Model based diagnostic tool for detection of gear tooth crack in a wind turbine gearbox under constant load

Abstract

Dynamic modelling of wind turbine drive-train system (WTDS) is important to analyse the behaviour of the vibration and load sharing characteristics of the gearbox. The WTDS mostly contains one planetary gear stage and two parallel gear stages. Sun gear is a significant component in the planetary gear stage and is prone to fail under fatigue loading due to the bending and shear effect. Crack is most likely to occur in the gear tooth root and it decreases the tooth thickness and load-carrying capacity of the gear. The presence of crack decreases the tooth contact number and tooth contact position during the time of meshing. This leads to a variation and quantified reduction in the time varying mesh stiffness (TVMS) of a meshing gear pair. Hence, a dynamic model is developed by including different crack depth levels in the sun gear. The governing equations for the model are derived using Lagrange’s formulation and dynamic responses are obtained numerically by step-by-step direct integration method. The dynamic responses like contact forces and angular velocities are estimated at 0%, 10%, and 30% of crack depth in the sun gear tooth and analysed to study the dynamic behaviour of the drive-train for un-cracked and cracked gears beforehand.

Appendix

$$K=\left[\begin{array}{ccccccccccc}{K}_{\mathrm{1,1}}& {K}_{\mathrm{1,2}}& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ {K}_{\mathrm{2,1}}& {K}_{\mathrm{2,2}}& {K}_{\mathrm{2,3}}& {K}_{\mathrm{2,4}}& {K}_{\mathrm{2,5}}& {K}_{\mathrm{2,6}}& 0& 0& 0& 0& 0\\ 0& {K}_{\mathrm{3,2}}& {K}_{\mathrm{3,3}}& 0& 0& {K}_{\mathrm{3,6}}& 0& 0& 0& 0& 0\\ 0& {K}_{\mathrm{4,2}}& 0& {K}_{\mathrm{4,4}}& 0& {K}_{\mathrm{4,6}}& 0& 0& 0& 0& 0\\ 0& {K}_{\mathrm{5,2}}& 0& 0& {K}_{\mathrm{5,5}}& {K}_{\mathrm{5,6}}& 0& 0& 0& 0& 0\\ 0& {K}_{\mathrm{6,2}}& {K}_{\mathrm{6,3}}& {K}_{\mathrm{6,4}}& {K}_{\mathrm{6,5}}& {K}_{\mathrm{6,6}}& {K}_{\mathrm{6,7}}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -{k}_{t2}& {K}_{\mathrm{7,7}}& {K}_{\mathrm{7,8}}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& {K}_{\mathrm{8,7}}& {K}_{\mathrm{8,8}}& {K}_{\mathrm{8,9}}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& {K}_{\mathrm{9,8}}& {K}_{\mathrm{9,9}}& {K}_{\mathrm{9,10}}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& {K}_{\mathrm{10,9}}& {K}_{\mathrm{10,10}}& -{k}_{t3}\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& -{k}_{t3}& {k}_{t3}\end{array}\right]$$

where, ${K}_{\mathrm{2,2}}={k}_{t1}+\left({k}_{rp1}{+k}_{rp2}+{k}_{rp3}\right){R}_{r}^{2}+\left({k}_{sp1}{+k}_{sp2}+{k}_{sp3}\right){R}_{s}^{2}+{k}_{bc}$, ${K}_{\mathrm{7,7}}=\left({k}_{t2}+{k}_{g12}{R}_{1}^{2}\right)$, ${K}_{\mathrm{6,7}}={K}_{\mathrm{7,6}}=-{k}_{t2}$, ${{K}_{\mathrm{1,2}}=K}_{\mathrm{2,1}}={-k}_{t2}$, ${{K}_{\mathrm{8,9}}=K}_{\mathrm{9,8}}={-k}_{t2}$, ${{K}_{\mathrm{10,11}}=K}_{\mathrm{11,10}}={-k}_{t3}$, ${{K}_{\mathrm{2,3}}=K}_{\mathrm{3,2}}=\left({k}_{rp1}{R}_{p}{R}_{r}-{k}_{sp1}{R}_{p}{R}_{s}\right)$, ${K}_{\mathrm{6,6}}={k}_{t2}+\left({k}_{sp1}{+k}_{sp2}+{k}_{sp3}\right){R}_{s}^{2}+{k}_{bs}$, ${K}_{\mathrm{10,10}}=\left({k}_{t3}+{k}_{g34}{R}_{4}^{2}\right)$, ${{K}_{\mathrm{2,4}}=K}_{\mathrm{4,2}}=\left({k}_{rp2}{R}_{p}{R}_{r}-{k}_{sp2}{R}_{p}{R}_{s}\right)$,${K}_{\mathrm{1,1}}={k}_{t2}$,${K}_{\mathrm{11,11}}={k}_{t3}$, ${{K}_{\mathrm{2,5}}=K}_{\mathrm{5,2}}=\left({k}_{rp3}{R}_{p}{R}_{r}-{k}_{sp3}{R}_{p}{R}_{s}\right)$, ${K}_{\mathrm{2,6}}={K}_{\mathrm{6,2}}=-\left({k}_{sp1}{+ k}_{sp2}+{k}_{sp3}\right){R}_{s}^{2}$,${{K}_{\mathrm{5,6}}=K}_{\mathrm{6,5}}={k}_{sp3}{R}_{p}{R}_{s}$, ${K}_{\mathrm{9,9}}=\left({k}_{t3}+{k}_{g34}{R}_{3}^{2}\right)$, ${K}_{\mathrm{4,4}}=\left({{k}_{rp2}+k}_{sp2}\right){R}_{p}^{2}$,${{K}_{\mathrm{7,8}}=K}_{\mathrm{8,7}}=\left({k}_{g12}{R}_{g1}{R}_{g2}\right)$, ${K}_{\mathrm{8,8}}=\left({k}_{t2}+{k}_{g12}{R}_{2}^{2}\right)$, ${K}_{\mathrm{5,5}}=\left({{k}_{rp3}+k}_{sp3}\right){R}_{p}^{2}$, ${{K}_{\mathrm{3,6}}=K}_{\mathrm{6,3}}={k}_{sp1}{R}_{p}{R}_{s}$, ${{K}_{\mathrm{9,10}}=K}_{\mathrm{10,9}}=\left({k}_{g34}{R}_{g3}{R}_{g4}\right)$, ${{K}_{\mathrm{4,6}}=K}_{\mathrm{6,4}}={k}_{sp2}{R}_{p}{R}_{s}$, ${K}_{\mathrm{3,3}}=\left({{k}_{rp1}+k}_{sp1}\right){R}_{p}^{2}$

$$M=\left[\begin{array}{ccccccccccc}{J}_{rot}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& {J}_{c}+3{J}_{p}+3m{R}_{c}^{2}& {J}_{p}& {J}_{p}& {J}_{p}& 0& 0& 0& 0& 0& 0\\ 0& {J}_{p}& {J}_{p}& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& {J}_{p}& 0& {J}_{p}& 0& 0& 0& 0& 0& 0& 0\\ 0& {J}_{p}& 0& 0& {J}_{p}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& {J}_{s}& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& {J}_{g1}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& {J}_{g2}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& {J}_{g3}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& {J}_{g4}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& {J}_{gen}\end{array}\right]$$

$$ \theta = \left[ {\begin{array}{*{20}c} {\theta_{rot} } & {\theta_{c} } & {\theta_{cp1} } & {\theta_{cp2} } & {\theta_{cp3} } & {\theta_{s} } & {\theta_{g1} } & {\theta_{g2} } & {\theta_{g3} } & {\theta_{g4} } & {\theta_{gen} } \\ \end{array} } \right]^{T} $$

$$ \Gamma = \left[ {\begin{array}{*{20}c} {\Gamma_{rot} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\Gamma_{gen} } \\ \end{array} } \right]^{T} $$