CN111488682A - Involute helical gear pair tooth width modification dynamic model establishing method - Google Patents

Abstract

The invention provides a method for establishing a dynamic model for tooth width modification of an involute helical gear pair, comprising the following steps: modifying the gear tooth width of the helical gear pair by using a polynomial function modification method; Discrete into N slice gear pairs of equal width; by calculating the dynamic meshing force of the meshed slice gear pair superposition and obtain the comprehensive dynamic meshing force of the helical gear pair; by calculating the yaw moment superposition of the meshed slice gear pair and obtain the comprehensive swing moment of the helical gear pair; establish the dynamic equation of the helical gear pair, and substitute the comprehensive dynamic meshing force of the helical gear pair and the comprehensive swing moment of the helical gear pair into the dynamic equation. The invention calculates by discretizing the helical gear pair into N slice gear pairs with the same width along the tooth width direction, thereby improving the accuracy of the dynamic characteristic of the gear pair with uneven tooth width direction.

Description

Establishment method of dynamic model for tooth width modification of involute helical gear pair

technical field

The invention relates to the technical field of mechanical dynamics, in particular to a method for establishing a dynamic model for tooth width modification of an involute helical gear pair.

Background technique

Helical gears are high-power and heavy-duty gears, which have relatively high requirements on the strength of the gears. For gear pairs with uneven load distribution in the tooth meshing deviation, the meshing center of the gear teeth is not in the middle of the gear in the direction of the tooth surface and tooth width, even in the gear movement. During the process, the meshing center fluctuates left and right in the tooth width direction. Therefore, it is difficult to analyze the dynamic characteristic law of the gear pair caused by the unevenness of the tooth width.

Therefore, how to improve the accuracy of the dynamic characteristics of the gear pair with uneven tooth width direction is a technical problem to be solved urgently by those skilled in the art.

SUMMARY OF THE INVENTION

In view of this, the purpose of the present invention is to provide a method for establishing a dynamic model of the tooth width modification of an involute helical gear pair, which can improve the accuracy of the dynamic characteristics of the gear pair with uneven tooth width directions.

In order to achieve the above object, the present invention provides the following technical solutions:

A method for establishing a dynamic model for tooth width modification of an involute helical gear pair, comprising the following steps:

The gear tooth width of the helical gear pair is modified by the polynomial function modification method;

Dispersing the helical gear pair into N slice gear pairs with equal widths along the tooth width direction;

By calculating the dynamic meshing force of the sheet gear pair in the meshing state, superimpose and obtain the comprehensive dynamic meshing force of the helical gear pair;

By calculating the yaw moment of the sheet gear pair in the meshing state, the yaw moment is superimposed and the comprehensive sway moment of the helical gear pair is obtained;

The dynamic equation of the helical gear pair is established, and the comprehensive dynamic meshing force of the helical gear pair and the comprehensive swing moment of the helical gear pair are substituted into the dynamic equation.

In a specific embodiment, the modification formula of the polynomial function is:

In the formula, Δ _d,i represents the modification amount at any position in the tooth width direction, d _i is the length of the helical gear tooth surface from the center position, s is the bending index of the polynomial function, b is the modification length of the tooth direction, Δ _d is the tooth modification amount, d is the tooth width, 1≤s≤5.

In another specific embodiment, the gear instantaneous pressure angle of the sheet gear pair in the meshing state is:

α _L <α _t,i <α _U ;

α _L is the minimum pressure angle of the helical gear, α _t,i is the instantaneous pressure angle of the sheet gear pair, i=1, 2 respectively represent the two helical gears in the helical gear pair, α _U is The maximum pressure angle of the helical gear.

In another specific embodiment, the comprehensive dynamic meshing force calculation formula of the helical gear pair is:

为所述斜齿轮副的综合动态啮合力，F_m,i为第i对薄片齿轮副在啮合线方向上的动态啮合力。In the formula,

is the comprehensive dynamic meshing force of the helical gear pair, and F _m,i is the dynamic meshing force of the i-th pair of sheet gear pairs in the direction of the meshing line.

In another specific embodiment, the dynamic meshing force of each pair of flake gear pairs in the direction of meshing line is calculated as:

为第i对薄片齿轮副啮合阻尼，m₁和m₂分别为斜齿轮副的模数，x₁、x₂、y₁、y₂、z₁、z₂分别为斜齿轮副两个斜齿轮在坐标系上的坐标位置，r_b1和r_b2分别为斜齿轮副两个斜齿轮的半径；In the formula, i=1,...,N, represents the ith pair of sheet gear pairs, _ke is the time-varying meshing stiffness of the ith pair of sheet gear pairs,

is the meshing damping of the ith pair of sheet gear pairs, m ₁ and m ₂ are the modules of the helical gear pair, respectively, x ₁ , x ₂ , y ₁ , y ₂ , z ₁ , and z ₂ are the two helical gears of the helical gear pair, respectively In the coordinate position on the coordinate system, r _b1 and r _b2 are the radii of the two helical gears of the helical gear pair;

为第i对薄片齿轮副的啮合线变形量的一阶导数，Δ_i(t)为第i对薄片齿轮副的动态传递误差，

为第i对薄片齿轮副的动态传递误差的一阶导数；δ _i (b,Δ _i t) is the deformation of the meshing line of the ith pair of sheet gear pairs,

is the first derivative of the meshing line deformation of the ith pair of sheet gear pairs, Δ _i (t) is the dynamic transmission error of the ith pair of sheet gear pairs,

is the first derivative of the dynamic transmission error of the ith pair of sheet gear pairs;

Γ is the sign function, Γ=1 represents the tooth surface meshing, Γ=-1 represents the tooth back meshing, α _i is the dynamic meshing angle of the ith pair of sheet gear pairs, γ _i is the relative position of the ith pair of sheet gear pairs at any time horn;

e _i is the comprehensive error of the ith pair of flake gear pairs and the tooth profile deviation caused by gear modification, es _,i is the tooth profile error of the ith pair of flake gear pairs, ep, _i is the ith pair of flake gear pairs The equivalent value of the projection of the assembly error of the flake gear pair on the meshing line, em _{, i} is the deformation of the bearing and box parts caused by manufacturing, installation and wear, resulting in the non-parallel transmission shaft and the error in the tooth width direction, e _{β, i} is the tooth profile deviation caused by the modification of the tooth direction of the i-th pair of sheet gear pairs.

In another specific embodiment,

In the formula, when Δ _i (t) ≥ bcosβ _b is the meshing state of the tooth surface, when Δ _i (t) ≤ -bcosβ _b is the meshing state of the tooth back, and the rest are the de-teeth state.

f_pd,i为齿轮基节偏差，f_f,i为齿形公差；In another specific embodiment _, the calculation formula of es,i is: es _,i =e _0,i +er _,i sin(2πωt),e _0,i =0,

f _pd,i is the gear base pitch deviation, f _f,i is the tooth profile tolerance;

The calculation formula of described ep,i is: ep _,i =A _{p ,i} sin(α _i +Γγ);

The calculation formula of the _em,i is:

The calculation formula of the e _β,i is:

In another specific embodiment, the calculation formula of the comprehensive swing moment of the helical gear pair is:

T _m,i =F _m,i ·d _i ;

T _m,i is the yaw moment of the ith pair of sheet gear pairs.

In another specific embodiment, the helical gear pair includes a first gear and a second gear;

The dynamic equation of the first gear is:

The dynamic equation of the second gear is:

T ₁ , T ₂ are the input and load torque of the system, respectively, k _ix , k _iy , k _iz and c _ix , c _iy , c _iz are the stiffness and damping of the center bearing of each gear, I _xi , I _yi , and I _zi are respectively is the moment of inertia of the gear about the x, y and z axes, i=1,2.

为齿面啮合摩擦力，μ为齿面摩擦系数。In another specific embodiment,

is the meshing friction force of the tooth surface, and μ is the friction coefficient of the tooth surface.

Various embodiments according to the present invention can be combined arbitrarily as required, and the embodiments obtained after these combinations are also within the scope of the present invention and are part of the specific embodiments of the present invention.

According to the above technical solutions, the method for establishing the dynamic model of the tooth width modification of the involute helical gear pair provided by the present invention adopts the polynomial function modification method to modify the gear tooth width of the helical gear pair, which can more accurately reflect the tooth width. Uneven loads at different positions in the direction lead to tooth surface deformation; in addition, the present invention superimposes the dynamic meshing force of the meshed sheet gear pairs by discretizing the helical gear pair into N sheet gear pairs with equal widths along the tooth width direction. The comprehensive dynamic meshing force of the helical gear pair is obtained, the yaw moment of the sheet gear pair in meshing state is superimposed to obtain the comprehensive swaying moment of the helical gear pair, and the comprehensive dynamic meshing force of the helical gear pair and the helical gear pair are combined. The comprehensive pendulum moment is substituted into the dynamic equation, and the dynamic model of the tooth width modification of the involute helical gear pair is obtained, which improves the accuracy of the dynamic characteristics of the gear pair with uneven tooth width.

Description of drawings

In order to explain the embodiments of the present invention or the technical solutions in the prior art more clearly, the following briefly introduces the accompanying drawings that need to be used in the description of the embodiments or the prior art. Obviously, the accompanying drawings in the following description are only It is an embodiment of the present invention. For those of ordinary skill in the art, other drawings can also be obtained according to the provided drawings without creative work.

Fig. 1 is a kind of involute helical gear pair tooth width modification dynamic model establishment method flow chart provided by the present invention;

2 is a schematic diagram of the modification parameters of the helical gear tooth width provided by the present invention;

3 is a schematic diagram of a gear sheet cutting provided by the present invention;

4 is a schematic diagram of the gear angle provided by the present invention;

5 is a three-dimensional dynamic model of the distributed meshing of the helical gear provided by the present invention;

Fig. 6 is the load distribution factor K _h curve diagram under different modification amounts provided by the present invention;

Fig. 7 is the load distribution factor K _h curve diagram under different bending indices provided by the present invention;

Fig. 8 is the relationship diagram of tooth dynamic load coefficient K _β with modification amount provided by the present invention;

Fig. 9 is the relationship diagram of tooth dynamic load coefficient K _β with rotational speed provided by the present invention;

FIG. 10 is a graph showing the relationship between the dynamic load coefficient K _β of the tooth direction and the load provided by the present invention;

11 is a three-dimensional tooth surface contact stress diagram provided by the present invention;

Fig. 12 is a tooth surface stress projection diagram provided by the present invention;

13 is a three-dimensional diagram of the stress distribution on the tooth surface when the tooth width modification amount Δ _d = 0 provided by the present invention;

Figure 14 is a two-dimensional contour diagram of tooth surface stress distribution when the tooth width modification amount Δ _d = 0 provided by the present invention;

15 is a three-dimensional diagram of the stress distribution on the tooth surface when the tooth width modification amount Δ _d = 8 μm provided by the present invention;

Fig. 16 is a two-dimensional contour diagram of tooth surface stress distribution when the tooth width modification amount Δ _d = 8 μm provided by the present invention;

17 is a three-dimensional diagram of the stress distribution on the tooth surface when the tooth width modification amount Δ _d = 16 μm provided by the present invention;

Fig. 18 is a two-dimensional contour diagram of tooth surface stress distribution when the tooth width modification amount Δ _d = 16 μm provided by the present invention;

FIG. 19 is a three-dimensional diagram of the stress distribution on the tooth surface when the tooth width modification amount Δ _d = 24 μm provided by the present invention;

Figure 20 is a two-dimensional contour diagram of tooth surface stress distribution when the tooth width modification amount Δ _d = 24 μm provided by the present invention;

Figure 21 is a three-dimensional diagram of tooth surface stress distribution when the bending index s=1.5 provided by the present invention;

Figure 22 is a two-dimensional contour diagram of tooth surface stress distribution when the bending index s=1.5 provided by the present invention;

Figure 23 is a three-dimensional diagram of tooth surface stress distribution when the bending index s=2 provided by the present invention;

Figure 24 is a two-dimensional contour diagram of tooth surface stress distribution when the bending index s=2 provided by the present invention;

Figure 25 is a three-dimensional diagram of tooth surface stress distribution when the bending index s=2.5 provided by the present invention;

Figure 26 is a two-dimensional contour diagram of tooth surface stress distribution when the bending index s=2.5 provided by the present invention;

27 is a three-dimensional diagram of the stress distribution on the tooth surface when the bending index s=3 provided by the present invention;

FIG. 28 is a two-dimensional contour diagram of the stress distribution on the tooth surface when the bending index s=3 provided by the present invention.

Detailed ways

In order to make those skilled in the art better understand the technical solutions of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

As shown in FIG. 1 , the present invention discloses a method for establishing a dynamic model for tooth width modification of an involute helical gear pair, comprising the following steps:

Step S1 : modifying the gear tooth width of the helical gear pair by using a polynomial function modification method.

如图2所示。It should be noted that the tooth width is comprehensively modified, and the same modification parameters are used on the left and right sides of the gear tooth width direction of the helical gear pair, that is, the tooth direction modification amount Δ _d , the tooth direction modification length b and the tooth direction are used. trim curve

as shown in picture 2.

Specifically, the present invention discloses that the modification amount formula of the polynomial function is:

In the formula, Δ _d,i represents the modification amount at any position in the tooth width direction, d _i is the length of the helical gear tooth surface from the center position, s is the bending index of the polynomial function, b is the modification length of the tooth direction, and Δ _d is the tooth To modify the amount, d is the tooth width, 1≤s≤5.

For the other meshing gear, the same modification parameters are usually used for modification. In this way, the tooth width modification of a single gear can be represented by a parameter set p containing 3 parameters, p∈{Δ _d ,b,s}.

Step S2: Discrete the helical gear pair into N sheet gear pairs with equal widths along the tooth width direction.

For gear pairs with uneven load distribution with tooth meshing deviation, the meshing center of the gear teeth is not in the middle of the gear in the tooth width direction of the tooth surface. The load response distribution of gear tooth meshing force in the tooth width direction cannot be accurately analyzed by conventional methods. The present invention analyzes the dynamic characteristic law of the gear system caused by the uneven direction of the tooth width by establishing a more accurate meshing model.

(i＝1,…,N)。The only fundamental difference between a helical gear pair and a spur gear pair is that the helical gear system design includes the helix angle parameter β. From another point of view, the spur gear pair is a special helical gear, that is, the helix angle value β=0°. Therefore, the helical gear is discretized into N slices with equal width along the tooth width direction. As shown in Figure 3, each slice is a pair of helical gears with smaller tooth width, the tooth width is Δl, the helix angle is β, each slice is Meshing stiffness of a lamella tooth

(i=1,...,N).

Step S3: by calculating the dynamic meshing force of the sheet gear pair in the meshing state, superimpose and obtain the comprehensive dynamic meshing force of the helical gear pair.

为定值，且

可以表示为：For a single pair of gear pairs, as long as the meshing position is determined, the meshing stiffness of the gears can be obtained. However, for helical gears, it is necessary to determine not only the meshing position of each flake, but also whether each flake is in meshing state for a series of flake helical gear pairs after flake cutting, so it is very important to determine whether each flake gear tooth is in meshing state. As shown in FIG. 4 , the second gear is driven by the input angular velocity _ω1 of the first gear as an example.

is a fixed value, and

It can be expressed as:

为

的初始最小角相位。where t is time,

for

The initial minimum angular phase of .

和

(i＝1,2分别为第一齿轮和第二齿轮)的大小可以分别表示为；in each slicing gear

and

(i=1, 2 are the first gear and the second gear respectively) The size can be expressed as;

Wherein, i is the number of gears in meshing, i=1,...,N. α' ₀ represents the face pressure angle of the gear. d _i indicates that the axial coordinate of each sheet gear is located in the middle of the sheet gear teeth, L is the center distance of the gear pair, and β indicates the helix angle of the helical gear pair.

处于啮合状态的薄片齿轮副的齿轮瞬时压力角为：α_L＜α_t,i＜α_U。α_L为斜齿轮的最小压力角，α_t,i为薄片齿轮副的齿轮瞬时压力角，i＝1,2分别代表斜齿轮副中的两个斜齿轮，α_U为斜齿轮的最大压力角。The instantaneous pressure angle of each lamella tooth is:

The gear instantaneous pressure angle of the sheet gear pair in the meshing state is: α _L <α _t,i <α _U . α _L is the minimum pressure angle of the helical gear, α _t,i is the instantaneous pressure angle of the helical gear pair, i=1, 2 respectively represents the two helical gears in the helical gear pair, α _U is the maximum pressure angle of the helical gear .

式中，

为斜齿轮副的综合动态啮合力，F_m,i为第i对薄片齿轮副在啮合线方向上的动态啮合力。Further, the present invention discloses that the comprehensive dynamic meshing force calculation formula of the helical gear pair is:

In the formula,

is the comprehensive dynamic meshing force of the helical gear pair, and F _m,i is the dynamic meshing force of the i-th pair of sheet gear pairs in the direction of the meshing line.

Specifically, the present invention discloses that the dynamic meshing force calculation formula of each pair of flake gear pairs in the meshing line direction is:

为第i对薄片齿轮副啮合阻尼，m₁和m₂分别为斜齿轮副的模数，x₁、x₂、y₁、y₂、z₁、z₂分别为斜齿轮副两个斜齿轮在坐标系上的坐标位置，r_b1和r_b2分别为斜齿轮副两个斜齿轮的半径。δ_i(b,Δ_it)为第i对薄片齿轮副的啮合线变形量，

为第i对薄片齿轮副的啮合线变形量的一阶导数，Δ_i(t)为第i对薄片齿轮副的动态传递误差，

为第i对薄片齿轮副的动态传递误差的一阶导数。Γ为符号函数，Γ＝1代表齿面啮合，Γ＝-1代表齿背啮合，α_i为第i对薄片齿轮副的动态啮合角，γ_i为第i对薄片齿轮副的任意时刻相对位置角。e_i为第i对式薄片齿轮副的综合误差及齿轮修形引起的齿形偏差，e_s,i为第i对式薄片齿轮副齿廓齿形误差，e_p,i为第i对式薄片齿轮副齿轮装配误差在啮合线上投影的等效值，e_m,i为制造加工、安装及磨损引起轴承、箱体零件变形导致传动轴不平行，引起齿宽方向的误差，e_β,i为第i对薄片齿轮副齿向修形引起的齿形偏差。In the formula, i=1,...,N, represents the ith pair of sheet gear pairs, _ke is the time-varying meshing stiffness of the ith pair of sheet gear pairs,

is the meshing damping of the ith pair of sheet gear pairs, m ₁ and m ₂ are the modules of the helical gear pair, respectively, x ₁ , x ₂ , y ₁ , y ₂ , z ₁ , and z ₂ are the two helical gears of the helical gear pair, respectively In the coordinate position on the coordinate system, r _b1 and r _b2 are the radii of the two helical gears of the helical gear pair, respectively. δ _i (b,Δ _i t) is the deformation of the meshing line of the ith pair of sheet gear pairs,

is the first derivative of the meshing line deformation of the ith pair of sheet gear pairs, Δ _i (t) is the dynamic transmission error of the ith pair of sheet gear pairs,

is the first derivative of the dynamic transmission error of the ith pair of flake gear pairs. Γ is the sign function, Γ=1 represents the tooth surface meshing, Γ=-1 represents the tooth back meshing, α _i is the dynamic meshing angle of the ith pair of sheet gear pairs, γ _i is the relative position of the ith pair of sheet gear pairs at any time horn. e _i is the comprehensive error of the ith pair of flake gear pairs and the tooth profile deviation caused by gear modification, es _,i is the tooth profile error of the ith pair of flake gear pairs, ep, _i is the ith pair of flake gear pairs The equivalent value of the projection of the assembly error of the flake gear pair on the meshing line, em _{, i} is the deformation of the bearing and box parts caused by manufacturing, installation and wear, resulting in the non-parallel transmission shaft and the error in the tooth width direction, e _{β, i} is the tooth profile deviation caused by the modification of the tooth direction of the i-th pair of sheet gear pairs.

The present invention fully considers various errors existing in the meshing of the helical gear pair, and further improves the modification accuracy.

式中，Δ_i(t)≥bcosβ_b时为齿面啮合状态，Δ_i(t)≤-bcosβ_b时为齿背啮合状态，其余为脱齿状态。Specifically, the present invention discloses

In the formula, when Δ _i (t) ≥ bcosβ _b is the meshing state of the tooth surface, when Δ _i (t) ≤ -bcosβ _b is the meshing state of the tooth back, and the rest are the de-teeth state.

f_pd，i为齿轮基节偏差，f_f，i为齿形公差；Further, the present invention specifically discloses that the calculation formula of es, _i is: es _,i =e _0,i +er _,i sin(2πωt), e _0,i =0,

f _{pd, i} is the gear base pitch deviation, f _{f, i} is the tooth profile tolerance;

The calculation formula of e p, _i is: e _p,i =A _p,i sin(α _i +Γγ);

The formula for calculating em _,i is:

The calculation formula of _eβ,i is:

Step S4 : by calculating the yaw moment of the sheet gear pair in the meshing state, superimpose and obtain the comprehensive yaw moment of the helical gear pair.

T_m,i＝F_m,i·d_i；T_m,i为第i对薄片齿轮副所受的偏摆力矩。The formula for calculating the comprehensive sway moment of the helical gear pair is:

T _m,i =F _m,i ·d _i ; T _m,i is the yaw moment received by the i-th pair of sheet gear pairs.

Step S5: establishing the dynamic equation of the helical gear pair, and substituting the comprehensive dynamic meshing force of the helical gear pair and the comprehensive swing moment of the helical gear pair into the dynamic equation.

As shown in Figure 5, specifically, the present invention discloses that the helical gear pair includes a first gear and a second gear, and the dynamic equation of the first gear is:

The dynamic equation of the second gear is:

T ₁ , T ₂ are the input and load torque of the system, respectively, k _ix , k _iy , k _iz and c _ix , c _iy , c _iz are the stiffness and damping of the center bearing of each gear, I _xi , I _yi , and I _zi are respectively is the moment of inertia of the gear about the x, y and z axes, i=1,2.

为齿面啮合摩擦力，μ为齿面摩擦系数。Further, the present invention discloses

is the meshing friction force of the tooth surface, and μ is the friction coefficient of the tooth surface.

In order to prove the accuracy of the method disclosed in the present invention, the present invention is verified by analyzing the dynamic load distribution state of the helical gear pair in the tooth width direction, and defining the dynamic load distribution factor of each sheet gear pair and the dynamic load coefficient of the gear system tooth direction, Each expression is as follows:

In the formula, (F _m,i ) _RMS is the root mean square value of the dynamic meshing force of the ith pair of sheet gears, [(F _m,i ) _RMS ] _max is the maximum value of the root mean square of the dynamic meshing force of the sheet gear pair value, F _n is the actual load on the tooth surface under quasi-static conditions, and N is the total number of flake gear pairs in meshing state after slicing. Among them, the dynamic load distribution factor is a parameter related to the dynamic meshing force on the tooth surface of each sheet gear pair, which can represent the distribution state of the dynamic load of the gear pair in the direction of the tooth width; the dynamic load coefficient of the tooth direction is related to the tooth surface of each sheet gear pair. The parameters related to the mean value of the dynamic load and the maximum load can be used to express the uneven distribution of the dynamic load of the helical gear pair in the direction of the tooth width.

Example 1

Under the condition that the input speed is 2000r/min and the load torque is 100Nm, the modification length b=d/2 (that is, the full tooth width modification), and the bending index of the s polynomial function is s=2. The distribution state of the dynamic load distribution factor in the tooth width direction under the modification amount is shown in Figure 6. Among them, the maximum modification amount Δd of the tooth width is respectively [0μm, 8μm, _16μm , 24μm]. It can be seen from the figure that in the unmodified state, due to errors such as gear installation and bearing deformation, the load in the direction of the tooth width is uneven in actual work. The maximum value (K _h,i =1.41) of the load is concentrated at one end of the tooth width direction, and the minimum value (K _h,i =0.62) is concentrated at the other end of the tooth width. With the increase of the modification amount, the maximum load in the tooth width direction gradually approaches the center of the tooth surface, and the maximum load distribution factor gradually decreases. The maximum load distribution factor is located at 4.2 mm from the center of the tooth width when the tooth width modification amount is _Δd = 24 μm. At this time, the maximum load distribution factor is K _h,i =1.19. Compared with the gear pair in the standard state, the load fluctuation amplitude in the tooth width direction is significantly reduced. It can be seen from the results that the tooth width modification method can improve the excessively concentrated state of the tooth surface load under the appropriate modification amount; at the same time, the tooth load distribution center can be changed, so that the tooth direction load center is gradually closer to the tooth surface center. Location.

Under the condition of modification length b= _d /2 and maximum modification of tooth width Δd = 24μm, the distribution of dynamic load distribution factor in the direction of tooth width under different bending exponents of polynomial function, the results are as follows As shown in Figure 7. Among them, the value of the bending index s is [1.5, 2, 2.5, 3] respectively. It can be seen from the figure that the change of the bending index will not significantly change the maximum and minimum value of the dynamic load in the tooth direction, nor will it have a great influence on the distribution center of the tooth direction load, but will affect the dynamic load in the tooth width direction. distribution shape. With the gradual increase of the bending index, the load changes near the center of the dynamic load distribution in the tooth direction are gradually gentle, and the load distribution is relatively uniform. The maximum value of the dynamic load in the tooth direction is slightly reduced compared to the modified curve with a smaller bending index. It can be seen from the results that using the tooth width modification method, under different bending exponents of the polynomial function, the dynamic load center position of the tooth direction will not change significantly, but it will affect the load change curvature near the load center position. That is to say, under this working condition, using a larger bending index of the polynomial function can make the load distribution near the center of the load more uniform.

Under the same rotational speed and load conditions, the influence of the tooth width modification amount on the dynamic load coefficient of the tooth direction under the condition of the bending exponent of different polynomial functions is shown in Fig. 8. Among them, the value of the bending index s is [1.5, 2, 2.5, 3] respectively, and the modification length b=d/2. It can be seen from the figure that for each bending index, there is an optimal modification amount, so that the dynamic load coefficient reaches the optimal value, and the optimal modification amount is about 23 μm. Secondly, it can be seen that under the optimal modification amount, the optimal dynamic load coefficient when the bending index s=3 can reach about 1.64; when the bending index s=1.5, the optimal dynamic load coefficient can reach about 1.81. This shows that under the optimal modification amount, a larger bending index can reduce the uneven distribution of the dynamic load in the tooth width direction of the helical gear pair to a certain extent. In addition, it can also be seen from the figure that when a smaller tooth width modification amount is adopted (modification amount Δ _d < 9μm), using a larger bending index will make the modification effect not as good as that corresponding to a smaller bending index. Correction gear. When a smaller modification amount is required, a modification curve with a lower bending index will make the modification effect better.

Under the condition of load torque of 100Nm, under the condition of bending index s=3 and modification length b=d/2, the dynamic load coefficient of gear system under different tooth width modification amount changes with the rotation speed as shown in the figure 9 shown. Among them, the maximum modification amount Δd of tooth width is respectively [0μm, 8μm, 16μm, _24μm ], and the range of rotation speed is [100r/min, 2500r/min]. It can be seen from the results that under different modification amounts, with the increase of the rotational speed, the dynamic load coefficient of the tooth direction increases gradually and finally tends to be stable. Compared with the standard gear system without modification, the gear pair after the modification of the tooth direction can achieve the effect of reducing the dynamic load in the whole speed working range. And with the increase of the speed, the dynamic load of the modified gear teeth decreases more obviously. When the rotation speed is 2500r/min, the modification effect of the gear pair with the tooth direction modification amount of Δ _d = 24μm is the most significant. At this time, the tooth direction dynamic load coefficient is 1.96. Compared with the result without modification (K _β = 2.21), the modification reduces the dynamic load coefficient of the tooth direction by 11.31%.

Under the condition that the input speed is 2000r/min, the modification of the tooth width is Δd = 24μm, and the modification length b = _d /2, the dynamic load coefficient of the gear system under the bending index of the polynomial function varies with The variation law of load torque results is shown in Figure 10. Among them, the bending index s of the polynomial function is respectively [1.5, 2, 2.5, 3], and the load variation range is [0Nm, 800Nm]. It can be seen from the results that when the load is large, the dynamic load coefficient of the tooth direction can be better reduced by using a smaller bending index. When the load is 800Nm, the optimal tooth dynamic load coefficient corresponding to the bending index s=1.5 is about 1.71, and the tooth dynamic load coefficient corresponding to the bending index s=3 is about K _β = 1.80. Compared with the bending index s=1.5 The corresponding trimming gear is optimized by 5.3% than the trimming gear corresponding to the bending index s=3. In addition, it is noted that the corresponding dynamic load factor is also larger under the lower load conditions. It can be seen from the figure that using a larger bending index at this time can better reduce the dynamic load coefficient of the tooth direction. Combined with the results shown in Figure 7, it can be analyzed that under lower load conditions, due to the small elastic deformation of the tooth surface under load, the use of a small bending index will make the load too concentrated at the center of the tooth surface load distribution, resulting in the tooth surface. The reduction in the dynamic load factor is not as pronounced as the larger bending index.

It can be seen that the optimal tooth width modification amount can be found under the bending exponents of different polynomial functions, so that the dynamic load coefficient of the helical gear pair is minimized. The method provided by the invention can not only reduce the degree of uneven distribution of the dynamic load in the tooth width direction, reduce the maximum value of the dynamic load in the entire tooth width direction, but also improve the load distribution center in the tooth width direction, so that the load center position gradually Close to the center of the tooth surface, thereby reducing the side yaw moment caused by the misalignment of the load center position. Under the optimal modification amount, when the load is constant, the corresponding modified gears under different bending exponents can reduce the dynamic load in the tooth width direction in the entire working speed range. When the modification of the tooth width is large, using a larger bending index will make the dynamic load change in the vicinity of the load center relatively smooth, which is better than the modification gear corresponding to the smaller bending index; when the modification of the tooth width is small, Using a smaller bending index will result in a better trimming effect. When the load is large, due to the large deformation of the tooth surface under load, the modification effect of using a smaller bending index at this time is due to the modified gear corresponding to the larger bending index; but when the load is small, the smaller bending index is used. The modification curve will make the load distribution too concentrated, resulting in the modification effect not as good as the modification gear corresponding to the larger bending index.

Embodiment 2

The sliced gear teeth are regarded as the contact of a pair of cylinders with the radius of curvature ρ ₁ and ρ ₂ at the contact point. (as shown in Figures 11 and 12), due to local elastic deformation, the contact line becomes a narrow and long contact strip with a width of H, and the contact width H can be obtained from the theory of elasticity and can be expressed as:

In the formula, F _n is the normal pressure on the cylinder, that is, the meshing force F _m,i on the sheet gear pair; l is the length of the contact line, that is, the sheet tooth width d/N in the meshing state; μ ₁ and μ ₁ is the Poisson's ratio of the two cylinder materials; E ₁ and E ₂ are the elastic moduli of the two cylinder materials; ρ ₁ and ρ ₂ are the radii of the two cylinders, that is, the instantaneous radius of the teeth of the flake gear pair.

第i对薄片齿轮副最大接触应力的基本公式为:

When the two gears are meshed and squeezed, the pressure on the contact area of the tooth surface is equal in magnitude and opposite in direction, so the stress and strain on the tooth surface are equal in magnitude and opposite in direction, as shown in Figure 12. The local stress generated on the surface of the contact area is called contact stress, and the maximum contact stress _δH occurs on the theoretical contact line with the largest deformation, which can be expressed as:

The basic formula for the maximum contact stress of the i-th pair of sheet gear pairs is:

According to the meshing dynamics model calculation process and the above-mentioned tooth surface contact dynamic stress analysis method provided by the present invention, first, under the steady-state working condition of the input speed of 2000r/min and the load torque of 500Nm, the modification length b=d/2( That is, full tooth width modification), under the condition of polynomial function bending exponent s=2.5, the dynamic contact stress distribution of the first gear under different tooth width modification amount was studied respectively. The results of the three-dimensional map and two-dimensional contour map of the tooth surface stress distribution under different tooth width modification amounts are shown in Figure 13-20. Among them, the maximum modification amount Δd of the tooth width is respectively [0μm, 8μm, _16μm , 24μm]. It can be seen from Figures 13 and 14 that when the tooth width modification amount Δ _d = 0, that is, the standard gear without modification, the tooth surface contact is uneven due to errors such as manufacturing and installation, bearing deformation, etc., resulting in tooth surface deviation. load phenomenon. Among them, the maximum tooth surface stress appears at one end of the tooth direction with a low degree of coincidence, and the maximum value is about 419MPa. It can be seen from the two-dimensional contour line that the contact dynamic stress of the tooth surface of the gear pair in the unmodified state is mainly concentrated in the range of the tooth surface width [-22.5mm, -10mm] at the contact position of the tooth surface exceeding 400MPa. It can be seen from Figure 15-16 that under the condition of modification amount Δ _d = 8μm, the value of the end face with large tooth width stress is reduced after modification, and the maximum stress on the tooth face is about 408MPa. It can be seen from the two-dimensional contour line that the contact dynamic stress of the gear pair tooth surface in this modified state is mainly concentrated in the range of the tooth surface width [-22.5mm, -3mm] at the tooth surface contact position exceeding 380MPa. It can be seen from Fig. 17-18 that under the condition of modification amount Δ _d = 16 μm, the maximum stress on the tooth surface after modification is about 379 MPa. It can be seen from the two-dimensional contour line that the contact dynamic stress of the gear pair tooth surface in this modified state is mainly concentrated in the range of the tooth surface width [-19mm, 3mm] at the tooth surface contact position exceeding 360MPa. It can be seen from Fig. 19-20 that under the condition of modification amount Δ _d = 24 μm, the maximum stress on the tooth surface after modification is about 368 MPa. It can be seen from the two-dimensional contour line that the contact dynamic stress of the gear pair tooth surface in this modified state is mainly concentrated in the range of the tooth surface width [-15mm, 5mm] at the tooth surface contact position exceeding 350MPa. It can be concluded from the results that the tooth width modification can effectively reduce the stress concentration phenomenon at the tooth end and reduce the maximum contact dynamic stress of the tooth surface. In addition, the tooth width modification can also reduce the distance between the load distribution center and the center of the tooth surface, and alleviate the side yaw phenomenon of the gear.

Under the steady working condition of input speed of 2000r/min and load torque of 500Nm, modification length b= _d /2 and tooth width modification amount of Δd=24μm, the first gear under different bending index The dynamic contact stress distribution state of the tooth surface. The results of the three-dimensional map and two-dimensional contour map of the tooth surface stress distribution under different bending indices are shown in Figure 21-28. Among them, the bending index s of the polynomial function is [1.5, 2, 2.5, 3] respectively. It can be seen from the three-dimensional diagram that no matter which bending index modification is adopted, the contact stress concentration at the end face of the tooth width can be significantly reduced under this modification amount, and the contact stress center of the tooth surface is near the center of the tooth surface. In addition, the maximum value of contact stress under the four bending indices is about 412MPa. Further comparing the two-dimensional contour diagrams of the contact stress under the four bending exponents, it can be seen that when the bending exponents are [1.5, 2, 2.5, 3] in turn, the tooth width ranges where the contact stress of the tooth surface exceeds 400 MPa are respectively are [-4.5mm, 3.5mm], [-7mm, 4mm], [-12.5mm, 5mm], [-15mm, 6mm], and the tooth width span ranges of the contact stress peak are 8mm, 11mm, 17.5mm, 21mm respectively . It can be seen that with the increase of the modification index, the contact stress peak distribution of the tooth surface under this working condition is relatively uniform. From the analysis results, it can be concluded that under the condition of a certain input working condition and modification amount, the change of the bending index of the polynomial function does not obviously change the peak value and the stress center position of the contact dynamic stress on the tooth surface, but it will affect the contact dynamic stress near the peak value. Evenly distributed state.

It should be noted that, in this document, relational terms such as first and second are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any relationship between these entities or operations. any such actual relationship or sequence exists.

The above description of the disclosed embodiments enables any person skilled in the art to make or use the present invention. Various modifications to the embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and inventive features disclosed herein.

Claims (10)

1. A method for establishing an involute helical gear pair tooth width modification dynamic model is characterized by comprising the following steps:

modifying the gear tooth width of the bevel gear pair by adopting a polynomial function modification method;

dispersing the helical gear pair into N sheet gear pairs with equal width along the tooth width direction;

calculating the dynamic meshing force superposition of the sheet gear pair in a meshing state and obtaining the comprehensive dynamic meshing force of the bevel gear pair;

calculating the deflection moment superposition of the sheet gear pair in a meshing state and obtaining the comprehensive swinging moment of the bevel gear pair;

and establishing a dynamic equation of the bevel gear pair, and substituting the comprehensive dynamic meshing force of the bevel gear pair and the comprehensive swinging moment of the bevel gear pair into the dynamic equation.

2. The method for establishing the involute helical gear pair tooth width modification dynamic model according to claim 1, wherein a modification quantity formula of the polynomial function is as follows:

in the formula,. DELTA._d,iRepresents the modification amount at any position in the tooth width direction, d_iIs the length of the tooth surface of the bevel gear from the center position, s is the bending index of a polynomial function, b is the tooth direction modification length, delta_dThe tooth direction modification amount is d, the tooth width is d, and s is more than or equal to 1 and less than or equal to 5.

3. The involute helical gear pair tooth width modification dynamic model building method according to claim 1, wherein the gear instantaneous pressure angle of the meshed thin plate gear pair is:

α_L＜α_t,i＜α_U；

α_Lat the minimum pressure angle of the bevel gear, α_t,iFor the instantaneous pressure angle of the gear of the thin-plate gear pair, i ═ 1,2 represents two bevel gears in the bevel gear pair, α, respectively_UIs the maximum pressure angle of the helical gear.

4. The method for establishing the involute helical gear pair tooth width modification dynamic model according to claim 1, wherein a comprehensive dynamic meshing force calculation formula of the helical gear pair is as follows:

in the formula (I), the compound is shown in the specification,

is the combined dynamic engagement force of said bevel gear pair, F_m,iIs the dynamic meshing force of the ith pair of thin-plate gear pairs in the direction of the meshing line.

5. The method for establishing the involute helical gear pair tooth width modification dynamic model according to claim 4, wherein a dynamic meshing force calculation formula of each pair of thin plate gear pairs in a meshing line direction is as follows:

e_i＝e_s,i+e_p,i+e_m,i+e_β,i；

where i is 1, …, N represents the ith pair of sheet gears, k_eFor the time varying mesh stiffness of the ith pair of foil gears,

damping of the i-th pair of thin-plate gear pairs mesh, m₁And m₂Respectively the modulus, x, of the bevel gear pair₁、x₂、y₁、y₂、z₁、z₂The coordinate positions r of two bevel gears of the bevel gear pair on a coordinate system_b1And r_b2The radiuses of two bevel gears of the bevel gear pair are respectively;

_i(b,Δ_it) is the deformation of the meshing line of the ith pair of the thin plate gears,

is the first derivative, Delta, of the distortion of the meshing line of the ith pair of thin-plate gear pairs_i(t) is the dynamic transmission error of the ith pair of slice gear pairs,

is the first derivative of the dynamic transmission error of the ith pair of the thin-plate gear pairs;

for the sign function, 1 stands for flank engagement, -1 stands for flank engagement, α_iIs the dynamic meshing angle, gamma, of the ith pair of laminar gear pairs_iIs the relative position angle of the ith pair of sheet gears at any time;

e_ithe tooth profile deviation caused by the combined error of the ith pair of sheet gear pairs and gear modification, e_s,iTooth profile error of ith pair of thin-plate gear pairs, e_p,iFor the projected equivalent value of the assembly error of the ith pair of thin-plate gear pairs on the meshing line, e_m,iDeformation of bearing and housing parts to cause non-parallelism of drive shaft and errors in tooth width direction for manufacturing, mounting and wear, e_β,iThe tooth shape deviation caused by the tooth direction modification of the ith pair of the thin-plate gear pairs.

6. The involute helical gear pair tooth width modification dynamic model building method of claim 5, wherein,

in the formula,. DELTA._i(t)≥bcosβ_bWhile being in tooth flank engagement, Δ_i(t)≤-bcosβ_bThe tooth back is engaged while the rest is disengaged.

7. The method for establishing the involute helical gear pair tooth width modification kinetic model of claim 6, wherein e is_s,iThe calculation formula of (2) is as follows: e.g. of the type_s,i＝e_0,i+e_r,isin(2πωt)，e_0,i＝0，

f_pd,iFor gear base pitch deviation, f_f,iIs a tooth profile tolerance;

said e_p,iThe calculation formula of (2) is as follows: e.g. of the type_p,i＝A_p,isin(α_i+γ)；

Said e_m,iThe calculation formula of (2) is as follows:

said e_β,iThe calculation formula of (2) is as follows:

8. the method for establishing the involute helical gear pair tooth width modification dynamic model according to claim 7, wherein a calculation formula of a comprehensive swinging moment of the helical gear pair is as follows:

T_m,i＝F_m,i·d_i；

T_m,ithe deflection moment borne by the ith pair of sheet gear pairs.

9. The involute bevel gear pair tooth width modification kinetic model creation method of claim 8, wherein the bevel gear pair comprises a first gear and a second gear;

the dynamic equation of the first gear is as follows:

the dynamic equation of the second gear is as follows:

T₁、T₂input and load torques, k, respectively, of the system_ix、k_iy、k_izAnd c_ix、c_iy、c_izRespectively the stiffness and damping of the central bearing of each gear I_xi,I_yi,I_ziThe rotational inertia of the gear around the x, y and z axes, i ═ 1 and 2, respectively.

10. The involute bevel gear pair tooth width modification dynamic die of claim 9A pattern creation method, characterized in that,

mu is a tooth surface friction coefficient.

Images