DYNAMIC LOADS IN SPUR GEAR PAIRS

Virgil ATANASIU, Dumitru LEOHCHI

Technical University "Gh. Asachi" Iaşi

Abstract: The paper presents a dynamic tooth load analysis of spur gears with addendum modification. The analytical model is developed to simulate the load sharing characteristics through a mesh cycle. The model takes into account the main internal factors of dynamic load as time-varying mesh stiffness and composite tooth profile errors. The complicated phenomenon of contact tooth pairs alternation during mesh cycle is integrated in this dynamic load modeling. A comparative study is included, which shows the effects of the factors with an important role in the way of the dynamic load variation.

Keywords: spur gears, dynamic loads, mesh stiffness, profile error, addendum modification

1. Introduction The level of vibration and noise of the gear pairs

is in correlation with the characteristics of dynamic load. Calculation of dynamic loads and determination of their variation during a mesh cycle for spur gears pairs has been considered a major aspect of gear design. An analytical model that covers the main influence factors with sufficient accuracy is not currently available.

This paper presents an analytical approach used in order to predict the characteristics of dynamic loads by considering the time-varying mesh stiffness and variable tooth profile errors. The calculus procedure of the mesh stiffness is found by using an exact analytical model.

2. Dynamic Load Analysis During the engagement cycle, the contact load

does not remain constant. The load variation is mainly caused by the following factors: (i) the alternating engagement of single and double pairs of teeth; (ii) the variation of the mesh stiffness along

the line of action; (iii) the deviation of the tooth profile from the theoretical involute profile. In order to build an accurate analytical model of the dynamic tooth load sharing, the parameters used in the model need to be estimated correctly.

2.1. Mesh Stiffness The gear tooth is modeled to be a nonuniform

cantilever beam supported by a flexible fillet region and foundation [1] as shown in Figure 1.

Fig.1

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The total deflection fj of a pair of meshing teeth is expressed as

∑ ∑= =

++=2

1

2

1j jHfjbjj ffff (1)

where fb - the deflection due to bending, shear and axial deformation of the tooth corresponding to the involute profile; ff - the deflection due to the flexibility of the tooth foundation and fillet; fH - the local compliance of the Hertzian contact. The individual tooth mesh stiffness is defined in the normal direction to the contact surface as

j

nj f

Fk = (2)

where Fn is the normal tooth load .

Fig.2

The teeth pairs in contact act like parallel springs. Therefore, the total mesh stiffness during each engagement cycle can be written as a function of the position of contact point on the action line

IIs

Ist KKK += , for double-tooth contact Ist KK = , for simple - tooth contact

where I and II are the mating points of the teeth pairs (Fig. 2). The time-varying mesh stiffness is mainly caused by the following factors: (i) the variation of the single mesh stiffness along the equivalent line of action; (ii) the fluctuation of the total number of total pairs in contact during the engagement cycle.

2.2. Tooth Profile Error The tooth profile error is defined as the distance

between the theoretical involute profile and the real tooth profile in the normal direction. The profile error function ei(t) due to manufacturing can be defined as

( ) ( )α+ω= tEte zii sin (3) where ωz is the mesh frequency and α is the phase angle. The composite error es is the sum of tooth errors of the pinion and gears. The gear tooth deflections can be considered as profile errors for the pinion and gear causing premature engagement and delayed disengagement. Tooth profile errors are added to the theoretical profile in normal directions .

2.3. Dynamic Model

The vibration model for one pair of teeth in contact is shown in Figure 3. In this model, the teeth are considered as springs and the gear blanks as inertia masses.

The internal forces of the model are the inertia force xm && , damping force xc& and the dynamic load Fdi.

In Figure 3, for a pair of contacting teeth i, the time-varying mesh stiffness ki(t) and the composite tooth profile error ei(t) act as parameter excitations. The total sum of internal forces must be equal to the external force Fn. Hence, the equation of motion can be expressed as

( ) nN

idi FtFxcxm =++ ∑

=1&&& (4)

where xd is the dynamic displacement and Nrepresents the number of simultaneous tooth pairs in mesh.

( ) ( ) ( )[ ]texxtktF idsidi ++= (5)

( ) ( )tFif,tF didi 00 <=

where xd is the dynamic displacement and xsrepresent the static displacement.

Fig.3

3. Numerical Results and Discussions

The design parameters of the analyzed gear pairs are chosen as: z1 =18; z2 = 42; m = 3 [mm], b=25 [mm], a = 90 [mm]; damping ratio, ξ = 0.12.Specific characteristics of these gear pairs are shown in Table 1.

00.20.40.60.81

1.21.4

0 5 10 15 20

1sc

y [ ]mm

1GAIdC

(c)

Fig.4

Table 1 Gear pair

x1 x2 εα km[N/µm]

fn[Hz]

GA1 0 0 1.63 546.5 3889 GA2 +0.8 -0.8 1.40 418.2 3398

(c)

Fig.5

0100200300400500600700

0 5 10 15 20

iK

µ mN

y

1K2K

mKtK

[ ]mm

1GA

(a)

0.40.60.81

1.21.41.61.8

0 5 10 15 20y [ ]mm

1GAdC

(b)

0100200300400500600

0 5 10 15 20

iK

y

µ mN

1K 2K

mKtK

[ ]mm

2GA

(a)

0.40.60.81

1.21.41.61.8

0 5 10 15 20y

2GA

[ ]mm

dC

(b)

00.20.40.60.81

1.21.4

0 5 10 15 20

1sc

y [ ]mm

2GAIdC

In the analysis of dynamic loads, the transmitting load is defined as W=q(Fn/b) (6)

where q represents the load factor. A numerical value Fn/b = 100 N/mm corresponding to a medium transmitting load is considered in the numerical analysis.

The dynamic factor is defined as the ratio of the dynamic load to the static load. Figures 4 and 5 show the effect of addendum modification coefficients on the mesh stiffness and dynamic load sharing characteristics through a mesh cycle. An addendum modification that decreases the average mesh stiffness increases the dynamic load factor Cdfor the total load. For a single tooth pair, dynamic load factor CdI decreases with decreasing average mesh stiffness.

The effects of damping coefficient and operational speed on the dynamic factor CdI are presented in Figures 6 and 7. The variation in mesh stiffness acts like a short duration impulse in gear pairs. The influence of this change on tooth dynamic load is more effective at lower speeds than at higher speeds. The number of load oscillation results as a

ratio of the meshing resonance frequency fn and the meshing frequency fz of the gear pair. The effect of both, the mesh stiffness and the composite profile error on the dynamic load variation is shown in Figure 8. The dynamic load fluctuation under different transmitting loads is shown in Figure 9.

4. Conclusions

An analytical investigation of the dynamic loads fluctuation in spur gears is presented. The dynamic model permits to predict the individual tooth load characteristics during the meshing cycle. Dynamic tooth loads along the line of action are computed by using the combined mesh stiffness, composite profile errors, and operating speed.

References

1. Atanasiu,V. An Analytical Investigation of the Time-Varying Mesh Stiffness of Helical Gears Bul. I.P.Iasi, Tomul XLIV, s.V, 1998, p.7-17. 2. Vedmar,L., Henriksson,B. A General Approach for Determining Dynamic Forces in Spur Gears.

Journal of Mech. Design, Trans. of the ASME, 1998, p.593-598.

00.20.40.60.81

1.21.4

4/1f/f nz =0eI =1q =

06.0=ξ

17.0=ξ

A E

2GAIdC

Fig.6

00.20.40.60.81

1.21.4

1q =4/1f/f nz =

0eI =

A E

2GAIdC

Fig.8

00.20.40.60.81

1.21.41.6

0eI =1q =

06.0=ξ

17.0=ξ

A E

2/1f/f nz =2GAIdC

Fig.7

00.20.40.60.81

1.21.41.6 4/1f/f nz =

0eI ≠

5.0q =

5.1q =

2GA

A E

IdC

Fig.9