Open Access Journal Journal of Sustainable Research in Engineering Vol. 3 (1) 2016, 1-7 Journal homepage:

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1 Open Access Journal Journal of Sustainable Research in Engineering Vol. 3 (1) 2016, 1-7 Journal homepage: A Development Quad-Band Fractal of a Dynamic Antenna Cutting with Defected Force Model Ground for Plane prediction Structure of chatter for S and vibration C-Bands in Applications pocket milling operations Edwin Kimani Miring u 1 *, Kibet Langat 1 and Dominic B. Onyango Konditi 2 Lucy W. Kariuki 1, Prof. Eng. Bernard W. Ikua 1 and Prof. George N. Nyakoe 1 1 Department of Telecommunication and Information Engineering, Jomo Kenyatta University of Agriculture and 1 Department Technology of Mechatronic (JKUAT), Engineering, P. O. Box JKUAT, P.O. Box Nairobi, , Kenya Nairobi. 2 Department of Electrical and Corresponding Electronic Engineering, Author - Technical lwanja@jkuat.ac.ke University of Kenya (TUK), P. O. Box Nairobi, Kenya Abstract The machining dynamics *Corresponding of a cutting process Author have - a direct miringu4edwin@gmail.com influence on the stability of the process, surface quality of the machined part, efficiency, and even life of the cutting tool. The modelling of the dynamic milling process is thus becoming increasingly important in order to determine the conditions under which machining productivity will be maximised without Abstract resulting This paper in instability proposes of a quad-band the process. fractal In this patch work, antenna mathematical with a defected modelsground for predicting plane structure static and for dynamic S and C-bands cutting forces applications. in pocket The milling designed were antenna developed consists in MATLAB of a radiating R andpatch verified with experimentally. fractals of circular In the shape dynamic and a cutting defected force ground model, plane the instantaneous structure (DGPS). undeformed The fractal chipgenerating thicknessprocess was modelled is done through to include an the iterative dynamic process modulations consisting caused of three bysteps. the tool Simulated vibrations results so that of antenna the dynamic parameters regeneration such as effect return which loss, leads gain, tovoltage chatter is Standing taken into Wave account. Ratio This (VSWR) model and wasradiation found topattern be more have accurate been inpresented. predictingthe cutting incorporation forces than of the the commonly fractals in used the patch static models. improves the defected ground plane structure (DGPS) design by Agenerating chatter stability an additional prediction fourth criterion band of is resonance. proposedthus, in which the proposed a force antenna ratio defined resonates as the at four ratiofrequencies maximum (quad-band), dynamic cutting which force are to maximum GHz, staticghz, cutting force GHz is employed and as GHz. an indicator The four of frequencies chatter occurrence. of resonance This can ratio be has used a for limit S value and C-bands beyond which applications the machining such as telecommunication, process becomes unstable commercial and and is satellite dependent applications. the cutter and workpiece materials. Experiments were carried out using an 8mm HSS helical cutter and Aluminium workpiece and the limiting value of force ratio was The Keywords experimental C-band, results Defected wereground in good plane agreement structure, with Fractal the patch prediction antenna, model. Quad-band, S-band. Keywords Chatter vibration,milling dynamics,modelling,cutting force low cost, light weight, ease of fabrication and 1. Introduction conformability to mounting surfaces [2]. In spite of these 1. The Introduction electromagnetic spectrum is very broad ranging advantages, Chatter is aconventional self-excited vibration micro-strip with antennas negative effects have from 3 Hz to 300 GHz. Of great interest is the range of disadvantages such as inability to radiate efficiently over Metal machining is the most dominant and most important operation in the manufacturing industry [1]. Among including poor surface quality, dimensional inaccuracy, frequencies from 300 khz to 140 GHz, which is further and multiple disproportionate frequencies tool of resonance, wear [3]. In which pocket in milling, turn limits the all divided the different into numerous metal frequency machining bands. processes, In the milling lower end is tool the paths band are of frequencies characterisedover by many which changes they can in direction operate one of of this therange most is widely the used medium due to frequency its flexibility (MF) inband producing ranging a from wide 300 range khz of products. 3 MHz, With whereas the competitive- at the upper Also, The due rapid to development the sometimes of frequent modern tool mobile retractions and wireless there and satisfactorily. velocity of the tool as the tool encounters corners. ness end of is the manufacturing F-band which industries ranges from today, 90 it is 140 a continuing GHz. The is communication bound to be systems variancehas in thus cutting necessitated forces, which the design may challenge S-band ranges to produce from 2 products 4 GHz of whereas the highest the possible C-band lead to chatter. Hence the need to study and predict its of micro-strip antennas with multiband capability [3]. In quality, ranges from with 4 the 8 least GHz. amount Each ofthese time bands atis aused low for cost. a occurrence. this scheme, a single micro-strip antenna is designed to In variety the machining of applications, industry, depending cost-effectiveness on the and particular product frequency. quality can For each be achieved of these through applications, optimization different of types the research alleviate but the the problem regeneration where conventional theory suggested patch by antennas Tobias operate Predicting at many its frequency occurrencebands is still (multi-band) the subjectand of helps much machining of antennas processes are employed [2]. [1]. [4] operate is still at a the single most frequency comprehensive band. explanation for it. A Since most decisions on CNC machining parameters wavy surface left by a previous tooth during milling and Micro-strip antennas are examples of antennas which Recently, various methods have been adopted to are made based on the intuition and experience of the removed by the successive tooth may result in the chip operator, have found theseincreased processesuse are in often these carried electromagnetic out at conditions frequency thatbands are not due to optimal. their inherent This results advantages in products such as vibrations. plane structure The(DGPS) forces on and the fractal toolgeometries. in that case increase thickness achieve multi-band growing, which operation in turn such results as defected in increasing ground of compromised surface quality, longer than necessary and may break the tool and produce a poor finish. The machining times and shortened tool life. One of the main8 dynamic cutting force acting on the cutter and workpiece cause of these challenges is chatter vibration. significantly determines the occurrence of self-excited 1

2 JSRE L. Kariuki et al., A Dynamic Cutting Force Model for Prediction of Chatter Vibrations df t,j (φ, z) = [K t,f h j (φ j (z)) + K te ]dz df r,j (φ, z) = [K r,f h j (φ j (z)) + K re ]dz (1) df a,j (φ, z) = [K a,f h j (φ j (z)) + K ae ]dz where dz is the height of the differential element and h j is the static chip thickness for the jth element at rotation angle φ. This chip thickness for a rigid system is given by Fig. 1. Principal forces on cutting edge in milling chatter vibration during cutting process [5]. The use of stability lobe diagram (SLD) is the most common method of chatter prediction. An SLD contains a series of intersected scallop-shaped borderlines showing the boundary between chatter-free machining operations and unstable processes, in terms of axial depth of cut and spindle speed. The process of developing stability lobes involves a lot of lengthy procedures and computations and also requires the user to have a deep understanding of machining dynamics. The dynamic milling process and chatter vibration are usually modelled in the time domain or in the frequency domain. Analysis done in the frequency domain enables identification of chatter-free cutting conditions such as depth of cut and spindle speed. With this method however, the amplitudes and profiles of the cutting force and the vibration cannot be observed. The non-linearities of the dynamic cutting process are also not taken into account. Time domain modeling enables prediction of milling forces, torque, power, dimensional surface finish and the amplitudes and frequency of vibration during a milling operation, but a clear chatter stability criterion in this method is still lacking. This paper presents the development of static and dynamic cutting force models. A simple chatter stability criterion that is based on the ratio of maximum dynamic cutting force to maximum static cutting force is proposed. This is verified experimentally and the results presented in latter sections. 2. Static and Dynamic Cutting Force Modelling h j (φ, z) = [f t sin φ j (z)]g(φ j ) (2) where f t is feed per tooth. The total cutting force is calculated by summing up the elemental forces for the entire part that is engaged with the workpiece. Cutting force is calculated based on the undeformed chip thickness, cutting conditions and specific cutting coefficients [6]. K t, K r and K a are model coefficients in tangential, radial and axial directions for the specific cutter-workpiece combination. These coefficients are determined from experiments approximating orthogonal cutting as done by Githu [7]. The cutter pitch angle φ p for a cutter with n t teeth is given by φ p = (2π)/n t. (3) In the dynamic force model, a predictive time domain model is presented for the simulation and analysis of the dynamic cutting process and chatter in milling. The instantaneous undeformed chip thickness is modelled to include the dynamic modulations caused by the tool vibrations so that the dynamic regeneration effect is taken into account as described by Altintas [8]. In this machining theory, the dynamic milling system is modelled as shown in Fig. 2 adopted from Altintas [8]. The governing equations are In the static force model, each tooth of a helical end milling cutter is discretised into a number of elements along the cutter axis. In milling, the principal forces acting on the cutting edge are Tangential(F t,j ), Radial(F r,j ) and Axial(F a,j ) forces. These are shown in Fig. 1. Axial force is parallel to the tool axis. Forces acting on a differential element of height dz in the axial direction, that is, df t,j, df r,j and df a,j are expressed as 2 Fig. 2. Dynamic milling model

3 JSRE Journal of Sustainable Research in Engineering Vol. 3 (1), 2016 m x ẍ + c x ẋ + k x x = m y ÿ + c y ẏ + k y y = F xj (φ j ) = F x (t) F yj (φ j ) = F y (t) (4) where m is the mass, c is the damping constant and k is the spring constant. This model was implemented in MATLAB R and a numerical method was employed to solve the differential equations governing the dynamic system. The model for the dynamic regenerative chip thickness adapted from Altintas [8] is illustrated in Fig. 3. F xj = F tj cos(φ j ) F rj sin(φ j ) F yj = F tj sin(φ j ) F rj cos(φ j ) (7) where F tj = K t ah(φ j ) and F rj = K r F t j. The total dynamic milling forces for all teeth are given by F x = F y = F xj (φ j ) F yj (φ j ) (8) where φ j is the rotational position of tooth j and φ j = φ + jφ p and φ p = 2π N. φ varies with time and φ = Ωt where Ω is the spindle angular speed in rad/s. 3. Chatter prediction criterion This work proposes that the ratio of the predicted maximum dynamic cutting force, max(f d ), to the predicted maximum static cutting force, max(f s ), be used as a criterion for the chatter stability, given as Fig. 3. Model for dynamic regenerative chip thickness The static undeformed chip thickness is determined from the feed per tooth as h = f t sin(phi j ), (5) As explained earlier, in the dynamic milling process, the undeformed chip thickness is comprised of two parts, a static part, h, and a dynamic component caused by the present tooth period(inner modulation)v j, and previous tooth period(outer modulation)v j o [9]. The static part does not contribute to the dynamic chip load, hence the dynamic undeformed chip thickness for tooth j at an angular position φ j is calculated using the following equation. h d (φ j ) = [ x sin(φ j ) + y cos(φ j )]g(φ j ), (6) where x = x x 0 and y = y y 0. (x, y) and (x 0, y 0 ) are the dynamic displacements at the present and previous tooth periods respectively. g(φ j )is a unit step function that determines whether the tooth is in cut or out of cut. The immersion angle varies with time, φ(t) = Ωt where Ω is the angular speed of the spindle. The tooth passing frequency ω = n t Ω and the tooth period T = 2π ω. The dynamic cutting forces on tooth j in the x and y directions are obtained as ρ = max(f d) max(f s ) This force ratio, ρ, is used to determine whether there is chatter or not. F d is the cutting force predicted using dynamic cutting force model which takes into account the regenerative effect on the undeformed chip thickness and F s is the cutting force predicted using the static cutting force model whereby the system is assumed to be rigid. A set of experiments and signal analysis were carried out in order to set the value of ρ. Since the measured cutting force is in agreement with modelled dynamic cutting force, the force ratio allows chatter prediction before actual milling is done. This method of chatter prediction is less costly in terms of time and computational work than the frequently used stability lobe diagrams. The MATLAB R model is designed based on three main blocks namely; milling forces, milling kinematics and system dynamics. The model inputs include: 1) Tool geometry parameters: number of teeth, tool diameter, helix angle, tool overhang length 2) Cutting parameters: spindle speed, feed per tooth, depth of cut, cutting coefficients 3) Dynamic parameters: natural frequency, mass, stiffness and damping coefficient for each mode of vibration 4) Simulation parameters: number of cycles, iterations per cycle and number of axial layers. To simulate the forces, the model inputs were as listed in Table 1. (9) 3

4 JSRE L. Kariuki et al., A Dynamic Cutting Force Model for Prediction of Chatter Vibrations Table 1. Table of model inputs predicted results. Input parameter Value Milling operation Up milling Axial depth(a) 2mm Nominal depth of cut(b) 2.4mm Feedrate(f) 0.02mm/tooth Spindle speed(n) 450rpm Helix angle 30 Number of teeth(n t) 4 Cutter diameter(d) 8mm 4. Experimental verification The experimental work in this study included the verification of the static and dynamic cutting force models and also the chatter stability criterion applied. The dynamic parameters of the end mill were also determined from modal tests by measuring the response of the system at the spindle and at the workpiece to an impact force. Pocket milling was done on a Fanuc Series Oi-MD machining center using 1-flute and 4-flute helical end mills with a diameter of 8mm.The cutting tool material was High Speed Steel (HSS) while the workpiece material was Aluminium alloy The measurement and data acquisition system comprised of a 3-axis piezoelectric dynamometer, two accelerometers model AC-102, three charge amplifiers, an NI-DAQ card USB 6008, LabVIEW software and the relevant connecting cables. Accelerometers were mounted on the workpiece in such a way as to capture vibration signals in the x and y directions, using threaded studs which provided a firm mounting. A schematic of this setup is shown in Fig. 4 and a photograph of the actual experimental setup in Fig. 5. The presence of chatter vibration was detected by FFT analysis of vibration measurements from accelerometer data. The accelerometers used give a reading of 100.6mV for an acceleration of 1g, which is equal to 9.8 ms 2. A digital low-pass filter was applied to the data from accelerometers to minimize high frequency noise signal which was observed. The readings were converted to ms 2 through LabVIEW R panel. The data processing was done in MATLAB R. 5. Results and Discussion Fig. 6 shows the waveforms of instantaneous cutting forces from measured and predicted forces in x and y directions. It can be seen from this waveforms that the predicted forces are generally in good agreement with experimental results. The dynamic model is found to be more accurate than the frequently used static model. The average percentage error when predicting using static model was found to be 16.86% while the average percentage error when using dynamic force model was 7.98%, thus the dynamic model is more reliable. The variations of cutting forces with feedrate and depth of cut is shown in Figs. 7 and 8 respectively. It can be seen that there is minimal deviation of experimental results from the Fig. 7. Fig. 8. Influence of feedrate on cutting force Influence of depth of cut on cutting force The parallel spiral tool path illustrated in Fig. 9 was used to machine the pocket. The tool path is derived parallel to the boundary of the pocket profile. The tool retractions are few hence minimizing machining time. It also allowed maintaining an up-cut throughout the process. Fig. 9. Parallel spiral tool path Fig. 10 shows the measured cutting force waveforms in x and y directions during pocket milling. Chatter vibration was not clearly evident in the cutting force waveforms, since the workpiece material is soft. Therefore, the vibration signals from the accelerometers were analysed in the frequency domain. In the frequency 4

5 JSRE Journal of Sustainable Research in Engineering Vol. 3 (1), 2016 Fig. 4. Schematic of experimental setup Fig. 5. Photograph of the actual setup Fig. 6. Force waveforms in x and y directions 5

6 JSRE L. Kariuki et al., A Dynamic Cutting Force Model for Prediction of Chatter Vibrations Fig. 10. Force waveform(f = 0.04, a = 1mm, n = 450rpm, parallel spiral tool path) Fig. 11. Plot of force ratios and respective frequency spectra spectrum, chatter is present when there are large peaks near the natural frequency of the system. A stable cut only has peaks due to natural frequency and tooth passing frequency in the frequency spectrum. Tests with this kind of spectrum are plotted as in Fig. 11(a). In the frequency spectrum for an unstable cut, plotted as, other frequency peaks are observed within the range of 20-27Hz, which are the vibration frequencies. This can be seen in Fig. 11(d). Test for which there are other small peaks which cannot be concluded to be chatter, such as in Fig. 11(c) are plotted as. This analysis was done for a set of experiments. To compare experimental occurrence of chatter vibration with model prediction, the ratio of maximum dynamic cutting force to maximum static cutting force for each set of parameters was established. Fig. 10a shows a plot of the data from 20 tests obtained when setting the value of force ratio,ρ, which was found to be This means that when a machinist selects milling parameters and applies in the chatter prediction model, the ratio of maximum 6

7 JSRE Journal of Sustainable Research in Engineering Vol. 3 (1), 2016 dynamic cutting force to maximum static cutting force is evaluated. For the cutter-workpiece combination used in this experiment, if the ratio is equal to or above then chatter vibration is expected to occur. Hence the machinist can change the parameters as desired. This force ratio was successfully applied to predict, and thus avoid, occurrence of chatter vibrations. 6. Conclusion This paper has proposed the use of force ratio to predict chatter vibration occurrence in pocket milling. This was tested on Aluminium 7075-A workpiece using HSS end mills. It was also shown that by using the proposed chatter stability criterion which is a force ratio of for this cutter-workpiece combination, the occurrence of chatter vibration could be predicted. It is recommended that the model be tested with other cutterworkpiece combinations. The suitability of integrating this model in CAD/CAM software also needs to be explored. Acknowledgement This work is supported by Jomo Kenyatta University of Science and Technology (JKUAT). We are grateful to National Youth Service-Engineering Institute (NYS- EI) for allowing access to their machining laboratory for experimental work. References [1] D. M. Sukru, Mechanics and Dynamics of Serrated End Mills. PhD thesis, jun [2] B. W. Ikua, H. Tanaka, F. Obata, S. Sakamoto, T. Kishi, and T. Ishii, Prediction of cutting forces and machining error in ball end milling of curved surfaces -II experimental verification, Precision Engineering, vol. 26, no. 1, pp , [3] G. Quintana and J. Ciurana, Chatter in machining processes: A review, International Journal of Machine Tools and Manufacture, vol. 51, pp , may [4] S. Fishwick and T. W., A theory of Regenerative Chatter, The Engineer-London, [5] M. Held, On the Computational Geometry of Pocket Machining. Springer Science+Business Media, [6] O. Omar, T. El-Wardany, E. Ng, and M. Elbestawi, An improved cutting force and surface topography prediction model in end milling, International Journal of Machine Tools and Manufacture, vol. 47, pp , jun [7] J. Githu, Optimization of machining process for freeform surfaces using an intelligent adaptive controller. PhD thesis, Jomo Kenyatta University of Agriculture and Technology, [8] Y. Altintas, Manufacturing Automation. Cambridge University Press, [9] S. Y. Liang and M. Shih, Albert J.Tools, Analysis of Machining and Machine Tools. Springer,