Abstract
The exact geometry of tooth meshing is incorporated into a dynamical non-linear model of the considered gear system, in consideration of the effect of pitch errors, tooth separation, DOF-coupling, and profile modifications. Various possible combinations of error distributions and profile corrections are applied to the gear model, which is simulated dynamically to calculate the load factor.
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Abbreviations
- \( \theta_{\text{i}} \) :
-
Angular position of gear i (additional subscripts: n, ref defined in the text)
- \( {\mathbf{s}}_{\text{i}} \) :
-
Deflection vector of DOF i
- \( \theta_{\text{s}} \) :
-
Slip angle
- \( \updelta_{\text{k}} \) :
-
Angular interference of tooth pair k
- \( {\text{i}}_{12} \) :
-
Transmission ratio
- \( {\text{I}}_{12} \) :
-
Directional index (equal to 1 for external gears, −1 for internal gears)
- \( {\mathbf{r}}_{\text{i}} , \) \( {\mathbf{r}}_{\text{i,k}} \) :
-
Position vector of a contact point in relation to centre of gear i (the optional k index refers to a specific tooth pair)
- \( {\mathbf{f}}_{\text{i}} \) :
-
Vector function of tooth profile of gear i
- \( {\mathbf{a}}_{12} \) :
-
Centre distance vector
- \( {\mathbf{R}} \) :
-
Generic rotary translation matrix
- \( {\hat{\mathbf{x}}}_{\text{i}} \) :
-
Unitary vector along the \( {\text{x}}_{\text{i}} \) direction, where i=1, 2, 3 (in Cartesian coordinates)
- \( {\mathbf{n}}_{\text{k}} \) :
-
Normal unitary vector at contact point of tooth pair k
- \( {\mathbf{m}}_{\text{k}} \) :
-
Unitary vector along the direction of instant sliding velocity of tooth pair k
- \( \sigma^{{({\text{j}})}} ,\,\sigma_{\text{o}} ,\,\sigma_{\text{or}} \) :
-
Anticipated indexing error of tooth j, maximum anticipated indexing error, maximum real indexing error
- \( {\text{U}}\sigma ,\,{\text{L}}\sigma \) :
-
Upper and lower tolerance for the maximum indexing error \( \sigma_{\text{o}} \)
- \( {\text{m,m}}_{\text{r}} \) :
-
Prescribed modification (equal to maximum slip angle \( \theta_{\text{s}} \)), actual modification
- \( {\text{Um,\,Lm}} \) :
-
Upper and lower tolerance for the modification \( {\text{m}} \)
- \( {\text{k}}_{\text{k}} \) :
-
Instant stiffness of individual tooth pair k
- \( {\text{c}}_{\text{hyst}} \) :
-
Damping coefficient due to tooth material hysterisis
- \( {\text{f}}_{\text{k}} \) :
-
Instant friction coefficient of individual tooth pair k
- \( {\text{F}}_{\text{k,elast}} \) :
-
Elastic component of the contact force of tooth pair k
- \( {\mathbf{F}}_{\text{k,hyst}} \) :
-
Hysteretic component of the contact force of tooth pair k
- \( {\mathbf{F}}_{\text{k,frict}} \) :
-
Frictional component of the contact force of tooth pair k
- \( {\mathbf{M}}_{\text{i}} \) :
-
Mass matrix of rotating element i
- \( {\mathbf{C}}_{\text{i}} \) :
-
Damping coefficient for bending of shaft i due to hysterisis
- \( {\mathbf{K}}_{\text{i}} \) :
-
Lumped bending stiffness matrix of DOF i (shaft with elastic supports: bearings/ housing)
- \( {\text{J}}_{\text{i}} \) :
-
Mass moment of inertia of rotating element i
- \( {\text{D}}_{\text{i}} \) :
-
Damping coefficient related to rotation of DOF i (i.e. windage)
- \( {\text{E}}_{{{\text{i}} - {\text{j}}}} \) :
-
Damping coefficient for torsion of shaft segment i − j due to hysterisis
- \( {\text{G}}_{{{\text{i}} - {\text{j}}}} \) :
-
Torsional stiffness of shaft segment i − j.
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Spitas, C., Spitas, V., Rajabalinejad, M. (2013). Dynamical Simulation and Calculation of the Load Factor of Spur Gears with Indexing Errors and Profile Modifications for Optimal Gear Design. In: Dobre, G. (eds) Power Transmissions. Mechanisms and Machine Science, vol 13. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6558-0_13
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DOI: https://doi.org/10.1007/978-94-007-6558-0_13
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