## Asymptotic Analysis of wave Propagation in Regular and Perturbed Helical Springs

Studenteropgave: Kandidatspeciale og HD afgangsprojekt

- Rasmus Nielsen

4. semester, Design af Mekaniske Systemer, Kandidat (Kandidatuddannelse)

This report presents how elastic wave propagation in infinite helical springs of regular and

irregular shape can be analysed. In the report focus is given to the understanding and insight

in the nature and mechanisms of wave propagation and modal coupling, and consequently

analytical tools are mainly utilised to derive explicit formulas and expressions.

Initially the theory of the dynamics of a rod is presented, starting from deriving the Frenet-

Serret equations of a spatial curve and a definition of the natural coordinate system. Next geometrical,

constitutive, and equilibrium relation are established which are combined to give

six governing linear differential equations of second order. The kinematics from the the Timoshenko

beam theory will be employed in the derivation, and finally the six general governing

equations are found.

Part one of the study constitutes an analysis of free vibrations in a regular helical spring. A

dispersion relation is derived which provides the relation between frequency and wave number.

In this context nondimensional variables are introduced to overcome difficulties on illconditioning.

The results presented are regarded as exact, since the governing equations are

solved in strong form, even though the roots of the dispersion polynomial is approximated by

a polynomial solver.

A study of modal coefficients is presented to identify differentmodes, and assess the amount

of coupling between them. It is found that a helical spring has coupling of modes at low frequencies,

but also that these couplings are broken at high frequencies. Additionally the dispersion

relation is subjected to an asymptotic analysis to retrieve approximate explicit solutions to

cut-on frequencies and wave numbers at high and low frequencies. The result from this analysis

facilitates the an assessment of dominant quantities. Consequently it is concluded that

asymptotic analysis is a strong tool to gain understanding of the influence of the parameters

on the wave propagation in the spring.

Part two presents a study of a corrugated helical spring which entails some peculiarities

on the geometrical description of the helix. Different candidate methods for solving the problem

is presented and shortly discussed. Namely the Method of Multiple scales and the WKB-approximation

both known from classical Perturbation Methods. It is found that certain restrictions

needs to be imposed on the corrugation for the problem to be solved by the method

of multiple scales. An approach for provoking a solvability condition is presented and a solution

is stated in general terms since the explicit version is much too cumbersome to be presented.

The solution to the regular helix appears in some sense as the leading order solution to

the corrugated helical spring. It is finally concluded that the solution, to the leading order, can

only be expected to approximate the real solution within a narrow perturbation range.

Part three briefly presents the basics in the Waveguide Finite Element Method (WFE method)

in which advantage is taken on the robustness of the FE method to determine stiffness and inertia

properties of geometrically complicated structures and how this is exploited to predict

the wave propagation in infinite periodic structures. The purpose is to make comparison to

the analytical solutions. A FE model is generated in ANSYS and the stiffness and mass matrix

is imported into MatLab where the calculations are performed. The results from the analytical

methods are lastly compared to the results from the WFE analysis. Good agreement is found

between the analytical method and the WFE for the conventional helix, while it turns out to be

more troublesome to assess the validity of the analytical solution for the perturbed helix due to

the narrow perturbation range.

irregular shape can be analysed. In the report focus is given to the understanding and insight

in the nature and mechanisms of wave propagation and modal coupling, and consequently

analytical tools are mainly utilised to derive explicit formulas and expressions.

Initially the theory of the dynamics of a rod is presented, starting from deriving the Frenet-

Serret equations of a spatial curve and a definition of the natural coordinate system. Next geometrical,

constitutive, and equilibrium relation are established which are combined to give

six governing linear differential equations of second order. The kinematics from the the Timoshenko

beam theory will be employed in the derivation, and finally the six general governing

equations are found.

Part one of the study constitutes an analysis of free vibrations in a regular helical spring. A

dispersion relation is derived which provides the relation between frequency and wave number.

In this context nondimensional variables are introduced to overcome difficulties on illconditioning.

The results presented are regarded as exact, since the governing equations are

solved in strong form, even though the roots of the dispersion polynomial is approximated by

a polynomial solver.

A study of modal coefficients is presented to identify differentmodes, and assess the amount

of coupling between them. It is found that a helical spring has coupling of modes at low frequencies,

but also that these couplings are broken at high frequencies. Additionally the dispersion

relation is subjected to an asymptotic analysis to retrieve approximate explicit solutions to

cut-on frequencies and wave numbers at high and low frequencies. The result from this analysis

facilitates the an assessment of dominant quantities. Consequently it is concluded that

asymptotic analysis is a strong tool to gain understanding of the influence of the parameters

on the wave propagation in the spring.

Part two presents a study of a corrugated helical spring which entails some peculiarities

on the geometrical description of the helix. Different candidate methods for solving the problem

is presented and shortly discussed. Namely the Method of Multiple scales and the WKB-approximation

both known from classical Perturbation Methods. It is found that certain restrictions

needs to be imposed on the corrugation for the problem to be solved by the method

of multiple scales. An approach for provoking a solvability condition is presented and a solution

is stated in general terms since the explicit version is much too cumbersome to be presented.

The solution to the regular helix appears in some sense as the leading order solution to

the corrugated helical spring. It is finally concluded that the solution, to the leading order, can

only be expected to approximate the real solution within a narrow perturbation range.

Part three briefly presents the basics in the Waveguide Finite Element Method (WFE method)

in which advantage is taken on the robustness of the FE method to determine stiffness and inertia

properties of geometrically complicated structures and how this is exploited to predict

the wave propagation in infinite periodic structures. The purpose is to make comparison to

the analytical solutions. A FE model is generated in ANSYS and the stiffness and mass matrix

is imported into MatLab where the calculations are performed. The results from the analytical

methods are lastly compared to the results from the WFE analysis. Good agreement is found

between the analytical method and the WFE for the conventional helix, while it turns out to be

more troublesome to assess the validity of the analytical solution for the perturbed helix due to

the narrow perturbation range.

Sprog | Engelsk |
---|---|

Udgivelsesdato | 31 maj 2011 |

Antal sider | 65 |