Tony Abbey
Instructor Noran Engineering
Westminster, Calif.

The diving board 

The cup holder 

Mention FEA and the first things that comes to an engineer's mind are stress and deflection. But FEA can do more than that. It can also show how structures shake when they vibrate, a topic that gets a lot less press. Yet examining a part or structure's modal vibrations can point the way to product longevity and design changes to prevent annoying or potentially hazardous vibrations.
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.
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 twopart 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 FEAgenerated 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, smalldisplacement 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 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 tradeoff studies to find an answer. Assume the cup is full, which gives a mass and position for the center of gravity. Also, assumethe 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 stainlesssteel thermally efficient vessel, but not so good if we are holding a milkshake from a drivein. 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 stiffcup scenario, a sister element (RBE2) assumes the cup is infinitely rigid. This makes the picture of the cupholder 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 expectall five to be important, but as yet we can't prove that.
The question now becomes:
What is the range of input or driving frequencies?
Suppose we are a subcontractor to an auto maker. 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.