Stress Analysis of Gear Meshing Impact Based on SPH Method

Based on the kinetic equations of the gear mesh impact, SPH discrete equations were established. Numerical simulation was carried out on the meshing impact process of the gear, and stress and strain of each discrete point were obtained. After data processing, stress propagationwas calculated, which shows stress distribution on tooth-profile surface. It is concluded that the stress concentrate mainly occurs in the pitch circle.The paper provides an effective new numerical simulation algorithm to gearmechanical properties analysis.


Introduction
In researches studying gear transmissions which bear heavy load and are responsible for dynamic transfer, analytical methods or simple numerical methods usually do a great deal of simplifications and could not properly reflect the actual situation [1].With the development of computer technology and the emergence of new numerical algorithms, computer simulation has shown a lot of advantages in gear transmission analysis.For example, with elastic contact finite element method, it is possible to analyze displacement and stress change in gear contact zone [2,3].Analytical methods and finite element method could be used to solve surface deformation and load distribution problem on tooth surface of meshing gear and also could be used to analyze effects of tooth surface deformation on the gear transmission [4] ANSYS/LSDYNA software can simulate gears meshing impact [5].However, with methods based on the meshing (finite difference method, finite element method, boundary element method, etc.), it is hard to track gear transmission process, especially when calculating large deformation and crack propagation because of mesh tangling and distortion.In this case, remeshing is needed in calculation process which is computationally expensive and affects calculation accuracy.
Smoothed particle hydrodynamics (known as SPH) is one of the meshless methods developed in recent years.The basic idea of SPH method is to discretize the gear model into a set of particles, each of them having properties like mass, position, velocity, and density [6,7].During computation, particles are linked by a kernel function.The process does not need meshing, is easy to program, can track gear transmission process every time, and guarantees calculation accuracy.
In this paper, meshing impact of a pair of gear teeth is analyzed using SPH method.Dynamic numerical simulation was conducted regarding meshing impact process in different rotation speeds, and the stress distribution on the tooth profile and changes was analyzed.The simulation results meet actual meshing rules.The results of this study show that SPH method has very good potential in analyzing impact of gear meshing.

SPH Description of Correct Gear
Meshing and Contact 2.1.Fundamentals of SPH Method.In the SPH method, the first step is to approximate a function and its gradient using integration of kernel function based on interpolation theories.Continuous partial differential equation is transformed into integral equations.At the second step continuous forms of integral equations are discretized into discrete equations using particle approximation method [6].

Mathematical Problems in Engineering
For any function (), the value at a point  can be approximated as where ⟨ ⟩ denotes kernel approximation of the function,   is position vector, and ( −   , ℎ) is kernel function which depends on distance | −   | and smoothed length ℎ.
The most frequently used kernel function in SPH method is the cubic B-spline kernel [8].It has the form as shown in (2); Figure 1 shows that choosing different smoothed length (ℎ = 1 and ℎ = 0.5, resp., and   = 0) affects influence radius and shape of cubic B-spline kernel.Consider where  is equal to 1/ℎ, 15/7ℎ 2 , and 3/2ℎ 3 in one-, two-, and three-dimensional cases, respectively;  is the relative distance between two points (particles) at points  and   ;  = /ℎ = | −   |/ℎ where  is the distance between the two points.
Using integration by parts, Gauss theorem, and property of kernel function, gradient ∇() can be approximated as Function () and its gradient ∇(), continuous equations ( 1) and (3) can be discretized using SPH method as where  and  are the particle indices,   and   are mass and density of the particles ,  is the number of particles in the influence domain of particle , In the above equation, the sum is calculated only in influence area of  and is not in the entire computational area; the influence area is determined by the radius of kernel function [9], as shown in Figure 2 in 2D case.

SPH Discretization of Impact Gear Meshing Equations
Based on Elastic Dynamics Theory.Based on continuum mechanics and linear elastic dynamics theory, the kinetic equation and constitutive equation of gear meshing impact can be written as follows [5].Conservation of mass:  Conservation of momentum: Strain-displacement equation: Constitutive equation: where V, , , , , and  are the velocity, density, stress, external force, elastic modulus, and Poisson ratio, respectively; Considering the artificial viscosity effect Π  , using (4), a set of SPH particle discreet equations such as conservation of mass, conservation of momentum, and strain-displacement equation could be obtained: where where   and   denotes the sound speeds at particles  and  respectively,   and   are coordinates of particles, V  and V  are velocities of particles, ℎ is smoothed length of kernel function, and , ,  are adjustable parameters [10].

SPH Model of Two Correct Meshing Gears.
Establishing SPH discrete particle model is to discretize the gear model into a set of particles.In this paper, by analyzing every curve of involute gear, tooth profile curve equation and tooth root transition curve equation were determined; gear 3D mathematical discrete particle model was established and programmed [11].Model data was visualized using open source software RASMOL, as shown in Figure 3.
After discretizing the gear model into discrete particles, it is difficult to introduce into whole body parameters like torque load (including resistance torque) and related conditions of fixed axis rotation; at the same time it will add additional computation work too.To save the modeling and calculation work, our numerical simulation considers only a pair of meshing impact gears as shown in Figure 4.

Initial Conditions.
At the beginning of the calculation, the driving gear was rotating with certain initial angler velocity and impact to the driven gear which in static state where  is angler velocity of the driving gear which rotates around gear center C;   is position vector of particle .

Boundary Conditions.
During calculation process particles on the driven gear root were fixed and kept motionless in the whole calculation process: 3.4.Interface Coupling.In particular, when a pair of gears contacts each other's, interaction of particles which belongs to different gears will happen near the contact surface and  the particles will participate in calculation.If the influence radiuses of all particles are the same as the original one, the interaction will occur between particles of two gears at contact area earlier, which will cause the two gears not directly contacting each other on the surface and still remains the gap during the meshing process as shown in Figure 5(a), so this will lead to errors on gear meshing and contact.
In general, the two problems occur when using the general SPH influence radius as mentioned above, one is correct gear meshing and the other one is correct contact with the tooth surface during the dynamic contact and meshing process.To describe dynamic contact phenomenon of gear meshing and impact process by using SPH method, defining different kernel radius method is used for correct coupling of the interaction and contact areas of gears.In this method, it not needed to define any special contact pair or contact area due to SPH kernel function characteristics, so the choice of the influence radius of kernel is very important.The influence radius affects the accuracy and efficiency of the calculation directly.
From the other point of view, different behaviors can be easily obtained by tuning the kernel effective radius ℎ.A larger value of ℎ will create averaged interactions of each particle in the body, so the body will be bonded into more strong continuous body such as evaluation of particles in the inside body of independent gear.And a small value of  will create very local interactions of each particle, so the body will be divided more easily into discrete pieces such as evaluation of particles near the two gears meshing at contact area.In this paper due to the above unique characteristics of SPH kernel, different kernel radiuses are defined by choosing different smoothed length as shown in Figure 6 and expressed as (13) to evaluate two gears correctly meshing and impact problems. Consider where ℎ  ,  = 1, 2 is larger kernel radius which is used to calculate and evaluate the interaction of particles inside gear and represent strong continuous bodies itself, the coefficient  is determined by corresponding kernel function which is used in calculation,  = 2 in cubic B-spline, smoothed length equals to 1.2 times of the distance between two particles [12].Then the ℎ  , ,  = 1, 2 ( ̸ = ) is smaller kernel radius which is used to calculate and evaluate the interaction between two gears at contact surface where two gears are not continuous and separated each other; generally choose an appropriate smoothed length which keeps kernel radius equal to one time of the distance between two particles.The errors caused by selection of influence radius were complemented by corrective SPH method [13].
The results were shown in Figure 5(b); the two gears correctly meshing and directly contacting each other's after different kernel radiuses are used which will effectively avoid interaction of noncontacting particles participating in calculation.

Numerical Simulation and Analyzing Results
In numerical simulation three different driving gear angular velocities,  = 60 r/min,  = 120 r/min, and  = 240 r/min, were used in calculation and analysis.Numerical calculation results were shown in Figure 7. Maximum stress and maximum equivalent stress change with    time after impact of two meshing gears.We can know from the figure that at the beginning of meshing, the biggest impact stress increases with the extension of engagement time.Meanwhile, enlarging impact speed will increase impact stress gradually.
Figure 8 shows selected consecutive figure of the equivalent stress contours.These stress variations in different time show the propagation of elastic waves in the meshing gears.
Figure 9 shows comparison of stress   and equivalent stress   at time  = 10s.It could be seen that impact stress increases with the increase of impact velocity; increase of driving gear initial impact velocity causes propagation of tooth impact stress.
Figure 10 shows comparison and calculation result of stress on the middle line along the contact faces of the driven gear and the driven gear's tooth profile at  = 10s in different rotation speed.The results indicate that maximum stresses appear near the pitch circle; when the impact velocity becomes bigger, the stress also becomes bigger.

Conclusion and Discussion
In this research, a SPH discrete model of impact of gear meshing was established, and automatic modeling program was developed.A simulation and analysis software of gear meshing which is capable of simulating gear mesh impact problem of different gear parameters, material property, and rotation velocity was programmed.
The results of numerical simulation comply with real gear behavior, which verified the correctness of the model and method.A new numerical simulation algorithm based on SPH method is provided for design and optimization of gear transmission.

Figure 1 :
Figure 1: Figure of cubic B-spline kernel function in 1D case.

Figure 3 :
Figure 3: Discrete model of gear contact.

Figure 4 :
Figure 4: Initial and boundary conditions of the gear meshing impact.

Figure 5 :
Figure 5: Comparison of selecting different influence radius.