- •Copyright Notice
- •Table of Contents
- •Chapter 1. Introduction
- •I. Basic Concepts
- •Examples:
- •Why Finite Element Method?
- •Applications of FEM in Engineering
- •Examples:
- •A Brief History of the FEM
- •FEM in Structural Analysis (The Procedure)
- •Example:
- •Available Commercial FEM Software Packages
- •Objectives of This FEM Course
- •II. Review of Matrix Algebra
- •Linear System of Algebraic Equations
- •Matrix Addition and Subtraction
- •Scalar Multiplication
- •Matrix Multiplication
- •Transpose of a Matrix
- •Symmetric Matrix
- •Unit (Identity) Matrix
- •Determinant of a Matrix
- •Singular Matrix
- •Matrix Inversion
- •Examples:
- •Solution Techniques for Linear Systems of Equations
- •Positive Definite Matrix
- •Differentiation and Integration of a Matrix
- •Types of Finite Elements
- •III. Spring Element
- •One Spring Element
- •Spring System
- •Checking the Results
- •Notes About the Spring Elements
- •Example 1.1
- •Chapter 2. Bar and Beam Elements
- •I. Linear Static Analysis
- •II. Bar Element
- •Stiffness Matrix --- Direct Method
- •Stiffness Matrix --- A Formal Approach
- •Example 2.1
- •Example 2.2
- •Distributed Load
- •Bar Elements in 2-D and 3-D Space
- •2-D Case
- •Transformation
- •Stiffness Matrix in the 2-D Space
- •Element Stress
- •Example 2.3
- •Example 2.4 (Multipoint Constraint)
- •3-D Case
- •III. Beam Element
- •Simple Plane Beam Element
- •Direct Method
- •Formal Approach
- •3-D Beam Element
- •Example 2.5
- •Equivalent Nodal Loads of Distributed Transverse Load
- •Example 2.6
- •Example 2.7
- •FE Analysis of Frame Structures
- •Example 2.8
- •Chapter 3. Two-Dimensional Problems
- •I. Review of the Basic Theory
- •Plane (2-D) Problems
- •Stress-Strain-Temperature (Constitutive) Relations
- •Strain and Displacement Relations
- •Equilibrium Equations
- •Exact Elasticity Solution
- •Example 3.1
- •II. Finite Elements for 2-D Problems
- •A General Formula for the Stiffness Matrix
- •Constant Strain Triangle (CST or T3)
- •Linear Strain Triangle (LST or T6)
- •Linear Quadrilateral Element (Q4)
- •Quadratic Quadrilateral Element (Q8)
- •Example 3.2
- •Transformation of Loads
- •Stress Calculation
- •I. Symmetry
- •Types of Symmetry:
- •Examples:
- •Applications of the symmetry properties:
- •Examples:
- •Cautions:
- •II. Substructures (Superelements)
- •Physical Meaning:
- •Mathematical Meaning:
- •Advantages of Using Substructures/Superelements:
- •Disadvantages:
- •III. Equation Solving
- •Direct Methods (Gauss Elimination):
- •Iterative Methods:
- •Gauss Elimination - Example:
- •Iterative Method - Example:
- •IV. Nature of Finite Element Solutions
- •Stiffening Effect:
- •V. Numerical Error
- •VI. Convergence of FE Solutions
- •Type of Refinements:
- •Examples:
- •VII. Adaptivity (h-, p-, and hp-Methods)
- •Error Indicators:
- •Examples:
- •Chapter 5. Plate and Shell Elements
- •Applications:
- •Forces and Moments Acting on the Plate:
- •Stresses:
- •Relations Between Forces and Stresses
- •Thin Plate Theory ( Kirchhoff Plate Theory)
- •Examples:
- •Under uniform load q
- •Thick Plate Theory (Mindlin Plate Theory)
- •II. Plate Elements
- •Kirchhoff Plate Elements:
- •Mindlin Plate Elements:
- •Discrete Kirchhoff Element:
- •Test Problem:
- •Mesh
- •III. Shells and Shell Elements
- •Example: A Cylindrical Container.
- •Shell Theory:
- •Shell Elements:
- •Curved shell elements:
- •Test Cases:
- •Chapter 6. Solid Elements for 3-D Problems
- •I. 3-D Elasticity Theory
- •Stress State:
- •Strains:
- •Stress-strain relation:
- •Displacement:
- •Strain-Displacement Relation:
- •Equilibrium Equations:
- •Stress Analysis:
- •II. Finite Element Formulation
- •Displacement Field:
- •Stiffness Matrix:
- •III. Typical 3-D Solid Elements
- •Tetrahedron:
- •Hexahedron (brick):
- •Penta:
- •Element Formulation:
- •Solids of Revolution (Axisymmetric Solids)
- •Axisymmetric Elements
- •Applications
- •Chapter 7. Structural Vibration and Dynamics
- •I. Basic Equations
- •A. Single DOF System
- •B. Multiple DOF System
- •Example
- •II. Free Vibration
- •III. Damping
- •IV. Modal Equations
- •V. Frequency Response Analysis
- •VI. Transient Response Analysis
- •B. Modal Method
- •Cautions in Dynamic Analysis
- •Examples
- •Chapter 8. Thermal Analysis
- •Further Reading
Lecture Notes: Introduction to Finite Element Method |
Chapter 7. Structural Vibration and Dynamics |
Chapter 7. Structural Vibration and
Dynamics
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Natural frequencies and modes |
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Frequency response (F(t)=Fo sinωt) |
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Transient response (F(t) arbitrary) |
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I. Basic Equations
A. Single DOF System
k
f=f(t) m 
c
ku
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f(t) |
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x, u
m - mass
k - stiffness
c - dampingf (t) - force
From Newton’s law of motion (ma = F), we have mu&&= f(t)−ku−cu&,
i.e.
mu+cu+ku= f(t), |
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and u&&= d |
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where u is the displacement, u = du / dt |
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u / dt |
© 1997-2003 Yijun Liu, University of Cincinnati |
157 |
Lecture Notes: Introduction to Finite Element Method Chapter 7. Structural Vibration and Dynamics
Free Vibration: f(t) = 0 and no damping (c = 0) |
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Eq. (1) becomes |
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mu+ku=0 . |
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(meaning: inertia force + stiffness force = 0) |
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Assume: |
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u(t) = U sin (ωt) , |
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where ω is the frequency of oscillation, U the amplitude. |
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Eq. (2) yields |
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−Uω2 m sin( ωt)+kU sin( ωt)=0 |
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i.e., |
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[−ω2 m+k]U =0 . |
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For nontrivial solutions for U, we must have |
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[−ω2 m+k]=0 , |
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which yields |
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ω = mk . |
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This is the circular natural frequency of the single DOF system (rad/s). The cyclic frequency (1/s = Hz) is
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© 1997-2003 Yijun Liu, University of Cincinnati |
158 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 7. Structural Vibration and Dynamics |
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u = U s in w t |
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T = 1 / f |
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U n d a m p e d F r e e V ib r a t io n |
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With non-zero damping c, where |
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0 < c < cc = 2mω = 2 k m (cc = critical damping) |
(5) |
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we have the damped natural frequency: |
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ωd =ω |
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where ξ = |
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cc |
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For structural damping: 0 ≤ξ < 0.15 |
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ωd ≈ω. |
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Thus, we can ignore damping in normal mode analysis. u
t
Damped Free Vibration
© 1997-2003 Yijun Liu, University of Cincinnati |
159 |
Lecture Notes: Introduction to Finite Element Method Chapter 7. Structural Vibration and Dynamics
B. Multiple DOF System
Equation of Motion
Equation of motion for the whole structure is
&& |
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(8) |
Mu |
+ Cu + Ku = f (t) , |
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in which: |
u nodal displacement vector, |
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M mass matrix, |
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C damping matrix, |
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K stiffness matrix, |
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f forcing vector. |
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Physical meaning of Eq. (8):
Inertia forces + Damping forces + Elastic forces = Applied forces
Mass Matrices
Lumped mass matrix (1-D bar element):
m1 |
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ρ,A,L 2 |
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Element mass matrix is found to be |
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m = |
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diagonal matrix
© 1997-2003 Yijun Liu, University of Cincinnati |
160 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 7. Structural Vibration and Dynamics |
In general, we have the consistent mass matrix given by
m = ∫V ρNT NdV |
(9) |
where N is the same shape function matrix as used for the displacement field.
This is obtained by considering the kinetic energy:
Κ = |
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(cf. |
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& T & |
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∫V ρN |
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14243 |
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m
Bar Element (linear shape function):
m = ∫V |
1 −ξ |
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ξ]ALdξ |
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ρ |
[1 |
−ξ |
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1/ 3 |
1/ 6 u |
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= ρAL |
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&& |
(10) |
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1/ 6 |
1/ 3 u2 |
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&& |
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Element mass matrices:
local coordinates to global coordinates
assembly of the global structure mass matrix M.
© 1997-2003 Yijun Liu, University of Cincinnati |
161 |