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Q: What is modal analysis?

Analyzing a structure’s normal modes or natural frequencies also finds its resonant frequencies.
Examples include a diving board after a diver leaps and a cup holder in a car rattling after the car
hits a bump. The terms natural frequency and normal mode tend to be synonymous. But the first
refers to frequency measured in cycles per second (Hz) or similar units. The second term refers to
the characteristic of a deflected structural shape as it resonates.
The mesh of solid elements defines a cup
holder. The point in the middle represents the
liquid center of mass that can be relocated to
simulate a liquid level.

In practice, you cannot know a structure’s natural frequencies until it is jolted, hit, or excited in some
way. As usual in physics, the system needs an input to get a response. Physical testing for normal
modes excites the system and measures the response. But theoretical analyses must take a different

The theoretical approach looks for frequencies of a system that perfectly balance internally stored
energy and kinetic energy. At these frequencies, exchange between the two energies is triggered by
any external input at the same driving frequency as the natural frequency. The diving board’s
resonance frequency matches the “spring” of the athlete’s jump. Similarly, the resonant frequency of
the cup holder matches a vibration input from the suspension.

In addition, the theoretical approach doesn’t need an external excitation. The energy balance is
calculated by considering inertial and stiffness terms in isolation. All structural frequencies and mode
shapes can be found this way. The downside to the approach is that the actual amplitude of the
calculated shapes cannot be determined.

This confuses many novice users and prompts a few questions. For example: How do you predict a
shape without knowing its magnitude? The question requires a two-part answer. First, predicting the
shape comes from balancing stored and kinetic energy. But the magnitude cannot be predicted without
defining an input frequency. For the diving board, needed inputs would be the diver’s weight or how
high he or she jumps. And we don’t know the speed of the car or the height of the bump that excites
the cup holder. All we know is the range of natural frequencies for the board and holder, and what their
deflected shapes will look like.

You’ve seen FEA-generated animations of vibrating structures and deformations, and their
deformations look large. Generating such images requires an arbitrary deformation value. They tend
to look large because the software scales the magnitude of the shapes for convenience and
comparisons. A postprocessor further scales deformations so users can see each mode shape clearly.
Software typically makes vibrations 10% of the maximum viewable dimension.

The second part of the answer is that we are dealing with linear, small-displacement theory. That
means vibration amplitudes in real life must be small relative to the size of the structure.

Frequency ranges
It’s also useful design information to find a range of resonant frequencies. It’s easy to picture the diving
board as having just one resonant frequency, and the cup holder two or three that may be noticeable.
But theoretically there are an infinite number of natural frequencies in any structure. We are usually
only interested in a system’s dominant or first few natural frequencies.

The first four modes of vibration for a diving
board show simple bending first order
bending through to third order bending,
together with a twisting or torsional mode.

The important question now becomes: How does one find the dominant natural frequencies?
The fact is, modal analysis on its own will not tell. Because analysis does not provide an input
excitation, it cannot reveal which modes will be excited. Modal analysis is really a first and vital
step to understanding the response of a structure. The next step is to apply an input excitation
in a response analysis, or make some assumptions about the input. Response analysis is
beyond the scope of this article.

In the meantime, we have to rely on engineering judgment. Suppose analysis shows that a
diving board has a first natural frequency of 5 Hz with a bending mode shape. That can be a
basic design input. The response for the cup holder may include bending and twisting of the
attachment arms and the cradle. To see what is important may require comparing test data, or
doing further response analysis. But the first few modes of a structure are vital to understanding
its dynamic characteristics.

Consider the two examples. The diving board shows a mesh of shell elements and the cup
holder, a mesh of solid elements. The board is straightforward, but the cup holder is more
interesting because the cup may be partially or completely filled. How it vibrates is more
important when it’s full. The mass of the cup and coffee is important, so simulating volume calls
for a few decisions. Should you assume it is full, half full, or somewhere in between? It’s
probably best to do trade-off studies to find an answer. Assume the cup is full, which gives a
mass and position for the center of gravity. Also, assume the center of gravity doesn’t shift
during vibration. This is a big assumption because fluid may slosh around. After investigating
vibrations and modes for their linear response, a second step or nonlinear analysis would add

After making assumptions about mass, think about the stiffness. Both are equally important. It’s
safe to assume the liquid is nonstructural and the cup is rigid. This is reasonable if we have a
stainless-steel thermally efficient vessel, but not so good if we are holding a milkshake from a
drive-in. Assume we have a flexible cup. In fact it’s so flexible we can ignore it.

But how is a nonstructural mass linked to the holder? Most FEA codes have elements that
distribute loads into a structure using “soft” connections. In the case of NEiNastran, it is an
element called RBE3.

An RBE3 is perfect for this case because it will not stiffen the holder. In a stiff-cup scenario, a
sister element (RBE2) assumes the cup is infinitely rigid. This makes the picture of the cup
holder mesh clearer. It shows a lumped mass for the liquid’s center of gravity, and RBE3
elements (shown as lines) represent the cup.

Analysis results of the diving board show its first four modes. It’s intuitive to expect the first mode
to dominate, but notice the twisting in mode four at 19.1 Hz. The twist might come from a heavy
diver that caught a corner of the board. If such loading is a reasonable concern, then include it
in later engineering work.

The cup holder shows complex shapes from 15.7 Hz upwards. It’s difficult to know what is
important. The first mode is a cantilever nodding mode, then two twisting modes, and the last
two are hogging modes which take the analysis up to 144.7 Hz. One might expect all five to be
important, but as yet we can’t prove that.

The first five modes of vibration for the
cup holder range from 15.7 to 144 Hz.
They show a first bending mode, a
sideways mode and then more complex
The question now becomes: What is the range of input or driving frequencies? Suppose we are
a subcontractor to an automaker. If we’re lucky, an auto engineer will tell us what range of
frequencies to expect at the attachment region on the dashboard. That figure becomes a
starting point to examine a range of interest. This given frequency range is not immediately
useful because it ignores complex interactions between harmonics of the system and other
factors. So it’s typical to take an upper bound of 1.5 or 2 times the range’s top limit. The lower
bound should go down to the lowest frequency found because it is difficult to provide a sensible
lower cut off here. That gives a set of modes of interest.

To go further we need to defined some type of loading on the structures so that we can
investigate a response to a real loading environment.

This can be of several types:

Shock Spectra
Transient Analysis
Frequency Response Analysis
Random Response Analysis
DDAM US Navy Analysis

In addition we can also make some assessment of the contribution of each mode from a base
excitation by looking at the Modal Effective Mass/

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