A curvature calculation method based on engagement of a trapezoidal thread coil spring and a bent mandrel

Abstract

This invention discloses a method for calculating the curvature of a bending mandrel mechanism based on a trapezoidal threaded coil spring engagement. The method comprises a bending mandrel, a trapezoidal threaded coil spring, a movable nut, a drive system, and a support and guide system. The trapezoidal threaded coil spring is sleeved on the outside of the bending mandrel, and the movable nut engages with the trapezoidal threaded coil spring. The curvature calculation method includes the following steps: determining the core parameters for curvature calculation, including the pitch P, the total thread height h, the thread angle α, and the design clearance α. c Spring mean diameter d2, nut effective length L0, minimum overlap coefficient ε min ; Calculate the minimum bending radius R under backlash constraint min1 ; Calculate the minimum bending radius R under the overlap coefficient constraint. min2 Step 4: Take R min1 and R min2 The maximum value in the range is taken as the minimum bending radius R of the trapezoidal threaded coil spring meshing bending mandrel mechanism. min R min =max(R min1 ,R min2 This addresses the limitations of traditional rigid screw drives and the lack of curvature calculation in flexible transmission mechanisms.

Description

Technical Field

[0001] This invention relates to the field of massager technology, and in particular to a method for calculating the curvature of a trapezoidal threaded coil spring meshing bending mandrel mechanism. Background Technology

[0002] In fields such as massage equipment and precision medical devices, transmission mechanisms often need to move along complex three-dimensional spatial curved trajectories. Traditional rigid screw transmission mechanisms are limited to straight lines or fixed arcs, which cannot adapt to complex curved trajectories such as the S-shaped curvature of the human spine. Furthermore, there is a lack of accurate curvature calculation methods for flexible transmission mechanisms, resulting in high design difficulty, poor bending adaptability, and insufficient transmission smoothness, making it difficult to meet the needs of practical applications.

[0003] The trapezoidal helical coil spring meshing bending mandrel mechanism, as a novel flexible transmission mechanism, decouples the "screw" and "thread" functions of the lead screw, employing a trapezoidal cross-section helical coil spring as the flexible thread, which, in conjunction with a pre-bent mandrel, enables precise movement along a three-dimensional spatial curve. However, the lack of a systematic curvature calculation method for this mechanism makes it impossible to accurately determine the minimum bending radius, thus limiting its engineering applications. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art by providing a curvature calculation method for a trapezoidal screw coil spring meshing bending mandrel mechanism, thereby solving the limitations of traditional rigid screw drives and the lack of curvature calculation for flexible transmission mechanisms.

[0005] This invention discloses a method for calculating the curvature of a bending mandrel mechanism based on a trapezoidal threaded helical spring engagement, characterized by comprising a bending mandrel, a trapezoidal threaded helical spring, a movable nut, a drive system, and a support and guide system; the bending mandrel is pre-bent, the trapezoidal threaded helical spring is sleeved on the outside of the bending mandrel, and the movable nut engages with the trapezoidal threaded helical spring; the curvature calculation method includes the following steps: Step 1: Determine the core parameters for curvature calculation, including pitch P, total thread height h, thread angle α, and design clearance a. c Spring mean diameter d2, nut effective length L0, minimum overlap coefficient ε min ; Step 2: Calculate the minimum bending radius R under backlash constraint. min1 ; Step 3: Calculate the minimum bending radius R under the overlap coefficient constraint. min2 ; Step 4: Take R min1 and R min2The maximum value in the range is taken as the minimum bending radius R of the trapezoidal threaded coil spring meshing bending mandrel mechanism. min R min =max(R min1 ,R min2 ).

[0006] A further provision of the present invention: Step two specifically includes: Determine the relationship between the relative deflection angle θ of adjacent thread teeth and the bending radius R when the mandrel is bent: ; Calculate the radial displacement of the tooth tip when the relative deflection θ of adjacent teeth is θ. : ; Based on the constraint that the radial displacement of the tooth tip does not exceed the design clearance ,Will Substituting into the radial displacement formula, the formula for the minimum bending radius under backlash constraint is derived: .

[0007] A further feature of the present invention is that the total thread height h is optimized by adjusting the tooth tip height coefficient, wherein the tooth tip height coefficient ranges from 0.35 to 0.40.

[0008] A further feature of the present invention: Step three specifically includes: Formula for determining the overlap coefficient in linear transmission: ; Calculate the effective contact length after the mandrel is bent. : ; Will Substituting the effective contact length formula, the overlap coefficient formula under bending conditions is derived: ; Based on transmission smoothness constraints The formula for the minimum bending radius under the overlap coefficient constraint is derived as follows: .

[0009] A further feature of the present invention: the minimum overlap coefficient The value is 1.3, and the preconditions must be met. .

[0010] The beneficial effects of this invention are as follows: This mechanism innovatively decouples the functions of the "screw" and the "thread," employing a trapezoidal cross-section helical coil spring as the flexible thread, which cooperates with a pre-bent mandrel to achieve precise movement of the massage mechanism along a three-dimensional spatial curve. This paper focuses on the core constraints of this mechanism, deriving the calculation formula for the minimum bending radius under backlash and overlap coefficient constraints. The study reveals that backlash constraints mainly restrict the feasibility of the path for curved movement, while overlap coefficient constraints mainly affect the transmission smoothness of the shaft rotation drive. The research results show that trapezoidal threads, compared with cylindrical cross-section threads, have better guidance and adaptability when dealing with curvature changes, and can significantly reduce the probability of "tooth collision." By rationally designing parameters such as the tooth tip height coefficient and nut length, a smaller bending radius can be achieved while ensuring transmission stability, meeting the needs of ergonomic curve massage. Attached Figure Description

[0011] Figure 1 This is a diagram showing the geometric relationship of the bent mandrel when it is bent according to the present invention. Figure 2 This is a comparison diagram of the geometric relationship of the bent mandrel when calculating the overlap coefficient during the bending of the mandrel in this invention. Detailed Implementation

[0012] The specific embodiments of the present invention are described in detail below with reference to the accompanying drawings: In the description of this invention, it should be understood that the terms "upper", "lower", "bottom", "top", "front", "rear", "inner", "outer", "left", "right", etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0013] I. Organizational Structure The trapezoidal threaded spiral spring meshing bending mandrel mechanism mainly consists of the following five parts: a) Bending mandrel: Pre-bent into shape using thermoplastic material or metal tubing, defining the motion trajectory of the massage mechanism.

[0014] b) Trapezoidal threaded coil spring: made of high-elasticity alloy steel wire, with a trapezoidal cross section and continuous helical threads.

[0015] c) Moving nut: Installed on the massager frame, its internal thread engages with the coil spring.

[0016] d) Drive system: including motor, reducer and worm gear.

[0017] e) Support and guidance system: ensures that the massage frame moves only along a predetermined trajectory.

[0018] II. Working Principle When the mechanism is working, the drive motor drives the helical coil spring to rotate around its own axis via a worm gear. Because the coil spring engages with the moving nut, and the coil spring is constrained by the curved mandrel into a curved shape, the nut is forced to move along the trajectory of the curved mandrel, thereby realizing the curved movement of the massage frame. At the same time, the end of the helical coil spring can directly or through a coupling drive the massage head to rotate, realizing the rotational massage function.

[0019] The innovation of this mechanism lies in utilizing the flexibility of a helical coil spring to transform rotational motion into translational motion along arbitrary spatial curves, overcoming the limitations of traditional rigid lead screws. The trapezoidal thread cross-section provides better guidance and load-bearing capacity, ensuring stable engagement even under bending conditions.

[0020] III. Theoretical Basis for Curvature Calculation When the mandrel bends, the coil spring bends accordingly, causing a relative deflection between adjacent thread teeth. Let the bending radius of the mandrel be R and the pitch be P, then the relative deflection angle θ between adjacent teeth is: (See Figure 1). From its geometric relationship, we can obtain:

[0021] In reality, because the bending angle of the shaft is generally small, the following relationship can be derived: …………………………………(1)

[0022] In bending drives, the shape of the thread cross-section has a significant impact on meshing performance. Trapezoidal threads have a clear advantage in bending drives, as their two sides provide a clear guiding effect, reducing the possibility of lateral slippage and interference, and are more suitable for applications with large curvature changes.

[0023] Calculation of minimum bending radius under backlash constraint 4.1 Formula Derivation The core of backlash constraint is to ensure that the spring tooth tip and the nut tooth root do not interfere with each other under bending conditions. When the mandrel bending radius is... R At that time, the relative deflection angle of adjacent thread teeth θ = P / R For trapezoidal threads, the slope of the tooth flank is tan(α / 2), where α is the tooth profile angle. When adjacent teeth deflect relative to each other... θ At that time, the radial displacement of the tooth tip Δr It can be approximated as: …………………………………(2)

[0024] In the formula, h This refers to the full height of the thread teeth. The condition for no interference is that Δr does not exceed the design clearance.a c, that is: ………………………………………………(3) Will θ = P / R Substituting into equation (2), we get: …………………………………(4) After simplification, the formula for the minimum bending radius under backlash constraint is obtained: …………………………………(5) Minimum bending radius (mm) under backlash constraint P Pitch (mm) h Thread height (mm) ac Design clearance (mm) α Tooth angle (°) 4.2 Parameter Influence Analysis As can be seen from formula (5), the minimum bending radius is related to the pitch. P and tooth height h Proportional to the design top gap ac It is inversely proportional to tan(α / 2). This means: 1. Increasing the pitch or tooth height can reduce the minimum permissible bending radius, thereby increasing the permissible curvature of the mandrel; 2. Increasing the design clearance or tooth angle can reduce the minimum bending radius; 3. Increase tooth height by increasing the tooth tip height coefficient. h This is an effective way to increase the permissible curvature of the mandrel.

[0025] 4.3 Example Verification Taking the trapezoidal thread Tr40×7 as an example, the parameters are: P=7mm, α=30°. ac =0.5mm. The calculation results for different tooth addendum coefficients are as follows: Table 1. Effect of tooth addendum coefficient on minimum bending radius (Tr40×7) Tooth tip height coefficient Total thread height (mm) min1(mm) Minimum radius (rounded down) Relative bending properties 0.30 2.4 62.7 ≥63 excellent 0.35 2.8 73.1 ≥73 excellent 0.40 3.2 83.6 ≥84 excellent 0.45 3.6 94.0 ≥94 good 0.50 4.0 104.5 ≥104 good 0.55 4.4 114.9 ≥115 middle 0.60 4.8 125.4 ≥125 middle The data in the table shows that when the tooth tip height coefficient decreases from 0.5 to 0.35, the minimum bending radius decreases from 104.5 mm to 73.1 mm, a reduction of 30%, and the bending performance is significantly improved.

[0026] Under the premise of meeting the requirements of strength and meshing stability, the tooth tip height coefficient is preferably taken in the range of 0 to 1, but the optimal range is tooth tip height coefficient (0.35 to 0.40) to obtain better bending performance.

[0027] For applications requiring small-radius bending, a small-module thread can be used, and the design clearance can be appropriately increased.

[0028] Calculation of minimum bending radius under overlap coefficient constraint 5.1 Formula Derivation The overlap factor is an important indicator for measuring the smoothness of threaded transmission, and is defined as the number of thread teeth actually in contact. For trapezoidal thread transmissions, the overlap factor ε is generally required to be ≥1.3 to ensure smooth transmission and load-bearing capacity.

[0029] In linear transmission, the overlap coefficient is defined by the following formula: ……………………………………(6) In the formula, L 0 represents the effective length of the nut. P The pitch is shown in Figure 2. When the mandrel bends, the upper and lower teeth are misaligned, causing the contact area between the coil spring and the nut to change.

[0030] From a geometric perspective, the effective contact length is shortened, and this shortening is related to the mean diameter of the spring. d 2, and its axial bending angle are linearly related. Assume the effective contact length after bending is... L e, which is related to the bending radius R The relationship can be represented as: ……………………………(7) In the formula, d 2 The mean diameter of the spring. θ = P / R This represents the relative deflection angle between adjacent teeth.

[0031] The overlap coefficient after bending is: ……………………………(8) To ensure smooth transmission, it is required ε e ≥ ε min ,Right now: …………………………(9) After simplification, the formula for the minimum bending radius under the overlap coefficient constraint is obtained: ……………………………(10) In the formula: Minimum bending radius (mm) under overlap coefficient constraint d 2: Spring mean diameter (mm) L 0: Effective length of nut (mm) ε min Minimum overlap coefficient (usually taken as 1.3) P Pitch (mm) Formula (10) holds true only if the denominator is greater than 0, i.e. L 0 > P × ε min .

[0032] It can be seen from formula (10): Minimum bending radius and spring mean diameter d 2 and pitch P Proportional; Minimum bending radius and effective nut length L The ratio is inversely proportional to 0; increasing the nut length can reduce the minimum bending radius. Minimum overlap coefficient ε min The larger the value, the larger the minimum bending radius.

[0033] 5.2 Example Verification Taking the trapezoidal thread Tr40×7 as an example, the parameters are: d 2 = 36.5mm, P =7mm, ε min =1.3 forms the following table 2: <![CDATA[Effective length L0 of nut (mm)]]> Overlap coefficient ε (direct axis state) <![CDATA[Overlap coefficient constrained minimum radius R min2 (mm)]]> Minimum permissible radius of curvature (mm) Design feasibility 20 2.86 23.4 ≥23 feasible 25 3.57 16.1 ≥16 feasible 30 4.29 12.2 ≥12 feasible 35 5 9.9 ≥10 feasible 40 5.71 8.3 ≥8 feasible 45 6.43 7.1 ≥7 feasible 50 7.14 6.2 ≥6 feasible The calculation results for different nut lengths are as follows: The effect of effective nut length on minimum bending radius (Tr40×7): As the nut length increases, the minimum bending radius gradually decreases. When the nut length increases from 20mm to 50mm, the minimum bending radius decreases from 23.4mm to 6.2mm, a reduction of 73.5%.

[0034] Constraint Correlation Analysis 6.1 The combined effect and design significance of the two constraints Table 3 Comparative Analysis of the Two Constraint Conditions Comparison Dimensions Backlash constraint Overlap coefficient constraint Physical essence Geometric interference problem (collision prevention) Transmission smoothness issues (ensuring contact) Mainly affects movement Curved movement (translational motion) Shaft-driven rotational motion Core design issues "Can it be approved?" (Path feasibility) "Can the transmission be smooth?" (Transmission quality) Key parameters <![CDATA[Tooth height h, designed backlash a c , thread angle α > <![CDATA[Nut length L 0. Mean diameter of spring d 2. Pitch P > Optimization direction Reduce tooth height and increase clearance Increase nut length and use multi-start thread Significance in flexible lead screw double meshing mechanisms Ensure the massage frame can move along complex curves Ensure the massage head can rotate smoothly to transmit power. In practical flexible lead screw double-meshing mechanisms, these two constraints are coupled and must be considered simultaneously: The movement process also requires smooth transmission: as the nut moves along a curve, the lead screw also rotates. If the overlap coefficient is insufficient, the movement process will become unstable and produce vibrations.

[0035] There are also geometric limitations to the shaft rotation process: even if the lead screw only rotates without axial movement, if the bending radius of the installation path is too small, it may still cause backlash interference, which will greatly increase the rotational resistance.

[0036] Therefore, the unified design criterion is to take the larger value between the two constraint calculation results: ……………………………(11) This analysis provides a clear physical picture for the design of flexible lead screw mechanisms: backlash constraints define the "spatial boundary" of curved movement, while overlap coefficient constraints define the "mass boundary" of shaft rotation drive. Together, they constitute two sides of the same coin in flexible transmission systems, defining the system's performance limits from the dimensions of "static geometry" and "dynamic transmission," respectively.

[0037] The following are design solutions for three different application scenarios, demonstrating how to balance two constraints based on motion requirements: Table 4 project Option A: High bending performance Option B: Balanced Design Option C: High load-bearing capacity design Application scenarios Precision medical devices Massage rehabilitation equipment Industrial heavy-duty equipment Thread specifications Tr20×4 Tr40×7 Tr60×9 Main exercise requirements Small radius curve movement Medium-curve movement + smooth rotation Large radius movement + high torque rotation Tooth tip height coefficient 0.35 0.45 0.50 Design clearance (mm) 0.30 0.50 0.80 Nut length 0 (mm) 25 35 50 min1(mm) 32.5 88.3 142.6 min2(mm) 41.2 85.4 135.0 Final min(mm) 41.2 88.3 142.6 Dominant constraints Overlap coefficient constraint Backlash constraint Backlash constraint Design Highlights Small module and low tooth height, sacrificing some load-bearing capacity to achieve high bending performance. Balancing bending performance and transmission smoothness, suitable for ergonomic curves Standard tooth height and large clearance ensure high load-bearing capacity, suitable for gentle curves. As can be seen from the examples: In Scheme A, the overlap coefficient constraint plays a dominant role, indicating that transmission smoothness becomes the main limiting factor when bending at small radii.

[0038] In schemes B and C, backlash constraints play a dominant role, indicating that geometric interference is the main limiting factor for medium and large radius bending.

[0039] Different application scenarios have different requirements for bending performance and load-bearing capacity, requiring the selection of design parameters accordingly.

[0040] In summary: This paper studies the trapezoidal threaded helical coil spring meshing bending mandrel mechanism and the curvature calculation method. The main conclusions are as follows: 1. A flexible transmission mechanism based on a trapezoidal screw thread helical coil spring is proposed. Through functional decoupling design, the massage mechanism can move precisely along a three-dimensional spatial curve, breaking through the limitations of traditional rigid lead screws.

[0041] 2. The formula for calculating the minimum bending radius under backlash constraint was derived: This formula reflects the influence of pitch, tooth height, design clearance, and tooth angle on bending performance. Reducing the tooth height by decreasing the addendum coefficient can significantly improve bending performance.

[0042] 3. The formula for calculating the minimum bending radius under the overlap coefficient constraint was derived: This formula reflects the influence of spring mean diameter, pitch, nut length, and minimum overlap factor on bending performance. Increasing the nut length can effectively improve bending performance.

[0043] 1. This study delves into the physical nature of two constraints and their relationship with the mode of motion: backlash constraints primarily restrict the feasibility of curved path movement (geometric interference problem), while overlap coefficient constraints mainly affect the transmission smoothness of shaft-rotor drives (contact quality problem). These two constraints define the performance boundaries of the flexible screw system from the two dimensions of "static geometry" and "dynamic transmission," respectively.

[0044] 2. Compared with cylindrical threads, trapezoidal threads have a significant advantage in bending transmission. Their two sides provide a clear guiding effect, reducing the possibility of lateral slippage and interference, and are more suitable for application scenarios with large curvature changes.

[0045] In practical design, both backlash constraints and overlap coefficient constraints need to be considered simultaneously, and the larger value of the two calculated results should be taken as the final minimum bending radius. For different application scenarios, the optimal balance between bending performance and load-bearing capacity can be achieved by adjusting the design parameters.

Claims

1. A method for calculating the curvature of a trapezoidal threaded coil spring meshing bending mandrel mechanism, characterized in that, It includes a bending mandrel, a trapezoidal threaded helical spring, a movable nut, a drive system, and a support and guide system; the trapezoidal threaded helical spring is sleeved on the outside of the bending mandrel, and the movable nut engages with the trapezoidal threaded helical spring; the curvature calculation method includes the following steps: Step 1: Determine the core parameters for curvature calculation, including pitch P, total thread height h, thread angle α, and design clearance a. c Spring mean diameter d2, nut effective length L0, minimum overlap coefficient ε min ; Step 2: Calculate the minimum bending radius R under backlash constraint. min1 ; Step 3: Calculate the minimum bending radius R under the overlap coefficient constraint. min2 ; Step 4: Take R min1 and R min2 The maximum value in the range is taken as the minimum bending radius R of the trapezoidal threaded coil spring meshing bending mandrel mechanism. min R min =max(R min1 ,R min2 ).

2. The curvature calculation method based on a trapezoidal threaded coil spring meshing bending mandrel mechanism according to claim 1, characterized in that: Step two specifically includes: Determine the relationship between the relative deflection angle θ of adjacent thread teeth and the bending radius R when the mandrel is bent: ; Calculate the radial displacement of the tooth tip when the relative deflection θ of adjacent teeth is θ. : ; Based on the constraint that the radial displacement of the tooth tip does not exceed the design clearance ,Will Substituting into the radial displacement formula, the formula for the minimum bending radius under backlash constraint is derived: .

3. The curvature calculation method based on a trapezoidal threaded coil spring meshing bending mandrel mechanism according to claim 2, characterized in that: The total thread height h is optimized by adjusting the tooth tip height coefficient, which ranges from 0.35 to 0.

40.

4. The curvature calculation method based on a trapezoidal threaded coil spring meshing bending mandrel mechanism according to claim 1, characterized in that: Step three specifically includes: Formula for determining the overlap coefficient in linear transmission: ; Calculate the effective contact length after the mandrel is bent. : ; Will Substituting the effective contact length formula, the overlap coefficient formula under bending conditions is derived: ; Based on transmission smoothness constraints The formula for the minimum bending radius under the overlap coefficient constraint is derived as follows: 。 5. The curvature calculation method based on a trapezoidal threaded coil spring meshing bending mandrel mechanism according to claim 4, characterized in that: The minimum overlap coefficient The value is 1.3, and the preconditions must be met. .

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