Load distribution of involute gears along time-varying contact line

LI Wen-liang，WANG Li-qin，GU-le李文良， 王黎钦， 古 乐

(School of Mechanical Engineering，Harbin Institute of Technology，Harbin 150001，China)

High－power gear transmission device is one of the important equipments of the ship turbine system.Large Width Helical Gears are widely adopted，which have the characteristics of complex structure，heavy load capacity，long life and high precision.Heavy gear teeth have larger deformation and stress，which have significant influence on dynamic characteristics.The numerical models are used to predict the important data such as load distribution and the gear transmission error can reduce the cost of gear design especially the cost of test in a real environment.Load distribution and meshing stiffness are function of position due to the change of meshing position and number of teeth.AGMA［1－2］and ISO［3－4］use simple equations which is given by the linear elasticity theory.Nevertheless，these equations are not in good consistent with experimental results if the load is considered as uniformly distributed along the time－varying contact line.So，it is necessary to calculate the load distribution along the contact line of gear pair，the deformation of the gear mesh stiffness，the contact stress and other dynamic characteristics.Hayashi［5］calculated the deflection of gear teeth according to the second kind Fredholm integral equation，and got the load distribution as the numerical solution of the integral equation.Ajmi and Velex［6］proposed an approach to calculate the deflection and load distributions under both quasi－static and dynamic conditions for spur and helical gears.Arafa and Megahed［7］adopted finite element model to calculate the mesh characteristics for spur gears，and discussed the load distribution among the meshing teeth in gear transmission system.

Base on the minimization elastic potential energy theory，the load distribution model is used in this paper.The elastic potential energy of the gear pairs is the function of instantaneous contact point of tooth profile and normal force.Spur gear calculates the load distribution by the minimization elastic potential energy.The same approach may be used for helical gear by dividing the helical gear into infinite slices along the contact lines.

1 Model of Load Distribution

The elastic potential of a spur gear is composed with bending component Ub，the compressive component Ucand the shear component Us:

The elastic potential is a function of profile geometric parameters.The geometry of involute teeth can be seen in Fig.1.

Fig.1 Geometrical parameters of involute tooth

The expression of bending component Ub，the compressive component Ucand the shear component Uscan be written［8］:

where F is normal load，N;αcis load angle of contact point;b is face width，mm;E is modulus of elasticity，MPa;C is shear potential correction factor，1.2［8］;ypand ycare the value of coordinate;G is transverse modulus of elasticity，MPa.

Load angle is the position function of involute profile.It can be written as follows［9］:

where ξcis profile parameter of contact point，which varies with meshing position.

where γbis central angle of base circle of involute profile［9］:

where z is number of teeth;rcis profile radius of contact point;rbis base radius;δ is rack shift coefficient;αtand αnare pressure angle of transverse and normal.

Other parameters expression can be written:

where γ(y)is the central angle of contact point y;r(y)is the profile radius of contact point y;v(y)is the polar angle of the profile.d point is another meshing point，and the relation of rdand rccan be expressed:

where pbis base pitch.

The relationship of two meshing point can be obtained by Eq.(1):

According to the relation between the reverse of elastic potential and contact load for a spur gear，the load distribution is calculated.When gear pair has n meshing teeth at the same time the load distribution formulation of the ith gear as follows［10］:

The load sharing ratio of the ith contact point of spur gear

The helical gear was divided into infinite slices，every slice was deemed to be a spur gear.lcis the length of contact line，fkis the load of per unit of length on slice k，we can write the expression as follows:

The load expression on slice k:

where P is transmitted power， W;ω1is angular velocity of pinion，rad/s;rb1is radius of base circle，mm.vkis the function of the inverse unitary potential of the considered slice k for a common slice，the previous equation was described by ξ and can be written as follows［10］:

where εβis axial contact ratio;βbis base helix angle.

2 Results and Discussion

Tab.1 is the base data of helical gear given by some institute.

Tab.1 Parameters of helical gear

2.1 Load Distribution of Spur Gear

The potential energy is the function of position and depends on the instantaneous meshing position.We obtain the relationship of profile parameter ξ and inverse unitary potential v(ξ)as shown in Fig.2.

Using Eq.(2)to calculate the load distribution，the results are shown in Fig.3.

Fig.2 Relation of Inverse unitary potential and profile parameter

Fig.3 Load sharing ratio for spur gears

The load distribution ratio is not completely symmetric in double teeth meshing zone，and the load distribution ratio is from 0.346 to 0.657 in dedendum and from 0.334 to 0.666 in addendum.The load distribution ratio is 1 in single tooth meshing zone.For a spur gear，the load distribution ratio depends on the meshing position，but not the base parameters such as module，pressure angle etc.From Fig.3，we know that from double meshing zone to single meshing zone，the shocking load will happen and the vibration will rapid increase.ISO 6336－2［11］use the load sharing ratio 0.5 to calculate the critical stress but not 0.3 which is obtained in this paper.

2.2 Load Distribution Along the Contact Line of Helical Gear

Because there are some contact lines when helical gear meshes at the same time，the load distribute depends on the meshing position and the length of contact line.Fig.4 shows the three typical meshing poisons and length of every contact line in one meshing period:the first contact line position is shown in Fig.4(a)，which begins to mesh，and the length is 0;the middle position is shown in Fig.4(b)，when the first contact line runs on middle occasion during one transverse pitch;the end position is shown in Fig.4(c)when the first contact line runs the distance for one transverse pitch，meanwhile the length of the first contact line is maximum.

The load distribution is calculated by Eq.(3)on three occasions.Fig.5 shows the load distribution of every contact line.From Fig.5，we know that the maximum value of load distribution appears at the pitch point(ξ=2.04)along the time－varying contact line.During one period，the length of the first contact line is 0，whose corresponding load is only a point in Fig.5 at initial time.As the length of the first contact line becomes longer，the total load capacity of contact line increases，when the contact line runs at the pitch point the value of load distribution is maximum.From the load distribution of other three contact lines，we know when the contact line locates at the under pitch point the load distribution becomes larger with the increase of the profile parameters.Above the pitch point the load becomes smaller with the increase of tooth profile parameters.From the change law，we know that load distribution depends on the meshing position and the length of contact line.The total load becomes larger with the increase of the length of contact line.

Fig.4 Positions of contact lines

Fig.5 Load distribution along the contact lines

3 Conclusions

Finally，some basic conclusions are made through analysis and studies above all.

1)Base on the theory of energy minimization，a numerical algorithm is used to calculate the load distribution along the time－varying contact line.

2)The load sharing ratio is obtained in double contact zone for a spur gear.The value varies from 0.33 to 0.67，which provides a base theory for calculating critical stress.

3)The maximum value of load distribution for a helical gear appeared at the pitch point.The change of load distribution is very smoothly along the contact line and no load mutation.The change law is good agreement with the conclusion that helical gear has lower vibration and noise than spur gear.

4)From the results obtained for helical gear，the load distribution depends on the meshing position and the length of contact line.According to the results we can calculate the bending and deformation and reduce the cost of experiment in real environment.

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