Steady-State Response of a Flexibly Mounted Stator Mechanical Face Seal Subject to Dynamic Forcing of a Flexible Rotor

2017 ◽  
Vol 139 (6) ◽  
Author(s):  
Philip Varney ◽  
Itzhak Green
Keyword(s):  
Steady State ◽  
Equations Of Motion ◽  
External Input ◽  
Face Seal ◽  
Flexible Rotor ◽  
State Equations ◽  
Fluid Forces ◽  
Flexible Rotors ◽  
Successful Design ◽  
Constitutive Components

Mechanical face seals are constitutive components of much larger turbomachines and require consideration of the system dynamics for successful design. The dynamic interplay between the seal and rotor is intensified by recent trends toward reduced clearances, higher speeds, and more flexible rotors. Here, the “rotor” consists of the flexible shaft and the rotating seal seat. The objective here is to, for the first time, determine how the rotor affects the seal performance and vice versa. Thresholds can then be established beyond which the rotor influences the seal but not vice versa (i.e., the rotordynamics can be sent to the seal analysis as an exogenous input). To this end, a model of a flexibly mounted stator face seal is provided including the coupled dynamics of the flexible rotor. The model accounts for axial and angular deflections of the rotor and seal. Coupled rotordynamics are modeled using a lumped-parameter approach including static and dynamic rotor angular misalignments. For expediency, linearized expressions for fluid forces are used, and the resulting steady-state equations of motion are solved analytically to investigate how rotor inertia, speed, and angular misalignment influence the coupled seal dynamics. Importantly, results from the study reveal that in some operating regimes, neglecting the rotordynamics implies healthy seal operation when instead intermittent rub exists between the faces. This work also shows that when the rotor inertia is much larger than the seal inertia, the rotordynamics can be solved separately and used in the seal model as an external input.

Dynamic modeling of an eccentric face seal including coupled rotordynamics, face contact, and inertial maneuver loads

2017 ◽  
Vol 232 (6) ◽  
pp. 732-748
Author(s):  
Philip Varney ◽  
Itzhak Green
Keyword(s):  
Dynamic Modeling ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Contact Friction ◽  
Face Seal ◽  
Transient Operation ◽  
Angular Misalignment ◽  
The Face ◽  
Mechanical Face Seal ◽  
Constitutive Components

Mechanical face seals are constitutive components of turbomachines, which in turn can be constitutive to other systems (e.g. aircraft). Furthermore, the rotating element of a face seal is inextricably coupled to the turbomachine via a flexible mount, and the stationary seal element is coupled to the rotating seal element via the fluid film existing between the seal faces. Consequentially, understanding interactions between the seal and turbomachine is important for quantifying seal performance and improving its design. With few exceptions, previous works study the face seal dynamics independent from the rotordynamics. In addition, most prior investigations consider only angular and axial seal dynamics and neglect eccentric (i.e. lateral) deflections of the seal element(s). For the first time, this work develops a comprehensive and novel model of a mechanical face seal in the inertial reference frame including coupled rotordynamics and inertial maneuver loads of the overall system. The model is developed for a general seal geometry where both seal elements, stationary and rotating, are flexibly mounted and allowed to undergo angular, axial, and eccentric deflections. In addition, the seal model presented here accounts for transient operation, fluid shear forces, seal face contact, friction, and thermoelastic deformation. Finally, various faults due to manufacturing imperfections, component flaws, and/or installation errors can be accounted for by incorporating static angular misalignment of both seal elements, dynamic angular misalignment of the rotating seal element, eccentric rotating imbalance, and axial offset of the rotating seal element center of mass. Throughout this work, the equations of motion developed are valid for both steady-state and transient operation. This comprehensive model significantly advances the state of the art in mechanical face seal dynamic modeling and represents a pivotal step towards analyzing seal performance regarding a broad diversity of realistic problems.

The Stokes-flow drag on prolate and oblate spheroids during axial translatory accelerations

1972 ◽  
Vol 52 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Robert Y. S. Lai ◽  
Lyle F. Mockros
Keyword(s):  
Steady State ◽  
Flow Field ◽  
Equations Of Motion ◽  
Mass Effect ◽  
Added Mass ◽  
Fluid Forces ◽  
Fluid Resistance ◽  
Oblate Spheroids ◽  
History Of ◽  
Flow Drag

Stokes's linearized equations of motion are used to calculate the flow field generated by a spheroid executing axial translatory oscillations in an infinite, otherwise still, incompressible, viscous fluid. The flow field, expressed in terms of spheroidal wave functions of order one, is used to develop general expressions for the drag on oscillating prolate and oblate spheroids. Formulae for the approximate drag, useful in making calculations, are obtained for small values of an oscillation parameter. These formulae reduce to the Stokes result in the limit when the spheroid becomes a sphere and the steady-state drag for a spheroid as the frequency of oscillation becomes zero. The fluid forces on spheroids of various shapes are compared graphically. The approximate formulae for the drag are integrated over all frequencies to obtain formulae for the drag on spheroids executing general axial translatory accelerations. The fluid resistance on the spheroid is expressed as the sum of an added mass effect, a steady-state drag and an effect due to the history of the motion. A table of added mass, viscous and history coefficients is given.

scholarly journals The formation of vortices from a surface of discontinuity

1931 ◽  
Vol 134 (823) ◽  
pp. 170-192 ◽  
Keyword(s):  
Steady State ◽  
Plane Surface ◽  
Equations Of Motion ◽  
Steady Motion ◽  
Small Oscillations ◽  
Approximate Form ◽  
History Of ◽  
State Increase ◽  
Liquid Surfaces

Helmholtz was the first to remark on the instability of those “liquid surfaces” which separate portions of fluid moving with different velocities, and Kelvin, in investigating the influence of wind on waves in water, supposed frictionless, has discussed the conditions under which a plane surface of water becomes unstable. Adopting Kelvin’s method, Rayleigh investigated the instability of a surface of discontinuity. A clear and easily accessible rendering of the discussion is given by Lamb. The above investigations are conducted upon the well-known principle of “small oscillations”—there is a basic steady motion, upon which is superposed a flow, the squares of whose components of velocity can be neglected. This method has the advantage of making the equations of motion linear. If by this method the flow is found to be stable, the equations of motion give the subsequent history of the system, for the small oscillations about the steady state always remain “small.” If, however, the method indicates that the system is unstable, that is, if the deviations from the steady state increase exponentially with the time, the assumption of small motions cannot, after an appropriate interval of time, be applied to the case under consideration, and the equations of motion, in their approximate form, no longer give a picture of the flow. For this reason, which is well known, the investigations of Rayleigh only prove the existence of instability during the initial stages of the motion. It is the object of this note to investigate the form assumed by the surface of discontinuity when the displacements and velocities are no longer small.

Sensitivity Analysis for the Steady-State Response of Damped Linear Elastodynamic Systems Subject to Periodic Loads

1990 ◽  
Author(s):  
Daniel A. Tortorelli
Keyword(s):  
Steady State ◽  
Vibration Amplitude ◽  
Equations Of Motion ◽  
Steady State Response ◽  
Analysis Techniques ◽  
Direct Differentiation ◽  
State Response ◽  
Need To Evaluate ◽  
The Time Domain ◽  
General Response

Abstract Adjoint and direct differentiation methods are used to formulate design sensitivities for the steady-state response of damped linear elastodynamic systems that are subject to periodic loads. Variations of a general response functional are expressed in explicit form with respect to design field perturbations. Modal analysis techniques which uncouple the equations of motion are used to perform the analyses. In this way, it is possible to obtain closed form relations for the sensitivity expressions. This eliminates the need to evaluate the adjoint response and psuedo response (these responses are associated with the adjoint and direct differentiation sensitivity problems) over the time domain. The sensitivities need not be numerically integrated over time, thus they are quickly computed. The methodology is valid for problems with proportional as well as non-proportional damping. In an example problem, sensitivities of steady-state vibration amplitude of a crankshaft subject to engine firing loads are evaluated with respect to the stiffness, inertial, and damping parameters which define the shaft. Both the adjoint and direct differentiation methods are used to compute the sensitivities. Finite difference sensitivity approximations are also calculated to validate the explicit sensitivity results.

Effect of Material and Geometric Parameters on the Steady-State Belt Stresses and Belt Slip for Flat Belt-Drives

2015 ◽  
Author(s):  
Cagkan Yildiz ◽  
Tamer M. Wasfy ◽  
Hatem M. Wasfy ◽  
Jeanne M. Peters
Keyword(s):  
Steady State ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Flexible Multibody Dynamics ◽  
Friction Model ◽  
Rigid Bodies ◽  
Geometric Parameters ◽  
Rubber Matrix ◽  
Axial Stiffness ◽  
Belt Drive

In order to accurately predict the fatigue life and wear life of a belt, the various stresses that the belt is subjected to and the belt slip over the pulleys must be accurately calculated. In this paper, the effect of material and geometric parameters on the steady-state stresses (including normal, tangential and axial stresses), average belt slip for a flat belt, and belt-drive energy efficiency is studied using a high-fidelity flexible multibody dynamics model of the belt-drive. The belt’s rubber matrix is modeled using three-dimensional brick elements and the belt’s reinforcements are modeled using one dimensional truss elements. Friction between the belt and the pulleys is modeled using an asperity-based Coulomb friction model. The pulleys are modeled as cylindrical rigid bodies. The equations of motion are integrated using a time-accurate explicit solution procedure. The material parameters studied are the belt-pulley friction coefficient and the belt axial stiffness and damping. The geometric parameters studied are the belt thickness and the pulleys’ centers distance.

Validation of a new method for determination of free fatty acid turnover

1987 ◽  
Vol 252 (3) ◽  
pp. E431-E438 ◽  
Author(s):  
J. M. Miles ◽  
M. G. Ellman ◽  
K. L. McClean ◽  
M. D. Jensen
Keyword(s):  
Steady State ◽  
Fatty Acid ◽  
Free Fatty Acid ◽  
Clearance Rate ◽  
Experimental Period ◽  
Metabolic Clearance ◽  
Metabolic Clearance Rate ◽  
State Equations ◽  
Tracer Methods

The accuracy of tracer methods for estimating free fatty acid (FFA) rate of appearance (Ra), either under steady-state conditions or under non-steady-state conditions, has not been previously investigated. In the present study, endogenous lipolysis (traced with 14C palmitate) was suppressed in six mongrel dogs with a high-carbohydrate meal 10 h before the experiment, together with infusions of glucose, propranolol, and nicotinic acid during the experimental period. Both steady-state and non-steady-state equations were used to determine oleate Ra ([3H]oleate) before, during, and after a stepwise infusion of an oleic acid emulsion. Palmitate Ra did not change during the experiment. Steady-state equations gave the best estimates of oleate inflow approximately 93% of the known oleate infusion rate overall, while errors in tracer estimates of inflow were obtained when non-steady-state equations were used. The metabolic clearance rate of oleate was inversely related to plasma concentration (P less than 0.01). In conclusion, accurate estimates of FFA inflow were obtained when steady-state equations were used, even under conditions of abrupt and recent changes in Ra. Non-steady-state equations, in contrast, may provide erroneous estimates of inflow. The decrease in metabolic clearance rate during exogenous infusion of oleate suggests that FFA transport may follow second-order kinetics.

Nonlinear Dynamics of Flexible Rotors Supported on Journal Bearings—Part I: Analytical Bearing Model

2017 ◽  
Vol 140 (2) ◽  
Author(s):  
Mohammad Miraskari ◽  
Farzad Hemmati ◽  
Mohamed S. Gadala
Keyword(s):  
Journal Bearing ◽  
Nonlinear Coefficient ◽  
Journal Bearings ◽  
Flexible Rotor ◽  
Dynamic Coefficients ◽  
Flexible Rotors ◽  
Rotor Bearing ◽  
Bearing Force ◽  
Derivatives Of ◽  
Bearing System

To determine the bifurcation types in a rotor-bearing system, it is required to find higher order derivatives of the bearing forces with respect to journal velocity and position. As closed-form expressions for journal bearing force are not generally available, Hopf bifurcation studies of rotor-bearing systems have been limited to simple geometries and cavitation models. To solve this problem, an alternative nonlinear coefficient-based method for representing the bearing force is presented in this study. A flexible rotor-bearing system is presented for which bearing force is modeled with linear and nonlinear dynamic coefficients. The proposed nonlinear coefficient-based model was found to be successful in predicting the bifurcation types of the system as well as predicting the system dynamics and trajectories at spin speeds below and above the threshold speed of instability.

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