Modal projections for synchronous rotor whirl

2008 ◽  
Vol 464 (2095) ◽  
pp. 1739-1760 ◽  
Author(s):  
Pradeep Mahadevan ◽  
C.S Jog ◽  
Anindya Chatterjee
Keyword(s):  
Finite Element ◽  
Dynamic Equilibrium ◽  
Mode Shapes ◽  
Simple Problem ◽  
Equilibrium Equations ◽  
Continuum Formulation ◽  
Finite Element Package ◽  
Whirl Speed ◽  
Gyroscopic Effects ◽  
The Continuum

We consider the synchronous whirl of arbitrary axisymmetric rotors supported on rigid bearings. Prior computational treatments of this problem were based on adding element-level gyroscopic terms to the governing equations. Here, we begin with a direct continuum formulation wherein gyroscopic terms need not be added on separately and explicitly: all gyroscopic effects are captured implicitly within the continuum elastodynamics. We present two new methods for obtaining the whirl speed, where we project the dynamic equilibrium equations of the rotor on to a few of its non-spinning vibration mode shapes. The first modal projection method is direct and more accurate, but requires numerical evaluation of more demanding integrals. The second method is iterative and involves a small approximation, but is simpler. Both the methods are based on one new insight: the gyroscopic terms used in other treatments are essentially the result of a prestress in the rotor caused by the non-zero spin rate, and may be incorporated as such in the continuum formulation. The accuracy of the results obtained, for several examples, is verified against detailed calculations with a commercial finite-element package, against our own nonlinear finite-element code or against analytical estimates. For further verification and illustration, a closed-form analytical solution for a simple problem, obtained using our method, matches the results obtained with explicit gyroscopic terms.

scholarly journals Effect of Location of Delamination on Free Vibration of Cross-Ply Conical Shells

2012 ◽  
Vol 19 (4) ◽  
pp. 679-692 ◽  
Author(s):  
Sudip Dey ◽  
Amit Karmakar
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Natural Frequencies ◽  
Dynamic Equilibrium ◽  
Twist Angle ◽  
Structural Instability ◽  
Numerical Results ◽  
Mode Shapes ◽  
Iteration Algorithm ◽  
Conical Shells

Location of delamination is a triggering parameter for structural instability of laminated composites. In this paper, a finite element method is employed to determine the effects of location of delamination on free vibration characteristics of graphite-epoxy cross-ply composite pre-twisted shallow conical shells. The generalized dynamic equilibrium equation is derived from Lagrange's equation of motion neglecting Coriolis effect for moderate rotational speeds. The formulation is exercised by using an eight noded isoparametric plate bending element based on Mindlin's theory. Multi-point constraint algorithm is utilized to ensure the compatibility of deformation and equilibrium of resultant forces and moments at the delamination crack front. The standard eigen value problem is solved by applying the QR iteration algorithm. Finite element codes are developed to obtain the numerical results concerning the effects of location of delamination, twist angle and rotational speed on the natural frequencies of cross-ply composite shallow conical shells. The mode shapes are also depicted for a typical laminate configuration. Numerical results obtained from parametric studies of both symmetric and anti-symmetric cross-ply laminates are the first known non-dimensional natural frequencies for the type of analyses carried out here.

scholarly journals Vibration Measurement of Rotating Blades Using a Root Embedded PZT Sensor

2008 ◽  
Vol 15 (5) ◽  
pp. 517-541 ◽  
Author(s):  
M. Sunar ◽  
B.O. Al-Bedoor
Keyword(s):  
Finite Element ◽  
Experimental Studies ◽  
Vibration Measurement ◽  
Mode Shapes ◽  
Rotating System ◽  
Blade Vibration ◽  
Wide Range ◽  
Finite Element Package ◽  
Steady State Responses ◽  
State Responses

Finite element and experimental studies are carried out to test the suitability of a piezoelectric (PZT) sensor in measuring vibrations of blades modeled as beams. The rotating system contains twelve blades mounted to the shaft through a rotor. The PZT sensor is secured in the root between the rotor and blade. First, finite element results are obtained using the finite element package ANSYS. A modal analysis is performed on the system to identify modes and mode shapes. Transient, harmonic and steady-state responses are then computed to test the ability of the PZT sensor in generating signals for blade vibrations. For the experimental part, the blade vibration signals are produced using the PZT sensor and a strain-gage, and the outputs are compared with each other. From both the finite element and experimental results, it is concluded that the root-embedded PZT sensor can be effectively used for blade vibration measurements in a wide range of cases.

A New Approach for Plate Vibrations: Combination of Transfer Matrix and Finite-Element Technique

1972 ◽  
Vol 94 (2) ◽  
pp. 526-530 ◽  
Author(s):  
M. A. Dokainish
Keyword(s):  
Finite Element ◽  
Transfer Matrix ◽  
Present Method ◽  
Degrees Of Freedom ◽  
Matrix Multiplication ◽  
Mode Shapes ◽  
Equilibrium Equations ◽  
Plate Vibrations ◽  
Left And Right ◽  
Element Technique

When the finite-element method is used in the vibration analysis of plates and shells, it results in large matrices requiring a large digital computer. A commonly used method of reducing the matrix size is to eliminate certain “slave” displacements by minimizing strain energy. The approach requires good judgement in the selection of the “master” displacements and involves additional approximations and some loss of accuracy. In the present method small matrices are obtained without any further approximations and without reducing the number of degrees of freedom. The transfer matrix technique, generally known as the Holzer-Myklestad method, is well known for beams and shafts. The present method is an extension of this idea to plates. The structure is divided into several strips, with a number of nodes on the left and right sections of each strip. Each strip is subdivided into elements and the stiffness and mass matrices are obtained for individual strips. The nodal equilibrium equations are rearranged to obtain a relation between the section variables of the left and the right sections. The section variables are the forces and the displacements of all the nodes on the section. Requirements of displacement continuity and force equilibrium at the nodes, on common sections of two adjacent strips, gives the transfer matrix relation. Successive matrix multiplication finally relates the variables of the left and right boundary of the structure. Boundary conditions require the determinant of a portion of the overall transfer matrix to vanish at the correct frequency. By calculating the determinant at various assumed values of frequency, the correct frequencies are obtained. The method also gives the corresponding mode shapes. The method as applied to several plate problems gives satisfactory results.

Experimental and Analytical Study of Coriolis Effects in Bladed Disk

2015 ◽  
Author(s):  
Valentina Ruffini ◽  
Christoph Schwingshackl ◽  
Jeff Green
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Measurement Data ◽  
Travelling Wave ◽  
Experimental Results ◽  
Mode Shapes ◽  
Element Analysis ◽  
Coriolis Effects ◽  
Gyroscopic Effects ◽  
Gyroscopic Coupling

Modern aero-engines have reached a high level of sophistication and only significant changes will lead to the improvements necessary to achieve the economic and environmental targets of the future. Open rotors constitute a major leap in this direction, both in terms of efficiency and of technological innovation. This calls for a revision of the accepted design practices, and a new focus on phenomena that have been little investigated in the past, such as the Coriolis effect, or the gyroscopic coupling of the blades with the shaft. Experimental results from modern fans, with large blades and strong stagger angles, are showing dependence on Coriolis gyroscopic effects already, an effect that is expected to be strongly enhanced with the proposed open rotor designs. For an accurate prediction of the Coriolis and gyroscopic effects in rotating assemblies a fully experimentally validated approach is needed. Today’s FE models can capture the basic physical phenomena, but experimental confirmation is still needed for the evolution of the mode shapes with angular speed, and the influence of damping and geometric nonlinearities when gyroscopic coupling is considered. To support this validation effort a new rotating test rig will be introduced, initial measurement data will be discussed, and a comparison with a finite element analysis presented. Different forcing patterns, including forward and backward travelling-wave engine order excitation could be experimentally excited in the new rig, Coriolis-induced frequency splits were found in the dynamic response, showing a significant change in the dynamic behaviour of the investigated dummy disk, and only a minor impact of the mistuning was observed on the frequency splits due to Coriolis effects. The experimental results have been compared to a finite element analysis, and after some updating a good agreement between the predicted and measured Campbell diagrams could be obtained, demonstrating the reliability of the modelling approach.

Free vibration response of carbon nanotube reinforced pretwisted conical shell under thermal environment

2019 ◽  
Vol 234 (3) ◽  
pp. 770-783 ◽  
Author(s):  
Pabitra Maji ◽  
Mrutyunjay Rout ◽  
Amit Karmakar
Keyword(s):  
Carbon Nanotubes ◽  
Finite Element ◽  
Carbon Nanotube ◽  
Free Vibration ◽  
Conical Shell ◽  
Dynamic Equilibrium ◽  
Thermal Environment ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Mode Shapes

Finite element procedure is employed to analyze the free vibration characteristics of rotating functionally graded carbon nanotubes reinforced composite conical shell with pretwist under the thermal environment. In this paper, four types of carbon nanotube grading are considered, wherein the distribution of carbon nanotubes are made through the thickness direction of the conical shell. An eight-noded isoparametric shell element is used in the present formulation to model the panel based on the first-order shear deformation theory. For moderate rotational speeds, the generalized dynamic equilibrium equation is derived from Lagrange’s equation of motion, neglecting the Coriolis effect. The finite element code is developed to investigate the effect of twist angle, temperature, aspect ratio, and rotational speed on natural frequencies. The mode shapes of a carbon nanotube reinforced functionally graded conical shell at different twist angles and rotational speeds are also presented.

scholarly journals Axisymmetric Free Vibration of Thick Orthotropic Hemispherical Shells Under Various Edge Conditions

1991 ◽  
Author(s):  
C. C. Chao ◽  
T. P. Tung ◽  
Y. C. Chern
Keyword(s):  
Free Vibration ◽  
Shear Strain ◽  
Transverse Shear ◽  
Natural Frequencies ◽  
Dynamic Equilibrium ◽  
Elastic Shell ◽  
Mode Shapes ◽  
Transverse Shear Strain ◽  
Equilibrium Equations ◽  
Hemispherical Shells

Abstract Axisymmetric free vibration of moderately thick polar orthotropic hemispherical shells are studied under the various boundary conditions of sliding, guided pin, clamped and hinged edges. Based on the improved linear elastic shell theory with the transverse shear strain and rotatory inertia taken into account, the dynamic equilibrium equations are formulated and transformed into the displacement form in terms of mid-surface meridian and radial displacements and parallel circle cross-section rotation. These partial differential equations are solved by the Galerkin method using proper Legendre polynomials as admissible displacement functions with the aid of the orthogonality and a number of special integral relations. Numerical results of the present theory compare well with existing data, which is available only in the isotropic theories. Good convergence is obtained for natural frequencies and mode shapes. Study of the effects of thickness and modulus ratio reveals higher frequencies for the thicker and/or stiffer shells with E\ oriented parallel to the meridians. Ranking of the natural frequencies descends in the order of guided pins, sliding, clamped and hinged edges in general. Also seen are the effects of transverse shear strain from the mode shapes with clamped and sliding edges on the slant. For the guided pin and sliding edges, frequencies increase fast as thickness increases so that new fundamental modes are generated in filling up the “frequency gap”. These are the new discoveries in the field of anisotropic shells, as a result of polar orthotropy of shell material and construction.

NUMERICAL ANALYSIS OF THE NON-LINEAR DYNAMIC BEHAVIOUR OF SUSPENDED CABLES UNDER TURBULENT WIND EXCITATION

2001 ◽  
Vol 01 (02) ◽  
pp. 207-233 ◽  
Author(s):  
L. MARTINELLI ◽  
F. PEROTTI
Keyword(s):  
Finite Element ◽  
Internal Resonance ◽  
Dynamic Behaviour ◽  
Dynamic Equilibrium ◽  
Numerical Procedure ◽  
Equilibrium Equations ◽  
Turbulent Wind ◽  
Linear Dynamic ◽  
Suspended Cables ◽  
Elastic Cables

In this paper, a numerical procedure is presented for the dynamic analysis of elastic cables subjected to turbulent wind excitation in quasi-steady conditions. The proposed methodology, which takes geometrical and aerodynamic non-linearities into account, is based on artificial simulation of turbulence, on finite-element modeling of the cables and on a step-by-step implicit procedure for the integration of the dynamic equilibrium equations. As a first application, the dynamic behaviour of a cable in 1 : 2 internal resonance conditions is studied, focusing on some aspects of the influence of wind turbulence on galloping oscillations.

Free vibration analysis of pretwisted delaminated composite stiffened shallow shells: A finite element approach

2017 ◽  
Vol 36 (8) ◽  
pp. 619-636 ◽  
Author(s):  
Mrutyunjay Rout ◽  
Tanmoy Bandyopadhyay ◽  
Amit Karmakar
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Dynamic Equilibrium ◽  
Twist Angle ◽  
Cylindrical Shells ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Shell Element ◽  
Mode Shapes ◽  
Shallow Shells

This paper presents the effect of stiffeners on the free vibration response of delaminated composite shallow cylindrical shells employing the finite element method. An eight-noded isoparametric shell element based on the first-order shear deformation theory is combined with a three-noded isoparametric curved beam element in the present formulation. The stiffeners follow the nodal lines of the shell wherein the stiffness and mass of the stiffeners are lumped at the corresponding nodal points of the shell elements considering curvature and eccentricity. The generalized dynamic equilibrium equation is derived from Lagrange’s equation of motion, wherein Coriolis effect for moderate rotational speeds is neglected. The multi-point constraint algorithm has been used to model delamination at the desired locations wherein the compatibility of deformation and equilibrium of stress resultants are ensured at the delamination crack front. Numerical results are presented for cantilevered long, intermediate and short cylindrical shells as defined by Aas-Jakobsen’s parameters, and the influence of important parameters like location of delamination, twist angle, rotational speed, number of layers and eccentricity of the stiffeners is studied. The mode shapes for a typical composite un-stiffened and stiffened long cylindrical shell at different rotational speeds and twist angles are also presented.

On the Compatibility Equations in Geometrically Exact Beam Finite Element

2016 ◽  
Author(s):  
Valentin Sonneville ◽  
Olivier A. Bauchau ◽  
Olivier Brüls
Keyword(s):  
Finite Element ◽  
Velocity Field ◽  
Spatial Interpolation ◽  
Dynamic Equilibrium ◽  
Beam Theory ◽  
Body Motion ◽  
Equilibrium Equations ◽  
Path Independence ◽  
Geometrically Exact Beam ◽  
Geometrically Exact Beam Theory

This paper discusses the compatibility equations which relate the velocity field and the strain field in geometrically exact beam theory. The analysis is carried out in the context of intrinsic equations, namely the dynamic equilibrium equations are formulated in terms of local velocities and strains only. In addition to the well established objectivity and path-independence requirements of the spatial discretization, these compatibility equations show that a consistent spatial interpolation of the velocity field should depend on the curvature of the beam, including initial curvature and curvature from the deformation, and it is shown that this consistency is connected to the ability of the finite element to represent exactly rigid body motion velocities. A two node interpolation scheme is studied and it appears that, as the element gets smaller under mesh refinement, the effect of this dependency reduces, leading eventually to the classical linear shape functions.

Effect of crack on free vibration of a pre-stressed curved plate

2021 ◽  
pp. 095440622199486
Author(s):  
Can Gonenli ◽  
Hasan Ozturk ◽  
Oguzhan Das
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Natural Frequency ◽  
Present Model ◽  
Mode Shapes ◽  
Load Parameter ◽  
Flexible Plate ◽  
Cantilever Plate ◽  
Finite Element Software ◽  
Aspect Ratios

In this study, the effect of crack on free vibration of a large deflected cantilever plate, which forms the case of a pre-stressed curved plate, is investigated. A distributed load is applied at the free edge of a thin cantilever plate. Then, the loading edge of the deflected plate is fixed to obtain a pre-stressed curved plate. The large deflection equation provides the non - linear deflection curve of the large deflected flexible plate. The thin curved plate is modeled by using the finite element method with a four-node quadrilateral element. Three different aspect ratios are used to examine the effect of crack. The effect of crack and its location on the natural frequency parameter is given in tables and graphs. Also, the natural frequency parameters of the present model are compared with the finite element software results to verify the reliability and validity of the present model. This study shows that the different mode shapes are occurred due to the change of load parameter, and these different mode shapes cause a change in the effect of crack.

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