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2 A Comparative Study on the Static and Dynamic Stiffness Evaluation Methods of Machine Tools

4 LIST OF FIGURES LIST OF TABLES NOMENCLATURE

5 ABSTRACT 58

6 LIST OF FIGURES Fig. 2.1 Elastic support and linear spring systems Fig. 2.2 Linear system input-output model(convolution time domain and multiplication in frequency domain) Fig. 2.3 Multi input and output system Fig. 3.1 Attachment of the cantilever beam Fig. 3.2 Connection of exciter drive rod Fig. 3.3 F.E.M. modeling of the cantilever beam Fig. 3.4 A Beam in bending Fig. 3.5 Exciter point of element model Fig. 3.6 Error ratio of static compliance by linear reduction method Fig. 3.7 Comparison of dynamic compliance by F.E.M. modeling Fig. 3.8 Block diagram of compliance measurement system Fig. 3.9 Comparison of frequency response functions Fig Comparison of compliance and coherence by impulse test Fig Comparison of compliance and coherence by exciter test Fig Evaluation of static compliance by linear reduction method Fig Comparison of compliance and coherence at resonance Fig Time and frequency domains of force pulse signal Fig Time and frequency domains of random signal Fig Rectangular window and exponential window at impulse test Fig Hanning window at exciter test Fig. 4.1 Design of a dummy tool Fig. 4.2 Prototype of a dummy tool Fig. 4.3 Prototype of hydraulic exciter(xcite System) Fig. 4.4 Prototype of machine tools Fig. 4.5 Measurement set-up for the stiffness evaluation

7 Fig. 4.6 Comparison of frequency response functions Fig. 4.7 Compliance and coherence of the point ( ) Fig. 4.8 Compliance and coherence of the point ( ) Fig. 4.9 Compliance and coherence of the point ( )

8 LIST OF TABLES Table 3.1 Specification and property of the cantilever beam Table 3.2 Comparison of natural frequency and error ratio Table 3.3 Comparison of static and dynamic compliances by finite element method Table 3.4 Specification of impulse and exciter tests equipments Table 3.5 Comparison of natural frequency and error ratio Table 3.6 Comparison of natural frequency Table 3.7 Comparison of static and dynamic compliance by finite element method Table 3.8 Comparison of static and dynamic compliance by experimental method Table 4.1 Specification of hydraulic exciter Table 4.2 Specification of machine tools Table 4.3 Measuring disposition for travel-stand milling machine Table 4.4 Measurement data of hydraulic exciter and FFT analyzer Table 4.5 Natural frequency and damping ratio Table 4.6 Comparison of static and dynamic compliance of the point Table 5.1 Comparison of natural frequency and error ratio by F.E.M. and experimental methods Table 5.2 Comparison of static and dynamic compliance by F.E.M. and experimental methods

9 NOMENCLATURE : Young's modulus : Moment of inertia : Vector : Matrix : Natural frequency : Amplitude : Ensemble average : Auto correlation function : Cross correlation function : Power spectrum density function : Cross spectrum density function : Coherence function : Bending moment : Shear force : External force per unit length : Cross-sectional area : Mass density : Normal mode or characteristic function : Stiffness matrix : Nodal displacement vector : Applied load vector : Reaction load vector : Nodal load vector : Acceleration load vector

10 : Element thermal load vector : Element pressure load vector : Structural damping matrix : Nodal acceleration vector : Nodal velocity vector : Nodal displacement vector : Maximum displacement : Square root of -1 : Imposed circular frequency : Displacement phase shift : Force phase shift : Vector of element inertia forces(real part) : Element mass matrix : Element real displacement vector : Vector of element inertia forces(imaginary part) : Element imaginary displacement vector : Vector of element of damping force(real part) : Element damping matrix : Vector of element of damping force(imaginary part) : Sampling frequency : Measurement maximum frequency

11 1 1.1., (, Mother machine). (Leonard Davinchi), (Head stock). (Henry Maudslay) (Josehp Whitworth) (Joseph Rogers Brown) (James Watt). (John Wilkinson) ,., 1 2

12 . 1960,,., 1970,.. [20][21][22].. 4,, AISI [14]. 0.. [1]

13 (Cutting test)... [1][2] Weck Teipel [1]. Tobias [5] (Electrodynamic exciter) Weck [2] Minis [3]. 200Hz [2][5],. [1][3]. [1][2][4] 1.3 (Finite element method) (Impulse test) (Exciter test).

14 (Dummy tool) (Hydraulic exciter). 5,.

15 2 2.1 Fig. 2.1 (2.1). (a) Elastic support Fig. 2.1 Elastic support and linear spring systems (b) Linear spring (2.1),.

16 (2.2) (2.2) (2.3) (2.4).. (2.3) (2.4). (2.5)

17 2.3 [9][10]. (Fourier transform).. Fig. 2.2., (Auto correlation function) (2.6). Fig. 2.2 Linear system input-output model(convolution time domain and multiplication in frequency domain) (2.6)

18 (Cross correlation function) (2.7), (Ensemble average) (Ergodic). (2.7) (Spectrum density function),. (Power spectrum density function). (2.8) (Cross spectrum density function). (2.9) (2.10), (Compliance). (2.10)

19 2.4 [9][16] (Coherence function). (2.10). (2.11) 1. 1, ,. Fig Fig. 2.3 Multi input and output system

20 ... x-, y- z-.. (2.12)..

21 3 (Added mass) (Tip) [6][18], [9][16],, (Drive rod or stinger) (Force transducer) [9][11] [ ] 30[ ] 5[ ] Table 3.1. Fig Table 3.1 Specification and property of the cantilever beam Size[ ] Young' modulus[ ] 207 Poisson's ratio 0.3 Density[ / ] 7,833

22 ... Fig. 3.1 Attachment of the cantilever beam....,. [9] Fig. 3.2 ISO [11].

Fig. 3.2 Connection of exciter drive rod 3.2 3.2.1 ) (Beam element), (Shell element), (Solid element).

23 Fig. 3.2 Connection of exciter drive rod ) (Beam element), (Shell element), (Solid element). 4 (Node) 3 (Element) [ ] 7.5[ ] [ ] 7.5[ ] 5[ ]. [17] Fig. 3.3.

24 (a) Beam element model (b) Shell element model

25 (c) Solid element model Fig. 3.3 F.E.M. modeling of the cantilever beam ) Fig. 3.4 Euler,,.. (3.1) z-. (3.2)

26 Fig. 3.4 A Beam in bending,. Figure 3.4 y-. (3.3) (3.2) (3.3). (3.4) (3.5) (3.5). (3.4)

27 (3.6). (3.7) (3.7) (3.6). (3.8),. (3.8) (3.8) [17] (3.8). (3.9).

28 (3.9). (3.10),. (3.11). (3.12) (3.9). (3.13). [5]. (3.14)

29 Table 3.2 y-. Table 3.2 Comparison of natural frequency and error ratio Exact sol. Beam element Shell element Solid element Freq.[ ] Freq.[ ] Error Freq.[ ] Error Freq.[ ] Error Note: Error ratio = (Exact solution-application solution)/exact solution (3.15) (3.16)

30 .. (3.17),,,. (3.15). (3.18) ,, (3.19),,,,,,.

31 (phase). (3.20),,., (3.20). (3.21) (3.22),.. (3.23) (3.24) (3.25). (3.22) (3.25) (3.19).

32 (3.26). (3.27) (3.28),, (3.29), (3.30). (3.29) (3.30). (3.31) (3.32).. (,, ).

33 (3.26). (3.33). (3.17). (3.34) 95[ ] 0.2[ ]. 270[ ]. Fig (a) Beam element model

34 (b) Shell element model Fig. 3.5 Exciter point of element model (c) Solid element model

35 Table Table 3.3 Comparison of static and dynamic compliances by finite element method Element Type Compliance Static compliance [ /N] by static by setting analysis at FRF Dynamic compliance [ /N] Beam element Shell element Solid element Figure % 10. Fig. 3.7,.

36 25 20 Error[%] BEAM Element SHELL Element SOLID Element Frequency[Hz] Fig. 3.6 Error ratio of static compliance by linear reduction method 10 BEAM Element Compliance[mm/N] 1 SHELL Element SOLID Element Frequency[Hz] Fig. 3.7 Comparison of dynamic compliance by F.E.M. modeling

37 3.3 Fig (Force Transducer).. Fig. 3.8 Block diagram of compliance measurement system FFT Medallion 16, 8, (Linear average). Table 3.4.

38 Table 3.4 Specification of impulse and exciter tests equipments Type Sensitivity Freq. range Force range Impulse hammer Kistler 9277A /N 9,300 2,500N Modal shaker LING LTM-100-2,000 - Accelerometer Kistler 8636C /N 22 - Force transducer Kistler 9712BE /N - 250N FFT analyzer IOtech Medallion 16Ch. - 10, Fig. 3.9 (Frequency response function), Table Magnitude[g/N] Exciter test (i=j) Impulse test (i=j) 1E Frequency[Hz] Fig. 3.9 Comparison of frequency response functions

39 Table 3.5 Comparison of natural frequency and error ratio Exact solution Exciter test Impulse test Natural freq.[ Hz ] Natural freq.[ Hz ] Error Natural freq.[ Hz ] Error Note: Error ratio = (Exact solution-application solution)/exact solution Fig Fig ( ) ( ), Coherence Compliance[mm/N] E-3 1E-4 1E-5 1E-6 1E-7 1E Frequency[Hz] Impulse test(i=j) Impulse test(i j) Fig Comparison of compliance and coherence by impulse test

40 1.0 Coherence Compliance[mm/N] E-3 1E-4 1E-5 1E-6 1E Frequency[Hz] Exciter test(i=j) Exciter test(i j) Fig Comparison of compliance and coherence by exciter test 0., ( ). Fig Fig

41 Fig Evaluation of static compliance by linear reduction method Coherence Exciter test(i=j) Impulse test(i=j) Exciter test(i j) Impulse test(i j) Compliance[mm/N] E-3 1E Frequency[Hz] Fig Comparison of compliance and coherence at resonance

42 3.3.3 (, Infinitesimal)... (Main lobe) (Side lobe).,. Fig Fig Force[N] PDS[N^2/Hz] Time[s] Frequency[Hz] Fig Time and frequency domains of force pulse signal

43 Force[N] PSD[N^2/Hz] Time[s] Frequency[Hz] Fig Time and frequency domains of random signal (Window function) (Weighting function).. (Leakage). Rectangular Exponential Hanning. Fig Fig

44 Fig Rectangular window and exponential window at impulse test Fig Hanning window at exciter test

45 %, 0.58% 0.45%. 4.65%, 5.33% 1.. Table 3.6. Table 3.6 Comparison of natural frequency Exact F.E.M. Exciter Impulse solution Beam Shell Solid test test Natural frequency [ Hz ] Table 3.7 Table

46 .. Table 3.7 Comparison of static and dynamic compliance by finite element method Element type Compliance Static compliance [ /N] by static by setting analysis at FRF Dynamic compliance [ /N] Beam Element Shell Element Solid Element Table 3.8 Comparison of static and dynamic compliance by experimental method Compliance Static compliance [ mm /N] Dynamic compliance [ mm /N] Method Type Impulse test Exciter test

47 4.... RMS(Root mean square) [1].. Fig. 4.1 BT30, Fig

48 Fig. 4.1 Design of a dummy tool Fig. 4.2 Prototype of a dummy tool

49 (Clearance). Table 4.1. Table 4.1 Specification of hydraulic exciter Total static & dynamic force range Stroke Rod Load cell force range Load cell sensitivity LVDT 4,450[N] 25[ ] 18[ ] 11,125[N] 250[lbs/V] 25[ ] Fig. 4.3 Prototype of hydraulic exciter(xcite System)

50 4.1.3 Fig. 4.4 x-, y-, z- 420[ ] 280[ ] 380[ ] 650[ ] 380[ ], 3.3. Fig. 4.4 Prototype of machine tools Table 4.2 Specification of machine tools [m/min] X / Y / Z 30 / 30 / 24 [ ] X / Y / Z 420 / 280 / 380 [ ] X / Y 650 / 300 [rpm] 8,000 Type BT30 [ ] 2,500

51 ,,... VDW WZL [1] Table 4.3. Table 4.3 Measuring disposition for travel-stand milling machine

52 4.2.2 Fig (Signal generator),.. Fig. 4.5 Measurement set-up for the stiffness evaluation. Shannon [10].

53 (4.1),. (Aliasing effect) 2.56 ~ , Hanning 50. 1,000N. 300N ~ 1,000N 500N 400N.... [1] Table 4.4 FFT. Table 4.4 Measurement data of hydraulic exciter and FFT analyzer Hydraulic exciter Static force Dynamic force Frequency range 500N 400N 800 FFT analyzer Linear average 50 Window function Hanning Block size 8,192

54 Fig. 4.6 Table Magnitude[g/N] E-3 1E-4 1E Frequency[Hz] FRF_xx FRF_yy FRF_zz Fig. 4.6 Comparison of Frequency response functions Table 4.5 Natural frequency and damping ratio Natural frequency[ ] Damping ratio[%] 1st nd rd th th

55 4.3.2 Figure 4.7 ~ Fig. 4.9 Table 4.3. x-, y-, z- ( ) Maxwell [3].. Compliance[um/N] E-3 1E Gxx Gxy Gxz 800 Coherence Frequency[HZ] Fig. 4.7 Compliance and coherence of the point ( )

56 Compliance[um/N] E-3 1E Gyx Gyy Gyz 800 Cohernce Frequency[Hz] Fig. 4.8 Compliance and coherence of the point ( ) Compliance[um/N] E Gzx Gzy Gzz 800 Coherence Frequency[Hz] Fig. 4.9 Compliance and coherence of the point ( )

57 Table 4.6 x-, y- z-.. x- y- y- x-. Table 4.6 Comparison of static and dynamic compliance of the point Measurement direction Static compliance [ /N] Dynamic compliance [ /N] x-axial excitation y-axial excitation z-axial excitation x-axial y-axial z-axial x-axial y-axial z-axial x-axial y-axial z-axial

58 5 Table 5.1. Table 5.1 Comparison of natural frequency and error ratio by F.E.M. and experimental methods 1st Natural freq.[ ] 2nd Natural freq.[ ] 3rd Natural freq.[ ] Exact solution Beam (1.09) (1.08) (8.99) F.E.M. Shell (0.58) (0.55) (0.68) Solid (0.45) (0.45) (0.62) Exciter test (4.65) (6.45) (5.47) Impulse test (5.33) (2.24) (2.31) Note: Error ratio=[(exact solution-application solution)/exact solution] 100[%], %, 0.58% 0.45%,. 1,..

59 ... Table 5.2. Table 5.2 Comparison of static and dynamic compliance by F.E.M. and experimental methods Method type F.E.M. Compliance Static compliance Dynamic compliance [ /N] [ /N] Beam Shell Solid Exciter test Impulse test

60 . Fig. 4.7 ~ Fig

61 6. 1),. 2).. 3)....

62 4)... 5).

63 1. Weck, M., Teipel, K., "Handbook of machine tools," Wiley, New York, Tlusty, J., Zamudio, S., "Evaluation of Cutting Performance of Machining Centers," Annals of the CIRP, Vol. 40, No. 1, pp , Minis, I., Magrab, E., and Pandelidis, I., "Improved Methods for the Prediction of Chatter in Turning, Part1: Determination of Structural Response Parameters," Transactions of the ASME, Vol. 112, pp. 12~20, Hanna, N. H., Tobias, S. A., "The Non-Linear Dynamics Behavior of a Machine Tool Structure," Int'l J. MDTR, Vol. 9, pp.293~307, Pergamon Press Tobias, S. A., "Machine Tool Vibration," Wiley, New York, Halvorsen, W. G., Brown, D. L., "Impulse Technique for Structural Frequency Response Testing," Sound and Vibration, Vol. 11, No. 11, pp. 8-21, November Weck, M., Eckstein, R., "An Examination Technique to Determine Static Weakpoints of Machine Tools," Annals of the CIRP Vol. 36, No. 1, pp , Xu, M., Birchmeier, J. R., "Dynamic Stiffness Testing and its Applications in Machine Tool," Sound and Vibration, Vol. 31, pp. 8-21, Ewins, D. J., "Modal Testing: Theory and Practice," Research Studies Press, London, 1986.

64 10., "," ISO 7626 Vibration and Shock 12. In-Ung Eun, "Development of Static and Dynamic Behavior of Machine Tools in the Period form ,", , ",,", American National Standards Institute 15. In-Ung Eun, "Development of Static and Dynamic Behavior of Machine Tools in the Period form ,", Kenneth G. McConnell, "Vibration Testing: Theory and Practice," John Wiley & Sons, Singiresu S. Rao, "Mechanical Vibrations," Addison-Wesley, D. Brown, G. Carbon and K. Ramsey, "Survey of Excitation Techniques Applicable to the Testing of Automotive Structures," Socity of Automotive Engineers, Inc., February 28-March 4, Swanson Analysis, "ANSYS User's Manual," Revision 5.5, , ",",

65 21., "," , "," "The Application Note for Sound & Vibration : using and application," Bruel & Kjær 24. "Frequency Analysis," Bruel & Kjær 25. Leonard Meirovitch, "Analytical Methods in Vibration," Macmillan, Demeter G. Fertis, "Mechanical and Structural Vibrations," John Wiley & Sons, J. R. Wright, J. E. Cooper and M. J. Desforges, "Normal-Mode Force Application - Theory and Application," Mechanical Systems and Signal Processing, Volume 13, No. 2, pp , N. Larbi and J. Lardies, "Experimental Modal Analysis of a Structure Excited by a Random Force," Mechanical Systems and Signal Processing, Vol. 14, No. 2, pp , , ",", Weck, M. & Teipel, K., "Acessing the Chatter Behavior of Machine-Tools," Annals of CIRP, Vol. 27, No. 1, pp , 1978.

66 30. Lee Jongkil and Kim Dae-Hwan, "Experimental modal analysis and vibration monitoring of cutting tool support structure," International Journal of Mechanical Sciences, Vol. 37, No. 11, pp , Najeh Tounsi and Alain Otho, "Dynamic cutting force measuring, International Journal of Machine Tools and Manufacture," Vol. 40, No. 8, pp , B. -T. Wang, "Structural Modal Testing with Various Actuators and Sensors," Mechanical Systems and Signal Processing, Vol. 12, No. 5, pp , , " -, " A, 20 7, pp , ,,, ",", 8 1, pp , ,,, ",", 10, 4, pp , ,, ",", 9 5, pp , ,, " 1,", 12 9, pp , Zatarain, M. ; Leizaola, J., "Improvement of the Dynamic Stiffness of Machine Tools by Adaptively Tuned Dynamic Absorbers," Noise & vibration worldwide, Vol. 29, No. 3, pp , 1998.

67 39. Z. Chen and C. K. Mechefske, "Machine Signature Identification by Analysis of Impulse Vibration Signals," Journal of Sound and Vibration, Vol. 244, No. 1, pp , Najeh Tounsi and Alain Otho, "Identification of machine-tool-workpiece system dynamics," International Journal of Machine Tools and Manufacture, Vol. 40, No. 9, pp , Tlusty, J., Ziegert, J. C. and Ridgeway, S., "Stiffness of Structures and Drives in Fast Milling Machines," Socity of Automotive Engineers, Vol. 108, , ",", J. A. Morgan, C. Pierre, G. M. Hulbert, "Calculation of Component Mode Synthesis Matrices From Measured Frequency Response Functions, Part1: Theory," ASME, Vol. 120, No. 2. pp , J. A. Morgan, C. Pierre, G. M. Hulbert, "Calculation of Component Mode Synthesis Matrices From Measured Frequency Response Functions, Part2: Application," ASME, Vol. 120, No. 2. pp , , " ( ),," 9 2, pp , Ronald F. Gibson, "Modal vibration response measurements for characterization of composite materials and structures," Composites Science and Technology, Vol. 60, No. 15, pp , 2000.

68 ABSTRACT A Comparative Study on the Static and Dynamic Stiffness Evaluation Methods of Machine Tools by Kang Young-Jin Department of Mechanical Design and Manufacturing Graduate School, Changwon National University Changwon, Korea Recent dynamic characteristics of machine tools which are requiring high speed and high precision cause an important factor to induce a manufacturing accuracy and production. This paper obtained an applicable stiffness evaluation method about a cantilever beam which has an exact solution analysing a natural frequency and evaluating a static and dynamic stiffness using a F.E.M., impulse test, and exciter test. It is impossible to analysis a whole machine tools' compliance completely. Because it is still difficult to understand damping characteristics and the stiffness of connecting parts. So when you evaluate a machine tools using the characteristics of a compliance frequency response function, you must depend on measurement. We evaluated a machine tools stiffness using a hydraulic exciter which can consider the exciting force's energy transfer and bonding parts and connecting parts' dynamic charateristics.

69 We modeled the cantilever beam using a beam, shell and solid element and analysed a static and dynamic stiffness using a finite element method. Thus we can get the smallest solid element model error. When we analysed cantilever's natural frequency using a experimental method, exciter test's errors were bigger than impulse test's. And when we evaluated a static stiffness, the exciter test could predict the exact static stiffness but the impulse test couldn't. And because of the lower coherence at resonance point, the dynamic compliance, the error coming into between the force transfer and structure, was also lower than the impulse test. Basing on the result of the stiffness evaluation method in cantilever beam, we evaluated the practical machine tools stiffness using the exciter test. In that case the different exciter and response direction presented the lower compliance and coherence than the same direction. When the direction was different, we can confirmed that the compliance error was relatively big.

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