Acoustic waveguide filters made up of rigid stacked materials with elastic joints

Abstract

The acoustic dispersion properties of monodimensional waveguide filters can be assessed by means of the simple prototypical mechanical system made of an infinite stack of periodic massive blocks, connected to each other by elastic joints. The linear undamped dynamics of the periodic cell is governed by a two degree-of-freedom Lagrangian model. The eigenproblem governing the free propagation of shear and moment waves is solved analytically and the two dispersion relations are obtained in a suited closed form fashion. Therefore, the pass and stop bandwidths are conveniently determined in the minimal space of the independent mechanical parameters. Stop bands in the ultra-low frequency range are achieved by coupling the stacked material with an elastic half-space modelled as a Winkler support. A convenient fine approximation of the dispersion relations is pursued by formulating homogenised micropolar continuum models. An enhanced continualization approach, employing a proper Maclaurin approximation of pseudo-differential operators, is adopted to successfully approximate the acoustic and optical branches of the dispersion spectrum of the Lagrangian models, both in the absence and in the presence of the elastic support.

Appendices

Appendix 1: The sixth order PDE governing the equivalent homogenized model

By applying the enhanced continualization procedure presented in Sect. 3 to the system of Eqs. (17) governing the motion of the block, a system of two PDEs is obtained, whose order depends on the order of truncation applied. Here, the equations derived by retaining terms up to the sixth order are given, having the following form

$$\left\{ \begin{aligned} & - \frac{{r_{kw} r_{k} }}{{r_{b} }}\Psi + \left( {1 + \frac{{r_{kw} r_{k} }}{{6r_{b} }}} \right)\ell^{2} \frac{{\partial^{2} \Psi }}{{\partial x^{2} }} - \left( {\frac{1}{12} + \frac{{7{\mkern 1mu} }}{360}\frac{{r_{kw} r_{k} }}{{r_{b} }}} \right)\ell^{4} \frac{{\partial^{4} \Psi }}{{\partial x^{4} }} \\ & + \left( {\frac{1}{120} + \frac{{31{\mkern 1mu} }}{15120}\frac{{r_{kw} r_{k} }}{{r_{b} }}} \right)\ell^{6} \frac{{\partial^{6} \Psi }}{{\partial x^{6} }} - \ell \frac{\partial \Phi }{\partial x} \\ & \quad = I_{\uppsi } \left( {\ddot{\Psi } - \frac{{\ell^{2} }}{6}\frac{{\partial^{2} \ddot{\Psi }}}{{\partial x^{2} }} + \frac{{7\ell^{4} }}{360}\frac{{\partial^{4} \ddot{\Psi }}}{{\partial x^{4} }} - \frac{{31\ell^{6} }}{15120}\frac{{\partial^{6} \ddot{\Psi }}}{{\partial x^{6} }}} \right) \\ & \ell \frac{\partial \Psi }{{\partial x^{{}} }} - \left( {1 + \frac{1}{12}\frac{{r_{k} r_{kw} }}{{r_{b}^{{}} }}} \right)\Phi + \frac{1}{12}\left[ {r_{k} r_{b}^{2} \left( {1 + \frac{{r_{kw} }}{{6r_{b}^{3} }}} \right) - 1} \right]\ell^{2} \frac{{\partial^{2} \Phi }}{{\partial x^{2} }} \\ & - \frac{1}{144}\left[ {r_{k} r_{b}^{2} \left( {1 + \frac{7}{30}\frac{{r_{kw} }}{{r_{b}^{3} }}} \right) - \frac{1}{60}} \right]\ell^{4} \frac{{\partial^{4} \Phi }}{{\partial x^{4} }} \\ & + \frac{1}{1440}\left[ {r_{k} r_{b}^{2} \left( {1 + \frac{31}{126}\frac{{r_{kw} }}{{r_{b}^{3} }}} \right) - \frac{1}{21}} \right]\ell^{6} \frac{{\partial^{6} \Phi }}{{\partial x^{6} }} \\ & \quad { = }\frac{1}{12}I_{\uppsi } \left[ {1 + \left( {\frac{b}{\ell }} \right)^{2} } \right]\left( {\ddot{\Phi } - \frac{{\ell^{2} }}{6}\frac{{\partial^{2} \ddot{\Phi }}}{{\partial x^{2} }} + \frac{{7\ell^{4} }}{360}\frac{{\partial^{4} \ddot{\Phi }}}{{\partial x^{4} }} - \frac{{31\ell^{6} }}{15120}\frac{{\partial^{6} \ddot{\Phi }}}{{\partial x^{6} }}} \right) \\ \end{aligned} \right.$$

(23)

Lower order formulations are derived retaining terms of fourth and second order, the latter being given in Eq. (20). It is worth to note that also for the higher order model the condition for the positive definiteness of the elastic potential energy turns out to be $r_{k} r_{b}^{2} \left( {1 + \frac{{r_{kw} }}{{6r_{b}^{3} }}} \right) > 1.$

Appendix 2: Standard continualization

Let us consider a standard continualization of (17) through the introduction of the shift operator previously introduced. The pseudo-differential problem of equilibrium for the i-th block takes the form

$$\left\{ \begin{aligned} & \left[ {\exp \left( {\ell D} \right) - \left( {2 + \frac{{r_{kw} r_{k} }}{{r_{b} }}} \right) + \exp \left( { - \ell D} \right)} \right]\uppsi_{i} \\ & - \frac{1}{2}\left[ {\exp \left( {\ell D} \right) - \exp \left( { - \ell D} \right)} \right]\upvarphi_{i} = I_{\uppsi } {\ddot{\uppsi }}_{i} \\ & \frac{1}{2}\left[ {\exp \left( {\ell D} \right) - \exp \left( { - \ell D} \right)} \right]\uppsi_{i} \\ & + \frac{1}{12}r_{k} r_{b}^{2} \left[ {\exp \left( {\ell D} \right) - \left( {2 + \frac{{r_{kw} }}{{r_{b}^{3} }}} \right)\text{ + }\exp \left( { - \ell D} \right)} \right]\upvarphi_{i} \\ & - \frac{1}{4}\left[ {\exp \left( {\ell D} \right) + 2\text{ + }\exp \left( { - \ell D} \right)} \right]\upvarphi_{i} = \frac{1}{12}I_{\uppsi } \left( {1 + r_{b}^{2} } \right)\ddot{\upvarphi }_{i} \\ \end{aligned} \right..$$

(24)

By expanding in a Taylor series the pseudo-differential operators a PDE system is obtained. In case the expansion is truncated to the fourth order one obtains

$$\left\{ {\begin{array}{*{20}l} { - \frac{{r_{kw} r_{k} }}{{r_{b} }}\Psi + \ell^{2} \frac{{\partial^{2} \Psi }}{{\partial x^{2} }} + \frac{1}{12}{\mkern 1mu} \ell^{4} \frac{{\partial^{4} \Psi }}{{\partial x^{4} }} - {\mkern 1mu} \ell \frac{\partial \Phi }{\partial x} - \frac{2}{3}\ell^{3} \frac{{\partial^{3} \Phi }}{{\partial x^{3} }} = I_{\uppsi } \frac{{\partial^{2} \Psi }}{{\partial t^{2} }}} \hfill \\ \begin{aligned} \ell \frac{\partial \Psi }{\partial x} + \frac{2}{3}\ell^{3} \frac{{\partial^{3} \Psi }}{{\partial x^{3} }} - \left( {1 + \frac{1}{12}\frac{{r_{k} r_{kw} }}{{r_{b}^{{}} }}} \right)\Phi + \frac{1}{12}\left( {r_{k} r_{b}^{2} - 3} \right)\ell^{2} \frac{{\partial^{2} \Phi }}{{\partial x^{2} }} \hfill \\ \quad + \frac{1}{144}\left( {r_{k} r_{b}^{2} - 3} \right){\mkern 1mu} \ell^{4} \frac{{\partial^{4} \Phi }}{{\partial x^{4} }} = \frac{1}{12}I_{\uppsi } \left( {1 + r_{b}^{2} } \right)\frac{{\partial^{2} \Phi }}{{\partial t^{2} }} \hfill \\ \end{aligned} \hfill \\ \end{array} } \right.,$$

(25)

and the corresponding elastic potential energy density turns out to be

$$\varPi_{e} = \frac{1}{2}\left[ \begin{aligned} & \frac{1}{6}\frac{{r_{k} r_{kw} }}{{r_{b}^{{}} }}\Phi^{2} + \left( {\ell \frac{\partial \Psi }{\partial x} - \Phi } \right)^{2} + \frac{1}{6}\left( {r_{k} r_{b}^{2} - 5} \right)\left( {\ell \frac{\partial \Phi }{{\partial x^{{}} }}} \right)^{2} \\ & + \frac{1}{6}\ell^{2} \left[ {\frac{\partial }{\partial x}\left( {\ell \frac{\partial \Psi }{\partial x} - \Phi } \right)} \right]^{2} + \frac{1}{6}{\mkern 1mu} \left( {\ell^{2} \frac{{\partial^{2} \Psi }}{{\partial x^{2} }}} \right)^{2} \\ & + \frac{1}{72}\left( {r_{k} r_{b}^{2} - 3} \right){\mkern 1mu} \left( {\ell^{2} \frac{{\partial^{2} \Phi }}{{\partial x^{2} }}} \right)^{2} \\ \end{aligned} \right].$$

(26)

The elastic potential energy density turns out to be positive defined for values of the ratios such that $r_{k} r_{b}^{2} > 5$, which is more restrictive than the corresponding one obtained through the enhanced homogenization $r_{k} r_{b}^{2} \left( {1 + \frac{{r_{kw} }}{{6r_{b}^{3} }}} \right) > 1$ related to the differential problem (20).