scholarly journals Stability Analysis for Unsymmetrical Shaft With Flexible Bladed-Disk

1997 ◽  
Author(s):  
Samer Masoud ◽  
Naim Khader
Keyword(s):  
Equations Of Motion ◽  
Rotor Speed ◽  
Governing Equations ◽  
Bladed Disk ◽  
Shaft System ◽  
Special Cases ◽  
Mass Moment ◽  
Wide Range ◽  
Mass Moment Of Inertia ◽  
Bearing Stiffness

The governing equations of motion for a rotating flexible blade-rigid disk-flexible shaft system are derived. The bladed disk is attached at one end of an asymmetric shaft with uniformly distributed mass, mass moment of inertia, and stiffness. The shaft is held by two isotropic supports; one at the far end from the bladed disk, modeled by two translational and two rotational springs, and an intermediate support, modeled by two translational springs only. The effect of shaft asymmetry on the dynamics behavior of the rotating bladed disk shaft system is examined over a wide range of rotational speed, and for different combinations of springs’ stiffness, which determines the type of shaft supports. The cantilever, and the simply supported shaft with an over hang can be looked upon as special cases of the described above shaft configuration, since the former is obtained by assigning large stiffness for both translational and rotational springs at the end support, and zero spring stiffness at the intermediate one, whereas the latter is obtained by assigning large stiffness for the translational springs at both supports and zero stiffness for the rotational springs. Stability boundaries are calculated, and presented in terms of shaft asymmetry and rotor speed for given bearing stiffness.

An Improved Transfer Matrix-Direct Integration Method for Rotor Dynamics

1986 ◽  
Vol 108 (2) ◽  
pp. 182-188 ◽  
Author(s):  
Jialiu Gu
Keyword(s):  
Transfer Matrix ◽  
Integration Method ◽  
Equations Of Motion ◽  
Rotor Dynamics ◽  
Direct Integration ◽  
Rotating System ◽  
Direct Integration Method ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
Matrix Iteration

A transfer matrix-direct integration combined method is proposed, which employs the transfer matrix method to derive the equations of motion of a “characteristic disk,” and uses the direct integration method to determine the critical speeds, modes and unbalance response of a rotor-bearing system, and to analyze its stability. Despite the complexity of the system, the number of governing equations is not greater than eight. For a single-spool rotating system, the number of equations is only four. A transfer matrix for a uniform shaft is derived to consider its distributed mass, moment of inertia and the effect of shearing force. An impedance matrix iteration method is proposed to consider the effect of a complicated bearing-supporting system on the rotor dynamics. Two examples are given, and the results agree satisfactorily with the experiments.

Vibration Frequencies for a Uniform Beam With One End Elastically Supported and Carrying a Mass at the Other End

1975 ◽  
Vol 42 (4) ◽  
pp. 878-880 ◽  
Author(s):  
D. A. Grant
Keyword(s):  
Normal Modes ◽  
Frequency Equation ◽  
The Other ◽  
Concentrated Mass ◽  
Mass Moment ◽  
Wide Range ◽  
Modes Of Vibration ◽  
Mass Moment Of Inertia ◽  
Elastically Supported ◽  
Range Of Values

In this paper the author obtains the frequency equation for the normal modes of vibration of uniform beams with linear translational and rotational springs at one end and having a concentrated mass at the other free end. The eigenfrequencies for the fundamental mode are given for a wide range of values of mass ratio, mass moment of inertia ratios, and stiffness ratios.

scholarly journals Influence of Variable Nonlocal Parameter and Porosity on the Free Vibration Behavior of Functionally Graded Nanoplates

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Pham Van Vinh ◽  
Le Quang Huy
Keyword(s):  
Free Vibration ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Closed Form Solution ◽  
Shear Deformation Theory ◽  
Form Solution ◽  
Functionally Graded ◽  
Governing Equations ◽  
Vibration Behavior ◽  
Special Cases

This paper studies the influence of the variable nonlocal parameter and porosity on the free vibration behavior of the functionally graded nanoplates with porosity. Four patterns of distribution of the porosity through the thickness direction are considered. The classical nonlocal elasticity theory is modified to take into account the variation of the nonlocal parameter through the thickness of the nanoplates. The governing equations of motion are established using simple first-order shear deformation theory and Hamilton’s principle. The closed-form solution based on Navier’s technique is employed to solve the governing equations of motion of fully simply supported nanoplates. The accuracy of the present algorithm is proved via some comparison studies in some special cases. Then, the effects of the porosity, the variation of the nonlocal parameter, the power-law index, aspect ratio, and the side-to-thickness ratio on the free vibration of nanoscale porous plates are investigated carefully. The numerical results show that the porosity and nonlocal parameter have strong effects on the free vibration behavior of the nanoplates.

Considerations in the Internal Force Analysis of Slider-Crank Machines

2005 ◽  
Author(s):  
Samuel Doughty
Keyword(s):  
Equations Of Motion ◽  
Angular Acceleration ◽  
Internal Force ◽  
System Of Equations ◽  
Engineering Analysis ◽  
Force Analysis ◽  
Connecting Rod ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
Internal Forces

In the analysis of internal forces for a slider–crank machine, writing the equations of motion is just the beginning. There are many engineering analysis decisions to be made regarding the handling of the crank angular acceleration, modeling of the connecting rod mass moment of inertia, and the approach to the solution of the resulting system of equations. Over the years, each of these have been handled in various ways, not all of then entirely correct. This paper looks at the various options and makes a critical review of the various practices. The intent is to encourage all engineering analysts to review the assumptions and methods of the software that they routinely use.

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

1986 ◽  
Vol 108 (4) ◽  
pp. 567-575 ◽  
Author(s):  
E. F. Crawley ◽  
E. H. Ducharme ◽  
D. R. Mokadam
Keyword(s):  
Analytical Model ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Disk System ◽  
Structural Dynamic ◽  
Nodal Diameter ◽  
Coupled Equations ◽  
Bladed Disk ◽  
Shaft System ◽  
The One

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.

Blade Mistuning Coupled With Shaft Flexibility Effects in Rotor Aeroelasticity

1989 ◽  
Author(s):  
Naim Khader ◽  
Robert G. Loewy
Keyword(s):  
Equations Of Motion ◽  
Governing Equations ◽  
Aeroelastic Stability ◽  
Stability Boundaries ◽  
Coriolis Forces ◽  
Bladed Disk ◽  
Out Of Plane ◽  
Shaft Flexibility ◽  
Rotor Aeroelasticity ◽  
Plane Deformations

The effect of bladed-disk polar dissymmetry, resulting from variations in mass from one blade to another, on aeroelastic stability boundaries for a fan stage is presented. In addition to both in-plane and out-of-plane deformations of the bladed-disk, bending of the supporting shaft in two planes is considered, and the resulting Coriolis forces and gyroscopic moments are included in the analysis. A quasi-steady aerodynamics approach is combined with the Lagrangian method to develop the governing equations of motion for the flexible bladed-disk-shaft assembly. Calculations are performed for an actual fan stage.

An Experimental Study of the Influence of Disk Flexibility and Rubbing on Rotordynamics

1993 ◽  
Author(s):  
Fangsheng Wu ◽  
George T. Flowers
Keyword(s):  
High Frequency ◽  
Analytical Study ◽  
Equations Of Motion ◽  
Frequency Component ◽  
High Amplitude ◽  
High Frequency Component ◽  
Rotor Speed ◽  
Governing Equations ◽  
Speed Range ◽  
Flexible Disk

Abstract This study is an experimental investigation of the influence of disk flexibility and rubbing on rotordynamics. The rotor rigs used in the experiments were designed to have included disk flexibility and a rubbing mechanism. The governing equations of motion are similar to those studied in analytical investigations. The system responses to imbalance were recorded and orbit trajectories were plotted at a set of different rotating speeds. The results show the rubbing response development from light forward bouncing, mixed forward bouncing, to high amplitude backward whirling heavy rubbing. The results also show that the flexible-disk rotor has the tendency of having a high frequency component at the upper rotor speed range of mixed bouncing. These agree very well with the results from an earlier analytical study.

The Assumed Mode Method in Structural Dynamics of Bladed-Disk-Shaft Systems

1990 ◽  
Author(s):  
Naim Khader ◽  
Sanier Masoud
Keyword(s):  
Equations Of Motion ◽  
Analytical Investigation ◽  
Assumed Mode Method ◽  
Bladed Disk ◽  
Wide Range ◽  
Out Of Plane ◽  
Mode Method ◽  
Inertia Parameters ◽  
Inertia Properties ◽  
Assumed Mode

Analytical investigation into the effect of transverse bending of continuous flexible shafts is presented. While the blades are allowed both in-plane and out-of-plane deformations, the considered disk is rigid, and the shaft is allowed to bend in two planes. The assumed mode method is used to express flexible blade and shaft deformations, and the Lagrangian approach is used to derive the governing equations of motion for the considered structure. Stiffness and inertia properties of an actual experimental rotor, typical of a fan stage, are used in the analysis. Calculations are performed for three different disk-shaft configurations, and results are presented for different shaft stiffness and inertia parameters, as well as for a wide range of rotational speed.

Vibration of Spinning Cantilever Beams Undergoing Coupled Bending and Torsional Motion

2013 ◽  
Author(s):  
Christopher G. Cooley ◽  
Robert G. Parker
Keyword(s):  
Original Problem ◽  
Natural Frequencies ◽  
Rotation Axis ◽  
Equations Of Motion ◽  
Vibration Modes ◽  
Cantilever Beams ◽  
Torsional Motion ◽  
Governing Equations ◽  
Helicopter Blade ◽  
Wide Range

This study investigates the vibration of a spinning cantilever beam with a rigid body attached to its free end undergoing coupled bending and torsional motion. The rotation axis is perpendicular to the beam (like a helicopter blade). The governing equations of motion are cast in a structured way using extended variables and extended operators. With this structure the equations represent a classical gyroscopic system and Galerkin discretization is readily applied where it is not for the original problem. The natural frequencies and vibration modes are investigated over a wide range of rotation speeds.

Natural Frequency Clusters in Planetary Gear Vibration

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Tristan M. Ericson ◽  
Robert G. Parker
Keyword(s):  
Natural Frequency ◽  
Natural Frequencies ◽  
Planetary Gear ◽  
The Other ◽  
Or Groups ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
Bearing Stiffness ◽  
Planet Gear ◽  
Central Member

This paper investigates how the natural frequencies of planetary gears tend to gather into clusters (or groups). This behavior is observed experimentally and analyzed in further detail by numerical analysis. There are three natural frequency clusters at relatively high frequencies. The modes at these natural frequencies are marked by planet gear motion and contain strain energy in the tooth meshes and planet bearings. Each cluster contains one rotational, one translational, and one planet mode type discussed in previous research. The clustering phenomenon is robust, continuing through parameter variations of several orders of magnitude. The natural frequency clusters move together as a group when planet parameters change. They never intersect, but when the natural frequencies clusters approach each other, they exchange modal properties and veer away. When central member parameters are varied, the clusters remain nearly constant except for regions in which natural frequencies simultaneously shift to different cluster groups. There are two conditions that disrupt the clustering effect or diminish its prominence. One is when the planet parameters are similar to those of the other components, and the other is when there are large differences in mass, moment of inertia, bearing stiffness, or mesh stiffness among the planet gears. The clusters remain grouped together with arbitrary planet spacing.

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