# Vibrational energy distributions in large built-up structures - a wave chaos approach

Lead Research Organisation:
University of Southampton

Department Name: Faculty of Engineering & the Environment

### Abstract

Predicting the response of a large, complex mechanical system such as a car or an aeroplane to high frequency vibrations is a remarkably difficult task. Still, obtaining good estimates for the distribution of vibrational energy in such structures, including coupling between sub-components, damping and energy loss in form of acoustic radiation, is of great importance to engineers. An increasing demand for low vibration, low noise products to meet performance specifications and to reduce noise pollution makes any improvement in predicting vibrations response characteristics of immediate interest for industrial applications. Demand for improved virtual prototyping, as opposed to the use of expensive and time-consuming physical prototypes, is another area of application in reducing development costs and time scales. Numerical tools are often based on 'Finite Element Analysis' (FEM). While these methods work well in the low frequency regime, that is, tackling wavelengths of the order of the size of the system, they become too expensive computationally in the mid-to-high frequency regime. In particular, FEM fails to describe accurately so-called mid-frequency problems where sub-components are characterized by a wide variation of wave-lengths. While FEM is suitable for handling 'stiff' elements such as the body frame in a car, it cannot routinely capture energy transport through 'soft' components such as thin, flexible plates coupled to stiff components. A common numerical tool for predicting the vibrational contribution of short wave length components is Statistical Energy Analysis (SEA); it is, however, based on a set of restrictive assumptions which, so far, are often hard to control and generally only fulfilled in the high frequency limit and for low damping. Thus, SEA can not deliver the degree of reliability necessary to make it attractive for a wider end user community in industrial R & D departments. It is suggested here that mathematical tools from wave or quantum chaos can considerably improve the situation sketched above. Recent results by the PI Tanner show that by combining methods ranging from operator theory, dynamical systems theory and small wavelength asymptotics, SEA can be embedded into a more general theory. The new approach is based on semiclassical expansions of the full Green function in terms of rays and describing the nonlinear ray-dynamics in terms of linear operators. The resulting method captures the full correlations in the ray dynamics and has such a much improved range of validity compared to SEA. The method could revolutionise the treatment of vibrations in complex mechanical systems. Not only does it allow (i) to give quantitative bounds for the applicability of SEA (of interest to SEA users); it will also (ii) improve predictive capability in situation where SEA does not apply at a moderate computational overhead; in addition, (iii) it can be easily combined with FEM methods thus making it an ideal candidate for tackling mid-frequency problems. The approximations made are well controlled by starting from a semiclassical approach which makes it possible (iv) to systematically incorporate wave interference effects (absent in standard SEA treatments) into the method.By tackling the issues addressed above we will be able to provide improved and conceptually completely new solution methods to the engineering community based on advanced mathematical methods. The proposed research evolved out of pump-prime EPSRC funding in terms of a Springboard Fellowship. The project is thus by default of interdisciplinary nature and will be tackled jointly by the PI Tanner (Nottingham, Mathematics) and PI Mace (Southampton, ISVR, Engineering) with industrial partners from the FEM/software side (inuTech) and an engineering consulting firm (DS2L) providing input about end-user demands.

## People |
## ORCID iD |

Emiliano Rustighi (Principal Investigator) |

### Publications

Chappell D
(2013)

*Solving the stationary Liouville equation via a boundary element method*in Journal of Computational Physics
Chappell D
(2012)

*Boundary element dynamical energy analysis: A versatile method for solving two or three dimensional wave problems in the high frequency limit*in Journal of Computational Physics
Chappell D
(2013)

*Discrete flow mapping: transport of phase space densities on triangulated surfaces*in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Chappell DJ
(2011)

*Dynamical energy analysis for built-up acoustic systems at high frequencies.*in The Journal of the Acoustical Society of America
Renno J
(2011)

*Calculating the forced response of two-dimensional homogeneous media using the wave and finite element method*in Journal of Sound and Vibration
Renno J
(2016)

*A Finite Element Method for Modelling Waves in Laminated Structures*in Advances in Structural Engineering
Renno J
(2013)

*Calculation of reflection and transmission coefficients of joints using a hybrid finite element/wave and finite element approach*in Journal of Sound and Vibration
Renno J
(2012)

*Vibration modelling of helical springs with non-uniform ends*in Journal of Sound and Vibration
Renno J
(2010)

*On the forced response of waveguides using the wave and finite element method*in Journal of Sound and Vibration
Renno J
(2014)

*Vibration modelling of structural networks using a hybrid finite element/wave and finite element approach*in Wave MotionDescription | In this project, we developed applications of the wave and finite element (WFE) method. This technique is suitable for modelling general waveguides, which are homogeneous in one dimension, but whose cross-section may have arbitrary complexity, or two-dimensional structures, such as plates and panels, which are homogeneous in two dimensions, but whose properties may vary in an arbitrary manner through the thickness. To resolve the wave behaviour of the whole structure (which can be arbitrarily long/large), the standard FE model of a small segment should be post-processed using periodic structure theory. The advantage of this method is two-folds: the FE model can be obtained using any commercial/in-house FE package and thus the full power of existing codes can be harnessed, and the model to be processed is very small which is computationally advantagoues. During this project, the WFE method was used to efficiently predict the response of structures subjected to time harmonic, arbitrarily distributed loads. It was also used to find the scattering properties of arbitrarily complicated joints, and to describe the vibrational behaviour of structural networks using waves only. These developments will contribute to bridging the mid-frequency gap by increasing the range of applicability of the FE method. Industrial applications included modelling the wave behaviour of a train floor panel, and predicting the resposne of a railcar cross-section. These are large structures, and modelling their vibration withing the audio-frequency range can lead to impractically large models. |

Exploitation Route | Outcomes of the research are still being developed by researchers and PhD students. The researcher, Dr Jamil Renno, is now working at Doosan Babcock employing the developed methodologies. Inutech (Innovative Numerical Technologies) is also using the technique. |

Sectors | Aerospace, Defence and Marine |

Description | The researcher employed in the grant, Dr Jamil Renno, is now an employer of Doosan Babcock, where he is applying Wave-FE models to plant integrity. |

First Year Of Impact | 2015 |

Sector | Energy,Other |

Impact Types | Economic |