Modal Analysis of a Cantilevered Beam

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Modal Analysis of a Cantilevered Beam

• An Application of Finite Element Methods and the
  Power Method for Symmetric Eigenvalue Problems
• Brett Williams
• Jake Lewis
• MSE 5984Advanced Engineering Mathematics for
  Material Scientists
• November 28th, 2000

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Finite Element Method

• Closed form solution for structural vibration
  problems with complex geometry is often not
  available
• Discretization of complex geometry results in
  multiple simple problems
• Well suited to computerized solution
• Element formulations available in literature

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Finite Element Model of Beam
Node
w(x,t)
Element
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Power Method

• Numerical solution to eigenvalue problem
• Computes eigenvalues and eigenvectors
  simultaneously
• Computes each eigenvalue and eigenvector
  iteratively from an initial guess of the
  eigenvector
• Solves for lowest eigenvalue first

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Power Method Equations
Iterative Process
Repeat for next set
Produces ?final and ?final
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Basics of Modal Analysis
u1
u2
k1
k2
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Modal Analysis
Coordinate Transformation
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Table of Results
Natural Frequencies, in Hertz