I am writing my own code to develop the dispersion curves for lamb waves in aluminum plate. I am having difficulty in finding the roots to my disperison equations. Can anybody please help me with that??

00:28 Jun-27-2008 Thomas Vogt R & D, - - Guided Ultrasonics Ltd, United Kingdom, Joined Apr 2007 ^{22}

Re: Lamb wave dispersion curves ----------- Start Original Message ----------- : Hi, : I am writing my own code to develop the dispersion curves for lamb waves in aluminum plate. I am having difficulty in finding the roots to my disperison equations. Can anybody please help me with that?? ------------ End Original Message ------------

Hi

try this paper for a start:

Lowe, M.J.S., "Matrix techniques for modelling ultrasonic waves in multilayered media", IEEE Trans. Ultrasonics, Ferroelectrics and Frequency Control, vol. 42, pp. 525-542, 1995.

If it doesn't help you'll have to be a bit more specific where your problem is.

Re: Lamb wave dispersion curves ----------- Start Original Message ----------- : : Hi, You can use thids book, wave propagation in solids By : Achenbach or wave propagation in layered anisotropic media By: Adnan h. Nayfeh if you get answer please aware me. : Regards, : Amir ------------ End Original Message ------------

I am writing my own code for plotting dispersion curves for guided waves propagating in two layered cylinder. I have derived the equation of motion and also applied the relevant boundary conditions. I have two questions to ask and would appreciate if any one can answer them

1) what boundary condition should be considered at the interface of the two layers. Should stress continuity and displacement continuity be considered OR strain and displacement continuities be considere. The cylinder is composed of two layers of two different materials.

2) I am unable to find the solution for layer matrix I derived from equation of motion. Please can anyone tell me how I can find the roots of this matrix (dispersion equation)

Thanks.

Regards, Neelima ---------- Start Original Message ----------- : : : Hi, : You can use thids book, : wave propagation in solids By : Achenbach : or : wave propagation in layered anisotropic media : By: Adnan h. Nayfeh : if you get answer please aware me. : : Regards, : : Amir ------------ End Original Message ------------

05:34 Aug-18-2008 James Barshinger Engineering General Electric Global Research Center, USA, Joined Aug 2000 ^{8}

Re: Lamb wave dispersion curves Hi.

Here are a couple of references that should help you out.

1. " Matrix techniques for modeling ultrasonic waves in multilayered media" by Mike Lowe. IEEE Trans on Ultrasonics Ferroelectrics and Frequency Control, vol 42, pp 525-542, 1995.

2. "Guided Wave Propagation in an Elastic Hollow Cylinder Coated with a Viscoelastic Material" by Barshinger and Rose. IEEE Trans on UFFC, vol. 51, pp 1547-1556, 2004.

To your questions.

1. Use continuity of stress and displacement for the boundary condition at the interface.

2. The easiest method to find the roots is to use a bi-section routine. As there are multiple roots, you need to test starting values in your solution space to find a sign change, then use bisection to find the root. This method works well, but can have trouble finding all of the roots when modes are close together. As I recall, Lowe's paper describes a routine that is more robust but probably more complicated to implement.

-Jim

----------- Start Original Message ----------- : -Hi, : : I am writing my own code for plotting dispersion curves for guided waves propagating in two layered cylinder. I have derived the equation of motion and also applied the relevant boundary conditions. I have two questions to ask and would appreciate if any one can answer them : 1) what boundary condition should be considered at the interface of the two layers. : Should stress continuity and displacement continuity be considered : OR : strain and displacement continuities be considere. The cylinder is composed of two layers of two different materials. : 2) I am unable to find the solution for layer matrix I derived from equation of motion. Please can anyone tell me how I can find the roots of this matrix (dispersion equation) : : Thanks. : Regards, : Neelima : ---------- Start Original Message ----------- : : : : Hi, : : You can use thids book, : : wave propagation in solids By : Achenbach : : or : : wave propagation in layered anisotropic media : : By: Adnan h. Nayfeh : : if you get answer please aware me. : : : Regards, : : : Amir ------------ End Original Message ------------

Re: Lamb wave dispersion curves ----------- Start Original Message ----------- : Hi. : Here are a couple of references that should help you out. : 1. " Matrix techniques for modeling ultrasonic waves in multilayered media" by Mike Lowe. IEEE Trans on Ultrasonics Ferroelectrics and Frequency Control, vol 42, pp 525-542, 1995. : 2. "Guided Wave Propagation in an Elastic Hollow Cylinder Coated with a Viscoelastic Material" by Barshinger and Rose. IEEE Trans on UFFC, vol. 51, pp 1547-1556, 2004. : To your questions. : 1. Use continuity of stress and displacement for the boundary condition at the interface. : 2. The easiest method to find the roots is to use a bi-section routine. As there are multiple roots, you need to test starting values in your solution space to find a sign change, then use bisection to find the root. This method works well, but can have trouble finding all of the roots when modes are close together. As I recall, Lowe's paper describes a routine that is more robust but probably more complicated to implement. : : -Jim : : -Hi, : : : : I am writing my own code for plotting dispersion curves for guided waves propagating in two layered cylinder. I have derived the equation of motion and also applied the relevant boundary conditions. I have two questions to ask and would appreciate if any one can answer them : : 1) what boundary condition should be considered at the interface of the two layers. : : Should stress continuity and displacement continuity be considered : : OR : : strain and displacement continuities be considere. The cylinder is composed of two layers of two different materials. : : 2) I am unable to find the solution for layer matrix I derived from equation of motion. Please can anyone tell me how I can find the roots of this matrix (dispersion equation) : : : : Thanks. : : Regards, : : Neelima : : ---------- Start Original Message ----------- : : : : : Hi, : : : You can use thids book, : : : wave propagation in solids By: Achenbach : : : or : : : wave propagation in layered anisotropic media : : : By: Adnan h. Nayfeh : : : if you get answer please aware me. : : : : Regards, : : : : Amir ------------ End Original Message ------------ Dear Neelima, At the interface different types of interface conditions can be imposed. They are (i) Perfect contact conditions (ii) Imperfect contact conditions In the perfect contact case, continuity of displacements and stresses are considered. In the imperfect contact case different varieties of conditions can be imposed leading to different imperfect contacts depending on the problem at hand. The basic problem consists in solving a determinantal equation with the two unknowns - wave number and frequency. Please attempting to solve the equation, non-dimensionalize the variables to avoid numerical inconsistencies. The best book is the one by J.L.Rose " Ultrasonic waves in solid media" All the best