A vibration model of a rotor system with the sinusoidal waviness by using the non-Hertzian solution

The nonlinear contact forces and deformations between the balls and raceways can cause very complex vibration behaviours of rotor systems with the waviness in the support bearings. However, almost all previous works that used sinusoidal waviness took the Hertzian solution as the calculation method, which is not an accurate method based on Johnson’s formulation since the changes in the curvature at the sinusoidal contact surfaces. To overcome this issue, a new dynamic model of a rigid rotor system with the waviness in the support bearings is proposed. To provide a more accurate nonlinear contact force formulation for the sinusoidal waviness profile, the model used the Johnson’s extended Hertzian contact model to replace Hertzian contact model. This model can consider the time-varying curvature between the mating sinusoidal surfaces. The lubricating condition in the support bearing is also considered. A comparative study on the effects of Hertzian contact model, simplified Hertzian contact model, and Johnson's extended Hertzian contact model on the nonlinear vibrations of the rotor system is developed. The effects of the waviness amplitude and orders on the vibrations of the rotor system are discussed. The comparative simulations show that the proposed model can provide a more reasonable approach for predicting the vibrations of the rigid rotor system. Moreover, the simulations give that the nonlinear contact forces in the support bearings can greatly affect the system vibrations.

Angular momentum and parity projected multidimensionally constrained relativistic Hartree-Bogoliubov model

The nuclear deformations are of fundamental importance in nuclear physics. Recently we developed a multi-dimensionally constrained relativistic Hartree-Bogoliubov (MDCRHB) model, in which all multipole deformations respecting the $V_4$ symmetry can be considered self-consistently. In this work we extend this model by incorporating the angular momentum projection (AMP) and parity projection (PP) to restore the rotational and parity symmetries broken in the mean-field level. This projected-MDCRHB (p-MDCRHB) model enables us to connect certain nuclear spectra to exotic intrinsic shapes such as triangle or tetrahedron. We present the details of the method and an exemplary calculation for $^{12}$C. We develop a triangular moment constraint to generate the triangular configurations consisting of three $\alpha$ clusters arranged as an equilateral triangle. The resulting $^{12}$C spectra are consistent with that from a triangular rigid rotor for large separations between the $\alpha$ clusters. We also calculate the $B(E2)$ and $B(E3)$ values for low-lying states and find good agreement with the experiments.

Modeling the Dynamics of a Gyroscopic Rigid Rotor with Linear and Nonlinear Damping and Nonlinear Stiffness of the Elastic Support

This study analytically and numerically modeled the dynamics of a gyroscopic rigid rotor with linear and nonlinear cubic damping and nonlinear cubic stiffness of an elastic support. It has been shown that (i) joint linear and nonlinear cubic damping significantly suppresses the vibration amplitude (including the maximum) in the resonant velocity region and beyond it, and (ii) joint linear and nonlinear cubic damping more effectively affects the boundaries of the bistability region by its narrowing than linear damping. A methodology is proposed for determining and identifying the coefficients of nonlinear stiffness, linear damping, and nonlinear cubic damping of the support material, where jump-like effects are eliminated. Damping also affects the stability of motion; if linear damping shifts the left boundary of the instability region towards large amplitudes and speeds of rotation of the shaft, then nonlinear cubic damping can completely eliminate it. The varying amplitude (VAM) method is used to determine the nature of the system response, supplemented with the concept of “slow” time, which allows us to investigate and analyze the effect of nonlinear cubic damping and nonlinear rigidity of cubic order on the frequency response at a nonstationary resonant transition.

Bogoliubov theory of a Bose-Einstein condensate of rigid rotor molecules

We consider a BEC of rigid rotor molecules confined to quasi-2d through harmonic trapping. The molecules are subjected to an external electric field which polarizes the gas, and the molecules interact via dipole-dipole interactions. We present a description of the ground state and low-energy excitations of the system including an analysis of the mean-field energy, polarization, and stability. Under large electric fields the gas becomes fully polarized and we reproduce a well known density-wave instability which arises in polar BECs. Under smaller applied electric fields the gas develops an in-plane polarization leading to the emergence of a new global instability as the molecules “tilt”. The character of these instabilities is clarified by means of momentum-space density-density structure factors. A peak at zero momentum in the spin-spin structure factor for the in-plane component of the polarization indicates that the tilt instability is a global phonon-like instability.

Numerical and experimental study on the dynamic bearing properties of a four-pad and eight-pad tilting pad journal bearings in a vertical rotor

In this paper, the dynamics of tilting pad journal bearings with four and eight pads are studied and compared experimentally and numerically. The experiments are performed on a rigid vertical rotor supported by two identical bearings. Two sets of experiments are carried out under similar test setup. One set is performed on a rigid rotor with two four-pad bearings, while the other is on a rigid rotor with two eight-pad bearings. The dynamic properties of the two bearing types are compared with each other by studying the unbalance response of the system at different rotor speeds. Numerically, the test rig is modeled as a rigid rotor and the bearing coefficients are calculated based on Navier-Stokes equation. A nonlinear bearing model is developed and used in the steady state response simulation. The measured and simulated displacement and force orbits show similar patterns for both bearing types. Compared to the measurement, the simulated mean value and range (peak-to-peak amplitude) of the bearing force deviate with a maximum of 16 % and 38 %, respectively. It is concluded that, unlike the eight-pad TPJB, the four-pad TPJB excite the system at the third and fifth-order frequencies, which are due to the number of pads, and the amplitudes of these frequencies increase with the rotor speed.

Experimental research on suppressing unbalanced vibration of rotor by integral squeeze film damper

As a novel structural damper, the unique structural characteristics of the integral squeeze film damper (ISFD) solve the nonlinear problem of the traditional squeeze film damper (SFD), and it has good linear damping characteristics. In this research, the experimental studies of ISFD vibration reduction performance are carried out for various working conditions of unbalanced rotors. Two ball bearing-rotor system test rigs are built based on ISFD: a rigid rotor test rig and a flexible rotor test rig. When the rotational speed of rigid rotor is 1500 rpm, ISFD can reduce the amplitude of the rotor by 41.79%. Under different unbalance conditions, ISFD can effectively improve the different degrees of unbalanced faults in the rotor system, reduce the amplitude by 43.21%, and reduce the sensitivity of the rotor to unbalance. Under different rotational speed conditions, ISFD can effectively suppress the unbalanced vibration of rigid rotor, and the amplitude can be reduced by 53.51%. In the experiment of the unbalanced response of the flexible rotor, it is found that ISFD can improve the damping of the rotor system, effectively suppress the resonance of the rotor at the critical speed, and the amplitude at the first-order critical speed can be reduced by 31.72%.

Wind Tunnel Studies on Hover and Forward Flight Performances of a Coaxial Rigid Rotor

The aerodynamic performance of a reduced-scale coaxial rigid rotor system in hover and steady forward flights was experimentally investigated to gain insights into the effect of interference between upper and lower rotors and the influences of the advance ratio, shaft tilt angle and lift offset. The rotor system featured by 2 m-diameter, four-bladed upper and lower hingeless rotors and was installed in a coaxial rotor test rig. Experiments were conducted in the Φ3.2 m wind tunnel at China Aerodynamics Research and Development Center (CARDC). The rotor system was tested in hover states at collective pitches ranging from 0° to 13° and it was also tested in forward flights at advance ratios up to 0.6, with specific focus on the shaft tilt angle and lift offset sweeps. To ensure that the coaxial rotor was operating in a similar manner to that of the real flight, the torque difference was trimmed to zero in hover flight, whilst the constant lift coefficient was maintained in forward flight. An isolated single-rotor configuration test was also conducted with the same pitch angle setting in the coaxial rotor. The hover test results demonstrate that the figure of merit (FM) value of the lower rotor is lower than that of the upper rotor, and both are lower than that of the isolated single rotor. Moreover, the coaxial rotor configuration can contribute to better hover efficiency under the same blade loading coefficient (CT/σ). In forward flight, the effective lift-to-drag (L/De) ratio of the coaxial rigid rotor does not monotonously change as the advance ratio increases. Increases in the required power and drag in the case with a high advance ratio of 0.6 leads to the decreasing L/De ratio of the rotor. Meanwhile, the L/De ratio of the rotor is relatively high when the rotor shaft is tilted backward. The increasing lift offset tends to result in reduced required rotor power and an increase in the rotor drag. When the effect of the reduced rotor power is greater than that of the increased rotor drag, the L/De ratio increases as the lift offset increases. The L/De ratio can benefit significantly from lift offset at a high advance ratio, but it is much less influenced by lift offset at a low advance ratio. The forward performance efficiency of the upper rotor is poorer than that of the lower rotor, which is significantly different from the case in the hover flight.