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IPS NASU

IPS NASU. DYNAMICAL ANALYSIS AND ALLOWABLE VIBRATION DETERMINATION FOR THE PIPING SYSTEMS. G.S. Pisarenko Institute for Problems of Strength of National Academy of Science of Ukraine Kiev , Ukraine. Software complex « 3D PipeMaster ». Method of calculation of piping at harmonical vibrations

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IPS NASU

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  1. IPSNASU DYNAMICAL ANALYSIS AND ALLOWABLE VIBRATION DETERMINATION FOR THE PIPING SYSTEMS. G.S. Pisarenko Institute for Problems of Strength of National Academy of Science of Ukraine Kiev, Ukraine

  2. Software complex«3D PipeMaster» • Method of calculation of piping at harmonical vibrations • Modeling of dynamical behavior of pipe bend as the beam as well as the shell • The abilities of the complex for vibrodiagnostics Accident of the oil pipeline

  3. «3D PipeMaster»Harmonical analysis y X1 X0 x dx stiffness matrix X10 X11 X20 X21 Xn-10 Xn-11 Xn0 Xn1 2 … n-1 n 1 x The sweeping procedure with method of initial parameters Dynamic stiffness method y

  4. «3D PipeMaster»Harmonical analysis Dynamic stiffness method • the equations of motion at transversal vibrations - frequency of vibration The inertial term • the equations of the method of initial parameters:

  5. «3D PipeMaster»Harmonical analysis 2 1 3 m The matrix of the turning element The algorithms for branched and curvelinear elements the conditions in the junctions equations for pipe bend

  6. «3D PipeMaster»Harmonical analysis y Xi-10 Xi-11 Xi0 Xn1 i-1 i x - natural frequency Method of the breaking of displacements for the determination of the natural frequencies and forms the criteria of the determination of the natural frequency The example of the graph for T –like frame

  7. «3D PipeMaster»Harmonical analysis Method of the breaking of displacements continuity The role of the estimator is essential !!! The additional frequency can be noticed only at very small step of frequency.

  8. «3D PipeMaster»Harmonical analysis =148 с-1 =212.4 с-1 =214.4 с-1 Method of the breaking of displacements continuity The examples of finding the natural frequencies and forms for T-like frame The forms given in the handbooks The additional form of vibration !!!

  9. «3D PipeMaster»Harmonical analysis Method of the breaking of displacements modeling of curvilinear element Example: frequencies of the circular ring vibrations in plane Out-of-plane Kang K.J., Bert C.W. and Striz A.G. Vibration and buckling analysis of circular arches using DQM // Computers and Structures. – 1996. –V.60, №1. – pp. 49-57. Е = 2∙106 МПа; G = 8∙105 МПа;  = 0.3;  = 8000 кг/м3; В0 = 2 м; R = 0.1 м

  10. «3D PipeMaster»Harmonical analysis Method of the breaking of displacements modeling of curvilinear element Example: frequencies of the circular arc 1. In-plane vibrations Austin W.J. and Veletsos A.S. Free vibration of arches flexible in shear // J. Engng Mech. ASCE. – 1973. – V.99. – pp. 735-753. 2. Out-of-plane Ojalvo U. Coupled twisting-bending vibrations of incomplete elastic rings // Int. J. mech. Sci. – 1962. – V.4. – pp. 53-72.

  11. «3D PipeMaster»Harmonical analysis Advantages 1. The strict analytical solutions are used. 2. The continuity is provided at transition from static to dynamic 3. The infinite number of natural frequencies can be obtained for finite number of elements. 4. The method of sweeping allows to speed up the calculation. 5. Analytical accuracy of modeling of curved element is attained. 6. Any complex spatial multibranched piping system can be treated. 7. The vibration direction (modes) of interest can be separated 8. The influence of the subjective factors are excluded (the breaking out on the elements)

  12. Dynamical model of pipe bend as the beam as well as the shell IPSNASU The curved beam element is strict but pipe bend have the increased flexibility! - parameter of curvature - flexibility parameter Physical equation is corrected  Equation of the transversal vibration with accounting of increased flexibility: Depends from the frequency !

  13. v u r w O1  O  y t B x R z Determination of the flexibility of the pipe bend IPSNASU Equation for bend as a shell Equilibrium equations: Physical equations

  14. Determination of the flexibility of the pipe bend IPSNASU Geometrical equations: deformations curvatures The simplifications: semimomentless Vlasov’s theory: geomtrical characteristics: restrictions on the wave length in the axial direction

  15. Determination of the flexibility of the pipe bend IPSNASU Solution for the cylindrical shell Salley L. and Pan J. A study of the modal characteristics of curved pipes // Applied Acoustics. – 2002. – V.63. – pp. 189-202.

  16. Determination of the flexibility of the pipe bend IPSNASU Решение для гиба The sought for solution : The resulting equations:

  17. if then we obtain : Determination of the flexibility of the pipe bend IPSNASU Assume: Results: - The coefficient of flexibilityat harmonical vibrations

  18. B l R h l Determination of the flexibility of the pipe bend IPSNASU L. Salley and J. Pan. A study of the modal characteristics of curved pipes // Applied Acoustics. – 2002. – V.63. – pp. 189-202. • Е = 2.07∙106 МПа; • = 0.3;  = 8000 кг/м3; R = 0.0806 м; h = 0.00711 м; В = 0.457 м l=0.2 м

  19. Abilities of «3D PipeMaster» for vibrodiagnostics 1 IPSNASU • Е = 2∙106 МПа; G = 8∙105 МПа; • = 0.3;  = 8000 кг/м3; l = 5 м; R = 0.1 м;h= 0.005 м. 1. The graph of bending moment in the central point of supported-supported beam 2.Restoration of the outer force from the known displacements in arbitrary point 1.25

  20. Abilities of «3D PipeMaster» for vibrodiagnostics IPSNASU The problems of vibrodiagnostics 1. The points of application of the outer forces, their directions and frequencies are unknown. 2. The gauges can measure the displacements of pipe points, their velocities and accelerations 3. The number of gauges is finite. The functions of the calculation software 1. The correct determination of the dynamical characteristics. 2. Correct modeling of the piping behavior when the correct measurement data are provided. 3. The best possible assessment of the behavior with restricted input data. 4. The best possible assessment of the dynamical stresses based on the incomplete measurements

  21. Abilities of «3D PipeMaster» for vibrodiagnostics • Е = 2.0689∙106 МПа; μ= 0.3; • = 7836.6 кг/м3; l = 6.096 м; Δl=0.3048м;R = 0.05715 м; t= 0.0188 м. 11.66 Гц, 37.65 Гц, 78.18 Гц. IPSNASU The frequency of outer force is given but the point of its application is unknown. The gauges measure the displacements • Input data are the results of excitation of beam by harmonical force applied at its center. The calculated values of transverse forces, bending moment, displacements in 21 points are recorded. This is so called «real case». • The system (beam) is loaded by «the real» displacements in a few (or one) points, the moments and displacements are calculated. • The calculated in 2 results are compared with «real data».

  22. Abilities of «3D PipeMaster» for vibrodiagnostics =21 Гц =8 Гц IPSNASU 2 points of measurements

  23. Abilities of «3D PipeMaster» for vibrodiagnostics =100 Гц =80 Гц =60 Гц IPSNASU

  24. Abilities of «3D PipeMaster» for vibrodiagnostics =140 Гц =60 Гц IPSNASU 2 points of measurements

  25. Abilities of «3D PipeMaster» for vibrodiagnostics 4points of measurements =100 Гц =60 Гц IPSNASU

  26. Abilities of «3D PipeMaster» for vibrodiagnostics IPSNASU All measurements in all points are used Complete coincidence Conclusions from modeling: 1. To evaluate stresses the most importance have the proximity of the points of measurements to the point of the force application. 2. The accuracy grows with the number of the points of measurement 3. The accuracy nonmonotically decrease with the frequency of the excitation

  27. Abilities of «3D PipeMaster» for vibrodiagnostics IPSNASU Determination of the maximal stresses based on the measurements of velocities For simply supported beam: dynamic susceptibility coefficient For a thin walled pipe: for a solid circular beam: For the real complex piping systems:

  28. Abilities of «3D PipeMaster» for vibrodiagnostics IPSNASU Examples of the piping configuration

  29. Abilities of «3D PipeMaster» for vibrodiagnostics IPSNASU Determination of the maximal stresses based on the measurements of velocities • Е = 2.06843∙106 МПа; • = 7834 кг/м3; l = 18 м; R = 0.1 м; t= 0.01 м. Theoretical value: J. C. Wachel, Scott J. Morton, Kenneth E. Atkins. Piping vibration analysis

  30. Abilities of «3D PipeMaster» for vibrodiagnostics IPSNASU Determination of the maximal stresses based on the measurements of velocities For parameters Е = 2.0689∙106 МПа; ρ=7836.6 кг/м3; R = 0.05715 м; t= 0.0188m The results of calculation: Theoretical value 11.66 hertz When the exciting frequency exceeds the first natural frequency the correlation between the vibrovelocity and maximal stresses is good

  31. Conclusion IPSNASU 1. Due to application of dynamical stiffness method the continuity between the static and dynamic solution is provided. 2. The procedure of the breaking of the displacements in any point and in any direction allow to find all natural frequencies and forms 3.In a first time in a literature the notion of dynamic coefficient of pipe bend flexibility is introduced and analytical expression for it is obtained. This allowed to perform calculation for the piping systems with a higher accuracy 4. The option of determination of exciting force in some point based on given displacement or velocity in any other point of the piping allows to efficiently perform the vibrodiagnostic analysis

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