by Sondipon Adhikari
MSc Thesis Abstract, Department of Civil Engineering,
IISc Bangalore.
Degree Awarded: September 1997.
Thesis Supervisor: Dr. C. S.
Manohar.
The field of stochastic structural mechanics is currently an active area of research. The primary concern here lies in the probabilistic modeling of the uncertainties in specifying elastic, mass and geometric properties of the structure. This is in contrast with the goals of the more traditional random vibration studies which primarily focus attention on probabilistic modeling of uncertainties associated with loads such as those caused due to earthquakes, wind and guideway irregularities. The motivation for the study of system randomness arises, on one hand, from the need to make the structural safety assessment more realistic, and, on the other hand, from the need to study associated phenomenological response features such as mode localization and system behavior in higher frequency regions. The latter class of problems are relevant in vibro-acoustic design of vehicles. Besides, the recent developments in the fields of robotics and composite structural materials have provided impetus towards developing generic procedures for handling structural randomness. Problems of linear deterministic structural dynamics have been extensively treated in the existing literature within the frameworks of finite element method using normal mode expansions, transfer matrix method and dynamic stiffness matrix approach. When these methods are to be extended to analyze structures with random material and/or geometric property variations, questions on solutions of random differential and algebraic eigenvalue problems, inversion of random matrix and differential operators and characterization of random matrix products arise. The present thesis aims at contributing new methodologies in achieving some of these extensions.
The thesis also considers the problem of vibration energy flow modeling in statistical ensemble of vibrating systems. When the frequency range of dynamical forces acting on engineering structures encompass several modes of vibrations, the prediction of vibrational response of the structures poses difficult challenges. These situations are widely encountered in vibro-acoustic design of aerospace, marine and automobile structures. In these problems, in order that higher mode responses are captured correctly, the discretization procedures, such as, finite element method, require very detailed idealization of the physical structures. This, not only, results in unmanageable computational difficulties, but also, fails to address a more fundamental problem, namely, the extreme sensitivity of higher mode response to minor changes in system parameters and modeling. The Statistical Energy Analysis (SEA) procedure has been developed in the literature to address these difficulties. A distinguishing feature of SEA modeling is that it treats the vibrating system as a member of a statistical ensemble of nominally identical systems; this being done to allow for the extreme sensitivity of higher mode characteristics to minor changes in system parameters and modeling. Consequently, the primary response variables are obtained as averages across ensemble of excitations and ensemble of vibrating systems. For these predicted averages to be useful in design, the knowledge of confidence bands associated with these averages is very essential. A major gap in the current state of art in the field of SEA has been the lack of generic procedures for estimating the confidence intervals associated with the SEA predictions. The study reported in this thesis also aims at developing stochastic dynamic stiffness matrix based finite element procedures to investigate the vibration energy flow in built up structures with an aim to gain insights into the variability in SEA predictions.
The scope of the study reported in this thesis is limited to linear vibrational behavior of skeletal structures. Specifically, we focus our attention on the extension of the method of dynamic stiffness matrix to study harmonic and stochastic steady state response analysis of skeletal structures with random spatial inhomogenieties in their properties. The thesis is divided into 5 Chapters. Firstly, the element stochastic dynamic stiffness matrix is developed for a general random beam element and, subsequently, applications on dynamics of built-up structures and vibration energy flow calculations in randomly parametered trusses are discussed.
A review of literature on the methods of linear structural dynamics with parameter uncertainties is presented in Chapter 1. The scope of the stochastic finite element method using normal mode expansions, transfer matrices and methods based on statistical energy analysis are critically examined. Limitations of the currently available techniques for the dynamic analysis of structures in certain application areas and some of the open questions requiring further research are brought out. The motivation for focusing attention on the method of dynamics stiffness matrix is highlighted.
In Chapter 2, a finite element based methodology is developed for the determination of the dynamic stiffness matrix of Euler-Bernoulli beams with randomly varying flexural and axial rigidity, mass density and foundation elastic modulus. The finite element approximation made employs frequency dependent shape functions, which give rise to stochastic dynamic stiffness matrix , characterized by a set of dynamic weighted integrals . This provides an effective means of discretization of random fields for dynamic problems. The analysis avoids eigenfunction expansion which, not only eliminates modal truncation errors, but also, restricts the number of random variables entering the formulations. Application of the proposed method is illustrated by considering two problems of wide interest in engineering mechanics, namely, vibration of beams on random elastic foundation and the problem of seismic wave amplification through randomly inhomogeneous soil layers. Satisfactory agreement between analytical solutions and a limited amount of digital simulation results is also demonstrated.
The element stochastic dynamic stiffness matrix developed in Chapter 2 is used to calculate the statistics of forced harmonic vibration response of portal frames in Chapter 3. The analysis is based on the assembly of the element stochastic dynamic stiffness matrices. The solution involves the inversion of the global dynamic stiffness matrix, which turns out to be a complex valued, symmetric, random matrix. Three alternative approximate procedures, namely, random eigenfunction expansion method, complex Neumann expansion method and combined analytical and simulation method are developed to invert the matrix. The performance of these approximate procedures is evaluated using Monte Carlo simulation results.
The vibration energy distribution in truss structures under harmonic excitations is considered in Chapter 4. A thirteen member truss is considered for the purpose of illustration. The truss is modeled within the frameworks of SEA and stochastic finite element method employing random dynamic stiffness matrices. The SEA model leads to a thirteen subsystem model with multiple coupling paths. The coupling loss factors at each junction is evaluated by using power injection method using dynamics stiffness matrix method. Subsequently, the spectrum of steady state vibration energy is evaluated. These energy spectra are obtained by frequency band averaging as is recommended in the traditional SEA procedures. To apply the stochastic finite element method, the geometry of the truss, mass and elastic properties of the truss members are treated as stochastic variables. The statistics of the spectra of the total steady state vibration energy in different truss members are evaluated by using dynamic stiffness matrix approach in conjunction with Monte Carlo simulation procedures. The results from SEA and stochastic finite element method are further compared to gain insights into the scope of different approximations made in the frequency band averaging used in SEA. The study helps in gaining insights into the use of finite element procedures in computing coupling loss factors used in SEA and also in understanding the influence of randomness in truss geometry and different damping models on the vibration energy distribution.
Chapter 5 presents the conclusions emerging from the above studies and makes a few suggestions for further research.
@mastersthesis{rp1,
author={S. Adhikari},
title={Stochastic Dynamic Stiffness Method for Vibration and Energy Flow Analyses
of Skeletal Structures},
school={Department of Civil Engineering},
year={1997},
month={July},
address={Indian Institute of Science, Bangalore, India}
}