Shaft-disk System Research Articles

DYNAMIC STABILITY OF ROTATING COMPOSITE SHAFTS UNDER PERIODIC AXIAL COMPRESSIVE LOADS

This paper describes the dynamic behavior of a rotating composite shaft subjected to axial periodic forces using the finite element method. Laminated composite shafts are modelled using Timoshenko beams. The numerical results show good agreement with the reported beams models. Effects of static and time dependent components of axial loads on the stability of the composite shaft are studied. This paper also investigates the effect of the rotational speeds and the disk on the unstable regions of the shaft. The numerical results show that for the same geometric parameters, a steel shaft has a lower frequency than that of the composite shafts; however, the steel shaft is more stable than composite shafts because the shaft-disk system is subjected to axial periodic forces at lower rotational speeds. Also, the effect of the gyroscopic moments makes the steel shaft more sensitive to the periodic axial load than the composite one.

STABILITY ANALYSIS FOR THE DYNAMIC DESIGN OF ROTORS

The dynamic behaviour of a cantilever flexible shaft, carrying a rigid disk at its free end is presented. The stability of the rotating shaft–disk system, subjected to periodic follower axial force and end moment is investigated. Due to the periodic nature of the external loads, the governing equations of motion contain periodic coefficients, which cannot be solved by the available classical eigenvalue routines. Floquet theory is used along with the Runge–Kutta method with Gill coefficients to calculate the transition matrix, whose eigenvalues provide the desired information about system stability. The effect of various system parameters on the calculated stability boundaries is examined. Such examination includes rotor speed, damping available in the system and associated with both translation and tilting of the shaft cross-sections, torque to damping ratio, and axial force to end torque ratio.

A modified semi-analytical approach towards the modelling of a shaft-disc system

A modified semi-analytical approach towards the modelling of a shaft-disc system

Stability of rotating shafts loaded by follower axial force and torque load

Abstract A simple structural model is adopted to study the stability of a rotating cantilever shaft with uniformly distributed mass, mass moment of inertia, and stiffness. The shaft carries a rigid disk at its free end, where it is subjected to follower axial force and torque load. The Lagrangian approach and the assumed mode method are employed to derive the governing equations of motion, with the free vibration mode shapes of a non-rotating cantilever shaft being used as admissible functions. The resulting eigenvalue problem is analyzed, and the stability boundaries are presented for different system configurations and different load combinations. Effects of the gyroscopic moment associated with the mass moment of inertia of the rotating shaft-disk system, and of external damping forces and moments, which restrain the translation and tilting of the shaft’s cross-section, are evaluated.

Dynamic Stability of a Cantilever Shaft-Disk System

The dynamic stability behavior of a cantilever shaft-disk system subjected to axial periodic forces varying with time is studied by the finite element method. The equations of motion for such a system are formulated using deformation shape functions developed from Timoshenko beam theory. The effects of translational and rotatory inertia, gyroscopic moment, bending and shear deformation are included in the mathematical model. Numerical results show that the effect of the gyroscopic term is to shift the boundaries of the regions of dynamic instability outwardly and, therefore, the sizes of these regions are enlarged as the rotational speed increases.

Dynamic stability of a shaft-disk system with flaws

The dynamic stability behavior of a cantilever shaft-disk system with flaws and subjected to axial periodic forces is studied by the finite element method. In the present analysis the effects of translational and rotatory inertia, gyroscopic moments, bending and shear deformation have been taken into consideration. Four parameters are employed to characterize the damaged zone: location, size, effective bending and shear stiffness of the damaged region. Numerical results show that due to the presence of the damaged zone, the regions of dynamic instability are shifted closer to the dynamic load factor axis and parametric resonance may occur at lower disturbance frequencies. Also, as the rotational speed increases, the sizes of the regions of dynamic instability will be enlarged, and therefore, the system becomes more unstable.

Unbalance Response of Flexible Rotors Coupled With Torsion

The effect of coupling with torsion on the unbalance response of flexible rotors, supported by isotropic flexible and damped bearings is investigated. Flexural vibrations of the shaft-disk system are coupled with torsional oscillations through mass eccentricity. The governing equations of motion of the continuous system are solved numerically with a modified Myklestad-Prohl method without the necessity of considering an equivalent lumped system. The cases of constant or harmonic torque applied to the disk are considered. Gyroscopic, rotary inertia, shear deformation, external and internal damping effects are taken into account.

Analytical and Experimental Investigation of the Coupled Bladed Disk/Shaft Whirl of a Cantilevered Turbofan

The structural dynamics of a rotating flexible blade-rigid disk-flexible cantilevered shaft system is analytically and experimentally investigated. A simple analytical model yields the equations of motion expressed in the rotating frame, which show that the blade one nodal diameter modes dynamically couple to the rigid body whirling motion of the shaft-disk system. The blade modes higher than one nodal diameter are uncoupled from the shaft-disk dynamics. Nondimensionalization of the coupled equations of motion yield the criteria for the propensity and magnitude of the interaction between the bladed disk and shaft-disk modes. The analytical model was then correlated with the results of a structural dynamic experiment performed on the MIT Aeroelastic Rotor, a fan similar in design to a modern high bypass ratio shroudless turbofan. A special whirl excitation apparatus was used to excite both forward and backward asynchronous whirl, in order to determine the natural frequencies of the system. The agreement between the predicted and experimental natural frequencies is good and indicates the possibility of significant interaction of the one nodal diameter blade modes with the shaft-disk modes.

On the Critical Speed of Continuous Shaft-Disk Systems

The critical speed of a shaft-disk system can be approximately determined from a single degree-of-freedom model. The errors in the critical speed predictions obtained from such a model are investigated. The percentage errors are plotted against disk to shaft mass ratio for different bearings and various disk locations.

The Natural Frequencies and Critical Speeds of a Rotating, Flexible Shaft-Disk System

An analytical investigation into the influence of disk flexibility on the transverse bending natural frequencies and critical speeds of a rotating shaft-disk system is presented. The geometric model considered consists of a flexible continuous shaft carrying a flexible continuous circular plate. The partial differential equations governing the system motion and the associated exact solution form are developed. Numerical solutions are presented covering a wide range of non-dimensional parameters and general conclusions are drawn.

Further remarks on the theory of three dimensional whirling

This is the last in a series of three papers by the authors dealing with the interaction among the effects of external damping, linear and non-linear material damping, linear and non-linear elastic restoring forces, and linear and non-linear inertia forces on the motion of a whirling elastic shaft-disc system. Using an energy technique a stronger stability theorem than any previously obtained for such systems is presented.

External and material damped three dimensional rotor system

This paper extends and improves previous work by the authors on the motion of a whirling elastic shaft-disc system. Using an energy technique the effects of external damping, linear and non-linear internal damping, linear and non-linear elastic restoring forces and non-linear inertia forces are isolated and investigated. The results are summarized in the form of stability theorems. In addition, using techniques previously developed by the authors, solutions to the linear damped system, for both the homogeneous and non-homogeneous cases, are presented. Apart from being of interest in their own right use is made of them in the non-linear analysis.

On the Critical Speeds of a Continuous Shaft-Disk System

The effect of gyroscopic moments on the critical speeds of a shaft-disk system mounted in short end bearings is investigated. Representation of the shaft as having continuously distributed mass and elasticity allows accurate determination of higher critical speeds. Frequency equations are obtained for the critical speeds associated with both backward and forward whirling modes. The first four critical speeds (backward and forward whirling modes) are given for a range of shaft and disk sizes and for various disk locations on the shaft. Experimental verification is given for the first and second critical speeds.