Modelling and Dynamic Analysis of an Unbalanced and Cracked Cardan Shaft for Vehicle Propeller Shaft Systems  
[PERSON]  
1 The vibrational behaviour of misaligned rotating machinery is described and analysed in this paper. The model, constructed based on the equations of vehicle dynamics, considered the dynamic excitation of a single Hooke's joint. The system adopted the breathing functions from a recent publication to approximate the actual breathing mechanism of a cracked driveshaft. The study aimed to understand the transmission of a nonlinear signal from the unbalanced and cracked driveshaft to an unbalanced driven shaft via a Hooke's joint. The governing equation of the system was established based on the energy principle and the Lagrangian approach. The instantaneous frequency (IF) identification of the cracked driveshaft was extracted based on the synchrosqueezing wavelet technique. To correlate the results, the nonlinear synchrosqueezing wavelet transforms combined with the classical waves techniques were experimentally used in various scenarios for dynamic analysis of the Cardan shaft system. The variations in the dynamic response in the form of a rising trend of higher harmonics of rotational frequency and increased level of sub-harmonic peaks in both shafts were presented as significant crack indicators. The synchrosqueezing response showed breathing crack excitation played a crucial role in the mixed faults response and caused divergence of the vibration amplitudes in the rotor's deflections. The simulation and test results demonstrated that the driveshaft damage features impacted the transfer motion to the driven shaft and the Hooke's joint coupling was the principal source of instability in the system. The proposed model offers new perspectives on vibration monitoring and enhancement analysis to cover complex Cardan shaft systems. 1 Theoretical  
may experience. Cardan joint has a wide range of applications in the automobile industry for transmitting rotary motion. However, when it comes to mechanical failures, faults such as unbalance and crack may contribute to premature wear of the joints if not early detected. Unfortunately, in modern machinery, due to high rotation and periodical fluctuation speed, the shafts system connected by a Hooke's joints is often challenging, making their analysis one of the most complex and most overlooked. The industry is continuously seeking scientific innovations and techniques to improve the lifespan of rotating machinery and equipment, as well as a new technology for monitoring and maintaining rotor systems. These systems composed of rigid or flexible misaligned shafts connected by joints, transmit the rotational motion to the driven shafts. In most published studies, the Cardan shaft systems are considered with a maximum of two degrees of freedom (DOF) lumped-mass models consisting of torsionally elastic, massless shafts. However, for reliable dynamic system analysis and prediction of the instability characteristics of these shaft systems, studies of a complex system cannot be expected to be successful using such approximate low-dimensional models. Many studies have investigated the dynamic characteristics of complex Cardan joint and misalignment in the transmission system. From a single to multiple DOF, a more realistic model of a multi-shaft system consisting of n shafts interconnected via Hooke's joints which results in the n-DOF system has been modelled for the offset shaft system [7; 8; 9; 10]. The mathematical model consisted of a set of coupled, linear partial differential equations with time-dependent coefficients. These proposed model systems demonstrated via numerical simulations that the coupled rotor system might display chaotic behaviour under specific conditions. Unlike the numerical studies mentioned, researchers have investigated numerically and validated experimentally the dynamic stability and vibration analysis of torsional vibrations of a shaft system connected via a Hooke's joint.