Ball screw

Abstract

By allowing the balls to pass smoothly through the circulation path, torque fluctuations and vibration noise are suppressed. [Solution] In a ball screw, when the lead angle of the rolling path is β and the distance between both ends of the circulation groove 9 in the direction parallel to the ball center trajectory C1 of the female screw groove 8 (y' direction) is ly, the maximum inclination angle φ of the ball center trajectory C2 of the circulation groove 9 formed on the inner circumferential surface of the nut is set to be greater than or equal to the allowable minimum value φ1 defined by the following formula. JPEG2026121110000020.jpg37170 JPEG2026121110000021.jpg14170

Description

[Technical Field]

[0001] This invention relates to a ball screw. [Background technology]

[0002] A ball screw is a motion conversion mechanism that converts rotational motion and linear motion into each other. For example, it is used to convert the rotational motion of an electric motor into linear motion. A ball screw comprises a screw shaft and a nut that rotate relative to each other via multiple balls, and a circulating member provided on the nut. When the screw shaft and nut rotate relative to each other, the balls circulate (revolve) along a loop-shaped ball passage that has a rolling path formed by a male screw groove formed on the outer circumference of the screw shaft and a female screw groove formed on the inner circumference of the nut, and a circulating section formed by the circulating member. Examples of circulating members include circulating balls, end caps, return tubes, and guide plates, which are used depending on the application of the ball screw (see Patent Document 1 below).

[0003] In ball screws using a circulating ball as a circulating component, the circulating groove formed in the circulating ball has a complex three-dimensional shape to guide the ball from the male screw groove on the outer surface of the screw shaft, over the screw threads, and back into the male screw groove. Various methods have been proposed to ensure that the ball passes smoothly through the circulating path formed by such a circulating groove.

[0004] For example, in Patent Document 2 below, in a ball screw of the spindle type, the range of the maximum inclination angle α of the circulation path is set by focusing on the variation in the path length (in / out variation) when the ball passes through the circulation path. [Prior art documents] [Patent Documents]

[0005] [Patent Document 1] Japanese Patent Publication No. 2003-240094 [Patent Document 2] Japanese Patent Publication No. 2019-190562 [Overview of the Initiative] [Problems that the invention aims to solve]

[0006] However, even if the maximum inclination angle α of the circulation path is set to the range shown in Patent Document 2, the ball may not pass through the circulation path smoothly, leading to increased torque fluctuations and vibration noise, which can hinder smooth operation.

[0007] Therefore, the present invention aims to suppress torque fluctuations and vibration noise in a ball screw by allowing the balls to pass smoothly through the circulation path. [Means for solving the problem]

[0008] To solve the above problems, the present invention provides a ball screw comprising: a screw shaft having a helical male screw groove formed on its outer circumference; a nut having a helical female screw groove opposite to the male screw groove and a circulation groove connecting the end and start of the female screw groove formed on its inner circumference; and a plurality of balls arranged in a loop-shaped ball passage having a rolling path formed by the male screw groove of the screw shaft and the female screw groove of the nut, and a circulation path formed by the circulation groove, The present invention provides a ball screw in which, when the lead of the rolling path is L, the lead angle of the rolling path is β, and the distance between both ends of the circulation groove in a direction parallel to the ball center trajectory of the female screw groove is ly, the maximum inclination angle φ of the ball center trajectory of the circulation groove is set to be greater than or equal to the allowable minimum value φ1 expressed by the following formula.

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[0009] Here, the maximum inclination angle φ of the ball center locus of the circulation groove refers to the state where the ball center loci of the circulation groove and the female thread groove are developed in a plane. Specifically, the ball center loci of the circulation groove and the female thread groove are projected onto a cylindrical surface provided from the nut axis to the outer periphery of the nut, and in the state where this cylindrical surface is developed in a plane, the maximum angle formed by the extending direction of the ball center locus of the circulation groove (in the case of a straight line, its straight line direction; in the case of a curve, the tangent direction) and the ball center locus of the female thread groove is what is meant.

[0010] By setting the maximum inclination angle φ of the ball center locus of the circulation groove to be not less than the above-mentioned allowable minimum value φ1, the load received by the nut from the ball when the ball passes through the curved portion of the circulation groove can be suppressed to a sufficient level, so that the ball can smoothly pass through the circulation path.

[0011] In a ball screw, the balls in the rolling path are sandwiched between the male thread groove of the screw shaft and the female thread groove of the nut. Therefore, when the nut and the screw shaft rotate relative to each other, the balls are driven by the frictional force between the male thread groove and the female thread groove and roll in the rolling path. On the other hand, the balls in the circulation path are not sandwiched between the male thread groove of the screw shaft and the female thread groove of the nut. Therefore, when the nut and the screw shaft rotate relative to each other, the balls are not driven. Rather, when the balls pass through the circulation path, energy loss occurs due to the friction caused by the contact between the balls and the circulation groove, and the balls decelerate. If this deceleration is significant, when the balls enter the rolling path again, the balls are suddenly accelerated, and torque fluctuations occur due to this sudden acceleration. In addition, such sudden acceleration of the balls induces slip between the balls and the rolling path, which also has an adverse effect on durability.

[0012] Therefore, it is preferable that the above-mentioned ball screw is set such that the maximum inclination angle φ of the ball center locus of the circulation groove is not more than the allowable maximum value φ2 represented by the following formula.

Equation

[0013] Here, in order to show the difference between the present invention and the prior art, Figure 15 shows the range of the maximum inclination angle α of the circulation path as defined in Patent Document 2, and the ranges of the lower limit (=φ1-β) and upper limit (=φ2-β) of the maximum inclination angle α calculated from the maximum inclination angle φ of the circulation path that satisfies the conditions of the present invention. The hatched area in the figure is the range defined in claim 1 of Patent Document 2. From the figure, it can be seen that, at least in the range where the ratio L / Da of the ball diameter Da to the lead L of the rolling path is 2 or less, and especially 1.6 or less, the maximum inclination angle of the circulation path that satisfies φ1≦φ is greater than the maximum inclination angle of the circulation path shown in claim 1 of Patent Document 2.

[0014] The circulation groove has, for example, a pair of curved sections connected to the end and start of the female screw groove, and an intermediate connecting section that connects the pair of curved sections. In this case, the curved sections can be formed as a single circular arc or a non-circular curve.

[0015] The female thread groove of a nut may have a load region that clamps a ball between itself and the male thread groove of the screw shaft, and non-load regions provided at both ends of the female thread groove that do not clamp a ball between themselves and the male thread groove. In this case, the nut may have a nut body and a circulating spool, with the load region of the female thread groove formed on the nut body and the circulating spool forming both the circulating groove and the non-load regions of the female thread groove.

[0016] Furthermore, a female screw groove and a circulation groove may be integrally formed on the inner circumferential surface of the nut. [Effects of the Invention]

[0017] As described above, the ball screw according to the present invention allows the balls to pass smoothly through the circulation path, thereby suppressing torque fluctuations and vibration noise. [Brief explanation of the drawing]

[0018] [Figure 1] This is a plan view of a ball screw according to one embodiment of the present invention. [Figure 2] This is a cross-sectional view taken along the line II-II in Figure 1. [Figure 3] This is a schematic plan view showing the screw shaft and circulating spool of the ball screw described above. [Figure 4] This is a view of the circulating valve from the inner diameter side. [Figure 5] This is a radial cross-sectional view along the trajectory of the ball's center in the spinning top. [Figure 6] This is a schematic diagram of a circulation channel unfolded in plan. [Figure 7] This graph shows the specific load f* applied to the ball as it passes through the circulation path, and the maximum inclination angle φ of the circulation groove. [Figure 8] This graph shows the calculation results for the specific load f* and the maximum inclination angle φ of the circulation groove when t=1.4. [Figure 9] This graph shows the approximate curve of the allowable minimum value φ1 for the maximum inclination angle φ of the circulation groove. [Figure 10] This is a schematic cross-sectional view illustrating the loads acting on a ball as it passes through the circulation path. [Figure 11] This graph shows the calculation results of the energy loss rate γ when the ball passes through the circulation path, as the parameter ξ is varied. [Figure 12] This graph shows the calculated energy loss rate γ as the ball passes through the circulation path, with respect to the parameter t. [Figure 13] This graph shows the calculated energy loss rate γ and the maximum inclination angle φ of the circulation groove when t=1.4. [Figure 14] This graph shows an approximate curve of the maximum allowable value φ2 for the maximum inclination angle φ of the circulation groove. [Figure 15] This graph shows the range of the maximum inclination angle of the circulation path as shown in claim 1 of Patent Document 2 and the range of the maximum inclination angle of the circulation path as defined in the present invention. [Modes for carrying out the invention]

[0019] Hereinafter, embodiments of the present invention will be described based on the drawings.

[0020] A ball screw 1 according to one embodiment of the present invention, as shown in Figures 1 and 2, comprises a screw shaft 2, a nut 3 fitted onto the screw shaft 2, and a plurality of balls 4 arranged between the screw shaft 2 and the nut 3. In this embodiment, the nut 3 has a cylindrical nut body 5 and a circulating ball 6 mounted in a window 5a that penetrates the nut body 5 radially. In the illustrated example, the window 5a and the circulating ball 6 are provided at two locations spaced apart in the axial direction of the nut body 5.

[0021] A continuous spiral male thread groove 7 is formed on the outer circumferential surface of the screw shaft 2. On the inner circumferential surface of the nut 3, a spiral female thread groove 8 of one turn or less is formed radially opposite to the male thread groove 7, and a circulation groove 9 is formed connecting the end and start of the female thread groove 8. In the illustrated example, the female thread groove 8 is formed on the inner circumferential surface of the nut body 5, and the circulation groove 9 is formed on the inner diameter side surface of the circulation ball 6. A loop-shaped ball passage is formed by the rolling path formed by the male thread groove 7 and the female thread groove 8, and the circulation path formed by the circulation groove 9, and a large number of balls 4 are filled into this ball passage. When the screw shaft 2 and the nut 3 rotate relative to each other, the balls 4 circulate in the loop-shaped ball passage. In other words, when the screw shaft 2 and the nut 3 rotate relative to each other, the large number of balls 4 roll along the rolling path, and when they reach the end of the rolling path, they are guided by the circulation path to overcome the threads 2a of the screw shaft 2 and return to the start of the rolling path, and roll along the rolling path again. In this embodiment, the cross-sections of the male screw groove 7 and female screw groove 8 (cross-sections perpendicular to the ball center trajectory of each groove) form a Gothic arch shape, and the cross-section of the circulation groove 9 forms a single circular arc shape. The cross-sectional shapes of the male screw groove 7, female screw groove 8, and circulation groove 9 are not limited to those described above; for example, the cross-section of the circulation groove 9 may be a Gothic arch shape.

[0022] Figure 3 schematically shows the screw shaft 2 and the circulating ball 6. In the figure, C1 is the ball center trajectory of the rolling path (male screw groove 7 and female screw groove 8), C2 is the ball center trajectory of the circulating groove 9, L is the lead of the rolling path, and β is the lead angle of the rolling path. In this embodiment, the ratio L / Da of the lead L to the diameter Da of the ball 4 is set to 2 or less. α is the maximum inclination angle of the ball center trajectory C2 of the circulating groove 9 with respect to a plane V perpendicular to the axis CS of the screw shaft 2. φ is the maximum inclination angle of the tangent to the ball center trajectory C2 of the circulating groove 9 with respect to the direction of the ball center trajectory C1 of the rolling path, and α = φ - β.

[0023] Figure 4 shows a view of the circulating ball 6 from the inner diameter side. As shown in the figure, the circulating groove 9 is formed in an S-shape and smoothly connects to the female screw groove 8. In the illustrated example, the circulating groove 9 consists of a pair of curved sections 9a and an intermediate connecting section 9b that connects them. In the cross-section shown in Figure 5, the groove depth of the circulating groove 9 increases towards the center. The groove depth of the circulating groove 9 is set so that the ball 4 passing through the circulating path can pass over the screw threads 2a of the screw shaft 2.

[0024] Figure 6 is a plan view of the area around the circulation groove 9 on the inner circumferential surface of the nut 3. More specifically, it is a plan view of the cylindrical surface formed by projecting the ball center trajectory C2 of the circulation groove 9 onto a cylindrical surface provided on the outer circumference of the nut 3 from the axis of the nut 3. The vertical direction (y' direction) in the figure is parallel to the ball center trajectory C1 of the female screw groove 8. The lines connecting points A to B and F to G are the ball center trajectory C1b of the second female screw groove 8b, and are straight lines provided on the extension of the ball center trajectory C1a of the first female screw groove 8a formed in the nut body 5. The line connecting points B to F is the ball center trajectory C2 of the circulation groove 9, and extends in a different direction from the ball center trajectory C1 of the female screw groove 8. Of these, the lines connecting points B to C and E to F are the ball center trajectory C2a of the curved portion 9a of the circulation groove 9, and consist of a single circular arc with a radius of curvature r. Furthermore, the line connecting points C and E is the ball center trajectory C2b of the intermediate connecting section 9b of the circulation groove 9, and is a straight line that smoothly connects the ball center trajectories C2a of the pair of curved sections 9a. Note that the shape of the curved section 9a is not limited to the above; for example, the curved section 9a may be a non-arc curve (for example, a curve formed by connecting multiple arcs, or a curve in which the radius of curvature changes continuously).

[0025] In this embodiment, most of the female screw groove 8, excluding both ends (first female screw groove 8a), is formed on the inner circumferential surface of the nut body 5, and the portions of the female screw groove 8 at both ends (second female screw groove 8b) and the circulation groove 9 are formed on the inner diameter surface of the circulation spool 6. The ball center trajectory C1a of the first female screw groove 8a and the ball center trajectory C1b of the second female screw groove 8b extend in the same helical direction (linear direction in Figure 6). The first male screw groove 8a is a load region that receives an axial load by clamping the ball 4 with the male screw groove 7, while the second male screw groove 8b is a non-load region that does not clamp the ball 4 with the male screw groove 7. Alternatively, part or all of the second male screw groove 8b may be a load region that clamps the ball 4 with the male screw groove 7. The entire circulation groove 9 is a non-load region that does not clamp the ball 4 with the male screw groove 7.

[0026] The inclination angle of the ball center trajectory C2 of the circulation groove 9, that is, the angle between the extension direction of the ball center trajectory C2 of the circulation groove 9 (the linear direction in the case of a straight line, and the tangential direction in the case of a curve) and the y' direction in the plan view of Figure 6, changes along the path and reaches the maximum inclination angle φ at the intermediate connecting section 9b. In the figure, ly is the distance in the y' direction between the two ends of the circulation groove 9, and as shown in Figure 6, if the ball center trajectory C2a of the curved section 9a is a circular arc with a single radius of curvature, it is the y' component of the distance between the centers of curvature. Also, lx is the distance in the x' direction between the two ends of the circulation groove 9, and in the illustrated example, it is the x' component of the distance between the centers of curvature of the ball center trajectory C2a of the pair of curved sections 9a. In the illustrated example, the ball center trajectory C2 of the circulation groove 9 is point-symmetric with respect to the midpoint D of the path length, which in the illustrated example is the midpoint D of the ball center trajectory C2b of the intermediate connecting section 9b.

[0027] Here, we will explain a method for obtaining the allowable minimum value φ1 of the maximum inclination angle φ of the ball center trajectory C2 in the circulation groove 9, based on the load that the nut 3 receives from the ball 4 as the ball 4 passes through the circulation groove 9.

[0028] Assuming that the translational velocity u at the center of ball 4 does not change while ball 4 passes through the circulation path, the centrifugal force f around the center of curvature acting on ball 4 when passing through the curved section 9a (centripetal force that ball 4 receives from the circulation groove 9) c2 The following equation holds true, where m is the mass of the ball. This centrifugal force f c2 This occurs when the ball 4 passes through the curved section 9a. In other words, a periodic load acts on the nut 3 as the ball 4 passes through the circulation path, resulting in this centrifugal force f c2 This can cause torque fluctuations and vibration noise. In order to suppress the torque fluctuations and vibration noise of the ball screw 1, this centrifugal force f c2 It is desirable to make it as small as possible.

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[0029] f c2 The representative load f0 = mu 2 / L * Specific load f divided by *It is represented by the following formula.

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[0030] In addition, the maximum inclination angle φ of the ball center locus C2 of the circulation groove 9 is given by the following formula.

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[0031] In FIG. 7, the specific gravity load f * (solid line) calculated by the above formulas (1) and (2) and the maximum inclination angle φ (dotted line) of the ball center locus C2 of the circulation groove 9 are shown. When t is small, φ can reach 90°, but when t is large, due to geometric constraints, φ cannot reach 90°, and its maximum value decreases as t increases. As shown in the figure, regarding the parameter s, the specific gravity load f * decreases monotonically, and the maximum inclination angle φ increases monotonically. From this figure, the larger φ is taken, the more f * can be reduced, but as s increases, the reduction rate of f * decreases. Therefore, if φ is set to a certain value or more, f * can be reduced to a sufficient level.

[0032] Next, a method for obtaining the minimum allowable value φ1 of φ that can sufficiently reduce f * and φ obtained from the above formulas (1) and (2) will be described. FIG. 8 shows f * and φ when t = 1.4. First, the maximum value of φ that can be obtained is determined (point P1), and f * corresponding to that maximum value is determined (point P2). This f * is the minimum value f * that can be obtained under this condition * min . And the value a·f * of the allowable f *min Find point P3, and then find the corresponding value of φ at point P4. This value of φ is the minimum acceptable value φ1. * A coefficient 'a' used to determine the tolerance value is preferably around 1.2 to 1.5.

[0033] Using the method described above, φ1 is calculated for several values ​​of t, and then an approximate function of φ1 with respect to t is found. Figure 9 and the following equation show the approximate function obtained by adopting a coefficient a of 1.3.

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[0034] As described above, by setting the maximum inclination angle φ of the ball center trajectory C2 of the circulation groove 9 to a value greater than or equal to φ1 shown in equation (3) above, the load that the circulation groove 9 receives from the ball 4 when the ball 4 passes through the circulation groove 9 can be reduced to a sufficient level, thereby reducing torque fluctuations and vibration noise caused by the ball 4 passing through the circulation groove 9. Note that the load that the circulation groove 9 receives from the ball 4 and the resulting torque fluctuations increase with increasing rotational speed of the ball screw. For this reason, the present invention is suitably applied to ball screws operated under high-speed conditions.

[0035] Next, we will explain a method for obtaining the maximum value φ2 of the maximum inclination angle φ of the ball's center trajectory C2 in the circulation groove 9, based on the energy loss when the ball 4 passes through the circulation groove 9.

[0036] Here, we examine the energy loss when ball 4 passes through the circulation groove 9 (curved section 9a and intermediate connecting section 9b). At this time, the following two types of inertial forces act on ball 4. (1) Centrifugal force f due to the rotation (revolution) of the ball 4 around the axis of the ball screw 1 c1 (2) Centrifugal force f due to the movement of the ball 4 along the curved portion 9a of the circulation groove 9 c2 In the following explanation, we will describe the case where the ball center trajectory C2a of the curved section 9a of the circulation groove 9 consists of a single circular arc. However, the same approach applies even if the ball center trajectory C2a of the curved section 9a has a different shape.

[0037] The balance of forces on the ball 4 as it passes through the circulation groove 9 is shown in Figure 10 and in the following equation. Note that when the ball 4 passes through the intermediate connecting section 9b, f c2 = 0. Note that R is the pitch circle radius of ball 4 of ball screw 1.

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[0038] From equations (4) and (5) above, the normal load F N The contact angle θ is obtained. For simplicity, we assume that the ball 4 does not rotate while passing through the circulation groove 9, and that its translational velocity is maintained at the velocity u before entering the circulation groove 9. The energy losses ΔE1 and ΔE2 when passing through the curved section 9a and the intermediate connecting section 9b are given by the following equations.

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[0039] Here, μ' is the coefficient of friction in the direction of ball movement, and Δt1 and Δt2 are the times required to pass through. The total energy loss ΔE is ΔE = 2ΔE1 + ΔE2, since there are two curved sections 9a.

[0040] The kinetic energy T of the ball 4 before it enters the circulation groove 9 is expressed by the following equation.

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[0041] If the ratio of the total energy loss ΔE to the kinetic energy T is denoted as the energy loss rate γ, then the energy loss rate γ is expressed by the following equation.

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[0042] Here, ξ is given by ξ = R / L * Therefore, ξ takes the range of 0.3 to 4 for a typical ball screw (screw shaft outer diameter 4 to 140 mm, lead 1 to 20 mm). Figure 11 shows the energy loss rate γ obtained from equation (6) above by varying the value of ξ. From the figure, it can be seen that as ξ increases, γ decreases. Here, since the objective is to minimize energy loss as much as possible, the value of ξ will be set to ξ = 4 from now on.

[0043] Figure 12 shows the energy loss rate γ obtained by equation (6) when ξ=4. Here, μ=μ'=0.1 is assumed. From the figure, it can be seen that to reduce energy loss, it is effective to make s as small as possible, that is, to make the inclination angle φ of the ball center trajectory C2 in the circulation groove 9 as small as possible. However, if the inclination angle φ is set to a certain value or less, it is possible to reduce the energy loss rate γ to a sufficient level.

[0044] Next, we will explain a method for determining the maximum allowable value of φ2 that allows γ to be sufficiently reduced, using γ and φ obtained from equation (6). Figure 13 shows γ and φ when t = 1.4. First, the minimum possible value of γ is γ min We find the value of (point P1). Next, we determine the acceptable value of γ, b·γ. min Find the value of φ corresponding to point P2 (point P3). This value of φ will be the maximum allowable value φ2. The coefficient b used to determine the allowable value of γ is preferably around 1.2 to 1.5.

[0045] Using the method described above, φ2 is calculated for several values ​​of t, and then an approximate function of φ2 with respect to t is found. Figure 14 and the following equation show the approximate function obtained by adopting a coefficient b of 1.3.

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[0046] Based on the above, by setting the maximum inclination angle φ of the ball center trajectory C2 in the circulation groove 9 to a value less than or equal to φ2 as shown in equation (7) above, it becomes possible to further reduce torque fluctuations when the ball 4 enters the rolling path (loaded area) from the circulation path (non-loaded area).

[0047] In this embodiment, the maximum inclination angle φ of the ball center trajectory C2 in the circulation groove 9 is set to a range of φ1 or more in equation (3) and φ2 or less in equation (7) above. However, if there is no problem with energy loss, the maximum inclination angle φ of the ball center trajectory C2 in the circulation groove 9 may be made larger than φ2.

[0048] The present invention is not limited to the embodiments described above. For example, in the embodiments described above, the nut 3 is shown to have a nut body 5 and a circulation valve 6, but the invention is not limited to this, and the circulation valve may be omitted, and the female screw groove 8 and the circulation groove 9 may be continuously and integrally formed on the inner circumferential surface of a cylindrical nut without a window.

[0049] The present invention is applicable to both a screw shaft rotation type ball screw, which rotates the screw shaft to move the nut linearly, and a nut rotation type ball screw, which rotates the nut to move the screw shaft linearly. Furthermore, the present invention is applicable to both a ball screw, which takes rotation as input and outputs linear motion, and a ball screw, which takes linear motion as input and outputs rotation. [Explanation of Symbols]

[0050] 1 Ball screw 2 Screw shaft 3 nuts 4 balls 5. Nut body 6 Circulating spinning top 7 Male screw groove 8 Female thread grooves 8a First female thread groove 8b Second female thread groove 9 Circulation groove 9a Curve section 9b Intermediate connection C1 Ball center trajectory of the rolling path (male screw groove and female screw groove) C1a Ball center trajectory of the first female screw groove C1b Ball center trajectory of the second female screw groove C2 Ball center trajectory in the circulation groove C2a Ball center trajectory of the curved section C2b Ball center trajectory of the intermediate connecting section L Lead β Lead angle

Claims

1. A screw shaft having a helical male screw groove formed on its outer surface, A nut having a spiral female thread groove on its inner surface opposite to the male thread groove, and a circulation groove connecting the end and start of the female thread groove, A ball screw comprising a plurality of balls arranged in a loop-shaped ball passage having a rolling path formed by the male screw groove of the screw shaft and the female screw groove of the nut, and a circulation path formed by the circulation groove, When L is the lead of the rolling path, β is the lead angle of the rolling path, and ly is the distance between both ends of the circulation groove in a direction parallel to the ball center trajectory of the female screw groove, the maximum inclination angle φ of the ball center trajectory of the circulation groove is the allowable minimum value φ, which is expressed by the following formula. １ The ball screw is set to the above specifications. [Math 1] [Math 2]

2. The maximum inclination angle φ of the ball center trajectory in the aforementioned circulation groove is the allowable maximum value φ, which is expressed by the following formula. ２ The ball screw according to claim 1, as set out below. [Math 3]

3. The ball screw according to claim 1 or 2, wherein the ratio L / Da of the diameter Da of the ball to the lead L of the rolling path is 2 or less.

4. The circulation groove has a pair of curved sections connected to the end and start of the female screw groove, and an intermediate connecting section connecting the pair of curved sections. The ball screw according to claim 1 or 2, wherein the ball center trajectory of each curved section consists of a single circular arc.

5. The circulation groove has a pair of curved sections connected to the end and start of the female screw groove, and an intermediate connecting section connecting the pair of curved sections. The ball screw according to claim 1 or 2, wherein the ball center trajectory of each curved section consists of a non-arc curve.

6. The ball screw according to claim 1 or 2, wherein the female thread groove of the nut has a load region that clamps the ball between itself and the male thread groove of the screw shaft, and non-load regions provided at both ends of the female thread groove that do not clamp the ball between itself and the male thread groove.

7. The ball screw according to claim 6, wherein the nut comprises a nut body in which the load region of the female screw groove is formed, and a circulating nut in which the circulation groove and the non-load region of the female screw groove are formed.

8. The ball screw according to claim 1 or 2, wherein the female screw groove and the circulation groove are integrally formed on the inner circumferential surface of the nut.

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