CEE M237A / MAE M269A Lecture 4: Dynamic SDOF Response to General Dynamic Loading Professor J. Woody Ju
DuhamelIntegral(undamped)
Duhamel Integral for undamped system = 0 Recall pure impulse (short duration) At time t = , short duration d impulse p ( ) d
CEE M237A / MAE M269A Lecture 1: Free Vibration Analysis of SDOF Equation of Motion Professor J. Woody Ju
SDOFSystems
Components of the basic dynamic system (SDOF)
Freebodydiagram forthemassm
2
E.O.M(Linearsystem)
Method 1: Direct Equilibrium by DAlembert
Part I
Fundamental Relations and Concepts
Our objective in this first and introductory part of the book is to provide a review of elementary force, stress, and strain concepts, which are useful in studying the integrity of structural members. The topics s
CEE M237A / MAE M269A Lecture 3: Dynamic SDOF Response to Periodic and Impulsive Loading Professor J. Woody Ju
ResponsetoPeriodicLoading
mv + cv + kv = p ( t ) non-harmonic loading period of excitation (loading) = Tp , satisfy: p ( t + nTp ) = p ( t )
per
CEE M237A / MAE M269A Lecture 2: Dynamic SDOF Response to Harmonic Loading Professor J. Woody Ju
HarmonicallyForcedVibration
Loading: E.O.M.:
p ( t ) = P0 sin t mv + cv + kv = P0 sin t Note:
= frequency of the forcing function
c = <1 ccr k = m
(Dampingra
MIT OpenCourseWare http:/ocw.mit.edu
Spring 2008
2.094 Finite Element Analysis of Solids and Fluids
For information about citing these materials or our Terms of Use, visit: http:/ocw.mit.edu/terms.
2.094 Finite Element Analysis of Solids and Fluids
Fall
Part I
Theory and methods
Paolo L.Gatti
Copyright 2003 Taylor & Francis Group LLC
1
Review of some fundamentals
1.1 Introduction
It is now known from basic physics that force and motion are strictly connected and are, by nature, inseparable. This is not a
4
Single-degree-of-freedom systems
4.1
Introduction
This chapter deals with the simplest system capable of vibratory motion: the single-degree-of-freedom (SDOF) system which, in its discrete-parameters form, is often called harmonic oscillator. Despite it
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11
Solution Methods for Linear Problems
11.1 NUMERICAL METHODS IN FEA 11.1.1 SOLVING THE FINITE-ELEMENT EQUATIONS: STATIC PROBLEMS
Consider the numerical solution of the linear system K = f, in
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10
Element and Global Stiffness and Mass Matrices
10.1 APPLICATION OF THE PRINCIPLE OF VIRTUAL WORK
Elements of variational calculus were discussed in Chapter 3, and the Principle of Virtual Wor
0749_Frame_C09 Page 121 Wednesday, February 19, 2003 5:09 PM
9
Element Fields in Linear Problems
This chapter presents interpolation models in physical coordinates for the most part, for the sake of simplicity and brevity. However, in nite-element codes,
0749_Frame_C08 Page 117 Wednesday, February 19, 2003 5:08 PM
8
Introduction to the FiniteElement Method
8.1 INTRODUCTION
In thermomechanical members and structures, nite-element analysis (FEA) is typically invoked to compute displacement and temperature e
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7
Thermal and Thermomechanical Response
ENERGY
7.1 BALANCE OF ENERGY AND PRODUCTION OF ENTROPY 7.1.1 BALANCE
OF
The total energy increase in a body, including internal energy and kinetic energy,
0749_Frame_C06 Page 95 Wednesday, February 19, 2003 5:06 PM
6
Stress-Strain Relation and the Tangent-Modulus Tensor
6.1 STRESS-STRAIN BEHAVIOR: CLASSICAL LINEAR ELASTICITY
Under the assumption of linear strain, the distinction between the Cauchy and Piola
0749_Frame_C05 Page 73 Wednesday, February 19, 2003 5:03 PM
5
Mechanical Equilibrium and the Principle of Virtual Work
5.1 TRACTION AND STRESS 5.1.1 CAUCHY STRESS
Consider a differential tetrahedron enclosing the point x in the deformed conguration. The a
0749_Frame_C04 Page 51 Wednesday, February 19, 2003 5:33 PM
4
Kinematics of Deformation
The current chapter provides a review of the mathematics for describing deformation of continua. A more complete account is given, for example, in Chandrasekharaiah an
0749_Frmae_C03 Page 43 Wednesday, February 19, 2003 5:01 PM
3
Introduction to Variational and Numerical Methods
3.1 INTRODUCTION TO VARIATIONAL METHODS
Let u(x) be a vector-valued function of position vector x, and consider a vectorvalued function F(u(x),
0749_Frame_C02 Page 25 Wednesday, February 19, 2003 5:00 PM
2
Mathematical Foundations: Tensors
2.1 TENSORS
We now consider two n 1 vectors, v and w, and an n n matrix, A, such that v = Aw. We now make the important assumption that the underlying informat
0749_Frame_C01 Page 1 Wednesday, February 19, 2003 4:55 PM
1
Mathematical Foundations: Vectors and Matrices
1.1 INTRODUCTION
This chapter provides an overview of mathematical relations, which will prove useful in the subsequent chapters. Chandrashekharaia
Mechanics of Structures Online by the SEAS
UCLA MS Engineering Online, msenrgol.seas.ucla.edu
Area Director: Prof. A. Mal, ajit@seas.ucla.edu Description: The main objective of this program is to provide students with the opportunity to develop the kno
CEE M237A / MAE M269A
Lecture 19: The Ritz Method II Professor J. Woody Ju
Example3 UseofMoreCoordinate FunctionsinExample1
Consider a series of sinusoidal coordinate functions: N n x w ( x, t ) = an ( t ) sin L n =1 The case for N = 2 is Example 1. Consi
CEE M237A / MAE M269A
Lecture 18: The Ritz Method I Professor J. Woody Ju
TheRitzMethod
Whereas Rayleighs quotient utilized only one trial function in the estimate of the lowest frequency, the Ritz method allows for the determination of a subset of the lo
CEE M237A / MAE M269A
Lect Lecture 17: 17 Global Approximation Methods Professor J. Woody Ju
Prof.J.W.Ju,2010
Gl GlobalApproximateMethods
Once a variational principle has been established for a particular problem, it constitutes a basis for seeking what a
CEE M237A / MAE M269A
Lecture 16: Hamiltons Principle on Dynamic Structural Systems IV: Equation of Motion via Hamiltons Principle Professor J. Woody Ju
Example5 ForcedVibrationofaBeam ColumnSupportingaMass
This column is in the presence of a gravity fiel
CEE M237A / MAE M269A
Lecture 15: Hamiltons Principle on Dynamic Structural Systems III: Equation of Motion via Hamiltons Principle Professor J. Woody Ju
CircularCylindersTorsionMembers
The primary kinematic variable for torsion of a circular cylinder is
CEE M237A / MAE M269A
Lecture 14: Hamiltons Principle on Dynamic Structural Systems II Professor J. Woody Ju
PotentialEnergyofExternalForces
Potential energy of external forces relates energy that can be recovered from the forces and/or the loads acting o
CEE M237A / MAE M269A
Lecture 13: Hamiltons Principle on Dynamic Structural Systems I Professor J. Woody Ju
HamiltonsPrinciple
Hamiltons principle is a potent structural dynamics tool for the formulation of the equations of motion of a complex structural
CEE M237A / MAE M269A
Lecture 12: Vibrations of Structural Systems Professor J. Woody Ju
VibrationofStructuralSystems
Some structures are composed of various members, whose vibrational characteristics are governed by different equations The natural vibrat
CEE M237A / MAE M269A
Lecture 11: Examples on Transverse Vibrations of Beams Professor J. Woody Ju
Example1 FreeVibrationsofaSimply SupportedUniformBeamonBothEnds
The homogeneous solution for Z ( x) is: Z ( x ) = C1 sin x + C2 cos x + C3 sinh x + C4 cosh
CEE M237A / MAE M269A Lecture 10: Examples on Rods, and Transverse Vibrations of Beams
Professor J. Woody Ju
Example1 FreeVibrationsofaBarwith OneEndFixedandOtherEndFree
Consider a uniform homogeneous, isotropic bar of length L. The homogeneous solution