Analytical Study of Nonlinear Vibration in a Rub-Impact Jeffcott Rotor

The purpose of this work is to explore the nonlinear vibration of a rub-impact Jeffcott rotor. In the first stage, the motion is not affected by the friction force, but in the second stage, the motion is influenced by the normal force and the friction force. The governing equations of the rotor of this model are derived in this paper. In consequence, there appears a difference between the two stages. We establish an approximate analytical solution for nonlinear vibrations corresponding to two stages with the mention of the location of jumps. The obtained results are compared with the numerical integration results. The steady-state response and the stability of the solutions are analytically determined for the two stages. The stability of a full annular rub solution is studied with the help of the Routh–Hurwitz criterion. Effects of different parameters of the system, the saddle-node bifurcation (turning points) and the Hopf bifurcation are presented. The main contribution lies in the analytical approximation solution based on the Optimal Auxiliary Functions Method.

Interrogating the Lead-up to a Critical Speed in Rotordynamics

This paper presents a new approach to predicting an incipient critical speed in a rotating shaft. Based on the classical governing equations of motion for an eccentric mass on a flexible shaft (the Jeffcott rotor model), the approach is centered on examining the behavior of small perturbations or random disturbances to infer the approach of a critical speed (resonance). Such disturbances, that may be based on intentional probing, or simply the result of naturally occurring fluctuations, cause small transients. It is the changing nature of these transients (as characterized by their associated eigenvalues) that is used to assess the proximity to a critical speed. In this paper the material developed is based on analysis, but generating the data from simulations or experiments will be the next step. The approach is a kind of stress-test, conceptually not dissimilar to structural health monitoring and damage detection, but here directed toward the lead-up to resonance.

Modelling and Analysis of a Virtual Prototype of a Rotary Machine in MSC.ADAMS

The article presents an analysis of the virtual model of Laval (Jeffcott) rotor in the software environment MSC.ADAMS. The parameters describing the stability of the rotor operation were monitored and evaluated, i. j. critical angular velocity and trajectory of the rotor center of gravity (orbit). The results were compared with the values measured on the experimental equipment, as well as with the values obtained by analytical calculation. The paper further presents a simulation in which the second critical velocity was reached. The paper further presents a simulation in which the second critical velocity was reached.

Effect of Coupled Torsional and Transverse Vibrations of the Marine Propulsion Shaft System

In this study, the coupled torsional–transverse vibration of a propeller shaft system owing to the misalignment caused by the shaft rotation was investigated. The proposed numerical model is based on the modified version of the Jeffcott rotor model. The equation of motion describing the harmonic vibrations of the system was obtained using the Euler–Lagrange equations for the associated energy functional. Experiments considering different rotation speeds and axial loads acting on the propulsion shaft system were performed to verify the numerical model. The effects of system parameters such as shaft length and diameter, stiffness and damping coefficients, and cross-section eccentricity were also studied. The cross-section eccentricity increased the displacement response, yet coupled vibrations were not initially observed. With the increase in the eccentricity, the interaction between two vibration modes became apparent, and the agreement between numerical predictions and experimental measurements improved. Given the results, the modified version of the Jeffcott rotor model can represent the coupled torsional–transverse vibration of propulsion shaft systems.

Periodic Response Analysis of A Jeffcott-Rotor System By The Integral Equation Method

In this paper, two modified nonlinear saturation-based controllers and negative velocity feedback controllers are integrated to suppress the horizontal and vertical vibrations of a horizontally supported Jeffcott-rotor system at primary resonance excitation and the presence of 1:1 and 1:2 internal resonances. The second order approximations and the amplitude equations are obtained by applying the integral equation method to analyze the nonlinear behavior of this model. The stability of the steady-state solutions is ascertained based on the Floquet theory. The necessity of adding a negative velocity feedback to the main system is stated. The effects of different control parameters on the frequency-response curves and the force-response curves are investigated. Time histories of the whole system are included to show the response with and without control. It is shown that the saturation-based controller can reduce the system response to almost zero and the negative velocity feedback can suppress the transient vibrations and prevent the main system having the large amplitude vibration. The analyses show that analytical solutions are in excellent agreement with the numerical simulations. Finally, a comparison with previously published works is included.

Natural Frequency Analysis of a Jeffcott Rotor Having Stepped Shaft

The Rotating shafts are mechanical elements used to transmit power or motion. A shaft with a step or steps is widely used instead of a shaft with a fixed (non-variable) diameter when operating at high speeds. The aim of this research is to study the effect of the step amount and its location in the shaft on the natural frequencies of the Jeffcott rotor model. Analytical methods are used to find an approximate formulation to obtain the natural frequencies of the Jeffcott rotor model neglecting the shaft mass. Lagrange equations are used to develop dynamic equations assuming elastic shaft with steps carrying a disk. The finite element method by using ANSYS is used to validate and compare the results obtained in the analytical method. The results obtained analytically and numerically were compatible and in good agreement. In addition, some parameters such as the step amount and the stepped shaft length are changed to check its effects on the natural frequencies. the results showed that the natural frequencies increase with an increase in the amount and length of the stepped part, while they decrease the closer the disc position to the center.

Bifurcation Analysis of Rotor/Bearing System using Third Order Journal Bearings Stiffness and Damping Coefficients

Journal bearings have many applications in industry due to its high load carrying capacity. In addition proper design of journal bearings enables safe operation at very high speeds. However, they are susceptible to oil whirl instability which may cause bearing failure. The fluid film pressure distribution inside the journal bearing is described by Reynolds equation. Many studies had been done to approximate the bearing performance using first order bearing coefficients. Although this analysis is stable for evaluating the threshold speed but it is insensitive to limit cycles above the threshold speed. Mush literature show that above the threshold speed, subcritical or supercritical bifurcations may be observed. Therefore, the aim of the present paper is to evaluate the third order bearing coefficients for a finite length journal bearing using finite perturbation method. The values of these coefficients are evaluated using infinitesimal perturbation analysis. These values are used to investigate the bifurcation stability of flexible Jeffcott rotor supported by two symmetric journal bearings. The effect of rotor stiffness ratio on the bifurcation stability of the system is investigated. The results of this work show that the third order parameters can be used to evaluate the type of bifurcation above the threshold speed.

Effect of Detuning of Clamping Force of Tie Rods on Dynamic Performance of Rod-Fastened Jeffcott Rotor

In view of the advantages of lightweight, high strength, easy cooling, and easy assembly, the rod-fastened rotor is widely used in the aeroengine and heavy gas turbine. However, because of assembly, stress relaxation, material creep, and other reasons, the clamping force of the tie rods will be out of tune during the long-term operation of the rotor. The detuning of the clamping force of the tie rods not only affects the contact stiffness of the contact interface but also causes the rod-fastened rotor with a certain residual shaft bow, which will affect the dynamic characteristics of the rod-fastened rotor. Based on the statistical model of rough surface contact (GW contact model), this paper presents a method to calculate the equivalent flexural stiffness of rough surface considering the detuning of the clamping force of the tie rods and gives the calculation method of the residual shaft bow deformation of the rod-fastened Jeffcott rotor with detuning of the tie rods. The effect of the preload, the rate of detuning of the tie rods, the number of detuning tie rods on the natural frequency, and the response of residual shaft bow of the rod-fastened Jeffcott rotor at a certain speed are investigated. The results show that the detuning of the tie rods makes the flexural stiffness of the rotor inconsistent along with two main stiffness directions of the rotor, which makes the natural frequency of the rotor divided into two. The negative detuning of the tie rods decreases the natural frequency of the rotor, while the positive detuning of the tie rods increases the natural frequency of the rotor. The smaller preload or the larger rate of detuning of the tie rods makes the detuning of the tie rods have a greater influence on the natural frequency of the rotor. These results will provide a theoretical reference for the dynamic analysis and design of the rod-fastened rotor.