Ritz Series Analysis of Rotating Shaft System: Validation, Convergence, Mode Functions, and Unbalance Response

2002 ◽  
Vol 124 (4) ◽  
pp. 492-501 ◽  
Author(s):  
Nicole L. Zirkelback ◽  
Jerry H. Ginsberg
Keyword(s):  
Equations Of Motion ◽  
Ritz Method ◽  
Natural Mode ◽  
Rotating Shaft ◽  
Disk System ◽  
Element Analysis ◽  
Cross Sectional ◽  
Energy Functionals ◽  
Unbalance Response ◽  
Gyroscopic Coupling

A shaft with attached rigid disks is modeled as a rotating Timoshenko beam supported by nonconservative, flexible bearing supports. The continuous shaft-disk system is described with kinetic and potential energy functionals that fully account for transverse shear, translational and rotatory inertia, and gyroscopic coupling. Ritz series expansions are used to describe the flexural displacements and cross-sectional rotations about orthogonal fixed axes. The equations of motion are derived from Lagrange’s equations and placed in a state-space form that preserves the skew-symmetric gyroscopic matrix as well as the full effects of the bearings. Both the general and adjoint eigenproblems for the nonsymmetric equations are solved. Bi-orthogonality conditions lead to the ability to evaluate dynamic response via modal analysis. Whirl speeds and logarithmic decrements calculated with the present model are verified with a finite element analysis. The present work provides two ways of evaluating the convergence of results to demonstrate an advantage of the Ritz method over other discretization methods. Natural mode functions and unbalance response are calculated for an example system.

Ritz Series Analysis of Rotating Machinery Incorporating Timoshenko Beam Theory

2001 ◽  
Author(s):  
Nicole L. Zirkelback ◽  
Jerry H. Ginsberg
Keyword(s):  
Timoshenko Beam ◽  
Space Form ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Cross Sectional ◽  
Energy Functionals ◽  
Orthogonality Conditions ◽  
Rigid Disks ◽  
Gyroscopic Coupling

A shaft with attached rigid disks is modeled as a rotating Timoshenko beam supported by general compliant, nonconservative bearing supports. The continuous shaft-disk system is described with kinetic and potential energy functionals that fully account for transverse shear, translational and rotatory inertia, and gyroscopic coupling. Ritz series expansions are used to describe the flexural displacements and cross-sectional rotations about orthogonal fixed axes. The equations of motion are derived from Lagrange’s equations and placed in a state-space form that preserves the skew-symmetric gyroscopic matrix, as well as the cross-coupling displacement and velocity coefficient matrices describing the effects of bearings. Both the general and adjoint eigenproblems for the nonsymmetric equations are solved. Bi-orthogonality conditions lead to the ability to evaluate dynamic response via modal analysis. Two examples, which show close agreement with prior analyses of critical speeds, demonstrate the ease with which the method may be applied.

Unbalance Response of Flexible Rotors Coupled With Torsion

1989 ◽  
Vol 111 (2) ◽  
pp. 179-186 ◽  
Author(s):  
H. Diken ◽  
I. G. Tadjbakhsh
Keyword(s):  
Shear Deformation ◽  
Equations Of Motion ◽  
Continuous System ◽  
Internal Damping ◽  
Governing Equations ◽  
Disk System ◽  
Torsional Oscillations ◽  
Unbalance Response ◽  
Flexible Rotors ◽  
Mass Eccentricity

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.

scholarly journals Analytical Solution for Whirling Speeds and Mode Shapes of a Distributed-Mass Shaft With Arbitrary Rigid Disks

2013 ◽  
Vol 81 (3) ◽  
Author(s):  
Jong-Shyong Wu ◽  
Foung-Tang Lin ◽  
Huei-Jou Shaw
Keyword(s):  
Free Vibration ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Natural Mode ◽  
Lumped Mass ◽  
Rotating Shaft ◽  
Disk System ◽  
Mass Moment ◽  
Rigid Disks ◽  
Equivalent Mass

The purpose of this paper is to present an approach for replacing the effects of each rigid disk mounted on the spin shaft by a lumped mass together with a frequency-dependent equivalent mass moment of inertia so that the whirling motion of a rotating shaft-disk system is similar to the transverse free vibration of a stationary beam and the technique for the free vibration analysis of a stationary beam with multiple concentrated elements can be used to determine the forward and backward whirling speeds, along with mode shapes of a distributed-mass shaft carrying arbitrary rigid disks. Numerical results reveal that the characteristics of whirling motions are significantly dependent on the slopes of the associated natural mode shapes at the positions where the rigid disks are located. Furthermore, the results obtained from the presented analytical method and those obtained from existing literature or the finite element method (FEM) are in good agreement.

Modeling and Finite Element Analysis of Drill Bit Vibrations

1989 ◽  
Vol 111 (2) ◽  
pp. 148-155 ◽  
Author(s):  
O. Tekinalp ◽  
A. G. Ulsoy
Keyword(s):  
Finite Element ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Helix Angle ◽  
Drill Bit ◽  
Element Analysis ◽  
Cross Sectional ◽  
Various Boundary Conditions ◽  
Drill Bits ◽  
Euler Bernoulli Beam

Drill bit vibrations are modeled using the Euler-Bernoulli beam theory. The model includes the most important properties of drill bits and of the drilling operation. These are: the drill bit cross sectional geometry, the drill helix angle, rotational speed of the drill bit, the thrust force, torque, and cutting forces generated during drilling. Equations of motion are derived in an inertial frame and then transformed to a rotating fluted frame for convenience in the solution. The transformed equations are discretized using finite element techniques. The finite element code developed is capable of solving the eigenvalue problem for various boundary conditions and drill cross sectional geometries. Finite element solutions are compared to known analytical, numerical, and experimental results from the literature and good agreement is obtained.

Evaluation of web reinforcements in prestressed concrete box girders through shear-transverse bending domains

2020 ◽  
pp. 136943322098170
Author(s):  
Michele Fabio Granata ◽  
Antonino Recupero
Keyword(s):  
Analytical Model ◽  
Prestressed Concrete ◽  
3D Analysis ◽  
Box Girders ◽  
Element Analysis ◽  
Cross Sectional ◽  
Interaction Domains ◽  
Global And Local ◽  
Resultant Forces

In concrete box girders, the amount and distribution of reinforcements in the webs have to be estimated considering the local effects due to eccentric external loads and cross-sectional distortion and not only the global effect due to the resultant forces of a longitudinal analysis: shear, torsion and bending. This work presents an analytical model that allows designers to take into account the interaction of all these effects, global and local, for the determination of the reinforcements. The model is based on the theory of stress fields and it has been compared to a 3D finite element analysis, in order to validate the interaction domains. The results show how the proposed analytical model allows an easy and reliable reinforcement evaluation, in agreement with a more refined 3D analysis but with a reduced computational burden.

scholarly journals Wellbore trajectory control tool seal system leakage analysis based on steady gap flow

2020 ◽  
Vol 12 (6) ◽  
pp. 168781402093046 ◽  
Author(s):  
Lei Shi ◽  
Keqiang Wang ◽  
Ding Feng ◽  
Hong Zhang ◽  
Peng Wang
Keyword(s):  
Prediction Method ◽  
Electronic System ◽  
Contact Length ◽  
Trajectory Control ◽  
Analysis Model ◽  
Rotating Shaft ◽  
Element Analysis ◽  
Gap Flow ◽  
Total Leakage ◽  
Gap Height

Lubricant leakage will inevitably occur during the working process of wellbore trajectory control tools. Even including the lubricant compensation system, serious leakage will still cause lacks lubrication of the internal mechanical structure as well as electronic system damaged by external infiltration fluid, especially when it comes to battery sub and other electronic equipment. Seal system leakage prediction method was presented based on the assumption of steady gap flow. It is assumed that there is a constant gap between the lip seal and the rotating shaft, the gap height is determined by oil film thickness, and the length of the gap was determined by the contact analysis using the Mooney–Rivlin constitutive model. The analysis results show that the contact length between the primary seal lip and the rotary shaft is about 0.1 mm under the condition of ensuring the contact between the deputy seal lip and the rotary shaft. The overall lubricant leakage finite element analysis model was established, and the relationship between the internal lubricant pressure of the tool and the total leakage was obtained. The results of analysis indicate that under the internal pressure of 0.03 MPa, the lubricant leakage is approximately 6 mL/h, which was verified by experiments.

scholarly journals Estimating Deterioration in the Concrete Tie-Ballast Interface Based on Vertical Tie Deflection Profile: A Numerical Study

2016 ◽  
Author(s):  
Hailing Yu
Keyword(s):  
Direct Detection ◽  
Numerical Study ◽  
Interface Condition ◽  
Vertical Deflection ◽  
Cohesive Elements ◽  
Material Loss ◽  
Element Analysis ◽  
Cross Sectional ◽  
Technical Requirements ◽  
Support Conditions

In ballasted concrete tie track, the tie-ballast interface can deteriorate resulting in concrete tie bottom abrasion, ballast pulverization and/or voids in tie-ballast interfaces. Tie-ballast voids toward tie ends can lead to unfavorable center binding support conditions that can result in premature concrete tie failure and possible train derailment. Direct detection of these conditions is difficult. There is a strong interest in assessing the concrete tie-ballast interface conditions indirectly using measured vertical deflections. This paper seeks to establish a link between the vertical deflection profile of a concrete tie top surface and the tie-ballast interface condition using the finite element analysis (FEA) method. The concrete tie is modeled as a concrete matrix embedded with prestressing steel strands or wires. The configurations of two commonly used concrete ties, one with 8 prestressing strands and the other with 20 prestressing wires, are employed in this study. All models are three-dimensional and symmetric about the tie center. A damaged plasticity model that can predict onset and propagation of tensile cracks is applied to the concrete material. The steel-concrete interface is homogenized and represented with a thin layer of cohesive elements sandwiched between steel and concrete elements. Strand- or wire-specific elasto-plastic bond models developed at the Volpe Center are applied to the cohesive elements to account for the interface bonding mechanisms. FE models are developed for both original and worn concrete ties, with the latter assuming hypothetical patterns of reduced cross sections resulting from abrasive interactions with the ballast. Static analyses of pretension release in these concrete ties are conducted, and vertical deflection gradients along tie lengths are calculated and shown to correspond well with the worn cross sectional patterns for a given reinforcement type. The ballast is further modeled with Extended Drucker-Prager plasticity, and hypothetical voids are applied toward the tie ends along the concrete tie-ballast interface to simulate center binding support conditions. The distance range over which the concrete tie is supported in the center is variable and yields different center binding severity. Static simulations are completed with vertical rail seat loads applied on the concrete tie-ballast assembly. The influences of various factors on the vertical deflection profile, including tie type, vertical load magnitude, center binding severity, cross sectional material loss and prestress loss, are examined based on the FEA results. The work presented in this paper demonstrates the potential of using the vertical deflection profile of concrete tie top surfaces to assess deteriorations in the tie-ballast interface. The simulation results further help to clarify minimum technical requirements on inspection technologies that measure concrete tie vertical deflection profiles.

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

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