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When a long ruler made of steel is grabbed by its ends and bent, the ruler flexes. If one continues to bend even after it flexes to an extreme degree, the ruler will eventually break. A person designing a ruler must determine how easy or difficult it will be to bend the ruler, while simultaneously determining how easy or difficult it will be to break.

Figure 1 Slightly bent ruler

For another example, take the case of a bridge. Multiple heavy trucks are crossing a bridge. People seeing this worry about whether the bridge will collapse. Although the bridge did not collapse, the heavy shaking that occurred when the truck went over the bridge scared the driver. Since the bridge didn't collapse even when a heavy truck went across, does that mean the design is acceptable? No, it does not. Just like in the previous example with the ruler, an engineer must reflect in the design how much of a load a bridge can handle as vehicles cross the bridge, and how much deformation will occur. Even though it did not collapse, a bridge that shakes excessively is difficult to view as being well designed.

Figure 2 Truck Crossing a Bridge

Here, whether or not a collapse will occur is a question of strength, and how much it will flex is a question of rigidity. While strength and rigidity are different types of standards, a design must satisfy both criteria.

1. Simulations and Flexible Bodies

Even without making a physical version of a designed product and conducting testing on it, computer simulations can be used to check strength (how much it can withstand before breaking) and rigidity (how much it flexes).

In the past, blueprints were created during the design phase by hand-drawing lines on drafting paper with a ruler and pencil. Then we began drawing 2 dimensional blueprints on computers, and now the process has developed to the point where 3 dimensional shapes are drawn directly on computers. Using computers in such a way to do design is called CAD (Computer Aided Design).

Simulations use the shape of a product that has already been created with CAD. But since the shapes created with CAD simply show exterior shapes and mass, they cannot be used to calculate how much it will flex. That is why additional work is needed for a simulation of flex. The method of allowing for flex is to cut the shape into very small pieces, and attach a spring between a cut piece and the pieces next to it. The individual cut pieces are called elements. When all of the cut pieces are drawn together, it looks very much like a net, so this is called a mesh. And a shape that can be flexed is called a flexible body.

When small cut pieces are used to form a curved shape, using smaller pieces allows for the curve to be followed more accurately. If infinitely small pieces could be used, the size of the pieces would be infinitely small and the curve would be accurate. However, since an infinite number of pieces cannot be used, a reasonable limit must be established in reality. And since a finite number (N) of elements are used instead of an infinite (∞) number, this is called the finite element method.

Figure 3 Shapes cut into different sized pieces

2. Vibration

Lewis Carroll Epstein’s 『Thinking Physics 』 explains vibrations in the following manner.

“A wiggle in space is a wave; a wiggle in time is a vibration. [omitted] A wave cannot exist in one place — it must extend from place to place. And a vibration cannot exist in one instant — it needs time to move to and fro. Besides extending through space and/or time, waves and vibrations are peculiar in yet another way. Unlike a rock, which will not share its space with any other rock, more than one wave or vibration can exist at the same time in the same place, like the voices of people singing at the same time in the same room. These vibrations and those of a whole symphony orchestra can be captured on the single wavy groove of a phonograph record, and amazingly, our ears discern the component vibrations as we delight in the intricate interplay of the various sources. We enjoy the vibes.”

(Lewis Carroll Epstein, physics professor at the City College of San Francisco and author of 『Thinking Physics』)

Figure 4 An orchestra playing music

This is an amusing way of explaining waves and vibrations using space and time. Two points are of interest.

1. Vibrations can exist simultaneously in the same space. (can overlap)
2. Our ears can sense the individual vibrations in a concerto. (sounds of individual instruments can be distinguished)

This means that vibrations can be separated into separate individual vibrations. For example, <Figure 5> shows the waveform of a vibrating tuning fork, and this can be separated into individual vibrations.

Figure 5 Acoustic waveform of a tuning fork

When an FFT is conducted on the vibrations of the acoustic waveform, the vibrations can be separated into individual frequencies. <Figure 6> shows the results of carrying out an FFT on the measured acoustic waveform, and it shows that the overall vibration can be separated into simple vibrations of 366Hz and 710Hz. Separating the overall vibration into separate vibrations at separate frequencies means that, when the separate vibrations at separate frequencies are combined together, this is identical to the overall vibration.

Figure 6 Results of FFT on the vibrations of a tuning fork

A layman’s explanation of the above can be provided by a quote from French physicist Fourier, who stated, “A single complex wave is actually the sum of multiple simple waves.” This statement means that no matter how complex the wave, it can always be divided up into multiple simple waves, and when these simple waves are combined together, they become identical to the complex wave.

<Figure 7> shows how a complex wave (C) is separated into the two simple waves (A) and (B).

Figure 7 Combination of waves

So even the most complex vibrations can be separated into multiple simple vibrations, and in the reverse, a combination of multiple simple vibrations can express a complex vibration.

This article 'Flexible Body Simulation and Reduced Model" will continue in the 2nd part.

Written by Taero Cha (Director of China Business Division)