Finite element analysis of single pair gear tooth root crack

: In the current research, the analysis about the stress and meshing stiffness for the single gear meshing process with two cases came out, which are case without tooth root cracking and with tooth root cracking. First, the solid assemble model for the two cases are set up with Pro/E software. With the format of parosolid, the model is imported into Ansys. In analysis, the contact, loading, and bonging condition are added. Then, computation came out. The results show that the stress in the compression position is greater than that in the tension position during gear engagement, and the stress increases when cracks occur. The research can provide theoretical basis for fault diagnosis to provide spectrum analysis and provide reference for theoretical analysis and meshing stiffness.


Introduction
Gear transmission is widely used in manufacturing with its advantages of compact structure, accurate transmission, high transmission efficiency, and strong bearing capacity. It is an irreplaceable mechanical transmission mechanism, to a certain extent, marks the level of mechanical engineering technology [1]. As of the position and function of gear transmission in the machinery industry and even in the whole national economy, gear is recognised as a symbol of industrialisation [2][3][4][5]. From the failure of parts, gear is also one of the most vulnerable parts. Gear transmission often occurs in gear breaking, tooth surface wear, pitting, tooth surface gluing, plastic deformation, and so on. Failure of transmission performance results in serious production accidents. According to statistics, in all kinds of mechanical failures, gear failure accounts for >60% of the total [2], of which tooth surface damage and tooth root fracture are the main causes of gear failure [6][7][8]. Therefore, to grasp the dynamic characteristics of the crack defect gear system is of certain theoretical significance and potential engineering application value for understanding the fault mechanism of the tooth root crack defects and realising the fault monitoring, identification, and diagnosis.
The complicated stress distribution and deformation mechanism of the transmission gear have become the main cause of the gear design difficulty. The finite element theory and the appearance of various finite element analysis software make the ordinary designers do not need to do a lot of analysis and research on the gear, so they can basically master the force and deformation of the gear and make use of the finite element. The result of metacomputation finds out the weak points in the design and achieves the purpose of gear design. ANSYS, a computer simulation engineering structure finite element analysis software developed by ANSYS company, has become the world's top finite element analysis software [9][10][11]. It integrates structure, heat transfer, fluid, electromagnetic, acoustic, and blast analysis. It has powerful function of pre-and post-processing and calculation and analysis [12,13]. At present, the ANSYS software is widely used in the fields of civil engineering, water conservancy and hydropower, automobile, machinery, mining, nuclear industry, ship, household electrical appliances, and so on [13][14][15]. As a general finite element analysis software, the powerful model, grid division, and analysis function greatly facilitate the user to analysis the product. [16] Here, the finite element analysis method is used to discuss and compare the stress and meshing stiffness changes in the process of toothless root crack and toothed root crack meshing.

Gear contact stress analysis method
As shown in Fig. 1, Pro/E was used to establish a model with or without two cracks and imported into ANSYS for analysis. In ANSYS, parameters such as contact type, load, and constraints are On the premise of not affecting the calculation results and improving the calculation efficiency, the contact stress analysis of the gear meshing tooth surface is based on the force condition of the gear transmission, and the contact stress model is assumed as follows.
i. The gear material is uniform, continuous, and linear. ii. Solving and analysis process, ignoring the influence of temperature and friction. iii. The force acting on the contact line is uniform force.

Gear modelling
The parameters and geometric parameters of the gear are determined, according to which: tooth number Z, modulus m, pressure angle, tooth top height coefficient, tooth bottom clearance coefficient, modification coefficient, and tooth width. Geometric parameters refer to parameters that can accurately and completely describe the geometric structure of the model. The geometric parameters of the cylindrical gear tooth are: the diameter of D = mz, the diameter of the base circle D b = Dcos α, the diameter of the tooth top circle D a = D + 2 h a + x m, the diameter of the tooth root circle D f = D − 2 h a + h f − x m and so on. In the analysis, the parameters of a pair of gears of a planetary transmission of the laboratory were model. The specific parameters are shown in Table 1. The gears are involute gears. The specific formula is: where r is the base circle radius; θ is involute rolling angle; the parameter t belongs to 0 to 1.
Assemble parts is completed by using Pro/E parameterised gear model, and a simple planet frame is established. Two gear meshing assembly is realised through pin connection and gear pair constraints, as shown in Fig. 2. To ensure that the tooth gap is fully meshed, a global interference analysis is carried out on the meshing point. As shown in Fig. 3, the results show that there is no interference at the meshing point and the gear is fully meshed.

Material properties
The gear material is 45 steel, the elastic modulus is E = 2.06 × 105 N/mm2, Poisson's ratio is 0.3. The density is 7.85 g/cm×3.

Load and contact
Here, the driver and driven wheel materials are the same, and the stiffness is similar. In all contact modes, surface-to-surface contact supports low-and high-order elements, which is suitable for contact problems with complex surface, large deformation, and friction, without the limitation of surface shape. The gear contact is a typical non-linear analysis, and the friction force plays an important role. Therefore, the contact surface of the gear is defined by the method of face-to-face contact.
As shown in Fig. 4, the tooth profile surface of the meshing sun gear and the tooth profile surface of the planetary gear is configured as a contact pair using the ANSYS contact guide, so that the tooth profile surface of the sun gear is the contact surface and the planetary gear tooth profile surface is the target surface. The specific parameters are shown in Table 2, the surface of the sun gear is 94 surfaces, the planet gears are 264 surfaces, and the contact type is frictional contact. The friction coefficient is 0.1. Second, add the rotation pair to the position of the two gear shafts Set the rotation loading, the driving wheel rotation loading is set as the rotation speed; to ensure the convergence, the rotation speed of the drive wheel is a constant rotation speed of 100 rmp. The driven wheel is set to torque, the torque is 1 N/m, and the load setting is shown in Fig. 5. The parameters are shown in Table 3.

Solution method
First, the imported model is meshed, and then the analysis parameters are set for solving. The ultimate goal of grid model is to divide grids into nodes and units. The grid partition process of the generated nodes and units consists of two steps: i. Definition of the element properties; ii. The grid generation control is defined and the grid is generated.
In the model, the element type of the grid is C3D8R, and the result is more accurate; the calculation time is short; when the grid is distorted, the accuracy of the analysis will not be greatly affected. After meshing, the number of generating units is 24,482 and the number of nodes is 66,409. The meshing model is shown in Fig. 6. Then, set the analysis parameters, set the calculation end time to 5.e-002 s, the initial substeps to 500, the minimum substeps to 5, the maximum substeps to 2000, and finally complete the calculation and analysis results.

For normal status
The overall equivalent stress distribution of the calculated meshing gears is shown in Fig. 7.
The Von Mises stress distribution at the contact tooth pair is shown in Fig. 8. The maximum stress point is located at the tooth pair contact point, and the maximum Von Mises stress is 2.0654 × 10 −2 Pa.
The stress distribution characteristic of the tooth root is very important for the gear transmission performance. The calculation results show that the stress of the tooth root is mainly expressed as the compressive stress on without loaded side of the tooth and the tensile stress on that side, and it can be seen that the stress of the meshing point is maximum, and the left side of the tooth root is drawn and the right side is pressed, which is in accordance with the actual situation. According to the distribution of the stress and deformation in the tooth meshing, the stress and deformation of the gear tooth are mainly distributed in the meshing tooth pair, and the gear and gear body adjacent to it, decrease rapidly with the increase of the distance of the meshing point. As shown in the picture 9, on the meshing gear pair, the stress and deformation at the meshing point are the largest. The stress value at the root of the gear pair is larger. Through the above analysis, we can see that the simulation model established by ANSYS is in accordance with the actual situation, which shows that the simplified finite element analysis model of the tooth root bending stress is correct. Therefore, this model can be applied to the stress analysis of the Cracked Gear in the following analysis. (Fig. 9)

For with root cracking status
The stress field and stress intensity factor near the crack tip are important parameters when the fracture safety analysis is carried out, which is an important basis for judging the failure of the cracked body under the load. For most parts, cracks are most likely to occur at dangerous sections and surface defects. For the tooth, the dangerous section can be determined by the tangent method, that is, the tangent line between the 30 angles of the symmetrical centre line of the tooth and the transition curve of the tooth root, and the cross-section which is parallel to the axis of the gear through the two tangent points. This section is the dangerous section of the tooth root. At the same time, it is assumed that the initial crack of the analysis is perpendicular to the tangent line and is constructed by Pro/e. The sun wheel crack model is shown in Fig. 10.
The contact stress distribution of the crack free gear has been obtained, and the maximum stress appears at the meshing place, which is 0.0020654 Pa. By the same amount method, the constant speed of 100 r/s is applied to the sun wheel with the crack defect, and the load of 1 N/M is applied to the normal planetary gear that is meshed with it, and the transient dynamic analysis is carried out on it. The stress distribution of the single tooth engagement of the wind turbine gearbox is shown in Fig. 11.
From the graph, the maximum stress appears at the root of the tooth and the maximum value is 2.5979 Pa. Compared with the crack-free gear, the maximum stress of the cracked gear is greater under the same constraint and load, and because of the existence of the crack, the crack produces the obvious stress concentration phenomenon, so the defect is easier to break the teeth and lead to the gear failure.  Here, the finite element analysis model of normal state and crack state with tooth root is established by using finite element software Ansys Workbench. First, the bending stress of the tooth root is analysed by the gear meshing of normal state, and the correctness of the simplified model is verified by comparing with the theoretical calculation results. Then, the stress distribution of meshing state with tooth root cracks is compared and analysed. The results show that the existence of cracks will result in stress concentration and stress at the crack tip. In view of the above conclusion, the machining quality of the tooth root is particularly important in the process of gear processing, and the vibration of the tool and gear can be reduced as much as possible in the process of processing, so as to avoid the production of small cracks. If the root cracks are detected in the work, the working load should be reduced to extend the service life or replace, thus increasing the safety factor. At the same time, as a method of gear fault diagnosis, it provides an effective theoretical basis for the safe use of equipment.