Reduced-order models for nonlinear vibrations of fluid-filled circular cylindrical shells: Comparison of POD and asymptotic nonlinear normal modes methods

TLDR: In this paper, the authors compare two different methods available for reducing the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell, which is a water-filled, simply supported circular cylindrical shell subjected to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency.

About: This article is published in Journal of Fluids and Structures.The article was published on 2007-08-01 and is currently open access. It has received 90 citations till now. The article focuses on the topics: Galerkin method & Nonlinear system.

Figures

Fig. 2. Time response at excitation frequency o=o1;n ¼ 0:991, for excitation of 3N; conventional Galerkin model, 16 dofs: (a) modal coordinate A1;nðtÞ associated to the driven mode; (b) modal coordinate B1;nðtÞ associated to the companion mode.

Fig. 8. Poincaré section in ðA1;5;A1;0Þ. Cloud of points: section of the orbits with the Poincaré plane, cases (b) and (c). POD differs very little from the original axis, whereas the cut of the 4-dimensional invariant manifold (curved hyperbolic line) goes right in-between the points (NNMs).

Fig. 7. Projection of the 32-dimensional state space onto the 3-dimensional subspace spanned by ðA1;5; _A1;5;A1;0Þ. Closed orbit: forced motion for o=o1;n ¼ 0:99 and for excitation of 3N, case (a). Curved surface: invariant manifold corresponding to the first NNM (driven mode).

Table 1 Coefficients of the main proper orthogonal modes (POMs)

Fig. 1. Maximum amplitude of vibration versus excitation frequency, for excitation of 3N; conventional Galerkin model, 16 dofs. (a) Maximum amplitude of A1;nðtÞ, driven mode; (b) maximum amplitude of B1;nðtÞ, companion mode; 1, branch ‘‘1’’; 2, branch ‘‘2’’; BP, pitchfork bifurcation; TR, Neimark–Sacker (torus) bifurcations. ——, stable periodic solutions; F’F, stable quasi-periodic solutions; ——, unstable periodic solutions.

Fig. 6. Maximum amplitude of vibration versus excitation frequency, for excitation of 8N; conventional Galerkin model, POD model with 3 modes and NNMs model with 2 modes. (a) Maximum amplitude of A1;nðtÞ, driven mode; (b) maximum amplitude of B1;nðtÞ, companion mode. ——, conventional Galerkin model (16 dofs); ——, POD model (3 dofs); ——, NNMs model (2 dofs).

Frequently asked questions

Q1. What are the contributions in "Reduced-order models for nonlinear vibrations of fluid-filled circular cylindrical shells: comparison of pod and asymptotic nonlinear normal modes methods" ?

The aim of the present paper is to compare two different methods available for reducing the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell. 

Q2. What is the result of the asymptotic NNMs method?

When increasing the nonlinearities by feeding more energy into the system, the results provided by the asymptotic NNMs method are expected to deteriorate. 

Q3. What is the norm of the basis functions ti in the present case?

The norm of the basis functions ti in the present case is pRL=2 for asymmetric modes and pRL for axisymmetric modes; without effect on the results, they can be assumed to be 0.5 and 1, respectively. 

Q4. What is the property of the invariant manifold that defines NNMs?

The nonlinear character of the invariant manifold that defines NNMs allows better reduction than the POD method, which is a linear decomposition. 

Q5. What is the irrotational theory of the fluid?

The contained fluid is assumed to be incompressible, inviscid and irrotational, so that potential flow theory can be used to describe fluid motion. 

Q6. What is the advantage of the proposed method?

The proposed method has the advantage of simplicity, quickness of computation, and allows deriving a differential model that could be used easily for parametric studies. 

Q7. What is the main advantage of the POD method?

The POD method, which consists in finding the best orthogonal hyper-planes that contain most information, is essentially a linear method. 

Q8. What is the main limitation of the POD method?

for very large vibration amplitude and large range of parameter variations, the POD method still performs better due to its global nature. 

Q9. What is the effect of the fluid on the shell?

Usually the inertial effect of the fluid is larger for axisymmetric modes, thus enhancing the nonlinear softening-type behaviour of the shell. 

Q10. What is the advantage of the POD method?

The POD method is global in the sense that it is able to capture any motion in state space and furnishes the adapted basis for decomposing it. 

Q11. What are the two popular methods used to build ROMs?

By far, the two most popular methods used to build ROMs are the proper orthogonal decomposition (POD) andthe nonlinear normal modes (NNMs) methods. 

Q12. What is the choice for discretizing the shell?

The linear modal base is the best choice for discretizing the shell, as these are the eigenfunctions of the linear operator of the PDE. 

Q13. What is the reason why two NNMs are necessary to recover the dynamics?

This explains why only two NNMs are necessary to recover the dynamics, as the four-dimensional manifold displays an important curvature in the direction of A1;0. 

Q14. What is the way to build a ROM of a water-filled shell?

Both the proper orthogonal decomposition (POD) and the nonlinear normal modes (NNMs) methods have been verified to be suitable for building ROMs of a water-filled shell.