针对滚珠丝杠副在高速、重载条件下的发热问题，设计了丝杠和螺母强制冷却结构，探讨了摩擦发热对变形量的影响程度，分析了滚珠丝杠副结构设计及使用条件对温升的影响因素，提供了滚珠丝杠副及冷却系统设计方案，降低了滚珠丝杠副温升对机床的精度影响。

　　Aiming at the heating problem of ball screw pairs under high-speed and heavy load conditions, a forced cooling structure for the screw and nut was designed. The influence of frictional heating on deformation was explored, and the factors affecting the temperature rise of the ball screw pair structure design and usage conditions were analyzed. A design scheme for the ball screw pair and cooling system was provided to reduce the impact of the temperature rise of the ball screw pair on the accuracy of the machine tool.

　　01

　　01

　　序 言

　　Preface

　　随着数控机床向高精、高速方向的发展，对机床的进给部分提出了更高的要求。滚珠丝杠副作为轴向驱动部件，在长时间运行下会产生温升，加之数控机床在运行中其他部分产生大量热量的传导，会使滚珠丝杠副由于温度变化而产生轴向热位移，直接导致数控机床进给轴的定位精度下降[1，2]。因此，需要采取相应的措施来控制温升，保证机床进给系统的定位精度，提高数控机床的加工精度。02

　　With the development of CNC machine tools towards high precision and high speed, higher requirements have been put forward for the feed part of machine tools. As an axial driving component, the ball screw pair will experience temperature rise during long-term operation. In addition, other parts of the CNC machine tool generate a large amount of heat conduction during operation, which will cause axial thermal displacement of the ball screw pair due to temperature changes, directly leading to a decrease in the positioning accuracy of the CNC machine tool feed axis [1,2]. Therefore, corresponding measures need to be taken to control temperature rise, ensure the positioning accuracy of the machine tool feed system, and improve the machining accuracy of CNC machine tools. 02

　　滚珠丝杠副冷却结构设计

　　Design of cooling structure for ball screw pair

　　滚珠丝杠副在高速、重载、连续运行的工况条件下，滚珠在螺母和丝杠之间高速滚动和滑动，产生大量摩擦热，随着热量增大，丝杠轴向和径向发生不均匀的热膨胀和热变形，严重影响到机床定位精度、预紧力波动，加速润滑脂失效，还会对滚珠和滚道造成热损伤。为提高机床的运行精度和精度保持性，降低温升对滚珠丝杠副精度的影响，针对应用领域及产品外部安装空间，采用滚珠丝杠副强制冷却的方式降低温升对滚珠丝杠副精度和性能的影响。空心滚珠丝杠副可分为丝杠和螺母内部强制冷却两种方式[3]，丝杠内部强制冷却采用单通道循环（见图1）和双通道循环，将丝杠设计为切削液循环通道，冷却介质在通孔循环流动，在入口和出口端增加专用旋转接头和密封装置，直接将丝杠的热量带走。螺母内部强制冷是在螺母壁设计若干冷却通道，通过优化冷却流道设计，使整体温度场更加均匀，极大地减少热梯度变形。该结构从根本上解决了精密传动系统性能与热变形问题，有效控制了丝杠在高速、高负荷运行时的温升，从而提升其精度、刚度和寿命。

　　Under high-speed, heavy load, and continuous operation conditions, the ball screw pair rolls and slides at high speed between the nut and the screw, generating a large amount of frictional heat. As the heat increases, the screw undergoes uneven thermal expansion and deformation in the axial and radial directions, seriously affecting the positioning accuracy and preload fluctuation of the machine tool, accelerating grease failure, and causing thermal damage to the ball and raceway. To improve the operational accuracy and precision retention of the machine tool, and reduce the impact of temperature rise on the accuracy of the ball screw pair, a forced cooling method is adopted for the ball screw pair to reduce the impact of temperature rise on the accuracy and performance of the ball screw pair in the application field and external installation space of the product. Hollow ball screw pairs can be divided into two methods: forced cooling inside the screw and nut [3]. Forced cooling inside the screw adopts single channel circulation (see Figure 1) and dual channel circulation. The screw is designed as a cutting fluid circulation channel, and the cooling medium circulates through the through-hole. Special rotating joints and sealing devices are added at the inlet and outlet ends to directly take away the heat of the screw. Forced cooling inside the nut is achieved by designing several cooling channels on the nut wall, optimizing the cooling channel design to make the overall temperature field more uniform and greatly reduce thermal gradient deformation. This structure fundamentally solves the performance and thermal deformation problems of precision transmission systems, effectively controlling the temperature rise of the screw during high-speed and high load operation, thereby improving its accuracy, stiffness, and lifespan.

　　图1　单通道滚珠丝杠副冷却循环结构

　　Figure 1 Cooling cycle structure of single channel ball screw pair

　　完整的空心冷却滚珠丝杠副系统主要包括：带有内部冷却通道的空心丝杠和专用螺母、连接静止冷却管路和旋转丝杠内部通道的旋转接头、提供恒定温度和压力的冷却单元等。空心丝杠结构减轻了质量，提高了临界转速，但为防止高速旋转时振动，加工需尽量保证丝杠和螺母壁厚均匀。由于丝杠空心，所以降低了丝杠抗弯和抗扭刚度，此问题可通过优化孔径和材料来弥补，采用特殊的热处理方式提高系统刚性。由于旋转接头长期与丝杠外圆高速摩擦，所以提高接触外圆的耐磨性和可靠密封是保证系统稳定运行的前提。将冷却系统与机床的数控系统联动，根据负载和速度实时调节冷却流量和温度，可实现智能热补偿。03

　　The complete hollow cooling ball screw system mainly includes: a hollow screw with internal cooling channels and dedicated nuts, a rotary joint connecting the static cooling pipeline and the internal channels of the rotating screw, a cooling unit providing constant temperature and pressure, etc. The hollow screw structure reduces mass and increases critical speed, but in order to prevent vibration during high-speed rotation, it is necessary to ensure that the wall thickness of the screw and nut is as uniform as possible during processing. Due to the hollow shape of the screw, the bending and torsional stiffness of the screw are reduced. This problem can be compensated for by optimizing the aperture and material, and using a special heat treatment method to improve the system rigidity. Due to the long-term high-speed friction between the rotating joint and the outer circle of the screw, improving the wear resistance and reliable sealing of the contact outer circle is a prerequisite for ensuring the stable operation of the system. Linking the cooling system with the CNC system of the machine tool, adjusting the cooling flow rate and temperature in real-time based on load and speed, can achieve intelligent thermal compensation. 03

　　滚珠丝杠副运转过程热量分析

　　Thermal analysis of the operation process of ball screw pairs

　　3.1 滚珠丝杠副温升分析空心滚珠丝杠副及冷却系统中循环通道孔径、数量、流量和压力等设计直接影响温升的高低和产品性能，而温升计算是设计中的关键环节，将直接影响产品的工作状态、胶合失效、定位精度以及传动效率。温升设计的本质是建立一个热平衡模型，使单位时间内摩擦产生热量等于系统向环境散发的热量。

　　3.1 Analysis of Temperature Rise in Ball Screw Pair Hollow ball screw pairs and cooling systems are directly affected by the design of circulation channel aperture, quantity, flow rate, and pressure, which directly affect the temperature rise and product performance. Temperature rise calculation is a key link in the design, which will directly affect the working state, adhesive failure, positioning accuracy, and transmission efficiency of the product. The essence of temperature rise design is to establish a thermal equilibrium model that ensures that the heat generated by friction per unit time is equal to the heat dissipated by the system to the environment.

　　滚珠丝杠副运转主要是滚动摩擦和滑移摩擦，建立热平衡方程[4]，求解温升ΔT

　　The operation of ball screw pairs mainly involves rolling friction and sliding friction. Establish a thermal equilibrium equation [4] and solve the temperature rise Δ T

　　ΔT＝Frv[1－exp(－αAt/Q)] /JαA? ? ? ? ? ? （1）

　　ΔT＝Frv[1－exp(－αAt/Q)] /JαA?  ? ?  ? ?  ? （1）

　　式中，ΔT是温升值（℃）；Fr是切线方向摩擦阻力（N）；v是丝杠线速度（m/s）；J是热功当量，一般为1；α是丝杠表面热传导率[W/（m2·℃）]；A是散热面积（m2）；Q是丝杠热容量（J/℃）；t是工作时间（s）。

　　In the formula, Δ T is the temperature rise value (℃); Fr is the tangential frictional resistance (N); V is the linear velocity of the screw (m/s); J is the thermal work equivalent, usually 1; α is the surface thermal conductivity of the screw [W/(m2 ·℃)]; A is the heat dissipation area (m2); Q is the heat capacity of the screw (J/℃); T is the working time (s).

　　温度饱和值与线速度v成正比，与热传导率α成反比。当工作时间t较长时，式（1）中v[1－exp（－αAt/Q）]的值趋近于零，此时ΔT接近于温升饱和值。

　　The temperature saturation value is directly proportional to the linear velocity v and inversely proportional to the thermal conductivity α. When the working time t is long, the value of v [1-exp (- α At/Q)] in equation (1) approaches zero, and Δ T approaches the temperature rise saturation value.

　　在丝杠转动过程中引起空气轴向流动时，丝杠和螺母表面的热传导率α计算如下

　　When air flows axially during the rotation of the screw, the thermal conductivity α on the surface of the screw and nut is calculated as follows

　　α＝cλ（vd/f）n/d ? ? ? ? ? ? ? ? ? ??（2）

　　α＝cλ（vd/f）n/d ?  ? ?  ? ?  ? ?  ? ?  ??（2）

　　式中，?d是丝杠外径（ m ） ；?v?是空气的流速（m/s）；λ为（t2＋t1）/2时的导热系数[W/（m2·℃）]，根据空气热物理性质确定；f为（t1＋t2）/2时的空气运动黏度（m2/s），根据空气热物理性质确定；t1是丝杠外表面温度（℃）；t2是空气温度（℃）；c、n是系数，c为经验常数，n值通常取0.5。

　　In the formula,? D is the outer diameter of the screw (m);? V? Is the velocity of air flow (m/s); The thermal conductivity [W/(m2 ·℃)] when λ is (t2+t1)/2, determined based on the thermophysical properties of air; The air motion viscosity (m2/s) when f is (t1+t2)/2, determined based on the thermophysical properties of the air; T1 is the temperature of the outer surface of the screw (℃); T2 is the air temperature (℃); c. N is the coefficient, c is the empirical constant, and the value of n is usually taken as 0.5.

　　温度饱和值在低速范围内与速度成线性关系；当达到一定速度后，又接近于与v1/2成正比。

　　The temperature saturation value is linearly related to the speed within the low-speed range; After reaching a certain speed, it approaches proportionality with v1/2.

　　综上所述，摩擦产生的热量与转速成正比，是的影响因素。有效润滑可有效降低摩擦系数，引入丝杠和螺母强制冷却，设计较好的循环回路，增加散热面积，可创造较好的散热条件。相同规格丝杠在相同转速下，滚珠丝杠副预加载荷越大，摩擦越大，丝杠温升越明显。

　　In summary, the heat generated by friction is directly proportional to the rotational speed and is the biggest influencing factor. Effective lubrication can effectively reduce the friction coefficient, introduce forced cooling of the screw and nut, design a better circulation circuit, increase the heat dissipation area, and create better heat dissipation conditions. At the same speed, the larger the preload and friction of the ball screw pair, the more significant the temperature rise of the screw.

　　3.2 滚珠丝杠副热变形分析

　　3.2 Thermal deformation analysis of ball screw pairs

　　滚珠丝杠副运行时因摩擦产生大量的热，引起材料热变形，直接影响高精数控机床的热稳定性能和加工精度，此外数控机床在运行中其他零部件的热量传导，会使滚珠丝杠副温度升高导致热变形，热变形量ΔLθ的计算如下

　　The ball screw pair generates a large amount of heat due to friction during operation, causing material thermal deformation and directly affecting the thermal stability and machining accuracy of high-precision CNC machine tools. In addition, the heat conduction of other components during the operation of CNC machine tools can cause the temperature of the ball screw pair to rise, resulting in thermal deformation. The calculation of the thermal deformation Δ L θ is as follows

　　ΔLθ＝ρθL? ? ? ? ? ? ? ? ? ? ? ? ?（3）

　　ΔLθ＝ρθL?  ? ?  ? ?  ? ?  ? ?  ? ?  ? ? （3）

　　式中，ΔLθ是热变形量（mm）；ρ是钢的热膨胀系数（12.0×10-6℃-1）；θ是丝杠轴（平均）温度上升值（℃）；L是丝杠轴长度（mm）。

　　In the formula, Δ L θ is the amount of thermal deformation (mm); ρ is the thermal expansion coefficient of steel (12.0 × 10-6 ℃ -1); θ is the (average) temperature rise value of the screw shaft (℃); L is the length of the screw shaft (mm).

　　通过式（3）计算可知，温度每上升1℃，1m丝杠轴就会伸长0.012mm，直接影响进给轴的定位精度。因此，采用强制冷却技术抑制丝杠副的热变形，对高精度滚珠丝杠副应用具有重要的意义。

　　According to equation (3), for every 1 ℃ increase in temperature, the 1m screw shaft will elongate by 0.012mm, directly affecting the positioning accuracy of the feed shaft. Therefore, adopting forced cooling technology to suppress the thermal deformation of the screw pair is of great significance for the application of high-precision ball screw pairs.

　　04

　　04

　　滚珠丝杠副运转过程热参数计算分析

　　Calculation and analysis of thermal parameters during the operation of ball screw pairs

　　以GQ50×20规格滚珠丝杠副为例，设计强制冷却结构[5]。已知转速n＝2000r/min，轴向载荷F＝15kN，螺母长度L＝120mm，支撑长度L2＝1000mm，摩擦生热的功率计算如下

　　Taking GQ50 × 20 specification ball screw pair as an example, design a forced cooling structure [5]. Given the speed n=2000r/min, axial load F=15kN, nut length L=120mm, support length L2=1000mm, the power generated by friction is calculated as follows

　　P＝0.12πnM ??? ? ? ? ? ? ? ? ? ? （4）M＝FPh/（2πη） ? ? ? ? ? ? ? ? （5）

　　P＝0.12πnM ???  ? ?  ? ?  ? ?  ? ?  ? （4）M＝FPh/（2πη） ?  ? ?  ? ?  ? ?  ? （5）

　　式中，P是丝杠副发热功率（W）；n是丝杠转速（r/s）；M是摩擦力矩（N·m）；Ph为丝杠导程（m）；η为滚珠丝杠副传动效率，一般为0.85～0.95。

　　In the formula, P is the heating power of the screw pair (W); N is the screw speed (r/s); M is the frictional torque (N · m); Ph is the lead of the screw (m); η is the transmission efficiency of the ball screw pair, generally ranging from 0.85 to 0.95.

　　基于能量守恒定律，为使滚珠丝杠副达到热稳定，丝杠、螺母冷却装置的总冷却功率应与发热功率相等，即P1=P。由于螺母结构及尺寸紧凑，冷却孔径尺寸受限，导致无法进行大流量冷却。故基于工程实际中滚珠丝杠副的结构特征，采用多源冷却控制，经计算，设置丝杠冷却装置冷却功率P2为螺母冷却装置冷却功率P3的4倍。

　　Based on the law of conservation of energy, in order to achieve thermal stability of the ball screw pair, the total cooling power of the screw and nut cooling device should be equal to the heating power, that is, P1=P. Due to the compact structure and size of the nut, the cooling aperture size is limited, making it impossible to perform high flow cooling. Therefore, based on the structural characteristics of ball screw pairs in engineering practice, multi-source cooling control is adopted. After calculation, the cooling power P2 of the screw cooling device is set to be four times the cooling power P3 of the nut cooling device.

　　丝杠冷却装置冷却功率P2计算如下

　　The cooling power P2 of the screw cooling device is calculated as follows

　　P2＝cm1ΔT/t1??? ? ? ? ? ? ? ? ??? ?（6）m1＝ρq1t1? ?? ? ? ? ? ? ? ? ? ? ? ?（7）

　　P2＝cm1ΔT/t1???  ? ?  ? ?  ? ?  ? ???  ?（6）m1＝ρq1t1?  ?? ?  ? ?  ? ?  ? ?  ? ?  ? ? （7）

　　式中，P2是中空丝杠冷却介质的冷却功率（W）；c是冷却介质的比热容[J/(kg·℃）]；m1是冷却介质质量（kg）；ΔT是冷却介质温升（℃）；t1是冷却介质通过螺母冷却通道所用时间（s）；ρ为冷却介质密度（kg/m3）；q1为冷却介质流量（m3/s）。为简化计算模型，忽略丝杠与螺母非接触区域的冷却，冷却时间为冷却介质通过螺母长度所用的时间，计算如下

　　In the formula, P2 is the cooling power (W) of the hollow screw cooling medium; C is the specific heat capacity of the cooling medium [J/(kg ·℃)]; M1 is the mass of the cooling medium (kg); Δ T is the temperature rise of the cooling medium (℃); T1 is the time (s) taken for the cooling medium to pass through the nut cooling channel; ρ is the density of the cooling medium (kg/m3); Q1 is the flow rate of the cooling medium (m3/s). To simplify the calculation model, the cooling of the non-contact area between the screw and nut is ignored. The cooling time is the time it takes for the cooling medium to pass through the length of the nut. The calculation is as follows

　　t1＝L/v1? ? ? ? ? ? ? ? ? （8）

　　t1＝L/v1?  ? ?  ? ?  ? ?  ? ?  （8）

　　式中，L是冷却通道长度（m）；v1是冷却介质的流速（m/s），v1=q1/A1，其中q1为冷却介质流量（m3/s），A1为冷却孔截面积（m2），A1＝πd12/4，d1为冷却孔直径（m）。

　　In the formula, L is the length of the cooling channel (m); V1 is the flow velocity of the cooling medium (m/s), v1=q1/A1, where q1 is the flow rate of the cooling medium (m3/s), A1 is the cross-sectional area of the cooling hole (m2), A1=π d12/4, d1 is the diameter of the cooling hole (m).

　　由式（6）可得P2=cρq1ΔT，即：q1＝P2/（cρΔT）。当冷却介质采用水时，其c＝4185J/（kg·℃），根据热稳定试验检测结果与丝杠热补偿量设置，设定冷却介质在丝杠副热平衡时整体温升约为3℃。故中空丝杠冷却流量为：q1＝0.042×10－3m3/s。

　　From equation (6), P2=c ρ q1 Δ T can be obtained, that is: q1=P2/(c ρ Δ T). When water is used as the cooling medium, its c=4185J/(kg ·℃). Based on the results of the thermal stability test and the setting of the screw thermal compensation amount, the overall temperature rise of the cooling medium at the thermal equilibrium of the screw pair is set to about 3 ℃. Therefore, the cooling flow rate of the hollow screw is q1=0.042 × 10-3m3/s.

　　为使丝杠内冷却介质流速合适，同时保证丝杠副轴向刚度，丝杠冷却通孔直径设计时应考虑冷却孔对丝杠刚度的影响，丝杠刚度与冷却孔直径关系如图2所示。

　　In order to ensure the appropriate flow rate of the cooling medium inside the screw and the axial stiffness of the screw, the influence of the cooling hole on the screw stiffness should be considered when designing the diameter of the screw cooling through-hole. The relationship between the screw stiffness and the diameter of the cooling hole is shown in Figure 2.

　　图2　丝杠副刚度与冷却孔直径的关系

　　Figure 2 Relationship between the stiffness of the screw pair and the diameter of the cooling hole

　　本型号滚珠丝杠副在满足刚度情况下，结合丝杠实际深孔加工工艺，设计冷却孔直径为10mm，则可得流速为v1＝q1/A1＝0.54m/s。

　　This model of ball screw pair is designed with a cooling hole diameter of 10mm based on the actual deep hole machining process of the screw, while meeting the stiffness requirements. The flow rate can be obtained as v1=q1/A1=0.54m/s.

　　螺母冷却装置冷却功率P3计算如下

　　The cooling power P3 of the nut cooling device is calculated as follows

　　P3＝P2/4＝cm2ΔT/t2?? ? ? ? ??（9）

　　P3＝P2/4＝cm2ΔT/t2??  ? ?  ? ?? （9）

　　同理可得，q2＝P2/（4cρΔT）＝0.0105×10－3m3/s。螺母冷却使用水为冷却介质，则冷却介质流速为

　　Similarly, q2=P2/(4c ρΔ T)=0.0105 × 10-3m3/s. If water is used as the cooling medium for nut cooling, the flow rate of the cooling medium is

　　v2＝q2/A2? ? ? ? ? ? ? ? ? ? ? （10）

　　v2＝q2/A2?  ? ?  ? ?  ? ?  ? ?  ? ?  （10）

　　式中，q2是冷却介质流量（m3/s）；A2是冷却孔的截面积（m2），A2＝πd22/4，d2为螺母冷却孔直径（m）。

　　In the formula, q2 is the flow rate of the cooling medium (m3/s); A2 is the cross-sectional area of the cooling hole (m2), A2=π d22/4, d2 is the diameter of the nut cooling hole (m).

　　基于螺母结构特点， 冷却孔径一般设计较小，螺母刚度影响忽略不计，为保证冷却介质流速合理，螺母通孔直径设计为5mm，则流速为v2＝0.54m/s。

　　Based on the structural characteristics of the nut, the cooling aperture is generally designed to be small, and the influence of nut stiffness is ignored. To ensure a reasonable flow rate of the cooling medium, the diameter of the nut through-hole is designed to be 5mm, and the flow rate is v2=0.54m/s.

　　05

　　05

　　结束语

　　Conclusion

　　对滚珠丝杠副从内部进行冷却，在丝杠或螺母内部通过有压力低温液体（或气体）的循环流动，降低丝杠或螺母本体的温升，使丝杠或螺母温度保持恒定，是提高数控机床精度、综合性能、精度保持性和可靠性的有效手段。参考文献：

　　Cooling the ball screw pair from the inside and circulating pressurized low-temperature liquid (or gas) inside the screw or nut to reduce the temperature rise of the screw or nut body and maintain a constant temperature is an effective means to improve the accuracy, comprehensive performance, accuracy retention, and reliability of CNC machine tools. References:

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