Static and dynamic analysis of spur and bevel gears using FEM

doi:10.1016/S0094-114X(97)00112-2

Mechanism and Machine Theory

Volume 33, Issue 8, November 1998, Pages 1177-1193

https://doi.org/10.1016/S0094-114X(97)00112-2 Get rights and content

Abstract

This paper presents the findings of three-dimensional stress analysis of spur and bevel gear teeth by finite element method using cyclic symmetry concept. The displacement of a tooth is computed for each Fourier harmonic component of the contact line load and all the components are added to obtain the total displacement. This displacement is used in the calculation of static stress in the teeth. The natural frequencies and mode shapes are obtained using the submatrices elimination scheme.

Zusammenfassung

Dieser Bericht stellt die Ergebnisse dreiachsiger Spannungsanalyse der Spur- und Tellerradzähne durch die Begrenztelementenmethode mittels des zyklischen Symmetriekonzepts dar. Die Verschiebung eines Zahnes wird für jede der Fourier-harmonischen-Teilschwingungen der Fahrleistungsbelastung berechnet, und alle Teilschwingungen werden summiert, um die gesamte Verschiebung zu erhalten. Diese Verschiebung wird dann zur Berechnung der Ruhepunktspannung der Zahnung benutzt. Die Eigenschwingungszahl und die Modenformen werden mittels des Eliminationsschemas der Submatrizen erhalten.

Section snippets

Nomenclature

a_jk	force acting on the kth degree of freedom of jth substructure
f_pk	pth Fourier harmonic of forces
m	number of degrees of freedom in each substructure
N	number of sectors
[K]	stiffness matrix
{δ}	displacement vector
{δ_j}	displacement vector of jth substructure
{δ^′_p}	displacement of the first substructure of pth harmonic

Static analysis

A gear wheel is made of substructures which have identical geometry and material properties and are connected to each other in the circumferential direction. Thus, a gear may be considered a rotationally periodic structure. If cyclic symmetry concept is used in the analysis of such structures, a significant reduction in computational effort can be effected.

In the present analysis, only one substructure (tooth) is assumed to have a contact line load. A line load on one substructure and zero load

Frequency analysis

The mathematical approach to the concept of cyclic symmetry used in the frequency analysis of rotationally periodic structures has been discussed in[7]. The use of cyclic symmetry concept effects a large saving in memory and reduction in computational effort. The method proposed by Balasubramanian and Ramamurti[8] is extended to the free vibration of the spur gear tooth. The simultaneous iteration scheme is modified to compute the Hermitian eigen value problem. An out of core technique is

Spur gear

A load of 500 N per mm width is applied first at the tip of the tooth and then at two other points along the profile. The stress distribution along the profile of the tooth for the three load cases is shown in Fig. 3. The points M and N indicate the stress concentration at the point of application of load. The stress distribution along the root section is shown in Fig. 4. As can be seen the nature of radial stress changes from tension on the loaded side to compression on the unloaded side along

Conclusion

This analysis demonstrates the use of cyclic symmetry concept in the finite element analysis of a rotationally periodic structure like spur gear. Application of this approach results in large saving in computer memory and reduction of computational effort. This efficient approach can also be used in the dynamic analysis of gear tooth utilising the geometrical periodicity and the submatrices elimination scheme.

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