Fluid-Structure Interaction Forces at Pump-Impeller-Shroud Surfaces for Rotordynamic Calculations

1989 ◽  
Vol 111 (3) ◽  
pp. 216-225 ◽  
Author(s):  
D. W. Childs
Keyword(s):  
Equations Of Motion ◽  
Perturbation Expansion ◽  
Momentum Equation ◽  
Governing Equations ◽  
Pump Impeller ◽  
First Order ◽  
Pressure Distributions ◽  
Pump Housing ◽  
Reaction Forces ◽  
Ring Seal

Governing equations of motion are derived for a bulk-flow model of the leakage path between an impeller shroud and a pump housing. The governing equations consist of a path-momentum, a circumferential-momentum, and a continuity equation. The fluid annulus between the impeller shroud and pump housing is assumed to be circumferentially symmetric when the impeller is centered; i.e., the clearance can vary along the pump axis but does not vary in the circumferential direction. A perturbation expansion of the governing equations in the eccentricity ratio yields a set of zeroth and first-order governing equations. The zeroth-order equations define the leakage rate and the circumferential and path velocity distributions and pressure distributions for a centered impeller position. The first-order equations define the perturbations in the velocity and pressure distributions due to either a radial-displacement perturbation or a tilt perturbation of the impeller. Integration of the perturbed pressure and shear-stress distribution acting on the rotor yields the reaction forces and moments acting on the impeller face. Calculated results yield predictions of possible resonance peaks of the fluid within the annulus formed by the impeller shroud and housing. Centrifugal acceleration terms in the path-momentum equation are the physical origin of these unexpected predictions. For normalized tangential velocities at the inlet to the annulus, uθ0(0) = Uθ0(0)/Riω of 0.5, the phenomenon is relatively minor. As uθ0(0) is increased to 0.7, sharp peaks are predicted. Comparisons for rotordynamic coefficient predictions with test results of Bolleter et al. show reasonable agreement for cross-coupled stiffness and direct damping terms. Calculated results are provided which make comparisons between seal forces and shroud forces for a typical impeller/wear-ring-seal combination.

Centrifugal-Acceleration Modes for Incompressible Fluid in the Leakage Annulus Between a Shrouded Pump Impeller and Its Housing

1991 ◽  
Vol 113 (2) ◽  
pp. 209-218 ◽  
Author(s):  
D. W. Childs
Keyword(s):  
Natural Frequency ◽  
Perturbation Expansion ◽  
Mode Shapes ◽  
Perturbation Equation ◽  
Governing Equations ◽  
Centrifugal Acceleration ◽  
Pump Impeller ◽  
First Order ◽  
Pressure Distributions ◽  
Reaction Forces

An analysis is presented for the perturbed flow in the leakage path between a shrouded-pump impeller and its housing. A bulk-flow model is used for the analysis consisting of the path-momentum, circumferential-momentum, and continuity equations. Shear stress at the impeller and housing surfaces are modeled according to Hirs’ turbulent lubrication model. The governing equations have been used earlier to examine rotordynamic reaction forces developed by lateral and axial impeller motion. A perturbation expansion of the governing equations in the eccentricity ratio yields a set of zeroth and first-order governing equations. The zeroth-order equations define the leakage rate, and the velocity and pressure distributions for a centered impeller position. The first-order equations define the perturbations in the velocity and pressure distributions due to axial or lateral motion of the impeller. Prior analyses by the author of the perturbation equation have examined the reaction forces on the shroud due to rotor motion. These analyses have produced “resonance” phenomena associated with the centrifugal-acceleration body forces in the fluid field. In the present analysis, an algorithm is developed and demonstrated for calculating the complex eigenvalues and eigenvectors associated with these resonances. First-and second-natural-frequency eigensolutions are presented for mode shapes corresponding to lateral excitation. First-natural-frequency eigensolutions are also presented for mode shapes corresponding to axial excitation.

Fluid-Structure Interaction Forces at Pump-Impeller-Shroud Surfaces for Axial Vibration Analysis

1991 ◽  
Vol 113 (1) ◽  
pp. 108-115 ◽  
Author(s):  
D. W. Childs
Keyword(s):  
Excitation Frequency ◽  
Perturbation Expansion ◽  
Momentum Equation ◽  
Shear Stresses ◽  
Governing Equations ◽  
Running Speed ◽  
Pump Impeller ◽  
First Order ◽  
Continuity Equations ◽  
Axial Forces

Solutions are presented for the dynamic axial forces developed by pump-impeller-shroud surfaces. A bulk-flow model of the leakage path between the impeller and the housing is used for the analysis consisting of the path-momentum, circumferential-momentum, and continuity equations. Shear stresses at the impeller and housing surfaces are modeled according to Hirs’ turbulent lubrication model. The governing equations were developed earlier to examine lateral rotordynamic forces developed by impellers. A perturbation expansion of the governing equations in the eccentricity ratio yields a set of zeroth and first-order governing equations. The zeroth-order equations define the leakage rate, velocity distributions, and the pressure distribution for a centered impeller position. The first-order equations define the perturbations in the velocity and pressure distributions due to axial motion of the impeller. Integration of the perturbed pressure and shear-stress distribution acting on the rotor yields the reaction forces acting on the impeller face. Calculated results yield predictions of resonance peaks of the fluid within the annulus formed by the impeller shroud and housing. Centrifugal acceleration terms in the path-momentum equation are the physical origin of these unexpected predictions. For normalized tangential velocities at the inlet to the annulus, uθo(0) = Uθo(0)/Riω of 0.5, the phenomenon is relatively minor. As uθo(0) is increased to 0.7, sharper peaks are predicted. The fluid modes are well damped in all cases. Numerical results are presented for a double-suction single-stage pump which indicate that the direct stiffness of the perturbed impeller shroud forces are negligible. Small but appreciable added-mass and damping terms are developed which have a modest influence on damping and peak-amplitude excitation frequency. The forces only became important for pumps with very low axial natural frequencies in comparison to the running speed, viz., ten percent of the running speed or lower.

Consistent section-averaged equations of quasi-one-dimensional laminar flow

2010 ◽  
Vol 656 ◽  
pp. 337-341 ◽  
Author(s):  
PAOLO LUCHINI ◽  
FRANÇOIS CHARRU
Keyword(s):  
Equations Of Motion ◽  
Stability Result ◽  
Momentum Equation ◽  
Dissipation Function ◽  
Dimensional Solution ◽  
First Order ◽  
Averaged Equations ◽  
Classical Stability ◽  
Momentum Integral ◽  
Free Falling

Section-averaged equations of motion, widely adopted for slowly varying flows in pipes, channels and thin films, are usually derived from the momentum integral on a heuristic basis, although this formulation is affected by known inconsistencies. We show that starting from the energy rather than the momentum equation makes it become consistent to first order in the slowness parameter, giving the same results that have been provided until today only by a much more laborious two-dimensional solution. The kinetic-energy equation correctly provides the pressure gradient because with a suitable normalization the first-order correction to the dissipation function is identically zero. The momentum equation then correctly provides the wall shear stress. As an example, the classical stability result for a free falling liquid film is recovered straightforwardly.

Dynamic Response of Cantilever FGM Cylindrical Shell

2011 ◽  
Vol 130-134 ◽  
pp. 3986-3993 ◽  
Author(s):  
Yu Xin Hao ◽  
Wei Zhang ◽  
L. Yang ◽  
J.H. Wang
Keyword(s):  
Cylindrical Shell ◽  
Volume Fraction ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Functionally Graded ◽  
Governing Equations ◽  
Temperature Dependent ◽  
Graded Materials ◽  
First Order ◽  
Two Degree Of Freedom

An analysis on the nonlinear dynamics of a cantilever functionally graded materials (FGM) cylindrical shell subjected to the transversal excitation is presented in thermal environment.Material properties are assumed to be temperature-dependent. Based on the Reddy’s first-order shell theory,the nonlinear governing equations of motion for the FGM cylindrical shell are derived using the Hamilton’s principle. The Galerkin’s method is utilized to discretize the governing partial equations to a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms under combined external excitations. It is our desirable to choose a suitable mode function to satisfy the first two modes of transverse nonlinear oscillations and the boundary conditions for the cantilever FGM cylindrical shell. Numerical method is used to find that in the case of non-internal resonance the transverse amplitude are decreased by increasing the volume fraction index N.

scholarly journals Nonlinear Vibration of the Blade with Variable Thickness

2020 ◽  
Vol 2020 ◽  
pp. 1-18 ◽  
Author(s):  
Xiaobo Jie ◽  
Wei Zhang ◽  
Jiajia Mao
Keyword(s):  
Variable Thickness ◽  
Conical Shell ◽  
Deformation Theory ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Dynamic Responses ◽  
Nonlinear Ordinary Differential Equations ◽  
Nonlinear Relationship ◽  
Governing Equations ◽  
First Order

In this paper, the nonlinear dynamic responses of the blade with variable thickness are investigated by simulating it as a rotating pretwisted cantilever conical shell with variable thickness. The governing equations of motion are derived based on the von Kármán nonlinear relationship, Hamilton’s principle, and the first-order shear deformation theory. Galerkin’s method is employed to transform the partial differential governing equations of motion to a set of nonlinear ordinary differential equations. Then, some important numerical results are presented in terms of significant input parameters.

Structural Dynamics Analysis of Rotating Blades Using Fully Intrinsic Equations, Part I: Formulation and Verification of Single-Load-Path Configurations

2013 ◽  
Vol 58 (3) ◽  
pp. 1-9 ◽  
Author(s):  
Zahra Sotoudeh ◽  
Dewey H. Hodges
Keyword(s):  
Structural Dynamics ◽  
Equations Of Motion ◽  
Turbine Blades ◽  
Rotor Blades ◽  
Load Path ◽  
Dynamics Modeling ◽  
Governing Equations ◽  
Wind Turbine Blades ◽  
First Order ◽  
Infinite Degree

As part of an ongoing investigation into potential advantages of so-called fully intrinsic formulations, this paper presents an application of the fully intrinsic equations of motion and kinematics for beams to rotor blades. A fully intrinsic formulation is devoid of displacement and rotation variables. Although the governing equations are geometrically exact, they are free of the attendant singularities and infinite-degree nonlinearities found in other types of formulations. These nonlinear, first-order partial differential equations are suitable for analyzing initially curved and twisted, anisotropic beams and thus are very attractive for analysis of both helicopter and wind turbine blades. This two-part paper is devoted to the structural dynamics modeling of rotor blades with a wide variety of boundary conditions—in particular hingeless and bearingless rotor configurations. In Part I, the theory and the formulation are presented, along with verification of single-load-path configurations. Part II is devoted to the verification of dual-load-path configurations.

Size-dependent free vibration analysis of three-layered exponentially graded nanoplate with piezomagnetic face-sheets resting on Pasternak’s foundation

2017 ◽  
Vol 29 (5) ◽  
pp. 774-786 ◽  
Author(s):  
M Arefi ◽  
MH Zamani ◽  
M Kiani
Keyword(s):  
Free Vibration ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Governing Equations ◽  
First Order ◽  
Size Dependent ◽  
Inhomogeneity Parameter ◽  
Exponentially Graded

This work is devoted to the free vibration nonlocal analysis of an elastic three-layered nanoplate with exponentially graded graphene sheet core and piezomagnetic face-sheets. The rectangular elastic three-layered nanoplate is resting on Pasternak’s foundation. Material properties of the core are supposed to vary along the thickness direction based on the exponential function. The governing equations of motion are derived from Hamilton’s principle based on first-order shear deformation theory. In addition, Eringen’s nonlocal piezo-magneto-elasticity theory is used to consider size effects. The analytical solution is presented to solve seven governing equations of motion using Navier’s solution. Eventually, the natural frequency is scrutinized for different side length ratio, nonlocal parameter, inhomogeneity parameter, and parameters of foundation numerically. The comparison with various references is performed for validation of our analytical results.

Structural Dynamics Analysis of Rotating Blades Using Fully Intrinsic Equations, Part II: Dual-Load-Path Configurations

2013 ◽  
Vol 58 (3) ◽  
pp. 1-9 ◽  
Author(s):  
Zahra Sotoudeh ◽  
Dewey H. Hodges
Keyword(s):  
Structural Dynamics ◽  
Equations Of Motion ◽  
Turbine Blades ◽  
Rotor Blades ◽  
Load Path ◽  
Dynamics Modeling ◽  
Governing Equations ◽  
Wind Turbine Blades ◽  
First Order ◽  
Infinite Degree

As part of an ongoing investigation into potential advantages of so-called fully intrinsic formulations, this paper presents an application of the fully intrinsic equations of motion and kinematics for beams to rotor blades. A fully intrinsic formulation is devoid of displacement and rotation variables. Although the governing equations are geometrically exact, they are free of the attendant singularities and infinite-degree nonlinearities found in other types of formulations. These nonlinear, first-order partial differential equations are suitable for analyzing initially curved and twisted, anisotropic beams and thus are very attractive for analysis of both helicopter and wind turbine blades. This two-part paper is devoted to the structural dynamics modeling of rotor blades with a wide variety of boundary conditions—in particular hingeless and bearingless rotor configurations. Part II is devoted to verification of certain dual-load-path configurations suggested by Bell Helicopter Textron.

Computational Reference Dynamical Model of a Multibody System With First Order Constraints

2017 ◽  
Author(s):  
Elżbieta Jarzębowska ◽  
Krzysztof Augustynek ◽  
Andrzej Urbaś
Keyword(s):  
Equations Of Motion ◽  
Computational Procedure ◽  
Dynamical Models ◽  
Constrained System ◽  
First Order ◽  
Performance Requirements ◽  
Control Goal ◽  
Order Constraints ◽  
Reaction Forces ◽  
And Control

The paper presents a development of a computational based procedure for generation of constrained system dynamical models. The constraints may be both holonomic and first order nonholonomic, either material or nonmaterial. The latter ones are referred to as programmed and they are imposed by a designer, a control engineer as a control goal, or may come from controlled system performance requirements. The procedure for generation of constrained dynamics provides then reference dynamical models, i.e. models whose solutions satisfy all the constraints put upon them. These models may serve as motion planners for control. The distinctions between the presented approach and the ones reported in the literature are that the constraints may be material or nonmaterial and the final equations of motion are derived in the reduced state form, i.e. constraint reaction forces are eliminated at the equations derivation level but not afterwards as in the case of the Lagrange approach. This is the essential advantage of our approach and this one computational procedure may serve both reference and control oriented dynamical models derivation. The procedure is applied to a manipulator model whose end effector is subjected to a programmed constrained.

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