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Summary of Masters Thesis
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Vibration Energy Flow Analysis in Randomly Parametered Coupled Plate Structures using Green’s Function Coupling Technique Structural
dynamic analysis aims to establish relationships between externally applied
loads and displacements/strains/stresses fields in the vibrating structures. The
most commonly used approach to achieve this consists of discretization of
displacement field using finite element method and subsequent solution of
equilibrium equations using normal mode expansion. This approach, although, has
been successfully applied in a wide variety of contexts, nevertheless, has not
been well received in some areas. For instance, the computational demands that
the method imposes increase significantly with increases in frequency range of
interest. In many engineering applications, such as building acoustics,
satellite structural dynamics, automobile and aircraft dynamics, it becomes
important to model high frequency vibrational response. It is, in principle,
possible to extend analysis to higher frequencies, at the expense of rapidly
increasing demands in terms of size of the model, and consequent analysis time
and cost. However, as frequencies increase, the results become increasingly
dependent upon fine structural detail, including structural connections, which
cannot be mathematically represented with sufficient accuracy. In addition, the
high frequency vibrational behaviour of individual physical realizations of
nominal structures are observed to differ, often greatly, because of the
influence of fabrication tolerances. It is clear that the high cost of
computational procedures based upon the deterministic models cannot be justified
if they do not yield reliable results. In such circumstances other methods may
be more appropriate. One
such approach is the Statistical Energy Analysis (SEA) formalism. Here, the
primary objective of the response analysis is to predict the space-wise and
frequency-wise distribution of total vibrational energy residing in the
structure. Detailed spatial distribution, however, is not sought. Instead,
vibrational energy fields integrated over certain spatial domains are considered
as being of interest. Such a macro-level response description is distinct from
the detailed description that finite element method aims to obtain. This method,
or more accurately, analysis philosophy, dates from the early 1960's, when
engineers sought new analytical tools for dealing with the problems of
predicting the response of launcher and payload structures to rocket noise at
launch. System parameters were expressed in probabilistic terms, and the
objective of an analysis was seen to be the prediction of the ensemble-average
behaviour of sets of grossly similar realizations of an archetypal system.
System response to vibrational inputs was characterized by time-average
vibration energy; energy flows between coupled subsystems were expressed in
terms of energy transfer coefficient; and vibration distribution was determined
from power balance equations. The basic principle was that the energy transfer
between two subsystems is proportional to the difference of their average
vibration energies. The proportionality constant was termed as the coupling loss
coefficient. This lead to the result that the input vibrational energies are
expressible as a linear combination of the averaged subsystem energies and the
coupling loss factors, i.e., {P}= [CLF]{E}.
Here {P} is the vector of input power averaged over one cycle, {E}
is the vector of time-averaged subsystem energy and [CLF]
is the matrix of coupling loss factors. This approach to modelling vibrational
behaviour can be shown to lead to acceptable solutions only if a set of
restrictive assumptions such as linear system behaviour, conservative coupling
mechanisms, random uncorrelated forcing, weak coupling between subsystems etc.,
are satisfied. The implications of these restrictions on the scope of SEA
continue to remain subject of current research. The objective of the present thesis is to examine the validity of some of the
mechanistic and probabilistic assumptions underlying SEA formalisms. The study
is conducted within the context of vibration energy flow (VEF) analysis of the
plate assemblies that are discretely coupled through a set of axially vibrating
rods; this class of structure being representative of satellite solar panels
tested in laboratory conditions. The thesis is organized into five chapters. The first chapter provides the details of SEA formalisms and critically examines
the underlying assumptions. Literature pertaining to theoretical basis of SEA
and methods for determining the SEA parameters are examined. VEF analysis that
employs substructure Green's function and a frequency domain coupling technique
is also discussed. The merits of A general formulation of VEF models for a set of coupled multi-modal structural
systems is outlined in chapter 2. These models make use of the uncoupled
subsystem Green's functions. At no stage in the analysis, the global normal
modes of the built-up structure are computed. A general-purpose
software on MATLAB platform has
been developed based on this formulation. This software accepts, as inputs, the
subsystem Green function over a given frequency range and produces estimates of
spectra of frequency band averaged estimates of the total subsystem energies.
The formulation is exemplified by considering VEF models of a structure that
consists of a set of three plates that are discretely coupled by a set of
axially vibrating rod elements. Results based on this formulation are compared
with FEM results (using NISA)
that require significantly more computational effort. The basic requirements on
vibration energy balance are shown to be satisfied, thereby providing a
validation of the software developed. The procedure developed in chapter 2 is extended in chapter 3 to obtain the
descriptors of VEF such as coefficients of coupling loss factor (CLF) matrix.
This extension employs the use of power injection method (PIM) that has been
developed originally in the context of experimental determination of CLF
coefficients. Specific issues on structure of CLF matrix that are associated
with far coupling effects in energy flow are investigated. Examples considered
again include coupled plate structures. The problems of characterizing stochastic variability in energy levels and also
on the nature of probability distribution function of energy levels are
considered in chapter 4. The elastic, geometric and inertial properties of the
plate subsystems are modelled as a vector of random variables. Influence of
varying damping properties on statistics of energy flow is examined. The study
in this part of the chapter employs Monte Carlo simulation (MCS) technique. A
new idea to define the satisfactory performance of SEA is considered in the
second part. This latter part of the chapter uses the ideas of reliability
indices borrowed from literature on structural reliability analysis and compares
the results with similar results obtained from MCS technique. The thesis concludes with chapter 5 wherein a summary of contributions made and
a few suggestions for future research are provided. Appendix A presents the definition reliability index and the algorithm that is used for computation of reliability index.
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