Prediction of Chatter Vibration and Stability Mapping In Cylindrical Turning of AISI 1045 Steel

Introduction Metal cutting process involves continuous removal of material from the work piece in the form of chips. Cutting process with a single point cutting tool like forming on a lathe, the heterogeneity of work piece material, the run-out or misalignment of the work piece may cause occasional disturbances to the cutting process resulting vibration of the work piece with respect to the cutting tool. If the cutting process is stable, the resulting vibration dies out quickly because of damping. However, under certain conditions, the magnitude of the ensuing vibration becomes ever increasing. This phenomenon is termed as chatter. In case of occurrence of chatter, the amplitude of the self-excited vibration increases until nonlinearity limits [1]. Results of chatter are rough surface finish, poor accuracy, shortened tool life and low metal-removal rate. Chatter becomes even more critical when machining materials that are difficult to cut. Some advanced cutting tool materials such as ceramic, silicon nitride and CBN require strict chatter control to prevent brittle breakage [2]. For high precision manufacturing, even mild vibration is undesirable. Furthermore, since modern machining systems, have become more flexible, frequently changing working conditions increase the possibility of bringing machining process into unstable operating regions [3].The productivity of expensive. Machining systems is often limited by chatter. Chatter is defined as self-generative vibrations that occur when the chip width is too great versus dynamic stiffness. This phenomenon leads to a bad surface aspect and high noise level. As it reduces tool life, it increases production costs. For instance, the cost due to chatter is estimated to be around 0.35 h per piece on a cylinder block. With such a cost, prediction of chatter becomes highly necessary and a chatter criterion has to be chosen. First evocations of chatter are due to Taylor in 1907 and then to Schlesinger in 1936. A first comprehensive study was led by Doi in 1937 [2] and then with Kato in 1956 [3]. Tlusty and Polacek published their criterion the next year [4] and Tobias proposed his chatter maps the year after [5]. During the early 1960s, Peters and Vanherck ran some tests and developed measurement techniques in order to discuss Tlusty and Tobias criterions [6]. The 1970s have shown some work on the dynamic parameters. Hanna and Tobias worked on the non-linearity of the stiffness [7] while the Peters and Vanherck team produces highly interesting thesis on the identification of dynamic parameters during the cutting 44 operations [8, 9]. At the end of 1970s, Tusty presented his CIRP keynote paper on the topic [10]. Up to now major developments have been designed for aeronautic industry where tools are mostly more compliant than work pieces. In this way, Altintas and Budak have proposed an analytic method for computing stability lobes corresponding to Tobias’s chatter maps in 1995 [11]. This work has been extended in 1998 [12] by taking the work piece’s behavior into account under the form of compliance-damping systems in two directions. A comprehensive summary of recent developments of the topic has been proposed by Altintas and Weck under the form of a CIRP keynote [13]. 2.0 Definition of Regenerative Chatter During a turning process, the heterogeneity of the work piece material causes variation of cutting forces and hence results in vibration (Lin, 1990). In most cases of practical interest, chatter observed in turning operation is due to the regenerative effect (Rao and Shin, 1998). As the single point cutting tool cuts a surface, the undulations generated in the previous revolution sustain the tool work piece vibration, which is coupled with the cutting force. Some external perturbations or a hard spot in the work piece material causes initial variation in cutting forces and results in vibration of the dynamic system. The vibration leaves a wavy tool path on the work piece surface. This wavy surface will affect subsequent chip thickness as a result variation in cutting force. Because of this uneven chip thickness, the system vibrates. If the magnitude of this vibration does not die out, the system becomes unstable. This phenomenon is known as the regenerative chatter. 3.0 Mathematical Modelling of Chatter Vibration Assume that a flat-faced orthogonal grooving tool is fed perpendicular to the axis of a cylindrical shaft held between the chuck and the tail stock center of a lathe (Fig.1). Fig.1: Turning Model As shown in Fig. 2, the initial surface of the shaft is smooth without waves during the first revolution but the tool starts leaving a wavy surface behind because of bending vibration of the shaft in feed direction ,when the second revolution starts ,the surface have waves in both inside the cut where tool is cutting(inner modulation y(t)and also outside surface of cut owing to vibrations during the previous revolution of cut(outer modulation y(t-T)).Hence the general dynamic chip thickness can be expressed. Fig.2: Regenerative Chatter Dynamics h(t) = h0 − [y(t) − y(t − T)].........(Eq.1) Where, h0 is intended chip thickness or feed rate, y(t) is inner modulation, y(t-T) is outer modulation The equation of motion of the system can be expressed as: [14] my(t) + c ̈ y(t) ̇ + ky(t) = Ff(t) = Kf ah(t) = Kf a[h0 + y(t − T) − y(t)]...........................( Eq.2) Where , F(t) is feed cutting force ,a is width of cut or depth of cut ,h(t) is dynamic chip thicknessThe fundamental equation put in laplas domain and gets a characterists equation 1 + (1 − e)Kf aФ(s) = 0 The root of the characteristic equation is s = σ + jωc .When the real part is zero, the system is critically stable and the work piece oscillates with constant vibration amplitude at chatter frequency. The chatter vibration frequency does not equal to natural frequency, is still close to the natural mode of the structure. For critical borderline stability analysis, the characteristic function becomes {1 + Kf alim[G(1 − cos ωcT) − H sin ωcT]} + J{Kf alim[G sin ωcT + H(1 − cos ωcT)]} = 0.......... (Eq.3) Where alim is the maximum axial depth of cut for chatter vibration-free machining, the critical axial depth of cut can be found by equating the real part of the characteristic equation to zero: 1 + Kf alim[G(1 − cos ωcT) − H sin ωcT] = 0 alim = −1 Kf G [(1 − cos ωcT) − ( H G ) sin ωcT] Substituting and rearranging this equation yields [14] H G = sin ωcT (cos ωcT − 1) and alim = −1 2Kf G(ωc) ... ... ... ... ... ... ... ... ... . (Eq. 4) Where G(ωc) = 1 k (1 − r) [(1 − r2) + (2ζr)2] The excitation to natural frequency ratio r = ω ωn , and ζ is Damping coefficient. The spindle speed and chatter vibration frequency have a relationship that affects on dynamic chip thickness, the no. of vibration waves left on the surface of the work piece is-2πfc T = 2kπ + ε 45 Where, K is integer no. of waves, ε-phase shift between inner and outer modulation, TSpindle revolution period T = 2kπ + ε 2πfc where , N = 60 T ... ... ... (Eq. 5) 4.0 Experimental Investigation Machining tests were carried out by the orthogonal wet turning. Medium carbon steel AISI1045 was cut into 70 cm long test specimens (shafts) with 32 mm outside diameter, performed on All Gear Lathe Machine. The cutting tool was taken as HSS tool. The cutting parameters that are selected for determination of the stability limits are given here. Spindle speeds [110,160,240,400,575 rev/min], the feed rate [0.625, 1.25, 2.5, 5,8mm/rev] depth of cut [0.15, 0.25, 0.35, 0.45, 0.6mm], while these are used for studying the regenerative effects. Instruments used arepiezoelectric Accelerometer, Signal Conditioner, and Analyzer (Picoscope-2202). The intensity of vibration was picked by accelerometer with the current and voltage sensitivity (1±1%) and (1±2%) respectively for Frequency Range (x1, x10 Gain) 0.15 to 100,000 Hz, accelerometer probe is fixed at a point on the tool holder close to cutting point to picked up the vibration frequency of tool in the feed direction, The calculation of frequency was taken using a portable vibration analyzer to investigate the vibration spectrum. Fig. 3: Experimental Set Up Table 1: Dynamic Cutting Coefficients, extracted from dynamic tests kt cutting stiffness(MPa) kf cutting constant (MPa) Dampingcoefficient (c) 5600 985 0.054 5.0 CATIA Model of the Beam The cutting tool assumed as a cantilever beam configuration with a rectangular cross –section and with a point loaded at the end. Beam Specifications are: Length 12.0cm, Width 2.5cm, Height 3.0cm, Material cast iron, Density 7800kg/m3, Young’s modulus 2.1x1011 N/m2 and Poisson’s ratio 0.3. 6.0 FEM Modeling and Modal Analysis After modelling, the cutting tool with CATIA model is exported to ANSYS-V13 environment. We have taken the model with 8721 elements and 1214 nodes and mechanical properties as stated above. Afterwards, boundary conditions on supporting are applied and finally modal analysis has-been done to obtain natural frequencies. Figure 4 and Figure 5 figures show the modal frequencies of the beam Fig. 4: 1st Modal Frequencies of the Beam Fig. 5: 2nd Modal Frequencies of the Beam Table 2: Modal frequencies of the beam 1st 2nd 3rd 4th 1136Hz 1397Hz 5480Hz 7025Hz The values of the above natural frequencies are required to calculate the limit of stability (ωc) –up to this frequency the system is dynamically stable, in different cutting conditions from equation 4&5 stated above. Table 3 Experimented and Simulated Results Seri al no. rp m Feed rate (mm/re v) Depth of cut(m m) Chatter frequen cy (Hz ) Natural frequen cy (Hz ) Max. limit of stabilit y (Hz ) 1 11 0 0.625 0.25 3254 5480 5425.2 2 11 0 1.25 0.25 200