Modal Analysis of Cylindrical Gears with Arcuate Tooth Trace

In this paper, the forming principle, meshing features and tooth surface equation were introduced. And the modal parameters distribution of cylindrical gears with arcuate tooth trace was researched. The results show: 1. The modulus was the biggest impact factor for modal and natural frequency of cylindrical gears with arcuate tooth trace, then tooth width, and the radius of tooth line have the minimum influence; 2. When the modulus increased, natural frequency of cylindrical gears with arcuate tooth reduced rapidly; 3. When the tooth width increased, natural frequency of cylindrical gears with arcuate tooth has a tendency to rise except for first-order modal; 4. The influence of radius of tooth line can be basic ignored; 5. The second-order modal and third-order modal, fifth-order modal and sixth-order modal was very close. The research on cylindrical gears with arcuate tooth trace in this paper has a certain reference value on gear design and selection.


Introduction
Gear drive is widely used in mechanical equipment and has the characteristic of compact structure.There are three form gear used regularly.But they all have some disadvantage because of the gear structure.So, the Japanese scholars, Kazuo Inoue successively have proposed finish machining method such as grinding teeth and burnishing teeth.Because of long contact line, high contact ratio, smooth transmission, high bearing capacity, good lubrication performance and a serial of advantages, cylindrical gears with arcuate tooth trace has been researched by many scholars all over the world [1][2][3][4][5][6][7][8][9][10].Thus, there are many difficult with the processing and design, the cylindrical gears with arcuate tooth trace has not been widely applied [2][3].
Tseng has built mathematical model of arc gear, but not studied its bending characteristics with his partners and Wilcox et al have made different gears' stress analysis by finite method, but they don't specifically research that how the important parameters such as tooth line radius influence [4][5][6][7].Professor Chen-min has made some breakthrough in analysis of arc gear forming principle, meshing performance and carrying capacity, but the modal was not analyzed [8][9].Song Aiping from Yangzhou university china researched the meshing mechanism and characteristics on cylindrical gears with arcuate tooth trace, and put forward the device the four connecting rod translational processing mechanism [1,[10][11][12][13].Shaojiang Wang and Huajun Xiao just derived mathematical equation of arc gear [14][15].
In this paper, how the modal distribution of cylindrical gears with curvilinear shaped teeth was researched.First, the 3-d model of cylindrical gears with curvilinear shaped teeth was established in UG NX8.0.Then, ANSYS14.0 Workbench was used to calculate 1-6 order modal of the gear under different parameters.At last, the analyze result was deal by Mat-lab2010a through curve fitting.Based on the analysis data and fitting curve, find cylindrical gears with curvilinear shaped teeth modal (frequencies) distribution.

Surface Equation and Modal Analysis 2.1 Forming principle and geometric parameters
The cylindrical gears with arcuate tooth trace can be processed by parallel linkage processing device as shown in Fig.
1 [13].Geometric parameters on ketch plane of cylindrical gears with arcuate tooth trace were processed by parallel linkage as shown in Fig. 2. As shown in Fig. 2, the ideal geometric parameters of cylindrical gears with arcuate tooth trace is: the tooth line radius must equal between convex tooth surface and concave tooth surface; then the circumferential tooth thickness, circumferential tooth width and pressure angle also must equal on reference circle [12].From Fig. 2: Because the radius of tooth line of convex tooth surface equal to concave tooth surface, a good wire mesh could achieve.The main geometric parameters of cylindrical gears with arcuate tooth trace were shown in Table 1.

Tooth surface equation of cylindrical gears with arcuate tooth trace
As shown in Fig. 3, tooth flank ∑ is formed since involute tooth profile T h of some radial cross-section scans along tooth line S of base cylinder.The coordinate systems S 1 (O 1 -X 1 Y 1 Z 1 ) is established, the plane X 1 O 1 Y 1 through the middle cross-section of base cylinder and the Z-axis through the axis of base cylinder.S h (O h -X h Y h Z h ) is location coordinates systems of tooth line, and h is the distance from some point on the tooth line to the middle cross-section.R b is the radius of base cylinder, and β is the position of arc tooth line angle.Section between plane X h O h Y h and tooth surface ∑ is shown in Fig. 3 [1-3].It's a involute.So the tooth surface equation could be represented by the following three steps.
, tooth surface equation of convex tooth surface could be obtained: x

y z M x y z =
The transformation matrix M 1h between coordinate S h and S 1 is: (2) Where: In the same way, tooth surface equation of concave tooth surface could get through transformation of coordinates.

Brief introduction to modal analysis
From the elastic mechanics, the differential equation of gear system is: Where: X ---the displacement vector, X = [x 1 , x 2 , ··· x n ] T ; X ̍ ---the velocity vector; X ̎ ---the acceleration vector; F(t) ---the vibration force vector; M ---the mass matrix; C ---the damping matrix; K ---the stiffness matrix; If there is not vibration force (F(t)=0), it's a free vibration system.Because the damping force can be ignored at this time, so the vibration equation can be written as: The corresponding characteristic equation is: Where, ω i is the natural frequency of the i-order modal for the system.
Vibration system generally has n individual natural frequency and main vibration mode.Each pair of frequency and vibration model represents a free vibration of single freedom system.The basic vibration characteristic of free vibration structure is called the modal of the structure.To multipledegree-of-freedom system, free vibration can be decomposed into n harmonic vibration of single degree of freedom.It means that multi degree of freedom system in general is not a natural frequency of free vibration, instead of doing multiple harmonic vibration of composite natural frequency vibration [16][17][18][19].

Modal Analysis of Cylindrical gears with arcuate tooth trace
To research the modal of cylindrical gears with curvilinear shaped teeth with arcuate tooth trace and its impact factor, principle of single variable was used.The influence between modal and tooth line radius was researched at first, the tooth width was the next, and the modulus was the last one.In the process of the finite element analysis, the three-dimensional model of gears was always established in the environment of UG8.0, then the three-dimensional model will be import to ANSYS14.0 Workbench for finite element analysis.In addition to the three research factor, the other parameters are selected as shown in Table 2. Poisson's ratio ν 0.3

The modal influence of cylindrical gears with arcuate tooth trace by tooth line radius
To research on the relationship between modal of cylindrical gears with arcuate tooth trace and tooth line radius, take the module of gear m = 4 , tooth width B = 46mm and the tooth line radius as shown in Table 3.
The 1-order modal to 6-order modal is shown in Fig. 4 where R=89mm.Gear modal distribute with tooth line radius is just shown in Table 4.

The modal influence of cylindrical gears with arcuate tooth trace by tooth width
To research on the relationship between the modal of cylindrical gears with arcuate tooth trace and tooth width, take the module of gear m = 4, radius of tooth line R T =127mm and the tooth width as shown in Table 5.The 1-order modal to 6-order modal is shown in Fig. 5 where B=24mm.Gear modal distribute with tooth width is just shown in Table 6.

The modal influence of cylindrical gears with arcuate tooth trace by modulus
To research on the relationship between the modal of cylindrical gears with arcuate tooth trace and the modulus, take the tooth width of gear B = 46mm, radius of tooth line R T =127mm and the modulus as shown in Table 7.The 1-order modal to 6-order modal is shown in Fig. 6, where m=6.Gear modal distribute with modulus is just shown in Table 8.

Data Analysis and Discussion
Curve fitting method was taken to deal with the data in Table 4, Table 6 and Table 8.Fitting result is shown in Fig. 7, Fig. 8 and Fig. 9. Fig. 7 use linear fitting, then Fig. 8 and Fig. 9 use 6 times polynomial fitting.In the figure, the 1-order modal was expressed by circle, the 2-order modal was expressed by cross symbols, the 3-order modal was expressed by fork symbols, the 4-order modal was expressed by prismatic, the 5-order modal was expressed by hexagram, and the 6-order modal was expressed by left triangle.Figure 7(a) shows that the natural frequency and modal parameters have a few changes with different tooth line radius, tooth line radius have limited influence on modal of cylindrical gears with arcuate tooth trace.From the Fig. 7(b) (c), we can find: when tooth line radius changes, there are very approximate between 2-order modal and 3-order modal, also 5-order modal and 6-order modal From the Fig. 8(a): 1-6 order modal of gears with arcuate tooth trace have a trend to larger with the tooth width increase.Tooth width has a big influence on 1-order modal when B<30mm.While the tooth width B>30mm, the tooth width has little influence on 1-order modal.The tooth width always has an obvious influence on 2-6 order modal.When B>45mm, the 5-order modal and 6-order modal no longer increases basic.From the Fig. 8(b)-(c), we also find: when tooth line radius changes, there are very approximate between 2-order modal and 3-order modal, also 5-order modal and 6-order modal.
From the Fig. 9(a): 1-6 order modal of gears with arcuate tooth trace have a trend to decline with the modulus increase.The modal falls faster where m<4, then the downward trend slow down while 4<m<8.The modal and natural frequency tends to stable while m>10.From the Fig. 9(b)-(c), we also find: when tooth line radius changes, there are very approximate between 2-order modal and 3-order modal, also 5-order modal and 6-order modal.(1) Processing forming method, tooth surface equation and the modal analysis of cylindrical gears with arcuate tooth trace was basiclly introduced.The parameters of ideal cylindrical gears with arcuate tooth trace were the radius of tooth line must equal between convex tooth surface and concave tooth surface; then the circumferential tooth thickness, circumferential tooth width and pressure angle also must equal on reference circle.(2) The 1-6 order modal was analyzed by finite element method for cylindrical gears with arcuate tooth trace with different parameters.And the Matlab2010a was used to deal the modal analysis data.(3) Analysis shows that: The modulus was the biggest impact factors for modal and natural frequency of cylindrical gears with arcuate tooth trace, then tooth width, and the tooth line radius have the minimum influence.When the modulus increased, natural frequency of cylindrical gears with arcuate tooth reduced rapidly.When the tooth width increased, natural frequency of cylindrical gears with arcuate tooth has a tendency to rise except for first-order modal.The influence of radius of tooth line can be basic ignored.There are very approximate between 2-order modal and 3-order modal, also 5-order modal and 6-order modal.

Fig. 3
Fig. 3 Tooth profile of cylindrical gears with arcuate tooth trace

Fig. 7 Fig. 6 Fig. 9
Fig. 7 The modal distribution figure with different tooth line radius

Table 1
Main geometric parameters of cylindrical gears with arcuate tooth trace t =S c =P/2 Circumferential tooth width e e=P c =P t =P/2 Tooth width B

Table 2
Parameter table of cylindrical gears with arcuate tooth trace

Table 3
Different tooth line radius of cylindrical gears with arcuate tooth

Table 4
Different modal distribution with tooth line radius

Table 5
Different tooth width of cylindrical gears with arcuate tooth trace

Table 6
Different modal distribution with tooth width

Table 7
Different modulus of cylindrical gears with arcuate tooth trace

Table 8
Different modal distribution with modulus