# Fundamentals of balancing

With rotating bodies, imbalance is an omnipresent phenomenon. A typical example are rotating tools on machine tools.

Because unbalance creates a centrifugal force, it increases linearly with the unbalance and squares with the number of rounds. The faster the rotor rotates, the more noticeable the unbalance. But how does unbalance arise, how can it be measured and how can it be eliminated by balancing?

On the following page we have put together the theoretical fundamentals of balancing, the basis for tool balancing.

## 1. Causes of unbalance

• Unsymmetrical design of the rotor (e.g. gripping groove on tool holders as specified in DIN 69871 or clamping screw on Weldon tool holders).
• Unsymmetrical distribution of mass due to concentricity errors caused by manufacturing tolerances, e.g. concentricity of the tool outer diameter with respect to the taper.
• Alignment errors during assembly of a rotor consisting of several components, e.g. milling spindle and tool holder, tool holder and tool.
• Concentricity errors in the bearings of a rotor, e.g. the spindle bearing.

## 2. What is unbalance?

### 2.1 Static unbalance

The centre of gravity of a rotor lies outside the axis of rotation.

• This can also be measured in stationary rotors, e.g. by means of unbalance scales for grinding wheels.
• Upon rotation this causes centrifugal forces perpendicular to the axis of rotation.
• It can be eliminated by balancing in one plane. Any balancing plane can be chosen. Normally there may still be couple unbalance after static balancing.

MU = unbalance mass (in g)
r = distance of the unbalance mass from the axis of rotation (in mm)
M = rotor mass (in kg)
e = distance of the centre of gravity from the axis of rotation (in μm)
S = center of gravity
FF = centrifugal force
Value of static unbalance: U = MU • r = M • e
Unit of unbalance: [U] = g • mm = kg • μm

### 2.2 Couple unbalance

The centre of gravity lies along the axis of rotation.

• This can only be measured in rotating rotors.
• It causes a moment of tilt during rotation.
• The centrifugal forces of both unbalance masses counterbalance each other (no lateral forces).
• It can only be eliminated by balancing in 2 planes.

MU1, MU2 = unbalance masses (in g)
S = center of gravity
r = distance of the unbalance masses from the axis of rotation (in mm)
M = rotor mass (in kg)
FF1, FF2 = centrifugal forces
MU1 = MU2
FF1 = FF2

### 2.3 Dynamic unbalance

Combination of static and couple unbalance

• This is the normal case for industrial rotors.

## 3. What is balancing?

Balancing is used to make compensation for the unsymmetrical distribution of mass in a rotor.
This is possible by:

• applying mass, e.g. a clamped weight to balance car tyres
• removing mass, e.g. by drilling a hole

### 3.1 Balancing in one plane (static)

Compensation for the static portion of an unbalance

• The centre of gravity of a rotor is brought back to the axis of rotation (eccentricity e=0).
• The couple unbalance of dynamic unbalance remains unchanged.

### 3.2 Balancing in two planes (dynamic)

Complete compensation for unbalance (static and couple unbalance)

• In principle, any balancing planes can be selected (it is best if they are as far apart as possible).

## 4. Measuring unbalance

### 4.1 Measuring principle

• The tool holder is inserted into the balancing spindle and made to rotate.
• Force sensors measure any centrifugal forces.
• The centrifugal forces are measured in two different planes on the support of the balancing spindle. A sinusoidal signal is produced as the direction in which the centrifugal forces act turns with the spindle. Both the magnitude of the signal and its angle to the spindle must be determined.
• The force signals are used to calculate the balances in relation to the balancing planes. If the position of the balancing planes changes, the unbalances calculated will also change.
• The unbalance compensation is calculated from the unbalance values.

## 5. Balancing tool holders

### 5.1 Balancing quality G

DIN ISO 1940-1 (previously VDI guideline 2060) defines the principles for measuring unbalance
and for balancing. The accuracy of balancing is expressed as balancing quality grade G (previously: Q).

The balancing quality grade is only ever valid for one particular rotation speed of the rotor.

The permissible residual unbalance is calculated from the balancing quality grade, the rotation
speed and the weight of the rotor.

Uper = (G•M)/n • 9549
Uper = Permissible residual unbalance of the rotor in gmm
M = weight of the rotor in kg
n = rotation speed of the rotor in rpm
9549 = a constant that is produced

Example

• A milling cutter is clamped in a collet chuck.
• The total weight is 0.8 kg.
• The milling cutter is to be used at a service speed of n = 15,000 rpm.
• The spindle manufacturer requires a balancing quality grade of G = 2.5.
• Permissible residual unbalance Uper = 1.3 gmm.

The permissible residual unbalance can be seen in the diagram.

### 5.2 Achievable accuracy

In the above example there is a permissible residual unbalance of 1.3 gmm. To illustrate
this value it is useful to convert the unbalance to eccentricity.

Uper = M • eper
eper = Uper/M =1.3 gmm/800g = 0.0016 mm = 1.6 μm

Therefore the centre of gravity of the tool holder can be offset by max. 1.6 μm from the axis of
rotation. During balancing the axis of rotation is assumed to be the axis of the taper or HSK.
However, in the milling machine the tool rotates about the axis of the spindle.

Even new spindles TIR of up to 5 μm (equivalent to eccentricity of e = 2.5 μm).

Further example:
Balancing quality G = 1
Rotation speed n = 40.000 rpm
Tool weight M = 0.8 kg
Uper = 0.2 gmm
eper = 0.3 μm

This permissible eccentricity cannot be achieved in practice.
Even good spindles have a repeatability of 1-2 μm when the tool is changed.
Small amounts of dirt worsen the result significantly.

The total unbalance of a milling spindle is affected by many factors:

• unbalance of the spindle itself
• unbalance due to concentricity errors in the spindle (The symmetry axis is not the axis of rotation.)
• concentricity errors in spindle accessories (opening for coolant, clamping device)
• lateral distortion of the clamping system upon tightening (springs, draw bar)
• concentricity error and inclination of the tool holder in the spindle
• unbalance of the tool holder itself
• concentricity error of the pullstud (offset)
• concentricity error in the tool
• unbalance of tool holder accessories (e.g. tightening nut)

Conclusion
A permissible residual unbalance of less than 1 gmm is unrealistic in practice.