Research On Some Fundamental Issues Of Modal Analysis And Its Generalization To Statistical Energy Analysis

Differential equations,which govern the motion of linear system,are always coupled inphysical coordinates,and can be generally decoupled in modal one through an appropriatetransformation.This is the essential of so-called modal analysis.The main object of modalanalysis is to identify system modal parameters,and it can be implemented in time andfrequency domain respectively.Considering its importance and some underlying drawbacks.fundarnental issues in modal analysis,including modal parameter identification as well asdata-fitting and synthesis of frequency response function (FRF),are improved in thisdissertation,and then generalized to statistical energy analysis (SEA) for determining thedynamical properties of modal dense systems.For modal parameter identification in frequency dornain,FRF is first constructed withrational fraction polynomial,and then modal parameters are extracted from FRFs throughlinear least square method.A new strategy to identify modal parameters for modalnon-dense multiple-degree-of-freedom systems is proposed here.In this strategy,a globalidentification process is separated into several parts of single mode identification and aniterative algorithm is applied to improve the precision.This strategy shows higher accuracyand has less restriction on the mode counts compared with traditional ones.Considering thesignificance of FRF's quality on modal parameter identification,data-fitting and synthesismethods of FRF are systematically discussed.An improved discrete orthogonal polynomialin Fourier domain and a FRF-based substructure synthesis method are presented.Theformer is used to replace power polynomial in rational fraction,and in this case thesimultaneous equations for modal parameter identification are decoupled.The latterconsiders the elastic connection between substructures in FRF synthesis of complicatedsystem.Numerical and experimental results show the efficacy of these two methods in FRFdata-fitting and synthesis.We notice that modal parameter identification methods is not appropriate for modal densesystem,but in practice complicated structures under broad-band random excitation areoften modal dense in high frequency.so exact studies of realistic structures are onlypossible for the lowest few vibration modes.For high frequency vibration analysis.SEA iscommonly adopted.Compared to modal analysis.SEA focuses on the rate of power flow among substructures and the spatial distribution of mean-square velocities,but not ondetailed modal behavior.In the light of these characteristics,conservatively-coupleddiscrete and continuous systems under stationary broad-band random excitation are studiedsystematically.The system responses expressed according to modal superposition are firstobtained by using the modal analysis.System state variables are then derived from systemresponses with probabilistic theory of structural dynamics.Based on the assumptions ofweak coupling,modal denseness and strongly modal overlapping,the approximateexpressions of power flow proportionality coefficient,which indicates the average powerflow between substructures,is acquired from the simplified state variables.To anotherimportant problem in SEA,i.e..structural mode counts estimation used for assessingenergy flows in certain frequency band,the exact expression of mode counts for simplysupported square plate is investigated by using corresponding theorem in number theory.