Modeling And Analysis Of Milling Process Dynamics With Time-varying Delay Effect

Machine tools are machines that make machines,and are used in almost all manufacturing sectors such as aerospace,rail and locomotives,automotive and marine,construction machinery,agricultural machinery,medical equipment,electronic communication equipment and consumer goods processing.Therefore,the machine tool is the key equipment to enhance the competitiveness of the manufacturing industry.Since neither the machine tool nor the workpiece is a rigid body,they will vibrate during the cutting process.The vibrations will cause the machine tool and the workpiece to deviate from their prescribed positional relationship,resulting in an error between the machining result and the required p art shape,and leaving vibration patterns on the machined surface,reducing the accuracy and smoothness of the machined part.Moreover,if the machining parameters,such as spindle speed or the cutting depth,are not set properly,the vibrations of the cutting may become unstable,leading to chatter,which may cause the vibration and the cutting force to increase sharply,accelerate tool wear and breakage,sometimes even damage to the machine and part in a very short time.Therefore,in order to avoid chatter in practice,it is often to reduce the spindle speed and cutting depth at the expense of limiting the performance of the machine tool.It is very important to study the cutting dynamics in depth,because not only can the machine tool manufacturers apply the principles of cutting process dynamics to analyze different design schemes when developing high-performance machine tools,but also the machine tool customers can profit from the knowledge of cutting process dynamics in the fierce competitions,by optimizing the machining parameters to obtain the most economical machining accuracy and efficiency.Calculating the milling force is an important interm.ediate step in solving the dynamic equation of h elical end milling.Usually,the integral of the cutting force of helical tool is approximated as follows:the tool is first divided into element disks,and then the elementary forces on each element disk are calculated as a corresponding straight teeth tool in turn,and finally the total cutting force is approximated by summing up all the elementary forces.This approximation approach was not only is time-consuming but also causes errors.An improved calculation method for the cutting force of the helical milling cutter is presented.An analytical expression for the linear milling force model is given.With this expression,the calculation process of the milling force can.be simplified,and there is no error caused by the axial discretization of the helical tool,so the solution efficiency and accuracy are improved.The semi-discrete method is an elegant and powerful method for analy zing the stability of delay differential systems.Based on the continuous Runge-Kutta numerical method,an improved semi-discrete algorithm,the continuous Runge-Kutta semidiscrete method,is proposed.By using the proposed iterative formulas,the precision and efficiency of milling stability analy sis can be greatly improved without changing the original semi-discretization algorithm structure,Using the dynamic relationship between the undeformed milling chip thickness and the milling force to studying the dynamic characteristics of the milling process is a fundamental approach in modeling milling process.In order to more accurately analyze the influence of dynamic milling force on the machined surface and machining stability,two new approaches are proposed to determine the chip thickness based on the true tool trajectory.The first approach converts the transcendental equation into an ordinary differential equation,then solving it numerically without recursive root-finding algorithms;Another approach simplifies the tool traj ectory transcendental equation by replacing the sine function with a liner function,and then solve it analytically.Then,A Scheme for completely replacing the traditional"circular"chip thickness model with the above "cycloidal" chip thickness model and method for the analysis of dynamical milling processes u sing is also presented.