Short Answer:

An eigenvalue in vibration analysis represents the square of the natural frequency of a vibrating system. It is obtained by solving the characteristic equation that relates the stiffness and mass of the system. Each eigenvalue corresponds to one mode of vibration, and the number of eigenvalues equals the number of degrees of freedom of the system.

In simple terms, eigenvalues are special numerical values that determine how a mechanical or structural system will vibrate naturally without any external force. They are essential for finding natural frequencies and mode shapes in vibration analysis.

Detailed Explanation :

Eigenvalue in Vibration Analysis

In vibration analysis, especially for systems with multiple degrees of freedom (MDOF), the concept of eigenvalues plays a central role in determining the natural frequencies and mode shapes of a mechanical or structural system. The eigenvalue is a mathematical quantity that defines the dynamic characteristics of the system — particularly how it vibrates when disturbed from its equilibrium position.

When a structure vibrates freely, it tends to do so at specific frequencies known as natural frequencies. These frequencies are directly related to the eigenvalues of the system. In fact, the eigenvalue is the square of the natural frequency (i.e., ).

Thus, understanding eigenvalues helps engineers predict how systems will behave dynamically and design them to avoid resonance or unwanted vibrations.

Definition

An eigenvalue in vibration analysis can be defined as:

“A scalar quantity that characterizes the natural frequency of vibration of a mechanical or structural system and is obtained by solving the characteristic equation derived from the system’s equations of motion.”

Mathematically, eigenvalues are found from the following standard form of the equation of motion for an undamped free vibration system:

Where:

= Mass matrix
= Stiffness matrix
= Displacement vector

Assuming a harmonic solution of the form , we get:

For a non-trivial solution (), the determinant of the coefficient matrix must be zero:

This is known as the characteristic equation, and solving it gives values of , which are the eigenvalues of the system.

Mathematical Representation

If we let

then the eigenvalue equation can be written as:

Where:

= Eigenvalue (equal to )
= Corresponding mode shape (eigenvector)

Hence, for an n-degree-of-freedom system, there will be n eigenvalues:

and each eigenvalue corresponds to a unique mode shape .

Physical Meaning of Eigenvalue

The eigenvalue represents the dynamic stiffness of the system and determines the rate at which the system oscillates naturally.

A larger eigenvalue means a higher natural frequency, indicating a stiffer mode of vibration.
A smaller eigenvalue corresponds to a lower natural frequency, indicating a more flexible mode of vibration.

Each eigenvalue defines one independent mode of vibration. When the system vibrates freely, it can vibrate in any of these modes, or as a combination of them.

For example, in a two-degree-of-freedom system:

The first eigenvalue corresponds to the first natural frequency.
The second eigenvalue corresponds to the second natural frequency.

Steps to Determine Eigenvalues in Vibration Analysis

Form the Mass and Stiffness Matrices:
Determine the matrices and based on the physical system’s parameters (mass, spring constants, etc.).
Write the Equation of Motion:
For free vibration (without damping or external force):

Assume Harmonic Motion:
Substitute into the equation.
Simplify the Equation:

Set the Determinant to Zero:

Solve for :
The solutions of this determinant give the eigenvalues ().

Each eigenvalue corresponds to one natural frequency and a specific mode shape (eigenvector).

Example

Consider a 2-degree-of-freedom spring-mass system with:

Masses and
Stiffness constants and

The stiffness and mass matrices can be written as:

The eigenvalue problem is:

Solving this determinant gives two eigenvalues ( and ), representing the squares of the system’s two natural frequencies ( and ).

Importance of Eigenvalues in Vibration Analysis

Determine Natural Frequencies:
Eigenvalues directly give the square of the natural frequencies () of a system.
Predict Resonance:
Knowing natural frequencies helps avoid resonance conditions, which can cause system failure.
Understand System Behavior:
Each eigenvalue reveals how stiff or flexible a vibration mode is.
Simplify Complex Systems:
Eigenvalue analysis helps decouple multiple degrees of freedom into independent vibration modes.
Used in Finite Element Analysis (FEM):
In FEM, eigenvalue analysis identifies vibration characteristics of large mechanical structures.

Applications of Eigenvalues

Mechanical Systems: Identifying vibration frequencies of machines and engines.
Structural Engineering: Determining the dynamic response of bridges, buildings, and towers.
Aerospace: Analyzing aircraft wing and fuselage vibrations.
Automotive: Studying suspension and chassis vibrations.
Finite Element Analysis: Used in modal and dynamic simulations.

Key Differences Between Eigenvalue and Eigenvector

Eigenvalue: Represents the square of the natural frequency ().
Eigenvector: Represents the mode shape associated with that eigenvalue.

Both together define the dynamic vibration behavior of the system.

Conclusion

In conclusion, the eigenvalue in vibration analysis represents the square of the natural frequency of a mechanical or structural system. It defines how the system naturally vibrates without external forces. Eigenvalues are obtained by solving the characteristic equation derived from the mass and stiffness matrices. Each eigenvalue corresponds to a unique vibration mode and determines how stiff or flexible that mode is. Eigenvalue analysis is essential in designing safe, stable, and efficient systems that can withstand vibrations without reaching resonance conditions.