針對滾珠絲杠副在高速、重載條件下的發熱問題，設計了絲杠和螺母強制冷卻結構，探討了摩擦發熱對變形量的影響程度，分析了滾珠絲杠副結構設計及使用條件對溫升的影響因素，提供了滾珠絲杠副及冷卻系統設計方案，降低了滾珠絲杠副溫升對機床的精度影響。

　　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

　　隨著數控機床向高精、高速方向的發展，對機床的進給部分提出了更高的要求。滾珠絲杠副作為軸向驅動部件，在長時間運行下會產生溫升，加之數控機床在運行中其他部分產生大量熱量的傳導，會使滾珠絲杠副由于溫度變化而產生軸向熱位移，直接導致數控機床進給軸的定位精度下降[1，2]。因此，需要采取相應的措施來控制溫升，保證機床進給系統的定位精度，提高數控機床的加工精度。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

　　滾珠絲杠副冷卻結構設計

　　Design of cooling structure for ball screw pair

　　滾珠絲杠副在高速、重載、連續運行的工況條件下，滾珠在螺母和絲杠之間高速滾動和滑動，產生大量摩擦熱，隨著熱量增大，絲杠軸向和徑向發生不均勻的熱膨脹和熱變形，嚴重影響到機床定位精度、預緊力波動，加速潤滑脂失效，還會對滾珠和滾道造成熱損傷。為提高機床的運行精度和精度保持性，降低溫升對滾珠絲杠副精度的影響，針對應用領域及產品外部安裝空間，采用滾珠絲杠副強制冷卻的方式降低溫升對滾珠絲杠副精度和性能的影響。空心滾珠絲杠副可分為絲杠和螺母內部強制冷卻兩種方式[3]，絲杠內部強制冷卻采用單通道循環（見圖1）和雙通道循環，將絲杠設計為切削液循環通道，冷卻介質在通孔循環流動，在入口和出口端增加專用旋轉接頭和密封裝置，直接將絲杠的熱量帶走。螺母內部強制冷是在螺母壁設計若干冷卻通道，通過優化冷卻流道設計，使整體溫度場更加均勻，極大地減少熱梯度變形。該結構從根本上解決了精密傳動系統性能與熱變形問題，有效控制了絲杠在高速、高負荷運行時的溫升，從而提升其精度、剛度和壽命。

　　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.

　　圖1　單通道滾珠絲杠副冷卻循環結構

　　Figure 1 Cooling cycle structure of single channel ball screw pair

　　完整的空心冷卻滾珠絲杠副系統主要包括：帶有內部冷卻通道的空心絲杠和專用螺母、連接靜止冷卻管路和旋轉絲杠內部通道的旋轉接頭、提供恒定溫度和壓力的冷卻單元等。空心絲杠結構減輕了質量，提高了臨界轉速，但為防止高速旋轉時振動，加工需盡量保證絲杠和螺母壁厚均勻。由于絲杠空心，所以降低了絲杠抗彎和抗扭剛度，此問題可通過優化孔徑和材料來彌補，采用特殊的熱處理方式提高系統剛性。由于旋轉接頭長期與絲杠外圓高速摩擦，所以提高接觸外圓的耐磨性和可靠密封是保證系統穩定運行的前提。將冷卻系統與機床的數控系統聯動，根據負載和速度實時調節冷卻流量和溫度，可實現智能熱補償。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

　　滾珠絲杠副運轉過程熱量分析

　　Thermal analysis of the operation process of ball screw pairs

　　3.1 滾珠絲杠副溫升分析空心滾珠絲杠副及冷卻系統中循環通道孔徑、數量、流量和壓力等設計直接影響溫升的高低和產品性能，而溫升計算是設計中的關鍵環節，將直接影響產品的工作狀態、膠合失效、定位精度以及傳動效率。溫升設計的本質是建立一個熱平衡模型，使單位時間內摩擦產生熱量等于系統向環境散發的熱量。

　　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.

　　滾珠絲杠副運轉主要是滾動摩擦和滑移摩擦，建立熱平衡方程[4]，求解溫升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是溫升值（℃）；Fr是切線方向摩擦阻力（N）；v是絲杠線速度（m/s）；J是熱功當量，一般為1；α是絲杠表面熱傳導率[W/（m2·℃）]；A是散熱面積（m2）；Q是絲杠熱容量（J/℃）；t是工作時間（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).

　　溫度飽和值與線速度v成正比，與熱傳導率α成反比。當工作時間t較長時，式（1）中v[1－exp（－αAt/Q）]的值趨近于零，此時ΔT接近于溫升飽和值。

　　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.

　　在絲杠轉動過程中引起空氣軸向流動時，絲杠和螺母表面的熱傳導率α計算如下

　　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是絲杠外徑（ m ） ；?v?是空氣的流速（m/s）；λ為（t2＋t1）/2時的導熱系數[W/（m2·℃）]，根據空氣熱物理性質確定；f為（t1＋t2）/2時的空氣運動黏度（m2/s），根據空氣熱物理性質確定；t1是絲杠外表面溫度（℃）；t2是空氣溫度（℃）；c、n是系數，c為經驗常數，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.

　　溫度飽和值在低速范圍內與速度成線性關系；當達到一定速度后，又接近于與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.

　　綜上所述，摩擦產生的熱量與轉速成正比，是的影響因素。有效潤滑可有效降低摩擦系數，引入絲杠和螺母強制冷卻，設計較好的循環回路，增加散熱面積，可創造較好的散熱條件。相同規格絲杠在相同轉速下，滾珠絲杠副預加載荷越大，摩擦越大，絲杠溫升越明顯。

　　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 滾珠絲杠副熱變形分析

　　3.2 Thermal deformation analysis of ball screw pairs

　　滾珠絲杠副運行時因摩擦產生大量的熱，引起材料熱變形，直接影響高精數控機床的熱穩定性能和加工精度，此外數控機床在運行中其他零部件的熱量傳導，會使滾珠絲杠副溫度升高導致熱變形，熱變形量ΔLθ的計算如下

　　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是熱變形量（mm）；ρ是鋼的熱膨脹系數（12.0×10-6℃-1）；θ是絲杠軸（平均）溫度上升值（℃）；L是絲杠軸長度（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).

　　通過式（3）計算可知，溫度每上升1℃，1m絲杠軸就會伸長0.012mm，直接影響進給軸的定位精度。因此，采用強制冷卻技術抑制絲杠副的熱變形，對高精度滾珠絲杠副應用具有重要的意義。

　　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

　　滾珠絲杠副運轉過程熱參數計算分析

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

　　以GQ50×20規格滾珠絲杠副為例，設計強制冷卻結構[5]。已知轉速n＝2000r/min，軸向載荷F＝15kN，螺母長度L＝120mm，支撐長度L2＝1000mm，摩擦生熱的功率計算如下

　　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是絲杠副發熱功率（W）；n是絲杠轉速（r/s）；M是摩擦力矩（N·m）；Ph為絲杠導程（m）；η為滾珠絲杠副傳動效率，一般為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，為使滾珠絲杠副達到熱穩定，絲杠、螺母冷卻裝置的總冷卻功率應與發熱功率相等，即P1=P。由于螺母結構及尺寸緊湊，冷卻孔徑尺寸受限，導致無法進行大流量冷卻。故基于工程實際中滾珠絲杠副的結構特征，采用多源冷卻控制，經計算，設置絲杠冷卻裝置冷卻功率P2為螺母冷卻裝置冷卻功率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.

　　絲杠冷卻裝置冷卻功率P2計算如下

　　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是中空絲杠冷卻介質的冷卻功率（W）；c是冷卻介質的比熱容[J/(kg·℃）]；m1是冷卻介質質量（kg）；ΔT是冷卻介質溫升（℃）；t1是冷卻介質通過螺母冷卻通道所用時間（s）；ρ為冷卻介質密度（kg/m3）；q1為冷卻介質流量（m3/s）。為簡化計算模型，忽略絲杠與螺母非接觸區域的冷卻，冷卻時間為冷卻介質通過螺母長度所用的時間，計算如下

　　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是冷卻通道長度（m）；v1是冷卻介質的流速（m/s），v1=q1/A1，其中q1為冷卻介質流量（m3/s），A1為冷卻孔截面積（m2），A1＝πd12/4，d1為冷卻孔直徑（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）。當冷卻介質采用水時，其c＝4185J/（kg·℃），根據熱穩定試驗檢測結果與絲杠熱補償量設置，設定冷卻介質在絲杠副熱平衡時整體溫升約為3℃。故中空絲杠冷卻流量為：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.

　　為使絲杠內冷卻介質流速合適，同時保證絲杠副軸向剛度，絲杠冷卻通孔直徑設計時應考慮冷卻孔對絲杠剛度的影響，絲杠剛度與冷卻孔直徑關系如圖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.

　　圖2　絲杠副剛度與冷卻孔直徑的關系

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

　　本型號滾珠絲杠副在滿足剛度情況下，結合絲杠實際深孔加工工藝，設計冷卻孔直徑為10mm，則可得流速為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.

　　螺母冷卻裝置冷卻功率P3計算如下

　　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。螺母冷卻使用水為冷卻介質，則冷卻介質流速為

　　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是冷卻介質流量（m3/s）；A2是冷卻孔的截面積（m2），A2＝πd22/4，d2為螺母冷卻孔直徑（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).

　　基于螺母結構特點， 冷卻孔徑一般設計較小，螺母剛度影響忽略不計，為保證冷卻介質流速合理，螺母通孔直徑設計為5mm，則流速為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

　　結束語

　　Conclusion

　　對滾珠絲杠副從內部進行冷卻，在絲杠或螺母內部通過有壓力低溫液體（或氣體）的循環流動，降低絲杠或螺母本體的溫升，使絲杠或螺母溫度保持恒定，是提高數控機床精度、綜合性能、精度保持性和可靠性的有效手段。參考文獻：

　　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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