IDENTIFICATION OF NONLINEAR CUTTING PROCESS MODEL IN TURNING

Transcription

1 ADVANCES IN MANUFACTURING SCIENCE AND TECHNOLOGY Vol. 33, No. 3, 2009 IDENTIFICATION OF NONLINEAR CUTTING PROCESS MODEL IN TURNING Bartosz Powałka, Mirosław Pajor, Stefan Berczyński S u m m a r y This paper presents a methodology of force model development. A special experimental stand has been used for cutting force identification to eliminate the regenerative phenomenon and improve the accuracy. The stand was used to carry out the cutting tests with the constant chip thickness. Also tests with sinusoidally varying chip thickness have been performed. During the tests cutting force and relative tool part vibration has been measured. The measured signals were used to estimate coefficients of the linear and nonlinear cutting force models. An approach similar to the restoring force method widely used in structural dynamics has been implemented for a dynamic force model identification Keywords: cutting force, nonlinear model, identification Identyfikacja nieliniowego modelu procesu skrawania podczas toczenia S t r e s z c z e n i e Artykuł prezentuje metodę kalibracji mechanistycznego modelu sił skrawania. Badania prowadzono na stanowisku eksperymentalnym umożliwiającym eliminację zjawiska regeneracji śladu i zwiększenie dokładności estymacji. Stanowisko stosowano do prowadzenia próby skrawania ze stałą grubością warstwy skrawanej. Wykonano również próby dla grubości warstwy skrawanej ulegającej zmianie w sposób sinusoidalny. Podczas próby skrawania prowadzono pomiary siły skrawania oraz drgań względnych narzędzia i przedmiotu obrabianego. Uzyskane sygnały użyto do estymacji współczynników liniowych i nieliniowych modeli siły skrawania. Do estymacji dynamicznego modelu siły skrawania przyjęto metodę sił resztkowych. Słowa kluczowe: siła skrawania, model nieliniowy, identyfikacja 1. Introduction Numerous research works investigate the influence of vibrations on the cutting forces [1-3]. Frequently a linear model is assumed. Such an assumption simplifies further analysis, for instance stability analysis can be performed using very efficient frequency domain methods. Also surface roughness and waviness prediction is significantly simplified. An important question, however, arises Address: Prof. Stefan BERCZYŃSKI, Bartosz POWAŁKA, D.Sc., Eng., Mirosław PAJOR, D.Sc., Eng., Szczecin University of Technology, Institute of Manufacturing Engineering, Al. Piastów 19, Szczecin, , Bartosz.Powalka@zut.edu.pl

2 18 B. Powałka, M. Pajor, S. Berczyński whether linear approximation of the dynamic force and chip thickness is justified. Cutting force may be determined using theory of orthogonal cutting. This theory assumes a thin shear plane. According to the orthogonal cutting theory [4] the cutting force consists of tangential and radial components: f ( βa αr ( + cos Ft = ah τs sinφc cos φc βa αr F ( βa αr ( + sin = ah τs sinφc cos φc βa αr Ff βa = α r + arctan, µ a = tan βa F t (1 where: µ a average Coulomb friction coefficient, β a friction angle, τ shear plane stress. rake angle, φ c shear angle, s α r Specific cutting pressure in tangential and radial directions is defined on the basis of equations (1 as: K K tc fc ( βa αr ( + cos = τs N/mm sinφc cos φc βa αr ( βa αr ( + sin = τs N/mm sinφc cos φc βa αr 2 2 (2 However defined specific cutting pressure coefficients may be used for the calculation of cutting forces, but such an analytical prediction is not very accurate. This is mostly due to the shear plane assumption. Work hardening that occurs in the thick shear plane causes an underestimation of the cutting forces predicted on the basis of yield shear stress obtained from the tensile test. Also, the temperature contributes to the hardening in the shear plane. Temperature variability depends strongly on chip thickness. This relationship, however, is difficult to be established analytically. Influence of the chip thickness on the shear and friction angles and stresses in the shear plane is expressed using simplified relationship:

3 Identification of nonlinear K K t f = K h T F p = K h q (3 Formulas (3 are basic nonlinearities in the cutting process. These formulas are frequently linearized and the components of the cutting force are expressed using mechanistic approach [5] as: F = K ah + K a t tc te F = K ah + K a f fc fe (4 Coefficients K, K are referred to as edge constants and represent te plowing/rubbing action of the cutting edge. fe Tangential force Ft, N Uncut chip area, mm 2 Fig. 1. Comparison of the linear and nonlinear models identified from the orthogonal cutting tests Coefficients K, K contribute to the shearing action of the cutting edge. tc fc These four coefficients are usually determined experimentally by means of cutting tests using wide range of cutting parameters. Figure 1 shows cutting forces obtained experimentally and forces synthesized using nonlinear (3 and linear (4 models. The differences between both experimentally identified models are not significant. Linear relationship is much more convenient than the nonlinear formula. It may be easily applied to the linear chatter stability analysis which, contrary to the nonlinear stability analysis, allows for the efficient

4 20 B. Powałka, M. Pajor, S. Berczyński generation of the stability lobe diagram. Also, prediction of the surface roughness using linear models is much simpler and efficient than using nonlinear models. Tangential and radial components of the cutting force may be expressed then as linear functions of vibration displacements: ( ( F = K ah + K a + K ax t t tc te tc ( ( F = K ah + K a + K ax t f fc fe fc (5 Where: h is a chip thickness and x( t are vibration displacements. Figure 1 compares nonlinear model (3, linear model (5 and experimental forces measured using experimental stand shown in Fig. 3. Coefficients of both models were estimated using least-squares approach. The goodness of fit is comparable for both investigated models. 2. Experimental stand The relationships (5 are valid as long as outer modulation of the chip thickness does not occur. In order to investigate the influence of vibrations displacements on the cutting force the experimental stand must be designed to exclude the possibility of occurrence of the regenerative effect. This demand is provided by the workpiece-tool system scheme shown in Fig. 2. Outer modulation Inner modulation Fig. 2. Scenario of the cutting test required to eliminate the regenerative effect In such cutting scenario only vibrations in X 1 direction influence the uncut chip area. Vibrations in X 2 have no effect on uncut chip area whereas vibrations

5 Identification of nonlinear in X 2 have a negligible effect on the uncut chip area. It must be noted that vibrations along X 1 axis may have some effect on the effective rake and relief angles and in consequences on the cutting force. This effect, however, is not significant for limited amplitudes of vibrations (<0.1 mm. The experimental scheme which is capable of providing cutting scenario shown in the scheme is presented in Fig. 3. The workpiece is a threaded shaft. Feed per tooth is set to be equal to the thread pitch. The tool holder is attached to the hydraulic exciter by the ball coupling to provide major vibration component along X 1 axis. Cutting forces are measured by calibrated strain gauges glued to the cutting tool. Vibrations of the tool excited by the hydraulic exciter are measured by means of inductive sensor. Inductive sensors are also used to measure vibrations of the workpiece. Fig. 3. Experimental stand for dynamic cutting force investigations Relative workpiece-tool vibrations x( t are calculated as: ( ( ( x t = x t x t (6 p During the test the nominal (mean value of the chip thickness was 0.1 mm. The amplitude of tool vibrations was 0.01 mm and the frequency was continuously varying in the range 40 to 120 Hz. Figure 4 shows fragment of the

6 22 B. Powałka, M. Pajor, S. Berczyński measured cutting force component F ( x1 t and relative tool-part (T-P vibrations. Mean values were subtracted from both signals to focus investigations on the relationship between the dynamic component of the cutting force and the dynamic component of the chip thickness (Fig Relative T-P vibrations, [mm] Dynamic component of of the the tangential force, [N] N Time, [s] s Time, [s] s Fig. 4. Experimental (a relative tool-part vibrations and dynamic component of the tangential cutting force (b Dynamic component of the tangential force, N Frequency, Hz Time, s Fig. 5. STFT of the dynamic component of the tangential cutting force

7 Identification of nonlinear Identification of nonlinear dynamic cutting force model Dynamic component of the cutting force may be influenced not only by vibrations displacements but also velocity. Linear term associated with the vibrations velocity is referred to as process damping. The cutting process damping leads to the increase of stability at low speed cutting operations. Dynamic cutting force can be expressed as a general function of vibrations: where: General function f ( x x& a i and ( d Fx1 = f x&, x& (7, can be modeled as a polynomial of n-th order: (, n1 n 2 f x x & = a x + b x & (8 i i= 1 j= 0 b j are polynomial coefficients that relate dynamic cutting force to displacements and velocities respectively. Linear terms a 1 and b 1 of expression (8 are specific cutting pressure and process damping. Such an approach is adopted from the restoring force method which is frequently used for the identification of nonlinear structural models. The coefficients of the polynomial are estimated using least squares approach. The goodness of fit is quantified using standard deviation of the error defined as: ( d x1, j ε = F f x x& (9 Figure 6 shows standard deviation of the error (9 for varying order of the displacement polynomial and for linear velocity term. It may be concluded from the Fig. 6 that model orders exceeding 2 do not result in significant improvement of the fit. Linear velocity term turned out to describe the process damping well enough. Figure 7 shows a comparison of the experimental force and the synthesized force for n1 = 2 and n2 = 1 displacement and velocity model orders. 4. Conclusions Identified cutting force model exhibit a weak nonlinearity with respect to vibrations displacements, i.e. quadratic term contributes only 5% to the dynamic cutting force at vibration amplitude of 0.03 mm. Also the process damping

8 24 B. Powałka, M. Pajor, S. Berczyński 7.7 Error Error standraddeviation, standard deviation N [N] Model order Fig. 6. Standard deviation of the error as a function of model order Dynamic component of the tangential force, N Velocity, mm/s Displacement, mm Fig. 7. Comparison of experimental force and synthesized force (n1 = 2, n2 = 1 (linear term that relates vibration velocity to the dynamic cutting force contributes less to the dynamic cutting force than specific cutting pressure. However its influence may vary for various cutting speeds. The cutting tests have been performed at single cutting speed and therefore the influence of the cutting speed on the process damping could not be observed. It is, however, expected that at higher cutting speeds the process damping may decrease due to the reduced friction at the flank face [2].

9 Identification of nonlinear Acknowledgement The authors are pleased to acknowledge the financial support of the State Committee for Scientific Research (grant 4 T07B References [1] W. WALLACE, C. ANDREW: Machining forces: some effects of tool vibration. Journal of Mechanical Engineering Science, 7(1965, [2] Y. ALTINTAS, M. EYNIAN, H. ONOZUKA: Identification of dynamic cutting force coefficients and chatter stability with process damping. Annals of the CIRP, 57(2008, [3] T. HOSHI: Cutting dynamics associated with vibration normal to cut surface. Annals of the CIRP, 21(1972, [4] M.E. MERCHANT: Mechanics of the metal cutting process. Plasticity conditions in orthogonal cutting. Journal of Applied Mechanics, 16(1945, [5] F. KOENIGSBERGER, A.J.P. SABBERWAL: An investigation of the cutting force pulsations during the milling process. Inter. Journal of Machine Tool Design and Research, 1(1961, Received in August 2009