# Book considering FEM analysis for complex eigenvalues (incl. damping)

• Engineering
• Arjan82
In summary, The conversation discusses recommendations for a book on complex eigenvalue problems, specifically in relation to FEM analysis and theory. The suggested books include "Theory of Vibration with Applications" and "Shock and Vibration Handbook," as well as checking the documentation of specific FEA software. Complex eigenvalue analyses are commonly used for problems such as brake squeal and dynamic stability. Zienkiewicz, Taylor, and Zhu briefly mention complex eigenvalues in their book.
Arjan82
Can anyone recommend a book in which complex eigenvalue problems are treated? I mean the FEM analysis and the theory behind it. These are eigenvalue problems which include damping. I think that it is used for composite materials and/or airplane engineering (maybe wing fluttering?).

You should find some information on this topic in W.T. Thomson’s book "Theory of Vibration with Applications" as well as in "Shock and Vibration Handbook" published by McGraw-Hill. Apart from such general books about vibrations, I would check the documentation of particular FEA software, theory behid the complex eigenvalue extraction procedure is usually discussed there.

Complex eigenvalue analyses are used for problems such as brake squeal and widely understood dynamic stability (rotor whirl).

berkeman and Arjan82
Zienkiewicz, Taylor and Zhu briefly mention complex eigenvalues in Ch. 16.6 - 16.8 where they treat the equation Mu" +Cu' + ku = 0

## 1. What is FEM analysis and how is it used in book considering complex eigenvalues?

FEM (Finite Element Method) analysis is a numerical technique used to solve complex engineering problems by dividing a continuous system into smaller, simpler elements. In the book considering complex eigenvalues, FEM analysis is used to study the behavior of structures under dynamic loading and determine their natural frequencies and damping ratios.

## 2. What are complex eigenvalues and why are they important in structural analysis?

Complex eigenvalues are solutions to a system of equations that involve both real and imaginary components. In structural analysis, they represent the natural frequencies and damping ratios of a structure, which are crucial in understanding its dynamic behavior and determining its stability.

## 3. How does damping affect the eigenvalues in FEM analysis?

Damping is a measure of energy dissipation in a structure, and it affects the eigenvalues by reducing the amplitude of the structure's response to dynamic loading. In FEM analysis, damping is incorporated into the equations of motion through damping matrices, which can change the magnitude and phase of the eigenvalues.

## 4. Can FEM analysis be used to analyze structures with complex geometries?

Yes, FEM analysis is highly versatile and can be used to analyze structures with complex geometries. This is because it allows for the discretization of the structure into smaller elements, making it easier to handle complex shapes and boundary conditions.

## 5. What are some practical applications of FEM analysis for complex eigenvalues?

FEM analysis for complex eigenvalues has a wide range of applications, including earthquake engineering, vibration analysis of mechanical systems, and structural health monitoring. It is also used in the design and optimization of structures, such as bridges, buildings, and aircraft, to ensure their safety and performance under dynamic loading.

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