Study On The Mechanism Of Temperature Rise In Ball Screw Subjected To A Reciprocating-motion Heat Source And Its Application

The development level of high-end manufacturing equipment is one of the most significant symbols to measure a country’s comprehensive national strength and international status.At present,compared with the developed countries,China’s overall level in this field is still in the catch-up stage.Independent research and development of powerful upmarket manufacturing equipment,as well as comprehensive improvement of the processing level,is the only way for us to leap from a manufacturing powerhouse"to a"manufacturing power".As a product of the combination of traditional machining and emerging electronic information technologies,geometric,control and force（deformation）errors during the operation of CNC machine tools have been effectively controlled,which increasingly highlights the impacts of thermal characteristics on machining accuracy.Under the background of development trend towards high speed and high precision,the thermal errors account for up to 75%of the total machining errors in high-end CNC machine tools.“Drive motor+ball screw”is the most commonly adopted transmission form of CNC machine tools.As a core components,thermal-induced deformation of the ball screw is one of the main inducements to result in positioning errors in transmission systems.Currently,scholars at home and abroad are carrying out researches on thermal error modelling and compensation of the machine tool feeding system in the light of static and quasi-static errors,most of which are based on the means of test as well as simulation methods to establish temperature rise-thermal deformation models.However,this kind of empirical models has great limitations for complex working status with strong time-varying operation conditions and the experimental processes are not only time-consuming but also laborious.Meanwhile,the finite element simulation method occupies a large amount of computing sources and the established models are with poor robustness,which is in difficulties to satisfy the design demands of high-speed and high-precision machine tools.Study on the evolution regularities of temperature field distribution and the dynamic features of structural thermal deformation from differential action mechanisms of heat sources in the feed system is being an important means to improve the machining accuracy.Research on the thermal characteristics,juxtaposed with the thermal error modelling methods,has also become a foundational and hot direction of thermal error compensation for the ball screw feeding system.This research was funded by the science and technology support program of Jiangsu Province（BE2014004-3）and the national key R&D Program（2019YFB2006402）.The reciprocating motion characteristics of heat sources has been taken into consideration to study on the temperature evolution mechanism of ball screws subjected to the nut pair heat source.Various relative-scale media are classified according to the motion stroke of heat sources and the diameter dimensions of ball screws.Through reasonable simplifications of the surface thread raceways,the mathematical models for heat transfer problems of infinite/finite cylinders subjected to reciprocating-motion one-dimensional plane/two-dimensional annular line heat sources are established,respectively.The integral solutions to corresponding temperature field distribution are derived,meanwhile,dimensionless control parameters are designed to discuss the influence on the time-average quantity and oscillation amplitude when the surplus temperature researches a stale fluctuating state.The radial distribution rules and evolution process of temperature gradient are revealed from mechanism.The hollow ball screw model with distinct boundary conditions is established and the surrogate model is utilized to reflect the mapping relationship between various structural,boundary condition parameters and the calculation data at temperature sensitive points.The multi-objective collaborative optimization provides guidance for the design of hollow ball screw in practical application.The main research contents and innovations of this work are as follows:1.The integral solution method for heat transfer problems in multi-scale media subjected to a reciprocating heat source.Focus on the reciprocating characteristics of moving heat sources,point heat source is taken as the object to carry out researches on the temperature field evolution rules in various relative-scale media.The mathematical models of three-dimensional heat transfer problems in an infinite/semi infinite medium excited by unidirectional constant velocity heat source is established.The reciprocating-motion feature of heat source decides that its coordinate position periodically changes with time so the integral domain can be segmented and classified according to the summarized change law.The obtained integral solutions are nondimensionalized and the designed dimensionless parameters not only assistant discuss their influence on temperature field evolution regulations but also establish a bridge for interpreting the differences and the connections between heat transfer problems subjected to a reciprocating motion heat source in various relative-scale media.The research results illustrate that the integral solutions to surplus temperature of multidimensional heat conduction problems in both infinite and semi infinite medium presents the similarity in forms.The periodic characteristics of the reciprocating motion point heat source decides that the expression of the integral solutions solved by sections to temperature fields in an infinite medium satisfies the form of generalized incomplete gamma function;Temperature rise curves of various relative-scale media exhibit the similar oscillation characteristics,which performs the periodical oscillation near a constant time-average quantity.The distributions of both surplus temperature values and fluctuation amplitudes along the direction of heat source movement are symmetrical aboutξ=β/4,but their changes are not synchronized;With the adiabatic boundary condition,integral solutions to temperature field in semi-infinite media subjected to a reciprocating point heat source degenerates into the similar expression with that of infinite media,however,heat source intensity doubles the original value.The integral solutions to the temperature field in various relative-scale media can realize the mutual transformation of the expression forms under specific conditions.2.Integral solutions to heat transfer problems in the infinite-length cylinder subjected to a reciprocating heat source.Neglecting the surface raceway of ball screws,straight cylinders are taken as the substituted research object and feasibility of the simplifications is verified.Mathematical model of the heat transfer problem in an infinite-length cylinder subjected to the small stroke reciprocating-motion heat source is established and the corresponding Green’s function is obtained through separation of variables.Combination of different physical quantities can eliminate the dependence of integral solutions to temperature fields on the unit system family,which is beneficial for the unity of expression forms.Temperature field calculation results based on the finite difference method illustrate that the temperature rise of the optical axis ignoring raceways in the non heat source parking area is only 2%different from that of the lead screw.The reasonable omission of threaded raceways can help simplify the calculation;Essence of one-dimensional axial heat conduction in the slender cylinders subjected to a moving heat source is the flat plate problem,and the integral solutions to the temperature field in each integral domain can also be expressed as the unified form of generalized incomplete gamma function;For slender cylinders with a fixed thermal diffusivity,when surplus temperature reaches a stable oscillation state subjected to a constant-intensity reciprocating-motion heat source,the time-average quantity and the fluctuation amplitude are mainly impacted by the dimensionless parameters that control the moving speed and stroke of the heat source and the convection boundary conditions,but the oscillation period is not affected by it;In a certain range,the value ofβandΨwill affect the heat accumulation on the cylinder bodies,resulting in the change of time-average quantity and oscillation amplitude of the surplus temperature.If the values exceeds a reasonable range,they will no longer affect the stable oscillation state characteristics.3.Integral solutions to heat transfer problems in the finite-length cylinder subjected to a reciprocating heat source.The mathematical model of corresponding heat transfer problem in a finite cylinder with adiabatic boundary conditions at both ends is established to obtain integral solutions to the temperature fields subjected to both fixed and periodic reciprocating heat sources.Z directional feeding drive system of a five axis gantry machining center is taken as the research object to measure temperature rise data of the fixed observational point on surface of the lead screw under a specific working condition.Test values are utilized to verify the accuracy of theoretical calculation results.The results show that the Green’s function of heat transfer problems in a finite cylinder performs as an infinite series,and the temperature field integral solutions with adiabatic boundary at both ends need to be respectively discussed and calculated according to if the terms withβ_mbeing zero or not;The motion characteristics of Z-axis feed drive system is a compound movement of the ball screw rotating around its own axis and reciprocally moving with the ram,so the convection heat transfer coefficient of the ball screw surface can only be estimated based on experience;The trend of the modified theoretical temperature rise curve is basically consistent with the test curve.The theoretical calculation results and the test data of the surplus temperature reach a stable fluctuation state at the nearly same time about 3500s,and the error of the temperature rise value is about 5.7%;The smaller estimated value of the surface convection heat transfer coefficient is one of the direct reasons for the higher theoretical calculated temperature rise curve compared with the experimental measured value.At the same time,it will also cause the fluctuation amplitude of the temperature around the average value to be smaller than the measured data when reaching the stable oscillation state.4.Evolution mechanism of the radial temperature rise in cylindrical bodies subjected to a reciprocating heat source.Considered in a cylindrical coordinate system,the mathematical model for heat transfer problems subjected to a reciprocating annular heat source is established and the corresponding integral solutions to temperature fields are derived.For the infinite series,appropriate term quantities m and n included in calculations can help to achieve the reasonable balance between convergence of analytical solutions and calculation efficiency.Based on M=N=50,temperature field calculations of different examples are carried out to study the influence of various heat source moving velocities U and surface convection heat transfer coefficients h on the radial temperature evolution law.The results show that the axisymmetric distribution of radial temperature fields are independent from circumference angleφso that the order of Bessel functionυ=0.Integral calculations of the axial temperature distributions for heat transfer problems in a finite-length thick rod with adiabatic boundary condition at both ends still need to be distinguished according to whether terms including the eigenvalueη_nequal zero or not,respectively;Temperature rise curve of the multi heat source model is very close to that of the single heat source model,and the difference between the average temperature rise when reaching the stable fluctuation state is only 0.5%;Different term quantities of the infinite series included in calculations have great impacts on the theoretical calculation accuracy and efficiency.Too large value of the term quantity included hardly improves the calculation accuracy,but significantly increases the calculation time;Both the moving velocity U of heat source and the surface convective heat transfer coefficient h affect the time-average quantity of the surplus temperature when it reaches the stable oscillation state,but the relationship related to the moving velocity of the heat source is not linear.Too large or too small velocities will result in the decrease of surplus temperature values;The radial temperature gradient of a ball screw is the synergistic action result of various parameters at different rotation rates and the radial heat balance is a evolution process from surface to center of the axis.For high rotation rate conditions,the moving velocity of nut heat source is the main factor affecting the steady-state temperature rise and heat balance rate,while the influence of surface convection heat transfer coefficient is more obvious under low rotation rate conditions;For same working conditions,the temperature rise of the large-diameter lead screw is lower when it reaches the stable state,but at the same time,more heat dissipation and greater radial thermal resistance lead to much more time consumption for the large-diameter lead screw to reach thermal equilibrium in the radial direction.5.Theoretical solutions to temperature field in hollow cylindrical bodies subjected to a reciprocating heat source and the parameter optimization of hollow screws.Fuzzy c-means clustering method is utilized for the temperature sensitive point selection,and temperature rise values at the corresponding positions under different working conditions and structural parameters are calculated to the construct the orthogonal test table for training the RBF proxy model.NSGA-Ⅱ genetic algorithm is adopted to obtain Pareto optimal solution to the specific optimization goal,so as to realize the optimization matching for various structural and boundary parameters of the hollow screw with constraint conditions.The results illustrate that the integral solutions to heat transfer problems of hollow cylinder subjected to reciprocating heat source is similar to these of solid cylinder and main difference lies in the more complex eigenfunctions and modes;Fuzzy c-means clustering method is utilized to reduce the number of temperature variables from 13 to 3,which helps avoid multicollinearity between temperature variables and improve modeling efficiency;Taking moving velocity U of the nut heat source,opening diameter a of the hollow screw and forced heat transfer coefficient h₁as the input,and temperature rise of inner and outer walls of the three temperature sensitive points as the output,a proxy model of the temperature field of the hollow ball screw based on RBF neural network is established.Taking into account the transmission efficiency and temperature rise suppression,moving velocity of nut heat source and the temperature rise of temperature sensitive points are taken as the optimization objectives,and NSGA-Ⅱ genetic algorithm is utilized to obtain Pareto optimal solution.The optimized structural parameters and boundary conditions significantly improve the temperature uniformity of the hollow ball screw,and the temperature rise value at the highest point is 32.3%lower than that of the original ball screw and ball screw,effectively reducing the thermal error in the operation of the ball screw feed system.This study reveals the temperature field evolution laws of heat transfer problems in the ball screw subjected to a reciprocating heat source from the mechanism,further enriches and perfects the theory of moving heat source.At the same time,mastering the characteristics of the thermal characteristics of the ball screw has guiding significance for understanding its thermal error and position,the temperature change law of sensitive points,and establishing the corresponding prediction model,which lays a good foundation for reducing the positioning error of the feeding system through the error compensation function in the future.