An Efficient Method for Calculating Damped Critical Speeds of Rotor-Bearing Systems

This paper describes an analytical approach for calculating the damped critical speeds of multi-degree-of-freedom rotor-bearing systems. It is shown that to calculate the critical speeds is equivalent to finding the roots of a proposed matrix algebraic equation. The technique employes a Newton-Raphson scheme and the derivatives of eigenvalues. The system left eigenvectors are used to simplify the calculations. Based on this approach, a general-purpose computer program was developed with a finite element model of rotor-bearing systems. The program automatically generates system equations and finds the critical speeds. The program is applied to analyze a turbomachine supported by two cylindrical oil-film Journal bearings. The results are compared with reported data and the agreements are very good.

Vibrations of Foundations Interacting With Subsoil Experimental Studies and BE Calculations

Calculation of the dynamic response of sensitive structures like foundations for vibrating machinery requires to take the interaction with subsoil into account. Structures and soil are discretized by boundary elements and coupled by a substructure technique. Viscoelastic constitutive equations contain fractional time derivatives. Surface waves generated by machine foundations and diffracted by embedded structures and soil inhomogeneities are analyzed by conventional and optoelectronic experimental techniques and calculated by the boundary element method (BEM).

A Transfer Matrix Technique for Evaluating the Natural Frequencies and Critical Speeds of a Rotor With Multiple Flexible Disks

Modern turbomachinery is used to provide power for a wide range of applications, from steam turbines for electrical power plants to the turbopumps used in the Space Shuttle Main Engine. Such devices are subject to a variety of dynamical problems, including vibration, rotordynamical instability, and shaft whirl. In order to properly design and evaluate the performance and stability of turbomachinery, It is important that appropriate analytical tools be available that allow for the study of potentially important dynamical effects. This research effort is concerned with developing a procedure to account for disk flexibility which can readily be used for investigating how such effects might influence the natural frequencies and critical speeds of practical rotor systems. In the present work, a transfer matrix procedure is developed in which the disk flexibility effects are accounted for by means of additional terms included in the transfer matrix formulation. In this development, the shaft is treated as a discrete system while the disk is modelled as a continuous system using the governing partial differential equation. Based on this governing equation, an equivalent inertial moment Mk*, which is the generalized dynamic force coupling between shaft and disk, is then derived. Analysis shows that only the disk modes of one nodal diameter contribute to the inertial moment, Mk*, and thus influence the natural frequencies of the rotor system. To determine the Mk*, the modal expansion method is employed and the governing partial differential equation of the disk is transformed to a set of decoupled forced vibration equations in the generalized coordinates. The Mk* are then calculated in terms of modal shapes, natural frequencies, and material and geometric parameters which can be found in the literature or can be obtained from experiments. Finally the Mk* are incorporated into the point transfer matrix. By so doing, the properties of quick computational speed and ease of use are retained and the complexity of solving partial differential equations is avoided. This allows the present procedure to be easily applied to practical engineering problems. This is especially true for multiple flexible disk rotor systems. As an example, three different cases for a simplified model of the Space Shuttle Main Engine (SSME) High Pressure Oxygen Turbo-Pump (HPOTP) rotor have been studied using this procedure. Some of the more interesting results obtained in this example study are enumerated below. 1.) Disk flexibility can introduce additional natural frequency(s) to a rotor system. 2.) Disk flexibility can cause shifting of some of the natural frequencies. 3.) As disk flexibility is increased, lower natural frequencies of the rotor system will be influenced. 4.) At certain rotor speeds, disk flexibility may cause the disappearance of a natural frequency. 5.) The axial position of the disk on the rotor shaft has a significant effect on the degree of this influence.

Analytical and Experimental Simulation of Loose Pedestal Dynamic Effects on a Rotating Machine Vibrational Response

This report presents experimental, analytical, and numerical results describing vibrational phenomena in a rotating machine with one loose pedestal. The loose-pedestal machine rotor vibrations represent unbalance-related excited vibrations of synchronous and fractional subsynchronous regimes. In this study the loose-pedestal machine is first simulated by a simple vibrating beam excited by a shaker mounted on it. The shaker simulates an unbalanced machine rotor. The beam occasionally enters in contact with the foundation. The excited vibrations are modified by impacting occurrences, and by periodic changes in system stiffness. A new model of the impact has been developed. The results of analytical and experimental studies stand in a good agreement. They illustrate the existence of the synchronous regime and several subsynchronous fractional regimes in various excitation frequency ranges. The analysis adequately predicts the occurrence of these regimes and determines the physical parameters affecting them. The analytical and experimental results are then compared with the responses of experimental rotor rig with one bearing pedestal looseness. They show the same qualitative pattern.

A Study of Control Strategy for Vehicle Semi-Active Suspension System

This paper investigates a reference control strategy for Vehicle semi-active suspension. The control is conducted by following the idea optimal active controller. The passive actuator is set to optimal whenever the active and passive actuators have the same signs; and set to zero output whenever the two signs are opposite. The simulation results of a 2DoF vehicle show that the semi -active suspension system can follow the ideal active system very well, both are superior to conventional passive systems. In this paper, a 2DoF vehicle model was also used to study a statistical optimal control strategy of the semi-active suspension system. The statistical optimal concept is the result of the combination of the nonlinear programming and controllable damper. A way of estimating statistical characteristics of road irregularities was also proposed. Vehicle active, suspension, due to its perfect v i bra t i on isolation performance, gets moreand more attention. Active suspension can be generally divided into two categories, totally active suspension system and semi-active suspension system. From the published results it is known that active suspension can surpass the performance limit of conventional passive suspension and greatly improve the vehicle riding comfort and steering ability. But active suspension has a critical disadvantage of less applicability, due to its high cost and low reliability. Also it consumes large amount of energy as it works. The idea of semi-active suspension was put forward to overcome the shortcoming of active suspension. It is a compromise between active suspension and passive suspension. Semi-active suspension has approximately the same behavior as active suspension, and almost consumes no energy as it works. So semi-active suspension possesses a great potential in application. At. present, in the field of suspension research over the world, a great deal of attention is paied to semi-active suspension. At present, for the cotrol of semi-active suspension the widely studied strategy is “on off” control [1] [2], which is first put forward by Karnopp. “On-off” control can eliminate the phenomenon of vibration amplification for passive suspension, thus it can improve the suspension performance to certain extent. At present, no substantive result has been obtained yet in the field of optimal control of semi-active suspension. This paper will investigate a reference control strategy on the basis of linear optimal control. The control is conducted by following the optimal ctive controller. The referrence control result is optimal when the outputs of the active and semi-active force generators have the same signs.

Response of a Rotating Beam Subject to an Accelerating Force

A method is given by which the response of a rotating Timoshenko beam subjected to an accelerating fixed direction force can be determined. The beam model includes the gyroscopically induced displacement transverse to the direction of the load. The solution for pinned supports is set up in general form using multi-integral transforms and the inversion is expressed in terms of convolution integrals. These are numerically integrated for a uniformly distributed load having an exponentially varying velocity function. Results are presented for the displacement under the load’s center as a function of position. Comparisons are made between the responses to a constant velocity load and a load which accelerates up to the same velocity.

Transient Response Technique Applied to Active Magnetic Bearing Machinery During Rotor Drop

The active magnetic bearing (AMB) is a relatively new technology which has many advantages compared with conventional bearing design. In an AMB system, the rolling-element back-up bearings are indispensable to protect the magnetic bearing rotor and stator, and other stationary seals along the rotor shaft. In this paper, a theoretical formulation is proposed and solved numerically to examine the transient response of the flexible rotor, from the time just previous to the AMB shuts down and including the rotor drop onto the back-up bearing. The backward whirl of the rotor, which may lead to the destructive damage of the machiner, has been analytically predicted at very light support damping and very high support damping. Also, the vibration due to the non-linearity of the contact point geometry has been included in the analysis. The influence of the support damping on the displacement of the disk and also the contact force between the journal and the inner-race of the back-up bearing have been computed for various rotor system parameters.

The Rocketdyne Multifunction Tester: Operation of a Radial Magnetic Bearing As an Excitation Source

The operation of the magnetic bearing used as an excitation source in the Rocketdyne Multifunction Tester is described. The tester began operation during the summer of 1990. The magnetic bearing can be used in two control modes: 1) open loop mode, in which the magnetic bearing operates as a force actuator, and 2) closed loop mode, in which the magnetic bearing provides shaft support. Either control mode can be used to excite the shaft; however, response of the shaft in the two control modes is different due to the alteration of the eigenvalues by closed loop mode operation. A rotordynamic model is developed to predict the frequency response of the tester due to excitation in either control mode. Closed loop mode excitation is shown to be similar to the excitation produced by a rotating eccentricity in a conventional bearing. Predicted frequency response of the tester in the two control modes is compared, and the maximum response is shown to be the same for the two control modes when synchronous unbalance loading is not considered. The analysis shows that the response of this tester is adequate for the extraction of rotordynamic stiffness, damping, and inertia coefficients over a wide range of test article stiffnesses.

Parameter Identification of Non-Linear Impact Damper

In this paper, we presented a mew mechanical model of an impact damper based on the theoretical analyses and experiments. We also presented a method of identifying the parameters of the non∼linear method of an impact system by means of microcomputer calculations. The method of identification presented in this paper has been proved byour experiments. These works supply the theoretical foundations of designing and calculating for impact damper.

Influence of Thermal Conductivity of Lubricant on the THD Characteristics of a Plain Journal Bearing

The viscous heat generation in the lubricant film of a hydrodynamic journal bearing causes a rise in temperature of the fluid film. Considering the influence of the temperature variation along and across the film, the performance of a journal bearing is investigated under adiabatic conditions for different values of thermal conductivity of the lubricant. In this analysis, the temperature of the journal surface has been chosen to ensure that there is no net heat transfer from the lubricant The results show that the variation of temperature across the film affects bearing performance significantly and that an increase in lubricant thermal conductivity enhances bearing performance.