Vibro-Acoustics of One-Dimensional Periodic Structures Using FEM

Vibro-Acoustics of One-Dimensional Periodic Structures Using FEM

Propagation constants of a fluid-loaded periodic beam using FEM

Abhijit Sarkar, Venkata. R. Sonti

Facility of Research in Technical Acoustics

Department of Mechanical Engineering

Indian Institute of Science, Bangalore- 560 012

Abstract

The finite element method is used to compute the propagation constants for a fluid loaded infinite periodically supported beam. At most frequencies there is at least a single propagating wave which corresponds to the acoustical pressure. At other frequencies there exist two or even more propagating waves because of the presence of coupling between multiple domains. The coupled natural frequencies of the periodic unit are closely associated with the critical points on the propagation constant curve.

1Introduction

Railway lines, ship hulls, multi-span bridges, aircraft fuselages are periodic structures where a basic unit (span or period) is repeated throughout the structure. Conveniently, one needs only to model the basic unit and impose periodicity conditions to obtain the system equations for the whole structure. There is extensive literature on periodic structures, a comprehensive review of which is presented in [1]. Although fluid loading has been investigated in the context of sound radiation from periodic structures [2, 3], most of the related literature has struggled with semi-analytical and semi-numerical methods and attempted mostly 1-D problems. With a final goal of attempting a large scale real world periodic structure problem (ship hulls and submarines) using only numerics (FEM/BEM), in this paper we present a preliminary study of a simple fluid-loaded infinite periodically supported beam.

The specific system considered involves an infinite beam with periodic supports. The beam is loaded all along its length with a column of water of height h (see figure 1). We use the Finite Element Method to formulate the equations and analyze this system.

Figure 1. Schematic of an infinite fluid loaded periodically supported beam.

2Theory

For an infinite periodically supported beam in free vibration, the displacement in one span is times that in the neighboring span [4]. The variable  is usually complex and is called the propagation constant. The real part of  is called the attenuation constant and the imaginary part the phase constant. For the system considered, with the fluid loading neglected, the propagation constant is presented in figure 2. The material of the beam considered is steel with a Young’s Modulus of 200Gpa and a density of 7840kg/. The beam has a square cross-section of side 0.1m and rests on simple supports placed at intervals of 1m. It was verified thateach propagation band starts from the natural frequency of the single span simply supported beam and terminates at the corresponding clamped-clamped natural frequency [4].

Figure 1: Analytical evaluation of propagation constant in the uncoupled case

The FE implementation for structural dynamics is given in any introductory textbook on finite element method [5]. The FE model in frequency domain (with circular frequency ) leads to the following equation

. (1)

The FE formulation of the acoustic helmholtz equation [6] gives a similar equation

. (2)

In equations (1) and (2), M and K denote the mass and stiffness matrices, respectively. The subscript s and a refer to the structural and acoustic quantities. F and v are the structural and acoustic load vectors, respectively. c is the sonic velocity in the acoustic medium. In structural acoustics, the acoustic pressures form the structural load, i.e., there is a relation of the form {F} = [ C ] { p }. Also, at the fluid-structure interface the acoustic velocities need to match the structural velocities. This would lead to a relation of the form {v} = [S]{x}. The exact form of the C and S matrices is as given in [7]. Combining these relations with equations (1) and (2) we get the following eigenvalue problem for the coupled structural acoustic system

. (3)

Periodicity conditions are then enforced in equation (3). Orris and Petyt [8] describes the methodology for this. As a result we have dependent stiffness and mass matrices. The resulting complex eigenvalue problem is

. (4)

The expressions for and are given in [8]. Solution of the eigenvalues () for purely imaginary values of (between - and ) gives the propagation bands. Every imaginary value of  results in n eigenvalues, where n is the system size of equation (4). Of these n eigenvalues, the ith one is a single point in the ith propagation band.

3Results and Discussion

Using the methodology elaborated, we compute the propagation constants for the fluid loaded infinite beam shown in figure 1. Figure 3 shows the propagation constant in the frequency range 0 to 1.1 KHz. This result contrasts with the earlier result of uncoupled analysis as shown in figure 2. There are a few salient features in this result that we wish to highlight in the following discussion.

The complex eigenvalue problem in equation (4) solved for imaginary values of between – and  results in real eigenvalues . The imaginary part of the calculated eigenvalues are about times the real part. This is in consonance with the physical understanding of the problem. In the absence of any dissipation mechanism in the model, purely imaginary should correspond to real frequencies. For an unbounded acoustic medium with non-reflecting boundary conditions at infinity the structure would encounter a radiation damping effect. The imaginary  values will result in complex frequencies and in turn real frequencies will result in complex  values. There will be no more a pure propagation band. Future work is planned in this direction. The mathematical reason for getting real eigenvalues on solving the complex eigenvalue problem of equation (4) is however not clear. The observation that imaginary values result in real eigenvalues for demands that there be a certain structure in the M and K matrices. It may be noted that due to the coupling terms these matrices are no longer symmetric. However symmetry is a sufficient condition for getting real eigenvalues and is not a necessary condition.

As seen in figure 3, for a given frequency, if is a propagation constant, then - is also a propagation constant. The sign reversal actually implies reversal of propagation direction of the free waves. From hereon, we focus on the positive propagation constant.

Figure 3: Imaginary part of propagation constant for the coupled problem

We have chosen imaginary values of  in solving equation (4). Thus, our focus is only on propagation zones for the system considered. It is observed that each frequency has at least one imaginary propagation constant and some have even more. The existence of more than one propagation constant for some frequencies seems reasonable as there are more than one coupling degrees of freedom between the periodic units, namely - beam rotation and acoustic pressure [4]. The acoustic pressure propagates at all frequencies. The straight line segment in the propagation constant curve in the figure between 0 and 750 Hz is indeed the acoustical dispersion curve (since the span length is 1m). The straight line with the negative slope beyond 750 Hz occurs because  values are phase wrapped around  and –. As seen from figure 2, the first propagation band for the uncoupled beam starts at 221Hz and ends at 519Hz. It is noted from figure 3 that within this band there are two propagation constants. Thus, in this band the energy is carried by both the acoustic pressure as well as the beam rotation.

The results of the uncoupled analysis in figure 2 showed that the natural frequencies of the simply supported and clamped-clamped periodic unit form critical points of the propagation constant curve. Two more cases have been analyzed with regard to the single fluid-loaded span in figure 1. In the first case, the beam was simply supported and zero velocity was assumed along the boundary. The first three natural frequencies were 229Hz, 682Hz and 750Hz. In the second case, the beam was clamped on both ends and all acoustic pressures at the left and right edges were constrained to zero. In this case, the first three natural frequencies were 519Hz, 750Hz and 1014Hz. The frequencies corresponding to the former case are presented in figure 3 with bold solid vertical lines while for the latter case they are in bold dash-dot vertical lines. As with the uncoupled analysis these frequencies form critical points of the graph of the propagation constant. Further investigations are underway to understand these results more clearly.

4Conclusion

Propagation constants for a fluid-loaded infinite periodic beam were presented using an FEM formulation. Although a very simple system was considered, physically meaningful results were obtained. It was found that there is a propagating acoustical wave at all frequencies, whose propagation curve matches the acoustical dispersion curve. At some frequencies there exist two or more propagating waves which are related to the coupled fluid-structure degrees of freedom.

The future plan would be to extend this analysis for the case of a fluid loaded two-dimensional plate. Also it is of greater interest to model an unbounded acoustic medium in the present setting. Thus, use of the infinite element method and the boundary element method could be explored. However, as noted earlier the unbounded acoustic medium would be equivalent to introducing a dissipation mechanism in the model in the form of radiation damping. This would complicate the simulation process. Also, practical structures are never infinite. Structures having a large number of periodic repetitions approximate the behaviour of infinite periodic structures. There is an easy and efficient method of arriving at the natural frequencies and mode shapes of such large, finite, periodic structures from the behaviour of its infinite counterpart. This has been covered in the context of uncoupled analysis by Sengupta [9]. It would be challenging to arrive at an analogous method for coupled structural acoustics.

References

[1] D. J. Mead 1996 Journal of Sound and Vibration 190(3), 495-524, Wave propagation in continuous periodic structures: Research contributions from Southampton, 1964-1995.

[2] B. R. Mace 1980 Journal of Sound and Vibration 73(4), 473-486, Periodically stiffened fluid loaded plates, 1: response to convected harmonic pressure and free wave propagation.

[3] B. R. Mace 1980 Journal of Sound and Vibration 73(4), 487-503, Periodically stiffened fluid loaded plates, 1: response to line and point forces

[4] D. J. Mead 1970 Journal of Sound and Vibration 11(2), 181-197, Free wave propagation in periodically supported infinite beams.

[5] T. R. Chandrupatla and A. D. Belegundu, Introduction to finite elements in Engineering, Prentice HallIndia.

[6] A. Craggs, Acoustic Modelling: Finite Element methods in Encyclopedia of Acoustics edited by Malcolm J. Crocker, John Wiley & sons Inc.

[7] A. Sarkar, V. R. Sonti and R. Pratap 2005 International Journal of Acoustics and Vibration 10(1), 1-17, A coupled FEM-BEM formulation in structural acoustics for imaging material inclusions.

[8] R. M. Orris and M. Petyt 1974 Journal of Sound and Vibration33, 223-236, A finite element study of harmonic wave propagation in periodic structures.

[9] G. Sengupta 1970 Journal of Sound and Vibration 13(1), 89-101, Natural flexural waves and the normal modes of periodically supported beams and plates.