Limits for the application of long protruding tools regarding the process stability and safety Prof. Dr.-Ing. H. Schulz Dipl.-Ing. W. Huerkamp Dipl.-Ing. U. Fiedler Dipl.-Ing. A. Versch Dr.-Ing. T. Würz Institute of Production Engineering and Machine Tools (PTW) Darmstadt University of Technology Darmstadt, Germany

KEYWORDS: High speed machining, long protruding tools, process safety Abstract: The use of long protruding tools reveals dynamic problems considering the process stability due to their functional shape. In addition to these technological problems the High Speed Cutting technology will also arise important safety problems. Different ways of excitation of the system of the long protruding tool and the spindle are presented, also the effects on the system behavior regarding working and process safety. Based on experimental results and analytical limitation analysis critical zones will be identified. The knowledge of the natural behavior of the combined dynamic system is of major importance. Several different strategies for the avoidance of process disturbances and system behavior which is critical for safety are recommended. Some procedures are presented to predict the limits of the system tool - spindle in advance of the cutting operation regarded.

1. Introduction Innovative technologies like high speed cutting (HSC) for production technology are more often applied in today’s industry. One basic characteristic opposite to conventional cutting technologies, is the increase of the cutting speed and feed rates by a factor of five to ten, which depends on the work piece material [SCH 1996]. HSC machining is used in die and mold making, the aerospace industry and the automotive industry for example. The highest cutting speeds of sometimes more than 5000 m/min are used today for the machining of aluminum and magnesium. Especially for milling of complex shapes in die and mold making, like for example deep pockets, long protruding tools are used. Based on their mechanical design (high mechanical compliance), which is mainly described by a high ratio of length to diameter (see figure 1), these tools cause even for the conventional cutting technology several problems. These are static deflection and the herewith resulting deviation in dimensional tolerances at the work piece [WER 1993, SCHW 1997, JAN 1996, HOC 1995].

Figure 1.

Design of a long protruding tool

Additionally to this independent or self exited vibrations can be easily caused by the occurring machining loads, due to the low mechanical stiffness of the tool or the whole tool - spindle system. These vibrations can be distinctively proven by a disturbance on the work piece surface [PUD 1997]. Due to the increasing application of HSC machining in industry, long protruding tools are more frequently used in combination with high frequency motor spindles. Besides the already known technological problems these tool also bear risks and hazards regarding safety technology [SCH 1999].

2. Kinds of excitation Basically long protruding tools are driven to vibration by several different excitations during their application on the machine tool. The highest influence on the system’s behavior is caused by free rotation or unbalance and the excitation by cutting edge engagement to the machine.

2.1 Excitation due to unbalance / free rotation

dyn. compliance

Unbalance of the rotor is caused by mass eccentricities. These can result out of asymmetries of the tool in the tool - spindle system, but also of clamping inaccuracies at the intersection of tool and clamping and also of the interface to the spindle. The unbalances of the rotor excite centrifugal loads at the rotational frequency, which will rise to the square of the rotational speed. As a primary result mainly the spindle bearings are loaded. Furthermore the machine structure is driven to vibration. But there is no general correlation between the rise in unbalance and a decrease in surface quality and tool life [SCH 1998]. This type of excitation can be reduced by balancing as far as permitted by the joining accuracy related to manufacturing technology of the combined tool - spindle system [SCH 1997]. Limits of balancing of tool systems are given in the FKM-guide “Requirements for balancing of fast rotating tool systems” [FKM 1999]. These balancing suggestions are applicable for short stout tools, because of their fulfillment of the requirement of rigid rotors according to ISO 1940, which also spindles do. Whereas the excitation due to unbalance onto long protruding tools can only be reduced within specific limits, because of dynamic compliance of the tools having a considerable influence on their behavior. This relation is displayed in figure 2.

0.6 x 1fb

1fb

area 1 fmax < 0.6 x 1fb rigid rotor

Figure 2.

Limits of rigid rotors

frequency area 2 fmax > 0.6 x 1fb elastic rotor

Until up to about 60 % of the first natural bending frequency (resonance), the tool can be considered as a rigid rotor, i.e. the excitation reduction by balancing is useful. Beyond the 60% limit the tool has to be considered flexural elastic. This means, that with increasing rotational speed the dynamic compliance of the rotor will increase to a maximum (resonance) to decrease after this. In the area of the maximum (resonance) the rotor is dynamically elastic, so that the smallest excitations will cause a huge deviation (vibration amplitudes). For a non damped case, the amplitudes would reach infinity. Depending on the bearing this can result into breaking or another failure of the tool (Figure 3).

Figure 3.

Failed tool due to free bending vibration

Those excitations by unbalance are immanently existing in a rotating system as an excitation (free rotation), which is dependent on the rotational frequency. A failure of the tool when reaching the critical natural frequency causes a direct hazard for the worker at the machine. This hazard is directly connected to the machining process.

2.2. Excitation due to machining The resulting cutting forces of the machining in an interrupted cut cause an additional excitation of the machine - spindle - tool system. The periodical cutting edge engagement produces an independent vibration, with the machine vibrating in the same frequency as the excitation [WEC 1977]. This means the machine system vibrates according to the frequency resulting out of the multiplication of the rotational frequency of the spindle and the number of teeth. In this type of vibration the locus curve of the cutter follows several loops during a spindle revolution, where the number of loops is defined by the engagement frequency of the cutting edges. Figure 4 displays the typical milling tool locus curve for a tool with three teeth

0

Shaft displacement z [µm]

Shaft displacement y [µm]

20

10

0

-10

-20

-10

-20

-30

-40 -20

-10

0

10

Shaft displacement x [µm] Technology: fz = 0.15 mm n = 22,500 rpm a =D e a = 8 mm p

Figure 4.

20

-20

-10

0

10

20

Shaft displacement x [µm]

Tool diameter D = 16 mm Teeth z = 3 L/D ratio = 2.8 Carbide

Locus curve for a milling tool with three teeth

If the excitation frequency is located close to a natural frequency of the spindle tool system, the result will be large vibration amplitudes. These can lead to a damage of the tool (tool breakage) or the spindle system (damage of the bearing), if there is an insufficient damping of the system. The typical sound of an unstable process, like for example when chatter occurs, cannot be heard, because the excitation is harmonic to the natural frequency. Therefore the sound suggests a stable process. Besides the indirectly excited vibrations, there are still more self excited vibrations when milling. The mostly known and main type of a self excited vibration in milling is the regenerative chatter. Even a stable cutting process is never free of distortion (shock because of tool engagement, force application through the machine base). Therefore a wavy surface is generated on the work piece resulting out of the relative movements between the tool and the workpiece. When this wavy surface is cut again, the modulation of the chip cross-section leads to dynamic cutting force variations. If the system damping is high enough, the process remains stable, but if the limiting chip cross-section is reached, the system will become unstable. Vibration will build up. In this case the frequency of the chatter is not in phase with the frequency of cutting or the rotational speed of the spindle, therefore this unstable process is accompanied by the typical chatter sound. Figure 5 shows a locus curve of a milling tool for regenerative chatter.

0

Shaft displacement z [µm]

Shaft displacement y [µm]

50

25

0

-25

-50

-50

-25

0

25

50

Shaft displacement x [µm]

-25

-50

-75

-100 -50

-25

0

25

50

Shaft displacement x [µm]

Technology: fz = 0.15 mm n = 12,000 rpm a =D e a = 10 mm p

Figure 5.

D = 16 mm z=3 L/D = 2.8 Carbide

Locus curve of a milling cutter for regenerative chatter

The locus curve does not follow a certain path, but follows a seemingly chaotic path build out of several natural shapes.

3. Effects to the system’s components These presented excitation types have different influences on the rotating components of the system. In the following the main effects will be clarified.

3.1 Tool Excited by the unbalance contained in the system, the tool does circular vibrations in free rotation. These can be seen as deviations of the center point of the shaft (dynamic concentricity). With increasing dynamic compliance of the tool at an increasing rotational speed, the size of the circular path of the tool grows. This can be seen distinctively in the following figure 6. The tested tool has a 12 mm diameter and a protrusion length of 180 mm (L/D=15), with a first critical natural frequency at 241 Hz. The tool is made out of steel. It is clamped in a collet chucking, which is fitted into the spindle using a HSK 63 E tool holder. The spindle supplies 40kW of power and a maximum rotational speed of nmax=24,000 rpm. It is obvious, that the measurement of the dynamic concentricity of the tool can be directly transferred to the measured compliance frequency response of the tool-spindle system at rest.

133 Hz

200 Hz 100

50

50

50

0

-50

-100 -100

-50

0

50

100

Y displacement [µm]

100

Y displacement [µm]

0

-50

0

-50

-100 -100

-50

0

50

100

-100 -100

X displacement [µm]

X displacement [µm]

16 Hz

133 Hz

-50

0

50

100

X displacement [µm]

200 Hz

180

180 Amplitude

amplitude [µm]

150

Phase

90

120 90

0

60

phase [°]

Y displacement [µm]

16 Hz 100

-90 30 0

-180 0

50

100

150

200

250

300

350

400

frequency [Hz]

Figure 6.

Changing of the locus curves of the tool by an increase of the dynamic compliance

With the excitation of the process forces during machining, the tool will be displaced statically by the teeth engagement. As described in 2.2 the tool follows a loop shaped locus curve, excited by the teeth engagement shocks. After the engagement the tool vibrates on an elliptic path back into it’s defined resting position before being displaced by another engagement of a cutting tooth. The structure of the amplitudes and the general shape of the locus curve for the milling tool depends on the machining process (stable or unstable/chatter), on the whole mechanical characteristics of the tool - spindle system, as well as on the cutting conditions, which can stabilize the system by their share of damping.

3.2 Tool clamping Different vibration modes can be seen for a long protruding tool clamped in a chuck. Here the frequency and the height of the amplitude of each vibration mode depends on the type of clamping and the used tool. In most cases a rigid body translation of the tool in the chuck will occur for low frequencies. With increasing protrusion length of the tool, the first critical bending vibration of the tool will decrease further more. The different vibration mode are displayed in figure 7. A 12 mm diameter inserted ball nose cutter with a steel shaft is clamped in a collet chuck for modal testing. On the left side of the figure at a frequency of 512 Hz the rigid

body motion of the tool in the chuck is clearly visible, whereas in the middle at 605 Hz the tool is subject to a typical bending vibration. The right side displays the second bending vibration of the tool. Mode 2: 605 Hz

Mode 3: 1.380 Hz

X

X

Chuck

Tool

Mode1: 521 Hz

X

Y

Figure 7.

Y

Y

Natural shapes of the milling tool

Therefore the influence of the clamping of the tool on the natural behavior (capability to vibrate and height of amplitude) of the system seems to be of neglectable importance. But in measurements and simulation it can be seen, that if the stiffness of the tool clamping related to the tool is lower the 0.5, the influence on the system characteristics cannot be neglected. This means, if the tool got a weaker clamping, the critical bending frequency of the tool will decrease, which will relate to a change of modal shapes.

3.3 Interface, Spindle Shaft, Bearing The interface can be seen as a spring with a high strength in the tool - clamping spindle system. The transition behavior is in that way, that the free rotation or the force amplitudes caused by the machining can be transmitted with slightest damping via the interface onto the spindle shaft and therefore also onto the bearings. The transition behavior of the steep taper interface (SK) differs from those of the hollow shaft taper interface (HSK) by the higher damping in the system caused by the taper fit of the SK. For large differences between the stiffness of the tool and the interface, the influence of the interface on the dynamic system behavior is not important and can be neglected. But if however the stiffness of the interface approaches the stiffness of the used long protruding tool, as it is the case for small interface size, for example HSK 32 and 25 and tools with large diameters, it can be expected that there will be changes

in the natural shapes. This means especially, that the first natural bending shape is shifted from normally a shaft bending between the bearings to a deflection of the whole applied tool in the clamping point [BEC 1999]. The amount of energy and forces which have to be dealt with by the interface and the bearing if one of the above described cases occur, are given by the following example. During a vibration analysis of a high frequency motor spindle with a HSK 25 E interface, this case of failure happened. A tool dummy made out of steel with 10mm diameter and a length to diameter ratio (L/D) of 11.3 was accelerated from 15,000 rpm to higher rotational speeds. At a rotational speed of about 17,000 rpm sudden buckling of the dummy occurred due to natural vibrations (figure 8, left side). The interface had to deal with a shearing force of about 13kN for this case of failure. The short time resulting bending moment on the interface was about 800 Nm. These extreme loads caused a tearing of the planar face at several points (Figure 8 right side). Tool material: Tool diameter Tool length Failure speed Spindle speed Spindle power Interface

Steel D = 10 mm L = 113 mm nf = 17,000 rpm nmax= 60,000 rpm P = 7 kW HSK 25 E

Effects on the interface: 1

1 Cracked planar shoulder

2

2 Grooved cone

Figure 8.

Tool failure due to vibration/cracked planar shoulder of a HSK 25 E

The shaft of the motor spindle is out of the mechanical point of view a rigid beam located on isotropic bearings with a mass agglomeration between the bearings due to the shrink fitted rotor. If the critical natural bending frequency of the spindle shaft or of the clamping system inside the spindle is excited by the cutting edge engagement or by a frequency of the unbalance, it can lead to non admissible large deformations of the parts of the spindle. In the most extreme case, there will be damages like for example an immediate destruction of the bearings. Just as also a touching of the rotor and stator can occur, which can lead to a destruction of the motor. Also a breakage of the drawbar of the clamping system can happen, which causes an immediate loss of clamping force and allows a loosening of the clamped tool. This will result into a big hazard for the human and the machine. Often roller bearings are used in today’s motor spindles. The bearings are right in the flow of forces, this means, that all forces, the cutting forces as well as also the forces due to unbalance are transmitted by these parts to the spindle housing. The

spindle bearings should have a sufficient mechanical fatigue strength according to the guideline and the life cycle calculation of the bearing manufacturers [FAG 1997]. In contrary to this spindle bearings never reach these given life cycle times in the real application. Most of the spindle manufacturers give a guarantee for their high frequency spindles for a life cycle of 5000h. The reason for this discrepancy is on the one hand a lack of knowledge of the load collective out of the machining operation required for dimensioning. And on the other hand a lack of knowledge of the damage of bearings due to vibrations.

4. Identification of problems involved Basically the mentioned problem zones of work safety and process safety for the idle and rotating system can be determined by theoretical analysis, calculations as well as experiments. For the evaluation of these values it is certainly necessary to consider carefully, that the occasionally existing changes in mechanical characteristics of the components have to be taken into account according to the rotational frequency.

4.1. System’s behavior in rest To determine the critical modes of the tool - spindle system out of the work safety point of view, which means in this case critical tool modes, the knowledge of the dynamic behavior of the system in rest is sufficient. A significant change of the stiffness of the intersection between tool and chuck is not to be expected over the application rpm range, as well as there will be no big change in the stiffness of the interface [WEC 1999]. However if the tool is clamped by a collet, a small loss of stiffness can be expected by the a widening of the taper fit at the intersection point due to centrifugal loads. For the other clamping types suitable for high speed cutting, like power shrinking chucks, shrink fits and hydraulic expansion chucks, the loss of clamping force related to centrifugal forces can be neglected. An increase of the stiffness of the safety critical tool - clamping system by gyroscopic effects can be also neglected because of the given relation of the moment of inertia of the respective main axes. The measurement of the critical tool mode can be done by frequency or modal analysis. Therefore the system is excited by a hammer impact and the excitation as well as the system’s response is measured and analyzed by a FFT. Critical areas will occur for large amplitudes which have at the same time a phase shift of -180°.

4.2. System’s behavior at work

First bending frequency of the spindle shaft

8 6 4 2 0

24000 21000 x 10

18000

-7

15000 12000

Spindle Speed [rpm]

Compliance [10 -7 m/N]

The dynamic behavior of the spindle - tool system can be described by the frequency response function, consisting of frequency response and phase. These transfer frequency response can be evaluated experimentally by the impulse hammer method or can be calculated by for example finite element methods. For the experimental evaluation of the frequency response function there are differences in the results between the spindle in rest and the spindle during rotation. The spindle used for the experimental research had a maximum power of 40kW and a maximum rotational speed of nmax=24,000 rpm. Figure 9 shows the amplitude response of this spindle evaluated for several different speeds.

9000 6000 3000 0 500

1000

1500

2000

2500

Frequency [Hz]

Figure 9.

Frequency response of different spindle speeds

It is clearly visible, that the first natural bending frequency of the spindle shaft in rest is located at 720 Hz. With increasing spindle speed up to the maximum speed of the spindle, the natural frequency decreases to 660 Hz. This decline of the natural frequency by 9% can be explained by an nonlinear decrease of the load stiffness with increasing speed. Influences of the gyroscopic effect on the location for the natural frequencies can only be expected for higher rpm. Regarding the examinations concerning stability, the effect of the lowering of the natural frequencies caused by the bearings cannot be neglected.

5. Solutions for process and work safety

The following will describe two methods to evaluate the above mentioned dynamic problems of the tool - spindle system for high speed milling, which help to avoid critical systems modes regarding process and work safety. The safe application of long protruding tools can be described by them and furthermore the stable work field of the spindle can be determined.

5.1. Calculation of the safe application speed The exact determination of the relevant critical tool frequency for work safety by measurement is related to a large effort and requires the availability of the appropriate measurement equipment. A changed configuration of the tool system causes a change in the dynamic characteristics of the system and requires another measurement to determine the natural frequency of the tool. Using a calculation model, a conservative prediction of the critical tool frequency is possible. Mechanical model of the tool L

Real tool geometry and assembly conditions

EI, A, ρ

I =

π ⋅D 4 π ⋅D 2 ; A = 64 4

Mathematical solution of the simplified problem

Experimental and theoretical identification of relevant influences

Dimensionless solution of the problem

Figure 10. Flow chart for the calculation of the critical tool frequency Based on a substitution model of a clamped Euler-Bernoulli beam a dimensionless relation will be derived for the first natural critical bending frequency f1b.

f1b =

π 1 E ⋅I ⋅ 2⋅ 8 L ρ ⋅A

(1)

32 ρ 1 ⋅ f1b ⋅d ⋅ = ⋅ π E L 2   d 

(2)

Using the dimensionless tool length L ΠL = d

(3)

and the dimensionless first critical bending frequency of the tool

Π f = f1b ⋅d ⋅

ρ E

,

(4)

the dimensionless critical bending frequency f1b of the long protruding tool results as

Πf =

1 π. ⋅ Π 2L 32

(5)

Therefore the maximum admissible limit speed can be calculated as:

nG =

Πf ρ d ⋅ tool Etool

⋅60

[rpm ]

(6)

Furthermore several other constraints influence the dynamic characteristics of the tool system. These are for example the stiffness of the interface and of the intersection points. A comparison between calculation and measurement showed, that equation (6) will provide a conservative and therefore safe estimation only for the field of long protrusion length, i.e. L/D >15 (measurement starts at tool clamping). In this area the calculated natural tool frequency is located below the real value (Figure 11). Because of the wide spread of the measured values for the respective test configuration, a correction of the calculation method by suitable factors is needed. These factors should mirror the influence of the real conditions. But still there are only geometrical and mechanical characteristics as well as material values of the tool or chuck available for the correction of this simple calculation method.

Dimensionless Frequency Πf 0,1

0,01

Πf

0,001

Πfc 4 ≤ Π L ≤11

0,0001 2

4

6

8

Π L ≥ 11

10

12

14

16

18

20

Dimensionless Length ΠL

Figure 11. Dimensionless natural frequency as a function of the dimensionless length For the function for the dimensionless first critical bending frequency (Equation 5) being more accurate for an increasing tool length as for short protrusion lengths, a dimensionless correction factor is introduced. It’s behavior is inverse proportional to the protrusion length of the tool.

k1 =

d 1 = L ΠL

(7)

To be able to reflect the influence of the tool clamping by the chuck a second dimensionless correction factor is needed. The second factor contains the relation of the effective resistant area against shear because of the lateral force. d k2 = − Π D =  D  m

2

   

(8)

This factor is calculated in its first approximation using the average chuck diameter Dm.

Dm =

Dmax + Dmin 2

(9)

The compliance behavior of the tool or the chuck due to bending is described by the third dimensionless factor.

k3 = Π H = 1 − e

HF − H  W

   

(10)

applying HF EF d4 L ⋅(6 ⋅L2 + 2 ⋅L ⋅LF + 2 ⋅( L + LF ) 2 ) = ⋅ 4 ⋅ F 4 H w 6 ⋅EW D m − d L3

(11)

Therefore the extended equation for the evaluation of the first natural tool frequency is 

Π fc



1

− 2 + Π − Π D + Π H   π π − (2 + k1 + k 2 + k 3 )  = ⋅Π L = ⋅Π L L . 32 32

(12)

Figure 12 proofs the increasing approaching of the calculation results to the measured values, regarding all the additional influencing factors. -1] Rotational Speed[min [rpm] Rotational Speed

60000 Interface: Chuck:

Simulation (k1,k2) Simulation (k1,k2,k3) Experiment

50000 40000 30000 20000

Tool material: Chuck material:

SK 40 Hydraulic expansion chuck Carbide Steel

Parameters: ET EC ρT ρC Dm D LC

600.000 N/mm² 210.000 N/mm² 14.000 kg/m³ 7.850 kg/m³ 50 mm 20 mm 65 mm

10000 0 2

4

6

8

10

12

14

16

18

20

Dimensionless Length ΠL

Figure 12.

Comparison between measurement and calculation

5.2. Stable working regions depending on the spindle speed As it could be seen in chapter 4.2 the frequency response functions are mainly evaluated for the spindle in rest. If these are used for the calculation of stability lobes, a deviation to the real behavior during machining is obvious.

Axial Depth of Cut [mm]

15

10

5

n=0 rpm n=24.000 rpm 0

0

0.6

1.2

1.8

2.4

3

3.6

4.2

4.8

5.4

6

Spindle Speed x Number of teeth [rpm]

6.6 7.2 x 104

Figure 13. Stability lobs dynamically or statically evaluated for a tool with three teeth (workpiece material aluminum) Obviously there is a change of the geometry and the position of the stability lobes at a rotational speed of n = 0 rpm and nmax=24,000 rpm. The reasons for this are located in the influence of the change of the system stiffness as well as non homogeneous change of the system damping described before in 4.2. To be able to predict a safe and stable process, it is absolutely necessary to consider the given constraints regarding their influence on the dynamic system behavior.

6. Conclusion This paper describes the basic problems of the application of long protruding tools in high speed machining. Based upon a systematic description of the types of excitation of vibrations as well as their effects of the system components tool and motor spindle, a solution method was derived for the calculation of safe application limits for long protruding tools.

Furthermore it was highlighted that the simple knowledge of the tool - spindle system in rest is not sufficient to make reliable predictions for a stable and therefore safe and chatter free milling process in high speed machining.

7. References [SCH 96] SCHULZ, H., Hochgeschwindigkeitsbearbeitung - High-Speed Machining, Carl Hanser Verlag, München, Wien: 1996 [WER 1993] WERNER, A., Prozeßauslegung und Prozeßsicherheit beim Einsatz von schlanken Schaftfräsern, Shaker Verlag, Aachen: 1993 [SCHW 1997] SCHWAB, J., Kompensation der Fräserabdrängung beim Schlichten von Hohlformen, Fortschrittberichte VDI Nr. 441, VDI-Verlag, Düsseldorf: 1997 [JAN 1996] JANOVSKY, D., Einfluß der Technologie auf Maßgenauigkeit und Prozeßsicherheit beim Hochgeschwindigkeitsfräsen im Werkzeug- und Formenbau, Shaker Verlag, Aachen: 1996 [HOC 1996] HOCK, S., Hochgeschwindigkeitsfräsen im Werkzeug und Großformenbau, Shaker Verlag, Aachen: 1996 [PUD 1997] PUDER, J., KÖHLER, B., „Prozeßeinflüsse an gefrästen Flächen diagnostizieren“, ZWF - Zeitschrift für wirtschaftlichen Fabrikbetrieb, 92 (1997) 9 [SCH 1999] SCHULZ, H., HUERKAMP, W., WÜRZ, T., Sichere Werkzeuge für die HSC-Fräsbearbeitung, Verlag Institut für Arbeitswissenschaft Kassel, Kassel: 1999 [SCH 1998] SCHULZ, H., WÜRZ, T., „Balancing requirements for fast rotating tools and spindle systems“, Annals of the CIRP, Vol. 47/1/1998 [SCH 1997] SCHULZ, H., WÜRZ, T, ASCHENBACH, B., „Sinnvoll Auswuchten“, HSC-Sonderteil 1997 S. 33-36, Supplement to Werkstatt und Betrieb 130 (1997) 9 [BEC 1999] BECK, J., „Lange Werkzeuge - hochtourige Spindelsysteme. Ein Widerspruch?“ in proceedings of the 7. Austrian HSC-Conference in Steyr, Austria, at June the 24. and 25. 1999 [WEC 1999] WECK, M., REINARTZ, T., Untersuchungen von Werkzeug- und Spannsystemen unter sicherheitstechnischen Aspekten, Verlag Institut für Arbeitswissenschaft Kassel, Kassel: 1999 [WEC 1977] WECK, M. , TEIPEL, K., Dynamisches Verhalten spanender Werkzeugmaschinen, Springer Verlag, Berlin, u.a.: 1977 [FKM 1999] n.n., Auswuchtanforderungen an schnelldrehende Werkzeugsysteme, VDMA Verlag, Frankfurt/Main: 1999

DIN/ISO 1940, T1: Mechanische Schwingungen. Anforderungen an die Auswuchtgüte starrer Rotoren. Bestimmung der zulässigen Restunwucht. Beuth Verlag, Berlin, Köln 1993 [FAG 1997] n.n.: Hochgenauigkeitslager, Publ.-Nr.: AC 41 130 – 3 DA, FAG Aerospace and Superprecision Bearings Division, Firmenschrift: May 1997

Definitions, Acronyms, Abbreviations

Indices

r A I E L d f1b nG LC D H ΠL Πf ΠfC

tool C m min max

density cross section of the tool deviation moment Young’s-modulus tool length diameter of the tool first critical bending frequency admissible limit speed chuck length diameter compliance non dimensional tool length non dimensional critical speed corrected non dimensional critical speed

chuck mean minimum maximum