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The change of the gear static transmission error is closely related to the vibration and noise of the gear transmission. Reducing the variation amplitude of the gear transmission error is an effective way to reduce the dynamic load of the gear transmission and reduce the gear vibration and system noise level. Through the shape modification of the gear, the variation of the amplitude of the static transmission error of the gear is reduced, and the control of the gear transmission error is realized, and the purpose of reducing the vibration of the gear transmission can be achieved.
The practical application of gear noise reduction technology is mainly reflected in the following three aspects:
(1) Geometric adjustment. It is to adjust the gear parameters: number of teeth, diameter, pressure angle and pitch circle. Geometric adjustment changes the properties of the gear, primarily for the first step in gear design.
(2) Geometric modification. It removes part of the metal from the gear surface, and the tooth profile is no longer a pure involute, so the shaping compensates for the deformation of the gear under load.
(3) Surface finish and precise tolerance manufacturing. It is the third method to reduce gear vibration and noise.
Surface finishing and precision tolerance manufacturing methods are too costly, and more importantly, their effects on vibration reduction and noise reduction are quite small. In summary, geometric reshaping has gained widespread attention in recent years.
Walker proposed the deformation of the gear after loading. He proposed that the gear after the tooth profile modification has a trapezoidal form of gear load distribution in one engagement period. Harris uses a special chart, the 'Harris Table', to illustrate the static transmission error after gear shaping.
Transmission error is a relatively complex mechanical property, usually with three factors that affect it:
(1) First order factor. This includes the tooth profile, backlash and running-in errors caused by the manufacturing process, geometric errors in the assembly, and gear shaping errors that will cause “rigid displacement†into the overall drive error.
(2) Higher order factors. Including elastic deformation of a single pair of teeth, bending deformation of the teeth, shear deformation, deformation of the teeth around the root and the tooth itself due to the transmission load, and the load transmitted to the shaft causes deformation of the shaft.
(3) Factors that depend on higher order factors. Relative contact slip is a first order factor, but this factor depends on the variation of the higher order factor. This type of factor can be classified as a load transfer error, and the corresponding other order factor can be regarded as a non-load transfer error. This factor also contains geometric errors due to static or dynamic elastic deformation of the bearing. The total transmission error generally includes both a first order factor and a high order factor.
For the straight bevel gear modification, this paper will carry out in-depth research using the finite element simulation method. The transmission error is defined as the relative displacement of the drive wheel hub and the passive wheel hub when the hub of the driven wheel is fixed.
TE = θ2 - z1z2θ1 (1) where: TE is the transmission error (rad); θ2, θ1 are the main gear and the gear rotation arc; z2, z1 are the main gear and the number of gear teeth.
The gear meshing stiffness can be defined as Km=TTM(2) where: T is the torque (Nm); TM is the angle of rotation (rad) when the hub of the driven wheel is fully fixed, the drive wheel hub is under load; Km is the gear Meshing stiffness (Nm/rad).
The gear load distribution ratio represents the ratio of the entire transfer load being distributed by each tooth during one engagement period. The load distribution ratio is the ratio of the load that one of the gear teeth receives to the gear bearing load in one engagement period. That is, K = N total × 100 (3) where: K is the load distribution ratio (); N is the load (N) of one of the gear teeth; N is always the total load (N) of the gear; gear-bearing The load is equal to the sum of the contact normal force and the contact tangential force vector of all contact nodes at the time of contact.
1 Establishment of finite element model 111 The geometric parameters of the gear pair are taken as the research object of the straight bevel gear on the differential of the Fukang sedan. The material is 20CrMoH steel. The basic parameters are as follows: z1=10, z2=14; big end modulus m =317792mm; tooth width b=9175297mm; high displacement coefficient x1=012117; x2=-012117; elastic modulus E=211×1011Pa; Poisson's ratio ν=01278; friction coefficient μ=011.
According to the literature [12], in the Cartesian coordinate system, the spatial spherical involute curve equation of the bevel gear can be expressed as x=l(sinφsinψ cosφcosψcosθ)y=l(-cosφsinψ sinφcosψsinθ)z=lcosφcosθ(4) :x2 y2 z2=l;ψ=φsinθ;l is the starting radius of the gear; θ is the base cone angle; φ is the angle between the starting line segment on the meshing surface and the instantaneous rotary axis, for the involute on the base cone The starting point φ is zero.
In the actual machining, since the spherical surface cannot be unfolded into a plane, the design and manufacture of the spherical involute is very close to the spherical involute with the spherical involute. When the ratio of the spherical radius R to the modulus of the gear is smaller, the error is larger. Since there is no involute curve inside the base circle, the tooth profile between the base circle and the root circle is a circular arc transition, so the arc transition function can be used to perform the arc transition between the circles. In the CAD, the theoretical spherical involute straight-toothed bevel gear is designed first, and it can be manufactured by the cold precision forging method.
The meshing of the 12-dimensional contact finite element model is completed because the gear transmission relies on the continuous meshing between the high-precision tooth conjugate tooth surfaces, and the meshing of the gear model must be strictly in accordance with the geometry of the gear teeth. In the 3D modeling Pro/Engineer development environment, the precise model of the meshing bevel gear pair was established by parameterization, and the igs data format file was exported.
To improve the calculation accuracy, an 8-node isoparametric hexahedral mesh is used.
Considering that the rim does not participate in the contact, a sparse grid is used at a distance from the gear teeth, and the complete mesh of the straight bevel gear meshing contact model is as shown.
13 Boundary conditions All the nodes on the hub of the passive wheel are fixed, and a node is added to the drive wheel axle, which is defined as the master control point. All the nodes of the master control point and the drive wheel hub are fully coupled in the axial direction, that is, the master control point and The hubs are aligned in the axial direction. Convert the master point and hub node coordinate system to a cylindrical coordinate system. The driving wheel boundary condition can be applied at the main control point, constraining the radial displacement and axial displacement of the main control point, and then applying torque at the main control point, so that the circumferential displacement of the main control point is the static transmission error. In order to simulate the meshing condition of the entire meshing zone, the position of the meshing point can be changed according to the gear ratio relationship. The useful results of the post-processing are imported into an Excel file, and key quantities such as transmission error, meshing stiffness, and gear load distribution ratio are calculated, and finally represented by a graph. Considering that the amount of calculation is relatively large, the entire meshing period is only 59 increments. It is handled in the MARC with a loop command.
2 Simulation results analysis 21 analysis of the unshaping gear meshing process 2 shows the unreformed gear transmission error. 2 It can be seen that the transmission error of the gear in the single-tooth meshing zone has abruptly changed, which is in accordance with the engineering practice, and is obtained under the condition that the large gear is the driving wheel. The basic rule is that the single-tooth transmission error is large, the double-tooth transmission error is small, the larger the torque is, the larger the transmission error is, and the more the single-tooth and double-tooth transmission error changes. Since the single tooth is much larger than the double tooth bearing load, the elastic deformation is relatively large, and the angular displacement of the driving gear is also large, so the transmission error is larger than the double tooth transmission error; the transmission error is almost proportional to the torque. Because the torque increases the contact area and increases the contact area when the meshing is in the same position. According to the elastic mechanics analysis, the displacement and the load can still be regarded as a proportional relationship.
The change of the gear meshing rigidity is as shown in the figure, and the basic rule is that the single tooth meshing rigidity is small, and the double tooth meshing rigidity is large. The torque is not the same, the stiffness is basically the same, and it does not change with the torque.
Since the straight bevel gear has a small degree of coincidence, the double-toothed meshing area is smaller than the single-toothed meshing region, from single tooth to double tooth or from double tooth to single tooth, the stiffness is suddenly increased from small to large or suddenly decreased from large to large. There is a transition zone from the double-toothed area to the single-toothed area instead of abruptly falling from the double-toothed area to the single-toothed area. This is because the coincidence of the straight bevel gear is only 1129, so the double-toothed meshing area is relatively small.
The variation of gear tooth load distribution is as shown in the figure. Since the degree of coincidence is small, the gear tooth load distribution rate on the gear pair still exhibits 4 mutations on the meshing line. Corresponding to the alternation of the single-tooth meshing and the double-toothed meshing, this sudden change causes vibration, and since the occurrence of these four mutations is extremely short, a strong impact of the teeth is caused.
It can be seen from the above analysis that the profile modification must be performed to make the load distribution relatively flat and reduce the impact and noise.
Analysis of the 22 meshing process of the spur gear The spur gear can be straightened, parabolic, arc, exponential function and the involute of the rotation for shaping, because the involute of the spur gear is a plane involute. For a straight bevel gear, since the involute is a space spherical involute, the trimming curve can only be modified by a straight line, an arc or a rotating involute. The method of shaping the tooth profile by rotating involute is to rotate around the S1S2 at the involute of the spherical surface near the top of the tooth, and the starting point S1S2 is an alternate point of single tooth engagement and double tooth engagement, and the amount of deformation is elastic. The sum of the amount of deformation and the knuckle error, and the degree of coincidence is greater than 1, to prevent the gear from slipping.
S1 is the alternate point of the small-end single-tooth engagement and the double-tooth engagement, and S2 is the alternate point of the big-end single-tooth engagement and the double-tooth engagement.
S1T1 is a part of the big endian involute close to the top of the tooth, S2T3 is a part of the big end theoretical involute close to the top of the tooth, T1T2 and T3T4 are the small end maximum profile amount Ca1 and the big end maximum profile amount Ca2; P1 And P2 are the nodes of the small end and the big end respectively; ΔLa1 and ΔLa2 are the maximum modification lengths, respectively. The three types of trimming curves are as follows.
(1) Rotating involute trimming:
S1T2 is a curve in which S1T1 is rotated around S1S2 with the dividing point S1 as a starting point, and S2T4 is a curve in which S2T3 is rotated around S1S2 at the starting point of the dividing point S2.
(2) Arc modification: S1T2 is the space arc between the starting point S1 and the ending point T2, and the space arc between the starting point S2 and the ending point T4 of S2T4.
(3) Straight line modification: S1T2 is a straight line starting point S1 and ending point T2, and S2T4 is a straight line starting point S2 and ending point T4.
In this paper, the gear profile is modified by three kinds of trimming curves. The maximum trimming amount Ca1 of the big gear small end and the maximum trimming amount Ca2 of the big end are 20167μm and 20138μm respectively, and the maximum trimming amount Ca1 and large of the pinion small end are large. The maximum profile amount Ca2 of the end is 21155μm and 18188μm, respectively.
The gear shaping model and the unreformed gear transmission error are as shown in ～, the double-toothed meshing zone is relatively reduced, and the single-toothed zone is relatively extended, which corresponds to an increase in the transition zone. When the load is small, the transmission error of the double-toothed area and the single-tooth area is relatively small for the modified gear and the unshaped gear; when the load is large, the transmission error of the double-toothed area and the single-tooth area of ​​the modified gear is relatively large. Since the coincidence degree is greater than 1, there is still a double-toothed meshing zone, and the transmission error still fluctuates.
The rotation of the involute is modified, the relative amplitude of the transmission error is reduced, but it is much smoother than the unshaped gear, which reduces the impact and noise; the arc curve shaping effect has no effect of rotating the involute shape modification.
When the straight line is modified, the rigid displacement of the double-toothed area is too large due to the large amount of trimming, which causes the transmission error of the gear double-toothed area to be larger than that of the single-tooth area. The fluctuation of the transmission error increases.
The meshing rigidity of the trimming gear and the unshaped gear is as shown in ～1. It can be seen from ~1 that the rotation involute shape is modified, the stiffness variation is not large, but it is relatively smooth. The meshing stiffness of the single tooth zone is almost equal to the meshing stiffness of the double tooth zone, avoiding the sudden change of the gear stiffness; the arc shape modification, The effect is not modified by the rotating involute, but it is better than the unreformed shape; the worst is the straight line modification, and the meshing stiffness fluctuates more than the unreformed shape.
Straight line shape modification model and unreformed gear load distribution ratio The load distribution ratio of the shape-removing gear is as shown in 2 to 4, and the rotation of the involute shape modification and the arc curve modification of the gear single-tooth bearing load range increases, and the double The tooth has a reduced load-bearing interval and changes almost linearly. No mutations have occurred, and if the gears are more coincident, the effect will be more pronounced. The straight line modification is more abrupt than the unsharpened load distribution rate. It can be seen from the above that the best shape-removing effect is the rotation involute shape modification, the second arc shape modification, the worst straight line shape modification, and it is preferable not to adopt the straight line shape modification.
In the past, the traditional shaping method that can not be realized by the traditional machining method can now be realized by the cold precision forging method. The complete straight bevel gear shaping pattern is constructed in the three-dimensional modeling software, and converted into the numerical control code input into the numerical control machine tool. The CNC machine tool directly processes the blank into a modified gear, a gear electrode or a gear mold for forging, and then directly forging the modified straight bevel gear from the mold. Now it is convenient to design and produce cold ingot forging with a rotating involute or arc curve. It is a straightened bevel gear that does not require subsequent processing and significantly reduces the cost.
3 Conclusions The finite element simulation of the untrimmed straight bevel gear shows that the transmission error and meshing stiffness are abrupt in the single tooth region and the double tooth region, and the gear loading rate on the gear pair appears 4 times on the meshing line. This will generate vibration and noise. This requires a tooth profile modification of the straight bevel gear to reduce vibration and noise.
(1) Even if straight line modification is the most economical method of shape modification, it is best not to use it for tooth top or tooth profile modification; from the perspective of transmission error, load distribution coefficient, etc., because the noise caused by straight line modification is the largest. .
(2) Rotating the involute tooth tip The modified gear will obtain a stable and smooth transmission error curve in one meshing period. The rotary involute profile modification will get the best benefit and easy to manufacture.
(3) Three kinds of curve modification, from the perspective of transmission error, it is best to rotate the involute, followed by the arc shape, and the straight line modification effect is the worst.