Rubbing Of a Bladed Disk Considering Coriolis Effect: A Reduced Model Based on Complex Modal Analysis

Due to the high demands of aerodynamic efficiency in modern aero-engines, tip clearances between blades and casings are becoming smaller. This increases the possibility of rubbing between rotating bladed disks and their surrounding casings. Rotational effects exhibit increased significance in the latest generation of fans, which currently have relatively long blade and elongated cantilevered shaft. Previous studies on the rotor dynamics during rub impact have mainly focused on simplified models. However, it is necessary to take both realistic blades and Coriolis effect into account. Based on an open-source bladed disk model, the impact of the Coriolis effect on rub-induced responses is investigated. A two-step model reduction method is adopted by combining the fixed interface reduction and cyclic symmetry reduction. Both centrifugal and gyroscopic effects are incorporated in the numerical model. Complex modal analysis, based on classical Craig-Bampton method, is used to improve the model reduction of the gyroscopic system. The response of a flexible bladed disk to a simplified pulse rubbing force is investigated. With the time and space Fourier transform, a Coriolis-induced frequency split is observed on some nodal diameter lines, which indicates the significance of the Coriolis effect in rub-induced responses. A complex model reduction has been successfully applied to the rub-impact problem of cyclic symmetric bladed disks. Compared with the classical model reduction, the numerical results obtained by complex modal analysis are more reasonable. This lays a solid foundation for further rub-impact research considering rotor dynamics.

Mechanism Analysis of a Low-Frequency Disc Brake Squeal Based on an Energy Feed-In Method for a Dual Coupling Subsystem

Brake squeal is a major component of vehicle noise. To explore the mechanism of the low-frequency brake squeal, a finite element model of an automobile disc brake was established, and a complex mode numerical simulation was performed. According to the unstable modes stemming from the complex modal analysis results, the low-frequency range brake squeal can be determined. Based on an energy feed-in method, the coupling subsystems of the piston-caliper and the disc-pad were established, and a calculation formula for the feed-in energy of the dual coupling subsystem was derived. The results showed that when the feed-in energy of the dual coupling subsystem is close to zero, the complex mode cannot be excited at the corresponding frequency. In addition, the difference in feed-in energy between the two coupling subsystems is positively correlated with the probability of the brake squeal, which can be used to determine the complex mode under which the brake squeal may occur. The greater the feed-in energy of a coupling subsystem is, the more likely it is that the maximum brake vibration mode will appear at this subsystem or its adjacent parts. The increase in brake oil pressure will eliminate some lower-frequency sounds but will not change the frequency of the original low-frequency brake squeals.

Free Vibration of a Taut Cable with Two Discrete Inertial Mass Dampers

Recently, inertial mass dampers (IMDs) have shown superior control performance over traditional viscous dampers (VDs) in vibration control of stay cables. However, a single IMD may be incapable of providing sufficient supplemental modal damping to a super-long cable, especially for the multimode cable vibration mitigation. Inspired by the potential advantages of attaching two discrete VDs at different locations of the cable, arranging two external discrete IMDs, either at the opposite ends or the same end of the cable is proposed to further improve vibration mitigation performance of the cable in this study. Complex modal analysis based on the taut-string model was employed and extended to allow for the existence of two external discrete IMDs, resulting in a transcendental equation for complex wavenumbers. Both asymptotic and numerical solutions for the case of two opposite IMDs or the case of two IMDs at the same end of the cable were obtained. Subsequently, the applicability of asymptotic solutions was then evaluated. Finally, parametric studies were performed to investigate the effects of damper positions and damper properties on the control performance of a cable with two discrete IMDs. Results showed that two opposite IMDs can generally provide superior control performance to the cable over a single IMD or two IMDs at the same end. It was also observed that attaching two IMDs at the same end of the cable had the potential to achieve significant damping improvement when the inertial mass of the IMDs is appropriate, which seems to be more promising than two opposite IMDs for practical application.

Complex modal analysis and coupled electromechanical simulation of energy harvesting piezoelectric laminated beams

In this paper, coupled electromechanical behavior of a vibrational energy harvesting system composed of a unimorph piezoelectric laminated beam with a large attached tip mass is investigated. To achieve this goal, first the electromechanically coupled partial differential equations governing the lateral displacement and output voltage of the harvester are extracted through exploiting the Hamilton’s principle. Considering vibration damping due to mechanical to electrical energy conversion, a complex modal analysis is performed to extract the complex eigenfrequencies and eigenfunctions of the system. Furthermore, an exact analytical solution is presented for the system response to the harmonic base excitations, including output voltage and harvested power. To validate the analytical results, at the next step a finite element simulation is conducted through ABAQUS software. To perform a fully-coupled analysis which brings into account the effect of harvesting circuit, user subroutine User-defined Amplitude (UAMP) is utilized to calculate the voltage–current relation and impose the correct electrical charge on the electrodes in each step by monitoring the output voltage of the system at previous time increments. Results of both analytical and numerical simulations are compared for a Micro-Electro-Mechanical Systems (MEMS) harvester as a case study, where a very good agreement is observed between them.

Effect of Dynamic Soft Elasticity on Vibration of Embedded Nematic Elastomer Timoshenko Beams

This paper addresses the transverse vibration of a nematic elastomer (NE) beam embedded in soft viscoelastic surroundings with the aim to clarify a new dissipation mechanism caused by dynamic soft elasticity of this soft material. Based on the viscoelasticity theory of NEs in low-frequency limit and the Timoshenko beam theory, the governing equation of motion is derived by using the Hamilton principle and energy method, and is solved by the complex modal analysis method. The dependence of vibration property on the intrinsic parameters of NEs (director rotation time, rubber relaxation time, anisotropic parameter) and foundation (spring, shear and damping constants) are discussed in detail. The results show that dynamic soft elasticity leads to anomalous anisotropy of energy transfer and attenuation. The relative stiffer foundation would restraint the rubber dissipation of viscoelastic beams, but has less influence on the director rotation dissipation, which is particular for NE beams. This study would provide a useful guidance in the dynamic design of NE apparatus embedded in soft viscous media.