Modified Lagrangian Formulation of Gear Tooth Crack Analysis using
  Combined Approach of Variable Mode Decomposition (VMD) and Time Synchronous
  Averaging (TSA)
Subrata Mukherjee Vikash Kumar Somnath Sarangi
This paper discusses the possible observation of an integrated gear tooth
crack analysis procedure that employs the combined approach of variable mode
decomposition (VMD) and time synchronous averaging (TSA) based on the coupled
electromechanical gearbox (CEMG) system. This paper also incorporates the
modified Lagrangian formulation to model the CEMG system by considering
Rayleigh's dissipative potential. An analytical improved time-varying mesh
stiffness (IAM-TVMS) with different levels of gear tooth crack depts is also
incorporated into the CEMG system to inspect the influence of cracks on the
system's dynamic behavior. Dynamic responses of the CEMG system with different
tooth crack levels have been used for further investigations. For the first
time, the integrated approach of variable mode decomposition (VMD) and
time-synchronous averaging (TSA) has been presented to analyze the dynamic
behaviour of CEMG systems at the different gear tooth cracks have been
experienced as non-stationary and complex vibration signals with noise. Based
on the integrated approach of VMD-TSA, two types of nonlinear features, i.e.,
Lyapunov Exponent (LE) and Correlation Dimension (CD), were calculated to
predict the level of chaotic vibration and complexity of the CEMG system at the
different levels of gear tooth cracks. Also, the LE and CD are used as chaotic
behaviour features to predict the gear tooth crack propagation level. The
results of the proposed approach show significant improvements in the gear
tooth crack analysis based on the chaotic features. Also, this is one of the
first attempts to study the CEMG system using chaotic features based on the
combined approach of VMD-TSA.