate, the value of A1 was calculated for a plate
without a coating, which turned out to be equal to 5.60 × 10−1 . The results of calculations indicate that, with the aid of an inhomogeneous coating, it is possible to reduce considerably the intensity of the acoustic field reflected by the plate. In particular, the laws of inhomogeneity of the coating material (3.2) make it possible to reduce the value of F that corresponds to an uncoated plate by three orders of magnitude.
Fig. 3.
Please cite this article in press as: Larin NV, et al. Modelling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.009
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For practical purposes, it is much more important to consider the case where the vibration frequency is not fixed but rather changes in a certain range from 1 to 2 . In this range we will construct the functional
determined for the class of parabolic functions (z), (z) and (z). Minimisation of the functional F was carried out using the algorithm proposed above. The integral was calculated numerically. Here, the frequency range was determined by changing the wave size of the plate in the range 5 ≤ k1 H ≤ 10, while the angle of incidence of the plane wave 0 was assumed to be equal to 25◦ . The minimum of the functional F was sought for the laws of inhomogeneity of the coating material that are presented in the right-hand part of Fig. 2. Comparison of the values of the integral characteristic of the reflected field that were calculated for a plate with a coating (F = 0.699) and for a plate without a coating (F = 0.852) indicates that, in the prescribed range of frequencies, an inhomogeneous coating with the mechanical properties
reduces the average value of sound reflection intensity by 18%. Acknowledgements This study was supported financially by the Russian Foundation for Basic Research and the government of the Tula Region (project number 16-41-710083r a) and the Ministry of Education and Science of the Russian Federation (state assignment number 1.1333.2014 K). References 1. Tolokonnikov LA, Yudachev VV. The reflection and refraction of a plane sound wave by an elastic plane layer with an inhomogeneous coating. Izv TulGU Yestestv Nauki 2015;(3):219–26. 2. Romanov AG, Tolokonnikov LA. The scattering of acoustic waves by a cylinder with a non-homogeneous elastic coating. J Appl Math Mech 2011;75(5):595–600. 3. Tolokonnikov LA. The scattering of a slanting incident plane sound wave by an elastic cylinder with an inhomogeneous coating. Izv TulGU Yestestv Nauki 2013;(2):265–74. 4. Tolokonnikov LA. The scattering of a plane sound wave by an elastic sphere with an inhomogeneous coating. J Appl Math Mech 2014;78(4):367–73. 5. Tolokonnikov LA. Diffraction of cylindrical sound waves by an elastic sphere with an inhomogeneous coating. J Appl Math Mech 2015;79(5):467–74. 6. Larin NV, Tolokonnikov LA. The scattering of a plane sound wave by an elastic cylinder with a discrete-layered coating. J Appl Math Mech 2015;79(2):164–9. 7. Brekhovskikh LM. Waves in Layered Media. New York: Academic; 1960. 8. Prikhod’ko VYu, Tyutekin VV. Calculation of the coefficient of reflection of sound waves from a layered inhomogeneous solid. Akust Zh 1986;32(2):212–8. 9. Skobel’tsyn SA, Tolokonnikov LA. The transmission of sound waves through a transversely isotropic inhomogeneous flat layer. Akust Zh 1990;36(4):740–4. 10. Tolokonnikov LA. The reflection and refraction of a plane sound wave by an anisotropic inhomogeneous layer. Prikl Mekh Tekh Fiz 1999;40(5):179–84. 11. Larin NV, Tolokonnikov LA. The transmission of a plane acoustic wave through a non-uniform thermoelastic layer. J Appl Math Mech 2006;70(4):590–8. 12. Vatul’yan AO, Satunovskii PS. Determining the elastic moduli in the analysis of the vibrations of an inhomogeneous layer. Dokl Ross Akad Nauk 2007;414(1):36–8. 13. Vatul’yan AO, Yavruyan OV, Bogachev IV. Identifying the elastic properties of an inhomogeneously thick layer. Acoust Phys 2011;57(6):741–8. 14. Vatul’yan AO, Yavruyan OV, Bogachev IV. Identifying the inhomogeneous properties of an orthotropic elastic layer. Acoust Phys 2013;59(6):702–8. 15. Vatyl’yan AO, Bogachev IV, Yavruyan OV. Reconstruction of inhomogeneous properties of orthotropic viscoelastic layer. Int J Solids Struct 2014;51(11–12):2238–43. 16. Larin NV, Skobel’tsyn SA, Tolokonnikov LA. Determination of the inhomogeneity laws for an elastic layer with preset sound-reflecting properties. Acoust Phys 2015;61(5):504–10. 17. Arushanyan OV, Zaletkin SF. Solution of the linear boundary-value problem for a system of ordinary differential equations by S.K. Godunov’s orthogonal sweep method. Vychisl Metody Programm 2001;2:41–8. 18. Vasil’ev FP. Numerical Methods for Solving Extremal Problems. Moscow: Nauka; 1988.
Translated by P.S.C.
Please cite this article in press as: Larin NV, et al. Modelling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.009
ARTICLE IN PRESS
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Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Modelling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties夽 N.V. Larin, S.A. Skobel’tsyn, L.A. Tolokonnikov Tula State University, Tula, Russia
a r t i c l e
i n f o
Article history: Received 29 December 2015 Available online xxx
a b s t r a c t The problem of determining the laws governing the inhomogeneity of the coating of a plane elastic plate that ensure the least reflection with a given angle of incidence of a plane sound wave is considered. On the basis of the direct problem solution, a functional representing the intensity of reflection is constructed, and an algorithm for its minimisation is proposed. Analytical expressions describing the mechanical parameters of the inhomogeneous coating are obtained. © 2016 Elsevier Ltd. All rights reserved.
The sound-reflecting characteristics of a body can be changed by means of a coating in the form of a continuous inhomogeneous elastic layer. The scattering of sound waves by flat, cylindrical, and spherical elastic bodies with through-thickness inhomogeneous coatings has been investigated.1–5 A continuous inhomogeneous coating can be modelled by a system of thin homogeneous elastic layers with dissimilar mechanical parameters (density and elastic constants).6 The present work is devoted to finding the laws governing the inhomogeneity of the coating of an elastic plate that would ensure the required sound reflection. The reflection and passage of sound through a plane homogeneous isotropic elastic plate have been investigated in many studies (see, for example, Brekhovskikh7 ). The problem of reflection and refraction of a plane sound wave by an inhomogeneous plane layer has been solved for an isotropic layer,8 for a transverse isotropic layer,9 for a layer with anisotropy of the general type,10 and for a thermoelastic layer.11 The recovery of elastic layer characteristics arbitrarily changing over the depth of the layer has been examined.12 Problems of the recovery of the properties of isotropic and orthotropic through-thickness inhomogeneous layers for a known displacement field at the layer boundary have been solved by analysing steady-state vibrations.13–15 The linear laws governing the inhomogeneity of a plane elastic layer having least reflection with a given angle of incidence of a plane sound wave have been determined.16 1. Statement of the problem Let us examine a homogeneous isotropic elastic plate of thickness H, the material of which is characterised by a density 0 and elastic constants 0 and 0 . The plate has a coating in the form of a through-thickness inhomogeneous isotropic elastic layer of thickness h (Fig. 1). We assume that the density (z) of the material of the inhomogeneous layer is described by a continuous function, and that the elastic moduli (z) and (z) are described by differentiable functions of the z coordinate. Here, a system of rectangular coordinates x, y, z is selected such that the x axis lies in the plane separating the plate and the coating, and the z axis is directed downwards along a normal to the plate surface. The external surfaces of the coating and plate are contiguous to ideal homogeneous liquids that have densities 1 and 2 and velocities of sound c1 and c2 respectively. Suppose that, from a half-space z < −h, onto a plate with a coating, there falls, at an arbitrary angle, a plane monochromatic sound wave of amplitude A0 , the velocity potential of which
夽 Prikl. Mat. Mekh. Vol. 80, No. 4, pp. 480–488, 2016. E-mail address: [email protected] (L.A. Tolokonnikov). http://dx.doi.org/10.1016/j.jappmathmech.2016.09.009 0021-8928/© 2016 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Larin NV, et al. Modelling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.009
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2
Fig. 1.
where k1x and k1z are projections of the wave vector k1 onto the x and z axes, k1 = /c1 is the wave number in half-space z < −h, is the angular velocity, and 0 is the angle of incidence of the plane wave, formed by the normal to the wave front with the z axis. Without limiting the generality, we assume that the wave vector k1 lies in the xz plane. The time factor exp(−it) is omitted below. The velocity potentials of a wave reflected from a plate with a coating 1 and of a wave passing through it 2 are written in the form1
where k2x and k2z are the projections of the wave vector k2 onto the x and z axes, and k2 = /c2 is the wave number in half-space z > H. The vector of displacement of particles of the elastic homogeneous plate is presented in the form1
where
Here, and are the scalar and vector potentials of dispacement, ey is the unit vector of the y axis, kl = /cl and k = /c are the wave
(0 + 20 )/0 and c = 0 /0 are the velocities of the longitudinal numbers of the longitudinal and transverse elastic waves, cl = and transverse waves, and klx , klz , and kx , kz are the projections of the wave vectors of the longitudinal kl and transverse k waves onto 2 = k2 − k2 , k2 = k2 − k2 . According to Snell’s law,7 k = k = k = k . The expressions for the coefficients A , B , C the coordinate axes; klz x 2x 1x j j j lx x l lx z (j = 1, 2) were given earlier.1 The projections of the displacement vector u onto the coordinate axes in the inhomogeneous coating are written in the form1
The functions U1 (z) and U3 (z) are the solution of the boundary-value problem for a system of linear, homogeneous, ordinary second-order differential equations: (1.1) (1.2) )T ,
where U = (U1 , U3 and the primes denote differentiation with respect to z. The expressions for the second-order matrices A, B, C, E1 , and E2 and the column vector D were given earlier.1 The linear boundary-value problem (1.1), (1.2) was solved by Godunov’s orthogonal sweep method.17 We will introduce the dimensionless quantities
˜ and where , ˜ , ˜ are characteristic quantities of the mechanical properties of the coating. Then, the dimensionless coefficient of reflection is defined by the expression
Please cite this article in press as: Larin NV, et al. Modelling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.009
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Using the solution of the direct problem, we will determine the laws of inhomogeneity of the coating material for which we will have the least reflection of sound with a given angle of incidence of a plane wave 0 . Below, the asterisk at all the dimensionless quantities will be omitted. 2. Determination of the laws governing the inhomogeneity of the coating We will assume that the functions (z), (z), and (z) are approximated by second-degree polynomials relative to variable z, i.e., we will consider the following parabolic laws of inhomogeneity of the coating material:
(2.1) We will construct the functional
determined for the class of parablic functions (z), (z) and (z), and expressing the coefficient of reflection with respect to energy. We will find the values of the coefficients (k) , (k) and (k) (k = 0, 1, 2) of functions (2.1) with which the functional F reaches its minimum value. For the functions (z), (z) and (z), determined on the segment [−1,0], we will introduce the constraints
(2.2) where ␣j , j and ␥j (j = 0, 1) are certain positive constants. Geometrically, each of inequalities (2.2) of the form
assigns in a rectangular system of coordinates with abscissa axis z and ordinate axis f an infinite set of curves lying in the rectangular region
shown in the left-hand part of Fig. 2. Here, the triads (a(0) , a(1) , a(2) ) stand for each of the triads ((0) , (1) , (2) ), ((0) , (1) , (2) ) and ((0) , (1) , (2) ), and b0 and b1 for the corresponding boundaries. In region , each parabola f(z) is determined in a unique way by three points
Substituting the coordinates of the points G0 , G1 and G2 into the expression for the function f(z) and solving the obtained system of equations relative to a(0) , a(1) and a(2) , we find (2.3)
Fig. 2.
Please cite this article in press as: Larin NV, et al. Modelling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.009
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Selecting from the segment [b0 , b1 ] the values for the ordinates f0 , f1 and f2 , and calculating by means of relations (2.3) the values of the coefficients a(0) , a(1) and a(2) , we obtain the parabolic (linear when a(2) = 0) laws of inhomogeneity of the coating material f(z). Here, not all the parabolic laws are to be considered. If the condition
(2.4) is fulfilled, this means that the abscissa of the apex of the parabola belongs to the segment [−1, 0]. In this case, the parabola should be considered only when the ordinate of its apex belongs to the segment [b0 , b1 ], i.e., when the condition
(2.5) is fulfilled. We will find the values of the unknown quantities (k) , (k) , (k) (k = 0, 1, 2) satisfying the condition (2.2) and minimising the function of nine variables (2.6) using the following algorithm. We will divide the segments MN, PQ, and RS in the region respectively into n0 , n1 , and n2 equal parts, i.e., on the segment [b0 , b1 ] we will construct three uniform grids with nodes (2.7) where lq = 0, 1, . . ., nq is the number of the node, and hq = (b1 −b0 )/nq is the step of the qth grid. Using relations (2.3) and conditions (2.4) and (2.5), we calculate the triad of coefficients a(0) , a(1) and a(2) , which determine the laws of inhomogeneity of the coating material
We will introduce grids (2.7) for each of the three regions (2.8) On these grids, we obtain the sets of values of the coefficients
The optimum set of parameters (k) , (k) and (k) (k = 0, 1, 2) is found using a procedure for seeking the minimum of the function of many variables (2.6). The calculation procedure was based on a combination of the methods of random search and coordinate descent18 and includes two stages. Here, the nine sought coordinates were not the parameters (k) , (k) and (k) themselves but the sets of quantities f0 , f1 and f2 present in expressions (2.3) that corresponded to them. At the first stage, the initial point – the set of nine values
is randomly selected from the set of permissible discrete combinations on the introduced grid. Here, the additional subscript indicates that the quantities f0 , f1 and f2 belong to the corresponding parameter. At the second stage, in a random order, one of the coordinates is selected, and a search is carried out for the minimum of function F with change in the values of this coordinate at all possible nodes with numbers lq . Here, the values of the other eight coordinates remain unchanged. This is repeated until all the coordinates have been selected. At the end of the second stage, we obtain the value of the local minimum of function F and the corresponding set of coordinates f, by which, using formulae (2.3), the sought parameters (k) , (k) and (k) are calculated. As, in the general case, the function F is not unimodal, the local minimum of F and the set of material parameters corresponding to it will also depend on the choice of the initial point and on the order of exhaustive search for the coordinates in coordinate descent. Therefore, the procedure for seeking the local minimum is repeated N times. As the final solution, the set of parameters (k) , (k) , (k) is selected that ensures the lowest value of F among the local solutions. The optimum solution obtained in this way is approximate, its accuracy depending on the choice of the step of the grid hq and the number N. 3. Results of calculations The parameters of the laws governing the inhomogeneity of the plate coating that ensure the least reflection at the given wave size of the plate and angle of incidence of the plane wave were calculated. In implementing the proposed algorithm, a homogeneous plate of thickness H = 0.05 m with a density 0 = 2700 kg/m3 and elastic moduli 0 = 5.3 × 1010 N/m2 and 0 = 2.6 × 1010 N/m2 (aluminium) was examined. It was assumed that the inhomogeneous coating has a thickness h = 0.005 m, a characteristic density ˜ = 1070 kg/m3 , and charac˜ = 3.9 × 109 N/m2 and teristic elastic moduli ˜ = 9.8 × 108 N/m2 (polyvinyl butyral). The coated plate lies in water (1 = 2 = 1000 kg/m3 , c1 = c2 = 1485 m/s). A plane sound wave of unit amplitude falls at an angle 0 = 25◦ . Please cite this article in press as: Larin NV, et al. Modelling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.009
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Table 1 (0)
(1)
(2)
(0)
(1)
(2)
(0)
(1)
(2)
F
1 0.5
2 −0.5
2 0
1.5 1.5
1 0
0 0
0.5 1
0 2
0 2
2.33 × 10−3 1.86 × 10−4
In the calculations it was assumed that
Such values of the boundaries of inequalities (2.2) ensure a fairly wide range of variation in the functions (z), (z) and (z), when the maximum possible values of the functions are 3 times greater than the minimum permissible values. Table 1 presents the results of calculating the sought parameters for a plate with a wave size k1 H = 10. The first row gives values of the optimum parameters (k) , (k) and (k) (k = 0, 1, 2) obtained when into each of the three regions (2.8) are introduced grids (2.7) with identical steps h0 = h1 = h2 = 0.5. The triads calculated on these grids determine in each of the regions all the permissible laws of inhomogeneity of the coating material, shown in the right-hand part of Fig. 2. From these laws, in accordance with the algorithm set out above, those laws are selected to which the minimum value of F corresponds. To the minimum value of F found with N = 65 and equal to 2.33 × 10−3 there corresponds a coating with the following properties: (3.1) The results given in the second row of Table 1 were obtained on condensed grids (h0 = h1 = h2 = 0.25). In this case, all permissible laws of inhomogeneity of the coating material are depicted in Fig. 3. Reduction in the size of the grids leads to a considerable increase in the number of permissible solutions. The least reflection with a value F = 1.86 × 10−4 obtained with N = 6800 is achieved with the following laws of inhomogeneity: (3.2) Thus, condensing the grids made it possible to find the laws governing the inhomogeneity of the coating (3.2) that led to a considerable reduction in sound reflection by comparison with the laws (3.1).
2
To assess the influence of an inhomogeneous coating on the sound reflection of a plate, the value of A1 was calculated for a plate
without a coating, which turned out to be equal to 5.60 × 10−1 . The results of calculations indicate that, with the aid of an inhomogeneous coating, it is possible to reduce considerably the intensity of the acoustic field reflected by the plate. In particular, the laws of inhomogeneity of the coating material (3.2) make it possible to reduce the value of F that corresponds to an uncoated plate by three orders of magnitude.
Fig. 3.
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For practical purposes, it is much more important to consider the case where the vibration frequency is not fixed but rather changes in a certain range from 1 to 2 . In this range we will construct the functional
determined for the class of parabolic functions (z), (z) and (z). Minimisation of the functional F was carried out using the algorithm proposed above. The integral was calculated numerically. Here, the frequency range was determined by changing the wave size of the plate in the range 5 ≤ k1 H ≤ 10, while the angle of incidence of the plane wave 0 was assumed to be equal to 25◦ . The minimum of the functional F was sought for the laws of inhomogeneity of the coating material that are presented in the right-hand part of Fig. 2. Comparison of the values of the integral characteristic of the reflected field that were calculated for a plate with a coating (F = 0.699) and for a plate without a coating (F = 0.852) indicates that, in the prescribed range of frequencies, an inhomogeneous coating with the mechanical properties
reduces the average value of sound reflection intensity by 18%. Acknowledgements This study was supported financially by the Russian Foundation for Basic Research and the government of the Tula Region (project number 16-41-710083r a) and the Ministry of Education and Science of the Russian Federation (state assignment number 1.1333.2014 K). References 1. Tolokonnikov LA, Yudachev VV. The reflection and refraction of a plane sound wave by an elastic plane layer with an inhomogeneous coating. Izv TulGU Yestestv Nauki 2015;(3):219–26. 2. Romanov AG, Tolokonnikov LA. The scattering of acoustic waves by a cylinder with a non-homogeneous elastic coating. J Appl Math Mech 2011;75(5):595–600. 3. Tolokonnikov LA. The scattering of a slanting incident plane sound wave by an elastic cylinder with an inhomogeneous coating. Izv TulGU Yestestv Nauki 2013;(2):265–74. 4. Tolokonnikov LA. The scattering of a plane sound wave by an elastic sphere with an inhomogeneous coating. J Appl Math Mech 2014;78(4):367–73. 5. Tolokonnikov LA. Diffraction of cylindrical sound waves by an elastic sphere with an inhomogeneous coating. J Appl Math Mech 2015;79(5):467–74. 6. Larin NV, Tolokonnikov LA. The scattering of a plane sound wave by an elastic cylinder with a discrete-layered coating. J Appl Math Mech 2015;79(2):164–9. 7. Brekhovskikh LM. Waves in Layered Media. New York: Academic; 1960. 8. Prikhod’ko VYu, Tyutekin VV. Calculation of the coefficient of reflection of sound waves from a layered inhomogeneous solid. Akust Zh 1986;32(2):212–8. 9. Skobel’tsyn SA, Tolokonnikov LA. The transmission of sound waves through a transversely isotropic inhomogeneous flat layer. Akust Zh 1990;36(4):740–4. 10. Tolokonnikov LA. The reflection and refraction of a plane sound wave by an anisotropic inhomogeneous layer. Prikl Mekh Tekh Fiz 1999;40(5):179–84. 11. Larin NV, Tolokonnikov LA. The transmission of a plane acoustic wave through a non-uniform thermoelastic layer. J Appl Math Mech 2006;70(4):590–8. 12. Vatul’yan AO, Satunovskii PS. Determining the elastic moduli in the analysis of the vibrations of an inhomogeneous layer. Dokl Ross Akad Nauk 2007;414(1):36–8. 13. Vatul’yan AO, Yavruyan OV, Bogachev IV. Identifying the elastic properties of an inhomogeneously thick layer. Acoust Phys 2011;57(6):741–8. 14. Vatul’yan AO, Yavruyan OV, Bogachev IV. Identifying the inhomogeneous properties of an orthotropic elastic layer. Acoust Phys 2013;59(6):702–8. 15. Vatyl’yan AO, Bogachev IV, Yavruyan OV. Reconstruction of inhomogeneous properties of orthotropic viscoelastic layer. Int J Solids Struct 2014;51(11–12):2238–43. 16. Larin NV, Skobel’tsyn SA, Tolokonnikov LA. Determination of the inhomogeneity laws for an elastic layer with preset sound-reflecting properties. Acoust Phys 2015;61(5):504–10. 17. Arushanyan OV, Zaletkin SF. Solution of the linear boundary-value problem for a system of ordinary differential equations by S.K. Godunov’s orthogonal sweep method. Vychisl Metody Programm 2001;2:41–8. 18. Vasil’ev FP. Numerical Methods for Solving Extremal Problems. Moscow: Nauka; 1988.
Translated by P.S.C.
Please cite this article in press as: Larin NV, et al. Modelling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.009