Modal analysis is a powerful engineering tool
used to
determine the most basic dynamic properties of a
structure. These properties include a
particular set frequencies and a corresponding set of deformation patterns that
can be used to determine how the structure will respond under
various dynamic loads. The deflection patterns are called
"mode shapes" and the frequencies are called "modal
frequencies" or "natural frequencies". Every
unique structure has its own set of modal frequencies and mode
shapes that define that structure dynamically just like
a fingerprint. These modal properties are the
frequencies and shapes that are seen
during free vibration in the absence of any dynamic
loads. When dynamic loads are applied at
frequencies near the modal frequencies then the phenomenon of dynamic
amplification can occur which is also called resonance.
The condition of resonance, particularly at lower
frequencies, usually generates the maximum dynamic deflections
and stresses within a structure. So the greatest
danger of structural failure under dynamic loading
usually involves resonance. Modal
analysis is an essential part of the overall dynamic
analysis process to ensure that a structure can
withstand its required dynamic loading environment and
that the effects of resonance can be effectively
mitigated.
The results from a modal analysis
typically include the list of relevant modal frequencies
and mode shapes along with other important results such
as the modal participation factors and modal masses.
These modal results are commonly used as input for a subsequent dynamic analysis in which the actual dynamic response of
the structure is predicted under specified dynamic input
loads. So a modal analysis is usually the first step
of an overall dynamic analysis to predict the structural
response under dynamic loads such as shock and vibration. The results from the
overall dynamic analysis will then include the
actual predicted stresses, strains, deflections,
velocities, and accelerations that can be used for
structural design requirement verification.
The modal properties of a structure depend primarily on two
features: the mass distribution and the
stiffness distribution. That is, the spatial
variation of mass and stiffness within the structure
determines its modal properties. In addition, the manner in which the
structure is supported or constrained at its boundaries
can also be important in a modal analysis.
So a major part of any modal analysis involves properly
specifying and computing the mass, geometry, and
stiffness properties of the structure along with the
proper boundary conditions. For the most
general 3D problems mass distribution really means the
full inertia distribution which includes the translational
mass and the rotational inertia tensor. Likewise, the
stiffness distribution really means all the 3D
translational and rotational stiffness tensor
components.
Modal analysis can thus be defined as the process of computing
the modal properties of a structure given the mass and
stiffness distributions within the structure and any
applicable boundary conditions. In the text that
follows I will attempt to provide a brief overview of
this process including the most common techniques along
with the underlying assumptions and limitations. I
will show several examples of how finite element
analysis (FEA) is used for doing modal analysis of
simple and complex structures. I will also show
how the results from a modal analysis can then be used
to predict structural responses under various dynamic
loading environments including shock and vibration.
Finally, I will present a list of essential engineering
references for the engineer interested in doing modal
analysis.
Doing a modal analysis can be simple or
complex depending on the design requirements and the
complexity of the structure being analyzed. The
simplest analyses can be done in a matter of minutes
using standard engineering formulas. The most
complex analyses can require months and can include
large FEA models supplemented by expensive and timeconsuming
modal test measurements. No matter how simple or complex,
all modal analyses will require as input the geometry, mass, and
stiffness of the structure. The analysis output will
include all the requested modal properties that depend on the
design requirements. For the simplest analysis the required
output
is only the first or lowest natural frequency in any single direction. For
the most complex analyses the required output may include hundreds of natural
frequencies, mode shapes, modal masses, and participation factors in 3D.
A modal analysis typically involves the following steps:

Review design requirements to determine the frequency range of interest based
on the shock and vibration environments that the structure must withstand.

Determine the structural domain to be analyzed.

Obtain drawings and select the proper dimensions and tolerances

Simplify the overall structure into a
collection of smaller substructures as necessary while properly accounting
for the materials and material boundaries within each substructure.

Measure or compute mass and stiffness properties.

Properly handle any internal joints and fittings.

Properly account for applicable boundary conditions.

If necessary build a single or multiple finite element models with
proper mesh densities based on the required frequency range of interest.

Select the proper formulas or numerical algorithms and then compute the modal results.
All of the above steps require considerable engineering
judgment and experience if the
analysis is to be done correctly. From circuit boards to bridges, the consequences
of incorrect modal analysis can include catastrophic structural failure and loss of life.
So modal analysis is a vital part of the overall design verification process
for most important and highvalue structures that are
subjected to dynamic environments.
Modal properties are computed from the dynamic equations of motion in the absence
of damping and without any external forcing functions. This results in a system of linear
homogeneous second order differential equations with a unique set of eigenvalues and eigenvectors
that are directly related to the desired modal properties. Computing
such eigensolutions for complex
structures cannot typically be done analytically with engineering formulas and instead
usually requires
intensive numerical techniques such as 3D finite element analysis (FEA). Several
sets of modal FEA results are presented below. Many
simple structures such as circular plates, pipes, and
beams do have closed form analytical formulas for their
modal properties. Ref. 1 provides an extensive
collection of modal analysis formulas for many simple
geometries.
Modal analysis and the modal dynamic analysis that is
done with the modal analysis results is based on a set
of assumptions. These assumptions include .
Linear problems only with constant load path.
No buckling.
Good for frequency domain problems such as random vibration
response, shock response, frequency response, and modal
transient response.
Direct transient analysis in time domain
Explicit or implicit time marching
Linear and nonlinear problems
Variable load path including contact and impact problems
Nonlinear material response
Nonlinear buckling
Nonlinear kinematics (large strain, large rotation, stress stiffening)
Thermomechanical coupling