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.

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.

Vibration characteristics of rotating truncated conical shells reinforced with agglomerated carbon nanotubes

2021 ◽  
pp. 107754632110004
Author(s):  
Hassan Afshari ◽  
Hossein Amirabadi
Keyword(s):  
Carbon Nanotubes ◽  
Differential Quadrature Method ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Governing Equations ◽  
Conical Shells ◽  
First Order ◽  
Effective Mechanical Properties ◽  
Comprehensive Study

In this article, a comprehensive study is conducted on the free vibration analysis of rotating truncated conical shells reinforced with functionally graded agglomerated carbon nanotubes The shell is modeled based on the first-order shear deformation theory, and effective mechanical properties are calculated based on the Eshelby–Mori–Tanaka scheme along with the rule of mixture. By considering centrifugal and Coriolis accelerations and initial hoop tension, the set of governing equations is derived using Hamilton’s principle and is solved numerically using the differential quadrature method Convergence and accuracy of the presented model are confirmed and the effects of different parameters on the forward and backward frequencies of the rotating carbon nanotube-reinforced truncated conical shells are investigated.

An Asymptotic, Thermo-Diffusive Ignition Theory of Porous Solid Fuels

1976 ◽  
Vol 98 (2) ◽  
pp. 269-275 ◽  
Author(s):  
Choong Se Kim ◽  
Paul M. Chung
Keyword(s):  
Nonlinear Partial Differential Equations ◽  
Porous Solid ◽  
Solid Fuels ◽  
Thermal Ignition ◽  
Mass Balances ◽  
Governing Equations ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
Time Space ◽  
Gaseous Oxidant

The governing equations of thermal ignition are analyzed for porous solid fuel, such as coal, of various two-dimensional and axisymmetric geometries by the Laplace asymptotic method. Mass diffusion of the gaseous oxidant through the porous fuel is included. The nonlinear partial differential equations of energy and mass balances in time-space coordinates containing the Arrhenius volumic chemical reaction terms are analyzed. By employing the Laplace asymptotic technique and by invoking a certain limit theorem, the governing equations are reduced to a first order ordinary differential equation governing the fuel surface temperature, which is readily solved numerically. Detailed discussion of the effects of the various governing parameters on ignition is presented. Because of the basically closed-form nature of the solutions obtained, many general and fundamental aspects of the ignition criteria hitherto unknown are found.

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.

Stability of ion–acoustic solitary waves in a multi-species magnetized plasma consisting of non-thermal and isothermal electrons

2009 ◽  
Vol 75 (5) ◽  
pp. 593-607 ◽  
Author(s):  
SK. ANARUL ISLAM ◽  
A. BANDYOPADHYAY ◽  
K. P. DAS
Keyword(s):  
Stability Analysis ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Perturbation Expansion ◽  
Magnetized Plasma ◽  
Wave Number ◽  
First Order ◽  
Zk Equation ◽  
Ion Acoustic Solitary Waves ◽  
Ion Acoustic

AbstractA theoretical study of the first-order stability analysis of an ion–acoustic solitary wave, propagating obliquely to an external uniform static magnetic field, has been made in a plasma consisting of warm adiabatic ions and a superposition of two distinct populations of electrons, one due to Cairns et al. and the other being the well-known Maxwell–Boltzmann distributed electrons. The weakly nonlinear and the weakly dispersive ion–acoustic wave in this plasma system can be described by the Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation and different modified KdV-ZK equations depending on the values of different parameters of the system. The nonlinear term of the KdV-ZK equation and the different modified KdV-ZK equations is of the form [φ(1)]ν(∂φ(1)/∂ζ), where ν = 1, 2, 3, 4; φ(1) is the first-order perturbed quantity of the electrostatic potential φ. For ν = 1, we have the usual KdV-ZK equation. Three-dimensional stability analysis of the solitary wave solutions of the KdV-ZK and different modified KdV-ZK equations has been investigated by the small-k perturbation expansion method of Rowlands and Infeld. For ν = 1, 2, 3, the instability conditions and the growth rate of instabilities have been obtained correct to order k, where k is the wave number of a long-wavelength plane-wave perturbation. It is found that ion–acoustic solitary waves are stable at least at the lowest order of the wave number for ν = 4.

Application of Generalized Differential Quadrature Method to the Bending of Thick Laminated Plates with Various Boundary Conditions

2006 ◽  
Vol 5-6 ◽  
pp. 407-414 ◽  
Author(s):  
Mohammad Mohammadi Aghdam ◽  
M.R.N. Farahani ◽  
M. Dashty ◽  
S.M. Rezaei Niya
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Simple Procedure ◽  
Shear Deformation Theory ◽  
Laminated Plates ◽  
Differential Quadrature ◽  
Governing Equations ◽  
First Order ◽  
Various Boundary Conditions ◽  
Generalized Differential

Bending analysis of thick laminated rectangular plates with various boundary conditions is presented using Generalized Differential Quadrature (GDQ) method. Based on the Reissner first order shear deformation theory, the governing equations include a system of eight first order partial differential equations in terms of unknown displacements, forces and moments. Presence of all plate variables in the governing equations provide a simple procedure to satisfy different boundary condition during application of GDQ method to obtain accurate results with relatively small number of grid points even for plates with free edges .Illustrative examples including various combinations of clamped, simply supported and free boundary condition are given to demonstrate the accuracy and convergence of the presented GDQ technique. Results are compared with other analytical and finite element predictions and show reasonably good agreement.

Theoretical and Experimental Study of the Nonlinearly Coupled Heave, Pitch, and Roll Motions of a Ship in Longitudinal Waves

1993 ◽  
Author(s):  
I. G. Oh ◽  
A. H. Nayfeh ◽  
D. T. Mook
Keyword(s):  
Large Amplitude ◽  
Degrees Of Freedom ◽  
Excitation Frequency ◽  
Force Response ◽  
Governing Equations ◽  
Response Curves ◽  
Varying Coefficients ◽  
Jump Phenomena ◽  
Time Varying Coefficients ◽  
Three Degrees Of Freedom

Abstract The loss of dynamic stability and the resulting large-amplitude roll of a vessel in a head or following sea were studied theoretically and experimentally. A ship model with three degrees of freedom (roll, pitch, heave) was considered. The governing equations for the heave and pitch modes were linearized and their harmonic solutions were coupled with the nonlinear equation governing roll. The resulting equation, which has time-varying coefficients, was used to predict the response in roll. The principal parametric resonance was considered in which the excitation frequency is twice the natural frequency in roll. Force-response curves were obtained. The existence of jump phenomena and multiple stable solutions for the case of subcritical instability was observed in the experiments and found to be in good qualitative agreement with the results predicted by the theory. The experiments also revealed that the large-amplitude roll is dependent on the location of the model in the standing waves.

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