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.

scholarly journals The Normal Modes of Nonlinear n-Degree-of-Freedom Systems

1962 ◽  
Vol 29 (1) ◽  
pp. 7-14 ◽  
Author(s):  
R. M. Rosenberg
Keyword(s):  
Linear System ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Minimum Problem ◽  
Infinite Class ◽  
Coupled Equations ◽  
Geometrical Problem ◽  
Maximum Minimum

A system of n masses, equal or not, interconnected by nonlinear “symmetric” springs, and having n degrees of freedom is examined. The concept of normal modes is rigorously defined and the problem of finding them is reduced to a geometrical maximum-minimum problem in an n-space of known metric. The solution of the geometrical problem reduces the coupled equations of motion to n uncoupled equations whose natural frequencies can always be found by a single quadrature. An infinite class of systems, of which the linear system is a member, has been isolated for which the frequency amplitude can be found in closed form.

scholarly journals Natural Frequncies of Coupled Blade-Bending and Shaft-Torsional Vibrations

2007 ◽  
Vol 14 (1) ◽  
pp. 65-80 ◽  
Author(s):  
B.O. Al-Bedoor
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Coupled Model ◽  
Small Deformation ◽  
Torsional Vibrations ◽  
Reduced Order ◽  
Rotating Speed ◽  
Coupled Equations ◽  
Stiffening Effect

In this study, the coupled shaft-torsional and blade-bending natural frequencies are investigated using a reduced order mathematical model. The system-coupled model is developed using the Lagrangian approach in conjunction with the assumed modes method to discretize the blade bending deflection. The model accounts for the blade stagger (setting) angle, the system rotating speed and its induced stiffening effect. The coupled equations of motion are linearized based on the small deformation theory for the blade bending and shaft torsional deformation to enable calculation of the system natural frequencies for various combinations of system parameters. The obtained coupled eignvalue system is ready for use as a reference for comparison for larger size finite element simulations and for the use as a fast check on natural frequencies for the coupled blade bending and shaft torsional vibrations in the design and diagnostics processes. Some results on the predicted natural frequencies are graphically presented and discussed pertinent to the coupling controlling factors and their effects. In addition, the predicted coupled natural frequencies are validated using the Finite Element Commercial Package (Pro-Mechanica) where good agreements are found.

Aerodynamic Damping Measurements in a Transonic Compressor

1982 ◽  
Author(s):  
E. F. Crawley
Keyword(s):  
Direct Measurement ◽  
Equations Of Motion ◽  
Dynamic Equations ◽  
Test Facility ◽  
Disk System ◽  
Test Analysis ◽  
Structural Dynamic ◽  
Aerodynamic Damping ◽  
Transonic Compressor ◽  
Reduced Frequency

A method has been developed and demonstrated for the direct measurement of aerodynamic damping in a transonic compressor. The method is based on the inverse solution of the structural dynamic equations of motion of the blade-disk system. The equations are solved inversely to determine the forces acting on the system. If the structural dynamic equations are transformed to multiblade or modal coordinates, the damping can be measured for blade-disk modes, and related to a reduced frequency and interblade phase angle. This method of damping determination was demonstrated using a specially instrumented version of the MIT Transonic Compressor run in the MIT Blowdown Compressor Test Facility. No regions of aeroelastic instability were found. In runs at the operating point, the rotor was aerodynamically excited by a controlled two-per-revolution fixed upstream disturbance. The disturbance was sharply terminated midway through the test. Analysis of the data in terms of multiblade modes led to a direct measurement of aerodynamic damping for three interblade phase angles.

scholarly journals Analysis of a Benchmark Building Installed with Tuned Mass Dampers under Wind and Earthquake Loads

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
S. Elias ◽  
R. Rupakhety ◽  
S. Olafsson
Keyword(s):  
Equations Of Motion ◽  
Equal Mass ◽  
Dynamic Responses ◽  
Acceleration Response ◽  
Tuned Mass Dampers ◽  
Response Control ◽  
Earthquake Loads ◽  
Coupled Equations ◽  
Wind Response ◽  
The One

This study presents analysis of a benchmark building installed with tuned mass dampers (TMDs) while subjected to wind and earthquake loads. Different TMD schemes are applied to reduce dynamic responses of the building under wind and earthquakes. The coupled equations of motion are formulated and solved using numerical methods. The uncontrolled building (NC) and the controlled building are subjected to a set of 100 earthquake ground motions and wind forces. The effectiveness of using different multiple TMD (MTMD) schemes as opposed to single TMD (STMD) is presented. Optimal TMD parameters and their location are investigated. For a tall structure like the one studied here, TMDs are found to be more effective in controlling acceleration response than displacement, when subjected to wind forces. It is observed that MTMDs with equal stiffness in each of the TMDs (usually considered for wind response control), when optimized for a given structure, are effective in controlling acceleration response under both wind and earthquake forces. However, if the device is designed with equal mass in every floor, it is less effective in controlling wind-induced floor acceleration. Therefore, when it comes to multihazard response control, distributed TMDs with equal stiffnesses should be preferred over those with equal masses.

Exact Frequency Analysis of a Rotating Cantilever Beam With Tip Mass Subjected to Torsional-Bending Vibrations

2011 ◽  
Vol 133 (4) ◽  
Author(s):  
Masoud Ansari ◽  
Ebrahim Esmailzadeh ◽  
Nader Jalili
Keyword(s):  
Frequency Analysis ◽  
Time Domain ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Bending Vibrations ◽  
Resonant Condition ◽  
Mass System ◽  
Coupled Equations ◽  
Tip Mass ◽  
Gyroscopic Systems

An exact frequency analysis of a rotating beam with an attached tip mass is addressed in this paper while the beam undergoes coupled torsional-bending vibrations. The governing coupled equations of motion and the corresponding boundary condition are derived in detail using the extended Hamilton principle. It has been shown that the source of coupling in the equations of motion is the rotation and that the equations are linked through the angular velocity of the base. Since the beam-tip-mass system at hand serves as the building block of many vibrating gyroscopic systems, which require high precision, a closed-form frequency equation of the system should be derived to determine its natural frequencies. The frequency analysis is the basis of the time domain analysis, and hence, the exact frequency derivation would lead to accurate time domain results, too. Control strategies of the aforementioned gyroscopic systems are mostly based on their resonant condition, and hence, acquiring knowledge about their exact natural frequencies could lead to a better control of the system. The parameter sensitivity analysis has been carried out to determine the effects of various system parameters on the natural frequencies. It has been shown that even the undamped systems undergoing base rotation will have complex eigenvalues, which demonstrate a damping-type behavior.

scholarly journals Dynamics of Flexible Rotor Systems with an Interim Mass Unbalanced Disk Using a Spectral Element Model

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Sangkyu Choi ◽  
Usik Lee
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Element Model ◽  
Spectral Element ◽  
Rotor System ◽  
Dynamic Responses ◽  
Disk System ◽  
Unbalanced Mass ◽  
Rotor Systems ◽  
Blade System

A frequency domain spectral element model is developed for a rotor system that consists of two spinning shafts and an interim disk or blade system. In this study, the shafts are represented by spinning Timoshenko beam models, and the interim disk system is represented by a uniform thick rigid disk with an unbalanced mass. In our derivation of the governing equations of motion of the disk system, the disk is considered to be wobbling about the geometric center of the disk at which the spinning shafts are attached. The high accuracy of the proposed spectral element model is evaluated by comparison with the natural frequencies obtained using the conventional finite element method (FEM). The spectral element model is then used to investigate the effects of the unbalanced mass on the natural frequencies and dynamic responses of an example rotor system.

scholarly journals A simple model to quantify rocking isolation

2018 ◽  
Vol 51 (1) ◽  
pp. 12-22 ◽  
Author(s):  
Sinan Acikgoz ◽  
Matthew J. DeJong
Keyword(s):  
Analytical Model ◽  
Equations Of Motion ◽  
Base Shear ◽  
Shear Forces ◽  
Coupled Equations ◽  
Seismic Shear ◽  
New Information ◽  
Excitation Mechanisms ◽  
Foundation Structure ◽  
Strong Ground

Rocking action at the foundation-structure interface has long been proposed to isolate structures from strong ground motion. In this paper, the fundamental concept of rocking isolation is examined in depth to guide further design efforts. This is achieved by first deriving an analytical model of a flexible structure freely rocking on rigid ground. Decomposing the coupled equations of motion of the model into their modal components provides new information on the mechanics of rocking isolation. After identifying the salient parameters needed to quantify rocking isolation, equations to predict the lateral accelerations, base shear and overturning moments arising during rocking are provided. The analytical model and the simplified equations are then validated using some of the earliest experiments on rocking structures, which were completed in New Zealand. These validations clarify poorly understood phenomena concerning rocking isolation, such as how rocking and vibrations of the structure couple, how this influences the excitation mechanisms of the structure, resulting in seismic shear forces and overturning moments larger than those required for uplift. The findings provide an analytical basis for designing efficient rocking systems that successfully limit force demands.

Small-amplitude free vibration analysis of active multistable shells under static multifield loading

2020 ◽  
pp. 107754632092393
Author(s):  
Dimitris Varelis
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Small Amplitude ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Shell Element ◽  
Curvilinear Coordinates ◽  
Coupled Equations ◽  
Modal Response

This study considers the small-amplitude free vibrational response performed on top of the quasi-static snap through buckling, which is accompanied by large displacements and rotations of shallow doubly curved laminated piezoelectric shells under multifield loading. The mechanics incorporate coupling between mechanical, electric, and thermal fields and encompass geometric nonlinearity effects due to large quasi-static displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear coordinates and combined with the kinematic assumptions of a mixed-field shear-layerwise shell laminate theory. Based on the above mechanics and adopting the finite element methodology, an eight-node nonlinear shell element is developed to yield the linearized discrete coupled small-amplitude dynamic equations of motion. Initially, the nonlinear coupled equations are linearized and solved quasi-statically using an extended cylindrical arc-length method in combination with the Newton–Raphson iterative technique, and subsequently the free vibration analysis is performed at each solution point. Validation and evaluation cases on laminated cylindrical shells demonstrate the accuracy of the present method and its robust capability to predict the modal response on top of the nonlinear quasi-static response of active multistable shells subject to combined thermo–piezo–electromechanical loads. Numerical cases show the feasibility to develop smart shell structures to detect, via the monitoring of natural frequencies, the onset of snap-through instability. The capability of smart shells to actively modify its natural frequencies such as to promote or mitigate snap-through instabilities is quantified. Additional results quantify the effect of thermomechanical loads on actuation capability. The influence of geometric parameters (curvature and thickness) on the modal response is finally investigated.

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.

Vibration Modes of Packeted Bladed Disks

1984 ◽  
Vol 106 (2) ◽  
pp. 175-180 ◽  
Author(s):  
D. J. Ewins ◽  
M. Imregun
Keyword(s):  
Natural Frequencies ◽  
Turbine Blades ◽  
Vibration Modes ◽  
Nodal Diameter ◽  
Vibrational Behavior ◽  
Bladed Disk ◽  
Methods Of Analysis ◽  
Receptance Coupling ◽  
Modal Interference ◽  
Series Of Experiments

This paper presents the results of investigating the vibrational behavior of turbine blades when grouped into packets. Two methods of analysis based on substructuring via receptance coupling have been developed and used with success to predict the natural frequencies of a 30-bladed disk with various packeting arrangements. A series of experiments have been conducted on a special testpiece to confirm these predictions. It is found that, unlike its continuously shrouded counterpart, the packeted bladed disk has modes which are always complex in shape, containing several nodal diameter components, a feature which can be predicted from the modal interference diagrams introduced in this work.

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