Torsional vibrations are hard to measure because of the rotating elements (gears, shafts) that are inside a closed gear case and not easily accessible. However, knowing the exciting torque, passing through the gear train and the gear data, critical vibrations such as resonance can be calculated and analyzed.

The following calculations and theoretical analysis are perhaps boring for most individuals dealing with carbide saws; but it shows that much can be theoretically analyzed, where practical limitations exist.

A well experienced carbide saw specialist will be able to analyze torsional vibration from the noise level, the cut surfaces, and the form of the chips. He will however, have difficulty pinpointing where the trouble starts and what to change in order to solve the problems.

In every gear train torsional vibrations and torsional natural frequencies can cause significant problems if not addressed properly.

One approach to realize basic effects and parameters is to discretize every shaft in stiff inertias and massless torsional springs which connect these inertias.

This has been done for a simple 1 stage gear train with 4 inertias and 3 degrees of freedom since inertia 2 and Inertia 3 are coupled with the ratio. After having modeled up the system in this fashion the dynamic equilibrium of the forces can be formulated for the free body diagrams.

The result is the following set of differential equations which has the structure: 

With the concept of eigenvalues and eigenvectors the natural frequencies and corresponding mode shapes can be calculated.

The determinant is set equal 0 and values below are used to calculate  λ^2

Example-Values based on a simplified portion of an AMSAW Gearbox :

The torsional natural frequencies and mode shapes are:

From this diagram one can see where the nodes which a standing still are and which parts are moving against each other. You can also see where torsional dampers will be effective, how a flywheel effects the system.

For reals world problem the use of numerical system such as Octave with necessary coding is needed to handle the large matrices efficiently.

One other approach is the use of Open Modelica which is quite intuitive to use for developing a gear train. Basically you drag and drop predefined objects, link them to represent your system and set the parameters.

Example: Input is a sine-signal with a certain frequency. See picture below.

If you mount an accelerometer and measure the angular acceleration you can see the vibration with its homogeneous and a particular solution. Due to damping the system is settled within a certain time frame as you can see in the following charts.

Afterwards you can measure the amplitude. You can repeat this for several frequencies. If you show the amplitude vs. frequency you get a chart like this:

This shows that at around 300 rad/s and 750 rad/s the amplitude has a peak. These are the two resonant frequencies.

How can you see if you experience torsional vibrations with your saw?
Just take a look at the chips and the cut surface.

The chips are evenly spiral curved with a clean surface. The cut surface is smooth showing no vibration pattern.

The sound of the saw blade and gear box measured with a sound level meter is smooth indicating no vibration.

A blade subject to torsional vibration will produce wrinkled uneven curled chips. The surface of the chips and the cut surface are rough and show the vibration pattern. The noise of the blade and the gearbox sounds rumbling. This is caused by the windup and relaxation of the gear train.

This fluctuation of the torque creates an uneven chip thickness and in consequence reduces the tool life. This forces the operator to slow down the machine and reduces the output. Furthermore the uneven surface requires more stock for finishing operation and increases material waste.

Instruments such a strobes and accelerometers allow us to pin point where trouble occurs inside of the gear box. This helps especially for trouble shooting when a gear box gets worn or an accident has damaged gears or bearings.

By Christian Mayrhofer, Manager R&D, AME &
Willy Goellner, AME

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