The herringbone gear is widely used in the main transmission of the ship. For the calculation of the contact strength of the spur gear, the traditional method is based on the Hertz model when the two cylinders with parallel axes are in contact with each other and pressed, and the approximate method is obtained by using the Hertz formula. The maximum contact stress of the tooth surface is difficult to solve accurately because of many approximations. For the analysis and calculation of the dynamic characteristics of the herringbone gear under contact, it is not possible to perform an accurate analytical solution. The traditional dynamics modeling method is based on the design pattern, making necessary assumptions and simplifications. The mathematical model is built according to the mechanical principle, which is expressed by the vibration differential equation. However, the assumptions and simplifications adopted by this theoretical modeling are difficult. Consistent with the reality, it is quite difficult to make the results of the system from theoretical modeling and dynamic analysis accurately match the actual situation. In recent years, many scholars have studied the contact and vibration of the gear pair based on the study of the meshing of the spur gear pair or the helical gear pair. The vibration characteristics of the gear pair system under the contact deformation and the equivalent stiffness simulation of the shaft end are not considered. Research on the contact and dynamic characteristics of the herringbone gear pair is still rare. In this paper, the finite element method is used to study the contact deformation and vibration characteristics of the herringbone gear system.
1 Meshing load distribution and meshing stiffness model Gear meshing state Due to the degree of coincidence, the number of meshing tooth pairs changes during gear meshing transmission, as discussed for the double meshing model, multi-engagement and so on.
The two meshing pairs share the normal load FL, and the displacement of the meshing point along the meshing line is x, and the load Fs1 carried by the tooth pair 1 and the load Fs2 of the tooth pair 2 can be obtained by the formula (1): Fs1 Fs2=FLFs1= Kc1xFs2 = Kc2x (1), that is, the load of the gear is distributed by the teeth participating in the meshing.
The distribution of the load on the teeth not only changes with the single-tooth or double-tooth engagement, but also the meshing stiffness Kci (i=1, 2) of the teeth changes due to the position of the meshing point. This change affects the load on the gear teeth. Distribution.
When calculating the meshing stiffness of the teeth, first calculate the deformation when the teeth are engaged. The deformation at the meshing point after the tooth is loaded is composed of three parts: the contact deformation H at the meshing point, the bending of the tooth and the displacement of the meshing point caused by the root shear, and the meshing point caused by the elastic deformation of the wheel body. Displacement A. The meshing stiffness of the meshing tooth pair is calculated by: Kcj = FLH T1 T2 A1 A2j = 1, 2 (2) These deformation amounts can be obtained by finite element in NXMaster FEM.
2 The establishment of the transmission vibration model and analysis of the vibration of the gear during the meshing process can be considered as a result of the continuous impact caused by the change of the gear meshing stiffness and the gear manufacturing error. Under the influence of simplifying manufacturing error, the dynamic model of gear transmission is established.
M1 and m2 are the equivalent masses of the gear 1 and the gear 2 on the meshing line, respectively, and can be calculated as follows: mi=Iir2bii=1, 2(3) where: the moment of inertia of the Ii gear i; the base of the rbi gear i The radius of the circle.
Kc1 and Kc2 are the meshing rigidity of the meshing tooth pair 1 and the tooth pair 2, respectively. X1, x2 represent the displacement of m1, m2 along the mesh line. The equation of motion of the system is: mx Kcx=FL(4) where: m is equivalent mass, m=m1m2m1 m2; relative displacement of xm1 and m2 along the meshing line, x=x1 x2.
Since the meshing stiffness is a function of the position of the meshing point or a function of the time t, in solving equation (3), one meshing period must be divided into sufficiently small intervals, the meshing stiffness is considered to be constant in each interval, and then iterative method is used. Solve.
Since the gear transmission system is a time-varying dynamic system with multi-stage transmission of torsional vibration, transverse 73-direction vibration and axial vibration, it is theoretically necessary to analyze the vibration differential equation according to the dynamic model. The model is greatly simplified, and the calculation results are difficult to match with the actual. For complex systems, the calculation becomes quite difficult and cannot be performed.
The finite element analysis (FEA) method developed in recent years with the support of computer technology and numerical analysis methods provides an effective way to solve these complex engineering analysis and calculation problems.
The NXMasterFEM analysis module performs structural response analysis under static or dynamic loads, including static response analysis, transient response analysis, frequency response analysis, and response spectrum analysis. In this paper, the analysis module is used to analyze the static contact stress and dynamic free mode of the gear mesh.
3 Response analysis process response analysis process.
4 Gearbox Herringbone Gear Pair Meshing Model Analysis 4.1 Static Response Contact Analysis 4.1.1 Physical Model A gearbox is used for deceleration of a gas turbine of 25,000 kW and outputs torque to the propulsion unit. The analysis object is a pair of herringbone gear pairs in the gearbox, the pinion gear is the driving wheel, the number of teeth is Z1, the input power of the shaft end is 25000kW, the stable speed is n1=1000r/min. The large gear is the passive wheel, the number of teeth Z2. Application IDEASNX software Building a solid model is shown in Figure 4.
4.1.2 Mathematical model The parameters of the solid model are rb1=0.20m, rb2=1.04m; I1=21.3kgm2, I2=4192.4kgm2. Calculate the tangential force of the tooth surface contact FL: FL=M1rb1=9.55 103Pn1rb1=1.19106N 3) Calculate the equivalent mass m: In the herringbone gear transmission system, since the two helical helix angles are equal in magnitude and the opposite direction helical gears are on the same axis, the structure is the same as the helical gear. However, since the axial component forces of the herringbone gears (two helical gears on the same shaft) are theoretically equal in magnitude and opposite in direction, they cancel each other out.
4.1.3 Establishing the classification quality of the finite element model mesh has a great influence on the calculation results of the finite element. The ideal mesh should be equilateral. Moreover, in general, the denser the mesh is, the higher the accuracy is. However, when the mesh density reaches a certain level, the contribution to the improvement of accuracy becomes small, and the calculation cost is sharply increased. Considering the above factors and the geometric characteristics of the gear. The hexahedral element is used to establish a finite element model by manual division and stretching, and local encryption processing of the tooth portion of the contact. In this way, the model finite element mesh comparison rule is obtained and the quality is high.
The finite element model of the large and small gears and their assemblies are built by the Simulation_meshing module of NXMasterFEM.
4.1.4 Boundary conditions 1) Setting of contact pairs.
In NXMasterFEM, contact pairs can be defined using point contact units or with face contact units. Considering that the face model can support large deformations with large sliding and friction, it can effectively deal with complex contact surfaces and dynamic contact problems, so the face contact pair model is adopted. Through the contact analysis of NXMasterFEM, a set of contact conditions is created, and a contact pair is established, in which 102 units of the collision area and 106 units of the target area.
2) Constraint and load setting.
Firstly, the local cylindrical coordinate system with the respective center points as the origin is established on the geometry of the large and small gears respectively. In the Mashing, the coordinates of the nodes on the respective rotating boundary surfaces are respectively converted into the corresponding local coordinates, then X, Y, Z represents the displacement of the nodes on the rotating boundary surface of the R, Z-constrained passive wheel, respectively, and the nodes on the rotating boundary surface of the driving wheel only leave the displacement. The load is converted into a tangential force applied to the swivel boundary surface node of the driving wheel, and the resultant force is equal to the torque applied to the driving wheel.
4.1.5 Results Analysis Finite element calculation results In the NXMasterFEM post-processing module, the relative size and location of the deformation can be qualitatively seen in a magnified deformation mode, and the deformation of each node can be quantitatively measured with a probe. the amount. From the deformation map, the location and size of the gear deformation can be visually understood. It also shows that the meshing stiffness has different values ​​in different parts. The stiffness matrix in the dynamic analysis refers to this result.
4.2 Vibration characteristics of the dynamic response structure (free mode) analysis 4.2.1 Establishment of the finite element model The finite element model of the dynamic response analysis differs from the static response analysis in the structural free modal analysis of the dynamic response analysis. Although the end does not have to consider the constraint, since the gear shaft is actually connected to the casing through the bearing, the bearing and the casing cannot be regarded as a rigid body in the vibration analysis, but should be applied as an elastic body to the gear shaft, otherwise the vibration The modal solution result will be quite different from the actual one. In this paper, four springs are used to simulate the bearing and the impact of the box on the gear shaft of each bearing on the shaft end of the gear. The spring stiffness is obtained by the theoretical modeling of the gear box and the spring stiffness value.
4.2.2 Structural Free Modal Solution The finite element model is built in NXMasterFEM, as shown in Figure 5.
In NXMasterFEM, there are three methods for solving the natural frequency and mode shape of the model: Lanczos method, Guyan method and synchronous vector iteration method (SVI).
Comparing the solution methods, combined with the actual situation of the analysis model, the Lanczos method is used to analyze the finite element model.
4.2.3 Analysis of free modal results The free modal calculation is performed on the finite element model of the gear pair dynamic response. The vibration response method is usually solved by the vibration mode superposition method. It is usually not necessary to find all the natural frequencies and modes. The lower the order, the more the influence Large, usually taking 510 steps, the accuracy is enough. In the analysis, in order to more clearly and intuitively represent the vibration of the gear pair, the first 50 modes are solved.
From the natural frequency and the vibration pattern, it can be seen that the large gear is about 32.9 Hz and the small gear starts to vibrate around 230.0 Hz. Therefore, it is necessary to avoid working under the excitation condition of 32.9427.9 Hz when designing and using the gear pair. If the condition of the gear system is in the low-order natural frequency range, the gear system must be modified to increase its stiffness to increase the natural frequency or increase the damping plate and reduce the influence of excitation on the amplitude.
5 Conclusion Through the establishment of a three-dimensional finite element model of a pair of herringbone gear pairs, static contact analysis and free modal analysis are performed on the gear pair respectively. The contact deformation map and the mode shape pattern can visually represent the contact parts of the gear pair. Dynamic characteristics, qualitative and quantitative accuracy, easy to understand the working characteristics of the gear system, easy to study and improve safety measures, reduce the failure of the gear system and extend its service life.
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