DYNAMIC ANALYSIS AND EIGENVECTORS

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DYNAMIC ANALYSIS AND EIGENVECTORS

• BY K. MOUHTAROV
• R. DUBAGUNTA

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INTRODUCTION

• The finite element method is a numerical analysis
  technique for obtaining approximate solutions to
  a wide variety of engineering problems.
• The basic idea in the finite element method is to
  find the solution of a complicated problem by
  replacing it by a simpler one.
• In the finite element, the actual continuum or
  body of matter like solid liquid or gas is
  represented as an assemblage of subdivisions
  called finite elements. These elements are
  considered to be interconnected at specified
  joints which are called nodes or nodal points.

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Finite element method

• Figure 1 shows a finite element model. The
  quadrilateral and triangular regions are finite
  elements. Black dots are nodes where elements are
  connected to one another. A mesh is an
  arrangement of nodes and elements. This
  particular mesh shows triangular and
  quadrilateral elements, some with side nodes and
  some with only corner nodes

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Finite element method
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Finite element method

• The solution of a general continuum problem by
  the finite element method always follows an
  orderly step by step process. With reference to
  static structural problems, the step by step
  procedure can be stated as follows
• Step (a) Discretization of the structure
• The first step in the finite element method is to
  divide the structure or solution region into
  subdivisions elements.

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Finite element method

• Step (b) Selection of proper interpolation or
  displacement model. Since the displacement
  solution of a complex structure under any
  specified load conditions can not be predicted
  exactly, we assume some suitable solution within
  an element to approximate the unknown solution.
• Step (c) Derivation of element stiffness
  matrices and load vectors. From the assumed
  displacement model, the stiffness matrix K and
  the load vector P , of element e is to be
  derived by using either equilibrium conditions or
  a suitable variational principle.

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Finite element method

• Step (d) Assemblage of element equations to
  obtain the overall equilibrium equations.Since
  the structure is composed of several finite
  elements, the individual element stiffness
  matrices and load vectors are to be assembled in
  a suitable manner and the overall equilibrium
  equations have to be formulated as
• Step (e) Solution for the unknown nodal
  displacements.The overall equilibrium equation
  has to be modified to account for the boundary
  condition of the problem.

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Finite element method

• After the incorporation of the boundary
  conditions, the equilibrium equations can be
  expressed as
• Step (f) Computation of element strains and
  stresses. From the known nodal displacements, if
  required, the element strains and stresses can be
  computed by using the necessary equations of
  solid or structural mechanics.

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STIFNESS AND FLEXIBILLITY. STIFFNES MATRIX

• Consider a uniform elastic spring subjected to a
  load P. This structure obeys Hooks law. If a
  force P is applied to a spring fixed at one end,
  to produce a displacement, then the linear
  force-displacement is u.

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STIFNESS AND FLEXIBILLITY. STIFFNES MATRIX

• k is called the stiffness of the spring
• where f is called the flexibility of spring
• Suppose the uniform elastic spring has nodal
  points 1 and 2 at its ends, and that the forces
  at these points are P1 and P2 with corresponding
  displacements u1and u2.

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STIFNESS AND FLEXIBILLITY. STIFFNES MATRIX

• Elemental spring
• From equilibrium considerations

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STIFNESS AND FLEXIBILLITY. STIFFNES MATRIX

• It is convenient to show the above in matrix form
  as follows
• simple system consisting of just two springs

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STIFNESS AND FLEXIBILLITY. STIFFNES MATRIX

• The system is in equilibrium

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STIFNESS AND FLEXIBILLITY. STIFFNES MATRIX

• The equations written in matrix form
• p- vector of external nodal loads acting on the
  structure
• K- system or structural stiffness matrix
• u-over-all nodal displacement vector

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. MASS MATRICIES

• The elemental mass matrix, which is always
  symmetrical, is a matrix of equivalent nodal
  masses that dynamically represent the actual
  distributed mass of the element.
• The element mass matrix is defined as

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DYNAMIC EQUATIONS

• The force equilibrium of a multi-degree-of-freedom
  lumped mass system
• vector of inertia forces acting on the node
  masses
• vector of viscous damping, or energy dissipation,
  forces
• a vector of internal forces carried by the
  structure
• vector of externally applied loads

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DYNAMIC EQUATIONS

• For many structural systems, the approximation of
  linear structural behavior is made in order to
  convert the physical equilibrium statement, to
  the following set of second-order, linear,
  differential equation

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VIBRATION ANALYSIS

• When loads are suddenly applied, or when the
  loads are of a variable nature, the mass and
  acceleration effects come into the picture. If a
  solid, such as an engineering structure, is
  deformed elastically and suddenly released, it
  tends to vibrate about its equilibrium position.
  This periodic motion due to the restoring strain
  energy is called free vibration. The number of
  cycles per unit time is called frequency. The
  maximum displacement from the equilibrium
  position is the amplitude.

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VIBRATION ANALYSIS

• Equation for damped forced vibration
• If there is no damping the equation become
• Free undamped vibration equation

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VIBRATION ANALYSIS

• The free undamped vibration equation is linear
  and homogeneous. Its general solution is a linear
  combination of exponentials. Under matrix
  definiteness conditions the exponentials can be
  expressed as a combination oftrigonometric
  functions sines and cosines of argument .
• A compact representation of such functions is
  obtained by using the exponential form

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THE VIBRATION EIGENPROBLEM

• Replace in equation
• The time dependence to the exponential is
  segregated
• Since is not identically zero, it can be dropped
  leaving the algebraic condition

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THE VIBRATION EIGENPROBLEM

• Because v cannot be the null vector, thisequation
  is an algebraic eigenvalue problem in .
  The eigenvalues are the roots of the
  characteristipolynomial be index by i
• Dropping the index i this eigenproblem is usually
  written as

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SOLVING THE VIBRATION EIGENPROBLEM

• Consider the mass-spring system illustrated in
  the figure.The displacements u are measured from
  references fixed at the locations where both
  springs are unstretched. F denotes an applied
  force.

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SOLVING THE VIBRATION EIGENPROBLEM

• The free-body diagrams

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SOLVING THE VIBRATION EIGENPROBLEM

• Newtons second law for mass 1 and 2
• The force-deflection equation for Hookes-law
  springs are

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SOLVING THE VIBRATION EIGENPROBLEM

• The two applications of the second law become
• The equations in matrix form

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SOLVING THE VIBRATION EIGENPROBLEM

• M is the mass matrix, K is the stiffnes matrix,
  and p is the external force vector.
• The solution of this matrix differential
  equation can be obtained under certain
  conditions. Consider the unforced case p0

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SOLVING THE VIBRATION EIGENPROBLEM

• Attempt a trial solution of the form
• Here, and are constants, the equation
  is called a pure or vibration mode response.
• The factor may be cancelled because
  this equation must be true at times when that
  factor is not zero.

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SOLVING THE VIBRATION EIGENPROBLEM

• The equation can be rearranged into the form
• The solution of this algebraic equation is called
  the generalized eigenvalue problem. If M has an
  inverse, we can convert the problem to the
  standard eigenvalue problem

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SOLVING THE VIBRATION EIGENPROBLEM

• If the generalized problem is shown as
• The modal solution is a solution if and only if
  ? is such that
• The roots of this polynomial equation are then
  substituted into the homogeneous equation to
  obtain

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SOLVING THE VIBRATION EIGENPROBLEM

• From here a method such as Gaussian elimination
  can be used to obtain the modal vectors .The
  final application of the homogeneous equation
  theory is the conclusion that for each , the
  number of independent modal vectors is equal to
  the rank defect of the matrix . In the case of
  vibration modes , it can be shown
  that this number is always the algebraic
  multiplicity of the eigenvalue.

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SOLVING THE VIBRATION EIGENPROBLEM

• We can obtain a physical interpolation of the
  generalized eigenvector. We have trial solution
  only if
• the shape is the generalized eigenvector

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SOLVING THE VIBRATION EIGENPROBLEM

• The velocity vector is

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Homework Problem

• For the example problem shown on the presentation
  if m1 and k1
• 1. Define the equations of motion,
• 2. Convert to matrix form (eigenvalue problem),
• 3. At t0 find the first eigenvalue and
  eigenvector. Consider the unforced case.

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REFERENCES

• 1. S. S. Rao, The Finite Element Method in
  Engineering, Second Edition, Pergamon Press plc,
  Headington Hill Hall, Oxford OX3 0BW, England,
  ISBN 0-08-033419-9, 1989.
• 2. R. D. Cook, Concepts and Applications of
  Finite Element Analysis, John Wiley Sons,
  Inc., Canada, ISBN 0-471-03050, 1981
• 3. J. W. Brewer, Engineering Analysis in Applied
  Mechanics, Taylor Francis Ins., New York, ISBN
  1-56032-932-7, 2001
• 4. http//www.csibekeley.com/Tech_info/12.pdf
• 5. T. R. Chandrupatla, and A. D. Belegundu,
  Introduction to Finite Elements in Engineering,
  Prentice-Hall, Inc. Simon Schuster/A Viacom
  Company, UPPER Saddle River, NJ 07458, ISBN
  0-13-207036-7, 1991.
• 6 .C. T. F. Ross, Advanced Applied Finite Element
  Methods, Horwood Publishing, International
  Publishers, Coll House, Westergate, Chichester,
  West Sussex, PO20 6QL England, ISBN 1-898563-61-9.