Flexural vibrations of the pipeline on elastic supports with moving fluid

2019 ◽  
Vol 14 (1) ◽  
pp. 10-16
Author(s):  
A.G. Khakimov ◽  
A.A. Yulmukhametov
Keyword(s):  
Natural Frequencies ◽  
Neutral Line ◽  
Unit Length ◽  
Fluid Pressure ◽  
Density Parameter ◽  
Liquid Density ◽  
Bending Vibrations ◽  
Technical Application ◽  
Elastic Supports ◽  
Flexural Vibrations

In the work investigated the flexural vibrations of the pipeline. Parts of the pipeline on both sides of the sagging section have elastic supports. It is assumed that a constant longitudinal force acts along the neutral line. An incompressible fluid flows through the pipe at a constant average speed. The influence of internal pressure in the pipe on these oscillations is taken into account. The direct problem of determining the eigenfrequencies of flexural vibrations of the pipeline by the Kirchhoff model using Ferrari formulas is solved. The frequency spectrum is determined depending on the fluid pressure, the elasticity of the supports, the velocity of the fluid through the pipe. Particular and limiting cases are considered, for example, when the stiffness of the supports is very large and when they are very small. Graphs of the dependence of the first and second eigenfrequencies on the velocity of the transported liquid at different values of the liquid density parameter are constructed. It is shown that with the growth of the velocity parameter there is a decrease in the natural frequencies of flexural vibrations of the pipeline, and the faster the higher the density parameter of the liquid. It is determined that with an increase in the mass of the liquid per unit length of the pipeline there is a decrease in the natural frequencies of bending vibrations of the pipe. It is found that with the increase in the mass flow through the pipe, the natural frequencies of bending oscillations also decrease. It is confirmed that the frequencies of flexural vibrations of the pipeline are the same for the cases of pipe fastening “rigid fixing — rigid fixing” and “free end — free end”. The results of the study will contribute to the development of methods of acoustic diagnostics and non-destructive testing and can find technical application for monitoring and diagnostics of pipeline systems.

Determination of the speed and density of the fluid inside the pipeline on elastic supports

2019 ◽  
Vol 14 (4) ◽  
pp. 262-267
Author(s):  
A.A. Yulmukhametov ◽  
A.G. Khakimov
Keyword(s):  
Mass Flow Rate ◽  
Mass Flow ◽  
Flow Rate ◽  
Natural Frequencies ◽  
Fluid Velocity ◽  
Diagnostic Methods ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Bending Vibrations ◽  
Elastic Supports

The natural frequencies of the bending vibrations of the pipeline are investigated. The pipe sags over the obstacle and is under the action of tensile force. Outside the sagging area, the pipe rests on elastic supports. The fluid transported through the pipeline is under pressure. The direct problem was solved earlier, in this article, the inverse problem of identifying the speed and density of the transported fluid by the known natural frequencies of bending vibrations is solved. The equation of bending vibrations of a pipeline is described by the Kirchhoff model. The characteristic equation is solved using Ferrari formulas. The general decision is determined. We substitute the general solution into the boundary conditions and obtain a system of equations. This system gives a frequency equation, which is solved numerically on a developed program in the Maple package. The method of successive approximations is applied, after the third iteration, the accuracy of calculating the parameters of the velocity and density of the liquid is approximately 10−3. Thus, it was found that with an increase in the oscillation frequency, the density of the liquid inside the pipe decreases. It is determined that with increasing natural frequencies of pipe bending vibrations, the fluid velocity parameter increases. It is shown that the two lower frequencies of bending vibrations of the pipeline can be used to determine the parameters of the velocity and density of the liquid. The dependence of the mass flow rate of the liquid on the first natural frequency of the pipe oscillations is given. It is shown that with increasing frequency, the mass flow rate decreases. The research results will help the development of acoustic diagnostic methods and non-destructive testing methods and will find technical application for monitoring and diagnosing the state of pipeline systems.

Determining the velocity of a moving rod and the thickness of its coating using natural frequencies of flexural vibrations

2016 ◽  
Vol 11 (1) ◽  
pp. 10-15
Author(s):  
A.G. Khakimov
Keyword(s):  
Coating Thickness ◽  
Textile Industry ◽  
Natural Frequencies ◽  
Neutral Line ◽  
Research Work ◽  
Longitudinal Force ◽  
Wire Drawing ◽  
Constant Length ◽  
Velocity Parameter ◽  
Flexural Vibrations

Research has been performed on natural transverse vibrations of a portion of a constant length in a straight thin coated rod moving along the neutral line of a non-deformed state. The movement takes place between two rigidly fixed coaxial guides (clamps), the distance between them equalling the length of the rod moving portion. The longitudinal force is assumed to constantly act along the neutral line. It has been found that a decrease in the natural frequencies of flexural vibrations of the rod occurs with an increase in the velocity parameter. It is also shown that a decrease in the natural frequencies of flexural vibrations of the rod occurs with an increase of the coating thickness. Using two frequencies of flexural vibrations, we can determine the velocity parameter of the moving rod and the thickness of its coating. The results of the research work can find technological use in the problems on dynamics and strength of machines and mechanisms in manufacturing the coated products: in textile industry, wire manufacturing process, metallurgy (especially for rolling metal bars and strips), wire drawing, plastic products and paper rolls manufacturing and can be used to determine the velocity and the coating thickness of a moving rod, strip or wire using two frequencies of flexural vibrations.

Determining the velocity of a moving three-layered plate and the thickness of its filler using natural frequencies of flexural vibrations

2017 ◽  
Vol 12 (1) ◽  
pp. 109-114
Author(s):  
A.G. Khakimov
Keyword(s):  
Natural Frequencies ◽  
Neutral Line ◽  
Research Work ◽  
Longitudinal Force ◽  
Layered Plate ◽  
Constant Length ◽  
Transverse Vibrations ◽  
Deformed State ◽  
Flexural Vibrations ◽  
Natural Transverse

Research has been performed on natural transverse vibrations of a portion of a constant length in a straight threelayered plate with a filler moving along the neutral line of a non-deformed state. The movement takes place between two rigidly fixed coaxial guides (clamps), the distance between them equalling the length of the vibrating portion. The longitudinal force is assumed to constantly act along the neutral line. It has been found that an increase in the natural frequencies of flexural vibrations of the plate occurs with an increase in the thickness of the filler. Using two frequencies of flexural vibrations, we can determine the velocity of the moving three-layered plate and the thickness of its filler. The results of the research work can find technological use in the problems on dynamics and strength of machines and mechanisms in manufacturing three-layered plate with fillers and can be used to determine the velocity of a plate and the thickness of its filler using two frequencies of flexural vibrations.

Algorithm for diagnosing fastenings of pipe with liquid

2007 ◽  
Vol 5 ◽  
pp. 96-100
Author(s):  
A.M. Akhtyamov ◽  
F.F. Safina
Keyword(s):  
Frequency Spectrum ◽  
Natural Frequencies ◽  
Algebraic Equations ◽  
Bending Vibrations ◽  
Narrow Tube

An algorithm is considered for diagnosing fastening of a narrow tube filled with a fluid by a spectrum of natural frequencies of its bending vibrations. The constructed algorithm, based on the solution of systems of algebraic equations, allows one to determine any pipe fastenings by 9 values from the frequency spectrum of its vibrations when the liquid is flowing through the pipe.

Free Bending Vibrations of a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) With a Gap in Mindlin Plates or Panels

2007 ◽  
Author(s):  
U. Yuceoglu ◽  
O. Gu¨vendik ◽  
V. O¨zerciyes
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Shear Stresses ◽  
Lap Joint ◽  
Bending Vibrations ◽  
Joint System ◽  
Adhesive Layers ◽  
Bonded Joint ◽  
Free Bending

In this present study, the “Free Bending Vibrations of a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) with a Gap in Mindlin Plates or Panels” are theoretically analyzed and are numerically solved in some detail. The “plate adherends” and the upper and lower “doubler plates” of the “Bonded Joint” system are considered as dissimilar, orthotropic “Mindlin Plates” joined through the dissimilar upper and lower very thin adhesive layers. There is a symmetrically and centrally located “Gap” between the “plate adherends” of the joint system. In the “adherends” and the “doublers” of the “Bonded Joint” assembly, the transverse shear deformations and the transverse and rotary moments of inertia are included in the analysis. The relatively very thin adhesive layers are assumed to be linearly elastic continua with transverse normal and shear stresses. The “damping effects” in the entire “Bonded Joint” system are neglected. The sets of the dynamic “Mindlin Plate” equations of the “plate adherends”, the “double doubler plates” and the thin adhesive layers are combined together with the orthotropic stress resultant-displacement expressions in a “special form”. This system of equations, after some further manipulations, is eventually reduced to a set of the “Governing System of the First Order Ordinary Differential Equations” in terms of the “state vectors” of the problem. Hence, the final set of the aforementioned “Governing Systems of Equations” together with the “Continuity Conditions” and the “Boundary conditions” facilitate the present solution procedure. This is the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials). The present theoretical formulation and the method of solution are applied to a typical “Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) with a Gap”. The effects of the relatively stiff (or “hard”) and the relatively flexible (or “soft”) adhesive properties, on the natural frequencies and mode shapes are considered in detail. The very interesting mode shapes with their dimensionless natural frequencies are presented for various sets of boundary conditions. Also, several parametric studies of the dimensionless natural frequencies of the entire system are graphically presented. From the numerical results obtained, some important conclusions are drawn for the “Bonded Joint System” studied here.

In-Plane Parametric Vibrations of Curved Bellows Subjected to Oscillating Internal Fluid Pressure Excitation

2004 ◽  
Author(s):  
Masahiro Watanabe ◽  
Eiji Tachibana ◽  
Nobuyuki Kobayashi
Keyword(s):  
Stability Analysis ◽  
Natural Frequencies ◽  
Parametric Instability ◽  
Fluid Pressure ◽  
Parametric Vibrations ◽  
Internal Fluid ◽  
Mathieu’S Equation ◽  
Stability Maps ◽  
Plane Direction ◽  
Pressure Excitation

This paper deals with the theoretical stability analysis of in-plane parametric vibrations of a curved bellows subjected to periodic internal fluid pressure excitation. The curved bellows studied in this paper are fixed at both ends rigidly, and are excited by the periodic internal fluid pressure. In the theoretical stability analysis, the governing equation of the curved bellows subjected to periodic internal fluid pressure excitation is derived as a Mathieu’s equation by using finite element method (FEM). Natural frequencies of the curved bellows are examined and stability maps are presented for in-plane parametric instability. It is found that the natural frequencies of the curved bellows decrease with increasing the static internal fluid pressure and buckling occurs due to high internal fluid pressure. It is also found that two types of parametric vibrations, longitudinal and transverse vibrations, occur to the curved bellows in-plane direction due to the periodic internal fluid pressure excitation. Moreover, effects of axis curvature on the parametric instability regions are examined theoretically.

Statistics of Cosmic Microwave Background Radiation with the Cosmic String Model

1997 ◽  
Vol 06 (05) ◽  
pp. 535-544
Author(s):  
Petri Mähönen ◽  
Tetsuya Hara ◽  
Toivo Voll ◽  
Shigeru Miyoshi
Keyword(s):  
Cosmic Microwave Background ◽  
Cosmic String ◽  
Large Scale ◽  
Background Radiation ◽  
Unit Length ◽  
Angular Size ◽  
Cosmic Microwave Background Radiation ◽  
Density Parameter ◽  
Microwave Background ◽  
Microwave Background Radiation

We have studied the cosmic microwave background radiation by simulating the cosmic string network induced anisotropies on the sky. The large-angular size simulations are based on the Kaiser–Stebbins effect calculated from full cosmic-string network simulation. The small-angular size simulations are done by Monte-Carlo simulation of perturbations from a time-discretized toy model. We use these results to find the normalization of μ, the string mass per unit length, and compare this result with one needed for large-scale structure formation. We show that the cosmic string scenario is in good agreement with COBE, SK94, and MSAM94 microwave background radiation experiments with reasonable string network parameters. The predicted rms-temperature fluctuations for SK94 and MSAM94 experiments are Δ T/T=1.57×10-5 and Δ T/T=1.62×10-5, respectively, when the string mass density parameter is chosen to be Gμ=1.4×10-6. The possibility of detecting non-Gaussian signals using the present day experiments is also discussed.

Free Flexural Vibrations Response of Composite Mindlin Plates or Panels With a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint)

2005 ◽  
Author(s):  
U. Yuceoglu ◽  
O. Gu¨vendik ◽  
V. O¨zerciyes
Keyword(s):  
Natural Frequencies ◽  
Mode Shapes ◽  
Solution Technique ◽  
Shear Stresses ◽  
Lap Joint ◽  
Adhesive Layers ◽  
Present Solution ◽  
Flexural Vibrations ◽  
Support Conditions ◽  
Interpolation Polynomials

The present study is concerned with the “Free Flexural Vibrations Response of Composite Mindlin Plates or Panels with a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint). The plate “adherends” and the plate “doublers” are considered as dissimilar, orthotropic “Mindlin Plates” with the transverse and the rotary moments of inertia. The relatively, very thin adhesive layers are taken into account in terms of their transverse normal and shear stresses. The mid-center of the bonded region of the joint is at the mid-center of the entire system. In order to facilitate the present solution technique, the dynamic equations of the plate “adherends” and the plate “doublers” with those of the adhesive layers are reduced to a set of the “Governing System of First Order ordinary Differential Equations” in terms of the “state vectors” of the problem. This reduced set establishes a “Two-Point Boundary Value Problem” which can be numerically integrated by making use of the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials)”. In the adhesive layers, the “hard” and the “soft” adhesive cases are accounted for. It was found that the adhesive elastic constants drastically influence the mode shapes and their natural frequencies. Also, the numerical results of some parametric studies regarding the effects of the “Position Ratio” and the “Joint Length Ratio” on the natural frequencies for various sets of support conditions are presented.

scholarly journals UNCOUPLED VIBRATIONS IN FUNCTIONALLY GRADED TIMOSHENKO BEAM

2016 ◽  
Vol 54 (6) ◽  
pp. 785 ◽  
Author(s):  
Nguyen Tien Khiem ◽  
Nguyen Ngoc Huyen
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Flexural Vibration ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Power Law Distribution ◽  
Material Parameters ◽  
Flexural Vibrations ◽  
Fgm Beam

Free vibration of FGM Timoshenko beam is investigated on the base of the power law distribution of FGM. Taking into account the actual position of neutral plane enables to obtain general condition for uncoupling of axial and flexural vibrations in FGM beam. This condition defines a class of functionally graded beams for which axial and flexural vibrations are completely uncoupled likely to the homogeneous beams. Natural frequencies and mode shapes of uncoupled flexural vibration of beams from the class are examined in dependence on material parameters and slendernes

Hydro-Elastic Vibration of Partially Liquid-Filled or Partially Liquid- Surrounded Composite Cylindrical Shells

2011 ◽  
Vol 462-463 ◽  
pp. 1127-1133
Author(s):  
Zhu Shan Shao ◽  
Guo Wei Ma ◽  
Zhan Ping Song
Keyword(s):  
Natural Frequencies ◽  
Cylindrical Shells ◽  
Hydrodynamic Pressure ◽  
Elastic Vibration ◽  
Vibration Characteristics ◽  
Radius Ratio ◽  
Liquid Density ◽  
External Work ◽  
Love’S Shell Theory ◽  
Composite Cylindrical Shells

Vibration characteristics of partially liquid-filled or partially liquid-surrounded composite cylindrical shells are investigated in this paper. Using Rayleigh-Ritz energy method and Love’s shell theory, eigenvalue equation of the problem is derived, and the polynomial for natural frequencies of such shells is further obtained. The external work by the hydrodynamic pressure, which is introduced by liquid sloshing, is taken into account in the energy function. Hydro-elastic vibration characteristics of a composite cylindrical shell are studied by using the present method. Effects of liquid level, liquid density, fiber orientation, length-to-radius ratio, and thickness-to-radius ratio on the natural frequencies are analyzed and graphically presented.

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