基于摆线刀齿轨迹的未变形铣屑厚度分析
|
摘要
利用动态变化的未变形铣屑厚度与铣削力之间的相互作用关系,研究铣削过程的动态特性是铣削过程建模的基本思路。未变形铣屑厚度的值取决于相继切入工件的刀刃在工件上留下的切削痕迹之间的相对位置关系。处于切削状态的刀刃与前一齿尖所经过的摆线运动轨迹相交,以其方位角与前一刀齿过同一交点时的方位角之差为辅助变量,建立满足铣屑形成条件的运动学超越方程。若该辅助变量已知,则可进一步求出铣削迟滞参数及未变形铣屑厚度。基于这一思路,给出两种新的求取铣屑厚度的方法:一种是将原方程转化为解描述辅助变量动态变化的微分方程,采用数值方法求得其数值解;另一种是以辅助变量的线性函数近似原方程中的三角函数,采用近似解析法解出其显式表达式。仿真结果表明:对于当前制造业普遍采用的铣削参数,所提近似解析法可以满足实际应用的精度要求,并且与现有摆线铣屑厚度模型相比数学表达式更为简洁;所提数值法不需要循环迭代求解超越方程,非常适合嵌入到对分析精度和运算效率有较高要求的铣削过程仿真或稳定性预测算法。
One of the key issues in the modeling of milling processes is to determine the response of the cutting forces to the dynamic variations of the uncut chip thickness. The uncut chip thickness depends on the relative position between the milled surfaces left by the successive cutting teeth of the tool. A cutting tooth intersects the path of its previous tooth, and forms an angle with the previous tooth passing through the same point of intersection. Given the intersection angle, both the variable time delay and the instantaneous uncut chip thickness can be calculated directly. Through the analysis of trochoidal tooth paths and using the intersection angle as an auxiliary variable, a transcendental equation is developed to model of the geometry of chip formation. Then two new approaches are proposed to determine the intersection angle. The first approach converts the transcendental equation into an ordinary differential equation of the intersection angle, then solving it numerically without recursive root-finding algorithms; assuming the intersection angle is infinitesimal, another approach approximates the transcendental equation by replacing the sine function with a liner function, and then solve it analytically. Case studies with different process parameters show that the analytical approach can provide a high accuracy in practical milling operations, with a simpler expression compared with other models. The proposed numerical method is suitable for embedding into the milling process simulation or stability prediction algorithms which are sensitive to accuracy and efficiency.
One of the key issues in the modeling of milling processes is to determine the response of the cutting forces to the dynamic variations of the uncut chip thickness. The uncut chip thickness depends on the relative position between the milled surfaces left by the successive cutting teeth of the tool. A cutting tooth intersects the path of its previous tooth, and forms an angle with the previous tooth passing through the same point of intersection. Given the intersection angle, both the variable time delay and the instantaneous uncut chip thickness can be calculated directly. Through the analysis of trochoidal tooth paths and using the intersection angle as an auxiliary variable, a transcendental equation is developed to model of the geometry of chip formation. Then two new approaches are proposed to determine the intersection angle. The first approach converts the transcendental equation into an ordinary differential equation of the intersection angle, then solving it numerically without recursive root-finding algorithms; assuming the intersection angle is infinitesimal, another approach approximates the transcendental equation by replacing the sine function with a liner function, and then solve it analytically. Case studies with different process parameters show that the analytical approach can provide a high accuracy in practical milling operations, with a simpler expression compared with other models. The proposed numerical method is suitable for embedding into the milling process simulation or stability prediction algorithms which are sensitive to accuracy and efficiency.
引文
[1]MARTELLOTTI M E.An analysis of the milling process[J].Transactions of the ASME,1941,63(8):677-700.
[2]SRIDHAR R,HOHN R E,LONG G W.A General formulation of the milling process equation:Contribution to machine tool chatter research-5[J].Journal of Engineering for Industry,1968,90(2):317-324.
[3]SABBERWAL A J P.Cutting forces in down milling[J].International Journal of Machine Tool Design and Research,1962,2(1):27-41.
[4]TLUSTY J,ISMAIL F.Basic non-linearity in machining chatter[J].CIRP Annals,1981,30(1):299-304.
[5]SMITH S,TLUSTY J.An overview of modeling and simulation of the milling process[J].Journal of Engineering for Industry,1991,113(2):169-175.
[6]BUDAK E,ALTINTAS Y,ARMAREGO E J A.Prediction of milling force coefficients from orthogonal cutting data[J].Journal of Manufacturing Science&Engineering,1996,118(2):216-224.
[7]BUDAK E.Analytical models for high performance milling.Part I:Cutting forces,structural deformations and tolerance integrity[J].International Journal of Machine Tools and Manufacture,2006,46(12):1478-1488.
[8]MONTGOMERY D,ALTINTAS Y.Mechanism of cutting force and surface generation in dynamic milling[J].Journal of Engineering for Industry,1991,113(2):160-168.
[9]ALTINTAS Y,MONTGOMERY D.Dynamic peripheral milling of flexible structures[J].Journal of Engineering for Industry,1992,114(2):137-145.
[10]CAMPOMANES M L,ALTINTAS Y.An improved time domain simulation for dynamic milling at small radial immersions[J].Journal of Manufacturing Science and Engineering,2003,125(3):416-422.
[11]SCHMITZ T L,COUEY J,MARSH E,et al.Runout effects in milling:Surface finish,surface location error,and stability[J].International Journal of Machine Tools and Manufacture,2007,47(5):841-851.
[12]LI H Z,LI X P.A numerical study of the effects of cutter runout on milling process geometry based on true tooth trajectory[J].The International Journal of Advanced Manufacturing Technology,2005,25(5-6):435-443.
[13]LI H Z,LIU K,LI X P.A new method for determining the undeformed chip thickness in milling[J].Journal of Materials Processing Technology,2001,113(1):378-384.
[14]KUMANCHIK L M,SCHMITZ T L.Improved analytical chip thickness model for milling[J].Precision Engineering,2007,31(3):317-324.
[15]闫雪,陶华,蔡晋,等.基于真实刀刃轨迹的立铣刀切削厚度模型[J].机械工程学报,2011,47(1):182-186.YAN Xue,TAO Hua,CAI Jin,et al.Model of the instantaneous undeformed chip thickness in milling based on real tooth trajectory[J].Journal of Mechanical Engineering,2011,47(1):182-186.
[16]LONG X H,BALACHANDRAN B.Milling model with variable time delay[C]//ASME 2004 International Mechanical Engineering Congress and Exposition.American Society of Mechanical Engineers,2004:933-940.
[17]LONG X H,BALACHANDRAN B,MANN B P.Dynamics of milling processes with variable time delays[J].Nonlinear Dynamics,2007,47(1):49-63.
[18]LONG X,INSPERGER T,BALACHANDRAN B.Systems with periodic coefficients and periodically varying delays:Semidiscretization-based stability analysis[M]//Delay Differential Equations.New York:Springer,2009.
[19]FAASSEN R,VAN DE WOUW N,NIJMEIJER H,et al.An improved tool path model including periodic delay for chatter prediction in milling[J].Journal of Computational and Nonlinear Dynamics,2007,2(2):167-179.
[20]SONG G,LI J,SUN J.Approach for modeling accurate undeformed chip thickness in milling operation[J].The International Journal of Advanced Manufacturing Technology,2013,68(5-8):1429-1439.
[21]ZATARAIN M,MUNOA J,PEIGNéG,et al.Analysis of the influence of mill helix angle on chatter stability[J].CIRP Annals-Manufacturing Technology,2006,55(1):365-368.
[2]SRIDHAR R,HOHN R E,LONG G W.A General formulation of the milling process equation:Contribution to machine tool chatter research-5[J].Journal of Engineering for Industry,1968,90(2):317-324.
[3]SABBERWAL A J P.Cutting forces in down milling[J].International Journal of Machine Tool Design and Research,1962,2(1):27-41.
[4]TLUSTY J,ISMAIL F.Basic non-linearity in machining chatter[J].CIRP Annals,1981,30(1):299-304.
[5]SMITH S,TLUSTY J.An overview of modeling and simulation of the milling process[J].Journal of Engineering for Industry,1991,113(2):169-175.
[6]BUDAK E,ALTINTAS Y,ARMAREGO E J A.Prediction of milling force coefficients from orthogonal cutting data[J].Journal of Manufacturing Science&Engineering,1996,118(2):216-224.
[7]BUDAK E.Analytical models for high performance milling.Part I:Cutting forces,structural deformations and tolerance integrity[J].International Journal of Machine Tools and Manufacture,2006,46(12):1478-1488.
[8]MONTGOMERY D,ALTINTAS Y.Mechanism of cutting force and surface generation in dynamic milling[J].Journal of Engineering for Industry,1991,113(2):160-168.
[9]ALTINTAS Y,MONTGOMERY D.Dynamic peripheral milling of flexible structures[J].Journal of Engineering for Industry,1992,114(2):137-145.
[10]CAMPOMANES M L,ALTINTAS Y.An improved time domain simulation for dynamic milling at small radial immersions[J].Journal of Manufacturing Science and Engineering,2003,125(3):416-422.
[11]SCHMITZ T L,COUEY J,MARSH E,et al.Runout effects in milling:Surface finish,surface location error,and stability[J].International Journal of Machine Tools and Manufacture,2007,47(5):841-851.
[12]LI H Z,LI X P.A numerical study of the effects of cutter runout on milling process geometry based on true tooth trajectory[J].The International Journal of Advanced Manufacturing Technology,2005,25(5-6):435-443.
[13]LI H Z,LIU K,LI X P.A new method for determining the undeformed chip thickness in milling[J].Journal of Materials Processing Technology,2001,113(1):378-384.
[14]KUMANCHIK L M,SCHMITZ T L.Improved analytical chip thickness model for milling[J].Precision Engineering,2007,31(3):317-324.
[15]闫雪,陶华,蔡晋,等.基于真实刀刃轨迹的立铣刀切削厚度模型[J].机械工程学报,2011,47(1):182-186.YAN Xue,TAO Hua,CAI Jin,et al.Model of the instantaneous undeformed chip thickness in milling based on real tooth trajectory[J].Journal of Mechanical Engineering,2011,47(1):182-186.
[16]LONG X H,BALACHANDRAN B.Milling model with variable time delay[C]//ASME 2004 International Mechanical Engineering Congress and Exposition.American Society of Mechanical Engineers,2004:933-940.
[17]LONG X H,BALACHANDRAN B,MANN B P.Dynamics of milling processes with variable time delays[J].Nonlinear Dynamics,2007,47(1):49-63.
[18]LONG X,INSPERGER T,BALACHANDRAN B.Systems with periodic coefficients and periodically varying delays:Semidiscretization-based stability analysis[M]//Delay Differential Equations.New York:Springer,2009.
[19]FAASSEN R,VAN DE WOUW N,NIJMEIJER H,et al.An improved tool path model including periodic delay for chatter prediction in milling[J].Journal of Computational and Nonlinear Dynamics,2007,2(2):167-179.
[20]SONG G,LI J,SUN J.Approach for modeling accurate undeformed chip thickness in milling operation[J].The International Journal of Advanced Manufacturing Technology,2013,68(5-8):1429-1439.
[21]ZATARAIN M,MUNOA J,PEIGNéG,et al.Analysis of the influence of mill helix angle on chatter stability[J].CIRP Annals-Manufacturing Technology,2006,55(1):365-368.