TWI672454B - Design method of point contact curved tooth cosine gear transmission mechanism with preset fourth-order transmission error - Google Patents

Abstract

A design method of a point contact curved tooth cosine gear transmission mechanism with a preset fourth-order transmission error, the point contact curved tooth cosine gear transmission mechanism comprising a curved tooth cosine pinion gear and a curved tooth cosine gear wheel, The design method is to first design two rotating surfaces of the grinding wheel or the cutting edge, and the axial section profile is a cosine function curve, and the curved tooth cosine pinion gear and the curved tooth cosine large gear are respectively created by the grinding wheels or the cutting edges. The point contact curved tooth cosine gear transmission mechanism designed by the design method has a preset fourth-order transmission error and has a relatively smooth motion curve, which is superior to the conventional second-order transmission error in anti-noise and anti-vibration capability. Gears do not weaken too much root strength.

Description

Design method of point contact curved tooth cosine gear transmission mechanism with preset fourth-order transmission error

The invention relates to a design method, in particular to a design method of a curved tooth cosine gear transmission mechanism.

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. In order for the gear set to have a preset second-order transmission error, the conjugate tooth surface can be modified by changing the tool geometry or changing the relative motion between the tool and the created tooth surface. However, the second-order transmission error weakens too much root strength, and the motion curve is not smooth enough, and there is still room for improvement in vibration suppression and noise reduction.

Accordingly, it is an object of the present invention to provide a method of designing a gear transmission having a predetermined fourth-order transmission error.

Therefore, the present invention has a design method of a point contact curved tooth cosine gear transmission mechanism with a preset fourth-order transmission error, the point contact curved tooth cosine gear transmission mechanism includes a curved tooth cosine pinion gear, and one can be combined with the music A curved tooth cosine gear that meshes with a cosine pinion.

The design method comprises: designing a grinding wheel or a blade, which rotates to generate a first rotating surface, the axial sectional profile of the first rotating surface is a cosine function curve, and the position vector function thereof is:

among them Is the root height coefficient, m is the modulus, and then the curved tooth cosine pinion is created by the grinding wheel or the blade, and the coordinate of the grinding wheel or the blade is , the coordinate system of the curved cosine pinion gear is , the grinding wheel or blade is in the coordinate system Function along The direction is linearly translated, and the curved cosine pinion is processed in the coordinate system Parameter Winding around Axis rotation, function And parameters The relationship between the four-order polynomials is as follows:

Where m is the modulus, T _{p is} the number of teeth of the cosine pinion of the tooth line, and C ₂ , C ₃ and C ₄ are the undetermined coefficients for generating the preset fourth-order transmission error;

Then another grinding wheel or blade is designed, which generates a second rotating surface when rotating, the axis profile of the second rotating surface is a cosine function curve, and its position vector function for:

,

among them Is the root height coefficient, m is the modulus, and then the curved gear cosine gear is created by the grinding wheel or the blade, and the coordinate of the grinding wheel or the blade is The coordinate system of the curved cosine gear is , the grinding wheel or blade is in the coordinate system Function along The direction is linearly translated, and the curved cosine gear is processed in the coordinate system Parameter Winding around Axis rotation, function And parameters The relationship between the first order polynomials is as follows:

Where m is the modulus and T _{g is} the number of teeth of the cosine gear of the tooth line.

The effect of the invention is that the point contact curved tooth cosine gear transmission mechanism designed by the design method has a preset fourth-order transmission error and has a relatively smooth motion curve, so that the noise resistance and vibration resistance are superior. Conventional second-order transmission error gears do not weaken too much root strength.

Referring to FIG. 1 , an embodiment of a design method of a point contact curved tooth cosine gear transmission mechanism with a preset fourth-order transmission error is suitable for designing and manufacturing a point contact curved tooth cosine gear transmission mechanism 1 . The toothed cosine gear transmission mechanism 1 includes a curved tooth cosine pinion 11 and a curved tooth cosine gear 12 engageable with the curved tooth cosine pinion 11.

Referring to FIG. 2, FIG. 3, and FIG. 4, in the design method, the curved tooth cosine pinion 11 and the curved tooth cosine large gear 12 are respectively created by two grinding wheels or blades, and the curved tooth line is created. The grinding wheel or the blade of the cosine pinion 11 is rotatably formed to form a first rotating surface 111, and the grinding wheel or the blade of the curved tooth cosine large gear 12 is created, and a second rotating surface 121 is formed by the rotation. The rotation axis of the first rotation surface 111 is A ₁ , and the radius parameter is The axis profile (Axial profile, hereinafter referred to as the first axis profile 1111) is a cosine function curve. The rotation axis of the second rotation surface 121 is A ₂ , and the radius parameter is The axis profile (hereinafter referred to as the second axis profile 1211) is also a cosine function curve.

Coordinate system Attached to the first axis profile 1111, in the coordinate system Below, the position vector function of the first axis profile 1111 for:

, (1)

Coordinate system Attached to the second axial profile 1211, in the coordinate system Below, the position vector function of the second axis profile 1211 for:

, (2)

Referring to FIG. 2, FIG. 4, and FIG. 5, in order to establish a position vector function of the first rotating surface 111, the coordinate system is used. Attached to the first rotating surface 111, the coordinate system And coordinate system The relationship is shown in Figure 5, in the coordinate system Next, the position vector function of the first plane 111 for:

(3)

At the coordinate system Next, the normal vector function of the first surface 111 Unit normal vector function They are:

(4)

(5)

Referring to FIG. 3, FIG. 4, and FIG. 6, in order to establish a position vector function of the second rotating surface 121, the coordinate system is used. Attached to the second rotating surface 121, the coordinate system And coordinate system The relationship is shown in Figure 6, in the coordinate system Next, the position vector function of the second surface 121 for:

(6)

At the coordinate system Next, the normal vector function of the second surface 121 Unit normal vector function They are:

(7)

(8)

Refer to Figure 2 and Figure 7, to make the coordinate system Coupling with the curved tooth cosine pinion 11 , when the first rotating surface 111 is creating the curved cosine pinion 11 , the coordinate system And coordinate system The relative motion relationship is shown in Figure 7, the coordinate system Function along Directional linear translation, coordinate system Parameter Winding around Axis rotation, function And parameters The relationship between the four-order polynomials is as follows:

(9)

Where m is the modulus, T _{p is} the number of teeth of the cogwheel pinion 11 of the tooth line, and C ₂ , C ₃ and C ₄ are the undetermined coefficients for generating the preset fourth-order transmission error.

Refer to Figure 3 and Figure 8, to make the coordinate system Coupling with the curved tooth cosine gear 12, when the second rotating surface 121 is creating the curved cosine gear 12, the coordinate system And coordinate system The relative motion relationship is shown in Figure 8, the coordinate system Function along Directional linear translation, coordinate system Parameter Winding around Axis rotation, function And parameters The relationship between the first order polynomials is as follows:

(10)

Where m is the modulus and T _{g is} the number of teeth of the cogwheel large gear 11 of the tooth line.

Referring to FIG. 1 , FIG. 2 , and FIG. 3 , the first rotating surface 111 forms a curved surface family on the curved tooth cosine pinion 11 . According to Coordinate transformation theory, the surface family Position vector function for:

(11)

among them:

(12)

Surface family Unit normal vector function for:

(13)

among them:

(14)

According to The theory of gearing, the surface family The necessary conditions for the existence of the envelope are:

(15)

among them:

(16)

(17)

(18)

The second rotating surface 121 forms a curved surface on the curved tooth cosine gear 12 According to the coordinate conversion theory, the surface family Position vector function for:

(19)

among them:

(20)

Surface family Unit normal vector function for:

(twenty one)

among them:

(twenty two)

According to the theory of gear meshing, the surface family The necessary conditions for the existence of the envelope are:

(twenty three)

among them:

(twenty four)

(25)

(26)

After substituting the equations (24), (25), and (26) into the equation (23), the parameters can be Decomposition into an explicit function :

(27)

According to the gear meshing theory, the position vector function of the surface family is linked with the necessary conditions of the surface family envelope, and the tooth surface position vector function of the gear can be obtained. By combining the unit normal vector function of the surface family with the necessary conditions of the surface family envelope, the tooth surface unit normal vector function of the gear can be obtained. Therefore, the tooth surface position vector function of the curved tooth cosine pinion 11 is:

(28)

The tooth surface unit normal vector function of the curved tooth cosine pinion 11 is:

(29)

The tooth surface position vector function of the curved tooth cosine gear 12 is:

(30)

The tooth surface unit normal vector function of the curved tooth cosine gear 12 is:

(31)

Referring to FIG. 1, FIG. 9, and FIG. 10, when the curved tooth cosine pinion 11 is meshed with the curved tooth cosine gear 12, the curved tooth cosine pinion 11 takes parameters. Winding around Axis rotation, the curved cosine gear 12 with parameters Winding around The axis rotates. Coordinate system It is a fixed coordinate system that is attached to the gearbox. The constraint conditions of the contact point of the curved tooth cosine pinion 11 and the curved tooth cosine gear 12 are:

(32)

among them:

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

Let the corner of the curved cosine pinion 11 As an input, the corner of the curved cosine gear 12 As an output, the transmission error of the point contact curved gear cosine gear transmission mechanism is:

(41)

The slope of the transmission error is:

(42)

among them:

(43)

(44)

(45)

10 is a preset fourth-order transmission error model of the point-contact curved tooth cosine gear transmission mechanism, so that when the curved tooth cosine pinion 11 and the curved tooth cosine large gear 12 are in contact at the R point, the following conditions are met: :

(46)

When the curved tooth cosine pinion 11 and the curved tooth cosine large gear 12 are in contact at the L point, the parameters have the following conditions:

(47)

Since the point R is the contact point, the constraint condition of the contact point of the equation (32) must also be established at the point R. Substituting the formula (46) into the equation (32) gives:

(48)

Since the point L is the contact point, the constraint condition of the contact point of the (32) type must also be established at the point L. Substituting the formula (47) into the equation (32) gives:

(49)

In addition, at the point L, there is a constraint condition that the slope of the transmission error is zero, and the equation (47) is substituted into the equation (42), and is made equal to zero.

(50)

According to Figure 10, the parameters The values are known as follows:

(51)

Since the length of the unit vector is known to be equal to 1, Therefore, there are thirteen independent nonlinear algebraic equations in equations (48), (49), and (50), but there are only ten unknowns in these equations: Therefore, the undetermined coefficients C ₂ , C ₃ and C _{4 can} also be regarded as unknown. After considering the three undetermined coefficients C ₂ , C ₃ and C ₄ as unknowns, equations (48), (49) and (50) form a thirteen equation and thirteen Unknown nonlinear joint equation system, as follows:

(52)

Then, using the Newton root method to solve the equation (52), you can find it. Wait for thirteen unknowns, so you can find three undetermined coefficients C ₂ , C ₃ and C ₄ . Referring to FIG. 2 and FIG. 3, finally, a blade or a grinding wheel which can form the first rotating surface 111 during rotation is designed to create the curved tooth cosine pinion 11 which is translated by the first rotating surface 111 of the cutting edge or the grinding wheel. The motion M1, the curved tooth cosine pinion 11 performs a rotational motion M2, and the three undetermined coefficients C ₂ , C ₃ and C ₄ in the translational motion M1 can be obtained according to the above steps. Then, a blade or a grinding wheel that can form the second rotating surface 121 is formed to create the curved tooth cosine large gear 12, and the second rotating surface 121 of the cutting edge or the grinding wheel performs a translational motion M3, the curved cosine The large gear 12 performs a rotational motion M4. The efficacy of this example is examined below by way of an experimental example.

Experimental example:

Referring to Figure 11, Figure 12, and Figure 13, together with Table 1, Table 2, and Table 3 below, Table 1 is the parameter setting used in the experimental example, and Table 2 is the first rotation surface 111 (see Figure 2). And the relative motion parameter data value of the curved cosine pinion 11 . Table 3 shows the relative motion parameter data values of the second rotary surface 121 (see FIG. 3) and the curved tooth cosine large gear 12. Under the parameter setting of Table 1, through the Tooth contact analysis (TCA), it can be verified that the real transmission error of this embodiment is indeed equal to the preset fourth order transmission error (Figure 12). Shown. At this time, the contact tooth 21 on the curved tooth cosine pinion 11 is a localized bearing contact, as shown in FIG. 13, the crank line cosine gear 12 is The contact tooth print 22 is also a regional contact tooth mark 22, whereby it can be clearly seen that the curved tooth cosine pinion 11 and the contact tooth marks 21 and 22 on the curved tooth cosine large gear 12 are concentrated on the teeth. The center of the face is far from the Tooth edge, so the point contact curved tooth cosine gear transmission mechanism 1 designed in this embodiment has no problem of edge contact.

Table 1: Parameter Name Symbol Sets the Unit Modulus Number of teeth of 10 mm curved tooth cosine pinion 11 20 -- Number of teeth of the curved tooth cosine gear 12 33 -- radius parameter of the surface 111 Radius parameter of 90 mm rotary surface 121 108 mm root height factor 1.25 Mm pinion 11 radius Mm large gear 12 pitch radius Mm presets the magnitude of the fourth-order transmission error 10 arcsec rotary surface 111 to create the motion parameters of the pinion 11 6164.5953e-4 The motion parameter when the rotary surface 111 is created as the pinion 11 -25977.59701e-4 The motion parameter when the rotary surface 111 is created as the pinion 11 24860.52367e-4

Table II: (rad) (mm) -0.095 -9.49201 -0.0636 -6.3568 -0.0322 -3.21927 0 0 0.0306 3.0605 0.062 6.20179 0.0934 9.34345 0.1248 12.4852 0.1562 15.6266 0.1876 18.7676

Table 3: (rad) (mm) -0.055 -9.075 -0.0362 -5.973 -0.0174 -2.871 0 0 0.0202 3.333 0.039 6.435 0.0578 9.537 0.0766 12.639 0.0954 15.741 0.1142 18.843

In this embodiment, a surface of revolution is used as a tool for creating a tooth, and the tooth line of the tooth is a Curvilinear tooth trace. The curved tooth line can make the transmission shaft only receive radial force and no axial force, so it is not necessary to use the bearing with the thrust function, the use cost of the bearing can be reduced, and the curved tooth cosine pinion 11 and the curved tooth line The gear teeth of the cosine large gear 12 are created by the cosine blade rotary surface. Compared with the gear teeth created by the traditional straight edge rotary surface, the cosine blade rotary surface creates a large tooth root thickness, and the root of the tooth The bending stress is low, the ability to resist fatigue fracture is also high, and the minimum number of teeth of the gear is not under-cutting. Therefore, the gear size can be reduced by reducing the number of teeth under the condition of sufficient strength. Therefore, it has the advantage of being lightweight and saving material costs.

In addition, the finished product of the embodiment has a preset fourth order transmission error, which is superior to the second order transmission error in suppressing noise and vibration through a relatively smooth motion curve (Second order transmission error) ), and does not weaken too much Tooth fillet strength like the second-order transmission error, and the fourth-order transmission error also has characteristics that are insensitive to assembly errors and manufacturing errors, so high-precision assembly adjustment is not required. Technology, therefore with lower assembly costs.

Finally, the manufacturer only needs to replace the vertical milling cutter with a face milling cutter on a common four-axis CNC milling machine. The blade curve of the blade is a cosine function curve, and the Y axis and the A axis are controlled by the NC program. With the coordinates, the curved tooth cosine pinion 11 and the curved tooth cosine large gear 12 of the present embodiment can be easily processed, and since it is not necessary to use a special dedicated machine, the manufacturing cost can be greatly reduced.

In summary, the point contact curved tooth cosine gear transmission mechanism 1 with a preset fourth-order transmission error can be obtained by the design method, and the curved tooth line can make the transmission shaft only receive radial force without axial force. Therefore, it is not necessary to use a bearing having a thrust function, so that the object of the present invention can be achieved.

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.

1‧‧‧ point contact curved wire cosine gear transmission

11‧‧‧ Curved cosine pinion

111‧‧‧First turning surface

1111‧‧‧First axis profile

12‧‧‧The curved cosine gear

121‧‧‧second turning surface

1211‧‧‧Second axis profile

21‧‧‧Contact tooth prints

22‧‧‧Contact tooth prints

M1‧‧‧ translational movement

M2‧‧‧ Rotating movement

M3‧‧‧ translational movement

M4‧‧‧ Rotating movement

Other features and effects of the present invention will be apparent from the embodiments of the drawings, in which:  1 is a perspective view showing an embodiment of a design method of a point contact curved tooth cosine gear transmission mechanism having a preset fourth-order transmission error;  Figure 2 is a perspective view showing the state of creating a curved cosine pinion;  Figure 3 is a perspective view showing the state of creating a curved cosine gear;  Figure 4 is a function graph illustrating the relationship between a first axial profile and a second axial profile;  Figures 5 to 9 are schematic diagrams of coordinate relationships, illustrating the relative motion relationship between coordinate systems;  10 is a function graph illustrating a fourth-order transmission error model preset at the point of contact with the curved tooth cosine gear transmission mechanism;  Figure 11 is a function graph illustrating the fourth-order transmission error model preset by the point contact curved gear cosine gear transmission mechanism in the experimental example;  Figure 12 is an incomplete perspective view showing the contact tooth marks on the curved cosine pinion; and  Figure 13 is an incomplete perspective view showing the contact tooth marks on the curved cosine gear.  

Claims (1)

A design method of a point contact curved tooth cosine gear transmission mechanism with a preset fourth-order transmission error, the point contact curved tooth cosine gear transmission mechanism comprising a curved tooth cosine pinion gear, and a cosine cosine A curved tooth cosine gear that meshes with a pinion gear, the design method includes:
Designing a grinding wheel or blade, when rotating, a first rotating surface is generated, the axial profile of the first rotating surface is a cosine function curve, and its position vector function for:
,
among them Is the root height coefficient, m is the modulus, and then the curved tooth cosine pinion is created by the grinding wheel or the blade, and the coordinate of the grinding wheel or the blade is , the coordinate system of the curved cosine pinion gear is , the grinding wheel or blade is in the coordinate system Function along The direction is linearly translated, and the curved cosine pinion is processed in the coordinate system Parameter Winding around Axis rotation, function And parameters The relationship between the four-order polynomials is as follows:

Where m is the modulus, T _{p is} the number of teeth of the cosine pinion of the tooth line, and C ₂ , C ₃ and C ₄ are the undetermined coefficients for generating the preset fourth-order transmission error;
Then another grinding wheel or blade is designed, which generates a second rotating surface when rotating, the axis profile of the second rotating surface is a cosine function curve, and its position vector function for:
,
among them Is the root height coefficient, m is the modulus, and then the curved gear cosine gear is created by the grinding wheel or the blade, and the coordinate of the grinding wheel or the blade is The coordinate system of the curved cosine gear is , the grinding wheel or blade is in the coordinate system Function along The direction is linearly translated, and the curved cosine gear is processed in the coordinate system Parameter Winding around Axis rotation, function And parameters The relationship between the first order polynomials is as follows:

Where m is the modulus and T _{g is} the number of teeth of the cosine gear of the tooth line.