lecture 6: bearings, main spindle systems - wzl.rwth …€¦ · lecture 6: bearings, main spindle...

Lecture 6:

Bearings, main spindle systems

Person-in-charge: Dipl.-Wirtsch.-Ing. Jens Rossaint

Steinbachstraße 53 B

Tel.: (80) 26 293

E-mail: [email protected]

Virtual machine tool – Modelling and Simulation Lecture: Bearings, main spindle system V6-1


Bearing function principles

Illustration 1 shows the main structure of the five different guide and bearing types:

• Hydrodynamic bearing,

• Hydrostatic bearing,

• Aerostatic bearing,

• Rolling bearing,

• Electromagnetic bearing.

In the hydrodynamic sliding principle, the lubricant is directed without pressure or only with less pressure to the contact points between cradle and bed or shaft and bearing shell. The lubricant film is built independently by collecting the oil during the relative motion of the guide elements. In hydrostatic and aerostatic systems, the lubricant film (oil or air) is maintained constantly by an external pressure system.

In rolling bearings, the partners in relative motion to one another are also separated by rolling bodies.

In the magnet bearings, a rotor equipped with ferromagnetic sheets are kept suspended by the tensile forces of the electromagnets arranged opposite in pairs. This bearing principle is rarely used in machine tool construction.

The bearing and guiding principle for the application is selected according to the requirements, e.g. the required rigidity, the lifetime, the damping and the price.


hydrodynamic bearing

+ high stiffness, high max. load+ high rotational accuracy

- high friction torque- expensive ancillary units- hydrodynamic: only applicable at certain rotanial speeds (stick-slip by start up)

spindle housing spindleaxial bearing

bearing bushes

© WZL / IPT


Hydrodynamic bearings

One comes across hydrodynamic bearing main spindle systems in such machine tools where operation of the bearing is ensured in the fluid friction area and no frequent starting is necessary. Only in individual cases, hydrodynamic sliding bearings are used in machine tool construction for slow-speed spindles, because operation in the mixed friction area is linked with relatively high wear.

Referring to the design, one can divide the hydrodynamic radial slide bearing in circular-cylindrical and multi-surface bearing, abbreviation MFL. Circular-cylindrical hydrodynamic slide bearing is rarely used in machine tool construction. Multi-surface bearings result in a better centring of the spindle due to its clamping effect. Therefore, they are largely used in machine tools. In these bearings, one differentiates between those with fixed sliding surfaces and those with self-aligning sliding surfaces.

The bearings with self-aligning sliding surfaces have the advantages to adjust themselves well to changing operating conditions. However, as they have no high rigidity owing to their aligning mechanism, they are rarely used in machine tool construction.

Multi-surface bearings have a higher rigidity in simple structures. In machine tool construction, they are used, e.g., for bearing grinding spindles and fine bore spindles as well as for spindle bearings of precision turning lathes and testing machines.

Special styles of slide bearings include the triple surface bearing according to Mackensen, which has one conical bearing bush outside in the longitudinal axis and one “triangular” bush in the cross-section. Due to the special shape of the bush, the bearing clearance can be defined accurately. Because of the small contact areas of the bearing bush to the housing bore hole, the rigidity drops a little. The Caro-

expansion bearing can conform to shaft deformations and temperature fluctuations through its ribbed, elastic, suspended bearing shell. The so-called tilt segment bearing has screw-adjustable sliding blocks. With that, the lubricant gap is automatically adjusted to the speed.

Axial slide bearings have worked-in lubricating wedges to support the hydrodynamic pressure formation. While designing them, the bend of the spindle on loading and thus a starting of the outer bearing edges on the running surfaces must be considered. Appropriately stiff spindles can prevent this.

Application example

Hydrodynamic slide bearings in machine tool construction are mainly used in fine processing machines. The grinding spindle bearings where hydrodynamic bearings can run under very favourable conditions form a focal point here. The speeds are sufficiently high and are almost constant. The grinding spindle does not need to be connected and disconnected frequently, rather runs continuously, e.g. during tool change.

Similar conditions are valid for fine boring spindle bearing systems. Illustration 2 shows the profile of a hydrodynamic bearing of a fine boring spindle. This bearing is distinguished by a very simple design. The bearing bushes are shrunk with interference in the location hole of the spindle housing. The compact design of the bearing together with its multi-surface ensures a very high rigidity.


Hydrostatic bearing - grinding spindle, suspended double eccentricly

© WZL / IPT


Hydrostatic bearings

In a hydrostatic bearing, the hydrodynamic effect described in the previous sections for creating the bearing pressure in the lubricated gap is largely replaced by the hydrostatic principle where the oil pressure is produced outside the bearing. Thereby, during the operation, one separates the contact surfaces of two machine parts sliding on one another by a permanent oil film.

The oil quantity required for producing and maintaining this oil film is delivered from an oil supply system arranged outside the bearing. Thus, the thickness of the oil film is practically independent of the sliding speed as opposed to the traditional sliding bearing. Besides, this type of oil supply ensures that hydrostatic bearings are wear-free and shows no starting friction, rather always pure fluid friction, whereby a stick-slip is ruled out even at low sliding speed.

Application example

Illustration 3 shows the spindle bearing system of a roll-grinding machine from Waldrich Siegen. During hollow and camber grinding as well as when grinding empirical supporting point curves, the bandwidth of the thickness changes of a roller is in the region of few millimetres. The demands for the accuracy of the grinding wheel advance in this area are very high.

In the illustrated hydrostatic spindle run on bearings, the radial feed motions of the grinding wheel are divided into two feed drives independent of each other. The rough delivery of the grinding wheel is accomplished by the drive of the upper support (X-axis).

For the radial fine adjustment (+/- 1.5 mm), a highly precise feed drive is provided, which acts on the eccentric journalled bush, in which the grinding spindle is journalled (U-axis). Both feed axes work

in the closed control loop and are coordinated by the CNC as function of the Z-axis. As the U-axis hardly turns, its hydrostatics can be designed for an optimum damping.

So that the entire grinding disc width is used while grinding, the grinding disc can be rotated through the B-axis during the grinding process based on the curve shape of the roller to grind (+/- 0.2°, repeating accuracy 0.2 µm). An independent servo drive is used for this purpose (like in the U-axis). Another reduction sleeve is twisted through a lever, which is likewise lodged hydrostatically and contains the eccentric bush of the U-axis. As seen in illustration 3, the axis of this intermediate bush is inclined vis-à-vis the spindle axis, so that the swivel (B-axis) is produced.

Another increase in the damping is achieved by a squeeze bearing which sits between spindle and U-axis.

Depending on the size, the belt driven spindle has over 100 kW driving power. This high power is necessary so that a high driving torque can be operated at low speeds. The cutting speeds move between standard 45 m/s and maximum 60 m/s, which corresponds to a spindle speed between 320 and 2,000 1/min. The work side bearings have a diameter of 120 mm, the drive side bearings a diameter of 100 mm.


aerostatic bearing - high frequency spindle

source: HAM

watercooling

spindle rear radial

bearing

axial

bearing

membranecylinder

front radial

bearing

2 mm shift

of the piston rod

power output: 900 W

max rot. speed: 120 000 min-1

radial load limit: 80 N

axial load limit : 245 N

radial stiffness: 5,78 N/µm

axial stiffness: 10 N/µm

pressure supply: 5 bar

air consumption: 75 l/min

axial bearing gap: 20 µm

radial bearing gap: 20 µm

piston rod(non

rotating)

statorspringchuck

tool

supply of the

aerostaticbearings

power supply

of the drive

rot. speed

signal

actuation of the

membranecylinder

© WZL / IPT


Aerostatic bearings

Gas-lubricated bearings work according to the same function principle as fluid-lubricated ones. The differences between these two applications lie in the lubricant properties. Usually, in gas-lubricated bearings, air is used as active medium. Therefore, often it is also called aerostatic or air bearing.

The viscosity of air is 2 to 3 power of ten lesser than that of oil. Due to the low shear forces of the gaseous medium, the resulting friction power is very little. In the very low mass flows and the very small specific heat capacity of air, however, the frictional heat cannot be fully dissipated through the lubricating medium, so that there is a temperature rise in the entire bearing system at high relative speeds. This is particularly true in very rapid spindles, which subsequently have to be cooled.

The compressibility of the air limits the feed pressure of aerostatic bearings for two reasons: the necessary safety measures are technically complex and expensive and can result in pneumatic instabilities. Therefore, aerostatic bearings are operated with feed

pressures of 4, maximum 10 105⋅ Pa. These low pressures set technical limits for the carrying force and the rigidity of air bearings. For the compensation of these disadvantages, in comparison to

hydrostatic, where pressures of 50 to 100 105⋅ Pa are normal, one essentially uses larger bearing surfaces.

In aerostatic bearings, the compressed air of about 4 10 105− ⋅ Pa is fed through many openings into the approx. 5 to 20 µm large bearing gap.

The bearing rigidity is produced either through a convergent gap shape or through restrictors that control the volume flow to the

bearing gap during air inlet. Due to the low air viscosity, no capillary resistances can be used, as the throttle effect is too low. Therefore, one uses very short nozzles or permeable sintered metal filling.

In bearings with gas permeable sintered metal bearing surfaces, the individual pores operate as micro-restrictors that contribute towards the rigidity.

Application example

Aerostatic spindle-bearing-systems are predominantly used in high precision machines and in machines with high speed drilling and grinding spindles.

Illustration 4 shows a high frequency spindle for speeds of up to 120,000 min

–1. Air bearing HF spindles are primarily used for boring,

milling and grinding operations in the printed circuit boards industry and in the precision mechanics. The vibration-less operation with radial run-outs enables machining operations accurate to dimension.

The spindle is water cooled for delivery of the motor heat and the friction power of the bearing. Axial and radial bearings are nozzle-regulated bearings. A special plastic coating provides for good emergency operating properties. The bearing gap for radial and axial bearing is 20 µm.


electromagnetic bearing

source: after S2M

+ active machine elements+ low friction torques- low maximal loads- expensive periphery needed- heat losses in the coils

touch down

bearing

radial bearing

axial

bearing

position

sensors

© WZL / IPT


Electromagnetic bearings

Electromagnetic bearing systems are suitable for very high speeds, as they have almost no friction torque. Particularly during high-speed chip removal, such systems are used sporadically.

An active electromagnetic bearing system that is suitable for bearing machine tool main spindles, comprises of two separate components:

• the bearing and

• the electronic control system.

Axial and radial bearings are used independent of one another for the spindle bearing.

The magnet bearings comprise of a rotor and a stator. The rotor is laminated with ferromagnetic material. It is attracted to the stator by opposite magnets and is thus suspended in a magnetic field. Absolute position transducers working mainly according to the induction principle measure the distance of the shaft from the magnets. Deviations from the centre position of the spindle are adjusted by the flow change for the magnets, i.e. by changing magnetic forces.

Application example

Illustration 5 shows the high-speed spindle with magnetic bearings. The spindle is guided by two radial bearings and an axial bearing electro-magnetically. A high frequency motor serves as spindle drive, which is integrated in the spindle unit. The spindle can be operated with a speed of up to 30,000 min

–1. The maximum driving power is 20

kW. As the air gap between rotor and stator is about 0.5 mm, one can expect only very minor friction losses. The electric power

requirement for the magnet coils is approx. 1 to 2 kW for the entire spindle system.

As the magnet bearing is controlled active, the system rigidity and damping depends exclusively on the closed loop dynamics.

The magnet bearing system is in the position to propel the rotor on controlled eccentric tracks up to high speeds. Because of this, a rotor balancing becomes possible through the bearing to a certain degree.

In addition to electromagnetic bearings, touch down bearings are also provided, which should prevent any disturbances in the event the magnet bearing fails, in case of power failure or overload of the bearing.

These touch down bearings are generally non-lubricated or grease lubricated rolling bearings that are not touched by the spindle during normal operation and subsequently do not revolve. The play between the shaft and the catch bearings is equal to half the air play in the electromagnetic radial bearings.


Rolling bearings

The most commonly supported bearing principle for spindle bearings, gear shafts and also for ancillary functions are the rolling bearings. The main reason for this development lies in the variety of positive properties of these bearings, especially in the international standardisation and the relatively simple calculation and selection by means of catalogues.

Special demands are made on spindle bearings, which are due to the fact that a high working precision should be achieved even under load in a wide speed range. From this, one can derive specific properties, such as high rotation precision and rigidity.

Therefore, special rolling bearings have been developed for the machine tool construction, which are distinguished by high running precision and rigidity as well as the avoidance of thermal problems through low friction torques.

Application example

Illustration 6 shows a high speed spindle that is designed as motor spindle. This spindle style is preferably used in high speed machining. As the bearings are increasingly brought close to their power limits during this application, one is trying to optimise its conditions of use if possible. The attempt lies in keeping the prestressing forces active in the bearings constant by an elastic adjustment of the bearing. The bearing of the high speed motor spindle comprises of two packages. The Tandem packages are prestressed against one another through springs. Subsequently, the rear tandem package also takes over the function of the movable bearing. The stabilisation of the initial stress and the compliance of the movable bearing functions make the displaceability of the rear bearing package in the housing necessary.

This should be ensured by a slide bush in many versions, which is placed in a hole in the housing and there, can be moved in axial direction. In order to reduce the influence of the friction between relatively moved parts on this motion, the movable bearing is often placed in a sleeve, on the outer diameter of which an axially movable rolling bearing is realised with the help of a ball bush.


Properties of different rolling bearings for machine tool spindles

unrestricted use

restricted use only

not usable

not applicable

1 2 3 4 5 6 7 8cyl. roller

bearing with flanges

needle rollerbearing

taper rollerbearing

barrel roller bearing

sphericalroller bearing

single rowthrust

ball bearing

double rowthrust

ball bearing

type name1 2 3 4 5 6 7

deep grooveball bearing

rad. angular contact

ball bearing

ax. angular contact thrustball bearing

double rowang. contactball bearing

self-aligningball bearing

single rowcyl. roller bearing

double rowcyl. roller bearing

nametype 8

α

α

α=15°α=25°

α=40°α=60°

1: accommodation of radial loads

3: accommodates shaft expansion since the rings and rolling elements move relative to each other

4: accommodates shaft expansion by a sliding seat between the inner or outer ring

5: assemblies, where the bearing needs to be disassembled

6: the bearings allow angle adjustment to compensate axis alignment errors

7: avaiable in ultra precision ranges

8: max. speed exceeds normal limits

2: accommodation of axial loads

© WZL / IPT


Rolling bearing styles

Illustration 7 shows a summary of the common rolling bearings and their typical properties. The crucial requirements for loading capacity, rigidity, axial angle adjustability, increased guide precision and high speeds based on the use are fulfilled by the individual bearing design very differently. Therefore, each bearing type is more or less suitable only for one characteristic application range.

Relatively simple physical considerations are enough as basis for appraising the bearing properties: on one hand, a large contact surface (line contact) provides for a good rigidity and high loading capacity, on the other hand, comparatively large frictional torques and low maximum speeds as well as problems during lubrication result from this. However, roller bearings are inferior to the ball bearings at combined radial and axial load and also with relation to the speed limits. The appropriate bearing type must be selected based on the requirements profile of a bearing (e.g. speed, loading capacity, load direction, life).


angular contact ball bearing

1) with constant preload 2) reached with WZL test rigs

7020C 7020C 71922C HS 7020C HCB 7020C HC 7020C

1,06 / 1,25 - 1 1,375 1,375 1,625

1,75 / 2,35 2,5 1,69 2,25 2,25 / 2,625 2,37

81,5 81,5 58,5 38 56 26,5

624 624 582 420 534 410

104 104 97 70 89 68

0,152 0,152 0,158 0,112 0,139 0,105

100% 110% 133% 113% 358% 194%

type of bearing

attributes

dynamic basic

load rating C [kN]

radial stiffness of abearing packet with low

preload [N/µm]

axial stiffness of abearing packet with lowpreload [N/µm]

friction torque at 10.000 1/min with greaselubrication (32cSt) [Nm]

relative price

WZL hybrid hybrid

speed factorn x d [10 mm/min]with oil-air-lubrication

m6

speed factorn x d [10 mm/min]with grease lubrication

m6

1)

3) calculation by WZL

3)

1)

2)2)

2)

2)

XCB 7020C

1,75

2,75

56

534

89

0,130

460%

HNS/hybrid

groove radius of theinner ring R

manufactured radial clearance

manufactured contactangle 0α

groove baseradius Ra

groove baseradius Ri

groove,i

groove radius of theouter ring Rgroove,a

e4

rolling elementdiameter D

w

semicirclediameter D

T

e4

)DRR(2

e1)αcos(

wa,groovei,groove

0−+

−=

© WZL / IPT


Angular ball bearings

Angular ball bearings are accepted for the rolling arrangement of spindles. In comparison to roller bearings, they are better suited for operation at high speeds because the contact surfaces are smaller and can be lubricated better, the friction power is lesser and the kinematics guide of the point-symmetric rolling bodies creates fewer problems.

Spindle bearings are angular ball bearings with a pressure angle between 12° and 25°. They are axially prestressed, which, in comparison to the adjustment of radial bearing play, is possible with relatively less effort with other bearings or with distance sleeves adjusted in length. In addition, angular ball bearings have the capability to absorb radial and axial loads and to develop only less friction. Consequently, they are particularly suitable for the arrangement of quick shafts.

Illustration 8 on the right shows the important cohesions between bearing play, fit and pressure angle for the angular ball bearing. Axial dislocation of the two bearing rings leads to contact between balls and bearing rings. The connection line of the contact points is

inclined to the pressure angle. The pressure angle α is determined by

the production radial play e of the bearing and by the osculations κ.

)DRR(2

e1cos

wa,groovei,groove −+−=α

(With Rgroove,i as groove basic radius of the inner ring, Rgroove,a as groove basic radius of the outer ring and Dw as rolling body diameter.)

w

wx,groove

R

RR −=κ

(With Rgroove,x as groove basic radius at the inner or outer ring and RW as rolling body radius.)

In production, the bearing is loaded with the desired prestress force, the existing axial projection of the inner ring vis-à-vis the outer ring is measured and worked off, so that the desired prestress force exists in the bearing when the rear end faces lie “on block”. With that, the prestress of the bearing through O-arrangement with or without distance rings and the correction of the prestress with different sized distance rings become possible.

The table in illustration 8, left, shows the various types of 7020 spindle bearings (100 mm inner hole, outer diameter 150 mm and 24 mm width), as compared to the characteristic rolling bearing characteristic values. A speed increase is especially possible by using ceramic balls made of Si3N4 as well as through special high nitride (HNS) bearing steel. While selecting the bearing, one must consider the extra price of bearings made of special materials. This surcharge must be justified by the relevant application.


Bearing kinematics: changes under influence of speed and centrifugal forces

Fv

Fvlowspeed

Fn

Fn

Fv

Fvhighspeed

Ff

Fn

Fn

elastically mounted bearing

Fv

Fv

Fv

Fv

∆xIR

lowspeed

high speed

and/or ∆ ∆ ∆ ∆ T

Ff

Ff

elastically mounted bearing

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Fv

Fv

Ff

Ff

Fv

Fv

Ff

Ff

Fv

Fv

highspeed

lowspeed

rigidly preloaded in a back to back arrangement

© WZL / IPT


Bearing kinematics

Through an excess temperature of the inner ring vis-à-vis the outer ring, a centrifugal conditional expansion of the inner ring or a press fit of the bearing inner ring on the spindle (necessary for high speeds), the bearing play is reduced for built-in and operating bearing play. In reduced bearing play, even the pressure angle decreases.

The bearing kinematics is altered by the speed and load:

• Through the centrifugal forces acting on the rolling bodies, the pressure angles on the bearing rings changes. At the outer ring, the pressure angle becomes smaller with increasing speed, at the inner ring larger (illustration 9, below).

• The centrifugal forces acting on the bearing inner ring produce tensile stresses that lead to an expansion of the inner ring (illustration 9, top left). This expansion becomes especially large when there is no compressive strain in the joint between inner ring and spindle and the inner ring no longer expands together with the spindle. The excess temperature of the inner ring rising with the load and speed has the same effect on the outer ring.

As shown in illustration 9, top left, both effects cause the inner ring to dislocate in the direction of the pressure angle peak vis-à-vis the outer ring at constant bearing loads. The outer dimensions of the machine element, angular ball bearings, are thus speed-dependent.

In a bearing package prestressed by rigid distance rings – e.g. an O-package, illustration 9, right, - this axial motion of the bearing rings to one another is eliminated by the relevant bearing adjusted opposite. The bearing responds to the constant axial position of the inner ring with higher inner deformations. In the rigid bearing package, this causes an increase in the prestress force dependent on the speed

and excess temperature of the inner ring. The prestress level can cause damage to the bearing under certain circumstances.


Axial strain diagrams for rigid and elastic arrangementaxia

l lo

ad

Fa

axialbearing suspension δδ(Fv)

Fv

Fa

δ(Fa)

Fa

axia

l lo

ad

Fa

axial bearing suspension δ

δ(Fv)

Fv

Fa

δ(Fa)

rigidly mounted bearing packet elastically mounted bearing packet

charact. curve ofthe loaded bearing

charact. curve ofbearing without load

caract. curve ofbearing

charact. curve ofspring

Fa

Fa Fa

© WZL / IPT


Diagrams of braced angular ball bearings

The advantage of a rigidly adjusted bearing package vis-à-vis an elastically adjusted package lies in the higher axial rigidity in motion and pressure direction. In the rigidly adjusted package, the bearings arranged in load direction are loaded by the applied axial power, while the opposite bearings are relieved (illustration 10, left). Consequently, all bearings participate in the load absorption. The initial stress in the relieved bearings is only eliminated when the axial load shows approx. triple initial stress.

With elastic adjustment, only the bearings arranged in load direction are essentially loaded by starting the prestress springs with their generally flat characteristic line, however, the opposite bearings are not relieved (illustration 10, right). Thereby, only the bearings oriented in load direction absorb the load, the spring action is accordingly higher, the axial rigidity of the package is accordingly lower.

These different properties of elastically and rigidly adjusted bearings imply that the spindle designer selects the type of bracing based on the desired spindle properties. Accordingly, three characteristic bearing styles are accepted for the bearing of machine tool spindles: fixed/movable bearing, fixed bearing and elastically adjusted bearing. These styles are explained below.


Principle of a rigid-moveable bearing system

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rigidly mounted spindle bearing packetin a back to back arrangement as a rigid bearing

cylindrical roller bearingas a moveable bearing

front side

© WZL / IPT


Principle of a fixed/movable bearing

The fixed/movable bearing is mostly used in lathes. High speeds are of secondary importance in these bearings, speed characteristic values of 1.0 x 10

6 mm/min are rarely exceeded. Anyhow, for an

accurate machining result, lathes require a rigid bearing. As the axial components of the cutting power can be effective in motion and in pressure direction, the spindle bearing must be axially rigid in both directions. The fixed bearing is arranged on the tool side in order to keep the effect of the thermal spindle extension on the machining as low as possible. Normally, two or more bearings arranged in O form the fixed bearing unit. As opposed to the X-arrangement, the O-arrangement offers a higher flexural strength due to its higher bearing distance.

The movable bearing in a fixed/movable bearing is not only faced with the task of absorbing radial loads play-less – it must permit axial dislocations between spindle and housing, because otherwise it would lead to high bracing between the bearing packages. This is especially true for motor spindles, where the stators and thus only the housings are cooled. For this task, cylindrical roller bearings are frequently used for this task, as an axial dislocation of the two bearing rings that are already in relative motion to one another is possible through the rotation without stick-slip and with less friction. The arrangement on the movable bearing side is conceivable even with a twisted angular ball bearing package, which sits on the outer ring movable in the housing.


principle of a rigid bearing

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front sidethermally compensatedbearing distance

δr(T)

δax(T)

γ

L *α*∆T = δax(T) = = tanγ

δr(T)erf

r*α*∆T

tanγ

L = r

tanγerf

© WZL / IPT


Principle of a fixed bearing

In the fixed bearing, the movable bearing is dropped and the fixed bearing stretches over the entire spindle. Such a style can only be realised with bearings that can absorb radial and axial loads simultaneously. Therefore, angular ball bearings are used for quick bearings, ball roller bearings for slow bearings. For the fixed bearing, the bearings are braced inside and outside with rigid distance sleeves.

The influence of the thermal expansion on the prestress force can be reduced by the thermal compensated bearing distance in an “O” bearing arrangement. In the so-called thermal compensated bearing distance, the axial and radial thermal expansions of the spindle are supplemented in such a way that the inner rings are shifted in the direction of the smallest rigidity and subsequently, prestress changes. The thermal compensated bearing distance corresponds to about twice the spindle diameter for bearings with 25° pressure angle, while for bearings 15° about four times the spindle diameter.

The pressure angle of an angular ball bearing is subject to fluctuations. It changes depending on speed, load and temperature. Thus, the thermal compensated bearing distance depends on the pressure angle of the bearing.

If the spindle is designed as motor spindle, the prestress adjustment is aggravated by the rigid distance sleeves, as the stator prevents the required precise length adjustment.

Appropriate spindle bearings are particularly used in milling machines up to speed characteristic values of 1.5 x 10

6 mm/min.


Principle of elastically mounted bearings

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frontside

preload-springsKugelbüchse

"rigid"-bearings "moveable"-bearingsballbush

axially tuned

gap

© WZL / IPT


Principle of an elastically adjusted bearing

The elastically adjusted bearing is especially used in spindles for high speed machining. Speed characteristic values of 2.5 x 10

6 mm/min

and higher are frequently achieved. Thereby, the bearings are advanced to their power limits. Nevertheless to maintain the fatigue life and reliability of the bearing, the prestress force must be kept constant through the elastic bearing adjustment. The bearing arranged on the tool side is fixed axial on the outer rings and is fixed with springs (or even a hydraulic piston) with the rear bearing. The rear bearing takes over the function of the movable bearing and must be movable in the housing. During its structural conversion, one must ensure that the elastic elements at axial load do not lie in the flow of force and subsequently do not drastically reduce the rigidity of the bearing system in this direction.

While designing elastic bearing prestress elements, one must keep in mind the possibility that the axial motion of the bearing rings is lost due to the formation of frictional corrosion, and the bearings thus have an undesired rigidity. One must therefore use bearing bushes made of nonferrous heavy metal, hole coatings or special shaft bases.

In order to reduce the influence of the friction between the parts in relative motion on this motion, the movable bearing is often placed in a sleeve, on the outer diameter of which, an axially movable rolling bearing is realised with the help of a spherical liner.

Particularly in elastically adjusted spindle bearings, the prestress force can be adjusted based on the machining task, e.g. through a variation of the hydraulic pressure in the prestress piston. With that, a high rigidity can be defined for the rough-machining through a high prestress force in the lower speed range and a low frictional torque in

the high speed range by a low prestress force for the finish-machining.


procedure of a machine tool spindle construction

© WZL / IPT


Procedural design sequence of a machine tool spindle

The sequence of a spindle design can be divided into many consecutive steps. Starting from the expected load collective (cutting forces, passive forces, speed, required rigidity), the tool interface (steep taper or drilled shank taper) suitable for the application is selected. Depending on the size, the inner diameter of the front spindle bearing is determined. Then follows the bearing calculation, the result of which is the required overlapping of the inner ring seat to obtain the maximum speed definitely. Depending on the rigidity requirement in motion and pressure direction and the required performance under load and temperature influences, the suitable bearing type is determined by specifying the necessary bearing prestress and bearing arrangement (for properties of various arrangements see illustrations 11 to 13).

In the next step, the static and dynamic optimisation of the bearing distance and the projection length on the tool side is carried out by taking into consideration the given boundary conditions, e.g. motor size and the specified dimensions.

The following pages provide an overview of the possible calculation steps and the optimisation possibilities by using calculation programs (YSPINDEL, SPILAD, LAGER6V4) while designing spindles with rolling bearings.


Static optimisation of the deflections at a spindle

© WZL

bearing distance b [mm]

yto

tal

[µm

]

40,0

50,0

60,0

70,080,0

90,0

100,0

110,0

120,0

130,0

140,0

150,0

160,0

170,0

100 200 300 400 500 600

spindle part ofthe deflection

bearing part ofthe deflection

calculation of the optimal bearing distancewith the WZL calculation program Y-Spindel

spindle part bearing part

y = F a (a + b) F (a + b) a

3 I E b c c2ar

2 2 2

brl+sp

r r ( )+ +

b -6 I E 6 I E (a + b)

c a cbr ar

-30 =

© WZL / IPT


Static system behaviour

Higher demands with regard to the static rigidity and dynamic properties are constantly made on the spindle-bearing-system subassembly due to the constantly increasing precision requirements.

Weak points of this subassembly immediately affect the quality of the products to be machined and the possible utilisation of the machine power.

For the optimum design of this subassembly, one always uses more frequent arithmetic techniques that enable a quick comparison of alternative design variants.

Decisive for the work precision of a main spindle system is the excursion y of the spindle at the outer point of application of force. The excursion is composed of the spindle part, the bearing part and

the housing part for the total deformation y y y ySp L k= + + (see

illustration 15, left). The objective is to minimise the excursion through a variation of bearing rigidity, bearing distance and projection length.

The equation shown in illustration 15, top right, specifies the total deformation at the spindle catch by neglecting the flexibility of the housing. If it involves a spindle system with two bearings, the static optimum distance can be determined analytically. By neglecting the normally very rigid bearing modified parts, an optimisation task can be defined wherein the above function is differentiated according to b and the zero positions are determined.

With more than two bearing points, the system is over defined and cannot be resolved analytically. These systems can be calculated by using the Finite Element methods.


Static optimisation influenced by the bearing distance

0

200

400

600

800

1000

1200

1400

1600

100 150 200 250 300 350 400 450 500

Lagerabstand [mm]

rad

ial ri

gid

ity a

t th

e s

pin

dle

nose

[N

/µm

]

spindle outer diameter 100mm

bearing stiffness 350 N/µm



bearing distance "b" [mm]

0

100

200

300

400

500

600

50 100 150 200 250 300

Auskraglänge [mm]

op

tim

al b

ea

ring

dis

tance

[m

m]

spindle bearing stiffness 175 N/µm




overhang "a" [mm]

Bk

ba

d a

kA

a: 50 mm - 300 mm

b: 100 mm - 500 mm

da: 100 mm

k: 175 N/µm - 3500 N/µm

Spindle nose

© WZL / IPT


Static optimisation influenced by the bearing distance

Illustration 16 shows the static rigidity of a spindle system through the

variation of the bearing distance and the projection length (i.e.

various distances between front bearing and tool point of contact) of

a spindle with 100 mm diameter equipped with two bearing points.

From the parameter calculations using the “SPILAD“ or the “YSPINDEL“ FEM calculation program, one can derive the following:

• If the bearing distance exceeds twice the spindle diameter, its effect in “normal” bearing rigidities is low. If the optimum bearing distance is exceeded, it leads to a minor drop in the rigidity of the spindle system than when it falls short.

• Only in very high bearing rigidities that can be achieved with hydrostatic or hydrodynamic bearings, the bearing distance has a larger influence.

• The influence of the projection length on the optimum bearing distance remains low.

• While designing a spindle, the exact observance of the optimum bearing distance has no priority.

The bearing distances optimised under static viewpoints do not match the dynamically optimised solution. Comparisons of calculated static and dynamic optimum bearing distances show that they are often relatively close to each other. Until now, it was not proven whether this similarity applies generally.


Static optimisation influenced by the tool projection

0

50

100

150

200

250

300

350

400

450

500

0 50 100 200 250 300tool overhang [mm]

sta

tic r

ad

ial stiff

ness

[N/µ

m]

at the cutting location, bearing stiffness 350 N/µm, 100 mm spindle diameter

at the spindle nose, bearing stiffness 350 N/µm, 100 mm spindle diameter

at the cutting location, bearing stiffness 350 N/µm, 70 mm spindle diameter

at the spindle nose, bearing stiffness 350 N/µm, 70 mm spindle diameter

bearing stiffness 280 mm

0

500

1000

1500

2000

2500

0 500 1000 2500 3000 3500radial bearing stiffness [N/µm]

sta

tic r

adia

l ri

gid

ity [N

/µm

]

stat. radial rigidity at the spindle nose

stat. radial rigidity at the spindle nose

spindle diameter 70 mm


radial rigidity of thespindle body: 6000 N/µm

radial rigidity of thespindle body: 1500 N/µm

Bk

ba

d a

kA

a: 0 mm - 300 mm

b: 280 mm

da: 70 mm und 100 mm

k: 0 N/µm - 3500 N/µm

Conclusion:An explicit increasement of thestatic rigidity is only possible withan abridgement of the tool overhangand an increasement of the spindleouter diameter.

Spindle noseCutting

location

Bearing distance: 280 mm

© WZL / IPT


Static optimisation influenced by the tool projection

Other calculations show the effect of the tool projection on the spindle rigidity on the tool lip. From the calculations, one can derive recommendations for the static optimisation of a spindle-bearing-system:

• The improvement of the radial spindle system rigidity by using more rigid bearings (higher preload) is also less without considering the tool. In addition, an increase in the initial stress means a drastic reduction of the speed suitability. The effect of the spindle bend dominates at the spindle nose in the total dislocation.

• The spindle diameter has a considerably larger influence on the radial total rigidity of the spindle system.

• By taking into consideration the tool projection and tool deformation, the spindle bearing system loses its radial rigidity very quickly.

As the static rigidity of the spindle-bearing system is hardly affected by a slight change in the bearing rigidity – because it can be achieved, e.g. without changing the bearing type and bearing principle through variation of the prestress force - the bearing preload must be selected from the viewpoint of speed suitability. At the same time, from the calculations one can conclude that a clear increase in the static rigidity succeeds only by reducing the tool projection and widening the spindle diameter. Instead of placing special significance on the observance of the optimum bearing distance while designing the spindle bearing, one must especially reduce the tool projection.


load bearing capacity for different bearing arrangements

1000N

1000N

1000N

685N 562N

120N 126N

766N 512N

130N 148N

627N 166N

146N

500N

145N

80mm

80mm

797N 597N 96N

257N 232N

80mm

150mm

80mm

spindle model of the FEM-program "SPILAD"

© WZL / IPT


Load suspension in different bearing arrangements

Illustration 18 shows a possible problem in the simplifying consideration of the spindle arrangement for the analytical calculation of the bearing system. The load suspension in the bearings of the bearings arranged at the ends of the spindle is uneven. This lies in the spindle bend, which, together with the axial distance of the force leakage points, causes an uneven load of the bearing.

The abstraction of a statically undefined spindle-bearing system on a statically defined one through the combination of the bearing points certainly simplifies the calculation, however also conceals errors. The combination of the bearing points prevents the determination of the actual bearing load. In principle, bearing reactions and spindle deformations can be determined only numerically for statically undefined systems.

Illustration 18 shows calculations with the “SPILAD“ calculation program for designing spindle-bearing systems in order to determine the load suspension.

The closer the load suspension points, the better the load distribution on the bearings. The initial bearings increasingly carry the power distribution with ascending crank of a lever.


dynamic optimisation of a spindle system with the belongingparameters

tool overhang

spindle diameter bearing distance

supplementary masses bearing stiffness

dynamic vibration behaviour of amachine tool

spindle

© WZL / IPT


Dynamic optimisation of a spindle bearing system

On one hand the objective of the dynamic optimisation is to lay out resonance frequencies of the spindle bearing system – which causes high ratios in the dislocation – from the speed of the driving motor or the cutting feed frequency. On the other hand, the resonance ratios themselves must be reduced. Here, the damping of the pendulous system plays a dominating role, which depends on the mass distribution, the rigidity as well as on the energy dissipation of the bearing point, the other joints and material damping.

To assess the dynamic system behaviour, the characteristic quantities – resonance frequency, resonance amplitude (resonance ratio xdyn/xstat), mode of vibration and system damping – are consulted.

Various calculation programs are used to calculate the dynamic behaviour of spindle bearing systems.

Resonance frequencies and amplitudes can also be extracted from the calculated flexibility frequency response, which represents the system flexibility in dependence on the frequency.

Five different points while optimising the dynamic properties of a spindle bearing system can primarily influence the designer:

1. Optimisation of the bearing distance,

2. Optimisation of the tool projection,

3. Optimisation of the spindle diameter,

4. Arrangement and minimisation of the additional masses,

5. Adjustment of the bearing rigidity by selecting bearings and preload.

The spindle bodies are calculated with the Finite Element Program – “SPILAD“ – for the static and dynamic analyses. The underlying calculation theory and model building are explained in the next few pages first, then the influence of the addressed parameters are discussed and recommendations for the spindle design are derived.


boring bar as an instance for the procedure of FEM-modelling

[ ] { } [ ] { } { }FuKuM =⋅+⋅ &&

mH

x1

x2β1

β2

1 2 .. n

q1 q2 q3 q4 q.. q.. q2n-1q2n

0 1 2 .. n

L

D

z

l1 l2 l... ln

mBK

mBK

D

d(z

)

tool shank

boring head

d2

dn

d...

d1

real structure

model consisting ofbeams and singlemasses

system'sdegree of freedom

element'sdegree of freedom

[ ] { } { }FuK =⋅

linear differential equation system

(static)

[ ] { } [ ] { } { }0=⋅+⋅ uKuM &&

homogeneous differential equation system

natural frequencies

© WZL / IPT


FEM modelling with the example of a boring bar

The SPILAD program designed specially for branched spindle bearing systems enables an accurate consideration of the real geometry of spindles and optionally of the housing as well as the exact acquisition of the bearing arrangement. To do this, the spindle is separated into individual (girder) sections, which show constant cross section data (outer and inner diameters) through their length. The bearing points are approximated by linear elastic spring elements, the rigidity levels of which correspond to those of the (rolling) bearings used. System loads can be considered at any number of application locations, whereby radial loads and bending moments are permitted at the model level and counterbalancing effects can be automatically combined. In addition, it is possible to calculate several load states in one calculation cycle independent of each other.

With the example of a boring bar, illustration 20 shows the calculated model building starting from the real model geometry. First, the abstraction of the geometry on an FEM calculated girder model is carried out. In the model, the steep taper plate is replaced by an ideal rigid gripping. The tool shank is formed by girder elements arranged in a row, which show a constant inner and outer diameter over the element length. The inner diameter of each individual girder element is determined in such a way that the original conical inside contour is approximated by a step profile as best as possible. As the drilling head cannot be presented as girder element due to its complex geometry, the drilling head mass is replaced by a point mass at the centre-of-gravity position.

In addition to elements, the dislocation degrees of freedom of the system must be determined in the nodal points. The degrees of

freedom must be selected in such a way that the geometric transfer conditions between the elements can be fulfilled.

Each loaded girder element has a girder element stressed on bending, two degrees of freedom at the end points respectively, the end point dislocation and the cross-section incline. The degrees of freedom of two girder elements adjacent to each other are identical in a node, so that two degrees of freedom are assigned to each nodal point in the complete model. At the node in the fixing point, there are neither dislocations nor torsions, so that even here, degrees of freedom need not be provided.

During the formulation of the dynamic behaviour (matrix rigidity procedure), a linear differential equation system is developed with the prerequisite of smaller oscillation amplitudes. The independent variables of this system correspond to the degrees of freedom of the structure to be considered. For a non-damped system, the differential equation system in matrix notation is

[ ] { } [ ] { } { }FuKuM =⋅+⋅ && . Equation (1)

Here, {u} means the displacement vector and {ü} the pertinent acceleration vector; [K] is the rigidity matrix applied in the static calculation and [M] the mass matrix. In the static case with {ü} = 0 and the vector of the external load {F}, the differential equation system (1) becomes the linear equation system

[ ] { } { }FuK =⋅ , Equation (2)

which must be resolved according to the dislocation vector {u}.

At first, the rigidity and mass matrices as well as the load vectors are fixed. Then, the static deformations are calculated by resolving the linear equation system (2).


Mathematical definition of a beam element in the FEA program„SPILAD“

beam element for unidimensional struktures

Fyi

Fyi+1

Fxi

Fxi+1

knot i+1Mi

z

x

y

Mi+1

knot il

the shear number Θ takes in account the shear

influence on the deformation and is calculated

with: .

ν: poisson's ratio of the material

κ: shear coefficient of the beam

κAI

J)ν+(124 = Θ

2⋅

⋅⋅

⋅

E: modulus of elasticity

A: cross sectional area of the beam

J: bending moment of inertia of the beam

l: length of a beam element

Θ: shear number

© WZL / IPT


Theoretical fundamentals of the “SPILAD” program

The homogeneous differential equation system must be solved for the dynamics calculation.

[ ] { } [ ] { } { }0uKuM =⋅+⋅ && Equation (3)

To create the rigidity and mass matrices, the structure to be calculated is divided into individual sections (finite elements), which are joined with each other through nodes. For these elements, an element rigidity matrix is fixed by taking into consideration the thrust and bend, wherein the rigidity properties of the element are reduced to the two corner nodes. The element mass matrix is created in the same manner, however the thrust influence is considered by an approximation according to Przeminiecki.

If one moves the system out of the balanced state and leaves it later without the effect of external forces, it executes free oscillations. The free oscillations are a superposition of the natural oscillations that one can pick up mathematically as the solution for the motion equation. With the solution

tωjevu ⋅= tωj2..

evωu ⋅⋅−= Equation (4)

if one sets the external forces to zero and later divides by the exponential term, the following expression ensues for the motion equation (Equation 4-2),

( )K M v− =ω2 0 Equation (5)

The vector u contains the system coordinates, the vector v the

amplitudes of the coordinates at a harmonic motion and ω describes the self angular frequency. Through further transformation, one can convert equation 5 to a special inherent value problem:

( ) 0vEKM 21 =ω−− Equation (6)

To be able to calculate with dimensionless quantities, one can use

the initiation of a reference angular frequency ω*. Moreover, one defines

λ ω ω= 2 2/ * Equation 7

and

21 */KMA ω= − Equation 8

thus, the inherent value problem from equation 6 is:

( )A E v− =λ 0 Equation 9

To solve this inherent value problem, one must first determine the

inherent values λι. They are produced from the characteristic equation

( )Det A E− =λ 0 . Equation 10

There are n inherent values (corresponds to the number of the system degrees of freedom) that by means of the equation 7 lead to the n natural frequencies of the system. For each individual inherent value, one can determine the vector vi from equation 9. For the

natural frequency ωI, the vector vi contains the amplitudes of the system coordinates that can be interpreted as mode of natural oscillation. The values of the vector vi are standardised to a specific scale, so that only the relative amplitude ratios and not the absolute ones can be indicated.


Dynamic optimisation of a spindle - bearing stiffness

The tilt rigidity of the tool interfaceis assumed with 10000 kNm/rad;that nearly conforms to the rigidity of a HSK80A.

0

200

400

600

800

1000

1200

1400

1600

1800

0 500 1000 1500 2000 2500 3000 3500

radial bearing stiffness [N/µm]

1st ra

dia

l eig

en

frequency

[H

z]

spindle diameter 100mm

The lowest radial bearing stiffness, for whichcalculations were carried out, can be reached with a 7020C spindle bearing with a low preloadof about 500 N.

0

200

400

600

800

1000

1200

0 500 1000 1500 2000 2500 3000 3500

radial bearing stiffness [N/µm]

1st axia

l eig

en

frequ

ency [H

z]

1st axial eigenfrequency

1st axial eigenfrequency, with tool interface (10000 kNm/rad),100 mm tool with 50 mm diameter

1st axial eigenfrequency, with tool interface (10000 kNm/rad),200 mm tool with 50 mm diameter


without tool

280 mm

SPILAD-calculation model

1st radial eigenfrequency

1st radial eigenfrequency, with tool interface (10000 kNm/rad),100 mm tool with 50 mm diameter

1st radial eigenfrequency, with tool interface (10000 kNm/rad),200 mm tool with 50 mm diameter

without tool

© WZL / IPT


Dynamic optimisation influenced by radial bearing rigidity

For the displayed calculations, a spindle bearing system with an outer diameter of 100 mm and an inner diameter of 40 mm was selected. The illustrated tool has a diameter of 50 mm and is joined with the spindle through a drilled shank taper HSK 80. The spindle bearings used are type 7020C (100 mm inner hole, 150 mm outer diameter, 24 mm width).

Illustration 22 shows that with an increase in the bearing rigidity, the axial and radial natural frequencies rise in tapering scale. In the axial natural frequency, it involves a uniform excursion of tool and spindle bodies. The radial natural frequency is marked by flexibility at the tool that becomes bigger with increasing projection length of the tool. A high increase in the bearing rigidities has a significant influence especially in large projection lengths. Anyhow, one must reckon with a minimisation of the maximum speed even in spindle bearings with an initial stress increase.


influence of the bearing distance

© WZL / IPT

0

500

1000

1500

2000

2500

220 280 340 400 460 520

bearing distance [mm]

1st

radia

l eig

enfr

equency

[Hz]

bearing stiffness 350 N/µm, 0 mm tool overhang




220 mm to 520 mm


0 and 200 mm

0

500

1000

1500

2000

2500

220 280 340 400 460 520

bearing distance [mm]

1st

radia

l eig

enfr

equency

[Hz]





220 mm to 520 mm


0 and 200 mm


Dynamic optimisation influenced by bearing distance

The bearing distance has a considerable influence on the position of the first radial natural frequency at high bearing rigidities. In lower rigidities, or while considering a projection, however, the influence of the bearing distance is lesser. The resulting radial natural frequency depends on the bearing distance in an almost linear manner.

For the spindle diameter of 100 mm, illustration 23 shows the dependence of the first radial natural frequency on the bearing distance with and without consideration of a tool of 200 mm length. It is discernible that only for very high bearing rigidities and low projections, an appreciable influence of the bearing distance can be ascertained on the dynamic properties of the spindle system in this case. In case of lower bearing rigidities, as they seem real for spindle bearings, or larger projections, bearing distances in the range between the two-fold and four-fold the spindle diameter hardly affect the oscillation behaviour of the spindle. Only at very large bearing distances (motor spindles), the influence of the bearing distance increases again.


influence of the tool overhang and the spindle outer diameter

280 mm


0 to 300 mm

0

200

400

600

800

1000

1200

1400

1600

1800

0 50 100 150 200 250 300

tool overhang [mm]

1st

rad

ial eig

en

freq

uen

cy

[Hz]

4x350 N/µm, 70 mm outer diameter4x1750 N/µm, 70 mm outer diameter4x3500 N/µm, 70 mm outer diameter4x350 N/µm, 100 mm outer diameter4x1750 N/µm, 100 mm outer diameter4x3500 N/µm, 100 mm outer diameter

rigid body vibration of thespindle in the bearings

bending/tilt vibrationsof the tool

bending vibration of thespindle in the bearings

280 mm


0 to 300 mm

0

200

400

600

800

1000

1200

1400

1600

1800

0 50 100 150 200 250 300

tool overhang [mm]

1st

rad

ial eig

en

freq

uen

cy

[Hz]

4x350 N/µm, 70 mm outer diameter4x1750 N/µm, 70 mm outer diameter4x3500 N/µm, 70 mm outer diameter4x350 N/µm, 100 mm outer diameter4x1750 N/µm, 100 mm outer diameter4x3500 N/µm, 100 mm outer diameter

rigid body vibration of thespindle in the bearings

bending/tilt vibrationsof the tool

bending vibration of thespindle in the bearings

© WZL / IPT


Dynamic optimisation influenced by tool projection and the spindle diameter

The improvement of the dynamic behaviour is particularly enduring when the increased bearing rigidity was low as compared to the spindle rigidity. As seen in illustration 24, this likewise applies only for small tool projections. As soon as the tool length exceeds a specific dimension, the dynamic behaviour of the spindle is no longer ascertained by a bending vibration or rigid body vibration of the spindle in its bearings, rather by a bending or relaxation oscillation of the tool on the spindle.

Noteworthy is the high level of the first radial natural frequency of the spindle system without tool (between 1000 and 1700 Hz) already at relatively low bearing rigidities. In the calculated spindles with 70 and 100 mm diameter – with lowered lever arms – one must reckon with the fact that the spindle speed lies in the range of these frequencies.


Influence of supplementary masses on the dynamic of spindle systems

basic model,

bearing distance: 280 mm

belt disk: 100 mm

rear overhang length: 125 mm

belt disk: 100 mm

overhang lenght: 225 mm

bearing system with four spindle

bearings

radial stiffness: 350 N/µm

axial stiffness (front): 60 N/µm

spindle diameter: 70 mm

axial eigenfrequency 415 Hz

radial eigenfrequency 563 Hz











© WZL / IPT


Dynamic optimisation influenced by additional masses

The illustration 25 shows the influence of a pulley with a diameter of 100 mm, as they are used, e.g. in belt driven spindles at the rear bearing. In comparison to the basic model, two versions are contemplated, which have the identical pulley, but different lever arm lengths of 125 mm and 225 mm at the rear bearing. For the basic variants without pulley, the initial radial natural frequency at 563 Hz with certainty lies outside the drive frequencies. Through the influence of the pulley with large lever arm of 225 mm, this initial radial natural frequency decreases to 268 Hz and thus comes absolutely in ranges of drive speeds (268 Hz correspond to 16080 1/min).

It generally remains to be ascertained that the mounting of oscillating relevant additional masses, like e.g. pulleys, deteriorates the dynamic properties of the spindle system. Anyway, the additional masses exercise a significant influence on the dynamic behaviour of the spindle only when the lever arms become big and/or the bearing rigidities are comparatively small.

For the dynamic optimisation, it can be ascertained that a remarkable increase in the spindle natural frequencies can be especially realised by

• a decrease in the tool projection, reduction of all lever arms

• an increase in the spindle diameter and only at

• high bearing rigidities and short projections by an optimisation of the bearing distance.

The influence of the bearing rigidities plays an ancillary role in the optimisation of the dynamics. Therefore, the selection of the bearing

and the initial stress must be made from the point of view of the speed suitability and carrying capacity.

lecture 6: bearings, main spindle systems - wzl.rwth …€¦ · lecture 6: bearings, main spindle...

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