Parabolic gear mechanism with end face circular arc and parabolic combined tooth profile

Abstract

The application provides a parabolic tooth trace gear mechanism with an end face circular arc and parabolic combined tooth profile, and belongs to the field of gear transmission. The parabolic tooth trace gear mechanism with the end face circular arc and parabolic combined tooth profile comprises an outer meshing pinion and a gear wheel with parallel axes. The tooth surfaces of the pinion and the gear wheel both have parabolic tooth trace structures and are formed by end face combined tooth profile curves moving with contact points. The contact lines are all parabolic after being developed along a cylindrical surface. The parabolic lines are determined by the basic design parameters of the meshing line parameter equation and the degree of coincidence, the number of teeth and the transmission ratio. When correctly installed, at least one pair of gear teeth of the pinion and the gear wheel realizes pure rolling meshing contact at the node. The pinion and the gear wheel rotate under the driving of a driver, and the transmission between two shafts is realized. The parabolic tooth trace gear mechanism with the end face circular arc and parabolic combined tooth profile has the advantages of low friction and wear, high transmission efficiency, large single-stage transmission ratio, strong bearing capacity and the like, and can be widely applied to the transmission system design of mechanical equipment.

Description

Technical Field

[0001] This invention relates to the field of transmission gear technology, and in particular to a parabolic toothed gear mechanism with a tooth profile combining an end face arc and a parabola. Background Technology

[0002] Gears are widely used in industrial equipment such as robot joint reducers, automotive gearboxes, wind turbine gearboxes, and machine tool headstocks to transmit motion and power, and are considered the "heart" of machines. Currently, conventional parallel-axis cylindrical gear drives, such as involute spur gears, helical gears, and circular arc gears, struggle to overcome transmission failures caused by relative sliding of tooth surfaces, including friction and wear, scuffing, plastic deformation, thermal deformation, vibration, and noise. Furthermore, gear lubrication systems increase the overall weight and cost of the machine, and in extreme environments such as high temperature, low temperature, high pressure, vacuum, and strong radiation, lubricants may fail, and their emissions cause irreversible environmental pollution. With the rapid development of the intelligent manufacturing industry, conventional gear products can no longer meet the precision transmission requirements of high-end equipment such as automotive automatic transmissions, robot reducers, wind turbine gearboxes, and high-speed rail transit. High-performance gear products are heavily reliant on imports. High-performance gear design and manufacturing technology has become a key factor restricting the development of the high-end equipment manufacturing field, and how to avoid relative sliding of tooth surfaces and improve gear transmission performance is one of the key problems that urgently needs to be solved in this field.

[0003] To address the various problems associated with parallel shaft gear transmissions, researchers both domestically and internationally have successively invented single-circular-arc gears, double-circular-arc gears, and circular-arc toothed cylindrical gears. For example, Chinese patent application number 202110318591.7 discloses "A double-circular-arc gear reduction transmission device with small tooth difference and a method for forming double-circular-arc teeth," and Chinese patent application number 202123012746.9 discloses "A variable hyperbolic circular-arc toothed cylindrical gear pair structure." However, the tooth profiles of the small and large gears in these double-circular-arc gears are based on the same hob cutter and cut using the generating method. Furthermore, to ensure correct meshing of the large and small gears, the pressure angles at the two meshing points of the hob tooth profile are set to equal values. Therefore, the limitation of the existing double circular arc gear mechanism is that, due to the constraint that the pressure angles of the two meshing points of the tooth profile are equal, its structure is not an optimal load-bearing design structure. When the equipped mechanical equipment is subjected to heavy-load transmission, it may cause tooth breakage and thus lead to accidents. The tooth surface design of the above-mentioned hyperbolic circular arc cylindrical gear pair is limited by the machining cutter head parameters, and the tooth tips at both ends become sharp. The effective contact area of ​​the tooth surface is only concentrated in a limited area at the center of the tooth width. Therefore, when applied to high-load transmission, there is a risk of tooth breakage. At the same time, the relatively large relative sliding of the tooth surface leads to severe friction and wear. Summary of the Invention

[0004] In view of this, in order to solve the problems in the prior art where the effective contact area of ​​the tooth surface in the gear mechanism is only concentrated in a limited area at the center of the tooth width, which poses a risk of tooth breakage, and where the relative sliding of the tooth surface is large and friction and wear are severe, the embodiments of the present invention provide a parabolic tooth gear mechanism with a tooth profile that combines an end face arc and a parabola.

[0005] Embodiments of the present invention provide a parabolic toothed gear mechanism with a combined end-face arc and parabolic tooth profile, comprising a pair of gears consisting of a small gear and a large gear with parallel axes. The small gear and the large gear engage in pure rolling meshing transmission. The end-face tooth profile curves of the small gear and the large gear are composed of end-face working tooth profile curves and tooth root transition curves, and the end-face tooth profile curves of the small gear and the large gear are symmetrical on both sides. The end-face working tooth profiles of the small gear and the large gear are a combined end-face arc and parabolic tooth profile. The tooth surfaces of the small gear and the large gear have a parabolic toothed structure. At least one pair of gear teeth of the small gear and the large gear are located at a node to achieve pure rolling meshing contact. The meshing points of the small gear and the large gear rotate relative to each other to form a meshing line, which forms two contact lines on the tooth surfaces of the small gear and the large gear respectively.

[0006] Furthermore, the tooth surface structure of the small wheel and the large wheel is formed by the tooth profile curves of the end face of the small wheel and the large wheel moving along the contact line of the tooth surface as the contact point moves, and the contact line is an axisymmetric parabola after being unfolded along the pitch cylinder surface of the small wheel and the large wheel.

[0007] Furthermore, the working tooth profile curves on the left side of the end faces of the small wheel and the large wheel are composed of both circular arcs and parabolic planar curves at the inter-tooth control point P. bi The wheels are smoothly connected, and the inter-tooth control point G on the right side of the tooth profile is used when installing the small and large wheels. bi With node P i Overlapping, control point G bi Control point P between teeth on the left working tooth profile curve bi Obtained by axisymmetry; the shape of the working tooth profile curve on the end face is determined by the tooth tip control point P. ai Inter-tooth control point P bi and tooth root control point P ci Specifically, the combination type of the working tooth profile curves of both the small and large gears from the tooth tip to the tooth root is CP, where C and P represent circular arc and parabola, respectively. The circular arc is the upper curve of the working tooth profile, and the parabola is the lower curve of the working tooth profile. The tooth root transition curve is controlled by the tooth root control point P. ci With tooth root control point P di The determined Hermite curve, and the root transition curve and the lower working profile curve at the tooth root control point P. ci Smooth connection.

[0008] Furthermore, the tooth tip control point P of the left working tooth profile of the small wheel and the large wheel aiFrom the tooth tip circle radius R ai and offset angle χ ai Confirmed, χ ai J is the reference point for the tooth tips of the small and large gears. ai The angle of clockwise rotation around the center; control point P at the root of the tooth. ci From the radius R of the tooth root circle ci and offset angle χ ci Confirmed, χ ci J is the reference point for the tooth root of the small and large wheels. ci The angle of clockwise rotation around the center; where J is the reference point for the tooth tips of the small and large gears. ai An involute with the same base circle radius, end face pressure angle, and radius R as the small and large wheels respectively. ai The intersection of the addendum circles; the reference point J of the root of the pinion and gear teeth. ci An involute with the same base circle radius, end face pressure angle, and radius R as the small and large wheels respectively. ci The intersection of the tooth root circles.

[0009] Furthermore, the contact line between the tooth surfaces of the small wheel and the large wheel is determined by the following method:

[0010] In o p -x p ,y p ,z p o k -x k ,y k ,z k and o g -x g ,y g ,z g In the three spatial coordinate systems, z p The axis of rotation of the shaft coincides with that of the small wheel, z g The axis of rotation of the shaft coincides with that of the large wheel, z k The shaft passes through the meshing point M a and M b The meshing lines KK coincide, and z k axis and z p z g The axes are parallel to each other, x p With x g Coincident axis, x k With x g The axis is parallel, o p o g The distance is a; coordinate system o1-x1,y1,z1 is fixed to the small wheel, and coordinate system o2-x2,y2,z2 is fixed to the large wheel. The coordinate systems o1-x1,y1,z1 and o2-x2,y2,z2 of the small wheel and the large wheel are respectively fixed to coordinate system o at their initial positions. p -xp ,y p ,z p and o g -x g ,y g ,z g When they coincide, the meshing point M is reached. a and M b The overlap is denoted as M, and the small wheel revolves around z at a uniform angular velocity ω1. p The shaft rotates clockwise, and the large wheel revolves around the z-axis at a uniform angular velocity ω2. g The axis rotates counterclockwise. After a period of time from the initial position, the coordinate system o1-x1,y1,z1 and o 2- x2, y2, z2 rotate respectively, and the small wheel revolves around z. p The shaft rotated Angle, large wheel around z g The shaft rotated horn;

[0011] When the small wheel and the large wheel mesh and drive, the meshing point M is set. a and M b Starting from the origin o k The motion begins along the line of engagement KK, moving up and down. The parametric equation describing the motion at the engagement point is:

[0012]

[0013] In equation (1), t is the meshing point M. a and M b The motion parameters are variables, 0≤t≤Δt; b is the tooth width; "+" corresponds to the meshing point M. a "-" corresponds to the engagement point M b ;

[0014] To ensure constant gear ratio meshing, the rotation angles of the pinion and gear and the motion of the meshing point must have a linear relationship, as shown in the following formula:

[0015]

[0016] In formula (2) i is the linear proportionality coefficient for the motion at the meshing point; 12 This is the transmission ratio between the small wheel and the large wheel;

[0017] When the meshing point M a and M b As they move along the line of engagement KK, they simultaneously form contact lines C on the pinion tooth surface and the gear tooth surface, respectively. p and C g Based on the coordinate transformation, the coordinate system o is obtained. p -x p ,y p ,zp o k -x k ,y k ,z k and o g -x g ,y g ,z g The homogeneous coordinate transformation matrix between o1-x1,y1,z1 and o2-x2,y2,z2 is:

[0018]

[0019] in,

[0020]

[0021]

[0022] In equations (4) and (5), R1 is the pitch cylinder radius of the smaller wheel, R2 is the pitch cylinder radius of the larger wheel, and α t The end face pressure angle at the meshing point;

[0023] The contact line C of the pinion tooth surface is obtained from equations (1) and (4). p The parametric equation is:

[0024]

[0025] The contact line C of the large gear tooth surface is obtained from equations (1) and (5). g The parametric equation is:

[0026]

[0027] Furthermore, the specific tooth profile structure of the left end face of the small wheel and the large wheel is determined by the following method:

[0028] Control points P between the teeth of the large and small gears respectively bi Establish a local coordinate system S pbi (o pbi -x pbi y pbi z pb i), i = 1, 2, where i = 1 represents the small gear and i = 2 represents the large gear. The parametric equation of the upper circular arc curve used for the combination of working tooth profile curves is obtained as follows:

[0029]

[0030] In equation (8), i = 1, 2, where i = 1 represents the small wheel and i = 2 represents the large wheel; ξ ai Let ξ be the angle parameter of the circular arc curve. aimin and ξ aimaxThey are ξ ai The minimum and maximum values ​​of ρ ai These are the radii of the small and large wheels, respectively, when the offset angle χ is determined. ai Tooth tip circle radius R ai , ρ ai ζ aimin and ζ aimax All of these can be solved, thereby determining the upper circular arc tooth profile curve;

[0031] The parametric equation for the lower parabolic curve used in the combination of working tooth profile curves is obtained as follows:

[0032]

[0033] In equation (9), i = 1, 2, where i = 1 represents the small wheel and i = 2 represents the large wheel; a pi Let x be the coefficient of the quadratic term of the parabolic curves of the small and large wheels. pi The x-axis is the coordinate axis. pbi The parameter, x pimin and x pimax They are x pi The minimum and maximum values ​​are taken; when the radius R of the tooth root circle is determined. ci , offset angle χ ci At that time, a pi x pimin and x pimax All of these can be solved to determine the lower parabolic tooth profile curve;

[0034] Based on the coordinate transformation, we can obtain the coordinate system S. pbi (o pbi -x pbi y pbi z pbi ) and S Invi (o Invi -x Invi y Invi z Invi The homogeneous coordinate transformation matrix between them is:

[0035]

[0036] Where, γ i For node P i The radial vector and the coordinate axis y Ihvi The acute angle enclosed by the positive directions;

[0037] Coordinate system S Inv1 (o Inv1 -x Inv1 y Inv1 z Inv1 ) and o p -x p ,yp ,z p The homogeneous coordinate transformation matrix between them is:

[0038]

[0039] Coordinate system S Inv2 (o Inv2 -x Inv2 y Inv2 z Inv2 ) and o g -x g ,y g ,z g The homogeneous coordinate transformation matrix between them is:

[0040]

[0041] Where, λ i The central angle corresponding to the pitch circle tooth thickness of the small and large wheels;

[0042] The transition curve of the left tooth root on the end face of the small and large wheels, i.e., the Hermite curve, is formed by point P. ci and P di and its tangent vector T ci and T di Decision, P di From the root circle radius R di and angle δ i Jointly decided, δ i Let P be the point di The radial vector and the x-axis k Given the acute angle, find the tooth root control point P. ci With the tooth root control point P di The parametric equation for the determined left tooth root transition curve, i.e., the Hermite curve, is as follows:

[0043]

[0044]

[0045] In equations (13) and (14), x p (P ci ), y p (P ci ), z p (P ci Points P and P are respectively. ci The three coordinate axis components, x p (P di ), y p (P di ), z p (P di Points P and P are respectively. diThe three coordinate axis components, x p (T ci ), y p (T ci ), z p (T ci Points P and P are respectively. ci The unit tangent vector T ci The three coordinate axis components, x p (T di ), y p (T di ), z p (T di Points P and P are respectively. di The unit tangent vector T di The three coordinate axis components, m t Here, b1, b2, b3, and b4 are the end face modulus, and T is the calculation parameter. H For the shape control parameters of the tooth root transition curve, 0.2≤T H ≤1.5, t H For the calculation parameters, 0≤t H ≤1;

[0046] In all the above formulas:

[0047] t—Meshing point M a and M b The motion parameter variables are t∈[0,Δt];

[0048] Δt—the maximum value of the motion parameter variable at the engagement point;

[0049] — is the linear proportionality coefficient for the motion at the meshing point;

[0050] m t —End face module;

[0051] Z1—Number of teeth on the pinion;

[0052] Z2—Number of teeth on the large gear;

[0053] a pi —The quadratic coefficients of the parabolic tooth profile curves of small and large wheels;

[0054] x pimin —x pi The minimum value;

[0055] x pimax —x pi The maximum value;

[0056] b—The width of the teeth of the small and large wheels;

[0057] α t—End face pressure angle;

[0058] J ai —Tilt reference point of small and large wheels

[0059] J ci —Reference point at the bottom of the gear teeth (small and large wheels)

[0060] χ a1 —The angle by which the reference point of the pinion teeth rotates clockwise around the center;

[0061] χ a2 —The angle by which the tooth tip reference point of the large wheel rotates clockwise around the center;

[0062] χ c1 —The angle by which the reference point of the tooth root of the small wheel rotates clockwise around the center;

[0063] χ c2 —The angle by which the reference point of the tooth root of the large wheel rotates clockwise around the center;

[0064] ρ a1 —Radius of the upper arc tooth profile on the end face of the small wheel;

[0065] ρ a2 —Radius of the upper arc tooth profile on the end face of the large wheel;

[0066] k c —Starting point P of the transition curve at the root of the small wheel and the large wheel ci The radius variation coefficient;

[0067] R1—is the pitch cylinder radius of the smaller wheel, R1=m t Z1 / 2; (15)

[0068] R2—is the pitch cylinder radius of the large wheel, R2=i 12 R1; (16)

[0069] i 12 —This represents the transmission ratio between the smaller wheel and the larger wheel.

[0070] a—Relative position of the axle of the small wheel and the large wheel: a = R1 + R2; (18)

[0071] r b1 —Radius of the base circle of the small wheel, r b1 =R1cosα t (19)

[0072] r b2 —Radius of the base circle of the large wheel, r b2 =R2cosα t (20)

[0073] Ra1 — Radius of the tip circle of the pinion teeth, R a1 =R1+m t ; (twenty one)

[0074] R c1 —The radius of the pinion tooth root circle, i.e., the starting point P of the root transition curve. c1 The radius R from the center of rotation of the small wheel c1 =R1-k c m t ; (twenty two)

[0075] R d1 — Radius of the pinion tooth root circle, R d1 =R1-1.25m t ; (twenty three)

[0076] R a2 — Radius of the tip circle of the large gear tooth, R a2 =R2+m t ; (twenty four)

[0077] R c2 —The radius of the root circle of the large gear tooth, i.e., the starting point P of the root transition curve. c2 The radius R from the center of rotation of the large wheel c2 =R2-k c m t (25)

[0078] R d2 — Radius of the root circle of the large gear tooth, R d2 =R2-1.25m t (26)

[0079] γ1—Radial vector of node P1 on the end face of the small wheel and coordinate axis y Inv1 The acute angle between the positive and negative directions,

[0080]

[0081] γ2—Radial vector of node P2 on the end face of the large wheel and the coordinate axis y Inv2 The acute angle between the positive and negative directions,

[0082]

[0083] λ1—The central angle corresponding to the pitch circle tooth thickness of the pinion.

[0084] λ2—The central angle corresponding to the pitch circle tooth thickness of the large wheel.

[0085] δ1—P, the tooth profile point on the left end face of the small wheel d1 The radial vector and the x-axis k The acute angle between them

[0086] δ2—P, the tooth profile point on the left end face of the large wheel d2 The radial vector and the x-axis k The acute angle between them

[0087] The overlap ratio of a parabolic gear mechanism with a combined end-face arc and parabolic tooth profile must be greater than 2. The formula for calculating the overlap ratio is:

[0088] Based on the overlap ratio ε, the linear scaling factor Given the number of teeth Z1 of the pinion, the maximum value of the kinematic parameter variable at the meshing point of the parabolic gear mechanism with the combined tooth profile of the end face arc and the parabola is obtained.

[0089] When the number of teeth Z1 of the pinion and the transmission ratio i are determined 12 End face module m t , overlap ε, linear scaling factor End face pressure angle α t Tooth width b, tooth root transition curve shape control parameter T H The angle χ of the tooth tip reference point of the small gear rotating clockwise around the center. a1 The angle χ of the tooth tip reference point of the large wheel rotating clockwise around the center. a2 The angle χ of the tooth root reference point of the small gear rotating clockwise around the center. c1 The angle χ of the tooth root reference point of the large wheel rotating clockwise around the center. c2 The starting point P of the transition curve at the root of the small wheel and the large wheel ci radius variation coefficient k c At that time, the maximum value of the motion parameter variable Δt at the meshing point, the contact line and the meshing line, the end face combined tooth profile of the pinion and the large gear and their correct installation distance are also determined accordingly. The parabolic tooth line structure of the tooth surface of the pinion and the large gear can also be determined, thus obtaining the parabolic tooth line gear mechanism with the end face arc and parabolic combined tooth profile.

[0090] Furthermore, the small wheel is used to connect the input shaft, and the large wheel is used to connect the output shaft.

[0091] Furthermore, the input and output shafts connected to the small wheel and the large wheel are interchangeable.

[0092] Furthermore, one of the small wheel and the large wheel is connected to an input shaft, the input shaft is connected to a driver, and the driver can drive the small wheel or the large wheel to rotate in both directions.

[0093] The beneficial effects of the technical solutions provided by the embodiments of the present invention are as follows:

[0094] 1. The parabolic gear mechanism of the present invention, with a combination of end-face arc and parabola tooth profile, is designed based on the active design of the motion law of the meshing point. It constructs a contact line for node meshing, and the contact line is an axisymmetric parabola after being unfolded on the pitch cylinder surface. This ensures that the theoretical relative sliding speed of all meshing points on the contact line is zero, thereby effectively reducing the relative sliding and frictional wear between the tooth surfaces. At the same time, the parabolic gear mechanism of the present invention with a combination of end-face arc and parabola tooth profile has no tooth tip sharpening phenomenon, and the contact area covers the entire tooth width. It can be designed to utilize a larger tooth width to transmit a larger load, resulting in better motion stability. In addition, the relative difference between the maximum contact stress on the tooth surface and the maximum bending stress at the tooth root during forward and reverse transmission of the parabolic gear mechanism of the present invention is extremely small.

[0095] 2. The parabolic toothed gear mechanism of the present invention, which combines the end face arc and parabola tooth profile, is theoretically a pure rolling meshing mechanism with low friction and wear, no axial force, good self-centering, easy installation, and low sensitivity to installation errors. Compared with the existing traditional involute herringbone gear transmission mechanism, the parabolic toothed gear mechanism of the present invention also has the advantages of not requiring a relief groove design, being able to be formed in one step, having a simple processing technology, and being easy to assemble.

[0096] 3. The end face tooth profile of the parabolic tooth gear mechanism of the present invention, which combines end face arcs and parabolic tooth profiles, is not a single circular arc or other planar curve, but a combination of multiple curves. This enables effective control of the contact ellipse and contact area, avoids edge contact, increases the relative radius of curvature, improves tooth surface contact strength and tooth root bending strength, and enhances load-bearing capacity.

[0097] 4. The parabolic toothed gear mechanism of the present invention, which combines the end face arc and parabola tooth profile, has a cylindrical surface of contact line that, when unfolded, becomes an axisymmetric parabola rather than an inclined straight line. Therefore, there is no axial force during transmission, the shaft system installation conditions are simpler, and the structure is simpler.

[0098] 5. The parabolic toothed gear mechanism of the present invention, which combines the end face arc and parabola tooth profile, has no undercut and a minimum number of teeth of 1. Compared with existing parallel shaft involute gear mechanisms and arc toothed cylindrical gear transmission mechanisms, it can achieve single-stage large transmission ratio and high overlap ratio transmission. At the same time, since the number of teeth can be designed to be smaller, a larger tooth thickness and module can be designed for the same gear pitch circle diameter, thereby having higher bending strength and greater load-bearing capacity. It is suitable for widespread application in the fields of micro / micro machinery, conventional mechanical transmission and high-speed heavy-duty transmission.

[0099] 6. The parabolic toothed gear mechanism of the present invention, which combines the end face arc and parabola tooth profile, can achieve similar tooth root bending strength for the small gear and the large gear by optimizing the design of the tooth root transition curve shape control parameters, thereby realizing the equal strength design of the transmission mechanism and further improving the service life of the equipment. Attached Figure Description

[0100] Figure 1 This is a schematic diagram of a parabolic toothed gear mechanism with a combined end-face arc and parabola tooth profile according to the present invention.

[0101] Figure 2 This is a schematic diagram of the spatial meshing coordinate system of a parabolic toothed gear mechanism with a combined end-face arc and parabola tooth profile according to the present invention.

[0102] Figure 3 For the present invention Figure 1 and Figure 2 The structure and coordinate system of the tooth profiles of the large and small wheels at their end faces.

[0103] Figure 4 This is a schematic diagram of the local coordinate system relationship of the combined tooth profile of the present invention.

[0104] Figure 5 This is a schematic diagram of the tooth tip reference point and its rotation angle of the combined tooth profile of the present invention.

[0105] Figure 6 For the present invention Figure 1 A three-dimensional spatial view of small and medium-sized wheels.

[0106] Figure 7 For the present invention Figure 1 A three-dimensional spatial view of the medium and large-sized ship.

[0107] Figure 8 This is a schematic diagram of the structure of the present invention when the large wheel is connected to the input shaft and drives the small wheel to increase speed.

[0108] In the above diagram: 1-Driver, 2-Coupling, 3-Input shaft, 4-Small gear, 5-Output shaft, 6-Large gear, 7-Meshing line KK, 8-Small gear pitch cylinder, 9-Small gear contact line Cp, 10-Large gear contact line Cg, 11-Large gear pitch cylinder, 12-Left tooth root transition curve of the large gear end face tooth profile, 13-Lower left parabola of the large gear end face working tooth profile, 14-Upper left arc curve of the large gear end face working tooth profile, 15-Left tooth root transition curve of the small gear end face tooth profile, 16-Lower left parabola of the small gear end face working tooth profile, 17-Upper left arc curve of the large gear end face working tooth profile. Detailed Implementation

[0109] To make the objectives, technical solutions, and advantages of the present invention clearer, the embodiments of the present invention will be further described below in conjunction with the accompanying drawings. The following description presents a preferred embodiment of the various possible embodiments of the present invention, intended to provide a basic understanding of the invention, but not intended to identify key or decisive elements of the invention or to limit the scope of protection sought.

[0110] In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values.

[0111] Techniques, methods, and equipment known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and equipment should be considered part of the specification.

[0112] It should be noted that similar labels and letters in the following figures indicate similar items; therefore, once an item is defined in one figure, it does not need to be discussed further in subsequent figures. Also, it should be understood that, for ease of description, the dimensions of the various parts shown in the figures are not drawn to actual scale.

[0113] In the description of this invention, it should be noted that the circuits, electronic components, and modules involved in this invention are all prior art, which can be fully implemented by those skilled in the art, and need not be elaborated upon. The content protected by this invention does not involve improvements to the internal structure and methods.

[0114] It should be further noted that, unless otherwise explicitly specified and limited, the terms "installation" and "connection" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0115] Example 1:

[0116] Please refer to Figure 1 This invention provides a parabolic gear mechanism with a combined end-face arc and parabolic tooth profile, applied to a reduction transmission with a transmission ratio of 3 between parallel shafts, designed with a contact ratio of ε = 2.4. Its structure is as follows: Figure 1 As shown, it includes a small gear 4 and a large gear 6, which form a pair of gears. The small gear 4 is connected to the input shaft 3, and the input shaft 3 is fixed to the drive motor 1 through the coupling 2. The large gear 6 is connected to the output shaft 5, that is, the large gear 6 is connected to the driven load through the output shaft 5. The axes of the small gear 4 and the large gear 6 are parallel to each other. Figure 2 This is a schematic diagram of the spatial meshing coordinate system of a parabolic toothed gear mechanism with a combined end-face arc and parabola tooth profile according to the present invention.

[0117] See Figure 1 , 2 3, 4, 5, 6, the pitch cylinder radius of the pinion is R1, and the addendum circle radius of the pinion is R. a1 The radius of the tooth root circle is R d1 The outer surface of the pinion tooth root cylinder is uniformly covered with parabolic tooth profiles. These profiles are formed by the movement of the pinion end face tooth profile curve along the contact line at the contact point. The contact line, when unfolded along the pinion pitch cylinder, is an axisymmetric parabola. The pinion tooth end face profile is axisymmetric, meaning the left and right side profiles are axially symmetrical. Taking the left side end face tooth profile as an example, from the tooth tip to the tooth root, it consists of the upper arc curve 17 of the left end face working tooth profile, the lower parabola 16 of the left end face working tooth profile, and the left end face tooth root transition curve, i.e., the Hermite curve 15.

[0118] See Figure 1 , 2 3, 4, 5, 7, the radius of the pitch cylinder 11 of the large wheel is R2, and the radius of the addendum circle of the large wheel is R. a2 The radius of the tooth root circle is R d2 The outer surface of the large wheel tooth root cylinder is uniformly covered with parabolic tooth line structures. These structures are formed by the movement of the large wheel end face tooth profile curve along the contact line at the contact point, and the contact line, when unfolded along the large wheel pitch cylinder, is an axisymmetric parabola. The end face tooth profile of the large wheel teeth is axisymmetric, meaning the left and right end face tooth profiles are axially symmetrical. Taking the left end face tooth profile of the large wheel as an example, from the tooth tip to the tooth root, it is successively composed of the upper arc curve 14 of the left end face working tooth profile of the small wheel tooth, the lower parabola 13 of the left end face working tooth profile, and the left end face tooth root transition curve, i.e., the Hermite curve 12.

[0119] The working tooth profiles on the end faces of the small wheel and the large wheel are a combination of end face circular arcs and parabolic curves, and are axially symmetrical on both sides. The tooth profile on the right side of the end face can be obtained by axially symmetrically aligning the tooth profile on the left side of the end face. The curve of the working tooth profile on the left side is formed by two planar curves, circular arc and parabola, at the inter-tooth control point P. bi The wheels are smoothly connected, and the inter-tooth control point G on the right side of the tooth profile is used when installing the small and large wheels. bi With node P i Overlapping, control point G bi Control point P between teeth on the left working tooth profile curve bi Obtained by axisymmetry; the shape of the working tooth profile curve on the end face is determined by the tooth tip control point P. ai Inter-tooth control point P bi and tooth root control point P ciSpecifically, the combination type of the working tooth profile curves of both the small and large gears from the tooth tip to the tooth root is CP, where "C" and "P" represent the circular arc (Cir) and parabola (Par), respectively. The circular arc is the upper curve of the working tooth profile, and the parabola is the lower curve of the working tooth profile. The tooth root transition curve is the tooth root control point P. ci With tooth root control point P di The determined Hermite curve (Her), and the root transition curve and the lower working profile curve at the tooth root control point P. ci Smooth connection.

[0120] The tooth tip control point P of the left working tooth profile of the small wheel and the large wheel ai From the tooth tip circle radius R ai and offset angle χ ai Confirmed, χ ai J is the reference point for the tooth tips of the small and large gears. ai The angle of clockwise rotation around the center; control point P at the root of the tooth. ci From the radius R of the tooth root circle ci and offset angle χ ci Confirmed, χ ci J is the reference point for the tooth root of the small and large wheels. ci The angle of clockwise rotation around the center; where J is the reference point for the tooth tips of the small and large gears. ai An involute with the same base circle radius, end face pressure angle, and radius R as the small and large wheels respectively. ai The intersection of the addendum circles; the reference point J of the root of the pinion and gear teeth. ci An involute with the same base circle radius, end face pressure angle, and radius R as the small and large wheels respectively. ci The intersection of the tooth root circles.

[0121] The small wheel 4 is connected to the input shaft 3. The input shaft 3 is fixedly connected to the drive motor 1 through the coupling 2. Under the drive of the drive motor 1, it rotates so that at least one pair of gear teeth of the small wheel and the large wheel are located at the node to achieve pure rolling meshing contact, thereby realizing the transmission of motion and power between parallel shafts. In this embodiment, the driver 1 is an electric motor.

[0122] The contact lines 9 and 10 of the tooth surfaces of the small wheel and the large wheel are determined by the following method: in o p -x p ,y p ,z p o k -x k ,y k ,z k and o g -x g ,y g ,z g In the three spatial coordinate systems, zp The axis of rotation of the shaft coincides with that of the small wheel, z g The axis of rotation of the shaft coincides with that of the large wheel, z k The shaft passes through the meshing point M a and M b The meshing lines KK coincide, and z k axis and z p z g The axes are parallel to each other, x p With x g Coincident axis, x k With x g The axis is parallel, o p o g The distance is a; coordinate system o1-x1,y1,z1 is fixed to the small wheel, and coordinate system o2-x2,y2,z2 is fixed to the large wheel. The coordinate systems o1-x1,y1,z1 and o2-x2,y2,z2 of the small wheel and the large wheel are respectively fixed to coordinate system o at their initial positions. p -x p ,y p ,z p and o g -x g ,y g ,z g When they coincide, the meshing point M is reached. a and M b The overlap is denoted as M, and the small wheel revolves around z at a uniform angular velocity ω1. p The shaft rotates clockwise, and the large wheel revolves around the z-axis at a uniform angular velocity ω2. g The axis rotates counterclockwise. After a period of time from the initial position, the coordinate system o1-x1,y1,z1 and o 2- x2, y2, z2 rotate respectively, and the small wheel revolves around z. p The shaft rotated Angle, large wheel around z g The shaft rotated horn;

[0123] When the small wheel and the large wheel mesh and drive, the meshing point M is set. a and M b Starting from the origin o k The motion begins along the line of engagement KK, moving up and down. The parametric equation describing the motion at the engagement point is:

[0124]

[0125] In equation (1), t is the meshing point M. a and M b The motion parameter variables are 0≤t≤Δt; b is the tooth width in millimeters (mm); "+" corresponds to the meshing point M. a "-" corresponds to the engagement point M b ;

[0126] To ensure constant gear ratio meshing, the rotation angles of the pinion and gear and the motion of the meshing point must have a linear relationship, as shown in the following formula:

[0127]

[0128] In formula (2) i is the linear proportionality coefficient for the motion at the meshing point, and its unit is radians (rad); 12 This is the transmission ratio between the small wheel and the large wheel;

[0129] When the meshing point M a and M b As they move along the line of engagement KK, they simultaneously form contact lines C on the pinion tooth surface and the gear tooth surface, respectively. p and C g Based on the coordinate transformation, the coordinate system o is obtained. p -x p ,y p ,z p o k -x k ,y k ,z k and o g -x g ,y g ,z g The homogeneous coordinate transformation matrix between o1-x1,y1,z1 and o2-x2,y2,z2 is:

[0130]

[0131] in,

[0132]

[0133]

[0134] In equations (4) and (5), R1 is the pitch cylinder radius of the smaller wheel, R2 is the pitch cylinder radius of the larger wheel, and α t The end face pressure angle at the meshing point;

[0135] The contact line C of the pinion tooth surface is obtained from equations (1) and (4). p The parametric equation is:

[0136]

[0137] The contact line C of the large gear tooth surface is obtained from equations (1) and (5). g The parametric equation is:

[0138]

[0139] The specific structure of the tooth profile on the left end face of the small wheel and the large wheel is determined by the following method:

[0140] Control points P between the teeth of the large and small gears respectively bi Establish a local coordinate system S pbi (o pbi -x pbi y pbi z pbi ), i = 1, 2, where i = 1 represents the small gear and i = 2 represents the large gear. The parametric equation for the upper circular arc curve used for the combination of working tooth profile curves is:

[0141]

[0142] In equation (8), i = 1, 2, where i = 1 represents the small wheel and i = 2 represents the large wheel; ξ ai Let ξ be the angle parameter of the circular arc curve. aimin and ξ aimax They are ξ ai The minimum and maximum values ​​of ρ ai These are the radii of the small and large wheels, respectively, when the offset angle χ is determined. ai Tooth tip circle radius R ai , ρ ai ξ aimin and ξ aimax All of these can be solved, thereby determining the upper circular arc tooth profile curve;

[0143] The parametric equation for the lower parabolic curve used in the combination of working tooth profile curves is obtained as follows:

[0144]

[0145] In equation (9), i = 1, 2, where i = 1 represents the small wheel and i = 2 represents the large wheel; a pi Let x be the coefficient of the quadratic term of the parabolic curves of the small and large wheels. pi The x-axis is the coordinate axis. pbi The parameter, x pimin and x pimax They are x pi The minimum and maximum values ​​are taken; when the radius R of the tooth root circle is determined. ci , offset angle χ ci At that time, a pi x pimin and x pimax All of these can be solved to determine the lower parabolic tooth profile curve;

[0146] Based on the coordinate transformation, we can obtain the coordinate system S. pbi (o pbi -xpbi y pbi z pbi ) and S Invi (o Invi -x Invi y Invi z Invi The homogeneous coordinate transformation matrix between them is:

[0147]

[0148] Where, γ i For node P i The radial vector and the coordinate axis y Invi The acute angle enclosed by the positive directions;

[0149] Coordinate system S Inv1 (o Inv1 -x Inv1 y Inv1 z Inv1 ) and o p -x p ,y p ,z p The homogeneous coordinate transformation matrix between them is:

[0150]

[0151] Coordinate system S Inv2 (o Inv2 -x Inv2 y Inv2 z Inv2 ) and o g -x g ,y g ,z g The homogeneous coordinate transformation matrix between them is:

[0152]

[0153] Where, λ i The central angle corresponding to the pitch circle tooth thickness of the small and large wheels;

[0154] The transition curve of the left tooth root on the end face of the small and large wheels, i.e., the Hermite curve, is formed by point P. ci and P di and its tangent vector T ci and T di Decision, P di From the root circle radius R di and angle δ i Jointly decided, δ i Let P be the point di The radial vector and the x-axis k Given the acute angle, find the tooth root control point P.ci With the tooth root control point P di The parametric equation for the determined left tooth root transition curve, i.e., the Hermite curve, is as follows:

[0155]

[0156]

[0157] In equations (13) and (14), x p (P ci ), y p (P ci ), z p (P ci Points P and P are respectively. ci The three coordinate axis components, x [ (P di ), y p (P di ), z p (P di Points P and P are respectively. di The three coordinate axis components, x p (T ci ), y p (T ci ), z p (T ci Points P and P are respectively. ci The unit tangent vector T ci The three coordinate axis components, x p (T di ), y p (T di ), z p (T di Points P and P are respectively. di The unit tangent vector T di The three coordinate axis components, m t Here, b1, b2, b3, and b4 are the end face modulus, and T is the calculation parameter. H For the shape control parameters of the tooth root transition curve, 0.2≤T H ≤1.5, t H For the calculation parameters, 0≤t H ≤1;

[0158] In all the above formulas:

[0159] t—Meshing point M a and M b The motion parameter variables are t∈[0,Δt];

[0160] Δt—the maximum value of the motion parameter variable at the engagement point;

[0161] — is the linear proportionality coefficient for the motion at the meshing point;

[0162] m t —End face module;

[0163] Z1—Number of teeth on the pinion;

[0164] Z2—Number of teeth on the large gear;

[0165] a pi —The quadratic coefficients of the parabolic tooth profile curves of small and large wheels;

[0166] x pimin —x pi The minimum value;

[0167] x pimax —x pi The maximum value;

[0168] b—The width of the teeth of the small and large wheels;

[0169] α t —End face pressure angle;

[0170] J ai —Tilt reference point of small and large wheels

[0171] J ci —Reference point at the bottom of the gear teeth (small and large wheels)

[0172] χ a1 —The angle by which the reference point of the pinion teeth rotates clockwise around the center;

[0173] χ a2 —The angle by which the tooth tip reference point of the large wheel rotates clockwise around the center;

[0174] χ c1 —The angle by which the reference point of the tooth root of the small wheel rotates clockwise around the center;

[0175] χ c2 —The angle by which the reference point of the tooth root of the large wheel rotates clockwise around the center;

[0176] ρ a1 —Radius of the upper arc tooth profile on the end face of the small wheel;

[0177] ρ a2 —Radius of the upper arc tooth profile on the end face of the large wheel;

[0178] k c —Starting point P of the transition curve at the root of the small wheel and the large wheel ci The radius variation coefficient;

[0179] R1—is the pitch cylinder radius of the smaller wheel, R1=m tZ1 / 2; (15)

[0180] R2—is the pitch cylinder radius of the large wheel, R2=i 12 R1; (16)

[0181] i 12 —This represents the transmission ratio between the smaller wheel and the larger wheel.

[0182] a—Relative position of the axle of the small wheel and the large wheel: a = R1 + R2; (18)

[0183] r b1 —Radius of the base circle of the small wheel, r b1 =R1cosx t (19)

[0184] r b2 —Radius of the base circle of the large wheel, r b2 =R2cosα t (20)

[0185] R a1 — Radius of the tip circle of the pinion teeth, R a1 =R1+m t ; (twenty one)

[0186] R c1 —The radius of the pinion tooth root circle, i.e., the starting point P of the root transition curve. c1 The radius R from the center of rotation of the small wheel c1 =R1-k c m t ; (twenty two)

[0187] R d1 — Radius of the pinion tooth root circle, R d1 =R1-1.25m t ; (twenty three)

[0188] R a2 — Radius of the tip circle of the large gear tooth, R a2 =R2+m t ; (twenty four)

[0189] R c2 —The radius of the root circle of the large gear tooth, i.e., the starting point P of the root transition curve. c2 The radius R from the center of rotation of the large wheel c2 =R2-k c m t (25)

[0190] R d2 — Radius of the root circle of the large gear tooth, R d2 =R2-1.25m t (26)

[0191] γ1—Radial vector of node P1 on the end face of the small wheel and coordinate axis y Inv1 The acute angle between the positive and negative directions,

[0192]

[0193] γ2—Radial vector of node P2 on the end face of the large wheel and the coordinate axis y Inv2 The acute angle between the positive and negative directions,

[0194]

[0195] λ1—The central angle corresponding to the pitch circle tooth thickness of the pinion.

[0196] λ2—The central angle corresponding to the pitch circle tooth thickness of the large wheel.

[0197] δ1—p, the tooth profile point on the left end face of the small wheel d1 The radial vector and the x-axis k The acute angle between them

[0198] δ2—p, the tooth profile point on the left end face of the large wheel d2 The radial vector and the x-axis k The acute angle between them

[0199] The overlap ratio of a parabolic gear mechanism with a combined end-face arc and parabolic tooth profile must be greater than 2. The formula for calculating the overlap ratio is:

[0200] Based on the overlap ratio ε, the linear scaling factor Given the number of teeth Z1 of the pinion, the maximum value of the kinematic parameter variable at the meshing point of the parabolic gear mechanism with the combined tooth profile of the end face arc and the parabola is obtained.

[0201] When the number of teeth Z1 of the pinion and the transmission ratio i are determined 12 End face module m t , overlap ε, linear scaling factor End face pressure angle α t Tooth width b, tooth root transition curve shape control parameter T H The angle χ of the tooth tip reference point of the small gear rotating clockwise around the center. a1 The angle χ of the tooth tip reference point of the large wheel rotating clockwise around the center. a2 The angle χ of the tooth root reference point of the small gear rotating clockwise around the center. c1 The angle χ of the tooth root reference point of the large wheel rotating clockwise around the center. c2 The starting point P of the transition curve at the root of the small wheel and the large wheelci radius variation coefficient k c At that time, the maximum value of the motion parameter variable Δt at the meshing point, the contact line and the meshing line 7, the end face combination tooth profile of the small gear and the large gear and their correct installation distance are also determined accordingly. The parabolic tooth line structure of the tooth surface of the small gear and the large gear can also be determined, thus obtaining the parabolic tooth line gear mechanism with the end face arc and parabolic combination tooth profile.

[0202] In the above formula: the axes of each coordinate system, a, b, m t , ρ a1 , ρ a2 x pimin x pimax R1 and R2 are both in millimeters (mm) as the unit of length, radius or distance. ξ ai ξ aimin ξ aimax ,δ1,δ2,χ a1 , χ c1 , χ a2 and χ c2 The unit for equal angles is radians (rad); pressure angle α t The unit is degrees (°).

[0203] In the above formula, the relevant parameters take the values ​​of: Z1 = 24, i 12 =3,m t =4 millimeters (mm), ε = 2.4, Radius (rad), b = 80 millimeters (mm), α t =20°, T H =0.5, χ a1 =0.08rad, χ a2 =0.04rad, substituting into equation (15)-(34), we get Δt =0.1, a =192 mm;

[0204] Then, by substituting the above values ​​into equations (1) to (14), we can obtain the contact line parameter equations and end face tooth profile parameter equations of the small wheel and the large wheel in this example. Then, based on the helical motion, we can obtain the tooth surface structure of the small wheel and the large wheel, and assemble them according to the correct center distance.

[0205] When the drive motor 1 drives the input shaft 3 and the small wheel 2 to rotate, due to the pre-set overlap ratio ε = 2.4 of the parabolic tooth profile pure rolling external meshing gears with the end face arc and parabolic combined tooth profile when the small wheel 2 and the large wheel 5 are correctly installed, both pairs of adjacent teeth are in a meshing state. Therefore, it is ensured that at any instant, at least two pairs of teeth participate in the meshing transmission simultaneously, thereby realizing the continuous and stable meshing transmission of the parabolic tooth gear mechanism with the end face arc and parabolic combined tooth profile during rotational motion. In this embodiment, the input shaft connected to the motor rotates clockwise, corresponding to the deceleration transmission mode of the parabolic tooth profile pure rolling external meshing gear with the end face arc and parabolic combined tooth profile, to realize the deceleration and torque increase transmission of the large wheel's counterclockwise rotation.

[0206] Example 2:

[0207] The parabolic toothed gear mechanism of the present invention, which combines the end face arc and parabola tooth profile, is applied to the speed-increasing transmission of a parallel shaft. For example... Figure 8 As shown, a large wheel 6 is connected to the input shaft 3, which is fixedly connected to the drive motor 1 via a coupling 2. A small wheel 4 is connected to the output shaft 5, meaning the small wheel 4 is connected to the driven load via the output shaft 5. The axes of the small wheel 4 and the large wheel 6 are parallel. In this embodiment, the large wheel 5 has 63 teeth, and the small wheel 2 has 21 teeth, with a designed overlap ratio ε = 2.4. When the input shaft 3 drives the large wheel 6 to rotate, since both pairs of adjacent teeth are in a meshing state when the large wheel 6 and the small wheel 4 are installed, the pre-set overlap ratio ε = 2.4 of the parabolic tooth profile pure rolling external meshing gear of the end face arc and parabola combination tooth profile ensures that at any instant, at least two pairs of teeth participate in the meshing transmission simultaneously, thus realizing continuous and stable meshing transmission of the parabolic tooth gear mechanism of the end face arc and parabola combination tooth profile during rotational motion. At this time, the speed ratio of the large wheel to the small wheel is 3, that is, the angular velocity ratio of the small wheel to the large wheel is 3.

[0208] The relevant parameters are respectively set to: Z1 = 21, i 12 =3,m t =3 millimeters (mm), ε = 2.4, Radius (rad), b = 80 millimeters (mm), α t =25°, T H =0.6, χ a1 =0.06rad, χ a2 Substituting 0.03 rad into equations (16)-(36), we obtain Δt = 0.1 and a = 126 mm.

[0209] Then, by substituting the above values ​​into equations (1) to (14), we can obtain the contact line parameter equations and end face tooth profile parameter equations of the small wheel and the large wheel in this example. Then, based on the helical motion, we can obtain the tooth structure of the small wheel and the large wheel, and assemble them according to the correct center distance.

[0210] In this embodiment, the input shaft connected to the driver rotates counterclockwise, corresponding to the speed-increasing transmission mode of the parabolic tooth gear mechanism with a combination of end-face arc and parabola tooth profile, in order to achieve clockwise rotation of the small wheel.

[0211] This invention discloses a parabolic gear mechanism with a combined end-face arc and parabolic tooth profile. Based on an active design method using the meshing line parametric equation, it employs a combination of arc curves and parabolas to form the end-face working tooth profile, achieving theoretically pure rolling meshing transmission. It also enables active control of the contact area and contact ellipse, reducing tooth surface friction, increasing the overall radius of curvature, and enhancing tooth surface contact strength and tooth root bending strength. This parabolic gear mechanism with a combined end-face arc and parabolic tooth profile exhibits no undercut and a minimum tooth count of 1. Compared to existing parallel-axis involute gear mechanisms, it can achieve single-stage high transmission ratio and high overlap ratio transmission. Furthermore, due to the small tooth count, it can achieve higher speeds for the same gear pitch circle diameter. The design features a larger tooth thickness, resulting in higher strength and greater load-bearing capacity, making it suitable for widespread application in micro / micro machinery, conventional mechanical transmissions, and high-speed heavy-duty transmissions. The parabolic gear mechanism of this invention, with its combined end-face arc and parabolic tooth profile, can further optimize the root transition curve parameters to achieve similar tooth root bending strength between the small and large gears, realizing equal strength design of the transmission mechanism and further extending the service life of the equipment. The parabolic gear mechanism of this invention exhibits extremely small differences between the maximum tooth surface contact stress and the maximum tooth root bending stress in both forward and reverse rotation, providing approximately bidirectional transmission strength. In practical use, one of the small gear 4 and the large gear 6 is connected to an input shaft, which is connected to a driver 1. The driver 1 can drive either the small gear 4 or the large gear 6 for forward or reverse rotation.

[0212] In this document, the directional terms such as front, back, top, and bottom are defined based on the position of the components in the accompanying drawings and their relative positions to each other, solely for the purpose of clarity and convenience in expressing the technical solution. It should be understood that these are relative concepts and can vary depending on different methods of use and placement; the use of these directional terms should not limit the scope of protection claimed in this application.

[0213] Where there is no conflict, the above embodiments and features described herein can be combined with each other.

[0214] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A parabolic gear mechanism with a tooth profile combining an end face arc and a parabola, comprising a pair of gears consisting of a small gear and a large gear with parallel axes, wherein the small gear and the large gear engage in pure rolling meshing transmission, characterized in that: The end face tooth profile curves of the small wheel and the large wheel are composed of the end face working tooth profile curve and the tooth root transition curve, and the end face tooth profile curves of the small wheel and the large wheel are symmetrical on both sides; the end face working tooth profile of the small wheel and the large wheel is a combination of end face arc and parabola; the tooth surface of the small wheel and the large wheel has a parabolic tooth line structure; at least one pair of tooth meshing points of the small wheel and the large wheel are located at the node to achieve pure rolling meshing contact, and the meshing points of the small wheel and the large wheel rotate relative to each other to form meshing lines, which respectively form two contact lines on the tooth surface of the small wheel and the large wheel; The tooth surface structure of the small wheel and the large wheel is formed by the tooth profile curves of the end face of the small wheel and the large wheel moving along the contact line of the tooth surface as the contact point moves. Moreover, the contact line is an axisymmetric parabola after being unfolded along the pitch cylinder surface of the small wheel and the large wheel. The working tooth profile curves on the left side of the end faces of the small wheel and the large wheel are composed of both circular arcs and parabolic planar curves at the inter-tooth control point P. bi The wheels are smoothly connected, and the inter-tooth control point G on the right side of the tooth profile is used when installing the small and large wheels. bi With node P i Overlapping, control point G bi Control point P between teeth on the left working tooth profile curve bi Obtained by axisymmetry; the shape of the working tooth profile curve on the end face is determined by the tooth tip control point P. ai Inter-tooth control point P bi and tooth root control point P ci Specifically, the combination type of the working tooth profile curves of both the small and large gears from the tooth tip to the tooth root is CP, where C and P represent circular arc and parabola, respectively. The circular arc is the upper curve of the working tooth profile, and the parabola is the lower curve of the working tooth profile. The tooth root transition curve is controlled by the tooth root control point P. ci With tooth root control point P di The determined Hermite curve, and the root transition curve and the lower working profile curve at the tooth root control point P. ci Smooth connection; The tooth tip control point P of the left working tooth profile of the small wheel and the large wheel ai From the tooth tip circle radius R ai and offset angle X ai Confirmed, X ai J is the reference point for the tooth tips of the small and large gears. ai The angle of clockwise rotation around the center; control point P at the root of the tooth. ci From the radius R of the tooth root circle ci and offset angle X ci Confirmed, X ci J is the reference point for the tooth root of the small and large wheels. ci The angle of clockwise rotation around the center; where J is the reference point for the tooth tips of the small and large gears. ai An involute with the same base circle radius, end face pressure angle, and radius R as the small and large wheels respectively. ai The intersection of the addendum circles; the reference point J of the root of the pinion and gear teeth. ci An involute with the same base circle radius, end face pressure angle, and radius R as the small and large wheels respectively. ci The intersection of the tooth root circles.

2. The parabolic toothed gear mechanism with a combined end-face arc and parabola tooth profile as described in claim 1, characterized in that: The contact line between the tooth surfaces of the small wheel and the large wheel is determined by the following method: In o p -x p y p , z p o k -x k y k , z k and o g -x g y g , z g In the three spatial coordinate systems, z p The axis of rotation of the shaft coincides with that of the small wheel, z g The axis of rotation of the shaft coincides with that of the large wheel, z k Shaft and through meshing point M a and M b The meshing lines KK coincide, and z k axis and z p z g The axes are parallel to each other, x p With x g Coincident axis, x k With x g The axes are parallel, and the distance between opog is a; coordinate system o1-x1, y1, z1 is fixed to the small wheel, and coordinate system o2-x2, y2, z2 is fixed to the large wheel. The coordinate systems o1-x1, y1, z1 and o2-x2, y2, z2 of the small and large wheels are respectively fixed to coordinate system o at their initial positions. p -x p y p , z p and o g -x g y g , z g When they coincide, the meshing point M is reached. a and M b Let M be the superposition and denoted as M. The small wheel moves at a uniform angular velocity. Around z p The shaft rotates clockwise, and the large wheel rotates at a uniform angular velocity. Around z g The axis rotates counterclockwise. After a period of time from the initial position, the coordinate systems o1-x1, y1, z1 and o2-x2, y2, z2 rotate respectively, and the small wheel revolves around z. p The shaft rotated Angle, large wheel around z g The shaft rotated horn; When the small wheel and the large wheel mesh and drive, the meshing point M is set. a and M b Starting from the origin o k The motion begins along the line of engagement KK, moving up and down. The parametric equation describing the motion at the engagement point is: ；（1） In equation (1), t is the meshing point M. a and M b Motion parameter variables, 0≤t≤Δt; b is the tooth width; "+" corresponds to the meshing point M. a "-" corresponds to the meshing point M b ; To ensure constant gear ratio meshing, the rotation angles of the pinion and gear and the motion of the meshing point must have a linear relationship, as shown in the following formula: ；（2） In formula (2) This is the linear proportionality coefficient for the motion at the meshing point; This refers to the transmission ratio between the small wheel and the large wheel; When the meshing point M a and M b As they move along the line of engagement KK, they simultaneously form contact lines C on the pinion tooth surface and the gear tooth surface, respectively. p and C g Based on the coordinate transformation, the coordinate system o is obtained. p -x p y p , z p o k -x k y k , z k and o g -x g y g , z g The homogeneous coordinate transformation matrix between o1-x1, y1, z1 and o2-x2, y2, z2 is: ；（3） in, ， ；（4） ， ；（5） In equations (4) and (5), R1 is the pitch cylinder radius of the small wheel, and R2 is the pitch cylinder radius of the large wheel; The parametric equation of the contact line Cp of the pinion tooth surface, obtained from equations (1) and (4), is as follows: ；（6） The contact line C of the large gear tooth surface is obtained from equations (1) and (5). g The parametric equation is: ；（7）。 3. The parabolic toothed gear mechanism with a combined end-face arc and parabola tooth profile as described in claim 1, characterized in that: The specific structure of the tooth profile on the left end face of the small wheel and the large wheel is determined by the following method: ， ；（8） In equation (8), i = 1, 2, where i = 1 represents the small wheel and i = 2 represents the large wheel; For the angle parameters of the arc curve, and They are The minimum and maximum values ​​are taken. These are the radii of the arcs of the small and large wheels, when the offset angle X is determined. ai Tooth tip circle radius R ai , , and All of these can be solved, thereby determining the upper circular arc tooth profile curve; The parametric equation for the lower parabolic curve used in the combination of working tooth profile curves is obtained as follows: ；（9） In equation (9), i = 1, 2, where i = 1 represents the small wheel and i = 2 represents the large wheel; a pi Let x be the coefficient of the quadratic term of the parabolic curves of the small and large wheels. pi It is the x-axis of the coordinate system pbi The parameter, x pimin and x pimax They are x pbi The minimum and maximum values ​​are taken; when the radius R of the tooth root circle is determined. ci Offset angle x ci At that time, a pi x pimin and x pimax All of these can be solved to determine the lower parabolic tooth profile curve; Based on the coordinate transformation, we can obtain the coordinate system S. pbi (o pbi -x pbi y pbi z pbi ) and S Invi (o Invi -x Invi y Invi z Invi The homogeneous coordinate transformation matrix between them is: ； （10） in, For node P i The radial vector and the coordinate axis y Invi The acute angle enclosed by the positive directions; Coordinate system S Inv1 (o Inv1 -x Inv1 y Inv1 z Inv1 ) and o p -x p ,y p ,z p The homogeneous coordinate transformation matrix between them is: ；（11） Coordinate system S Inv2 (o Inv2 -x Inv2 y Inv2 z Inv2 ) and o g -x g ,y g ,z g The homogeneous coordinate transformation matrix between them is: ；（12） in, The central angle corresponding to the pitch circle tooth thickness of the small and large wheels; The transition curve of the left tooth root on the end face of the small and large wheels, i.e., the Hermite curve, is formed by point P. ci and P di and its tangent vector T ci and T di Decision, P di From the root circle radius R di and angle Joint decision, Let P be the point di The radial vector and the x-axis k Given the acute angle, find the tooth root control point P. ci With tooth root control point P di The parametric equation for the determined left tooth root transition curve, i.e., the Hermite curve, is as follows: ；（13） ；（14） In equations (13) and (14), x p (P ci ), y p (P ci ), z p (P ci Points P and P are respectively. ci The three coordinate axis components, x p (P di ), y p (P di ), z p (P di Points P and P are respectively. di The three coordinate axis components, x p (T ci ), yp(T ci ), zp(T ci Points P and P are respectively. ci The unit tangent vector T ci The three coordinate axis components, x p (T di ), yp(T di ), z p (T di Points P and P are respectively. di The unit tangent vector T di The three coordinate axis components, m t Here, b1, b2, b3, and b4 are the end face modulus, and T is the calculation parameter. H For the shape control parameters of the tooth root transition curve, 0.2≤T H ≤1.5, t H For the calculation parameters, 0≤t H ≤1; In all the above formulas: t-meshing point M a and M b The motion parameter variables are t∈[0,Δt]; Δt - the maximum value of the motion parameter variable at the engagement point; - is the linear proportionality coefficient for the motion at the meshing point; m t - End face module; Z 1- Number of teeth on the pinion; Z2 - Number of teeth on the large gear; a pi - The quadratic coefficients of the parabolic tooth profile curves of small and large wheels; x pimin -x pi The minimum value; x piman -x pi The maximum value; b - the width of the teeth on the small and large wheels; α t - End face pressure angle; J ai - Reference points for the tooth tips of the small and large gears; J ci - Reference points at the bottom of the small and large gear teeth; x a1 - The angle by which the tooth tip reference point of the small wheel rotates clockwise around the center; x a2 - The angle by which the tooth tip reference point of the large wheel rotates clockwise around the center; x c1 - The angle by which the reference point of the tooth root of the small wheel rotates clockwise around the center; x c2 - The angle by which the reference point of the tooth root of the large wheel rotates clockwise around the center; - Radius of the upper arc tooth profile on the end face of the small wheel; - Radius of the upper arc tooth profile on the end face of the large wheel; k c -Starting point P of the transition curve at the root of the small wheel and the large wheel ci The radius variation coefficient; R1 is the pitch cylinder radius of the smaller wheel, R1 = m t Z1 / 2; (15) R2 is the pitch cylinder radius of the large wheel, R2 = i 12 R1; (16) i 12 - represents the transmission ratio between the small wheel and the large wheel. (17) a - Relative installation positions of the axles of the small wheel and the large wheel: a = R1 + R2; (18) r b1 - Radius of the base circle of the small wheel, r b1 =R1cos (19) r b2 - Radius of the base circle of the large wheel, r b2 =R2cos (20) R a1 - Radius of the tip circle of the pinion teeth, R a1 =R1+m t ;(twenty one) R c1 - The radius of the pinion tooth root circle, i.e., the starting point P of the root transition curve. c1 The radius to the center of rotation of the small wheel, R c1 =R1-k c m t ;(22) R d1 - Radius of the pinion tooth root circle, R d1 =R1-1.25m t ;(twenty three) R a2 - Radius of the tip circle of the large gear tooth, R a2 =R2+m t ;(twenty four) R c2 - The radius of the bottom circle of the large gear tooth, i.e., the starting point P of the root transition curve. c2 The radius to the center of rotation of the large wheel, R c2 =R2-k c m t ;(25) R d2 - Radius of the root circle of the large gear teeth, R d2 =R2-1.25m t (26) - The radial vector of node P1 on the end face of the small wheel and the coordinate axis y Inv1 The acute angle between the positive and negative directions, ；（27） - The radial vector of node P2 on the end face of the large wheel and the coordinate axis y Inv2 The acute angle between the positive and negative directions, ；（28） - The central angle corresponding to the pitch circle tooth thickness of the pinion. (29) -The central angle corresponding to the pitch circle tooth thickness of the large wheel. (30) -P, the tooth profile point on the left end face of the small wheel d1 The radial vector and the x-axis k The acute angle between them ；（31） -P, the tooth profile point on the left end face of the large wheel d2 The radial vector and the x-axis k The acute angle between them ；（32） The overlap ratio of a parabolic gear mechanism with a combined end-face arc and parabolic tooth profile must be greater than 2. The formula for calculating the overlap ratio is: (33) Based on the overlap value linear scaling factor Given the number of teeth Z1 of the pinion, the maximum value of the kinematic parameter variable at the meshing point of the parabolic gear mechanism with the combined tooth profile of the end face arc and the parabola is obtained. ；（34） When the number of teeth Z1 of the pinion and the transmission ratio i are determined 12 End face module m t overlap ratio ε, linear scaling factor End face pressure angle α t Tooth width b, tooth root transition curve shape control parameter T H The angle X of the clockwise rotation of the reference point of the pinion tooth around the center. a1 The angle X of the clockwise rotation of the tooth tip reference point of the large wheel around the center. a2 The angle X of the clockwise rotation of the reference point of the pinion's teeth around the center. c1 The angle X of the clockwise rotation of the reference point at the root of the large wheel around the center. c2 The starting point P of the transition curve at the root of the small wheel and the large wheel ci radius variation coefficient k c At that time, the maximum value of the motion parameter variable Δt at the meshing point, the contact line and the meshing line, the end face combined tooth profile of the pinion and the large gear and their correct installation distance are also determined accordingly. The parabolic tooth line structure of the tooth surface of the pinion and the large gear can also be determined, thus obtaining the parabolic tooth line gear mechanism with the end face arc and parabolic combined tooth profile.

4. The parabolic toothed gear mechanism with a combined end-face arc and parabola tooth profile as described in claim 1, characterized in that: The small wheel is used to connect the input shaft, and the large wheel is used to connect the output shaft.

5. The parabolic toothed gear mechanism with a combined end-face arc and parabola tooth profile as described in claim 4, characterized in that: The input and output shafts connecting the small wheel and the large wheel are interchangeable.

6. The parabolic toothed gear mechanism with a combined end-face arc and parabola tooth profile as described in claim 1, characterized in that: One of the small wheel and the large wheel is connected to an input shaft, and the input shaft is connected to a driver. The driver can drive the small wheel or the large wheel to rotate in both directions.

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