Resumen de Vibroacoustic coupling and transmission paths
Francisco Javier Magrans Fontrodona
This dissertation deals with four topics. The first three are in the same environment, the transmission paths. The fourth refers to the synthesis of subsystems and more specifically to two subsystems linked by any number of elastic elements.
In the first topic it is proved that the solution of any linear mechanical system can be expressed as a linear combination of signal transmission paths. This is done in the framework of the Global Transfer Direct Transfer (GTDT) formulation for vibroacoustic problems. Transmission paths are expressed as powers of the transfer matrix. The key idea of the proof is to generalise the Neumann series of the transfer matrix ,which is convergent only if its spectral radius is smaller than one, into a modified Neumann series that is convergent regardless of the eigenvalues of the transfer matrix. The modification consists in choosing the appropriate combination coefficients for the powers of the transfer matrix in the series. A recursive formula for the computation of these factors is derived. The theoretical results are illustrated by means of numerical examples. Finally, we show that the generalised Neumann series can be understood as an acceleration of Jacobi iterative method.
For complex geometries, the definition of the subsystems is not a straightforward task. We present as a second topic a subsystem identification method based on the direct transfer matrix, which represents the first-order paths. The key ingredient is a cluster analysis of the rows of the powers of the transfer matrix. These powers represent high-order paths in the system. Once subsystems are identified, the proposed approach also provides a quantification of the degree of coupling between subsystems. This information is relevant to decide whether a subsystem may be analysed independently of the rest or subsystems or not. The two features (subsystem identification and quantification of the degree of coupling) are illustrated by means of numerical examples: plates coupled by means of springs and rooms connected by means of a cavity.
In the third work, Advanced Transfer Path Analysis (ATPA) is applied to a cuboid-shaped box. The simplicity of this vibroacoustic system helps to make a detailed analysis of the ATPA method in a more controlled environment than in situ measurements in trains, wind turbines or other mechanical systems with complex geometry, big dimensions and movement. At the same time, a numerical model (based on finite elements) of the box is developed. The agreement between the experimental measurements and the numerical results is good. The numerical model is used in order to perform tests that cannot be accomplished in practise. The results are helpful in order to verify hypotheses, provide recommendations for the testing procedures and study some aspects of ATPA such as the reconstruction of operational signals by means of direct transfer functions or to quantify and understand which are the transmission mechanisms in the box.
The fourth topic introduces a method to synthesize the modal characteristics of a system from the modal characteristics of its subsystems. The interest is focused on those systems with elastic links between the parts which is the main feature of the proposed method. An algebraic proof is provided for the case of arbitrary number of connections. The solution is a system of equations with a reduced number of degrees of freedom that correspond to the number of elastic links between the subsystems.
In addition the method is also interpreted from a physical point of view (equilibrium of the interaction forces). An application to plates linked by means of springs shows how the global eigenfrequencies and eigenmodes are properly computed by means of the subsystems eigenfrequencies and eigenmodes.
