9 Finite Element Applications in Dynamics Mohamed S. Gadala

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The University of British Columbia

9.1 Problem and Element Classification ................................. 9-2

Geometric Modeling † Discrete Element Types in FE

Programs † Truss and Beam Problems † Two-Dimensional

Problems † Shell and Plate Problems † Three-Dimensional

Solid Problems † Synopsis of Problem Classification and

Element Choice

9.2 Types of Analysis ................................................................. 9-20

9.3 Modeling Aspects for Dynamic Analysis .......................... 9-23

Model Size and Choice of Master Degrees of Freedom †

Lumped and Consistent Mass Modeling † Use of Symmetry

9.4 Equations of Motion and Solution Methods ................... 9-27

Equation of Motion † Direct Integration Method † Modal

Superposition Method † Damping Formulation

9.5 Various Dynamic Analyses ................................................. 9-33

Modal Analysis † Transient Dynamic Analysis † Harmonic

Response Analysis † Response Spectrum Analysis

9.6 Checklist for Dynamic FE Analysis ................................... 9-41

Summary

This chapter discusses the use of the finite element (FE) method in problems of vibrations and structural dynamics.

The first three sections outline the main steps in modeling a physical problem for a specific dynamic analysis.

Section 9.1 concentrates on the finite element aspect of the modeling process. The basis for geometric modeling is

outlined and an overview of commonly used types of elements in typical commercial programs is presented.

A summary of element capabilities is given at the end of the section (Table 9.1). Section 9.2 discusses the basis for

classifying and choosing a particular type of dynamic analysis, including modal, harmonic or frequency response,

transient and shock, and random analysis. Section 9.3 discusses some special aspects in modeling that are pertinent

to dynamic system analysis; namely, choice of master and slave degrees of freedom (DoF), lumped vs. consistent

mass modeling, and use of symmetry in dynamic analysis.

The second part of the chapter discusses solution methods and damping considerations, and outlines basic

steps for performing various dynamic analyses. Section 9.4 briefly presents the theory and equations for various

dynamic analyses. The analysis types included in the discussion are direct integration and modal superposition.

In the direct integration analysis, both implicit and explicit schemes are discussed and an example of each is

presented. Most of the theory and equations are presented in a summarized form without rigorous mathematical

proofs. Section 9.5 describes details of various dynamic analyses and provides a brief discussion of the choice of

the solution method for each analysis. Emphasis is placed on the basic steps required to perform a particular

analysis type. Modal analysis, transient analysis (direct integration and mode superposition approaches),

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frequency response harmonic analysis, and random response analysis are presented. The various methods of

combination of modal responses are also discussed. Finally, Section 9.6 provides some general guidelines for a

typical dynamic analysis using the FE method.

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