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 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.

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.

A New Solution Method for Vibration Analysis of Circular Cylindrical Thin Shells with a Circumferential Surface Crack

2013 ◽  
Vol 639-640 ◽  
pp. 1003-1009 ◽  
Author(s):  
Tao Yin ◽  
Dian Qing Li ◽  
Hong Ping Zhu
Keyword(s):  
Boundary Conditions ◽  
Natural Frequency ◽  
Thin Shell ◽  
Solution Method ◽  
Cylindrical Shells ◽  
Axial Direction ◽  
Mode Shapes ◽  
Governing Equations ◽  
Simply Supported ◽  
Initial Trial

In this paper, a new solution method is proposed for determining the natural frequency of a given mode for a finite-length circular cylindrical thin shell with a circumferential part-through crack. The governing equation of the cracked cylindrical shell is derived by integrating the line-spring model with the classical thin shell theory. The proposed method calculates the natural frequency from an initial trial to satisfy both the governing equations and appropriate boundary conditions through an optimization process. The initial trial is proposed to satisfy the governing equations by using the beam modal function to determine the modal wavenumbers and mode shapes of cylindrical shells in the axial direction, assuming the flexural mode shapes of cylindrical shells in the axial direction to be of the same form as that of a flexural vibration beam with the same boundary conditions. Four representative sets of boundary conditions are considered: simply supported (SS-SS), clamped-clamped (C-C), clamped-simply supported (C-SS), and clamped-free (C-F). Compared with the finite element (FE) method, the proposed solution method is verified to provide an accurate and efficient way to calculate the dynamic characteristics of both intact and cracked cylindrical shells.

Nonlinear Thermally Buckled Piezoelectric Energy Harvester

2016 ◽  
Author(s):  
M. H. Ansari ◽  
M. Amin Karami
Keyword(s):  
Natural Frequency ◽  
Energy Harvester ◽  
Microelectromechanical Systems ◽  
Energy Content ◽  
The Body ◽  
Mode Shapes ◽  
Governing Equations ◽  
Biomedical Sensors ◽  
Piezoelectric Energy ◽  
Bimorph Beam

A thermally buckled piezoelectric energy harvester is designed to power biomedical devices inside the body. The energy harvester (EN) uses the vibrations inside the body to generate the electricity needed for powering biomedical sensors and devices. The piezoelectric beam consists of a brass substrate and two piezoelectric patches attached to the top and the bottom of the substrate. The bimorph beam is inside a rigid frame. The bimorph beam is buckled due to the difference in the coefficient of the thermal expansion of the beam and the frame. Inside the body, most of the energy content come from the low-frequency vibrations (less than 50 Hz). Having high natural frequency is a major problem in Microelectromechanical systems (MEMS) energy harvesters. Considering the small size of the EN, 1 cm3, the natural frequency is expected to be high. In our design, the natural frequency is lowered significantly by using a buckled beam. A mass is also used in the middle of the beam to decrease the natural frequency even more. Since the beam is buckled, the design is bistable and nonlinear which increases the output power. In this paper, the natural frequencies and mode shapes of the EN are analytically derived. The geometric nonlinearities are included in the electromechanical coupled governing equations. The governing equations are solved and it is shown that the device generates sufficient electricity to power biomedical sensors and devices inside the human body.

scholarly journals Wet Modal Analyses of Various Length Coaxial Sump Pump Rotors with Acoustic-Solid Coupling

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Guangjie Peng ◽  
Zhuoran Zhang ◽  
Ling Bai
Keyword(s):  
Modal Analysis ◽  
Natural Frequency ◽  
Shaft Length ◽  
Fluid Medium ◽  
First Order ◽  
Modal Characteristics ◽  
Working Environments ◽  
Reaction Forces ◽  
Pump Shaft ◽  
Vibration Modal

The dynamic characteristics of the rotor components were determined using a first-order modal model of the rotor components for various sump pump shaft lengths for actual working environments. By employing ANSYS-Workbench software, this paper uses a fluid-solid coupling analysis to calculate the reaction forces of the fluid on the rotor with results, which is then used in dry and wet modal analyses of the rotor parts to calculate the vibration modal characteristics with and without prestresses. The differences between the wet and dry modal characteristics were compared and investigated by ANSYS. The results show that increasing the sump pump shaft length reduces the first-order natural frequency of the prestressed rotor components. The structure also experiences stress stiffening, which is more obvious in the high-order modes. The natural frequency of the rotor in the wet mode is about 16% less than that in the dry mode for the various shaft lengths due to the added mass of the water on the surface which reduces the natural frequency. In the wet modal analysis, when the structure is in a different fluid medium, the influence of its modal distribution will also change, this is because the additional mass produced by the fluid medium of different density on the structure surface is different. Thus, the wet modal analysis of the rotor is important for more accurate dynamic analyses.

scholarly journals A Novel Dynamic Method to Improve First-order Natural Frequency for Test Device

2018 ◽  
Vol 18 (5) ◽  
pp. 183-192 ◽  
Author(s):  
Jun Zhang ◽  
Yu Tian ◽  
Zongjin Ren ◽  
Jun Shao ◽  
Zhenyuan Jia
Keyword(s):  
Dynamic Model ◽  
Natural Frequency ◽  
Design Method ◽  
Mode Shapes ◽  
Test Device ◽  
Dynamic Design ◽  
First Order ◽  
Improve Measurement ◽  
Trial And Error Method ◽  
Six Degree Of Freedom

Abstract It is important to improve the natural frequency of test device to improve measurement accuracy. First-order frequency is basic frequency of dynamic model, which generally is the highest vibration energy of natural frequency. Taking vector force test device (VFTD) as example, a novel dynamic design method for improving first-order natural frequency by increasing structure stiffness is proposed. In terms of six degree-of-freedom (DOF) of VFTD, dynamic model of VFTD is built through the Lagrange dynamic equation to obtain theoretical natural frequency and mode shapes. Experimental natural frequency obtained by the hammering method is compared with theoretical results to prove rationality of the Lagrange method. In order to improve the stiffness of VFTD, increase natural frequency and meet the requirement of high frequency test, by using the trial and error method combined with curve fitting (TECF), stiffness interval of meeting natural frequency requirement is obtained. Stiffness of VFTD is improved by adopting multiple supports based on the stiffness interval. Improved experimental natural frequency is obtained with the hammering method to show rationality of the dynamic design method. This method can be used in improvement of first-order natural frequency in test structure.

scholarly journals Natural Frequency Sensitivity Analysis of Fire-Fighting Jet System with Adaptive Gun Head

Processes ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 808 ◽  
Author(s):  
Xiaoming Yuan ◽  
Xuan Zhu ◽  
Chu Wang ◽  
Lijie Zhang ◽  
Yong Zhu
Keyword(s):  
Sensitivity Analysis ◽  
Natural Frequency ◽  
Natural Frequencies ◽  
Fire Fighting ◽  
Mode Analysis ◽  
Mode Shapes ◽  
Design Parameters ◽  
Parameter Method ◽  
First Order ◽  
Typical Design

The gun head is the end effector of the fire-fighting jet system. Compared with a traditional fixed gun head, an adaptive gun head has the advantages of having an adjustable nozzle opening, a wide applicable flow range, and a high fire-extinguishing efficiency. Thus, the adaptive gun head can extinguish large fires quickly and efficiently. The fire-fighting jet system with an adaptive gun head has fluid-structure interaction and discrete-continuous coupling characteristics, and the influence of key design parameters on its natural frequencies needs to be determined by a sensitivity analysis. In this paper, the dynamic model and equations of the jet system were established based on the lumped parameter method, and the sensitivity calculation formulas of the natural frequency of the jet system to typical design parameters were derived. Natural frequencies and mode shapes of the jet system were determined based on a mode analysis. The variation law of the sensitivity of the natural frequency of the jet system to typical design parameters was revealed by the sensitivity analysis. The results show that the fluid mass inside the spray core within a certain initial gas content is the most important factor affecting the natural frequency of the jet system. There was only a 0.51% error between the value of the first-order natural frequency of the jet system determined by the modal experiment and the theoretical one, showing that good agreement with the first-order natural frequency of the jet system was found. This paper provides a theoretical basis for the dynamic optimization design of the adaptive gun head of the fire water monitor.

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.

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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