Modal analysis tells you how a structure naturally wants to vibrate. This Book Byte introduces modal analysis, the eigen equation of solid mechanics, the numerical methods used to solve it, what a mode is and what it is not, and why computing a modal analysis is useful.
Introduction to modal analysis
- Modal analysis is the study of the dynamic properties of linear structures using finite element simulations.
- These dynamic properties are the natural frequencies and the structural modes.
- They depend on the mass and stiffness of the structure, and they determine its vibration behavior under operational loads.
- Modal-based calculations are very effective and allow efficient evaluation of structural changes.
The eigen equation of solid mechanics
The starting point is the general equilibrium equation:
[M]{ü} + [B]{u̇} + [K]{u} = P(t)
with [M] the mass matrix, [B] the damping matrix, [K] the stiffness matrix, {ü}, {u̇} and {u} the acceleration, velocity and displacement vectors, and P(t) the applied load.
For a real eigenvalue analysis (normal mode analysis), damping and external loads are removed, and a harmonic solution is assumed:
- [M]{ü} + [K]{u} = 0
- {u} = {Φ} sin(ωt)
- ([K] − ω²[M]){Φ} = 0
The square of ω is the eigenvalue, which gives the natural frequency f = ω / 2π. {Φ} is the associated eigenvector, which gives the mode shape.
Numerical methods for solving the eigen equation
The two most popular methods are:
- The enhanced inverse power method
- The Lanczos method
The choice depends on:
- The total number of degrees of freedom of the problem
- The number of eigenvalues to compute
- The available computer memory
- How well you can estimate the range of eigenvalues for the problem
| Enhanced inverse power method | Lanczos method (my favorite) |
|---|---|
| Use it only if few modes are needed. | Efficient for a large number of modes. |
| You need a reasonable idea of the eigenvalue range. | No initial estimate of the eigenvalues is needed. |
What a mode is and what a mode is not
Natural frequency
An object's natural frequency is the frequency at which it vibrates naturally when disturbed.
What a mode is
A mode is a deformed shape plus the natural frequency at which this shape occurs. If we deform a structure into one of its mode shapes and release it, it oscillates continuously at the natural frequency of that mode.
So a mode is a shape with a corresponding natural frequency, at which the structure absorbs all the energy provided by an excitation.
What a mode is not
Mode shapes have nothing to do with displacements. The displacements, stresses and strains computed by a modal analysis are arbitrarily scaled and cannot be used as physical results.
Why compute a modal analysis?
Modal analysis provides very useful information in various situations:
- Defining the natural frequencies at which a structure tends to vibrate, and the associated mode shapes, to avoid resonance.
- Identifying possible modeling mistakes, such as missing connections between components or incorrectly defined constraints.
- Determining the behavior of the system under dynamic loads.
Go further
This Book Byte is taken from Chapter 24 of Practical Finite Element Analysis for Mechanical Engineers by Dominique Madier. The topic is covered in depth, with worked examples, in the on-demand course Practical Basic FEA.
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