CN111625758B - A time-varying mesh stiffness calculation method for planetary gears based on tooth profile correction method - Google Patents

Abstract

The invention discloses a time-varying meshing stiffness calculation method of a planetary gear based on a tooth profile correction method, which belongs to the technical field of mechanical dynamics, fully considers the influence of tooth profile on the calculation of comprehensive meshing stiffness, and improves the calculation accuracy of the time-varying meshing stiffness of the planetary gear, and comprises the following specific steps of: firstly, basic parameters of the sun wheel, the planet wheel and the inner gear ring are determined, and then the radius r of the sun wheel and the planet wheel is determined_fThe method has the beneficial effects that the principle is simple, the calculation is easy, and the practicability is strong.

Description

A time-varying mesh stiffness calculation method for planetary gears based on tooth profile correction method

technical field

The invention belongs to the technical field of mechanical dynamics, and relates to a time-varying mesh stiffness calculation method for gear meshing, in particular to a gear time-varying mesh stiffness calculation method based on a tooth profile correction method.

Background technique

The planetary gear transmission system is mainly composed of a ring gear, a number of planetary gears, a sun gear and a planet carrier. It is more complex in structure than the ordinary fixed-axis gear system. The planetary gear meshes with the sun gear and the ring gear at the same time, and also has a Multiple pairs of sun gear-planet gear, planetary gear-ring gear mesh, because the planetary gear has two rotation modes of rotation and revolution at the same time, the transmission path of the vibration signal generated by meshing changes at any time, resulting in the planetary gear transmission system. The vibration signal is extremely complex and difficult to decompose.

Planetary gear transmission systems are widely used in various power transmission processes due to their compact structure, strong bearing capacity, and easy realization of large transmission ratios. Studying its dynamic characteristics can deepen the research on its failure mechanism and provide a simulation experiment basis for the fault diagnosis of planetary transmission system. The most important thing to study the dynamic characteristics of planetary transmission system is to calculate the meshing stiffness of gear teeth accurately. However, the existing calculation methods of tooth meshing stiffness have disadvantages such as large amount of calculation, complicated formula derivation and calculation process, and inaccurate calculation results. The tooth shape of the actual gear needs to consider the influence of the basic parameters of the gear such as the displacement coefficient, the number of teeth, and the modulus, so the calculation method of the tooth meshing stiffness of the gear with different basic parameters will be different. If the gear teeth are directly simplified to the trapezoidal section cantilever beam model, the influence of the involute tooth profile of the gear will be ignored, and the excessive arc of the tooth root will also have an important impact on the accuracy of the meshing stiffness of the gear teeth. Therefore, the most accurate method is to follow the The actual shape of the gear teeth is used to calculate the mesh stiffness.

The calculation method of the time-varying mesh stiffness of planetary gears has important influence and value for studying the dynamic characteristics and failure mechanism of planetary transmission system.

SUMMARY OF THE INVENTION

The purpose of the present invention is to solve the above problems, and design a method for calculating the time-varying meshing stiffness of planetary gears based on the tooth profile correction method.

The technical scheme of the present invention to achieve the above object is, a method for calculating the time-varying meshing stiffness of planetary gears based on the tooth profile correction method, the specific steps of which are as follows:

Step 1: Determine the basic parameters of the sun gear, planetary gear and ring gear, including: number of teeth z, module m, tooth width B, tip height coefficient h _a *, tip clearance coefficient c*, indexing circle pressure Angle α, displacement coefficient is x (positive value for positive displacement, negative value for negative displacement), pitch circle pressure angle α _w , shear modulus G of material, elastic modulus E of material, Poisson Ratio μ, rotational speed n, transmission power P;

Step 2: Determine the relative positional relationship between the sun gear and the planetary gear with the radius of r _f +c*m and the base circle, and clarify the relative positional relationship between the tooth tip circle and the base circle of the inner gear ring, which are as follows:

时，该齿轮基圆半径r_b小于r_f+c*m，当满足条件

时，该齿轮基圆半径r_b大于r_f+c*m，对于内齿圈，内齿圈齿顶圆直径d_a始终大于基圆直径。Verify the number of teeth of the sun gear and planetary gear respectively, when the conditions are met

When the gear base circle radius r _b is less than r _f +c*m, when the conditions are met

When , the gear base circle radius r _b is greater than r _f + _c *m, and for the inner gear ring, the inner gear ring tooth tip circle diameter da is always larger than the base circle diameter.

Step 3: Calculate the meshing stiffness of a single gear tooth of the sun gear, planetary gear and ring gear, including: shear stiffness ks , axial tensile and compressive stiffness ka , bending stiffness _{k b} , contact _{stiffness kh} and gear base flexible deformation The stiffness k _f is calculated as:

时，剪切刚度k_s、轴向拉压刚度k_a和弯曲刚度k_b的计算公式为：1) For the sun gear and planetary gear, the radius of the base circle is less than r _f +c*m, including the case where the radius of the base circle is less than the radius of the root circle, that is

When , the calculation formulas of shear stiffness k _s , axial tensile and compressive stiffness _{ka and bending stiffness k b are} :

r _cx represents the distance from the current meshing point of the gear teeth (the meshing point always changes between the tooth profile working sections) to the gear axis, M _cx represents the bending moment of the meshing force at the meshing point to the tooth root, and its calculation formula is:

h_ix表示渐开线齿廓部分任意位置的弦齿厚，h_x表示齿廓上任意位置的弦齿厚，计算公式为：h _f is the thickness of the chord tooth on the root circle, r _f is the radius of the root circle, r _x is the radius of any position on the tooth profile, α _cx is the pressure angle at the current meshing point,

h _ix represents the chord tooth thickness at any position on the involute tooth profile, h _x represents the chord tooth thickness at any position on the tooth profile, and the calculation formula is:

Δh表示根过渡圆弧处的半弦齿厚，h_ix和Δh的计算公式为：where α _x represents the pressure angle at any position on the tooth profile,

Δh represents the half-chord tooth thickness at the root transition arc, and the calculation formulas of h _ix and Δh are:

where a and b are calculated as:

b ² =(0.5m(z-2h _a *-2c*+2x)+0.38m) ² -a ² (8)

invα=tanα-α (10)

The formula for calculating the contact stiffness k _h is:

ρ₁、ρ₂分别表示接触点位置两轮齿齿廓的曲率半径，E₁、E₂分别表示两轮齿材料的弹性模量，μ₁、μ₂分别表示两轮齿材料的泊松比，b_c表示两齿轮轮齿啮合接触线长。ρ₁、ρ₂的计算公式为：in,

ρ ₁ , ρ ₂ respectively represent the radius of curvature of the tooth profile of the two gear teeth at the contact point position, E ₁ , E ₂ respectively represent the elastic modulus of the two gear teeth materials, μ ₁ , μ ₂ respectively represent the Poisson’s ratio of the two gear teeth materials , b _c represents the length of the meshing contact line between the two gear teeth. The calculation formulas of ρ ₁ and ρ ₂ are:

Among them, r _x1 and r _x2 respectively represent the radius of any point on the involute tooth profile of the two gears, and r _b1 and r _b2 respectively represent the radius of the base circle of the two gears.

The calculation formula of the flexible deformation stiffness k _f of the gear base is:

Among them, u _fx represents the shortest distance from the intersection of the extension line of the meshing force and the radial symmetry line of the gear tooth to the root circle, and the calculation formula is:

The coefficients L*, M*, P* and Q* are represented by X _i *, and the calculation formula is:

h _fi =r _f /r _in (17)

h_fi＝r_f/r_in。Among them, θ _f represents the central angle corresponding to the upper half tooth thickness of the tooth root circle,

h _fi =r _f /r _in .

时，且啮合点在基圆与齿顶圆之间变化，剪切刚度k_s、轴向拉压刚度k_a和弯曲刚度k_b的计算公式为：2) For sun gears and planetary gears, the radius of the base circle is greater than r _f +c*m, i.e.

When , and the meshing point varies between the base circle and the addendum circle, the formulas for calculating the shear stiffness k _s , the axial tensile and compressive stiffness _ka and the bending stiffness k _b are:

When the meshing point varies between the base circle and the r _f +c*m circle, the formulas for calculating the shear stiffness k _s , the axial tensile and compressive stiffness _ka and the bending stiffness k _b are:

M _cx represents the bending moment of the meshing force at the meshing point to the tooth root, h _b represents the chord tooth thickness on the base circle of the gear, and its calculation formula is:

b ² =(0.5m(z-2h _a *-2c*+2x)+0.38m) ² -a ² (32)

The formula for calculating the contact stiffness k _h is:

The calculation formula of the flexible deformation stiffness k _f of the gear base is:

The coefficients L*, M*, P* and Q* are represented by X _i *, and the calculation formula is:

h _fi =r _f /r _in (40)

3 ₎ For the inner gear ring, the calculation formulas of shear stiffness k _s , axial tensile and compressive stiffness ka and bending stiffness k _b are:

The formula for calculating the contact stiffness k _h is:

The calculation formula of the flexible deformation stiffness k _f of the gear base is:

h _fi =r _f /r _g (53)

r _g represents the radius of the base body of the ring gear, and the coefficients L*, M*, P* and Q* are represented by X _i *, and the calculation formula is:

h _fi =r _f /r _in (56)

Step 4: Calculate the comprehensive time-varying meshing stiffness, including the time-varying meshing stiffness k _spn of external meshing between the sun gear and the planetary gear and the time-varying meshing stiffness k _rpn of internal meshing of the ring gear meshing with the planetary gear.

Using the technical scheme of the present invention, a method for calculating the time-varying meshing stiffness of planetary gears based on the tooth profile correction method is made. The method firstly defines the basic parameters of the sun gear, planetary gear and ring gear, including: number of teeth z, module m, tip height coefficient h _a *, tip clearance coefficient c*, indexing circle pressure angle α, variable The displacement coefficient is x (positive value for positive displacement, negative value for negative displacement), pitch circle pressure angle α _w , shear modulus G of material, elastic modulus E of material, Poisson's ratio μ, rotational speed _n , transmission power P, etc; The meshing stiffness of the individual gear teeth of the wheel, planetary gear and ring gear, including: shear stiffness k _s , axial tensile and compressive stiffness ka , bending stiffness k _b , contact stiffness k _h and gear base flexible deformation stiffness _{k f ;} finally Calculate the comprehensive time-varying meshing stiffness, including the external meshing meshing stiffness k _spn of the sun gear meshing with the planet gear and the internal meshing time-varying meshing stiffness k _rpn of the ring gear meshing with the planet gear. Using this method, different gear types can be reasonably divided according to the number of teeth and the meshing method, and the meshing stiffness of the gears in the fixed number of teeth can be calculated reasonably, so that the calculation results are more accurate, the assumptions are reduced as much as possible, and the shape of the actual gear teeth is as close as possible. to calculate.

The beneficial effects of the present invention are:

(1) The present invention accurately divides the number of teeth of the gear according to the relationship between the number of teeth, the coefficient of displacement and the pressure angle, thereby accurately dividing the tooth profile of the gear, and provides a solid foundation for accurately calculating the time-varying meshing stiffness of the gear;

(2) The present invention makes necessary corrections to the tooth shape to make it more in line with the actual tooth shape, changes the previous method of simplifying the tooth root into a straight line, and improves the accuracy of mesh stiffness calculation;

(3) The present invention fully considers the influence of the gear displacement coefficient on the meshing stiffness of the gear teeth. At present, the application range of the displacement gear is very wide. Due to the existence of the gear displacement coefficient (positive displacement will make the gear teeth thicker, and negative displacement will Thinning the gear teeth) will change the shape of the gear teeth, so that the bearing capacity of the gear teeth will increase or decrease, and the meshing stiffness of the gear teeth will change. This method can accurately calculate the time-varying meshing of the displacement or non-displacement planetary gears stiffness;

(4) The present invention derives the calculation formula of the gear time-varying meshing stiffness from the angle of the chord tooth thickness, which reduces the calculation complexity, and the calculation principle is simpler and more intuitive and easy to understand.

Description of drawings

FIG. 1 is a working flow chart of a method for calculating the time-varying meshing stiffness of planetary gears based on the tooth profile correction method according to the present invention.

Figure 2 is a schematic diagram of the structure of the experimental planetary transmission system.

Figure 3 is a schematic diagram of calculating the chord tooth thickness of the planet gear teeth.

Figure 4 is a schematic diagram of the calculation of the chord tooth thickness at the excessive arc of the tooth root of the sun gear and the planet gear.

FIG. 5 is a schematic diagram showing the parameters of the flexible deformation stiffness of the gear base of the planetary gear and the sun gear.

FIG. 6 is a schematic diagram of calculating the chord tooth thickness of the sun gear teeth.

FIG. 7 is a schematic diagram of calculating the chord tooth thickness of the ring gear teeth.

Fig. 8 is a schematic diagram of calculating the chord tooth thickness at the excessive arc of the tooth root of the ring gear.

FIG. 9 is a schematic diagram showing the parameters of the flexible deformation stiffness of the gear base of the ring gear.

FIG. 10 is a schematic diagram of the initial position of each planetary gear in the planetary transmission system.

Figure 11 is a stiffness numerical diagram of the time-varying meshing stiffness k _spn of the sun gear meshing with the planet gears.

Figure 12 is a stiffness numerical diagram of the time-varying meshing stiffness k _rpn of the ring gear meshing with the sun gear.

Detailed ways

The present invention will be specifically described below in conjunction with the accompanying drawings. As shown in accompanying drawing 1, a method for calculating the time-varying meshing stiffness of planetary gears based on the tooth profile correction method includes the following specific steps:

Step 1: Determine the basic parameters of the sun gear, planetary gear and ring gear, including: number of teeth z, module m, tooth width B, tip height coefficient h _a *, tip clearance coefficient c*, indexing circle pressure Angle α, displacement coefficient is x (positive value for positive displacement, negative value for negative displacement), pitch circle pressure angle α _w , shear modulus G of material, elastic modulus E of material, Poisson Ratio μ, rotational speed n, transmission power P;

Table 1 Basic parameters of gears

Step 2: Determine the relative positional relationship between the sun gear and the planetary gear with the radius of r _f +c*m and the base circle, and clarify the relative positional relationship between the tooth tip circle and the base circle of the inner gear ring;

Step 3: Calculate the meshing stiffness of a single gear tooth of the sun gear, planetary gear and ring gear, including: shear stiffness ks , axial tensile and compressive stiffness ka , bending stiffness _{k b} , contact _{stiffness kh} and gear base flexible deformation stiffness k _f ;

Step 4: Calculate the comprehensive time-varying meshing stiffness, including the external meshing time-varying meshing stiffness k _spn of the sun gear meshing with the planetary gear and the internal meshing time-varying meshing stiffness k _rpn of the ring gear meshing with the planetary gear.

The present invention is described in detail below by a specific embodiment, and the following is a specific implementation process:

Step 1: Determine the basic parameters of the sun gear, planetary gear and ring gear, including: number of teeth z, module m, tooth width B, tip height coefficient h _a *, tip clearance coefficient c*, indexing circle pressure Angle α, displacement coefficient is x (positive value for positive displacement, negative value for negative displacement), pitch circle pressure angle α _w , shear modulus G of material, elastic modulus E of material, Poisson Ratio μ, rotational speed n, transmission power P;

The basic structure of the planetary transmission system used in this specific embodiment is shown in Figure 2, with 3 planetary gears evenly distributed, the basic parameters of each gear are shown in Table 1, the input speed n of the sun gear is 1445r/min, and the transmission The power is 4000W.

Step 2: Determine the relative positional relationship between the sun gear and the planetary gear with the radius of r _f +c*m and the base circle, and clarify the relative positional relationship between the tooth tip circle and the base circle of the inner gear ring, which are as follows:

时，该齿轮基圆半径r_b小于r_f+c*m，当满足条件

时，该齿轮基圆半径r_b大于r_f+c*m，对于内齿圈，内齿圈齿顶圆直径d_a始终大于基圆直径。Verify the number of teeth of the sun gear and planetary gear respectively, when the conditions are met

When the gear base circle radius r _b is less than r _f +c*m, when the conditions are met

When , the gear base circle radius r _b is greater than r _f + _c *m, and for the inner gear ring, the inner gear ring tooth tip circle diameter da is always larger than the base circle diameter.

The number of teeth of the sun gear z _s = 12, the number of teeth of the planetary gear z _pn = 31, the number of teeth of the ring gear z _r = 75, the displacement coefficient of the sun gear x _s = 0.4152, the displacement coefficient of the planet gear x _pn = 0.2724, the displacement of the ring gear The coefficient x _r = 0.4, and the three pressure angles are all 20 degrees, then:

Therefore, the relative positional relationship between the circle with the radius of r _f +c*m and the base circle of each gear is as follows:

Sun gear: the radius of the base circle is greater than r _f +c*m;

Planetary gear: the radius of the base circle is less than r _f +c*m;

Ring gear: the radius of the base circle is less than r _f +c*m.

The direct method is used to verify the following:

The formula for calculating the root circle radius of the external gear is:

r _f =0.5m(z-2h _a *-2c*+2x) (2)

The formula for calculating the radius of the base circle of the external gear is:

r _b = 0.5mz cosα (3)

The formula for calculating the root circle radius of the ring gear is:

r _f =0.5m(z+2h _a *+2c*+2x) (4)

The formula for calculating the base circle radius of the ring gear is:

r _b = 0.5mz cosα (5)

Then the sun gear root circle radius r _fs = 0.0103, r _fs +0.25m = 0.0108, the base circle radius r _bs = 0.0113; the planet gear root circle radius r _fpn = 0.0290, r _fpn +0.25m = 0.0295, the base circle radius is r _bpn =0.0291; the root circle radius of the ring gear is r _fr =0.0783, r _fr +0.25m = 0.0788, and the base circle radius is r _br =0.0705.

The verification calculation results are consistent with the conclusions based on the verification conditions for the number of teeth.

Step 3: Calculate the tooth meshing stiffness of the sun gear, planetary gear and ring gear, including: shear stiffness ks , axial tensile and compressive stiffness ka , bending stiffness _{k b} , contact _{stiffness kh} and gear base flexible deformation stiffness k _f ;

According to the conclusion in step ₂ , the planetary gear should be calculated according to the following calculation formulas. The calculation formulas of the planetary gear tooth shear stiffness k _s , axial tensile and compressive stiffness ka and bending stiffness k _b are:

M_cx表示啮合点处的啮合力对齿根部分的弯矩，其计算公式为：Among them, h _f represents the chord tooth thickness on the tooth root circle, r _f represents the radius of the tooth root circle, and r _cx represents the distance from the current meshing point of the gear teeth (the meshing point always changes between the tooth profile working sections) to the gear axis, r _x represents the radius of any position on the tooth profile, α _cx represents the pressure angle at the meshing point,

M _cx represents the bending moment of the meshing force at the meshing point to the tooth root, and its calculation formula is:

h _x represents the thickness of the chord tooth at any position on the tooth profile. The schematic diagram of the calculation is shown in Figure 3. The calculation formula is:

Δh表示齿根过渡圆弧处的半弦齿厚，h_ix和Δh的计算公式为：where α _x represents the pressure angle at any position on the tooth profile,

Δh represents the half-chord tooth thickness at the transition arc of the tooth root, and the calculation formulas of h _ix and Δh are:

Wherein, the specific representation of a and b is shown in accompanying drawing 4 (a), and its calculation formula is:

b ² =(0.5m(z-2h _a *-2c*+2x)+0.38m) ² -a ² (13)

invα=tanα-α (15)

The formula for calculating the contact stiffness k _h is:

ρ₁、ρ₂分别表示接触点位置两轮齿齿廓的曲率半径，E₁、E₂分别表示两轮齿材料的弹性模量，μ₁、μ₂分别表示两轮齿材料的泊松比，b_c表示两齿轮轮齿啮合接触线长。ρ₁、ρ₂的计算公式为：in,

ρ ₁ and ρ ₂ respectively represent the radius of curvature of the tooth profile of the two gear teeth at the contact point, E ₁ and E ₂ respectively represent the elastic modulus of the two gear teeth materials, and μ ₁ and μ ₂ respectively represent the Poisson’s ratio of the two gear teeth materials. , b _c represents the length of the meshing contact line between the two gear teeth. The calculation formulas of ρ ₁ and ρ ₂ are:

Among them, r _x1 and r _x2 respectively represent the radius of any point on the involute tooth profile of the two gears, and r _b1 and r _b2 respectively represent the radius of the base circle of the two gears.

The calculation formula of the flexible deformation stiffness k _f of the gear base is:

Among them, u _fx represents the shortest distance from the intersection of the extension line of the meshing force and the radial symmetry line of the gear tooth to the root circle, and the calculation formula is:

The coefficients L*, M*, P* and Q* are represented by X _i *, and the calculation formula is:

h_fi＝r_f/r_in，A_i、B_i、C_i、D_i、E_i、F_i为常系数，其值如表2所示：Among them, θ _f represents the central angle corresponding to the upper half tooth width of the tooth root circle,

h _fi =r _f /r _in , A _i , B _i , C _i , _Di , E _i , F _i are constant coefficients, and their values are shown in Table 2:

Table 2 Values of the coefficients in the formula

The specific representation of each parameter in the flexible deformation stiffness of the gear base of the planetary gear is shown in FIG. 5 .

According to the conclusion in step 2, the sun gear should be calculated according to the following formulas. When the meshing point changes between the base circle and the addendum circle, the shear stiffness k _s , the axial tensile and compressive stiffness _ka and the bending stiffness k _b are calculated The formula is:

When the meshing point varies between the base circle and the r _f +c*m circle, the formulas for calculating the shear stiffness k _s , the axial tensile and compressive stiffness _ka and the bending stiffness k _b are:

Among them, the calculation methods of parameters such as Δh, h _x , a, b ² , M _x and h _b are as follows, the calculation diagram of the chord tooth thickness h _x is shown in Figure 6, and h _b represents the chord on the gear base circle Tooth thickness.

The specific representations of a and b in the calculation of the meshing stiffness of a single tooth of the sun gear are shown in Figure 4(b). The calculation formulas of the contact stiffness k _h and the flexible deformation stiffness k _f of the gear base are the same as those of the planetary gear, and are not repeated here. A schematic diagram of the calculation of the flexible deformation stiffness of the gear base of the sun gear teeth is shown in FIG. 5 .

According to the conclusion in step 2, the formulas for calculating the shear stiffness k _s , the axial tensile and compressive stiffness _ka and the bending stiffness k _b of the ring gear teeth are:

A schematic diagram of the calculation of the chord tooth thickness of the ring gear is shown in Figure 7, and the specific representations of a and b in the calculation of the meshing stiffness of a single gear tooth are shown in Figure 8.

The calculation formula of the flexible deformation stiffness k _f of the gear base needs to be corrected u _fx and h _fi as:

h _fi =r _f /r _g (41)

Among them, r _g represents the radius of the ring gear base. The specific representation of each parameter in the flexible deformation stiffness of the gear base of the ring gear is shown in FIG. 9 .

The calculation method of the contact stiffness k _h and other parameters of the ring gear teeth is the same as that of the planetary gear, and will not be repeated here.

Step 4: Calculate the comprehensive time-varying meshing stiffness, including the external meshing time-varying meshing stiffness k _spn of the sun gear meshing with the planetary gear and the internal meshing time-varying meshing stiffness k _rpn of the ring gear meshing with the planetary gear. The specific calculation process is as follows:

First, find the coincidence degree of the meshing of the two gears. The calculation method is as follows:

In the formula, α _a represents the pressure angle of the tooth tip circle of the gear; when the two gears are externally meshed, ± takes the positive sign; when the two gears are internal meshing, the ± takes the negative sign, at this time, 2 represents the internal gear ring.

A pair of gear pairs always alternate between single and double pairs of gear teeth during the meshing process. Here, the following method is used to calculate the single-tooth meshing interval and the double-tooth meshing interval of the gear tooth meshing. The single-tooth meshing interval is expressed as follows:

The double-tooth meshing interval is expressed as follows:

In the formula, n represents the n-th single-double-tooth alternating period of the meshing pair, ε rpn and ε _spn represent the coincidence degree of internal meshing and external meshing, respectively, _{singlek spn} and singlek _rpn represent the nth planetary gear and the sun gear respectively. , the meshing interval of a single pair of gear teeth meshing with the ring gear, doublek _spn and doublek _rpn respectively represent the meshing interval of the double pair of gear teeth meshing with the sun gear and the ring gear respectively.

In this embodiment, since there are three planetary gears in the planetary transmission system, there will be a certain meshing phase difference when each planetary gear meshes with the sun gear and each planetary gear meshes with the ring gear. Therefore, it is also necessary to determine the difference between the three planetary gears. The phase difference between, the phase difference between each external meshing and the phase difference between each internal meshing are calculated using the following formula:

Among them, φ _spne and φ _rpne represent the phase difference between the nth outer and inner meshing stiffness and the initial phase, respectively.

The phase difference between the inner meshing and the outer meshing of the same planetary gear is calculated by the following formula:

Among them, φ _spn and φ _rpn represent the initial phase of the comprehensive meshing transmission error of the nth planetary gear meshing with the sun gear and the ring gear, respectively, and φ _spnr represents the phase difference between the external meshing and the internal meshing.

ψ _pn =ψ _pn0 ±ω _c t (48)

ω _c represents the rotation speed of the planet carrier, ψ _pn0 represents the initial position angle of the nth planetary gear, n is the number of the planetary gear, and N is the number of the planetary gear. When the sun gear rotates clockwise, ± takes the negative sign, and when the sun gear rotates counterclockwise, ± takes the positive sign. In this embodiment, the sun gear rotates clockwise, and the negative sign is taken here.

The initial positions of the planetary gears of the planetary transmission system in this embodiment are shown in FIG. 10 .

Table 3 Meshing phase difference of planetary transmission system

Finally, the time-varying mesh stiffness k _spn of external meshing and the time-varying meshing stiffness k _rpn of internal meshing are obtained. Since gear meshing is always performed alternately between single and double teeth, the formula for calculating the time-varying mesh stiffness when a pair of gear teeth meshes is:

The formula for calculating the time-varying mesh stiffness of the meshing of two pairs of gear teeth is:

The results of k _spn are shown in Fig. 11. In the figure, k _sp1 represents the time-varying meshing stiffness of the gear meshing between the planetary gear 1 and the sun gear, k _sp2 represents the time-varying meshing stiffness of the gear meshing between the planetary gear 2 and the sun gear, and k _sp3 means The gear meshing stiffness of the planetary gear 3 and the sun gear is time-varying. The results of k _rpn are shown in Fig. 12, in the figure k _rp1 represents the time-varying meshing stiffness of the gear meshing with the ring gear 1 and the ring gear, k _rp2 represents the time-varying meshing stiffness of the gear meshing with the ring gear 2 and the ring gear, k _rp3 represents the time-varying meshing stiffness of the gear of the planetary gear 3 meshing with the ring gear.

The above technical solutions only represent the preferred technical solutions of the technical solutions of the present invention, and some changes that those skilled in the art may make to some parts of them all reflect the principles of the present invention and fall within the protection scope of the present invention.

Claims (1)

1. A planetary gear time-varying meshing stiffness calculation method based on a tooth profile correction method is characterized by comprising the following specific steps:

the method comprises the following steps: the method is characterized in that basic parameters of the sun gear, the planet gear and the inner gear ring are defined, and the basic parameters comprise: number of teeth z, modulus m, tooth width B, tooth crest height coefficient h_aTop clearance coefficient c, pitch circle pressure angle α, displacement coefficient x (positive value for positive displacement and negative value for negative displacement), pitch circle pressure angle α_wShear modulus G of the material, elastic modulus E of the material, Poisson ratio mu, rotation speed n and transmission power P;

step two: the radiuses of the sun wheel and the planet wheel are determined to be r_fThe relative position relation between the + c m circle and the base circle, the relative position relation between the addendum circle and the base circle of the inner gear ring, and r_fRepresenting the root circle radius;

step three: computingThe single tooth meshing rigidity of sun gear, planet wheel and ring gear includes: shear stiffness k_sAxial tension and compression stiffness k_aBending stiffness k_bContact stiffness k_hAnd gear base flexible deformation rigidity k_f；

Step four: calculating comprehensive time-varying meshing stiffness including external meshing time-varying meshing stiffness k of the sun wheel and the planet wheel_spnInner gearing time-varying meshing rigidity k meshed with inner gear ring and planet gear_rpn；

In the second step, the radiuses of the sun wheel and the planet wheel are determined to be r_fThe relative position relationship between the + c m circle and the base circle defines the relative position relationship between the addendum circle and the base circle of the inner gear ring, and specifically comprises the following steps:

respectively verifying the tooth number of the sun wheel and the planet wheel when the conditions are met

The gear base radius r_bLess than r_f+ c + m when the condition is satisfied

The gear base radius r_bGreater than r_f+ c × m, for the inner gear ring, the addendum circle diameter d of the inner gear ring_aIs always larger than the diameter of the base circle;

in the third step, calculating the meshing stiffness of the single gear teeth of the sun gear, the planet gear and the inner gear ring comprises the following steps: shear stiffness k_sAxial tension and compression stiffness k_aBending stiffness k_bContact stiffness k_hAnd gear base flexible deformation rigidity k_fThe specific calculation method is as follows:

1) for sun and planet gears, the base radius is less than r_f+ c m, including the case where the radius of the base circle is smaller than the radius of the root circle, i.e.

Time, shear stiffness k_sAxial tension and compression stiffness k_aAnd bending stiffness k_bIs calculated by the formula：

Δ h denotes the half chord tooth thickness at the root transition arc, α_xIndicating the pressure angle at any position on the tooth profile,

h_ixrepresenting the chordal tooth thickness, r, of any position of the involute profile part_xRepresenting radius at any position on the tooth profile, r_cxRepresenting the distance, M, from the current tooth meshing point (which always varies between the tooth profile working sections) to the gear axis_cxThe bending moment of the meshing force at the meshing point to the tooth root part is represented by the following calculation formula:

h_fdenotes the chord tooth thickness, alpha, on the root circle_cxIndicating the pressure angle at the current point of engagement,

h_xthe chord tooth thickness of any position on the tooth profile is represented by the following calculation formula:

wherein h is_ixAnd Δ h are calculated as:

wherein, the calculation modes of a and b are as follows:

b²＝(0.5m(z-2h_a*-2c*+2x)+0.38m)²-a² (8)

invα＝tanα-α (10)

contact stiffness k_hThe calculation formula of (2) is as follows:

wherein,

ρ₁、ρ₂radius of curvature of two tooth profiles at the contact point, E₁、E₂Respectively, the modulus of elasticity, μ₁、μ₂Respectively representing the Poisson's ratio of the material of two teeth, b_cThe length of the meshing contact line of the gear teeth of the two gears is shown by r_x1、r_x2Respectively represents the radius of any point on the involute profiles of the two gears by r_b1、r_b2Respectively representing the base radii of the two gears, then ρ₁、ρ₂The calculation formula of (2) is as follows:

gear matrix flexible deformation rigidity k_fThe calculation formula of (2) is as follows:

wherein u is_fxThe shortest distance from the intersection point of the meshing force extension line and the gear tooth radial symmetry line to the tooth root circle is represented by the following calculation formula:

the coefficients L, M, P and Q are represented by X_iExpressed, the calculation formula is:

wherein A is_i、B_i、C_i、D_i、E_i、F_iIs a constant coefficient, θ_fThe central angle corresponding to the upper half tooth thickness of the tooth root circle of the gear tooth is represented by the following specific calculation formula:

h_fi＝r_f/r_in (17)

2) for sun and planet gears, the base radius is greater than r_f+ c x m, i.e.

And the meshing point varies between the base circle and the addendum circle, the shear stiffness k_sAxial tension and compression stiffness k_aAnd bending stiffness k_bThe calculation formula of (2) is as follows:

h_brepresenting the chordal thickness on the gear base circle with the mesh point between the base circle and r_fShear stiffness k when varying between + c m circles_sAxial tension and compression stiffness k_aAnd bending stiffness k_bThe calculation formula of (2) is as follows:

M_cxthe bending moment of the meshing force at the meshing point to the tooth root part is represented by the following calculation formula:

b²＝(0.5m(z-2h_a*-2c*+2x)+0.38m)²-a² (32)

contact stiffness k_hThe calculation formula of (2) is as follows:

gear matrix flexible deformation rigidity k_fThe calculation formula of (2) is as follows:

the coefficients L, M, P and Q are represented by X_iExpressed, the calculation formula is:

h_fi＝r_f/r_in (40)

3) for ring gear, shear stiffness k_sAxial tension and compression stiffness k_aAnd bending stiffness k_bThe calculation formula of (c) is:

contact stiffness k_hThe calculation formula of (2) is as follows:

gear matrix flexible deformation rigidity k_fThe calculation formula of (2) is as follows:

h_fi＝r_f/f_g (53)

r_grepresenting the radius of the annular gear base body, the coefficients L, M, P and Q being represented by X_iExpressed, the calculation formula is:

h_fi＝r_f/r_in (56)

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