248019_ALIS0006.hwp

2002 12

A Comparative Study on the Static and Dynamic Stiffness Evaluation Methods of Machine Tools 2002 12

. 2002 12

LIST OF FIGURES LIST OF TABLES NOMENCLATURE 1 1 1.1 1 1.2 2 1.3 3 2 5 2.1 5 2.2 6 2.3 7 2.4 9 2.5 9 3 11 3.1 11 3.2 13 3.2.1 13 3.2.2 19 3.3 27 3.3.1 28 3.3.2 29 3.3.3 32 3.3.4 33 3.4 35

4 37 4.1 37 4.1.1 37 4.1.2 39 4.1.3 40 4.2 41 4.2.1 41 4.2.2 42 4.3 44 4.3.1 44 4.3.2 45 5 48 6 51 53 ABSTRACT 58

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. 3.10 Comparison of compliance and coherence by impulse test Fig. 3.11 Comparison of compliance and coherence by exciter test Fig. 3.12 Evaluation of static compliance by linear reduction method Fig. 3.13 Comparison of compliance and coherence at resonance Fig. 3.14 Time and frequency domains of force pulse signal Fig. 3.14 Time and frequency domains of random signal Fig. 3.15 Rectangular window and exponential window at impulse test Fig. 3.16 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 1100-7 System) Fig. 4.4 Prototype of machine tools Fig. 4.5 Measurement set-up for the stiffness evaluation

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 ( )

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

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

: 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

1 1.1., (, Mother machine). (Leonard Davinchi), (Head stock). (Henry Maudslay) 1797. (Josehp Whitworth) 1821 1862. 1862 (Joseph Rogers Brown). 1769 (James Watt). (John Wilkinson) 1775. 19. 19. 20,., 1 2

. 1960,,., 1970,.. [20][21][22].. 4,,. 1.2.. AISI [14]. 0.. [1]

(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).

.. 2. 3. 4 (Dummy tool) (Hydraulic exciter). 5,.

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),.

2.2 1. (2.2) (2.2) (2.3) (2.4).. (2.3) (2.4). (2.5)

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)

(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)

2.4 [9][16] (Coherence function). (2.10). (2.11) 1. 1, 0. 2.5,. Fig. 2.3. Fig. 2.3 Multi input and output system

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

3 (Added mass) (Tip) [6][18], [9][16],, (Drive rod or stinger) (Force transducer) [9][11].. 3.1 300[ ] 30[ ] 5[ ] Table 3.1. Fig. 3.1. Table 3.1 Specification and property of the cantilever beam Size[ ] 200 30 5 Young' modulus[ ] 207 Poisson's ratio 0.3 Density[ / ] 7,833

... 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). 4 (Node) 3 (Element). 210 164 7.5[ ] 7.5[ ]. 420 164 7.5[ ] 7.5[ ] 5[ ]. [17] Fig. 3.3.

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

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

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)

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

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

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 46.214 45.71 0.989 46.48 1.006 46.42 1.004 289.619 292.75 1.011 291.22 1.006 290.93 1.005 810.942 883.82 1.090 816.49 1.007 815.93 1.006 Note: Error ratio = (Exact solution-application solution)/exact solution. 3.2.2 0... (3.15) (3.16)

.. (3.17),,,. (3.15). (3.18). 1.. 2.,,. 3.. 4... (3.19),,,,,,.

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

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

(3.26). (3.33). (3.17). (3.34) 95[ ] 0.2[ ]. 270[ ]. Fig. 3.5. (a) Beam element model

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

Table 3.3.. 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 0.01695 0.01695 2.73452 Shell element 0.01625 0.01625 2.80444 Solid element 0.01631 0.01631 3.26958 Figure 3.6. 5% 10. Fig. 3.7,.

25 20 Error[%] 15 10 5 0 BEAM Element SHELL Element SOLID Element 0 5 10 15 20 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 0.1 44.0 44.5 45.0 45.5 46.0 46.5 47.0 47.5 48.0 Frequency[Hz] Fig. 3.7 Comparison of dynamic compliance by F.E.M. modeling

3.3 Fig. 3.8. (Force Transducer).. Fig. 3.8 Block diagram of compliance measurement system FFT Medallion 16, 8,912 0.3125. 5 20 (Linear average). Table 3.4.

Table 3.4 Specification of impulse and exciter tests equipments Type Sensitivity Freq. range Force range Impulse hammer Kistler 9277A2000 2 /N 9,300 2,500N Modal shaker LING LTM-100-2,000 - Accelerometer Kistler 8636C50 99.5 /N 22 - Force transducer Kistler 9712BE250 16.94 /N - 250N FFT analyzer IOtech Medallion 16Ch. - 10,000-3.3.1 Fig. 3.9 (Frequency response function), Table 3.5. 100 10 Magnitude[g/N] 1 0.1 0.01 Exciter test (i=j) Impulse test (i=j) 1E-3 0 100 200 300 400 500 600 700 800 900 1000 Frequency[Hz] Fig. 3.9 Comparison of frequency response functions

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 46.214 44.063 0.953 43.75 0.947 289.619 270.938 0.935 283.125 0.978 810.942 766.563 0.945 792.188 0.977 Note: Error ratio = (Exact solution-application solution)/exact solution 3.3.2 Fig. 3.10 Fig. 3.11. ( ) ( ),. 1.0 0.8 Coherence Compliance[mm/N] 0.6 0.4 0.2 0.0 10 1 0.1 0.01 1E-3 1E-4 1E-5 1E-6 1E-7 1E-8 0 100 200 300 400 500 600 700 800 900 1000 Frequency[Hz] Impulse test(i=j) Impulse test(i j) Fig. 3.10 Comparison of compliance and coherence by impulse test

1.0 Coherence Compliance[mm/N] 0.8 0.6 0.4 0.2 0.0 10 1 0.1 0.01 1E-3 1E-4 1E-5 1E-6 1E-7 0 100 200 300 400 500 600 700 800 900 1000 Frequency[Hz] Exciter test(i=j) Exciter test(i j) Fig. 3.11 Comparison of compliance and coherence by exciter test 0., ( ). Fig. 3.12. Fig. 3.13..

Fig. 3.12 Evaluation of static compliance by linear reduction method 1.0 0.8 Coherence 0.6 0.4 0.2 0.0 10 Exciter test(i=j) Impulse test(i=j) Exciter test(i j) Impulse test(i j) Compliance[mm/N] 1 0.1 0.01 1E-3 1E-4 42 43 44 45 46 Frequency[Hz] Fig. 3.13 Comparison of compliance and coherence at resonance

3.3.3 (, Infinitesimal)... (Main lobe) (Side lobe).,. Fig. 3.14 Fig. 3.15. Force[N] 100 80 60 40 20 0-20 PDS[N^2/Hz] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.012 Time[s] 0.010 0.008 0.006 0.004 0.002 0.000 0 200 400 600 800 1000 Frequency[Hz] Fig. 3.14 Time and frequency domains of force pulse signal

Force[N] PSD[N^2/Hz] 15 10 5 0-5 -10-15 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.25 Time[s] 0.20 0.15 0.10 0.05 0.00 0 200 400 600 800 1000 Frequency[Hz] Fig. 3.14 Time and frequency domains of random signal 3.3.4 (Window function) (Weighting function).. (Leakage). Rectangular Exponential Hanning. Fig. 3.15 Fig. 3.16.

Fig. 3.15 Rectangular window and exponential window at impulse test Fig. 3.16 Hanning window at exciter test

3.4 1 1.09%, 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 ] 46.214 45.71 46.48 46.42 44.063 43.75 289.619 292.75 291.22 290.93 270.938 283.125 810.942 883.82 816.49 815.93 766.563 792.188 Table 3.7 Table 3.8...

.. 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 0.01695 0.01695 2.73452 Shell Element 0.01625 0.01625 2.80444 Solid Element 0.01631 0.01631 3.26958 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 0.020 0.0005 2.4570 0.1272 Exciter test 0.018 0.0003 2.1052 0.0794

4.... RMS(Root mean square)... 4.1 4.1.1. [1].. Fig. 4.1 BT30, Fig. 4.2 4.

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

4.1.2. 1.. (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 1100-7 System)

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

4.2 4.2.1,,... VDW WZL [1] Table 4.3. Table 4.3 Measuring disposition for travel-stand milling machine

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

(4.1),. (Aliasing effect) 2.56 ~ 4. 6 800. 8,192 0.25. 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

4.3 4.3.1 Fig. 4.6 Table 4.5. 0.1 Magnitude[g/N] 0.01 1E-3 1E-4 1E-5 0 100 200 300 400 500 600 700 800 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 33 0.40 2nd 64 1.59 3rd 100 2.05 4th 119 0.13 5th 488 0.82

4.3.2 Figure 4.7 ~ Fig. 4.9 Table 4.3. x-, y-, z- ( ) Maxwell [3].. Compliance[um/N] 0.1 0.01 1E-3 1E-4 1.0 0.8 Gxx Gxy 0 100 200 300 400 500 600 700 Gxz 800 Coherence 0.6 0.4 0.2 0.0 0 100 200 300 400 500 600 700 800 Frequency[HZ] Fig. 4.7 Compliance and coherence of the point ( )

Compliance[um/N] 0.1 0.01 1E-3 1E-4 1.0 0.8 Gyx Gyy 0 100 200 300 400 500 600 700 Gyz 800 Cohernce 0.6 0.4 0.2 0.0 0 100 200 300 400 500 600 700 800 Frequency[Hz] Fig. 4.8 Compliance and coherence of the point ( ) Compliance[um/N] 0.1 0.01 1E-3 1.0 0.8 Gzx Gzy 0 100 200 300 400 500 600 700 Gzz 800 Coherence 0.6 0.4 0.2 0.0 0 100 200 300 400 500 600 700 800 Frequency[Hz] Fig. 4.9 Compliance and coherence of the point ( )

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 0.07 0.2148 y-axial - 0.0597 z-axial - 0.0427 x-axial - 0.0369 y-axial 0.035 0.2293 z-axial - 0.0948 x-axial - 0.0289 y-axial - 0.0347 z-axial 0.04 0.1275

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 46.214 289.619 810.942 Beam 45.71 (1.09) 292.75 (1.08) 883.82 (8.99) F.E.M. Shell 46.48 (0.58) 291.22 (0.55) 816.49 (0.68) Solid 46.42 (0.45) 290.93 (0.45) 815.93 (0.62) Exciter test 44.063 (4.65) 270.938 (6.45) 766.563 (5.47) Impulse test 43.75 (5.33) 283.125 (2.24) 792.188 (2.31) Note: Error ratio=[(exact solution-application solution)/exact solution] 100[%], 1 46.214 1.09%, 0.58% 0.45%,. 1,..

... 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 0.01695 2.73452 Shell 0.01625 2.80444 Solid 0.01631 3.26958 Exciter test 0.018 2.1052 Impulse test 0.020 2.4570...

. Fig. 4.7 ~ Fig. 4.9...

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

4)... 5).

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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.

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.

..,.,.,,,.,,,,,. 2,.,,.,,,,,.,,,,,,.

B&K. RRC.,,., 10,,,,,. 30.,,.. 2002 12