JP3521744B2 - Eccentric tolerance determination method and meshing phase restriction method in gear train - Google Patents

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention is used for constructing a gear drive train formed by meshing a plurality of gears including at least an input gear and an output gear, and more specifically, for forming a gear drive train. The present invention relates to a method for determining an eccentricity tolerance in manufacturing each gear, and a method for regulating a gear meshing phase in a gear drive train.

[0002]

2. Description of the Related Art In a color image forming machine called a tandem type, four electrophotographic photosensitive drums for each color of black, yellow, cyan and magenta are arranged. An image of each color is formed on each of the photosensitive drums, and an image of each color is transferred to a common transfer member, thus forming a color image on the transfer member.
In such a color image forming machine, in order to avoid so-called color misregistration in the generated color image, it is important to make the rotational error of the low cycle between the photosensitive drums sufficiently small. Therefore, usually, each of the photosensitive drums is provided with a rotary drive source such as a pulse motor,
By electronically controlling these rotary drive sources, the rotational error between the photosensitive drums is kept below a required value.

As described above, if a rotary drive source is attached to each of the photosensitive drums, as will be easily understood, the manufacturing cost of the color image forming machine becomes considerably high.
In order to reduce the manufacturing cost, a single common rotary drive source is provided instead of the four rotary drive sources, and each of the four photosensitive drums is formed by meshing a plurality of gears. It is intended to be connected to a common rotary drive source via a drive train. Thus, when a gear drive train is arranged between the rotary drive source and each of the photosensitive drums, a rotation error due to the eccentricity of the gears forming the gear drive train is generated in each of the photosensitive drums. To be done. Therefore, in manufacturing and assembling the gear drive train, it is necessary to properly grasp the relative relationship between the eccentricity of the gears and the rotation error generated on each of the photosensitive drums, and determine the rotation error generated on each of the photosensitive drums to the required value. The following are necessary.

Regarding the interrelationship between the eccentricity of two gears meshed with each other, that is, the drive gear and the driven gear, and the rotational error of the driven gear, the Japan Society of Mechanical Engineers, Vol. Issued monthly) pp. 148-157
Yuichi Asano, "Analytical Study on Influence of Eccentricity on Involute Gear Meshing" (hereinafter referred to as "Asano Paper")
It is described in. According to the calculation formula described in the Asano paper, it is possible to obtain the rotation error of the driven gear due to the eccentricity of the two gears. However, the calculation formula described in the Asano paper is remarkably complicated, and it is considerably difficult to calculate the rotation error of the driven gear even when the gear drive train is composed of two gears. If the train is made up of three or more gears, it is practically very difficult, if not impossible, to calculate the rotational error of the final or output gear.

[0005]

SUMMARY OF THE INVENTION The present invention has been made in view of the above facts, and its basic problem is that even when the gear drive train is composed of three or more gears, The present invention is to devise a calculation formula capable of relatively easily and approximately calculating the rotation error of the output gear due to the eccentricity of the gears forming the gear drive train.

The technical problem in one aspect of the present invention is to determine the eccentricity tolerance in the manufacture of each gear so that the maximum rotation error of the output gear calculated based on the above-mentioned calculation formula is set to a predetermined value or less, Thus, the eccentricity tolerance can be set to be extremely small and the rotational error of the output gear can be kept below a predetermined value without excessively increasing the manufacturing cost of the gear.

A technical problem in the second aspect of the present invention is to mesh at least one pair of mutually meshing gears when the maximum rotation error of the output gears calculated based on the above-mentioned calculation formula exceeds a predetermined value. The phase is regulated so that the rotation error of the output gear is equal to or less than the above predetermined value, and thus the eccentricity tolerance in the production of each gear is considerably reduced to increase the production cost of the gear without increasing the rotation error of the output gear. That is, it is possible to reduce the value to a predetermined value or less.

[0008]

Means for Solving the Problems The inventors of the present invention have conducted extensive studies on the relationship between the eccentricity of each gear forming the gear drive train and the rotation error of the output gear, and have made the meshing positions between the gears meshing with each other. When the rotation error of the output gear is calculated by solving the rotation angle ratio of the gears meshing with each other by approximating the radius of gyration at the cosine theorem and solving the rotation angle ratio as a differential equation of the variable separation type, the gear drive train is It has been found that the rotational error of the output gear can be relatively easily obtained even when the rotational error of the output gear is composed of three or more gears, and the obtained rotational error is sufficiently precise in practical use.

Therefore, according to the first aspect of the present invention, in the manufacture of each of the gears used in the gear drive train configured by meshing a plurality of gears including at least an input gear and an output gear. As an eccentricity tolerance determination method, approximate the radius of gyration at the meshing position between mutually meshing gears by the cosine theorem to find the rotation angle ratio of the mutually meshing gears, and solve the rotation angle ratio as a differential equation of variable separation type. Based on the above, the maximum rotation error of the output gear with respect to the input gear is obtained, and the eccentricity tolerance in manufacturing each of the gears is determined so that the maximum rotation error is equal to or less than a predetermined value. An eccentricity tolerance determination method in a row is provided.

According to the second aspect of the present invention, there is provided a meshing phase regulation method in a gear drive train which is constituted by meshing a plurality of gears including at least an input gear and an output gear, and the gears are meshed with each other. The radius of gyration at the meshing position of is approximated by the cosine theorem to obtain the rotation angle ratio of the gears meshing with each other, and based on solving the rotation angle ratio as a variable separation type differential equation, the output gear of the input gear If the maximum rotation error is found and the maximum rotation error exceeds a predetermined value, the meshing phase of at least one pair of gears that mesh with each other is adjusted so that the rotation error of the output gear with respect to the input gear becomes equal to or less than the predetermined value. There is provided a meshing phase regulation method characterized by:

The eccentricity tolerance determining method and the meshing phase regulating method provided by the present invention are not limited to the gear drive train used in the above-described tandem type color image forming machine, and the rotation error of the output gear is not more than a predetermined value. It goes without saying that the invention can be applied to gear drive trains in various machines that need to be squeezed.

[0012]

BEST MODE FOR CARRYING OUT THE INVENTION Preferred embodiments of the present invention will be described in more detail below with reference to the accompanying drawings.

FIG. 1 shows an input gear 2 meshed with each other.
A gear drive train is shown which is composed of an output gear 4 and an output gear 4. The input gear 2 is connected to a rotary drive source (not shown) such as a pulse motor, and the output gear 4 is connected to an appropriate driven rotary member (not shown). Input gear 2
Although pitch circle 6 of a true circle having a radius r _1, not a perfect circle pitch circle 6 and 8 in the pitch circle 8 of the output gear 4 be regarded as is true circle of radius r ₂ (strictly, There is no practical problem even if it is regarded as a perfect circle). A point A indicates the center of rotation of the input gear 2 and is eccentric by e _{1 with} respect to the center C of the pitch circle 6 due to a manufacturing error. A point B indicates the center of rotation of the output gear 4 and is eccentric by e _{2 with} respect to the center D of the pitch circle 8 due to a manufacturing error. Point G is a mutual meshing point between the input gear 2 and the output gear 4. r ′ ₁ is the radius of rotation of the input gear, that is, the distance from the center of rotation A to the meshing point G.
r ′ ₂ is the radius of gyration of the output gear and is the distance from the center of rotation B to the meshing point G.

[0014] The ratio of the input gear 2 angular velocity omega ₁ and the angular velocity omega ₂ of the output gear 4, i.e. the angular velocity ratio omega ₂ / omega ₁ can be expressed by the following formula (1).

[0015]

[Equation 1]

When the angle formed by the line AC and the line AG is θ ₁ and the angle formed by the line BD and the line BG is θ ₂ ,
From the cosine theorem,

[0017]

[Equation 2]

Because it is [0018], of the input gear 2 rotating radius r _'1
And the radius of rotation r ′ ₂ of the output gear 4 is

[0019]

[Equation 3]

[0020] Since in _{^{e 1 2 «r 1 2, e}} 2 2 «r 2 2, when approximately the _{^{e 1 2 ≒ 0, e 2}} 2 ≒ 0,

[0021]

[Equation 4]

It becomes Then, Θ ₂ = θ ₂ + P ₂ . P ₂ is the angle by which θ ₂ = 0 when θ ₁ = 0, and therefore the output gear 4 with respect to the maximum eccentric angular position of the input gear 2.
Is the phase angle of the maximum eccentric angle position of, that is, the eccentric phase angle. Then, the above formula (2) becomes

[0023]

[Equation 5]

It becomes Solving the above equation (9) as a variable separation type differential formula,

[0025]

[Equation 6]

Since θ ₂ = 0 when θ ₁ = 0,

[0027]

[Equation 7]

[0028] Let Δθ _{2 be} the rotation angle error of the output gear 4 with respect to the rotation angle of the output gear 4 when there is no eccentricity,

[0029]

[Equation 8]

It is From the above equations (10), (11) and (12), Δθ ₂ is

[0031]

[Equation 9]

It becomes Also, θ ₂ is

[0033]

[Equation 10]

It can be expressed as Displaying the rotation error of the output gear 4 as the circumferential length h ₂ on the pitch circle 8,

[0035]

[Equation 11]

It becomes

The above equation (14) devised using the approximation method.
In order to verify the adequacy of, the rotation angle error Δθ ₂ of the output gear 4 in the case of r ₁ = 25 mm r ₂ = 25 mm e ₁ = 0.07 mm e ₂ = 0.06 mm is obtained according to the above equation (14). Was calculated according to the calculation formula described in the above Asano paper (the basic circle radius of the input gear 2 in the calculation formula described in the Asano paper is 2
3.49 mm, the output gear basic circle radius is also 23.49
mm). The results are shown in FIGS. 2 and 3. FIG. 2 is a diagram showing the value obtained according to the above equation (14) and the value obtained according to the calculation formula described in the above Asano paper, and FIG. 3 is a diagram showing the difference between the two. For example, considering a case where a photosensitive drum having a radius of 15 mm is connected to the output gear 4, the difference between the value obtained according to the above equation (14) and the value obtained according to the calculation equation described in the above Asano paper is θ ₁ Is maximum near 3 (rad.),
The maximum value is about 4.5 μm, which is practically negligible. Therefore, it is understood that the rotation angle error Δθ ₂ of the output gear 4 obtained according to the above equation (14) is sufficiently precise and practically causes no problem.

Next, as shown in FIG. 4, three gears,
That is, consider a gear drive train composed of an input gear 12, an intermediate gear 14, and an output gear 16. Input gear 12
The pitch circle 18 is a perfect circle having a radius r ₁ and the intermediate gear 14
Of the output gear 16 is a perfect circle having a radius r _2.
It is considered that the pitch circle 22 is a perfect circle having a radius r ₃ (strictly speaking, the pitch circles 18, 20 and 22 are not perfect circles, but it can be regarded as a perfect circle in practice). Point A is the input gear 12
Indicates the center of rotation, are eccentric by e ₁ with respect to the center C of the pitch circle 18 due to manufacturing error. Point B is the center of rotation of the intermediate gear 14 and is eccentric by e _{2 with} respect to the center D of the pitch circle 20 due to a manufacturing error.
Indicates the center of rotation of the output gear 16, which is eccentric by e _{3 with} respect to the center F of the pitch circle 22 due to a manufacturing error. A point G is a mutual meshing point between the input gear 12 and the intermediate gear 14, and a point H is a mutual meshing point between the intermediate gear 14 and the output gear 16.

The angle formed by the line AC and the line AG is θ ₁ ,
The angle formed by the line BD and the line BG is Θ ₂ , and the line EF and the line E are
Angle theta ₃ to H form, when the angle formed by the line BG and line BH and P _'3, the ratio of the angular velocity omega ₂ of the intermediate gear 14 and the angular velocity omega ₃ of the output gear 16, i.e. the angular velocity ratio omega ₃ / Ω
₂ is, as understood from the above formula (8),

[0040]

[Equation 12]

It becomes P ₂ is the eccentric phase angle of the intermediate gear 14 with respect to the input gear 12, P ₃ is the eccentric phase angle of the output gear 16 with respect to the intermediate gear 14, and Θ ₃ = θ ₃
Since it is + P ₃ , the above formula (17) is

[0042]

[Equation 13]

It becomes When θ ₁ = 0 and θ ₂ = 0, θ ₃ =
When the above equation (18) is solved as a variable separation type differential equation with 0,

[0044]

[Equation 14]

It becomes Let Δθ _{3 be} the rotation angle error of the output gear 16 with respect to the rotation angle of the output gear 16 when there is no eccentricity,

[0046]

[Equation 15]

From equation (19) and equation (20),
Δθ ₃ is

[0048]

[Equation 16]

It becomes Also, θ ₃ is

[0050]

[Equation 17] It can be expressed as. When the rotation error of the output gear 16 is displayed in the circumferential length on the pitch circle,

[0051]

[Equation 18]

It becomes

Rotational error Δθ _n of the output gear in a gear drive train composed of four or more n gears
Alternatively, h _n can be similarly obtained.

Thus, in one aspect of the present invention, for example, when the gear drive train is composed of three gears, that is, the input gear 12, the intermediate gear 14, and the output gear 16 as shown in FIG. In the above formula (21) or formula (23)
By using the eccentric phase angle P ₂ of the intermediate gear 14 with respect to the input gear 12 and the eccentric phase angle P ₃ of the output gear 16 with respect to the intermediate gear 14 as parameters, the maximum value of the rotation error of the output gear 16, that is, the maximum rotation error. Ask for. And
The eccentricity tolerance (that is, the tolerance of e ₁ , e ₂ and e ₃ ) in manufacturing each of the input gear 12, the intermediate gear 14 and the output gear 16 is determined so that the obtained maximum rotation error is within the allowable range. The maximum rotation error can be solved as a Max-Max value of an optimization problem in mathematical programming, and can be formulated as follows (for mathematical programming, for example, "Mathematical Programming" issued by Iwanami Shoten Co., Ltd.) And Kyoritsu Publishing Co., Ltd. "Basics of Mathematical Programming").

[0055]

[Formula 19]

R ₁ = 25 mm r ₂ = 25 mm r ₃ = 25 mm e ₁ = 0.08 mm e ₂ = 0.06 mm e ₃ = 0.07 mm P ′ ₃ = 3.93 rad. Calculating in the case of, the eccentric phase angle P ₂ = 1.52 rad. , The eccentric phase angle P ₃ of the output gear 16 with respect to the intermediate gear 14 =
3.14 rad. When, the rotation error of the output gear 16 becomes maximum, and the maximum rotation error h ₃ (max) = 0.319 mm
Becomes In FIG. 5, P ₃ is set to 3.14 rad. Θ when
_The rotation error due to the change between ₁ and P ₂ is shown three-dimensionally, and the vertical plane indicated by a large number of parallel lines indicates the maximum rotation error h ₃
It has passed (max). Fig. 6 shows the maximum rotation error h
The relative relationship among the input gear 12, the intermediate gear 14, and the output gear 16 when ₃ (max) is generated is shown, and the symbol * indicates the maximum eccentric angle position.

In one aspect of the present invention, the eccentricity tolerance in manufacturing each of the input gear 12, the intermediate gear 14, and the output gear 16 is determined so that the maximum rotation error is equal to or less than the allowable value. For example, if the maximum rotational error obtained for the specific eccentricity tolerances e ₁ , e ₂ , and e ₃ exceeds the allowable value, at least one value of the eccentricity tolerances e ₁ , e ₂ , and e ₃ should be set to Reduce appropriately.

In another aspect of the present invention, when the maximum rotation error exceeds the allowable value, the meshing phase of at least one pair of mutually meshing gears (that is, the input gear 12).
Eccentricity phase angle P ₂ of the intermediate gear 14 with respect to and / or eccentricity phase angle P ₃ of the output gear 16 with respect to the intermediate gear 14)
Is appropriately regulated so that the rotation error is within a predetermined value. The meshing phase regulation of a pair of mutually meshing gears is
It can be easily realized by providing an appropriate mark at a predetermined angular position of each gear and matching the marks when mounting the gear. The minimum value of the rotation error of the output gear 16, that is, the minimum rotation error, with the eccentric phase angle P ₂ of the intermediate gear 14 with respect to the input gear 12 and the eccentric phase angle P ₃ of the output gear 16 with respect to the intermediate gear 14 as parameters,
If the minimum rotation error is equal to or less than the predetermined allowable value, the eccentric phase angle P ₂ of the intermediate gear 14 with respect to the input gear 12 and / or the eccentric phase angle P ₃ of the output gear 16 with respect to the intermediate gear 14 are appropriately regulated. Input gear 12,
When the intermediate gear 14 and the output gear 16 are mounted at required positions and sequentially meshed with each other, the mutual meshing phase between the input gear 12 and the intermediate gear 14 and / or the intermediate gear 14 and the output gear 16
It is confirmed that the rotation error of the output gear 16 can be made equal to or less than a predetermined allowable value by appropriately controlling the mutual meshing phase with the. The minimum rotation error can be solved as the Min-Max value of the optimization problem in mathematical programming, and can be formulated as follows.

[0059]

[Equation 20]

Calculating for the case where r ₁ , r ₂ , r ₃ , e ₁ , e ₂ , e ₃ and P ′ ₃ have the above values, the eccentric phase angle P ₂ of the intermediate gear 14 with respect to the input gear 12 = ₂
0.39 rad. Eccentric phase angle P ₃ of output gear 16 with respect to intermediate gear 14 = 3.14 rad. Output gear 1 at
The rotation error of 6 becomes the minimum, and the maximum value h of the minimum rotation error
₃ (min) = 0.039 mm. In FIG. 7, P ₃ is set to 3.14 rad. The rotation error due to the change of θ ₁ and P ₂ is expressed three-dimensionally, and the vertical plane indicated by many parallel lines passes through the maximum value h ₃ (min) of the minimum rotation error. . FIG. 8 illustrates the relative relationship between the input gear 12, the intermediate gear 14, and the output gear 16 when the minimum rotation error is generated, and the symbol * indicates the maximum eccentric angular position.

[0061]

According to the present invention, the rotation error of the output gear due to the eccentricity of each gear in the gear drive train can be calculated based on the calculation formula that can be easily calculated in practice.

Therefore, according to one aspect of the present invention, the maximum eccentricity error of the output gear is set without setting the eccentricity tolerance in manufacturing the gear to be smaller than necessary, in other words, as large as possible. Can be set to a predetermined value or less.

According to the second aspect of the present invention, the meshing phase of at least one pair of gears meshing with each other is regulated so that the rotation error of the output gear is equal to or less than the above-mentioned predetermined value, thus Even if the eccentricity tolerance is considerably large, the rotation error of the output gear can be kept below a predetermined value.

[Brief description of drawings]

FIG. 1 is a schematic diagram for explaining a calculation formula for calculating a rotation error of an output gear when a gear drive train is composed of two gears.

FIG. 2 is a diagram showing a rotation error value calculated by a calculation formula devised by the present invention and a rotation error value calculated by a remarkably complicated calculation formula conventionally proposed.

FIG. 3 is a diagram showing a difference between a rotation error value calculated by a calculation formula devised by the present invention and a rotation error value calculated by a remarkably complicated calculation formula conventionally proposed.

FIG. 4 is a schematic diagram for explaining a calculation formula for calculating a rotation error of an output gear when the gear drive train is composed of three gears.

5 is a three-dimensional diagram showing a rotation error of an output gear due to a change in a rotation angle θ ₁ of an input gear and an eccentric phase angle P ₂ of an intermediate gear in a gear drive train shown in FIG.

FIG. 6 is a schematic diagram showing relative rotation phases of an input gear, an intermediate gear, and an output gear when the rotation error is maximum in the gear drive train shown in FIG.

7 shows three-dimensionally the rotation error of the output gear due to the change in the rotation angle θ ₁ of the input gear and the eccentric phase angle P ₂ of the intermediate gear in the gear drive train shown in FIG. 4, similar to FIG. Line diagram.

8 is a schematic diagram showing relative rotation phases of an input gear, an intermediate gear, and an output gear when a rotation error is minimum in the gear drive train shown in FIG.

[Explanation of symbols]

2: Input gear 4: Output gear 6: Pitch circle of input gear 8: Pitch circle of main gear 12: Input gear 14: Intermediate gear 16: Output gear 18: Pitch circle of input gear 20: Pitch circle of intermediate gear 22: Pitch circle of output gear

─────────────────────────────────────────────────── ─── Continuation of the front page (56) References Japanese Patent Laid-Open No. 9-53702 (JP, A) Japanese Patent Laid-Open No. 7-208578 (JP, A) Special Table 10-506172 (JP, A) (58) Field (Int.Cl. ⁷ , DB name) F16H 55/00-55/30 G03G 13 / 00,15 / 00 G03G 15/00 101 G03G 15/00 550-556

Claims (2)

(57) [Claims] 1. A method for determining an eccentricity tolerance in the manufacture of each gear used in a gear drive train configured by meshing a plurality of gears including at least an input gear and an output gear, and gears meshing with each other. The output gear for the input gear is based on solving the rotation angle ratio of the gears meshing with each other by approximating the radius of gyration at the meshing position between them by the cosine theorem, and solving the rotation angle ratio as a variable separation type differential equation. Is determined, and the eccentricity tolerance in manufacturing each of the gears is determined so that the maximum rotation error is equal to or less than a predetermined value. 2. A cosine theorem for a radius of gyration at a meshing position between mutually meshing gears by using a meshing phase regulation method in a gear drive train configured by meshing a plurality of gears including at least an input gear and an output gear. To obtain the maximum rotation error of the output gear with respect to the input gear based on solving the rotation angle ratio of the gears meshing with each other, and solving the rotation angle ratio as a differential equation of the variable separation type. When the error exceeds a predetermined value, the meshing phase of at least one pair of mutually meshing gears is regulated so that the rotation error of the output gear with respect to the input gear is equal to or less than the predetermined value. Engagement phase regulation method.