Modelling machine tool dynamics using a distributed parameter tool holder joint interface

Transcription

1 International Journal of Machine Tools & Manufacture 47 (27) Moelling machine tool ynamics using a istribute parameter tool holer joint interface Keivan Ahmai, Hami Ahmaian School of Mechanical Engineering, Iran University of Science an Technology, Narmak, Tehran 6844, Iran Receive 7 October 26; receive in revise form 5 March 27; accepte 3 March 27 Available online 9 March 27 Abstract Increasing prouctivity in machining process emans high material removal rate in stable cutting conitions an epens strongly on ynamic properties of machine tool structure. Combine analytical experimental proceures base on receptance coupling substructure analysis (RCSA) are employe to etermine the stability of machine operating conitions at ifferent tool configurations. The RCSA employs holer spinle experimental mobility measurements in conjunction with an analytical moel for the tool to preict the ynamics of ifferent sets of tool an holer spinle combinations without the nee for repeate mobility measurements. In this paper an alternative approach using the concept of tool on resilient support is aopte to preict the machine tool ynamics in various tool configurations. In the propose moel the tool, represente by an analytical moel, is partly resting on a resilient support provie by the holer spinle assembly. The support ynamic flexibility is measure by performing vibration tests on the holer spinle assembly. Tool holer joint interface characteristics are inclue in the moel by consiering a istribute elastic interface layer between the holer spinle an the tool shank part. The istribute interface layer takes into account the change in normal contact pressure along the joint interface an comparing with the lumpe joint moel use in RCSA it allows more etaile representation of the joint interface flexibility an amping which have crucial roles in machine ynamics. Experiments are conucte to emonstrate the efficiency of propose moel in preiction of milling operation ynamics an it is shown that the moel is capable of accurately preicting the ynamic absorber effect of spinle in a tool tuning practice. r 27 Elsevier Lt. All rights reserve. Keywors: Chatter vibration; Machining ynamics; Stability lobes; Tool tuning. Introuction Machine tool regenerative chatter is the main obstacle in high-spee machining []; it prouces unstable vibration of tool ue to the feeback between subsequent cuts, an leas to unwante effects such as poor imensional accuracy an tool or work piece amage. Regenerative chatter suppression is mainly performe by choosing proper epth of cut in each cutting spee. Knowlege of machine tool frequency responses is a primary requirement in etermining the stable cutting conitions. Frequency responses of machine tool structure, commonly obtaine by experimental mobility measurements, have been use by many researchers to etermine the stable cutting conition Corresponing author. Tel.: ; fax: aress: ahmaian@iust.ac.ir (H. Ahmaian). [2 7]. The process of stability analysis becomes expensive an time consuming when experimental measurements are neee for each iniviual combination of spinle holer an tool configurations. This creates a eman for preictive moels that are capable of etermining the ynamics of ifferent tool/spinle/holer configurations without the nee for vibration measurement repetitions. For slener tools where the system ynamics is mostly ominate by the tool moes, flexible tool an rigi holer moels are commonly use [8 ]. For tools with consierable lateral stiffness the ynamics of machining process is strongly epenent on spinle behavior. Schmitz et al. [2,3] treate the tool holer spinle assembly as two separate substructures, i.e. the tool an the holer spinle, an employe receptance coupling substructure analysis (RCSA) to preict the tool tip frequency responses. They preicte tool frequency responses by /$ - see front matter r 27 Elsevier Lt. All rights reserve. oi:.6/j.ijmachtools

2 K. Ahmai, H. Ahmaian / International Journal of Machine Tools & Manufacture 47 (27) combining the experimentally measure holer spinle frequency responses an the frequency responses of the tool which are obtaine from an analytical moel. Employing RCSA enables one to preict the changes in the tool tip frequency responses ue to any variation in the tool configuration, an removes the nee for repeate frequency response measurements in practices such as tool tuning. The RCSA requires the response of the substructures at all egrees of freeom of the joint interface. Direct measurement of moments an rotational eformations at the joint interface is impractical. Due to the ifficulties involve in measurement of rotational motions an moments, Schmitz et al. [2,3] consiere only translational motion of the spinle substructure. In practice tool bens at the holer joint interface an ignoring the rotational flexibilities in moelling prouces inaccurate preictions of machine ynamics. Park et al. [4] propose algorithms for etermination of rotational responses of tool/holer joint interface using two ajacent translational vibration experiments. Movahhey et al. [5] propose an equivalent lumpe joint moel between the tool an holer replacing the common translational rotational spring sets by two parallel linear springs, avoiing rotational receptance measurements in RCSA. Duncan an Schmitz [6] use RCSA to preict the ynamics of tool/holer/spinle assembly an inclue the rotational flexibility of spinle assuming that each spinle moe at the joint interface can be approximate locally using an equivalent clampe free Euler Bernoulli beam. They ientifie the moal parameters of spinle an base on the obtaine information geometrical specifications of the equivalent beam with similar moal properties are use to preict the rotational ynamics of the support. Later Schmitz et al. [7 9] employe the concept of multi-point coupling to provie more accurate moels of the joint interfaces stiffness an amping. More recently, Erturk et al. [2,2] propose an analytical metho that uses Timoshenko beam theory to calculate the tool point response in a given combination by using RCSA. They use the moel in examining the effects of iniviual bearing an contact parameters on tool tip response an the effects of spinle, holer an tool parameters on chatter stability. In this paper the tool ynamics is moelle consiering the tool inserte shank is resting on a resilient support. The support, provie by the spinle holer, is represente by a ampe-elastic founation capable of simulating the ominant translational an rotational eformations of spinle in each iniviual frequency. The support receptance functions are measure using a set of mobility experiments on spinle holer assembly an are irectly employe in the analytical solution to encounter ynamic properties of the support. The joint interface between the tool an the holer is moelle using an elastic interface layer. Introuction of this layer enables one to take into account the variation in contact stiffness ue to tool changes, interface contact pressure istribution, etc. The interface stiffness is assume to be a complex value function to inclue the joint interface amping effects. The interface stiffness can be efine as a function along the tool inserte shank length. This enables the analyst to introuce the contact stiffness in more etail taking into account variations of normal pressure between the tool an holer. The stiffness an amping parameters of the tool holer joint interface represente using a istribute elastic layer is ientifie by performing an impact test on the tool holer spinle assembly. These parameters are ientifie by minimizing the ifference between the measure an calculate receptance curves of the assembly. An analytical solution is evelope for ynamic response of the assembly an using the obtaine solution its ynamic behavior an stability are investigate in ifferent machining conitions. The remaining of the paper runs as follows. In Section 2 the propose moel consisting of a tool partly resting on a flexible support is introuce an its frequency response at the tool tip is etermine using an analytical approach. Experimental proceures are conucte in Section 3; holer spinle frequency responses are measure using impact hammer test an are use to calculate the support ynamic stiffness. Also in this section the tool holer interface characteristics are ientifie by comparison of measure an calculate receptance curves at the tool tip. In Section 4 the presente moel is use to preict the milling ynamics an to fin the optimum tool length for a stable cutting conition. A tool tuning practice is performe using the presente moel an its preictions are valiate by experimentally measure responses. Finally, some conclusions are mae in Section Moel evelopment In evelopment of a ynamic moel for machining operations, tool is moelle using continuous beam theory partly resting on a resilient support; the support resembles spinle/holer ynamic effects. In this moel, shown in Fig., tool is represente using a step beam with two sections, namely inserte shank part an the overhung portion. A tool with more complicate geometry can be represente using a beam moel with variable crosssections or a finite element moel. The joint interface between the inserte shank portion of the tool an the holer is represente by a zero thickness elastic layer. The ynamics of the tool inserte shank part is consiere as an Euler Bernoulli beam resting on an elastic support an is efine using the following governing equation: q 4 U ðx; tþ EI qx 4 pxpl, þ m q 2 U ðx; tþ qt 2 ¼ KðxÞ½vðx; tþ U ðx; tþš, where U ðx; tþ is lateral isplacement of the tool inserte shank, vðx; tþ is the lateral isplacement of the tool holer, E is Young moulus of the tool material, L, I ; m are, ðþ

3 98 K. Ahmai, H. Ahmaian / International Journal of Machine Tools & Manufacture 47 (27) compare to the viscous amping moel commonly aopte in other machine tool ynamic moels. The ynamics of overhung portion of the tool is also efine using Euler Bernoulli beam theory as q 4 U 2 ðx; tþ q 2 U 2 ðx; tþ EI 2 qx 4 þ m 2 qt 2 ¼ ; L pxpl, (3) where U 2 ðx; tþ is lateral isplacement of the tool overhung portion, an L L,I 2 ; m 2 are the length, secon moment of inertia an mass per unit length of overhung portions of the tool. The propose moel escribe in Eqs. () an (3) etermines the tool ynamics uner ifferent configurations an loaings. In tool ynamic analysis one requires the tool tip frequency response function. This function can be obtaine by calculating the tip response to a unit harmonic excitation applie at the same location. The bounary conitions at the tool tip in the presence of unit harmonic excitation are unit harmonic shear force an zero moment; this is expresse as q 3 U 2 ðl; tþ EI 2 qx 3 ¼ e iot, ð4aþ q 2 U 2 ðl; tþ qx 2 ¼. ð4bþ The bounary conitions on the other en of the tool are zero shear force an moment, i.e.: Fig.. Schematic view of tool holer spinle assembly an its representative moel. respectively, the length, secon moment of inertia an mass per unit length of the tool inserte shank an KðxÞ is the elastic interface layer stiffness coefficient. Stiffness of the layer is assume to vary along the tool holer joint interface. The elastic layer stiffness coefficient is a function of normal pressure between holer an the tool inserte shank, the contact surfaces martial, the surface finish, etc. It is assume that there is no slip between these two parts uring machining operations an stiffness coefficient remains constant. Schmitz et al. [8,9] an Hanna et al. [22] have obtaine typical functions for the normal pressure istribution between tool, holer an spinle through numerical simulations. The stiffness of elastic interface layer is proportional to the normal pressure, therefore a parametric stiffness istribution base on the joint type (shrink fit, etc.) is selecte an the stiffness parameters can be ientifie by matching experimentally recore responses. These coefficients are assume to be complex value to inclue a isplacement-epenent energy issipation mechanism in the joint interface i.e.: KðxÞ ¼ kðxþð þ izþ, (2) where kðxþ an Z are the joint interface stiffness an its structural amping factor. The selecte amping mechanism is consistent with physics involve in the joint an prouces a more representative moel of amping q 2 U ð; tþ qx 2 ¼, ð4cþ q 3 U ð; tþ qx 3 ¼. ð4þ The tool is moelle using a step beam with two sections. The compatibility requirements for the solutions of these two parts at the interface are continuity of isplacements, slopes, moments an shear forces. These requirements set the following conitions: U ðl ; tþ U 2 ðl ; tþ ¼, ð5aþ qu ðl ; tþ qu 2ðL ; tþ ¼, ð5bþ qx qx q 2 U ðl ; tþ q 2 U 2 ðl ; tþ EI qx 2 EI 2 qx 2 ¼, ð5cþ q 3 U ðl ; tþ q 3 U 2 ðl ; tþ EI qx 3 EI 2 qx 3 ¼. ð5þ Steay-state solutions for the escribe linear system expresse using the partial ifferential Eqs. (), (3) are of the following form: U ðx; tþ ¼FðxÞe iot, ð6þ U 2 ðx; tþ ¼CðxÞe iot, ð7þ where the tool eforme shapes FðxÞ an CðxÞ are complex value functions ue to non-proportional amping nature of the system. These eforme shapes are obtaine by satisfying the bounary conitions (4) an compatibility requirements (5).

4 The holer spinle assembly motion in the inserte shank region vðx; tþ acts as a forcing function in governing Equation () an uner unit harmonic excitation it can be expresse using translational isplacements of two neighboring points on the tool support with the istant, shown in Fig., as vðx; tþ ¼ v 2 þ L x ðv v 2 Þ e iot ; oxol, (8) where v an v 2 are the eformation amplitues at points an 2 on the tool support ue to single harmonic excitations at tool tip. The complex frequency-epenent eformation amplitues v an v 2 are in fact cross-frequency responses between these points an the tool tip. Alternatively, one may express eformations v an v 2 in terms of effective forces applie to points an 2 from tool inserte shank: ( ) ( ) v f ¼½GŠ. (9) v 2 f 2 In the above force-isplacement relations the 2 2 symmetric frequency-epenent matrix ½GŠ contains G ij ðoþ, i; j ¼ ; 2, the measure irect an cross-frequency response functions at points an 2, an f ; f 2 are the effective applie forces on the holer ue to movement of tool shank part. The istribute force applie on the holer KðxÞðFðxÞ vðxþþ is replace by the concentrate forces f i, i ¼ ; 2, that prouce equivalent shear an bening effects on the holer spinle assembly: Z L f 2 ¼ L x KðxÞðFðxÞ vðxþþ x, f ¼ Z L L x KðxÞðFðxÞ vðxþþ x. ðþ An explicit relationship between v ; v 2 an tool inserte shank part motion FðxÞ is efine using Eqs. (9) () that is: ( ) 8 R 9 v < L ¼ðI v 22 þ½gš½jšþ ½GŠ ðl x ÞKðxÞFðxÞ x = 2 : ð L x ÞKðxÞFðxÞ x ;, where ½JŠ ¼ Z L 2 KðxÞ4 L x 2 L x L x L x 2 L x L x () 3 5 x. Employing the frequency responses obtaine from measurement irectly into the analysis introuces the true nature of mass, stiffness an amping effects of the K. Ahmai, H. Ahmaian / International Journal of Machine Tools & Manufacture 47 (27) a n ¼ K n 6 þ K n 5L v þ K n 6 holer spinle assembly. Having obtaine the holer motion ue to tool tip excitation, we may solve Eq. () an obtain the response of tool inserte shank part. Response of the tool inserte shank moelle as a beam on elastic founation with variable support stiffness is expresse using a power series [23]. One may introuce the interface layer stiffness function in a polynomial form: KðxÞ ¼ XP p¼ K p x p. (3) Such a polynomial can be viewe as Taylor series expansion of the actual interface stiffness function. Corresponingly, the eforme shape for the tool inserte shank is assume in the following power series format: FðxÞ ¼ XN a n x n, (4) where N is the power series orer, efine by the analyst, epening on the esire accuracy, an a n, n ¼ ; 2;...; N, are complex coefficients. As will be shown only four of these coefficients are inepenent an the rest of the coefficients are expresse as a function of these four inepenent parameters. Relations between the coefficients of inserte shank part of the tool solution are obtaine by substituting the power series solution into the governing equation (). This leas to the following relations: X N EI ðn Þðn 2Þðn 3Þðn 4Þa n x n 5 þ XP p¼ K p x p mo 2! X N a n x n x L ¼ KðxÞ v 2 þðv 2 v Þ. ð5þ Eq. (5) must be true for all values of x; this leas to the following recursive relations: a 5 ¼ K L v þ K L v 2 ðk m o 2 Þa, 24EI a 6 ¼ K L K 2EI v þ K þ K L ðk m o 2 Þa 2 K a, (2) þ K n 5 L v 2 ðk m o 2 Þa n 4 Pn 5 K j a n 4 j j¼ ; n46. ðn Þðn 2Þðn 3Þðn 4ÞEI v 2 (6) The polynomial coefficients, a n, 5pnpN, are linearly epenent on a ; a 2 ; a 3 ; a 4, v an v 2 through recursive relations (6). These recursive relations can be written in

5 92 K. Ahmai, H. Ahmaian / International Journal of Machine Tools & Manufacture 47 (27) matrix notation as 8 9 >< >: a a 2.. a N >; ( ) v ¼½T v Š N2 v 2 8 >< þ½t a Š N4 >: a a 2 a 3 a 4 9. (7) >; Eq. () can be further simplifie an a irect relation between v, v 2 an coefficients a n, n ¼ ;...; N, is obtaine as 8 9 ( ) v >< a 2 ¼ðI 22 þ½gš½jšþ ½GŠ½RŠ.. (8) v 2 >: >; Components of matrices ½T v Š, ½T a Š an ½RŠ are erive in Appenix A an I is an ientity matrix. Eqs. (7) an (8) form a set of matrix equations from which the coefficients of the assume polynomial a n,pnpn, can be obtaine as follows: >< >: a a 2.. a N >< ¼½SŠ N4 >; >: a a 2 a 3 a 4, >; ½SŠ ¼ðI NN ½T v ŠðI 22 þ½gš½jšþ ½GŠ½RŠÞ ½T a Š. a a N ð9þ Having obtaine a parametric solution for the tool inserte shank part in terms ofa n, n ¼ ;...; 4, we turn our attention to the solution for the overhung part of the tool. Deforme shape of the tool overhung portion can be obtaine by substituting the solution form efine in Eq. (7) into the governing equation for this portion of the tool expresse in Eq. (3). This results in the following orinary ifferential equation in terms of the function CðxÞ: C IV ðxþ l 4 CðxÞ ¼; l 4 ¼ m 2o 2. (2) EI 2 Solution to the above orinary ifferential equation is of the following form with the complex coefficients C i, i ¼ ;...; 4: CðxÞ ¼C e ilx þ C 2 e ilx þ C 3 e lx þ C 4 e lx ; L pxpl. (2) There are eight inepenent complex value coefficients namely C i, a i, i ¼ ;...; 4, which nee to be specifie in orer to etermine the tool response. They are specifie by satisfying the bounary conitions (4) an the compatibility requirements (5). The given bounary conitions an compatibility requirements form a set of eight linear equations in terms of these eight inepenent coefficients an in general may be written in the following form: C. >< C 4. ><. ½ZðoÞŠ ¼., (22) a.... >: >; >: >; a 4 where ½ZðoÞŠ is a full rank square matrix (its entries are given in Appenix A) an entries of right-han sie vector are all zero except for the first entry which is one, ue to the fact that only Eq. (4a) is non-homogenous an the rest of bounary conitions (4b) (4) an compatibility requirements (5) are homogenous. The unknown coefficients vector can be etermine by inversion of matrix ½ZðoÞŠ at any require frequency. Obtaining the solution for tool ynamics, one may express the tool tip frequency response function which is necessary in constructing the stability lobes as G T ðoþ ¼C e ill þ C 2 e ill þ C 3 e ll þ C 4 e ll, l 4 ¼ m 2o 2. ð23þ EI 2 The obtaine moel preicts the machining ynamics in ifferent tool configurations. The mobility measurements are performe once to obtain the support flexibility functions G ij ðoþ, i; j ¼ ; 2, an to ientify the joint interface stiffness KðxÞ an as long as the support ynamic flexibility functions an the joint interface stiffness remain constant the moel is capable of preicting the machining ynamics at ifferent tool configurations. Any change in tool configurations such as the change in its length, its iameter, etc. can be implemente in the moel. In the evelope metho, the support ynamic flexibility matrix ½GðoÞŠ an the stiffness an amping properties of elastic layer KðxÞ are use as input information to preict the tool ynamics. The following provies an example on how the measurements are performe to obtain matrix GðoÞ an the complex stiffness KðxÞ. 3. Experimental moel ientifications Experimental setup use to obtain the support ynamic flexibility matrix ½GðoÞŠ an also to ientify the stiffness an amping properties of elastic layer KðxÞ is shown in Fig. 2. In orer to measure the support ynamic flexibility of a vertical three axis milling machine, two measurement points are selecte on the holer. Fig. 3 an the irect an cross-mobility response functions are measure. The system is excite using an instrumente hammer (B&K822) an the responses are measure using a uniaxial piezoelectric accelerometer (B&K4393). The time omain measurements are collecte an transforme into frequency omain via a ual channel signal analyzer

6 K. Ahmai, H. Ahmaian / International Journal of Machine Tools & Manufacture 47 (27) (B&K232). The measure receptance curves are shown in Fig. 4. The coefficients of the joint interface polynomial are ientifie by conucting a separate set of experiments on the tool/holer/spinle assembly. In this experiment an extra long HSS DIN 889/ en mill with geometric properties, tabulate in Table, is mounte using a collettype holer. An effective iameter for the flute portion is use in moelling following Schmitz s proposal [24], assuming that the equivalent cylinrical section in the moel an the actual flute portion have the same mass; the equivalent cylinrical section s iameter is consiere to be 3.4 mm. A low-mass accelerometer is attache to the tool tip as shown in Fig. 2 an irect mobility measurements are performe using an impact test. The tool tip measure frequency response is employe in ientifying the joint interface parameters, i.e. polynomial coefficients of the elastic joint interface layer. The ientification proceure in obtaining the elastic joint interface layer is base on minimizing the ifference between the observe frequency response an the preictions of the moel evelope in Section 2. In this stuy, the joint interface moel is ientifie in two steps: first a zero orer moel is aopte to approximate the joint interface stiffness function, next the orer of function is increase to achieve improve correlation with the observe behavior of the machine tool. 3.. Homogenous joint interface moel Fig. 2. The test setup. Initially, the joint interface stiffness is assume to be homogenous. This correspons to a KðxÞ efine using a zero orer polynomial function. Ientification of such a function is straightforwar an the obtaine function can be use as a starting point in ientifying the stiffness functions with higher orer polynomial coefficients. Using the measure frequency response curves of the support, shown in Fig. 3, an an initial estimate of the joint stiffness parameter the irect mobility at the tool tip is calculate. The tool tip frequency response is obtaine using the proceure efine in Section 2 an the orer of polynomial representing the inserte shank eformation is selecte to be 5. The selection of the orer is base on the fact that increasing it to a higher value oes not have any effect on the tool response. Next an objective function is forme comparing the ifference between measure G m ðoþ an the estimate Fig. 3. Support frequency response measurements, irect (a) an cross (b).

7 922 K. Ahmai, H. Ahmaian / International Journal of Machine Tools & Manufacture 47 (27) Fig. 4. Measure frequency response curves on the support points. Fig D plot of the objective function. Table Geometry of the tool (mm) Total length 23 Shank length 4 Flute iameter 8 Shank iameter 6 G c ðoþ tool tip frequency responses: min k log jg m ðoþ G c ðoþjk 2. (24) K The estimate for the joint stiffness parameter is ajuste in such a way that maximum agreement between measure an calculate frequency response curves is obtaine an the objective function is minimize. A 3-D plot of the objective function versus real an imaginary parts of the K, i.e. stiffness an amping coefficient of the joint interface layer, is shown in Fig. 5.As shown in this figure the error is minimize when the contact parameters in the joint interface are set as k ¼ 3:5 N=m 2, Z ¼ :4. The calculate tool tip frequency response curves using the optimum parameters an corresponing measure values are compare in Fig. 6. These parameters provie a starting point in optimization proceures use in obtaining the higher orer stiffness istribution polynomial coefficients First orer joint interface moel Next a first orer polynomial form is selecte for the joint interface stiffness function KðxÞ. The parameters of Fig. 6. The receptance curves at the tool tip, homogenous joint interface moel (soli line), measurements (ashe line). first orer joint interface moel are obtaine by minimizing the objective function efine in Eq. (24). A nonlinear least square algorithm is use to minimize the objective function by tuning the parameters of joint interface. The zero orer polynomial coefficients obtaine in the previous section are employe as starting point in the optimization algorithm. The polynomial coefficients that minimize the objective

8 K. Ahmai, H. Ahmaian / International Journal of Machine Tools & Manufacture 47 (27) Table 2 The ientifie first orer polynomial coefficients K ðn=m 2 Þ K ðn=m 3 Þ 6:9ð þ :35iÞ 8:25ð þ :35iÞ frequency response require for generating the stability lobes is as follows:. Development of an analytic moel for tool similar to the one presente in Section Measuring the irect an cross-frequency response functions G ij ðoþ, i; j ¼ ; 2, at points an 2 on the holer, shown in Fig. 3, within frequency range of interest. 3. Ientifying the elastic joint interface layer between tool an holer using proceures explaine in Sections 3. an Incorporating the above information into Eqs. (7) (2) to form matrix ½ZðoÞŠ an to etermine the unknown C i, i ¼ ;...; Obtaining the tool tip frequency response from Eq. (23). 6. Evaluating the ynamics of the machining process using the obtaine tool tip frequency response. Fig. 7. The receptance curves at the tool tip, linearly varying joint interface moel (soli line), measurements (ashe line). function are tabulate in Table 2. The ientifie stiffness function shows a ecreasing tren in contact stiffness between the tool an holer as x varies from zero to L which is consistent with our expectation of behavior of such a joint. Increasing the orer of polynomial from zero to one prouces 7% ecrease in the minimum value of the objective function in previous section which inicates further improvement of the moel. Using the obtaine first orer joint interface stiffness function the tool tip frequency response is calculate an is compare with the measure function in Fig. 7. An improvement in the moel preictions especially in the vicinity of tool secon moe at 492 Hz is consierable. This emonstrates the efficiency of the propose moel in preiction of tool ynamics. The stability iagrams are generate using the proceure presente by Buak an Altintas [2,3]. The require frequency responses for this proceure are generate using the evelope moel in Section 3.2. The obtaine stability lobes are shown in Fig. 8. Next a tool tuning practice is performe using the evelope moel; the tool overhung length is varie an its ynamic is investigate at ifferent configurations. The change in critical epth of cut, i.e. the stable epth of cut in a range of cutting spees, is stuie in terms of tool overhung length. The stability ecreasing tren by increasing the tool length is observe in this practice, as shown in Fig. 9. Also the local increasing in stability iagram can be recognize in the figure in two specific tool overhung lengths i.e. 3 an 59 mm. At these two specific lengths the funamental moes of 4. Results an iscussions An accurate moel for preiction of tool ynamics is evelope in previous sections. In the present section the performance of the moel is investigate consiering the tool ynamics in milling operation of an aluminum work piece. A step-by-step proceure to construct the tool tip Fig. 8. The stability lobe iagram (tool overhung length.9 m).

9 924 K. Ahmai, H. Ahmaian / International Journal of Machine Tools & Manufacture 47 (27) Fig. 9. Critical epth of cut versus overhung tool length. Fig.. Measure (ashe) an preicte (soli) receptance curves near the first moe. Fig.. Measure (ashe line) an preicte (soli line) receptance curves; overhung length.59 m. tool overhung part are 428 an 34 Hz, respectively. The local increases in stability are justifie consiering the ynamic absorber effect of spinle moes on tool vibration [2,3]; when the tool ominant moe correspons with one of spinle natural frequencies, stability of the system increases. By inspecting the frequency responses of the holer spinle assembly shown in Fig. 3 one can etect the resonances at 34 an 428 Hz. There are two specific tool lengths within range of interest in which the tool ominant moe of vibration an one of the spinle moes match; the tool length can be tune so that the ynamic vibration absorbing phenomenon comes into effect, hence obtaining the optimum tool overhung length. To valiate the preictions of the moel, the tool length is change to 59 mm an frequency response function is measure at the tool tip. Preicte an measure frequency response curves are shown in Fig. ; there is an excellent agreement between the two responses inicating the accuracy of presente moel in preiction of tool ynamics in ifferent configurations. Fig. compares the tool tip frequency responses at two ifferent tool lengths: 9 an 59 mm; by ecreasing the tool length the first moe of spinle at 34 Hz interacts with the first moe of tool overhung length an prouces a ouble peck in the frequency response plot at this frequency ue to ynamic vibration absorption effect. This experimental case stuy reflects the fact that the evelope moel is capable of accurately preicting the milling ynamics, incluing the ynamic vibration absorber effect ue to spinle moes of vibrations, emonstrate by a tool tuning practice.

10 K. Ahmai, H. Ahmaian / International Journal of Machine Tools & Manufacture 47 (27) Conclusions A new approach in moelling high-spee machining ynamics is presente using the measure ynamic flexibility of the holer spinle assembly an an analytic moel for the tool. This enables the analyst to moel the machining ynamics in various tool combinations without the nee for repeate measurements. The joint between the tool an holer unlike other commonly use moels which employ lumpe stiffness an viscous ashpots is represente using a zero thickness istribute interface layer with variable stiffness. The propose tool holer istribute parameter joint interface takes into account the change in normal stiffness along the tool shank part. This variation is ue to the change in normal pressure along the joint interface. The moel also uses a isplacementepenent amping mechanism to account for structurally ampe characteristics of the interface. The interface layer parameters are ientifie using experimentally observe behavior of the tool holer spinle assembly. An experimental stuy is performe to verify the effectiveness of the propose moel an its capability in preicting the structure ynamics. A physically justifiable linearly varying tool holer interface stiffness function is ientifie from the measure results an the tool ynamics is preicte using the propose moel. The linear variation of the interface stiffness is ue to the mechanism use to join the tool an holer. Employing such an interface layer in moelling significantly improves preictions of machining ynamics. It is shown using an experimental case stuy that the evelope moel is capable of accurately preicting the milling ynamics, incluing the ynamic vibration absorber effect ue to spinle moes of vibrations, emonstrate by a tool tuning practice. The accuracy of moel in preicting the frequency responses of the machine at various tool configurations an its ability to emonstrate the physical phenomena involve in the machine structure such as ynamic vibration absorbing effect are emonstrate in the case stuies. The results reflect the fact that the assumptions mae in constructing the moel are base on vali engineering jugments an the moel can be use to preict the machine ynamics at working conitions beyon those consiere in this stuy. Appenix A In this Appenix the entries of matrices ½T v Š, ½T a Š, ½RŠ an ½ZŠ are erive. Base on the recursive equations (6) the entries of matrices ½T v Š, ½T a Š when the interface stiffness function KðxÞ is assume to be linear take the following forms: 2 3 m o 2 K 24 K m o 2 K ½T a Š¼ 2 2 EI K m o 2 K. (A.) K m o 2 K m o 2 K m o 2 K K

11 926 K. Ahmai, H. Ahmaian / International Journal of Machine Tools & Manufacture 47 (27) The first few rows of matrix ½T a Š are erive an the rest can be obtaine from Eq. (6). In case of matrix ½T v Š entries of the rows 5 7 are non-zero an the rest are zeros: 2 ½T v Š¼ EI K L K ð L =Þ ðk = þ K ð L =ÞÞ. (A.2) 2 K K ðk L = K =Þ 2 Entries of matrix ½RŠ, use in Eqs. (8) an (9), are obtaine by introucing the inserte shank eflection shape, Eq. (4), into the right-han sie of Eq. (): 8 K R L KðxÞ L 9 x >< FðxÞ x KðxÞ L x >: FðxÞ x >; 8 L KðxÞ >< ¼ >: 2 ¼ 6 4 KðxÞ PN a n x n x R L L KðxÞ KðxÞx KðxÞ KðxÞL KðxÞx N a n x n x R L x þ KðxÞx L KðxÞx N KðxÞL x R L KðxÞxN KðxÞL x N KðxÞ a n x n x þ R L x þ KðxÞxN KðxÞx a n x n x KðxÞ L KðxÞx KðxÞx2 9 a n x n x >; x KðxÞL x þ KðxÞx2 x a a >< ><.. 7 ¼½RŠ. ða:3þ 5 x >: >; >: >; a N a N

12 The matrix ½ZŠ introuce in Eq. (22) has the following form: 2 EI 2 il 3 e ill EI 2 il 3 e ill EI 2 l 3 e Ll EI 2 l 3 e ll l 2 e ill l 2 e ill l 2 e Ll l 2 e ll 2S 3 6S 4 e ill e ill e ll e ll L j S j j¼ ile ill ile ill le ll le ll ðj ÞL j 2 S j j¼2 EI 2 l 2 e ill EI 2 l 2 e ill EI 2 l 2 e ll EI 2 l 2 e ll P EI N ðj Þðj 2ÞL j 3 S j j¼3 6 4 EI il 3 e ill EI il 3 e ill EI l 3 e ll EI l 3 e ll P EI N ðj Þðj 2Þðj 3ÞL j 4 j¼2 j¼ j¼4 S j 2S 32 2S 33 2S 34 6S 42 6S 43 6S 44 L j S j2 j¼ L j S j3 j¼ L j S j4 ðj ÞL j 2 P S N j2 ðj ÞL j 2 P S N j3 ðj ÞL j 2 j¼2 j¼2 S j4 P EI N ðj Þðj 2ÞL j 3 P S j2 EI N ðj Þðj 2ÞL j 3 P S j3 EI N ðj Þðj 2ÞL j 3 j¼3 j¼3 j¼3 S j4 P EI N ðj Þðj 2Þðj 3ÞL j 4 P S j2 EI N ðj Þðj 2Þðj 3ÞL j 4 P S j3 EI N ðj Þðj 2Þðj 3ÞL j 4 j¼4 K. Ahmai, H. Ahmaian / International Journal of Machine Tools & Manufacture 47 (27) where S ij are entries of matrix ½SŠ efine in Eq. (9). j¼4 j¼4 S j4 3, ða:4þ 7 5 References [] S.A. Tobias, Machine Tool Vibration, Blackie, Lonon, 965. [2] E. Buak, Y. Altintas, Analytical preiction of chatter stability in milling part I: general formulation, Transactions of the ASME, Journal of Dynamic Systems, Measurement, an Control 2 (998) [3] E. Buak, Y. Altintas, Analytical preiction of chatter stability in milling part II: application of general formulation to common milling systems, Transactions of the ASME, Journal of Dynamic Systems, Measurement, an Control 2 (998) [4] J. Tlusty, Dynamics of high-spee milling, Transactions of the ASME Journal of Engineering for Inustry 8 (986) [5] N.H. Hanna, S.A. Tobias, A theory of nonlinear regenerative chatter, Transactions of the ASME Journal of Engineering for Inustry (974) [6] H.E. Merritt, Theory of self-excite machine-tool chatter, Transactions of the ASME Journal of Engineering for Inustry (965) [7] R. Srihar, R.E. Hohn, G.W. Long, A general formulation of milling process equation, Transactions of the ASME Journal of Engineering for Inustry (968) [8] E.B. Kivanc, E. Buak, Structural moeling of en mills for form error an stability analysis, International Journal of Machine Tools an Manufacture 44 () (24) 5 6. [9] E. Buak, Analytical moels for high performance milling. Part I: cutting forces, structural eformations an tolerance integrity, International Journal of Machine Tools an Manufacture 46 (2 3) (26) [] E. Buak, Analytical moels for high performance milling. Part II: process ynamics an stability, International Journal of Machine Tools an Manufacture 46 (2 3) (26) [] H. Ahmaian, M. Salahshour, Preicting chatter in milling using a beam moel on elastic support, in: Proceeings of the Tehran International Congress on Manufacturing Engineering, Tehran, Iran, December 2 5, 25. [2] T.L. Schmitz, Preicting high-spee machining ynamics by substructure analysis, Annals of the CIRP (2)

13 928 K. Ahmai, H. Ahmaian / International Journal of Machine Tools & Manufacture 47 (27) [3] T.L. Schmitz, M.A. Davies, M.D. Kenney, Tool point frequency response preiction for high-spee machining by RCSA, Transactions of the ASME, Journal of Manufacturing Science an Engineering 23 (2) [4] S.S. Park, Y. Altintas, M. Movahhey, Receptance coupling for en mills, International Journal of Machine Tools an Manufacture 43 (23) [5] M.R. Movahhey, J.M. Gerami, Preiction of spinle ynamics in milling by sub-structure coupling, International Journal of Machine Tools an Manufacture 46 (26) [6] G.S. Duncan, M.F. Tummon, T.L. Schmitz, An investigation of the ynamic absorber effect in high-spee machining, International Journal of Machine Tools an Manufacture 45 (25) [7] T.L. Schmitz, G.S. Duncan, Receptance coupling for ynamic preiction of assemblies with coincient neutral axes, Journal of Soun an Vibration 289 (26) [8] T.L. Schmitz, K. Powell, D. Won, G.S. Duncan, W.G. Sawyer, J.C. Ziegert, Shrink fit tool holer connection stiffness/amping moeling for frequency response preiction in milling, part : receptance moel, in: Proceeings of the secon CIRP-HPC 26, Vancouver, Canaa, 26. [9] T.L. Schmitz, K. Powell, D. Won, G.S. Duncan, W.G. Sawyer, J.C. Ziegert, Shrink fit tool holer connection stiffness/amping moeling for frequency response preiction in milling, part 2: finite element moel, in: Proceeings of the secon CIRP-HPC 26, Vancouver, Canaa, 26. [2] A. Erturk, H.N. Ozguven, E. Buak, Effect analysis of bearing an interface ynamics on tool point FRF for chatter stability in machine tools by using a new analytical moel for spinle tool assemblies, International Journal of Machine Tools an Manufacture 47 (27) [2] A. Erturk, H.N. Ozguven, E. Buak, Analytical moelling of spinle tool ynamics on machine tools using Timoshenko beam moel an receptance coupling for the preiction of tool point FRF, International Journal of Machine Tools an Manufacture 46 (26) [22] I.M. Hanna, J.S. Agapiou, D.A. Stephenson, Moeling the HSK toolholer spinle interface, Transactions of the ASME, Journal of Manufacturing Science an Engineering 24 (22) [23] M. Eisenberger, Vibration frequencies for beams on variable one- an two-parameter elastic founations, Journal of Soun an Vibration 76 (5) (994) [24] T.L. Schmitz, T.J. Burns, J.C. Ziegert, B. Dutterer, W.R. Winfough, Tool length-epenent stability surfaces, Machining Science an Technology 8 (3) (24)