Design method of the pitch curve of the fastest-rotating non-circular gear driven by arc length constraint

Abstract

The present invention discloses a design method for the pitch curve of a fastest-rotating non-circular gear driven by arc length constraint. In this design method, according to the actual application scenarios of the non-circular gear in mechanical equipment, a designer selects an appropriate arc length of the pitch curve as a constraint condition, and takes the shortest time of the non-circular gear's rotary motion as the design goal. Based on the principles of differential calculus and kinematics, a pitch curve of a non-circular gear with the fastest-rotating characteristic driven by arc length constraint is designed. Then, according to the meshing principle of the non-circular gear and its actual application in mechanical equipment, a non-circular gear pair with the fastest-rotating characteristic that meets various application scenarios can be designed, which expands the non-linear transmission relationship between two shafts to a certain extent and broadens the application of non-circular gears in various mechanical industries such as light industry and textile, instrumentation, automobiles, and machine tools, enabling the designed non-circular gear drive system to meet the requirements of modern mechanical equipment design for performance aspects such as high speed, heavy load, light weight, high precision, and automation.

Description

Technical Field

[0001] The invention belongs to the technical field of non-circular gear design, and relates to an innovative design method for the pitch curve of a non-circular gear, in particular to a design method for the pitch curve of a brachistochrone non-circular gear driven by arc length constraint. Background Art

[0002] Due to the high transmission performance of non-circular gears, in the design process of mechanical equipment, non-circular gears have gradually begun to replace linkages or cams to achieve the non-linear motion relationship between the main and driven mechanisms, and this non-linear motion relationship is mainly determined by the shape of the pitch curve of the non-circular gear. For example, non-uniform continuous swinging or intermittent motion can be achieved by a non-circular gear transmission system with a non-closed pitch curve (or called an open pitch curve); while continuous one-way non-uniform rotary motion can be achieved by a non-circular gear transmission system with a closed pitch curve.

[0003] Currently, most of the non-circular gears in existing non-circular gear transmission systems are designed and verified with conventional planar curves such as elliptical curves, higher-order elliptical curves or eccentric circular curves for the pitch curve, making the existing non-circular gears have certain limitations in transmitting the non-linear motion relationship between two shafts and unable to meet the requirements of modern mechanical equipment design for performance aspects such as high speed, heavy load, light weight, high precision and automation, which to a certain extent limits the application of non-circular gears in modern mechanical equipment. Summary of the Invention

[0004] The purpose of the invention is to expand the non-linear transmission relationship between two shafts and expand the application of non-circular gears in various mechanical industries such as light industry and textile, instrumentation, automobiles and machine tools. Considering the problems existing in the existing design methods, according to the constraint conditions for the pitch curve design of non-circular gears, the invention discloses a design method for the pitch curve of a brachistochrone non-circular gear driven by arc length constraint.

[0005] This design method is that the designer selects an appropriate pitch curve arc length as the constraint condition according to the actual application scenario of the non-circular gear in the mechanical equipment, takes the shortest time of the non-circular gear rotary motion as the design goal, designs the pitch curve of the non-circular gear with the brachistochrone rotation characteristic driven by arc length constraint based on the principles of differential calculus and kinematics, and then can design a non-circular gear pair with the brachistochrone rotation characteristic that meets various application scenarios according to the meshing principle of non-circular gears and its actual application in the mechanical equipment.

[0006] The non-circular gear pair with the characteristic of the fastest return can form a new non-linear transmission relationship, which to a certain extent expands the non-linear transmission relationship between two shafts and broadens the application of non-circular gears in various mechanical industries such as light industry and textile, instrumentation, automobiles, and machine tools, enabling the designed non-circular gear transmission system to meet the requirements of modern mechanical equipment design in terms of performance such as high speed, heavy load, light weight, high precision, and automation.

[0007] The design method of the pitch curve of the fastest-return non-circular gear driven by arc length constraint includes the following steps:

[0008] Step 1: Through the analysis and calculation of the non-circular gear transmission relationship in the mechanical equipment, obtain the angular velocity of the non-circular gear and the constraint conditions such as the arc length of the pitch curve, the left and right boundaries, etc.

[0009] Step 2: According to the principles of kinematics and differential calculus, taking the arc length of the non-circular gear pitch curve as the constraint driving condition and the shortest return time of the non-circular gear rotation as the design goal, establish a design model and method for the pitch curve of the non-circular gear with the characteristic of the fastest return driven by arc length constraint.

[0010] Step 3: According to the rotation requirements output by the non-circular gear transmission relationship in the mechanical equipment, taking the maximum polar angle of the non-circular gear output rotation as the input condition, design and calculate the center distance of the external or internal meshing non-circular gear pair that meets specific requirements.

[0011] Step 4: According to the meshing principle of the non-circular gear, derive the external or internal meshing non-circular gear pitch curve conjugate to the pitch curve of the non-circular gear with the characteristic of the fastest return driven by arc length constraint in Step 2, and realize the modeling and design calculation of the non-linear transmission relationship with the characteristic of the fastest return driven by arc length constraint.

[0012] Further, the fixed coordinate system Γ(o-xy) is rigidly connected to the rotation center o of the non-circular gear pitch curve r(θ) (θ ∈ [0, 2π]). The polar angle θ is measured counterclockwise along the positive x-axis. The left and right boundary points of the pitch curve r(θ) are a and b respectively, and the polar angles corresponding to the left boundary point a and the right boundary point b are θ a and θ b , where 0 ≤ θ a < θ b ≤ 2π. Assuming that d S is a small arc length on the non-circular gear pitch curve r(θ), then according to the principle of differential calculus, the expressions of the small arc length ds and the arc length S are respectively:

[0013]

[0014] According to the kinematic principle, when the non-circular gear rotates counterclockwise at an angular velocity ω and passes through a micro-arc length ds on the pitch curve r(θ), the rotation time dt used can be expressed as:

[0015]

[0016] Integrating both sides of the above equation, the rotation time T used when the non-circular gear rotates from the left boundary point a to the right boundary point b of the pitch curve r(θ) can be obtained as:

[0017]

[0018] Furthermore, it can be seen from equations (1) to (4) that under the given arc length constraint conditions, for different non-circular gear pitch curves r(θ), the rotation time T used by the non-circular gear to pass through the pitch curve r(θ) is also different. Then the minimum value T[r(θ)] of the rotation time min can be expressed as:

[0019]

[0020] In order to obtain the non-circular gear pitch curve r(θ) that satisfies the above equation, according to the variational principle, an auxiliary functional as described in equation (6) is established:

[0021]

[0022] In the formula, λ is the undetermined Lagrange multiplier;

[0023] It can be seen from the above formula that the differential variable θ is not explicitly included in the integrand F. Therefore, the non-circular gear pitch curve r(θ) to be solved should satisfy the following constraint conditions:

[0024]

[0025] Rearranging the above formula, we get:

[0026]

[0027] Integrating both sides of the above formula, we can obtain:

[0028]

[0029] In the formula, c 1 is the undetermined integration constant;

[0030] Substituting the F expression in equation (6) into equation (9) and simplifying, we can obtain:

[0031]

[0032] Assume:

[0033]

[0034] In the formula, μ is a parameter to be determined;

[0035] Substituting Equation (11) into Equation (10), the parametric equation of the pitch curve r(θ) with respect to the parameter μ can be obtained as:

[0036]

[0037] Differentiating both sides of the above equation with respect to θ, we can get:

[0038]

[0039] Combining Equation (11), the differential of the polar angle θ with respect to the parameter μ can be obtained:

[0040]

[0041] Integrating both sides of Equation (14) simultaneously, the parametric equation of the polar angle θ with respect to the parameter μ can be obtained as:

[0042]

[0043] In the formula, c 2 is an integration constant to be determined;

[0044] Assume:

[0045]

[0046] In the formula, μ a and μ b are parameters of μ corresponding to the polar angles θ a and θ b to be determined;

[0047] Combining Equation (12) and Equation (15), the parametric equation of the pitch curve r(θ) of the non-circular gear with the fastest-return characteristic driven by the arc length constraint with respect to the parameter μ can be obtained as:

[0048]

[0049] Assume that when the left and right boundary conditions a(θ a , r(θ a )) and b(θ b , r(θ b )) of the pitch curve r(θ) of the non-circular gear and the arc length S are given, the constraint conditions satisfied by the pitch curve r(θ) of the non-circular gear with the fastest-return characteristic driven by the arc length constraint can be obtained as:

[0050]

[0051] Furthermore, according to the meshing principle of non-circular gears, the conjugate external or internal meshing non-circular gear pitch curve r e (θ e )、r i (θ i ) of the non-circular gear conjugate to the pitch curve r(θ) of the non-circular gear with the fastest-return characteristic driven by arc length constraint:

[0052]

[0053] wherein, E e and E i are the center distances of the pitch curves of the conjugate external and internal meshing non-circular gear pairs respectively;

[0054] Then, according to the application scenario of the designed non-circular gear, the maximum values θ e and θ i of θ emax and θ imax are given, and the center distances E e 、E i of the conjugate external or internal meshing non-circular gear pair to the non-circular gear pitch curve r(θ) described in Equation (17) are solved through Equation (19) or Equation (20).

[0055] Furthermore, according to the solved center distances E e 、E i of the conjugate external and internal meshing non-circular gear pairs to the non-circular gear pitch curve r(θ), the conjugate external or internal meshing non-circular gear pitch curves r e (θ e )、r i (θ i ) to the non-circular gear pitch curve r(θ) described in Equation (17) are designed and calculated through Equation (19) or Equation (20), and the non-linear transmission relationships i e (θ)、i i (θ) of the external or internal meshing non-circular gear with the fastest-return characteristic driven by arc length constraint are calculated by Equation (21) or Equation (22);

[0056]

[0057] wherein, r(θ) is the non-circular gear pitch curve satisfying the fastest-return characteristic driven by arc length constraint described in Equation (17). Brief Description of the Drawings

[0058] Figure 1 This is the design model of the non-circular gear pitch curve with the fastest-return characteristic driven by arc length constraint in the embodiment of the present invention.

[0059] Figure 2This is the design flow block diagram of the non-circular gear pitch curve r(θ) with the fastest return characteristic driven by arc length constraint and its conjugate non-circular gear pitch curve in the embodiment of the present invention.

[0060] Figure 3 This is the non-circular gear pitch curve r(θ) with the fastest return characteristic driven by arc length constraint in the embodiment of the present invention.

[0061] Figure 4a This is for the embodiment of the present invention Figure 3 The conjugate external meshing non-circular gear pitch curve r e (θ e ) of the non-circular gear pitch curve r(θ) with the fastest return characteristic.

[0062] Figure 4b This is for the embodiment of the present invention Figure 3 The conjugate internal meshing non-circular gear pitch curve r i (θ i ) of the non-circular gear pitch curve r(θ) with the fastest return characteristic.

[0063] Figure 5a This is for the embodiment of the present invention Figure 4a The non-linear transmission relationship i e (θ) corresponding to the external meshing non-circular gear pitch curve with the fastest return characteristic described in

[0064] Figure 5b This is for the embodiment of the present invention Figure 4b The non-linear transmission relationship i i (θ) corresponding to the internal meshing non-circular gear pitch curve with the fastest return characteristic described in

[0065] Next, in conjunction with the accompanying drawings, the technical content and design principle of the design method of the non-circular gear pitch curve with the fastest return characteristic driven by constraint of the present invention will be introduced in detail. Specific embodiments

[0066] Figure 1 This is the design model of the non-circular gear pitch curve with the fastest return characteristic driven by arc length constraint. In the figure, the fixed coordinate system Γ(o-xy) is rigidly connected to the rotation center o of the non-circular gear pitch curve r(θ) (θ ∈ [0, 2π]). The polar angle θ is measured counterclockwise along the positive x-axis. a and b are respectively the left and right boundary points of the pitch curve r(θ). θ a and θ b are respectively the polar angles corresponding to the left boundary point a and the right boundary point b (0 ≤ θ a < θ b ≤ 2π). Assume d SLet \(ds\) be an infinitesimal arc length on the pitch curve \(r(\theta)\) of a non-circular gear. According to the principles of differential calculus, the expressions for the infinitesimal arc length \(ds\) and the arc length \(S\) are as follows:

[0067]

[0068] According to the principles of kinematics, when the non-circular gear rotates counterclockwise with an angular velocity \(\omega\) through the infinitesimal arc length \(ds\) on the pitch curve \(r(\theta)\), the rotational time \(dt\) used can be expressed as:

[0069]

[0070] Integrating both sides of the above equation, the rotational time \(T\) used when the non-circular gear rotates from the left boundary point \(a\) to the right boundary point \(b\) of the pitch curve \(r(\theta)\) can be obtained as:

[0071]

[0072] From equations (1) to (4), it can be seen that under the given arc length constraint conditions (i.e., the arc length \(S\) is set by the designer according to different application scenarios of the non-circular gear), for different non-circular gear pitch curves \(r(\theta)\), the rotational time \(T\) used by the non-circular gear to pass through the pitch curve \(r(\theta)\) is also different. Then the minimum value \(T[r(\theta)]\) of the rotational time min can be expressed as:

[0073]

[0074] In order to obtain the non-circular gear pitch curve \(r(\theta)\) that satisfies the above equation, according to the principles of the calculus of variations, an auxiliary functional as described in equation (6) is established:

[0075]

[0076] where \(\lambda\) is an undetermined Lagrange multiplier.

[0077] From the above equation, it can be seen that the integrand \(F\) does not explicitly contain the differential variable \(\theta\). Therefore, the non-circular gear pitch curve \(r(\theta)\) to be solved should satisfy the following constraint conditions:

[0078]

[0079] Rearranging the above equation gives:

[0080]

[0081] Integrating both sides of the above equation gives:

[0082]

[0083] where \(c\) 1 is an undetermined integration constant.

[0084] Substitute the F expression in Equation (6) into Equation (9) and simplify to obtain:

[0085]

[0086] Assume:

[0087]

[0088] where μ is a parameter to be determined.

[0089] Substitute Equation (11) into Equation (10) to obtain the parametric equation of the pitch curve r(θ) with respect to the parameter μ as:

[0090]

[0091] Differentiate both sides of the above equation to obtain:

[0092]

[0093] Combine Equation (11) to obtain the differential of the polar angle θ with respect to the parameter μ:

[0094]

[0095] Integrate both sides of Equation (14) simultaneously to obtain the parametric equation of the polar angle θ with respect to the parameter μ as:

[0096]

[0097] where c 2 is an integration constant to be determined.

[0098] Assume:

[0099]

[0100] where μ a and μ b are parameters μ corresponding to the polar angles θ a and θ b respectively, which are parameters to be determined.

[0101] Combine Equation (12) and Equation (15) to obtain the parametric equation of the pitch curve r(θ) of the non-circular gear with the fastest return characteristic driven by the arc length constraint with respect to the parameter μ as:

[0102]

[0103] Assume that the left and right boundary conditions a(θ a , r(θ a )) and b(θ b , r(θb ) and the arc length S are given, the constraint condition satisfied by the pitch curve r(θ) of the non-circular gear with the fastest return characteristic driven by the arc length constraint can be obtained as follows:

[0104]

[0105] In addition, according to the meshing principle of non-circular gears, the pitch curves r e (θ e ) and r i (θ i ) that are conjugate to the pitch curve r(θ) of the non-circular gear with the fastest return characteristic driven by the arc length constraint:

[0106]

[0107] In the formula, E e and E i are the center distances of the pitch curves of the conjugate external and internal meshing non-circular gear pairs respectively.

[0108] Figure 2 is the design process of the pitch curve r(θ) of the non-circular gear with the fastest return characteristic driven by the arc length constraint and its conjugate non-circular gear pitch curve, where the known conditions are the angular velocity ω of the non-circular gear, the given arc length S of the pitch curve r(θ), and the left and right boundary conditions a(θ a , r(θ a )) and b(θ b , r(θ b ). The values of the undetermined parameters c 1 , c 2 , λ, μ a and μ b in the pitch curve r(θ) of the non-circular gear with the fastest return characteristic driven by the arc length constraint can be determined by equations (17) and (18). Then, the designer gives the maximum values θ e or θ i of θ according to the application occasion of the designed non-circular gear. Finally, the center distances E emax , E imax of the conjugate external or internal meshing non-circular gear pairs with the pitch curve r(θ) of the non-circular gear are solved through equation (19) or equation (20), and the corresponding pitch curves r e , E i of the conjugate external or internal meshing non-circular gears r e (θ e ), r i (θ i ).

[0109] Figure 3It is the pitch curve r(θ) of a non-circular gear driven by arc length constraint and having the characteristic of the quickest return. The non-circular gear pitch curve r(θ) described in the figure is the non-circular gear pitch curve with the characteristic of the quickest return solved by using Equations (17) and (18) according to the known conditions given in the first column of Table 1, and its related design parameters are listed in Table 1. The units of all parameters in Table 1 adopt the international standard unit system. In particular, for all content related to design parameters involved in the present invention, if there is no additional explanation, the units of the design parameters described therein adopt the international standard unit system.

[0110] Table 1 Parameter equations and design parameters of the pitch curve r(θ) of a non-circular gear driven by arc length constraint and having the characteristic of the quickest return

[0111]

[0112] Figure 4a and Figure 4b are respectively the external meshing non-circular gear pitch curve r Figure 3 (θ e ) and the internal meshing non-circular gear pitch curve r e (θ i ) that are conjugate to the non-circular gear pitch curve r(θ) described in i . The equations of the non-circular gear pitch curve r e (θ e ) and the non-circular gear pitch curve r i (θ i ) and their related design parameters have been listed in Table 2, where the maximum polar angles θ emax , θ imax of the external meshing or internal meshing non-circular gear pitch curve can be preferentially given by the designer according to the actual application of the non-circular gear pair in the mechanical equipment, and then the center distances E e , E i of the external meshing or internal meshing non-circular gear pair and the pitch curve equations r e (θ e ), r i (θ i ) are determined by Equations (19) and (20) respectively.

[0113] Table 2 External meshing non-circular gear pitch curve r Figure 3 (θ e ) and internal meshing non-circular gear pitch curve r e (θ i ) that are conjugate to the non-circular gear pitch curve r(θ) described in i

[0114]

[0115] Figure 5a andFigure 5b are respectively related to Figure 4a and Figure 4b the non - linear transmission relationships i e (θ) and i i (θ) corresponding to the pitch curves of the external - meshing and internal - meshing non - circular gears described in e (θ) and i i (θ) shown in the figure. The non - linear transmission relationships i

[0116]

[0117] In the formula, the specific values of r(θ) and E e 、E i can be determined by the data in Table 1 and Table 2 respectively.

[0118] From the above design calculations, it can be seen that a design method for the pitch curve of a fastest - rotating non - circular gear driven by arc - length constraint disclosed by the present invention can, according to the actual requirements of mechanical equipment design, design a pitch curve of a non - circular gear with the fastest - rotating characteristic and its conjugate external - meshing or internal - meshing non - circular gear pitch curve that meet various application scenarios by taking different pitch - curve arc lengths as constraint - driving conditions. Thus, the pitch - curve design method disclosed by the present invention can, to a certain extent, expand the non - linear transmission relationship between two shafts and expand the application of non - circular gears in various mechanical industries such as light industry and textile, instrumentation, automobiles, and machine tools, so that the designed non - circular gear transmission system can meet the requirements of modern mechanical equipment design for performance aspects such as high speed, heavy load, light weight, high precision, and automation.

[0119] For those skilled in the art, it is obvious that the present invention is not limited to the details of the above - mentioned exemplary embodiments, and can be implemented in other specific forms without departing from the spirit or basic characteristics of the present invention. Therefore, from any point of view, the embodiments should be regarded as exemplary and non - restrictive. The scope of the present invention is defined by the appended claims rather than the above description. Therefore, it is intended to encompass all changes falling within the meaning and scope of the equivalent elements of the claims in the present invention.

Claims

1. Design method for pitch curve of non-circular gear with the fastest rotation characteristic driven by arc length constraint, characterized in that, it includes the following steps: Step 1: Through the analysis and calculation of the transmission relationship of non-circular gears in mechanical equipment, obtain the angular velocity of the non-circular gear, the arc length of the pitch curve, and the constraint conditions such as the left and right boundaries; Step 2: According to the principles of kinematics and differential calculus, taking the arc length of the pitch curve of the non-circular gear as the constraint driving condition and the shortest rotation time of the non-circular gear as the design goal, establish a design model and method for the pitch curve of the non-circular gear with the fastest rotation characteristic driven by arc length constraint; Step 3: According to the rotation requirements output by the transmission relationship of non-circular gears in mechanical equipment, taking the maximum polar angle of the output rotation of the non-circular gear as the input condition, design and calculate the center distance of the external or internal meshing non-circular gear pair that meets specific requirements; Step 4: According to the meshing principle of non-circular gears, derive the external or internal meshing non-circular gear pitch curve conjugate to the non-circular gear pitch curve with the fastest rotation characteristic driven by arc length constraint in Step 2, and realize the modeling and design calculation of the non-linear transmission relationship with the fastest rotation characteristic driven by arc length constraint; The specific content of Step 1 is as follows: The fixed coordinate system Γ(o-xy) is rigidly connected to the center of rotation o of the non-circular gear pitch curve r(θ) (θ ∈ [0, 2π]). The polar angle θ is measured counterclockwise along the positive x-axis. The left and right boundary points of the pitch curve r(θ) are a and b respectively, and the polar angles corresponding to the left boundary point a and the right boundary point b are θ a and θ b , where 0 ≤ θ a < θ b ≤ 2π, d S is a small arc length on the non-circular gear pitch curve r(θ). When the non-circular gear rotates counterclockwise at an angular velocity ω and passes through the small arc length ds on the pitch curve r(θ), the rotation time dt used can be expressed as: Integrating both sides of the above formula, the rotation time T used when the non-circular gear rotates from the left boundary point a to the right boundary point b of the pitch curve r(θ) can be obtained as: The parametric equation of the non-circular gear pitch curve r(θ) with the fastest rotation characteristic driven by arc length constraint about the parameter μ is: Among them, c 1、 c 2 is an integral constant to be determined, λ is a Lagrange multiplier to be determined, μ is a variable parameter to be determined, μ a and μ b are the parameters μ corresponding to the polar angles θ a and θ b respectively, which are parameters to be determined; The specific content of Step 3 is as follows: The pitch curve r(θ) of the external meshing non-circular gear conjugated with the pitch curve r(θ) of the non-circular gear with the characteristic of the fastest rotary motion driven by arc length constraint e (θ e ), the pitch curve r(θ) of the internal meshing non-circular gear i (θ i ): where E e and E i are the center distances of the conjugate external and internal meshing non-circular gear pair pitch curves, respectively; Then, according to the application scenario of the designed non-circular gear, θ is given. e and θ i The maximum value of θ emax and θ imax are used to solve the center distance E of the external meshing non-circular gear pair conjugate to the pitch curve r(θ) of the non-circular gear e or the center distance E of the internal meshing non-circular gear pair i ; The specific content of Step 4 is as follows: According to the center distances \(E\) of the external and internal meshing non-circular gear pairs conjugated with the pitch curve \(r(\theta)\) of the non-circular gear solved e 、\(E\) i , design and calculate the pitch curve \(r\) e (\(\theta\) e ) of the external meshing non-circular gear conjugated with the pitch curve \(r(\theta)\) described by Equation (17) or the pitch curve \(r\) i (\(\theta\) i ) of the internal meshing non-circular gear through Equation (19) or Equation (20), and calculate the non-linear transmission relationship \(i\) e (\(\theta\)) of the external meshing non-circular gear with the fastest rotation characteristic driven by arc length constraint or the non-linear transmission relationship \(i\) i (\(\theta\)) of the internal meshing non-circular gear by Equation (21) or Equation (22); In the formula, r(θ) is the non-circular gear pitch curve with the fastest rotation characteristic driven by arc length constraint as described in formula (17).

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