TWI672448B - Design method of point contact cosine helical gear transmission mechanism of fourth-order transmission error - Google Patents

Abstract

A design method of a point contact cosine helical gear transmission mechanism of a fourth-order transmission error is obtained by substituting actual parameters into a surface position vector function, and then drawing a coronal cosine helical pinion and a cosine helical gear, respectively, followed by two The crown cosine spiral pinion and the cosine spiral gear are respectively produced by a grinding wheel. The coronal cosine helical pinion and the cosine helical gear have a fourth-order transmission error superior to the second-order transmission error in noise and vibration, and have better root strength, and the cosine helical gear has a larger helical gear than the involute helical gear. It has the advantages of large root thickness, low root bending stress, high resistance to fatigue fracture, and minimum number of teeth without overcutting, and the point contact design avoids the problems caused by edge contact.

Description

Design method of point contact cosine helical gear transmission mechanism of fourth-order transmission error

The invention relates to a design method, in particular to a design method of a point contact cosine helical gear transmission mechanism with a fourth-order transmission error.

The main purpose of the gear mechanism is to transmit the motion and power between the two shafts. Theoretically, except for the non-circular gears of non-uniform speed ratio, the other constant speed ratio gear mechanism, the rotational speed of the driven gear and the rotational speed of the driving gear are always expected to be a fixed ratio relationship. However, in practice, due to the inevitable manufacturing and assembly errors, the actual speed of the driven gear often cannot match the expected speed, but there is a transmission error. If the transmission error is a linear error, the meshing teeth face will collide with each other, causing the gear mechanism to generate strong vibration and noise. In order to eliminate the adverse effects of linear transmission errors on the system, Litvin proposes to apply a pre-designed Second-Order Transmission Error to absorb the linear transmission error, so that the error curve of the mechanism motion becomes discontinuous and continuous. Greatly reduce the vibration and noise of the system, but the second-order transmission error weakens too much root strength, and the motion curve is not smooth enough, and it is not ideal for vibration suppression and noise reduction. In addition, the common involute spiral The gear has room for improvement due to problems such as a small root thickness and low bending stress.

Accordingly, it is an object of the present invention to provide a method of designing a point contact cosine helical gear transmission that overcomes the above problems.

Therefore, the design method of the point contact cosine helical gear transmission mechanism of the fourth-order transmission error of the present invention, the point contact cosine helical gear transmission mechanism comprises a coronal co-rotating helical pinion gear, and a mesh cosine helical pinion gear can be meshed with the coronal cosine helical pinion gear Cosine spiral gear, the design method includes: substituting the actual parameters into the following surface position vector function to draw the coronal cosine spiral pinion:

Then substitute the actual parameters into the following surface position vector function to draw the cosine spiral gear:

Finally, the crown cosine spiral pinion and the cosine spiral gear are respectively processed by two grinding wheels.

The effect of the invention is that the coronal cosine helical pinion and the cosine helical gear are fourth order transmission errors superior to second order transmission errors in noise and vibration (Second order transmission) Error), the fourth-order transmission error does not weaken too much Tooth fillet strength like the second-order transmission error. Furthermore, the fourth-order transmission error also has characteristics that are insensitive to assembly errors and manufacturing errors, and therefore does not require high-precision assembly adjustment techniques, and thus has a low assembly cost. In addition, compared with the involute helical gear, the cosine helical gear has the following advantages: (1) the root thickness is large (2) the root bending stress is low (3) the ability to resist fatigue fracture is high (4) does not occur The minimum number of teeth of the non-undercutting is small (5). The strength can be reduced to reduce the size of the gear box by reducing the number of teeth, so that it has the advantages of light weight and material cost saving.

1‧‧‧ point contact cosine helical gear transmission

11‧‧‧Coronal co-rotating spiral pinion

111‧‧‧First turning surface

1111‧‧‧First Cosine Curve

12‧‧‧ Cosine spiral gear

121‧‧‧second turning surface

1211‧‧‧Second cosine curve

21‧‧‧Contact tooth prints

22‧‧‧Contact tooth prints

A ₁ , A ₂ ‧‧‧ rotary axis

ρ ₁ , ρ ₂ ‧‧‧ radius parameter

h _f ‧‧‧ root height coefficient

M1‧‧‧Parabolic movement

M2‧‧‧ translational movement

M3‧‧‧Rotary movement

M4‧‧‧ translational movement

M5‧‧‧ translational movement

M6‧‧‧Rotary movement

Other features and effects of the present invention will be apparent from the following description of the drawings. FIG. 1 is a perspective view illustrating one of the design methods of the point contact cosine helical gear transmission mechanism of the fourth-order transmission error of the present invention. 2 is a perspective view showing a state of creating a coronal cosine helical pinion; FIG. 3 is a perspective view showing a state of creating a cosine helical large gear; FIG. 4 is a function graph showing a first The relationship between the cosine curve and a second cosine curve; Figure 5 to Figure 11 are schematic diagrams of the coordinate relationship, indicating the relative motion relationship between the coordinate systems; Figure 12 is a function graph showing the fourth-order transmission error model preset by the point contact cosine helical gear transmission mechanism; Figure 13 is a function curve diagram illustrating the preset contact of the point contact cosine helical gear transmission mechanism in the experimental example The fourth-order transmission error model; Figure 14 is an incomplete perspective view illustrating the contact tooth marks on the coronal cosine helical pinion; and Figure 15 is an incomplete perspective view illustrating the contact tooth marks on the cosine helical bull gear.

Referring to FIG. 1, an embodiment of a design method of a point contact cosine helical gear transmission mechanism for a fourth-order transmission error of the present invention is suitable for designing a point contact cosine helical gear transmission mechanism 1 as shown in FIG. The point contact cosine helical gear transmission mechanism 1 includes a coronal co-rotating helical pinion 11 and a cosine helical bull gear 12 engageable with the coronal cosine helical pinion 11 .

Referring to FIG. 2, FIG. 3, and FIG. 4, in the design method, an Axial profile of a first rotating surface 111 is defined as a first cosine curve 1111, and the first rotating surface 111 is rotated. The axis is A ₁ and the radius parameter is ρ ₁ . The coordinate system S ₁ ( x ₁ , y ₁ , z ₁ ) is fixed to the first cosine curve 1111, and the first cosine curve 1111 is at the position of the coordinate system S ₁ ( x ₁ , y ₁ , z ₁ ) The vector function r ₁ ( u ₁ ) is:

Where h _f is the root height coefficient (1.25 m), m is the modulus, and m _n is the normal modulus.

The axis profile of the second rotation surface 121 is further defined as a second cosine curve 1211. The rotation axis of the second rotation surface 121 is A ₂ and the radius parameter is ρ ₂ . The coordinate system S ₂ ( x ₂ , y ₂ , z ₂ ) is fixed to the second cosine curve 1211, and the second cosine curve 1211 is at the position of the coordinate system S ₂ ( x ₂ , y ₂ , z ₂ ) The vector function r ₂ ( u ₂ ) is:

Where h _f is the root height coefficient (1.25 m), m is the modulus, and m _n is the normal modulus.

Referring to FIG. 4, 5, and 6, to create the helical pinion crown wheel or cosine blade 11 may be formed of the first surface of revolution 111, which is a rotation about the rotation axis A ₁ is formed by the first cosine curve 1111 Surface of revolution. The coordinate system S ₃ (x ₃ , y ₃ , z ₃ ) is fixed to the first rotating surface 111. Since the coordinate system S ₁ (x ₁ , y ₁ , z ₁ ) is fixed to the first cosine curve 1111, the coordinate system S ₁ (x ₁ , y ₁ , z ₁ ) and the coordinate system S ₃ (x ₃ , The relationship between y ₃ and z ₃ ) is shown in Fig. 5. The position vector function r ₃ ( θ ₁ , u ₁ ) of the first plane 111 below the coordinate system S ₃ (x ₃ , y ₃ , z ₃ ) can be obtained by the following coordinate conversion formula:

Where M ₃₁ ( θ ₁ ) is a coordinate transformation matrix that converts the coordinates from the coordinate system S ₁ ( x ₁ , y ₁ , z ₁ ) to the coordinate system S ₃ ( x ₃ , y ₃ , z ₃ ), which is:

Substituting the formula (1) into the formula (3) gives:

Under the coordinate system S ₃ ( x ₃ , y ₃ , z ₃ ), the normal vector function N ₃ ( θ ₁ , u ₁ ) of the first plane 111 and the unit normal vector function n ₃ ( θ ₁ , u ₁ ) Can be expressed as follows:

n ₃ ( θ ₁ , u ₁ )= N ₃ /| N ₃ | (7)

A grinding wheel or blade that creates the cosine helical bull gear 12 can form the second turning surface 121, which is a Surface of Revolution formed by the second cosine curve 1211 rotating about the A ₂ axis. The coordinate system S ₄ ( x ₄ , y ₄ , z ₄ ) is fixed to the second rotating surface 121 because the coordinate system S ₂ ( x ₂ , y ₂ , z ₂ ) is solidified with the second cosine curve 1211 Therefore, the relationship between the coordinate system S ₂ ( x ₂ , y ₂ , z ₂ ) and the coordinate system S ₄ ( x ₄ , y ₄ , z ₄ ) is as shown in FIG. 6. The position vector function r ₄ ( θ ₂ , u ₂ ) of the second plane 121 below the coordinate system S ₄ ( x ₄ , y ₄ , z ₄ ) can be obtained by the following coordinate conversion formula:

Where M ₄₂ ( θ ₂ ) is a coordinate transformation matrix that converts the coordinates from the coordinate system S ₂ ( x ₂ , y ₂ , z ₂ ) to the coordinate system S ₄ ( x ₄ , y ₄ , z ₄ ), which is:

Substituting the formula (2) into the formula (8) gives:

Under the coordinate system S ₄ ( x ₄ , y ₄ , z ₄ ), the normal vector function N ₄ ( θ ₂ , u ₂ ) of the second surface 121 and the unit normal vector function n ₄ ( θ ₂ , u ₂ ) Can be expressed as follows:

n ₄ ( θ ₂ , u ₂ )= N ₄ /| N ₄ | (12)

Referring to FIG. 2 and FIG. 7 and FIG. 8 , when the crown-shaped cosine helical pinion 11 is created by the first rotating surface 111, the first rotating surface 111 performs a parabolic motion M1 as shown in FIG. 2 . And a translational motion M2, the crown cosine helical pinion 11 performs a rotational motion M3. So that coordinate system _{S p (x p, y p} , z p) and the crown cosine helical pinion 11 fixedly connected, coordinates _{S 3 (x 3, y 3} , z 3) and the coordinates S _p (x _p, The relative motion relationship of y _p , z _p ) is shown in FIG. 7 and FIG. 8. FIG. 7 shows that the coordinate system S ₃ (x ₃ , y ₃ , z ₃ ) is in the coordinate system S ₅ (x ₅ , y ₅ , z _{5 ).} On the parabolic motion M1, the parameter is t ₁ . Figure 8 shows that the coordinate system S ₇ (x ₇ , y ₇ , z ₇ ) performs a translational motion M2 along the ( -x ₇ ) direction with the parameter s _p ( ). The coordinate system S _p (x _p , y _p , z _p ) is rotated around the (- z _p ) axis M3, and the parameter is . There is no relative motion between the coordinate system S ₅ (x ₅ , y ₅ , z ₅ ) and the coordinate system S ₇ (x ₇ , y ₇ , z ₇ ), and the coordinates are from the coordinate system S ₃ (x ₃ , y ₃ , z ₃ The coordinate transformation matrix M ₅₃ ( t ₁ ) converted to the coordinate system S ₅ (x ₅ , y ₅ , z ₅ ) can be expressed as:

Converting the coordinates from the coordinate system S ₅ ( x ₅ , y ₅ , z ₅ ) to the coordinate transformation matrix M _{p 5 of the} coordinate system S _p ( x _p , y _p , z _p ) ) can be expressed as:

Where β is a helix angle, ρ _p = m _t T _p /2, m _t is the end face modulus, and T _{p is} the number of teeth of the coronal cosine helical pinion 11 . The parameter of the translational motion M2 s _p ( ) and the parameters of the rotational motion M2 The relationship can be expressed as a fourth-order polynomial function as follows:

Wherein, C _2, C ₃ and C ₄ coefficients to be determined, the group {Σ _111} surfaces of the first surface of revolution 111 is formed under the coordinates _{S p (x p, y p} , z p) position vector function ( θ ₁ , u ₁ , t ₁ , ) can be obtained by the following coordinate conversion formula:

Substituting the formula (5) into the formula (16), you can obtain:

{Σ _111} surface set unit normal vector of the function of the surface 111 under the first rotating coordinate system _{S p (x p, y p} , z p) of the formed ( θ ₁ , u ₁ , ) can be obtained by the following coordinate conversion formula:

among them:

The necessary conditions for the envelope surface of the surface family {Σ ₁₁₁ } are:

According to the formula (20), the parameter t ₁ can be solved as:

From the equations (17), (21) and (22), the surface position vector function of the coronal cosine helical pinion 11 can be obtained as follows:

The tooth surface unit normal vector function of the coronal cosine helical pinion 11 can be obtained by the equations (18) and (20) as follows:

Referring to FIG. 3 together with FIG. 9 and FIG. 10, when the cosine helical bull gear 12 is created by the second rotating surface 121, the second rotating surface 121 performs two translational movements M4 and M5, and the cosine helical large gear 12 then performs a rotary motion M6. Let the coordinate system S _g (x _g , y _g , z _g ) be fixed to the cosine helical gear 12, the coordinate system S ₄ (x ₄ , y ₄ , z ₄ ) and the coordinate system S _g (x _g , y The relationship between _g and z _g ) is shown in Fig. 9 and Fig. 10. Fig. 9 shows that the coordinate system S ₄ (x ₄ , y ₄ , z ₄ ) is made on the coordinate system S ₆ (x ₆ , y ₆ , z ₆ ). The translational motion M4 has a parameter of t ₂ , and FIG. 10 shows that the coordinate system S ₈ (x ₈ , y ₈ , z ₈ ) performs a translational motion M5 along the (− x ₈ ) direction, and the parameter is s _g ( ), the coordinate system S _g (x _g , y _g , z _g ) is rotated around the z _g axis M6, the parameter is There is no relative motion between the coordinate system S ₆ (x ₆ , y ₆ , z ₆ ) and the coordinate system S ₈ (x ₈ , y ₈ , z ₈ ). The coordinate transformation matrix M ₆₄ ( t ₂ ) that converts the coordinates from the coordinate system S ₄ (x ₄ , y ₄ , z ₄ ) to the coordinate system S ₆ (x ₆ , y ₆ , z ₆ ) can be expressed as:

Converting the coordinates from the coordinate system S ₆ (x ₆ , y ₆ , z ₆ ) to the coordinate transformation matrix M _{g 6 of the} coordinate system S _g (x _g , y _g , z _g ) ( ) can be expressed as:

Where β is the helix angle, ρ _g = m _t T _g /2, m _t is the end face modulus, and T _{g is} the number of teeth of the cosine helical bull gear 12 . The parameter s _{g of the} translational motion M5 ( ) and the parameters of the rotational motion M6 The relationship can be expressed as a first-order polynomial function as follows:

The position vector function of the surface family {Σ ₁₂₁ } formed by the second rotating surface 121 under the coordinate system S _g (x _g , y _g , z _g ) ( θ ₂ , u ₂ , t ₂ , ) can be obtained by the following coordinate conversion formula:

Substituting the formula (10) into the formula (28), you can obtain:

The unit normal vector function of the surface family {Σ ₁₂₁ } formed by the second rotation surface 121 under the coordinate system S _g (x _g , y _g , z _g ) ( θ ₂ , u ₂ , ) can be obtained by the following coordinate conversion formula:

among them:

The necessary conditions for the existence of the envelope of the surface family {Σ ₁₂₁ } are:

According to the formula (32), the parameter θ ₂ can be solved as: θ ₂ =0 (34)

According to the equations (33) and (34), the parameter θ ₂ can be solved:

The tooth surface position vector function of the cosine helical bull gear 12 can be obtained from the equations (29), (34), and (35):

The tooth surface unit normal vector function of the cosine helical bull gear 12 can be obtained by the equations (30), (34), and (35):

Referring to Figures 1, 4, and 11, the undetermined coefficients C ₂ , C _{3 ,} and C ₄ in the translational motion M2 are then determined. As shown in FIG. 11, when the coronal cosine helical pinion 11 and the cosine helical bull gear 12 are meshed and transmitted, the coronal cosine helical pinion 11 rotates about a z _p axis with a parameter φ _p , and the cosine helical large gear 12 takes a parameter φ _{The g} -rotating ( -z _g ) axis rotates, and the coordinate system S _f ( x _f , y _f , z _f ) is a fixed coordinate system fixed to the gear box, and the coronal cosine helical pinion 11 and the cosine helical large gear 12 The constraint conditions for the tooth surface contact points are:

among them:

When the rotation angle φ _p of the coronal cosine helical pinion 11 is input and the rotation angle φ _g of the cosine helical large gear 12 is output, the transmission error of the gear mechanism is:

The slope of the transmission error is:

among them:

Referring to FIG. 1, FIG. 11, and FIG. 12, FIG. 12 is a preset fourth-order transmission error model of the point contact cosine helical gear transmission mechanism 1. When the coronal cosine helical pinion 11 and the cosine helical large gear 12 are at point R When contacting, φ _p and φ _g have the following conditions:

When the coronal cosine helical pinion 11 and the cosine helical large gear 12 are in contact at point L, φ _p and φ _g have the following conditions:

When the coronal cosine helical pinion 11 and the cosine helical bull gear 12 are in contact at point R, let θ ₁ , u ₁ , , u ₂ , t ₂ conditions are as follows:

When the coronal cosine helical pinion 11 and the cosine helical bull gear 12 are in contact at point L, let θ ₁ , u ₁ , , u ₂ , t ₂ conditions are as follows:

Since the point R is the contact point, the constraint type (38) of the contact point must also be established at the point R, so that:

Since the point L is the contact point, the constraint type (38) of the contact point must also be established at the point L, so that:

In addition, there is a constraint condition at the point L where the slope of the motion error is zero, so there are:

due to There are thirteen independent nonlinear algebraic equations in equations (54), (55), and (56). Since there are only ten unknowns in the equation, namely θ _1R , u _1R , , θ _1L , u _1L , , u _2L , t _{2 L} , u _2R , t _{2 R} , so the extra three equations can be used to solve the three undetermined coefficients C ₂ , C ₃ and C _{4 of} the translational motion M2 , in other words, the translational motion M2 After the three undetermined coefficients C ₂ , C ₃ and C ₄ are also considered to be unknown, equations (54), (55), and (56) form a thirteen equation and thirteen Unknown nonlinear bifurcated program system:

The three undetermined coefficients C ₂ , C ₃ and C _{4 of} the translational motion M2 can be obtained by solving the nonlinear simultaneous equation (57) using the Newton's root finding method.

According to actual needs, the parameters are substituted into the equations (23) and (36), that is, the position vector function of the coronal cosine helical pinion 11 and the position vector function of the cosine helical gear 12, and then can be drawn in CAD. The crown cosine helical pinion 11 and the cosine helical bull gear 12 are completed, thereby completing the design method, and finally, a workpiece to be processed which performs a rotational motion M3 by a grinding wheel or a blade capable of forming the first rotary surface 111 by rotation is performed. Processing, the grinding wheel creates the coronal cosine helical pinion 11 through the translational motion M2 and the rotational motion M3. In addition, a grinding wheel or a blade capable of rotating the second rotating surface 121 is used to process another workpiece to be processed by a rotating motion M5, and the grinding wheel or the blade creates the cosine spiral large tooth through two translational movements M4 and M5. Wheel 12. The efficacy of this example is examined below by way of an experimental example.

Experimental example: Referring to Fig. 13, Fig. 14, and Fig. 15, this experimental example takes the values listed in Table 1 below as parameters, and substitutes these parameters into equations (23) and (36) to draw out The coronal cosine helical pinion 11 and the cosine helical bull gear 12. Then, the true transmission error of the point contact cosine helical gear transmission mechanism 1 is analyzed by the Tooth contact analysis (TCA) technique. As can be seen from FIG. 13, the actual transmission error of the embodiment is indeed equal to the pre-preparation. A fourth-order transmission error (Predesigned fourth order transmission error), at this time, the contact contact on the coronal cosine helical pinion 11 is a localized contact toothing 21 as shown in FIG. 14 (Localized bearing) Contact). The contact tooth marks on the cosine helical bull gear 12 are also localized contact tooth marks 22 as shown in FIG. As is apparent from FIGS. 14 and 15, the crown cosine helical pinion 11 and the contact tooth marks 21, 22 on the cosine helical bull gear 12 are concentrated in the center of the tooth surface and away from the Tooth edge. Therefore, the problem of stress concentration caused by tooth edge contact and crushing of the tooth edge can be avoided.

In summary, the point contact cosine helical gear transmission mechanism 1 with fourth-order transmission error can be designed by the design method. The cosine helical gear has the following advantages: (1) the root thickness is large (2) the root root bending stress is low ( 3) The ability to resist fatigue fracture is high (4) The minimum number of teeth without non-cutting is small (5) Under the premise of sufficient strength, the gear size can be reduced to achieve the purpose of reducing the size of the gear box. The advantages of light weight and material cost saving, and the design of point contact can avoid the problems caused by edge contact, and the fourth-order transmission error can improve the root strength, and can also improve the ability of vibration suppression and noise reduction, so it can achieve this. The purpose of the invention.

However, the above is only the embodiment of the present invention, and the scope of the invention is not limited thereto, and all the equivalent equivalent changes and modifications according to the scope of the patent application and the patent specification of the present invention are still The scope of the invention is covered.

Claims (1)

A design method of a point contact cosine helical gear transmission mechanism with a fourth-order transmission error, the point contact cosine helical gear transmission mechanism comprising a coronal co-rotating helical pinion, and a cosine spiral that can mesh with the coronal cosine helical pinion Gear, the design method includes:
Substituting the actual parameters into the following surface position vector functions to draw the coronal cosine spiral pinion:

Then substitute the actual parameters into the following surface position vector function to draw the cosine spiral gear:

Finally, the crown cosine spiral pinion and the cosine spiral gear are respectively processed by two grinding wheels or blades.