What is resonance? How do I prevent resonance from ruining my machine?

Resonance occurs when a vibratory system is subject to an external pulsing force and the excitation frequency is the same as the natural frequency of the system. When this happens, and there is no damping in the system, amplitude continues to grow infinitely. Typically, machines are designed with some damping in the system so that the amplitude reaches a finite peak value. Without proper damping, the displacements can escalate to a point where the system can no longer support its function and this can lead to complete destruction of the
system. Think of your machine. The base, normally a heavy casting or weldment, has a certain natural frequency depending on mass and stiffness. Experienced machine designers will try to create a sufficient spread between the natural frequencies of the base structure and the exciting frequency. But, even in the case of minor resonance, the tool life will be affected.  The problem can be significantly reduced by filling the base with a compound, which dissipates vibration energy as thermal energy to dampen the system. Consider the tool, in this case, a circular carbide-tipped saw blade. These blades are very stiff in the cutting direction (torsional stiffness), but laterally, 90° to the blade plane, the blades are very weak. To demonstrate this yourself, hit the blade body with an object when it is mounted on the drive spindle and see how long it will vibrate if it is not restrained by other means. Imagine the effect this can have on each cut. In extreme cases, when sawing hard, high alloy steel, the carbide tooth can have an impact force of up to 4,500N when it contacts the material. The harder the material, the harder the carbide tooth must be to resist wear and obtain an acceptable tool life. On the other hand, the harder the carbide tooth, the more brittle it becomes and, of course, brittle materials are debilitated by vibration forces. Smaller diameter saw blades are not as challenged, because the vibration amplitudes are smaller and the natural frequency is higher. The amplitudes of the vibration increase proportionally with the blade diameter, so the larger the saw blade, the more challenging it becomes to suppress the vibration amplitudes.

The magnification factor of the amplitude as a function of the frequency ratio.
The curve parameter is the dampening ratio.

History teaches some great examples of how important the
knowledge of resonance is. In 1940 the Tacoma Narrows Bridge
collapsed due to strong wind that caused the bridge to vibrate in
a torsional resonance mode.

Experimental Modal Analysis (Impact Test)

You hit the blade with an impact hammer. Accelerometers will track the transfer function with the help of a data acquisition device (DAQ). Hint: In case you don’t have a DAQ available, you can also use an oscilloscope and do the signal transformation (Fast Fourier Transformation) in Excel. As a result, you can see the lowest natural frequency, for example at 23Hz in the chart below.

Finite Element Modal Analysis

The equations that arise from the modal analysis are the same that can be found when solving eigenvalue problems.

1. Every eigenvalue (natural frequency) has a corresponding eigenvector (mode shape).
2. The benefit when using FEA is that you do not only get the frequency value, but you can visualize the mode shape easily.
3. The end result is very close to the measurement of 23Hz.
4. This mode, which is represented by node diameter 2 and node circle 0, is one which causes most damage to the teeth.

Calculation Using the Kirchhoff Plate Theory

One solution to this problem is to make the blade thicker, but thicker blades with wider kerfs create more waste material and, thus, make the sawing process more expensive. Thicker blades also require more horsepower to cut through the material, demanding heftier, more expensive carbide saws.

It is useful to take a closer look at a saw blade. It is essentially a circular plate from a structural standpoint. In our last piece, we explained that only lower resonant frequencies have a damaging effect on sawing. So how can you measure the critical resonant frequency of a blade in cycles/sec (Hz)? 

Now, since we know the methods to obtain the natural frequency of the blade, we can compare it with the tooth pass frequency of the blade. You must keep in mind that the blade mounted on the drive hub usually has different boundary conditions and therefore a different natural frequency to the free annular plate we analyzed before.
Still, let us assume you are cutting some alloy steel with a cutting speed (vc) of 82m/min and a saw blade (as mentioned before) of 1,120mm diameter with 60 teeth.

If the saw blade with 60 teeth will run at 23rpm you will have a tooth pass frequency of 23Hz. Matching frequencies – that is your problem, if it matches the natural frequency f1 of your carbide-tipped saw blade. A slight change of the saw blade RPM will spread the frequencies and improve your machine performance without compromising your tool life.
When you next order saw blades you can also increase or decrease the number of teeth a small amount and get a better performance.

It is much easier to make a change in RPM than it is to repair a poorly designed machine, but without the knowledge of the damaging effect of resonance and how to make the appropriate adjustments to avoid it, you can expect a downtime crisis.

Conclusion

• Carbide saws are relatively simple machines, but modern engineering practices are still used to uncover hidden performance-robbing factors, such as resonance.
• Modern engineering aids like data acquisition devices (DAQ) and finite element analysis (FEA) features in CAD software are used to uncover issues during the design of industrial machinery.
• There is no substitute for practical engineering and industrial machinery experience. Modern technology only serves to facilitate quicker calculations.
• A solid knowledge of the dangers of resonance allows you to know the important parameters that need to be adjusted to benefit from longer tool life and higher productivity.
• Experienced machine designers analyze all vibration sources using stabilizing and damping aids to improve the sawing process.

by Willy Goellner, chairman and founder – Advanced Machine & Engineering/AMSAW

Download this article as seen in the Tube & Pipe Journal

Read Part 3 in this Series on the topic of Minimizing Vibration and Resonance

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