Balancing theory

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(Rev. 2.1)

Ing. G. Manni

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CEMB S.p.A. - Via Risorgimento, 9 - MANDELLO DEL LARIO (LC) - ITALY - tel. 0341/706111

The following text reports the main arguments discussed during the balancing courses proposed by CEMB and are a record for the participants and a guiding path for the teacher .

During the courses ,the different arguments are more widely explained and enriched with practical examples.

The explanation ,even if correct ,is simple and full of useful examples , and so understandable to all the people (with different culture level ),interested in the balancing technology ..

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1.2 Balancing requirement 1.3 Unbalance(definition) 1.4 Unbalance measuring unit 1.5 Centre of mass (definition) 1.6 Mass eccentricity (definition) 1.7 Axis of inertia (definition) 1.8 Unbalance classification 1.9 Static unbalance (definition) 1.10 Couple unbalance (definition) 1.11 Dynamic unbalance

1.12 Equivalent total unbalances (equal) 1.13 Vector relationship between unbalances 1.14 Dynamic balancing

1.15 Examples of dynamic balancing 1.16 Unbalance effect

1.17 Balancing speed 1.18 Common frequent words

1.19 Criteria for deciding the number of balancing planes ( 1 or 2 ) for rigid rotors 1.20 Static balancing without the use of a balancing machine

CHAPTER 2 BANANCING TOLERANCES

1.21 Foreword

1.22 Balance quality grades for various groups of representative rigid rotors 1.23 Balancing tolerance

1.24 Examples of calculation of the residual unbalance according to ISO 1940/1 Standards for rigid rotors .

1.25 Evaluation of the balancing quality G (The total residual unbalance is known) 1.26 Balancing tolerances according to API 610 standards

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3.1 Foreword

3.2 Coupling accuracy evaluation

3.3 Basic principles to design a mounting adapter 3.4 Examples of mounting adapters

3.5 Common errors caused by the Adapters

3.6 Electronic compensation for mounting adapters errors.(eccentricity compensation) 3.7 Manual compensation for mounting adapter errors (eccentricity correction) 3.8 Example for evaluating the error caused by a coupling sleeve mounted eccentric 3.9 Basic concepts for adapter eccentricity correction

3.10 Balancing of rotors shafts without fitments ;rotor shaft key convention

3.11 Balancing the fitment (flywheel ,coupling , etc.) with an adapter having a full key

CHAPTER 4 ON FIELD BALANCING

Foreword

Necessary equipment Theory

Test mass calculation method

Two planes balancing on service conditions Not linear response

Manual unbalance calculation with the graphic vector method

Evaluation of the optimum angle position of the test mass during calibration Manual balancing with the use of a simple vibration meter

CHAPTER 5 FLEXIBLE ROTORS BALANCING

Foreword

Shaft critic (natural ) speed evaluation methods Calculation of the critic (natural ) speed Natural frequencies of a beam calculation Rotors classification

Rotor flexibility measurement on a balancing machine Basic criteria for flexible rotors balancing

Rotors classification according to balancing requirements Quasi rigid rotors

Examples of low speed balancing Flexible rotors

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Balancing tolerances for flexible rotors Flexible shaft bending evaluation (Whirl)

CHAPTER 6 THE BALANCING MACHINES

Industrial balancing machines classification Unbalance transducers and support mechanics Horizontal axis balancing machine support

Horizontal axis hard bearing balancing machine support equipped with piezoelectric transducers Vertical axis dynamic balancing machine equipped with piezoelectric pick ups.

Unbalance calculation mode

Main differences between hard and soft balancing technology

Error occurring when using a soft bearing machine for static unbalance measuring Hard bearing balancing machine proper use

Working range of a variable speed hard bearing balancing machine

Specific calibration balancing on a hard bearing machine (Self learning of influence coefficients) Different types of cradles used for rotors balancing

CHAPTER 7 BALANCING METHODS FOR MOS COMMON CASES

Crankshafts Propeller shafts

Propeller shaft body balancing (No flexible joints) Fan impellers

Pump impellers Paper rolls

Vehicle turbo chargers Hydraulic couplings

Tools and toolholder balancing Car wheels

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Balancing machine test according to ISO 2953

Balancing machine control according to ISO 9000 standards .

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CHAPTER 1 BASIC PRINCIPLES

1.1 Balancing requirement

Unbalance control and measure of rotating bodies is today more and more important for different reasons:

1) Higher and higher operating speeds (more production) 2) Lighter frames (lower production costs)

3) Service speed near to critical speeds (technologic or space reasons do not allow more rigid frames) 4) Longer life for each parts (bearings for instance) for a reduced load

5) Lower maintenance costs (for repair and change) 6) Longer machines availability (less production stops)

It is important to point out that the measure of the unbalance is an overall control placed at the end of the production line (it reveals errors on dimension tolerances,casting faults,uneven parts) and it is an index for the quality of the final product.

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CEMB S.p.A. - Via Risorgimento, 9 - MANDELLO DEL LARIO (LC) - ITALY - tel. 0341/706111

1.2 Unbalance(definition)

Not uniform mass distribution around the axis of rotation

A Rotor is unbalanced when its mass is not evenly distributed around the axis of rotation

From definition it is clear that it makes no sense to speak of unbalance without defining the axis of rotation, that is the ideal line around which the mass distribution is considered

Example:

Balanced section Unbalanced section

Every rotor can be divided into different sections (perpendicular to the axis of rotation) having each one its own unbalance.

As consequence we call local unbalance (of the section i) the value

j j

i

m

r

U

=

∑

⋅

where

U

i is the unbalance of the section

i

(described by a vector normal to the axis of rotation), j

m

are the single masses belonging to the section i j

r

are the distances of the component masses to the axis of rotation The symbol

∑

means vectors addition.

From definition it is clear that the unbalance of a section is the mass static moment calculated with reference to the axis of rotation

Total unbalance

U

t is the set of local unbalances and is mathematically described by the following formula

{ }

i

t

U

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1.3 Unbalance measuring unit

Please refer to the following drawing which shows a perfectly balanced section

(U = 0)

, on which a disturbing mass

m

has been added on point

P

at a distance from the axis of rotation equal to r Added mass

m

causes an unbalance

U

, (vector with direction

P-O

and value equal to

m·r

.)

Unbalance measuring unit is:

gr mm

⋅

mass distance from the axis of rotation

U

= ⋅ =

m r

10 gr

⋅

100 mm

=

1000

gr mm

⋅

Same value for

U =

1000 gr·mm can be obtained with a mass of 20 gr on a radius of 50 mm (placed in the same angular position)

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1.4 Centre of mass (definition)

Point around which the mass static moment is equal to zero.

With regard to the centre of mass following relationship is valid

m r

_{i i}

∑

= 0

where

m

r

i i

=

generica massa

distanza massa - centro di massa

Calculation example

We obtain:

m

1

⋅ r

1

=

3 ×

25

(Vettoreorientatoasinistra)=75gr⋅mm mm gr 75 destra) a orientato (Vettore

75

1

2 2

⋅ r

=

×

= ⋅

m

(The words mass centre or gravity centre are used indifferently

The centre of mass of a system is important because its motion can be described as the sum of the mass centre plus the motion of the single parts around it.

From unbalance and centre of mass definitions it follows that , if the mass centre of a section lays on the axis of rotation, the section is perfectly balanced;, that is:

U = 0

.

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1.5 Mass eccentricity (definition)

Distance between the centre of mass and the axis of rotation

Please refer to next picture where an unbalance

U

, mounted on a perfectly balanced section ,moves the position of the mass centre .

The added mass moves the centre of mass position ,which was originally on the geometric centre (axis of rotation), to the right side

The distance between the centre of mass and the axis of rotation (Eccentricity) is calculated with the following formula

[

]

[

_{[ ]}

]

E

U

M

microns gr mm kg

μ

:

=

⋅

=

1000

=

μ

10

100

where

M

= massa in kg del rotante,

U

= squilibrio in gr·mm,

E

= eccentricità in microns (To be more precise value

M+m

should be placed in the denominator)

From the previous formula it is clear that:

1) The unbalance of a body

U

[gr·mm] is equal to the product of its mass

M

[kg] times its eccentricity

E

[μ] A pulley which is mounted not concentric (eccentric) on the motor shaft, generates ,under service conditions , high vibrations caused by the unbalance;).

Following formula is valid

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1.6 Axis of inertia (definition)

Line around which the mass static moment is equal to zero

From the definition it follows:

m r

_{i i}

∑

= 0

where:

m

i

=

generica massa elementare

r

i

=

distanza della generica massa elementare dall' asse di inerzia

From the definitions of axis of inertia and unbalance of a rotor it follows that a rotor is perfectly balanced (

U

_t

= 0

) if its axis of rotation is the same axis as the axis of inertia

the meaning is that a rotor is balanced if its mass is evenly distributed around the axis of rotation which is at the same time axis of inertia

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1.7 Unbalance classification

The unbalance of a rotor (set of local unbalances) can be drawn as a set of parallel vectors starting from the axis of rotation

{ }

i t

U

=

where sezioni) varie (delle locali squilibri totale squilibrio

=

i t

U

Each vector of the above figure describes the unbalance of a single section of the rotor.

It is worth to point out that it is impossible to measure the total unbalance of a rotor ,because it requires the measure of the unbalances for each section (which ,in the most cases it is not possible)

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1.8 Static unbalance (definition)

The total unbalance is called static if it is equivalent to a single unbalance vector placed in a section which contains also the centre of mass of the rotor.

(The axis of inertia is parallel to the axis of rotation)

if the equivalent vector

U

_t is not located in one section containing also the centre of mass , we call it quasi-static unbalance.

(In the practice most people call static unbalance the total equivalent unbalance when it is placed in a single plane only)

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1.9 Couple unbalance (definition)

The total unbalance is called as couple unbalance if the equivalent unbalance is made by two vectors,placed on two different planes. having equal values (amplitudes) and opposite directions

(The axis of inertia cuts the axis of rotation passing through the centre of mass)

The measuring unit for couple unbalance

U

_c is by definition equal to U d⋅ =

[

gr mm mm⋅ ⋅ = ⋅gr mm2

]

Of course values

U

s e

U

d (unbalance value in the two sections) are equal.

For example ,if the declared couple unbalance value is 6000 gr.cm.cm ,and the distance between the two balancing planes is 15 cm , then the unbalance per plane is 6000/15 =400gr.cm (4000 g.mm ) .If ,the balancing radius on each plane is 20 cm ,then the unbalance per plane is 400/20=20grams .(the two unbalances on each plane are equal in value ,but opposite in the angle position )

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1.10 Dynamic unbalance

It is possible to demonstrate that total unbalance

U

_t

=

{ }

U

i (set of local unbalances

U

_i) is always equivalent to two vectors

U

1

+

U

2 placed in two different and arbitrary planes.

The set of two vectors

U

₁

+

U

₂ is called dynamic unbalance (amplitudes of

U

₁ e

U

₂ depend on the position of the planes where they are applied)

The simple demonstration of the above sentence is obtained by considering that a rigid rotor ,with different unknown unbalances in each section,,rotating free in the space ,can be kept fixed by placing only two bearings at arbitrary axial positions.on it

Each one of the bearings generates a rotating force .The two reacting forces at the bearing position compensate all the unknown rotating forces (inertia forces) which are generated by the distributed local unbalances along the rotor

The load on the bearings is a function of their axial position (distance) ;in the same way the values of the dynamic unbalances

U

1 e

U

2 depend on the axial position for the balancing planes.

Please note that balancing machines are called dynamic ,because they are capable of measuring the dynamic unbalance of a rotor (it is almost impossible to measure the distributed local unbalances)

Considering the rules of vectors summing ,the demonstration is still simpler

One vector can always be split into two parallel vectors which are properly positioned according to the lever law. As consequence each local vector can be substitued by two parallel vectors placed in the same two arbitrary planes.The unbalance components on the two planes can after be summed to originate the dynamic unbalance

U

1 e

U

2 as it is shown by following figure.

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Example Nr.1: Equivalent dynamic unbalance placed on narrow planes at the same side

The rotor is mounted overhang and cosequently high load on the bearing is generated

. The dynamic unbalance equivalent to

U

t located on the two selected planes is

U

1

= 2U

_t

, U

2

= 3U

_t

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1.11 Equivalent total unbalances (equal)

The total unbalance of a rotor

U

t is the complete set of local unbalances The rules for vector summation (composition) are valid

We can say that two total vectors

U

_ta (unbalance of rotor a) and

U

_tb (unbalance of rotor b) are equivalent (equal) if:

1) they have the same resultant vector , placed in the centre of mass (static unbalance) and the same couple vector (couple unbalance) or , which is the same:

2) they have the same dynamic unbalance (two vectors) placed on two same planes

The rule 2 is equivalent to the rule 1 because the dynamic unbalance

U

₁

+

U

₂ is on its side composed by a static unbalance plus a couple unbalance

The above mentioned concepts are described by the following mathematic relationships 1)

∑

U

ia

=

R

=

∑

U

ib

2)

∑

M

ia

=

M

=

∑

M

ib

This means that two total unbalances

U

tb e

U

ta are equivalent if they have the same vector risultant

(equation Nr. 1) and the same moment (equation Nr. 2) of local unbalances

U

i with reference to the same

arbitrary point.

Since the dynamic unbalance

U

1,

U

2 is equivalent to the total unbalance

U

t,the consequence is that:

R

U

1

+

2

=

M

₁

+

₂

=

where:

R

= Resultant vector ( sum)

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1.12 Vector relationship between unbalances

The next figure shows the vector composition of the two vectors

U

₁,

U

₂ (dynamic unbalance). The resultant vector

U

_S ,sum of the two vectors is the static unbalance.

Following the rules of composition of vectors, acting on different planes, it comes out that:

1) the static unbalance (resultant vector) is not dependent from the plane where it is placed (Normally the static unbalance is placed in the same plane containing also the centre of mass

2) Couple unbalance value depends on the position where the static unbalance (resulting vector) is placed In the following example of vector calculation the static unbalance is placed in an intermediate plane between two planes containing the two vectors forming the dynamic unbalance.

where

U

1

, U

2 = Dynamic unbalance applied in the planes 1, 2

U

s = Resulting unbalance (static unbalance if positioned on the centre of mass plane)

U

c = Couple unbalance (arm equal to the distance between planes 1 e 2)

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1.13 Dynamic balancing

• Dynamic balancing a rotor means to reduce its dynamic unbalance to zero or better to acceptable levels

• The dynamic unbalance is by definition

U

₁

+

U

₂, ;so it is necessary to operate on two different planes

• Since the dynamic unbalance equivalent to the total unbalance

U

_t can be calculated with reference to two arbitrary planes, the consequence is that the two balancing planes (where material can be added or removed)can be arbitrary chosen

What above reported is valid only for rigid rotors , where mass distribution (local unbalances) does not vary with the speed.

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1.14 Examples of dynamic balancing

a) Original unbalance placed in one plane only

a).1 One plane bala ncing

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CEMB S.p.A. - Via Risorgimento, 9 - MANDELLO DEL LARIO (LC) - ITALY - tel. 0341/706111 b) Couple unbalance balancing

b).1 Balancing on t wo distant planes

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,calculated with reference to an arbitrary plane, are equal to zero

From the above reported examples ,it is clear that a rotor can be balanced in different ways depending on the elected balancing planes

In order to balance doing the minimum effort two rules are valid

1) To choose balancing planes as far as possible 2) To choose balancing radius as large as possible

Important note: by the dynamic balancing, acting on two different planes, the total unbalance ( set of local unbalances )is not reduced to zero ; only the dynamic unbalance (on two planes ) equivalent to the total unbalance

U

T is reduced to zero

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1.15 Unbalance effect

An unbalanced rotor generates an inertial force (centrifugal) which increases with the square speed.

F

= ⋅ ⋅

m r

ω

2

= ⋅

U

ω

2 where

ω

=

2 π

⋅

60 N

where minute s revolution

=

N

F

= Centrifugal force in Newton

The vector unbalance

U

(multiplied by the factor

ω

2, square of the angular speed ) originates the centrifugal force F ; this means that the load caused by the unbalance increases with the square of the speed (doubling the running speed the centrifugal force ( inertia force ) becomes four times greater);

Note: In the MKSA system distance is measured in meters [m] ;as a consequence the unbalance should be measured in kg·m. following relationship is valid 1 kg·m = 106 gr·mm.

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1.16 Balancing speed

The unbalance of a rotor is caused by the radial distribution of its masses along its axis of rotation; the consequence is that ,if the rotor is rigid and this means that the values and relative positions of its masses do not change ,the unbalance does not change with the speed.

In a rigid rotor the operating speed does not modify mass distribution and consequently has no influence on the unbalance.

By adding a 20 gr mass at a defined radial position on a perfectly balanced disc an unbalance is generated ; this unbalance does not change with the speed because in order to reset the original conditions , it is just necessary to remove the added 20 gr mass, and this independently on rotor speed

For rigid rotors the balancing speed is not to be specified ; because it is related only to machine sensitivity and not to the rotor unbalance which is under measurement.

Modern hard bearing balancing machines have the capability to measure the dynamic unbalance starting from 70 RPM

The unbalance effect (centrifugal force) increases with the speed ;the electric signal increases at the samr time , so machine sensitivity tends to increase, because of a better ratio signal to noise

Depending on the model and manufacturer ,optimum sensitivity values are obtainable starting from 400 600 RPM.

Note Not expert people make confusion between the cause (unbalance) with its effect (centrifugal force or vibration).

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1.17 Common frequent words

Static balancing : Unbalance measuring and correction is done in one plane only.

Dynamic balancing : Unbalance measuring and correction is done in two different planes.

Correction planes : è It is the section (plane? Normal to rotor axis where unbalance correction is performed by adding or removing masses.

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1.18 Criteria for deciding the number of balancing planes ( 1 or 2 ) for rigid rotors

From the previous explanation it is clear that the total unbalance of a rotor is equivalent to a dynamic unbalance ( two unbalances placed on two arbitrary planes) ; only in special cases the total unbalance is equivqlent to a single unbalance placed in one plane (static unbalance).). The consequence is that a rotor is to be balanced dynamically on two planes). Notwithstanding , in the practical application ,good results are obtained sometimes acting on one plane only. The selection (one or two planes )is made according to the following table ..

With reference to the following table , where _{l and d are respectively rotor length and diameter reported} criteria are valid .Exceptions are possible according to the acquired experience . Please note that the speed plays a big role ; higher the speed better balancing (dynamic) is requested

Useful table to decide ,(as function of the speed and rotor geometric dimensions) the necessity of balancing in one plane (static ) or in two planes (dynamic

Service speed (RPM)

l

d

Number of balancing planes

< 200 whichever 1 da 200 a 1200 < 0,5 1 da 200 a 1200 > 0,5 2 da 1200 a 3600 < 0,15 1 da 1200 a 3600 > 0,15 2 > 3600 > 0,05

Disc shaped rotors 2 1

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1.19 Static balancing without the use of a balancing machine

The static balancing can be obtained by simply supporting the rotor on two free rollers (or flat knives) having low friction values.

The heavy part of the rotor moves by gravity ito the lower position; for static balancing it is sufficient or adding masses on rotor upper side or removing material from its lower side

A good static balancing level is obtained when trying to slowly rotate the rotor it maintains its position (does not rotate any more by gravity)

When a dynamic balancing machine is not available ,the above mentioned operation may grant acceptable service conditions, exemption made when big couple unbalances are present,; only the static unbalance is corrected, couple unbalance still remains.

In order to reduce to a minimum the residual couple unbalance ,the balancing: plane or planes are to be properly chosen with following criteria :

1) distributing the unbalance on two planes symmetrical with respect to the centre of mass position

2) distributing the correction over the rotor length, especially when the original unbalance is uniformly balancing plane containing the centre of mass

3) balancing plane where we know that the most of original unbalance is concentrated

4) correcting the unbalance on two planes , located at the same distance with respect to the axis of rotation , distributed along the rotor axis of rotation

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CHAPTER 2 BALANCING TOLERANCES

2.1 Foreword

The balancing of a rotating body has different goals: 1) reduced load on the bearings (low centrifugal forces) 2) long bearings life

3) acceptable vibration levels (a good vibration level does not create any problems to the comfort or to component life.

From previous point 3 , it is clear that the optimum value for the residual unbalance can be evaluated in an experimental mode , by considering that:

a) The inertia force generated by the unbalance can be calculated using the formula reported on paragrath 1.15;

b) On service vibrations levels can be easily measured with a simple vibrometer.

For each application an acceptable value for the admitted residual unbalance (which grants good performances ) can be defined..

ISO 1940 standards gives a rule in order to calculate an acceptable residual unbalance ,having following features:

1) gross unbalance deficiencies are avoided,

2) useless and excessive balancing works are avoided

For each rotor type,(depending on its maximum service speed) the acceptable total residual unbalance per unit of mass is calculated _⎥

⎦ ⎤ ⎢ ⎣ ⎡ ⋅ kg mm

gr _{(specific residual unbalance).}

The calculated value is the same mass eccentricity defined on paragraph 1.5; so following relationship is valid:

[ ]

M

U

E

μ

=

where:

E

= Mass eccentricity [microns]

U

= Unbalance [gr·mm]

M

= Rotor mass [kg]

According to ISO 1940 standards ,all rotors are classified (grouped) ,depending on their balancing

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2.2 Balance quality grades for various groups of representative rigid rotors

Note: Some groups of rotors ,not included in official ISO table , are added and reported in Italic types ..

Balanci ng quality grade G mm/s ROTOR TYPES 0,4 Gyroscopes

Spindles, discs and armatures of precision grinders

Textile fuses

1,0 Small electric armatures with special requirements

Tape recorder and phonograph (gramophone) drives, cine projectors Grinding machine drives

Turbines and Compressors with special requirements

2,5 Gas and steam turbines, including marine main turbines (merchant service) Turbine driven pumps

Rigid turbo generator rotors Turbo compressors

High speed compressors and aeronautic compressors

Medium and large electric armatures with special requeriments

High quality household electric armatures ,dentist drills .textile components

Small electric armatures not qualifying for one or both of the conditions specified for small electric armatures of balance quality grade G6,3

Machine tool drive

Air conditioning fans for Hospitals and concert halls High speed gears(over 1000 RPM) of marine turbines .

Computer memory drums and discs

6,3 Small electric armatures ,often mass produced , in vibration insensitive applications and / or with vibration isolating mountings

Medium and large electric armatures (of electric motors having at least 80 mm shaft height ) without special requirements

Machine tool and general machinery parts

Parts of process plant machines , Centrifuge drums, decanters, washers

Hydraulic machine rotors

Fly wheels , Fans ; Pump impellers

Marine main tuebine gears (merchant service ) Paper machinery rolls ; print rolls

Assembled aircraft gas turbine rotors

Individual components of engines under special requirements

16 Drive shafts(propeller shafts , cardan shafts ) with special requirements Parts of agricultural machinery, parts of crushing machines

Individual components of engines (gasoline or diesel) for cars ,trucks and locomotives Crankshaft / drives of engines with six or more cylinders under special requirements

Low speed separators Light boat impellers) Motor bicycle and car wheels Normal transmission pulley Wood machine tools

40 Car wheels ,wheel rims ,wheel sets .drive shafts

Crankshaft / drives of elastically mounted fast four cycle engines (gasoline or diesel ) with six or more cylinders (pistons speed greater than 9 m/s

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2.3 Balancing tolerance

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CEMB S.p.A. - Via Risorgimento, 9 - MANDELLO DEL LARIO (LC) - ITALY - tel. 0341/706111 the previous table defines the required balancing quality G according to each rotor type.

The maximum service speed is reported on the orizontal x axis , while the acceptable specific unbalance (acceptable unbalance per unit of mass or acceptable residual mass eccentricity ) is reported on the vertical y axis The following formula can be used instead of previous diagram:

( )

G

N

E

_t

μ

=

9550

⋅

where:

E

t [μ] = total acceptable mass eccentricity

N

[RPM] = Maximum service rotor speed

G

[mm/s] = Balancing quality or grade Total residual accepted unbalance:

U

[gr·mm] =

E

t

·M

where:

M

[kg] = Rotor mass

Total residual admitted unbalance in grams is

R

U

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2.4 Examples of calculation of the residual unbalance according to ISO 1940/1

Standards for rigid rotors .

Example N°1 – Fun impeller

Maximum service speed = 1500 RPM Mass M = 200 kg

Left , right side correction radius Rs = Rd = 800 mm Balancing quality G = 6,3

From previous diagram we obtain:

Tatal acceptable residual eccentricity et = 40 μ

Total acceptable residual unbalance Ut = M·e = 200 kg x 40 μ = 8000 gr x mm

8000 gr x mm (Total acceptable unbalance)

4000 gr x mm 4000 gr x mm

(acceptable unbalance for left plane)

(acceptable unbalance for right plane)

Per plane acceptable unbalance in grams

5gr

4000/800

5gr

4000/800

=

〈

=

⋅

=

R

mm

gr

Note: The acceptable unbalance per plane has been calculated by simply dividing by two the total acceptable unbalance ; this operation is correct because the two balancing planes have almost the same distance from the centre of mass position .,which is at the same time almost in the centre of the rotor.

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CEMB S.p.A. - Via Risorgimento, 9 - MANDELLO DEL LARIO (LC) - ITALY - tel. 0341/706111 Example N°2 – Turbine

Maximun service speed = 3000 RPM Rotor mass

M

= 500 kg

Left side balancing radius

R

s = 500 mm

Right side balancing radius

R

d = 400 mm

Balance quality

G

= 2,5

Total acceptable residual eccentricity

e

t = 8 μ

By using the formula

( )

G

N

E

t

μ

=

⋅

9550

we obtain:

=

⋅

2 .

5 ≅

8 μ

3000

9550

t

E

The total acceptable unbalance

U

t

= M·e

= 500 kg x 8 μ = 4000 gr x mm

4000 gr x mm (Squilibrio totale ammissibile)

2000 gr x mm 2000 gr x mm

squilibrio ammissibile piano sinistro

squilibrio ammissibile piano destro

The accepted unbalance value on the left plane is

U

_s

=

2000

=

( )

500 4gr 1,7

The accepted unbalance value for the right plane is

U

_d

=

2000

=

( )

400 5gr 2

Values within brackets are valid for the quality G = 1 (quality g 1 is nowadays commonly required for turbines )

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Maximum service speed = 6000 RPM Mass

M

= 10 kg

Balancing radius R = 100 mm Required balancing quality

G

= 6.3

e

t = 10 μ

By using the formula

( )

G

N

E

_t

μ

=

9550

⋅

we obtain:

=

⋅

6 .

3 ≅

10 μ

6000

9550

t

E

The total acceptable unbalance

U

t

= M·e

= 10 kg x 10 μ = 100 gr x mm

The total acceptable unbalance in grams (for the correction radius of 100 mm) is ₁_gr mm 100 mm gr 100 ⋅ ₌ = = R U

Note: Since the impeller is thin (reduced axial dimentions ) it is balanced in one plane only ( Static balancing)

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CEMB S.p.A. - Via Risorgimento, 9 - MANDELLO DEL LARIO (LC) - ITALY - tel. 0341/706111 Example N°4 – Tool holder dynamically balanced

The tool holder has a useful length

L

bigger than

2D

(where

D

is the cone diameter ). Considering its length it is advisable to balance it on two planes.

Maximum service speed = 24'000 RPM Tool holder mass M = 5 kg

Correction radius on balancing plane 1 R1 40 mm

Correction radius on balancing plane 2 R2 20 mm

Required balancing quality G = 2,5

(ISO standards specify quality G=2.5 for machine tools spindles and driving systems)

E

= 1 μ

Total acceptable residual unbalance

U

t =

M·E

= 5 kg x 1 μ = 5 gr x mm

5 gr x mm (Squilibrio totale ammissibile)

2,5 gr x mm 2,5 gr x mm

(squilibrio ammesso nel piano sinistro)

(squilibrio ammesso nel piano destro)

Acceptable unbalance on plane 1

U

1 (in grams)

gr

mm

gr

06 ,

0

40

5 ,

2 ⋅

₌

=

Acceptable unbalance on plane 2

U

2 (in grams)

gr

mm

gr

125 ,

0

20

5 ,

2 ⋅

₌

=

Note: The total acceptable unbalance has been divided by two because we assumed that tool holder mass is more or less symmetrical with regard to the centre of mass position ,and that the two correction planes contain the centre of mass almost in the middle position.

(37)

Let us consider a tool holder which is to be balanced in one plane (static balancing).

Normally the tool holder is balanced in one plane only , if its length

L

is lower than 2

D

.(D is cone diameter)

Maximum service speed = 12'000 RPM Tool holder mass M = 1 kg

Balancing radius = 20 mm Balancing quality G = 1

(ISO standards specify quality G 1 for grinding machine spindles)

Total acceptable eccentricity

E

= 2 μ

Total acceptable residual unbalance

U

t =

M·E

= 1 kg x 2 μ = 2 gr x mm

Total acceptable unbalance in the correction radius

U

(in grams)

gr

mm

gr

1 ,

0

20

2 ⋅

₌

=

(38)

2.5 Evaluation of the balancing quality G (The total residual unbalance is known)

In the assumption that the total residual unbalance is known , we it is possible to calculate the corresponding value for the balancing quality G according to ISO standards 1940/1.

Example of calculation: Rotor mass M [kg] = 6

Maximum service speed N [RPM] = 5000 Total residual unbalance U [gr mm] = 180 Total residual eccentricity E [μ] = 180/6 = 30

Using the diagram at paragrath 2.2 two lines are drawn ;one line ,normal to the x axis ,passing through the maximum service speed value ,(5000 in the example) the second line ,normal to y axis, passing through the residual eccentricity (30 in the example).

The inclined line , passing through the intersection point of the two drawn lines , defines the balancing quality (grade)..

As option ,the following formula can be used:

mm/s

7 .

15 9550

5000

30 9550

=

×

=

⋅

=

E

N

G

(39)

2.6 Balancing tolerances according to API 610 standards

The following formula is valid:

U

W

N

= 6350

where:

U [gr mm] =

Admitted residual unbalance referred to the bearing journals

W [kg] =

Static load on the considered bearing(mass)

N [RPM] =

Maximum service speed Modifying previous formula , we obtain:

[ ]

U

W

=

E

μ

=

N

6350

(total acceptable eccentricity = acceptable unbalance per mass unity)

The equivalent ISO formula is :

[ ]

N

G

E

μ

=

⋅

9550

Important notes:

1) Unbalance tolerance according to API standards is more severe than ISO grade G=1;it is 1,5 more precise and it seams sometimes not obtainable.

2) It is important to point out that the required tolerance ,according to API standards, is referred to the bearing journals and not to the two balancing planes ,(look at the paragrath 2.8)

3) The unbalance tolerance measured in microns, (Eccentricity = unbalance per unit of mass) is related to the required mechanical precision , especially when adapters are necessary to mount the rotor on the machine spindle.(the used adapter shall have a mounting precision below the required tolerance

4) For balancing qualities equal or below G 1 ISO standad recommends to balance the rotor complete with its own bearings .(The eccentricity between the inside and the outside bearing race can be of the same level as the requested eccentricity)).

(40)

2.7 Balancing tolerances calculated according to the maximum admitted load on the

bearings

The goal of balancing is to reduce loads /vibrations on the supporting frames , in order to achieve an acceptable life. The unbalance introduces internal couples and rotating forces on the bearings As a

consequence , the residual acceptable unbalance can be calculated by stating a maximum acceptable value for the rotating (centrifugal forces )generated by the unbalance in service conditions

A possible rule is to state that the rotating force is kept below 10 percent of the static load.(USA navy standards)

F

r [N] (Rotating force caused by the unbalance) = 6 2 2

10 ω

⋅

=

ω

⋅

r

U

m

F

g [N] (Static load on the bearing) =

M

⋅

g

where

M

= body mass related to the bearing [kg];

g

= gravity acceleration = 9.8 m/s2

According to the above mentioned rule g r

F

10

1 =

that is

8 .

9

10

1

60

2

10

2 6

⎟

=

⋅

⎠

⎞

⎜

⎝

⎛ π

⋅

M

N

r

m

it follows: 6 2 2 6 2

10

1

896

10

1

10

4 3600

10

8 .

9

) or ity (eccentric

⋅

=

⋅

=

⋅

N

E

M

r

m

unbalance specific acceptable

π

It is worth to point out that according to API and to ISO standards the accepted residual eccentricity (unbalance) varies with

N

1

;,the relationship is linear while with the last rule (USA navy standards ) it varies with the inverse of the square of the speed.(as the speed increases the accepted residual unbalance decreases rapidily)

(41)

2.8 Allocation of permissible residual unbalance to each correction plane according

to ISO 1940/1

ISO 1940/1 standards calculate the total acceptable unbalance of a rotor (static unbalance ) referred to the plane (rotor section)containing the centre of mass The acceptable residual unbalance on the two balancing planes (dynamic unbalance) is calculated taking care of the position of the centre of mass with regard to the position of the correction planes..

a) Distance between correction planes less than the bearing span

It is valid:

L

<

b

<

L

3

(the balancing planes are sufficiently spaced within rotor supports)

2

1

h

≅

(the balancing planes have the same distance from the centre of mass)

Following relationships are valid:

0 3

.

⋅

U

≤

U

₁

=

U

⋅

h

2

≤

0 7

.

⋅

b

U

t t t

0 3

2

0 7

1

.

⋅

U

≤

U

=

U

⋅

h

≤

.

⋅

b

U

t t t

(42)

CEMB S.p.A. - Via Risorgimento, 9 - MANDELLO DEL LARIO (LC) - ITALY - tel. 0341/706111 b) Distance between the balancing planes much greater than the bearing distance

In this case the couple unbalance on balancing planes 1, 2 has the bigger effect on the supports. The acceptable unbalance value on planes 1, 2 is given by following formula :

U U

U

L

b

t 1

,

2

=

⋅

where

U

_t

=

total acceptable unbalance according to ISO 1040/1.

(43)

c) Balancing planes distance lower than 1/3 supports distance (

b

is lower than

L

/3)

An arbitrary plane is chosen for the static unbalance (it can be plane 1 or 2).

st

U

(referred to plane 3)

C

L

U

t

2

2 ⋅

=

The acceptable residual unbalance is kept between

2

t

U

e

4

t

U

c

U

(referred to planes 1, 2)

b

L

U

_t

⋅

=

4

3

2

The couple unbalance is greater than

U

_t

In other terms the acceptable unbalance for each plane 1, 2 is bigger than

U

t

2

;its value is ::

U

L

b

L

C

t 1 2

2

3

4

2

,

=

+

⎛

⎝

⎜

⎞

⎠

⎟

Under the condition that the static unbalance is lower than

U

L

C

st t

=

⋅

(44)

CEMB S.p.A. - Via Risorgimento, 9 - MANDELLO DEL LARIO (LC) - ITALY - tel. 0341/706111 A frequent application of the above mentioned rules happens with pump and fun impellers (over hang mounted.)

Exemple: One stage over hang pump

L = 400 mm Service speed = 1500 RPM b = 50 mm Rotor mass = 50 kg

C = 500 mm G = 6,3

From ISO diagram ( G = 6,3 ) we obtain Et = 40 μ

U

t

=

40 50

×

=

2000 gr mm

⋅

U

L

C

s t

=

⋅

=

⋅

=

⋅

2

2 2000

2

400 2 500

400 gr mm

U

L

b

c t

=

⋅ ⋅ =

⋅ ⋅

=

⋅

2

3

4 2000

2

3

4

400

50 6000 gr mm

If the static unbalance

U

_s

≤

400 gr mm

⋅

, the acceptable value on the two balancing planes 1, 2 can be

≤

6400 gr mm

⋅

U

L

b

L

C

t 1 2

2

3

4

2

,

≤

+

⎛

⎝

⎜

⎞

⎠

⎟

(45)

2.9 Static / couple unbalance with narrow balancing planes

When balancing on narrow planes , it is necessary to distinguish between static and couple unbalance, because the two types of unbalances have a different effects on the supports..

Example 1: Pure static unbalance

The following figure shows the effects ,on the rotor supports ,generated by a static unbalance applied on a over hang pump impeller. Support loads are calculated according to the laws of static

M

= 0 ;

R

= 0 (The conditions for equilibrium are that the momentum and the resultant of all forces are zero).

The load on the support nearer to the impeller is bigger and its value is

⎟

⎠

⎞

⎜

⎝

⎛ +

400

100

1

st

U

The load on the farther support is lower and its value is

400

100

st

U

(46)

Example 2: Couple unbalance

The next figure shows the effect generated by a couple unbalance on the supports of an over hang impeller

The effect of couple unbalance is reduced by the ratio of the arms

U

_supporto

=

U

_c

⋅

40 =

U

c

400

10

For the above mentioned reason different values for static and couple unbalances are specified ;for instance: Static unbalance tolerance = 1 gr mm

Dynamic unbalance tolerance(couple) = 4 gr mm per plane

For instance , for axial fun impeller (width 30÷40 mm and an external diameter of

300÷400 mm ) the normal required tolerance on the static unbalance is 30÷50 gr mm while a couple unbalance of 100÷200 gr mm .is accepted .

(47)

2.10 Balancing tolerance / balancing planes

Let us consider a rotor having a pure couple unbalance of 15 gr mm placed on two different planes with 100 mm distance

U

c

=

15 gr mm

⋅

×

100 mm

=

1500

gr mm

⋅

2

Taking as reference the previous figure , it is clear that , depending on the position (distance ) of the two selected balancing planes ,the measured unbalance which is to be corrected varies (30, 15, 10 gr mm). If the acceptable balancing value per plane is 15 gr mm ,then the rotor is considered within tolerance only if the two balancing planes are placed on the supporting position or at a distance of 100 mm; for shorter distances balancing planes the rotor is no more within tolerance

Now ,a rotor should be considered properly balanced (within tolerance ) indifferently of the two selected balancing planes.

As a consequence a correct unbalance tolerance can be specified in two ways by defining .

1) A tolerance on the static unbalance (referred to a specified plane) and a tolerance on the couple unbalance 2) A dynamic tolerance

U

1a e

U

2a specifying also the two balancing planes .

ISO 1940/1 specifies a total tolerance

U

t placed on a balancing plane which contains the centre of mass. API standard specifies the admitted dynamic unbalance (on two planes ) placed on the bearing journals.

(48)

CEMB S.p.A. - Via Risorgimento, 9 - MANDELLO DEL LARIO (LC) - ITALY - tel. 0341/706111 Defining a limit value (balancing tolerance) for the unbalance referred to the bearing journal directly gives a limitation to the rotating forces which exert on it.

This is particularly useful ,because an acceptable residual unbalance calculated with the above mentioned rule , is valid whichever are the two selected balancing planes.

API 612 e 613 standards use this rule and calculate the residual acceptable unbalance with the following formula

U

m

n

1 2 6 1 2 2

89 451 10

, ,

.

]

=

×

[gr mm

⋅

where:

n

m

=

numero di giri al minuto

massa gravante sul supporto 1,2 [kg] 1 2,

The calculated value for the acceptable residual unbalance grants that the rotating centrifugal force ,acting on the support, is lower than ten percent of the static load(weight).

For calculating the residual acceptable eccentricity ,the following formula is valid :

E

n

t

=

×

89 451 10

6 2

.

per 2,5) G e 1 G tra intermedio (valore ISO) secondo 2,5 G (circa

5 .

2 RPM

6000

10 RPM

3000

= = =

μ

≅

=

μ

≅

=

t t

E

n

E

n

it is very important:

To avoid any confusion between the actual balancing planes (where we act ) and the two planes where the unbalance tolerance is specified.

To specify always. in a clear way , the two planes where the acceptable residual unbalance is valid. With the use of a modern microprocessor measuring unit,it is possible to specify the tolerance on the two balancing planes or on the two rotor supports..

To specify the unbalance tolerance on the two rotor supports ,it is sufficient to set the parameters A = C = 0 and the parameter B = Supports distance (look chap. 6, par. 3).

If

U

_t is the total acceptable residual unbalance (calculated according to an accepted standard, ISO 1940 f. i.) in the most of cases ,when the two supports are similar ,the acceptable residual unbalance per each support is:

U

₁_A

U

₂_A

U

t

2 =

=

(49)

2.11 Balancing certificate

In order to verify / certify the balancing quality ,a good balancing certificate must contain the following informations:

1) ROTOR TYPE

It is useful to define the necessary balancing quality 2) ROTOR MASS and SERVICE SPEED

They are useful to calculate the residual unbalance and to verify if the rotor is a rigid or a flexible one 3) BALANCING METHODS

Resting position on the balancing machine (they define the actual axis of rotation), position of actual balancing planes , correction radius, balancing by adding or removing.

4) UNBALANCE DATA

Original and residual unbalance on the two balancing planes 5) USED BALANCING MACHINE AND SPEED

Useful data to verify the machine sensitivity and if it is suitable.

In the following page an example for a balancing certificate is reported The certificate can be completed with the following data:

• Original unbalance • Unbalance angular position

(50)

(51)

CHAPTER 3 MOUNTING ADAPTERS

3.1 Foreword

Many types of rotors ,produced also on big volumes , (for instance pulleys, flying wheels, pumps and funs impellers etc..) , on service conditions are connected with a key to their driving shafts . In order to be bal-anced they require a proper adapter to mount them on a balancing machine .

The balancing machine measures the unbalance of the rotating part (rotor plus adapter ); as a consequence an ideal adapter should :

- Have a very low unbalance (equal to zero )

- Reproduce ,in the balancing machine ,the same axis of rotation existing in service conditions

The compliance of the above mentioned criteria is limited mostly by mechanical problems (geometric toler-ances and centring accuracy).

It is not rare the case when perfectly balanced rotors , are mounted eccentric in service conditions and conse-quently generate high unbalances and vibrations.

To avoid this problem and in order to not destroy the achieved balancing conditions , same rotors are centred, on service conditions , with a conic shaft (centrifugal separators .).

A cylinder type mounting , always causes centring errors (unbalances ) because of the mechanic coupling shaft / hole (different geometric values within the specified mechanic tolerance )

The balancing machine is responsible for the unbalance of the rotor and the balancing condition can be di-rectly verified on the machine itself , by measuring the unbalance with the rotor mounted in different angular positions.

Good service conditions of balanced rotors are possible only if : 1) Centring is correct

(52)

3.2 Coupling accuracy evaluation

Type of rotor = Pulley

Max service speed = 3000 RPM Balancing quality

G

= 6,3

Total acceptable eccentricity according to ISO 1940 = 20

μ

m

In order to grant a residual eccentricity of 20 μm , the mounting adapter must :

1) Centre the rotor with a mechanic accuracy lower than 20 microns (the electronic compensation for tool er-ror is necessary )

2) Be able to centre rotors having different diameters values caused by the manufacturing process (machining tolerances ) which may cause random eccentricities in the mounting .

Now, considering that:

1) The required balancing quality is equal or even better than Q = 6,3 (2,5)

2) The rotors , to be balanced, have necessarily a geometric tolerance (variation ) on the centring diameters . The natural consequence is that the mounting adapters must have :

A conical tape centring , or

An expanding type cylinder centring .

The expanding type cylinder mounting makes it easier the loading / unloading process ,no interference oc-curs.

The conical centring may require an additional device to unlock the rotor and dismount it after the balancing process .

The conical mounting ,also on the service conditions , has the advantage of not destroying the previously reached balancing state .

The goodness of a mounting adapter (centring accuracy and repeatability) can be easily verified by measur-ing the unbalance of the rotor mounted each time in a different angular position .(a good adapter grants readings with a small variations ).

(53)

3.3 Basic principles to design a mounting adapter

A suitable mounting adapter must have :

1) Perfect centring on the balancing machine axis , without any clearances . 2) Easy rotor mounting / dismounting.

3) Safety against rotor unlocking during the measuring spin ..

4) Hardened and resistant to wear surfaces , above all the ones responsible for rotor locking and centring. 5) As light as possible weight

6) Rotor supporting surface , granting that the rotor is kept perfectly normal to the axis of rotation..

Note: It happens sometimes ,when a fine accuracy is requested ,that different measurements are obtained by mounting the rotor in different angular positions on the adapter ,even using the electronic func-tion to compensate mounting adapter errors.. This happens because the centring hole is not normal to the resting (supporting ) surface and the centring is not repeatable .

A small rubber ring placed between the rotor and the supporting adapter can solve the problem in a simple way ;the result is that the centring hole is no t disturbed by the supporting surface. This example clearly shows that ,in a good adapter ,the centring and the supporting surfaces must

be perfectly perpendicular ;otherwise a random interference in between (not compensated by the electronic) is generated .

(54)

3.4 Examples of mounting adapters

Different types of mounting tools are reported in the following a) Rotor centring on the shaft

Simple cylinder shaft

Tapered shaft

Shaft with a cut washer

Shaft with a key

(55)

b) Rotor centring on the external cylinder surface (tool with a round hole)

Cylinder / tapered mounting

Locking by radial movable jaws

(It is also used to lock from the inside big rotors diameters)

(56)

d) Expanding type mounting tool ;to be used on a vertical axis machine.

Important features of an expanding mounting tool are ::

1) Threaded holes ,each 30 degrees , to be used to perfectly balance the tool itself.. 2) Hardened tool centring shaft on the machine spindle (coupling tolerance H7).

3) Rotor resting surface ; it grants that the rotor is mounted perfectly normal to the centring hole. 4) Interchangeable expanding bush.

5) Holes (3 at 120°) used to connect the tool to the machine spindle. 6) Mobile drawbar used for a quick locking / unlocking.

(57)

e) Mounting tool with double centring tapered shafts (on rotor and spindle side )

Main feature of a tapered tool are :

1) Centring cone to fix the tool on the machine spindle. In the microprocessor type modern machines . the conic part can be eliminated , because the electronic is capable of correcting any eccentricity error in the

(58)

f) Expansion type adapter to be used on a horizontal axis balancing machine

In many practical applications with horizontal axis balancing machines , the rotors (pump or fun impellers ecc.) are balanced in an over hang position , keeping the mounting adapter shaft on the balancing machine This way the balancing operation is quicker ,the parts are easily mounted and dismounted

Two different mounting adapter shafts , based on the same principle , are.shown

The main feature of this adapter shaft are:

1) Body having a mass and a length capable of keeping the centre of mass of the assembly (tool plus rotor ) within machine supports.

2) Cylinder surface granting tool centring.

3) Tool base ,interchangeable ,in order to cover a wide diameter range.. 4) Expanding type bush ,interchangeable , to cover a wide diameter range

(59)

and rotor mounting dismounting..

The main features of this adapter shaft are :

1) Shaft body having a mass and a length in order to maintain the centre of mass of the assembly (rotor plus adapter ) within machine supports. As option, a reverse thrust roller cradle ,capable of sustaining an up-ward force ) can be used. (look at. 6.11).

2) Tapered seat ,to centre the tool on the shaft.

(60)

g) Adapter shaft ,with conical centring ,to be used on an horizontal axis balancing machine

Main features of the adapter are :

7) Cylinder shaped end side for connection to the balancing machine cardan drive .

8) Shaft body having a mass and a length capable to maintain the centre of mass (adapter plus rotor ) within machine supports . Two sets of threaded holes ,placed near machine supports ,are used to balance the shaft.

9) Hardened and ground surfaces Shaft supported positions on machine rollers. 10) Tapered part to centre the rotor.

Note: Special roller cradles (reverse thrust rollers or four roller cradle) mounted on the opposite machine support are necessary for the use of lighter mounting adapters shafts (type f or g) where the rotor ,to be balanced in over hang position , generates a force in the upward direction

(61)

The above reported adapter is currently used to balance car wheels . The main features are :

1) Tool body with a cylinder centring to the machine spindle . 2) Rim resting surface (it grants the ortogonality).

(62)

i) Segment type adapter

The mounting tools , using expanding bushes sliding on a conical shaft , have an accuracy (mechanic repeat-ability) lower than 0.01 mm (10 microns). The expanding bush has a geometric run out ,even small , between the inside and the outside diameter . This run out (constant error) can be compensated by a modern measur-ing unit ,under the condition that the bush does not change its angular position on tool shaft (pls.refer to next sketch ).

An expanding type tool with reduced errors (~ 5 microns) is made by expanding segments. The movable segments are placed in repetitive positions , as a consequence , the related errors can be measured and com-pensated by the measuring unit . The segment type tool is moreover safer against the possible entrance of small chips . (To avoid the same problem ,the bush slots are filled up with rubber )

1) Tool body with its centring and connecting part to the machine spindle.

2) Rotor supporting system. It is composed by three supports at 120 degrees , with same open space in be-tween , in order to allow chips outgoing .

3) N° 5 radial moving segments

4) Air operated draw bar whose movement causes the opening of the segments.

Two reference pins at 180° avoid bush rotation keeping it in a fixed position (no centring errors

(63)

j) Hydraulic expanding tool

The always higher accuracy (grade G = 2,5 e 1) , now required for high speed rotors (tool cutters.) , can be obtained only by using mounting adapters having an high mechanic accuracy (centring repeatability). The hydrauli expanding tool has following features :

• It is suitable for rotors having a very precise centring hole /shaft (< H7) • It can be used to centre inside / outside diameters

• Centring accuracy < 3 μ

• Expanding range ≈ 3/1000 on the centring diameter

1) Tool body with a cylinder centring on to machine spindle. 2) Pressurized oil . The pressure increase causes the expanding. 3) Screw type or system which increases the pressure.

(64)

3.5 Common errors caused by the Adapters

The balancing machine measures the unbalance of the complete rotating part (rotor plus mounting adapter). We are interested to measure the unbalance of the rotor only ,as better as possible .

The errors caused by mounting adapters can be divided in :

a) Repeatable (constant ) errors ,due to : • Balancing machine spindle unbalance

• Adapter unbalance

• Adapter mounted eccentric on the balancing machine spindle (it is equivalent to an unbalance) • Rotor mounted eccentric von the adapter (it is equivalent to an unbalance)

b) Not repeatable errors (random) due to :

• Mechanic clearances rotor / adapter

• Normal mechanic variation on centring diameters (production tolerances)

Constant errors can be totally compensated by the measuring unit ; while not repeatable errors can only be reduced (not completely eliminated ) by using high precision type adapters , which can be of two types : Expanding type ,cylinder shape (accuracy : 2 ÷ 5 μm)