A Method for Predicting the Influences of Bearing Support Stiffness and Position on the Vibrations of a Flexible Rotor System

The bearing support stiffness and position can greatly affect the vibrations of flexible rotor systems (FRSs). However, most previous works only focused on the effect of the bearing support stiffness on the critical speeds or modal characteristics including the natural frequencies and mode shapes of rigid rotor systems (RRSs). The previous studies missed the combined effects of the bearing support stiffness and position. To overcome this issue, an analytical method of a FRS based on the finite element (FE) method is proposed. Our model considers the bearing support stiffness and rotational inertia of FRS. The frequency equation of FRS is established for solving the critical speeds. The critical speeds and modal deformations of FRS from our model and the numerical model based on a commercial software are compared to verify the effectiveness of the presented method. The effects of the bearing support stiffness and position on the critical speeds of FRS are analyzed. The results show that the critical speeds are positively correlated with the bearing support stiffness. The critical speeds of FRS are also greatly affected by the bearing position. This study can provide some guidance for the optimization design method of bearing support stiffness and position in FRSs.

Harmonic Analysis of the High Speed Direct Coupled SCO2 Turbo-Generator

The rotordynamics and harmonic characteristics of the rotor assembly designed for 40-kW high-speed sCO2 direct-coupled turbo-generator pair have been evaluated numerically using finite element solver “ANSYS Mechanical”. First, the shaft geometry and dimensions have been optimized using lumped mass-inertia-based AxSTREAM RotorDynamics module followed by the bearing selection analysis using SKF SimPro expert to ensure enough separation margin from the nearby critical speeds. Equivalent 2D geometry has been used with an FEA-based ANSYS general axisymmetric model to reduce the computation time. The effect of the damping on the forces transmitting to the bearings and shaft deflection at the critical speeds are analyzed by performing harmonic analysis under various damped and undamped conditions (ζ = 0, 0.005, 0.01, and 0.02).

Sensitivity of the Spool Rate on the Dynamic Response of the System

In rotodynamic systems, the rotor is spooled up from zero speed to its operating speed during the engine start. One of the considerations in design of rotating systems is the placement of rotor critical speed. It is vital to ensure that the rotor critical speeds are not close to the engine operating speed. However, it is not always possible to isolate all the frequencies above the operating speed. So, during the engine start to full speed, rotor system does travel through the mode. Therefore, to avoid a large system response, the rotor is spooled up quickly through the critical speed. In addition to the rotor critical speeds, the natural frequencies of the static structures may also get excited during the rotor spool up and spool down. The static structure response is one of the important considerations in designing a system for dynamic loading condition. It has been observed that the rotor spool rate affects the static structure response. This paper focuses on effect on system’s response under various spool rate. It has also been shown that the natural frequency of the system and damping in the system are two of the major factors towards sensitivity of system response with spool rate. Additionally, it has been observed that the presence of non-linearities shifts the peak response away from the natural frequency depending on the spool rate and spool direction.

Host--Parasitoid Dynamics and Climate-Driven Range Shifts

Climate change has created new and evolving environmental conditions, impacting all species, including hosts and parasitoids. I therefore present integrodifference equation (IDE) models of host--parasitoid systems to model population dynamics in the context of climate-driven shifts in habitats. I describe and analyze two IDE models of host--parasitoid systems to determine criteria for coexistence of the host and parasitoid. Specifically, I determine the critical habitat speed, beyond which the parasitoid cannot survive. By comparing the results from two IDE models, I investigate the impacts of assumptions that reduce the system to a single-species model. I also compare critical speeds predicted by a spatially-implicit difference-equation model with critical speeds determined from numerical simulations of the IDE system. The spatially-implicit model uses approximations for the dominant eigenvalue of an integral operator. The classic methods to approximate the dominant eigenvalue for IDE systems do not perform well for asymmetric kernels, including those that are present in shifting-habitat IDE models. Therefore, I compare several methods for approximating dominant eigenvalues and ultimately conclude that geometric symmetrization and iterated geometric symmetrization give the best estimates of the parasitoid critical speed.

Optimization of Nonlinear Unbalanced Flexible Rotating Shaft Passing Through Critical Speeds

This work studies the nonlinear oscillations of an elastic rotating shaft with acceleration to pass through the critical speeds. A mathematical model incorporating the Von-Karman higher-order deformations in bending is developed to investigate the nonlinear dynamics of rotors. A flexible shaft on flexible bearings with springs and dampers is considered as rotor system for this work. The shaft is modeled as a beam and the Euler–Bernoulli beam theory is applied. The kinetic and strain energies of the rotor system are derived and Lagrange method is then applied to obtain the coupled nonlinear differential equations of motion for 6 degrees of freedom. In order to solve these equations numerically, the finite element method (FEM) is used. Furthermore, for different bearing properties, rotor responses are examined and curves of passing through critical speeds with angular acceleration due to applied torque are plotted. Then the optimal values of bearing stiffness and damping are calculated to achieve the minimum vibration amplitude, which causes to pass easier through critical speeds. It is concluded that the value of damping and stiffness of bearing change the rotor critical speeds and also significantly affect the dynamic behavior of the rotor system. These effects are also presented graphically and discussed.

STABILITY AND VIBRATION ANALYSES OF AN INTERNALLY DAMPED TAPERED COMPOSITE DRIVESHAFT USING THE FINITE ELEMENT METHOD

The present study considers the linear vibration and stability analyses of an internally damped rotating tapered composite shaft supported on rolling bearings. The Timoshenko beam theory is utilized to model the tapered drive-shaft based on Equivalent Single Layer Theory (ESLT). The ESLT considers a laminated driveshaft that consists of several lamina with different fiber orientations. Since the bearings are considered as rolling element bearings, the bearings stiffnesses are modeled using linear translational springs and dampers. The equations of motion are derived by applying Lagrange’s equation, including the hysteretic internal damping of composite material, and then finite element formulation is utilized to solve the equations. The effects of various system parameters on the natural frequencies and instability threshold are investigated. An extensive parametric study has been carried out to determine the effects of various system parameters including hysteresis internal damping, fiber orientation, stacking sequence, taper angle, rotational velocity, and bearings stiffness and damping on the natural frequencies, critical speeds, and instability thresholds of internally damped tapered composite drive-shafts. Furthermore, Campbell and critical speed map diagrams are depicted to present the effects of rotational velocity and bearings stiffness on natural frequencies and critical speeds. It is shown that the stability of the driveshaft is enhanced by increasing the damping of the bearings, whereas increasing the internal damping of the composite driveshaft may reduce the instability threshold.

Mud Motor PDM Dynamics: An Analytical Model

In a fast drilling environment, suchas shale drilling, refining advanced technologies for preventing downhole toolfailures is paramount. Challenges are still very much associated with complex bottom-hole assemblies and the vibration of the drill string when used with a downhole mud motor. The mud positive displacement motor with various lobe configurations and designs becomes an additional excitation source of vibration. Further, it affects the transient behavior of the performance mud motor. Unbalanced force exists because the center ofmass of the motor rotor does not coincide with the axis of rotation.Further, the vector of full acceleration of the center of the rotor can be decomposed into two perpendicular projections—tangent and normal—which aretaken into account and integrated intothe full drill string forced frequency modelas force and displacement at the motorlocation. The paper includes two models, first one to predict the critical speeds and the second one to see the transient behavior of the downhole parameters when the mud motor is used.The model also considers the effect of the stringspeed. The unbalanced force is more pronounced at the lower pair or lobe configuration as compared to the higher pairlobe configuration because of the larger eccentricity. The unbalance is modeled in terms of an equivalent mass of therotor with the eccentricity of the rotor. Also, the analysis provides an estimation of relative bending stresses, shear forces, lateral displacements and transient bit rpm, bit torque, and weightone bit for the assembly used. Based onthe study, severe vibrations causing potentially damaging operating conditionswhen transient downhole forcing parametersare used for the vibration model.It has been found that when a mud motor isused using static forcing parameters may not provide the conservative estimation of the critical speeds as opposed totransient parameters. This is because coupled oscillations fundamentally can create new dynamic phenomena, which cannot be predicted from the characteristics of isolated elements of the drilling system.