2 Basic principles of balancing techniques Balancing means improving the mass distribution of a rotating object so that it rotates in its bearings

without any effect from free centrifugal forces, and the bearings are not subjected to excessive

periodic forces at the fundamental frequency.

In general the concept of "balancing" involves two separate actions:

• measuring the unbalance and

• correcting the unbalance.

A rotor is completely balanced when the centre of gravity axis coincides with the rotational axis as

prescribed by construction. If these do not coincide, then the weight distribution is not even and the rotor is not balanced.

2.1 Types of unbalance

Depending on the distribution of the unbalance over the length of the rotor, a distinction is made

between two types of unbalance in rigid rotors1):

• Static unbalance

If an unbalance is added to a symmetrical, completely balanced rotor in the same radial plane as

the centre of gravity, this constitutes a static unbalance. This unbalance causes a parallel displacement of the centre of gravity axis from the rotational axis equal to the displacement of the

centre of gravity of the complete rotor.

• Moment unbalance

If, in the case of a symmetrical, completely balanced rotor, two equal unbalance weights are applied at the same radius in two different planes at phase positions exactly opposite one another,

they constitute a moment unbalance. In this case the centre of gravity axis is inclined to the

rotational axis and intersects it at the centre of gravity of the rotor.

1) Most rotors are constructed such that the unbalance condition does not change, or changes only insignificantly, throughout

the entire service speed range. Rotors of this type are called "rigid rotors".

Figure 2.1: Types of unbalance - Static unbalance, moment unbalance and dynamic unbalance

In general, because of manufacturing reasons, almost all rotors possess both static and moment

unbalance. This means that the centre of gravity axis is inclined to the rotational axis but does not pass through the centre of gravity of the rotor. For this type of unbalance the term

dynamic unbalance

is used.

2.2 Unbalance effects

The unbalance effect will be explained using a rotating disc as an example.

If an unbalance mass ‘u’ is added to a balanced disc at radius ‘r’ the centre of gravity point S will be

displaced from the rotational axis by the distance ‘e’ . According to the conditions of equilibrium:

e x m = u x r The product u x r is called unbalance U, and the value ‘e’ is called the specific unbalance, or

centre of gravity eccentricity. U is described in units of g.mm or g.ins, and ‘e’ in units of g.mm/kg and µm, or g.ins/lb and mils.

The nomogram in Fig. 2.3 shows the relationship between centre of gravity eccentricity, rotor mass and static unbalance.

Figure 2.2: A rotating disc with an unbalance mass at radius ‘r’, has its centre of gravity displaced by a value ‘e’

which creates a

centrifugal force F

Figure 2.3: Nomogram to determine the centre of gravity eccentricity when the rotor mass and static unbalance are known.

Example 2.1: A pulley with mass of 20 kg has a static unbalance mass u = 5 g added at a radius r = 100 mm. Calculate the magnitude of the center of gravity eccentricity ‘e’.

Solution: e = u � r m e = 5 g � 100 mm

m

e = 25 µm

Figure 2.4: Extract from the nomogram in Fig. 2.3 showing the example 2.1

During rotation the unbalance mass ‘u’ creates a centrifugal force F, according to the formula

Fu = u x r x ω2 in N when ‘u’ is in kg and ‘r’ is in meters)

where the angular velocity of the rotating object is equal to ω = 2πf. The centrifugal force caused by

the unbalance can also be simply determined from the diagram in Fig. 2.5.

Figure 2.5: Relationship between unbalance, rotational speed and centrifugal force.

Example 2.2: The disc in example 2.1 with the static unbalance u = 5 g at a radius r = 100 mm rotates at a speed

of n = 6, 000 rpm (f = 100 cps). Calculate the centrifugal force in N induced by the unbalance.

Solution: Fu = u x r x ω2 N

Fu = 5g x 100mm x (2π x 100) 2

Fu = 197 N

Figure 2.6: Diagram from figure 2.5 showing example 2.2 With a moment unbalance a centrifugal force moment is created which consists of two equal

centrifugal forces in opposite directions at the two radial planes. The two centrifugal forces Fu1 and Fu2 can be calculated according to the following relationships:

Fu1 = U1 x ω2

Fu2 = U2 x ω2

As long as the rotor is supported in rigid bearings, the centrifugal forces must be completely absorbed

by the bearing elements, bearing supports and the foundations. These centrifugal forces caused by the unbalances excite the complete machine and its surroundings into

vibrations at the fundamental frequency of the rotor.

How large the vibrations will be depends not only on the bearing forces, but also on the dynamic behavior of the entire machine and in particular the dynamic stiffness. The basic principles and

methods of determining the dynamic behavior of machines are described in detail in the Machine/Rotor Dynamics seminar.

The size of the bearing forces can be calculated using the following equations (see also Fig. 2.7). Static unbalance: Us = u x r = e x m Centrifugal force:

Fu = Us x ω 2 Bearing forces:

FA = FB = Fu 2 Moment unbalance: Um = U1 x l = U2 x l Centrifugal forces:

Fu1 = U1 x ω 2 Fu2 = U2 x ω 2 Bearing forces:

FA = l x Fu1 FB = l x Fu2 L L

Figure 2.7: Centrifugal forces and bearing forces with static unbalance and moment unbalance

2.3 Causes of unbalance

Unbalance always exists when the mass distribution of the rotor is not symmetrical with reference to

the shaft rotational axis. The causes of unbalance can take many forms and can be combined into

four groups:

§ Construction and drawing errors e.g. components not symmetric, un-machined surfaces on the rotor, variations in roundness and

construction because of coarse tolerances.

§ Material faults

e.g. blow-holes in cast components, non-homogeneous material density, uneven material

thickness

§ Manufacturing and assembly errors

e.g. distortion from welding and casting errors, stress errors caused by work procedures,

permanent deformation caused by relieved stress, shrinking after welding or soldering, stress

caused by uneven tightening of bolts or screws.

§ Faults during operation

e.g. erosion and corrosion of the rotor, material build-up on impellers, thermal deformation of

hot-gas exhaust fans, blade fracturing on turbine rotors, wear on grinding wheels, displacement

of rotor parts caused by centrifugal force, general wear.

Many of these errors and faults can be controlled, but can never be totally eliminated to the extent

that balancing becomes unnecessary. Faults which occur during operation over a long period of time

are especially unavoidable.

Figure 2.8: Build-up on an impeller is a typical fault which occurs during operation. Large unbalances can suddenly be caused by pieces of the buildup breaking away.

2.4 Measuring and correcting the unbalance

To eliminate the unbalance it is necessary to restore the mass symmetry of the rotor by adding or removing mass at the correct position. Depending on the type of unbalance in the rotor, the act of

correcting the unbalance is also called static or dynamic balancing.

2.4.1. Errors caused by correction

The aim of unbalance correction is to oppose an existing unbalance vector with an

equally large correction vector located exactly 180º to the existing unbalance. The sum

of the initial unbalance vector and the correction vector gives rise to a residual unbalance

vector, which does not have to be "0", but whose magnitude should remain within the

prescribed tolerance.

Figure 2.9: The aim of unbalance correction

The more accurately the magnitude (the weight of the correction mass) and the direction

(the angular position at which the correction mass is applied or removed) can be

determined, the greater the accuracy with which this goal can be reached.

The further the original unbalance is from the circle of tolerance, generally speaking, the

more accurately both magnitude and angular position for the correction weight must be

determined.

2.4.2 Correction weight tolerances

If we assume at the outset that the angle can be determined with great accuracy by

calculation, the question arises "How accurate must the magnitude of the physical

correction weight be to ensure that the residual unbalance after the first correction lies

within the tolerance?"

The answer is illustrated in figure 2.10. The correction vector can be smaller or greater

than the initial unbalance vector by the magnitude of the tolerance circle. The residual

unbalance will then always lie just within the tolerance area.

However, the greater the initial unbalance is in relation to the residual unbalance, the

smaller the error which can be allowed for the magnitude of the correction weight, which

means that the correction weight must be determined more accurately.

Figure 2.10: The relationship between a large initial unbalance and the tolerance

If the initial unbalance lies just outside the area of tolerance, the correction weight can

have any magnitude between almost "zero" and twice the tolerance, and the resultant

residual unbalance will always lie within the circle of tolerance.

Figure 2.11: The relationship between a small initial unbalance and the tolerance (diagram 204291)

2.4.3 Angular errors in correcting unbalance As in all manual procedures, the unbalance correction can never be achieved with 100% accuracy.

The angle at which the correction weight is attached or removed may deviate from the angle calculated by the balancing program by 5 - 10 degrees when the correction is actually done. The

consequence is apparent in figure 2.12.

Even if the magnitude of the correction unbalance is determined with 100% accuracy, if

there is an angular error, a residual unbalance will remain. This residual unbalance will lie

at an angle of approximately 90º from the initial unbalance. Conversely, from a 90º

displacement of the residual unbalance from the initial unbalance, one can often conclude

that an angular error has been made.

The magnitude of the residual unbalance is dependent of the size of the angular error. As

shown in the diagram, the magnitude of the residual unbalance reaches a size

corresponding to approximately 10% of the magnitude of the initial unbalance for an

angular error of just 6º.

Figure 2.12: The result of making an angle error when correcting unbalance (diagram 203981)

If the ratio of the tolerance to the initial unbalance is very small, an angular error may under certain

circumstances no longer have any significance.

It is therefore worthwhile to determine the correction angle as accurately as possible mainly in the

case of a large initial unbalance, to reduce the number of corrections and consequently the cost of the balancing task.

Depending on the type of unbalance in the rotor, the act of correcting the unbalance is called static or dynamic balancing.

2.5 Static / Dynamic balancing 2.5.1 Static balancing Only the static component of unbalance can be corrected by static balancing. As a rule this type of

balancing is only sufficient with narrow rotors which have no axial swash motion. In the simplest case

the static unbalance can be determined by rolling the rotor on knife-edges or roller stands to allow the heavy spot to be determined by gravity. However there is a limit to the sensitivity of this procedure

and far greater accuracy is obtained by measuring the unbalance during rotation.

This type of unbalance is corrected in a single radial plane of the rotor, as close as possible to the

plane of the center of gravity. Therefore this method of unbalance correction is also called

single-plane balancing

Typical rotors which often require only single-plane balancing, especially in the assembled condition, are:

• fans, ventilators and air separators,

• grinding discs, die chucks, • pulleys, flywheels, clutches,

• gears, • impellers, atomizer discs

Figure 2.13: The static unbalance can be determined by rolling the rotor on roller stands. The balance quality achieved is only satisfactory for coarse tolerances.

Figure 2.14: Narrow grinding wheels are normally only statically balanced. Measuring the unbalance during rotation leads to more sensitive balancing and the achievement of tighter tolerances. (new pic with VT60?)

2.5.2 Dynamic balancing

On more elongated rotors the moment unbalance can not be neglected. Therefore generally in these

cases it is not possible to avoid dynamically balancing the rotor.

In contrast to static unbalance, the effects of moment unbalance only appear during rotation.

Therefore measurement is exclusively possible when the rotor is rotating and not on roller stands.

Dynamic balancing is the most effective method of balancing because the static and dynamic

unbalance can be simultaneously determined. The unbalance correction on a rotor with two bearings

requires at least two radial correction planes. Therefore the method is also known as

two-plane balancing

Figure 2.15: A typical elongated rotor which requires balancing in two planes. Typical rotors which should be balanced in two planes are e.g.:

• Paper machine rolls,

• centrifuge drums and decanter rotors,

• electric motor and generator armatures,

• crushing and cutting rotors,

• machine tool spindles,

• grinding rolls,

• fans and blowers with longer distances between the sides and

• compressor rotors.

2.6 Balancing methods There are two methods available for accurately balancing rotors:

• Balancing on a permanently installed balancing machine, and

• balancing the rotors in their fully assembled, operational state.

Both methods have specific fields of application. The permanently installed balancing machine is the most

economic solution for balancing during the manufacturing stage.

Operational balancing (often called field balancing) provides a practical efficient method for test facilities,

assembly and plant maintenance to balance or trim balance completely assembled machines where the

rotor can be accessed for unbalance correction.

Some important advantages of field balancing are:

• dismantling the machine and transporting the rotor to a balancing machine are not necessary (time

and cost saving),

• lower capital investment (with a portable, electronic balancing instrument),

• operationally induced changes can be measured and compensated (e.g. thermal influence, centrifugal

force, assembly-induced unbalance),

• rotors of almost any weight and dimensions can be balanced using the same instrument.

Only operational or field balancing of rotors will be described in the following material.

2.7 Particularities of field balancing

The unbalance of a rotor can only be determined by measuring the effect of the unbalance and not by

direct measurement. To do this when balancing a rotor in its operational state, the absolute bearing

vibrations or the relative shaft vibrations which occur at the bearing positions are measured using portable or permanently installed vibration sensors.

Because unbalance only causes vibrations at rotational frequency, these rotational frequency vibrations must be very accurately filtered out of the vibration mixture which occurs, and must be

measured and displayed according to their amount and phase angle by the balancing or diagnosis instrument.

As already described in the section on tracking analysis in the Advanced Vibration seminar, modern

balancing instruments use the discrete Fourier Transformation (DFT). By digital multiplication of the measured signal by a sine and cosine function, the fundamental frequency vibrations are determined

and displayed according to their amount and phase angle. The rotational speed and angle reference for the rotor is provided by a reference generator, e.g. a photo-electric sensor.

Figure 2.16: Measuring arrangement for field balancing in two planes. The relationship which exists between the fundamental vibrations v and the rotor unbalance U is given by

v = k x U

The constant ‘k’ is mainly dependent on the dynamic behavior of the machine, i.e. on the dynamic stiffness, damping factor and rotational speed. Naturally the dynamic behavior of the machine is not

known in advance, and must be determined for the purposes of field balancing by test runs with known trial or test weights.

Because the balancing instrument only measures the unbalance effect and not the unbalance directly, the results of the measurement only give direct information about the amplitude and phase angle of

the unbalance vibration. It is only possible to obtain information for the unbalance correction, i.e.

amount and phase angle of the unbalance, after a subsequent graphic or computerized evaluation of the measurements taken during the measuring run or test runs.

In addition to the discrete Fourier Transformation of the vibrations, modem balancing instruments

offer immediate computerized calculation of the correction weights for single-plane and two-plane

balancing. The built-in microprocessor executes the necessary calculation in the shortest possible time, controls the operating dialogue and prints out the balancing report.

Field balancing has become faster, more accurate, convenient and simple with this innovation.

Figure 2.17: A modem analysing and balancing instrument with built-in printer and computerized 1 and 2-plane balancing programs. (new pic with VT60?)