Calculation of the Combined Torsional Mesh Stiffness of Spur Gears ...

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)11, 810-8183.1.1 Stiffness of the Gear BodyThe stiffness of the gear body is assumedto only depend on the following parameters: shaftradius r s dedendum radius r d , face width w andYoung’s modulus E.A simplified model (see Fig. 8) is usedto analyse the influence of the above mentionedparameters on the body’s stiffness. Differentcombinations of parameters were used to come tothe following Eq.:( ) ⋅16 . 16 .BP , B d s sK = c ⋅E⋅w⋅ln r − r r ,(3)where c B is a coefficient which was found out tobe 9.555e -4 .The results from this equation reproducethe results from the FE model within an accuracyof about 5% for gear bodies with a outer radiusbetween 10 and 200 mm and various inner radii.where c T is a coefficient which was found out tobe 3.2e -5 . The results from this Eq. are within 7%of the results from the FE model for the parameterrange analysed.3.1.3 Stiffness of the Teeth ContactIn contrast to the stiffness mentionedabove, the stiffness of the contact between themeshing teeth is highly non-linear with the loadas it is a Hertzian contact between two curvedsurfaces. Therefore, the load torque T needs to beconsidered in the equation describing the contactstiffness. In addition, the following parametershave been found out to have a significantinfluence: modulus m and number of teeth z,contact radius, Young’s modulus E and face widthw. The contact stiffness is approximated by:K = c ⋅E⋅w⋅m ⋅z ⋅TCP ,C185 . 2 0.105,(5)where the coefficient c C is 7.937e -5 . The resultsfor the contact stiffness were found to be within10% of the results from the FE model.3.2 Stiffness for the Driven Gear with a SinglePair of Teeth in ContactFig. 8. Simplified model of the gear body withapplied constraints and forces3.1.2 Bending Stiffness of the TeethAs the teeth basically bend under load,the assumption is made that the parameters thatinfluence the stiffness of the teeth are the sameas those of a bending beam. These parametersare: height and width w of the teeth and Young’smodulus E. The influence of the radius at whichthe tooth is located and the tooth height was takeninto account with the parameters modulus m andnumber of teeth z. This results in the stiffness ofthe tooth K T,P :K = c ⋅E⋅w⋅m ⋅ zTP ,T2 22 .,(4)The stiffness values determined aboveare related to the driving gear (index P). Withthe given gear ratio u, the driven gear’s stiffness(index G) can be directly deduced taking intoaccount that both load torque and radius of contactare u-times the respective torque and contactradius of the driving gear. So the body, teeth andcontact stiffnesses are calculated by:2 zPKiG , = KiP , ⋅ u = KiP, ⋅ ⎛ ,⎝ ⎜ ⎞⎟ (6)zG⎠with i being the indices B, T and C for body,teeth and contact. It has to be mentioned that theparameters of the driven gear have to be used forthe calculation of its stiffness (e.g. z G , E G ).3.3 Stiffness in the Double Contact ZoneThe body’s stiffness value in the doublecontact zone is obtained by adding the factor f B .The only difference between single and doublecontact for the body stiffness is the width of the2Calculation of the Combined Torsional Mesh Stiffness of Spur Gears with Two- and Three-DimensionalParametrical FE Models815