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Introduction to FE Analysis

Tutorial CD

Introduction to FE Analysis

Tutorial CD

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

approach.

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

information.

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 modes |

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/

Follow the links for further information

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/

Follow the links for further information

If you still need some guidance, try clicking the hyperlinks for fuller definitions or sign up for the Introduction to FE Analysis Tutorial CD |