Object:Dynamic balance is calculated. It is required on high speed milling.
Milling cutters for alminum require high cutting speed.
High cutting speed require high rotational speed.
When the center of grabity of the milling cutter is not on the axis ot rotation, the high rotational speed causes the vibration which is caused by centrifugal forces.
The vibration damages the main spindle or the workpiece surface.

The vibration is the dynamic phenomenon, not the static one.
Thus, when the cutter is not rotated, the vibration can not be found.
This kind of vibration is caused by the dynamic unbalance or the static unbalance.
The gap between the center of grabity of the milling cutter and the axis ot rotation is quantified as the dynamic balance.
Before using high speed milling cutter, the dynamic balance should be checed and modified.
Thus, this calculation function is made for analysing the dynamic balance.

Quality of the dynamic balance is shown as the dynamic balance grade "G".
The equation for the dynamic balance grade is shown as follows.
The dynamic balance grade equals to the gap between the center of grabity of the milling cutter and the axis ot rotation times the rorational speed.
Thus, if the dynamic balance grade is the same, the gap must be controlled more closely with higher rotational speed.
The definition of "high speed milling" is written in ISO 15641.
The rotational speed for high speed milling depends on the tool diameter.

\(\displaystyle G = \cfrac{ \varepsilon \times N}{9550} \)

\( G \)：Dynamic Balance Quality[mm/s]
\( \varepsilon \)：Eccentricity[um]
\( N \)：Rotational speed[rpm]

Pysical sense of the dynamic balance grade shows the maximum value of vibration speed during rotation.
However, currently, I can not understand the reason why the maximum value of vibration speed during rotation is used.

On JIS 0905, reference values of the dynamic balance grade for each product are shown.
For example, the main spindle of the machining center require G2.5.
Parts for machining center require G6.3.
According to JIS 0905, G2.5 or G6.3 is enough for the high speed milling cutter.
Reference values of the dynamic balance grade for the high speed milling cutter are shown in the catalogue which is made by milling cutter companies.
However, reference values of each company are different.
Thus, I guess that there are no collective view.

Residual unbalance is also used to check the dynamic balance.
Residual unbalance equals to the gap between the center of grabity of the milling cutter and the axis ot rotation times the rotor weight.

\( U = \varepsilon_{per} \times m \)

\( U \)：Residual Unbalance[um kg = g mm]
\( m \)：Rotor weight[kg]

Centrifugal force equals to the residual unbalance times rotational speed.

\(\displaystyle F_{r} = U \times \omega^2 = U \times ( 2\pi \cfrac{ N }{60})^2 = \varepsilon \times m \times ( 2\pi \cfrac{ N }{60})^2 \)

\( F_{r} \)：Centrifugal force

The most difficult problem is how to measure and control the dynamic balance.
The dedicated measurement instrument is required for measuring the dynamic balance.
And the process to controll the dynamic balance is also difficult.
For example, attaching counter weights or removing some parts of the cutter should be done.
Some high speed cutter has screw holes for controlling the dynamic balance.
However, most of cutters do not have these.
Thus, if we would like to controll the dynamic balance, we have to remove some parts of the cutter.
However, this method can not be done many times.

Controlling the dynamic balance is one of vibrational science and it is very difficult.
If you would like to understand them, you have to read documents yourself.
I think that it is better.
This calculation function is only used for static unbalance.
If the overhang of the cutter is long, dynamic unbalance become larger.
This calculation function can not be used for dynamic unbalance.