Solution of Eigenvalue Problems for Non-Proportionally Damped Systems with Multiple Frequencies

1
Solution of Eigenvalue Problems for
Non-Proportionally Damped Systems with Multiple
Frequencies
Fourth World Congress on Computational
Mechanics Buenos Aires, Argentina June 30, 1998
Session V-C Structural Dynamics I

• In-Won Lee, Professor, PE
• Structural Dynamics Vibration Control Lab.
• Korea Advanced Institute of Science
  Technology
• Korea

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OUTLINE

• Introduction
• Objectives and scope
• Current methods
• Proposed method
• Newton-Raphson technique
• Modified Newton-Raphson technique
• Numerical examples
• Grid structure with lumped dampers
• Three-dimensional framed structure with lumped
  dampers
• Conclusions

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INTRODUCTION
Objectives and Scope

• Free vibration of proportional damping system
• (1)
• where Mass matrix
• Damping matrix
• Stiffness matrix
• Displacement vector

4

• Eigenanalysis of proportional damping system
• where Real eigenvalue
• Natural frequency
• Real eigenvector(mode shape)
• Low in cost
• Straightforward

(2)
5

• Free vibration analysis of non-proportional
  damping system

, then
(6)
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(7)
where
Eigenvalue(complex conjugate)
Eigenvector(complex conjugate)
(8)
(9)
Orthogonality of eigenvector
Solution of Eq.(7) is very expensive.
Therefore, an efficient eigensolution technique
for non-proportional damping system is required.
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Current Methods

• Transformation method Kaufman (1974)
• Perturbation method Meirovitch et al (1979)
• Vector iteration method Gupta (1974 1981)
• Subspace iteration method Leung (1995)
• Lanczos method Chen (1993)
• Efficient Methods

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PROPOSED METHOD

• Find p smallest multiple eigenpairs

Solve
Subject to
For
and
multiple
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• Relations between and vectors in the
  subspace of

(7)
where
(8)
(9)

• Let be the
  vectors in the subspace of ,
  and be orthonormal with respect to ,
  then

(10)
(11)
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• Introducing Eq.(10) into Eq.(7)

(13)

• Let

where
Symmetric

• Then

(14)
or
(15)
or
(16)
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Newton-Raphson Technique
(19)
(20)
(21)
where
(22)
(23)
unknown incremental values
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• Introducing Eqs.(21) and (22) into Eqs.(19) and
  (20)

and neglecting nonlinear terms
(23)
(24)
where
residual vector

• Matrix form of Eqs.(23) and (24)

(25)
Coefficient matrix Symmetric Nonsingular
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Modified Newton-Raphson Technique
(25)
Introducing modified Newton-Raphson technique
(26)
(21)
(22)
Coefficient matrix Symmetric Nonsingular
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Step

• Step 1 Start with approximate eigenpairs

• Step 2 Solve for and

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• Step 4 Check the error norm

Error norm
If the error norm is more than the tolerance,
then go to Step 2 and if not, go to Step 5
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NUMERICAL EXAMPLES

• Structures
• Grid structure with lumped dampers
• Three-dimensional framed structure with lumped
  dampers
• Analysis methods
• Proposed method
• Subspace iteration method (Leung 1988)
• Lanczos method (Chen 1993)

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• Comparisons
• Solution time(CPU)
• Convergence
• Error norm
• Convex with 100 MIPS, 200 MFLOPS

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Grid Structure with Lumped Dampers
? Material Properties Tangential Damper c
0.3 Rayleigh Damping ? ? 0.001 Youngs
Modulus 1,000 Mass Density 1 Cross-section
Inertia 1 Cross-section Area 1 ? System
Data Number of Equations 590 Number of Matrix
Elements 8,115 Maximum Half Bandwidths
15 Mean Half Bandwidths 14
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Table 1. Results of proposed method for grid
structure
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Table 2. CPU time for twelve lowest eigenpairs of
grid structure
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Lanczos method (48 Lanczos vectors)
Fig.3 Error norms of grid model by
subspace iteration method
Fig.4 Error norms of grid model by
Lanczos method
Fig.2 Error norms of grid model by
proposed method
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Three-Dimensional Framed Structure with Lumped
Dampers
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? Material Properties Lumped Damper c
12,000.0 Rayleigh Damping ? -0.1755 ?
0.02005 Youngs Modulus 2.1E11 Mass Density
7,850 Cross-section Inertia 8.3E-06 Cross-sect
ion Area 0.01 ? System Data Number of
Equations 1,128 Number of Matrix Elements
135,276 Maximum Half Bandwidths 300 Mean Half
Bandwidths 120
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Table 3. Results of proposed method for
three-dimensional framed structure
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Table 4. CPU time for twelve lowest eigenpairs of
three-dimensional framed structure
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Lanczos method (48 Lanczos vectors)
Fig.7 Error norms of 3-D. frame model by
subspace iteration method
Fig.8 Error norms of 3-D. frame model by
Lanczos method
Fig.6 Error norms of 3-D. frame model by
proposed method
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CONCLUSIONS

• Proposed method
• converges fast
• guarantees nonsingularity of coefficient matrix

Proposed method is efficient
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• Thank you for your attention.