Modelling Wave Behaviour of Elastic Helical Waveguides

Abstract

Curved and straight waveguides are ubiquitous in engineering structures. For modelling these structures in the mid-frequency and high-frequency range, the knowledge of the wave behaviour can be greatly beneficial. This paper addressed the modelling of such structures using the wave and finite element (WFE) method. In particular, power flow is formulated and investigated, and two numerical examples are presented to demonstrate how the method can be used to analyse structures that comprise straight and helical waveguides.

Appendix 1. Matrices of Analytical Eigenvalue Problem

As per the notation of [16], we will use the following:

$$ q= \cos ^2 (\psi ), \ r=\sin (\psi ) \cos (\psi ), \ \epsilon =\frac{d}{R}, $$

and $\kappa$ is the shear coefficient. The matrices of Eq. (2) are scaled to the left and right with a diagonal matrix having a diagonal (1, 1, 1, 1/d, 1/d, 1/d):

$$ \mathbf {A}_0= \left[ \begin{array}{cccccc} -\frac{\epsilon ^2 \left( 2 (\nu +1) q^2+r^2 \kappa \right) }{2 (\nu +1)} &{} 0 &{} 0 &{} -\frac{r \epsilon \kappa }{2 \nu +2} &{} 0 &{} 0 \\ 0 &{} -\frac{r^2 \epsilon ^2 \kappa }{2 (\nu +1)} &{} \frac{q r \epsilon ^2 \kappa }{2 \nu +2} &{} 0 &{} -\frac{r \epsilon \kappa }{2 \nu +2} &{} 0 \\ 0 &{} \frac{q r \epsilon ^2 \kappa }{2 \nu +2} &{} -\frac{q^2 \epsilon ^2 \kappa }{2 (\nu +1)} &{} 0 &{} \frac{q \epsilon \kappa }{2 \nu +2} &{} 0 \\ -\frac{r \epsilon \kappa }{2 \nu +2} &{} 0 &{} 0 &{} -\frac{q^2 \epsilon ^2+r^2 (\nu +1) \epsilon ^2+8 \kappa }{16 (\nu +1)} &{} 0 &{} 0 \\ 0 &{} -\frac{r \epsilon \kappa }{2 \nu +2} &{} \frac{q \epsilon \kappa }{2 \nu +2} &{} 0 &{} -\frac{r^2 (\nu +1) \epsilon ^2+8 \kappa }{16 (\nu +1)} &{} \frac{1}{16} q r \epsilon ^2 \\ 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{16} q r \epsilon ^2 &{} -\frac{1}{16} q^2 \epsilon ^2 \\ \end{array} \right] , $$

$$ \mathbf {A}_1= \imath \left[ \begin{array}{cccccc} 0 &{} -\frac{r \epsilon \kappa }{\nu +1} &{} \frac{q \epsilon (\kappa +2 \nu +2)}{2 (\nu +1)} &{} 0 &{} -\frac{\kappa }{2 (\nu +1)} &{} 0 \\ \frac{r \epsilon \kappa }{\nu +1} &{} 0 &{} 0 &{} \frac{\kappa }{2 \nu +2} &{} 0 &{} 0 \\ -\frac{q \epsilon (\kappa +2 \nu +2)}{2 (\nu +1)} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} -\frac{\kappa }{2 (\nu +1)} &{} 0 &{} 0 &{} -\frac{r \epsilon }{8} &{} \frac{q \epsilon (\nu +2)}{16 (\nu +1)} \\ \frac{\kappa }{2 \nu +2} &{} 0 &{} 0 &{} \frac{r \epsilon }{8} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -\frac{q \epsilon (\nu +2)}{16 (\nu +1)} &{} 0 &{} 0 \\ \end{array} \right] , $$

$$ \mathbf {A}_2= \left[ \begin{array}{cccccc} 0 &{} -\frac{r \epsilon \kappa }{\nu +1} &{} \frac{q \epsilon (\kappa +2 \nu +2)}{2 (\nu +1)} &{} 0 &{} -\frac{\kappa }{2 (\nu +1)} &{} 0 \\ \frac{r \epsilon \kappa }{\nu +1} &{} 0 &{} 0 &{} \frac{\kappa }{2 \nu +2} &{} 0 &{} 0 \\ -\frac{q \epsilon (\kappa +2 \nu +2)}{2 (\nu +1)} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} -\frac{\kappa }{2 (\nu +1)} &{} 0 &{} 0 &{} -\frac{r \epsilon }{8} &{} \frac{q \epsilon (\nu +2)}{16 (\nu +1)} \\ \frac{\kappa }{2 \nu +2} &{} 0 &{} 0 &{} \frac{r \epsilon }{8} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -\frac{q \epsilon (\nu +2)}{16 (\nu +1)} &{} 0 &{} 0 \\ \end{array} \right] , $$

and finally

$$ \mathbf {B}= \left[ \begin{array}{cccccc} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \frac{1}{16} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{16} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{8} \\ \end{array} \right] . $$