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 G.S. Pisarenko Institute for Problems of Strength of National Academy of Science of Ukraine Kiev, Ukraine

IPS NASU 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

IPS NASU «3D PipeMaster» Harmonical analysis Dynamic stiffness method x y dx X0X0 X1X1 stiffness matrix y with method of initial parameters x X10X10 2…n-1n X11X11 X20X20 X21X21 X n-1 0 Xn0Xn0 X n-1 1 Xn1Xn1 1 The sweeping procedure

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

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

IPS NASU «3D PipeMaster» Harmonical analysis Method of the breaking of displacements for the determination of the natural frequencies and forms x i-1i X i-1 0 Xi0Xi0 X i-1 1 Xn1Xn1 y the criteria of the determination of the natural frequency - natural frequency The example of the graph for T – like frame

IPS NASU «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.

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

IPS NASU «3D PipeMaster» Harmonical analysis Method of the breaking of displacements modeling of curvilinear element Example: frequencies of the circular ring Е = 2∙10 6 МПа; G = 8∙10 5 МПа;  = 0.3;  = 8000 кг/м 3 ; В0 = 2 м; R = 0.1 м Е = 2∙10 6 МПа; G = 8∙10 5 МПа;  = 0.3;  = 8000 кг/м 3 ; В0 = 2 м; R = 0.1 м n = 2n = 3n = 4n = 5 Vibration in the plane of circular ring theoretical Our results Out-of-plane vibration of circular ring теоретическое наши результаты Kang K.J., Bert C.W. and Striz A.G. Vibration and buckling analysis of circular arches using DQM // Computers and Structures. – –V.60, №1. – pp vibrations in plane Out-of-plane

IPS NASU «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. – – V.99. – pp Out-of-plane Ojalvo U. Coupled twisting-bending vibrations of incomplete elastic rings // Int. J. mech. Sci. – – V.4. – pp In-plane vibrations Austin W.J. and Veletsos A.S. Free vibration of arches flexible in shear // J. Engng Mech. ASCE. – – V.99. – pp Out-of-plane Ojalvo U. Coupled twisting-bending vibrations of incomplete elastic rings // Int. J. mech. Sci. – – V.4. – pp

IPS NASU «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)

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

IPS NASU Equation for bend as a shell r R O  B O1O1 t  x y z vu w Equilibrium equations: Physical equations Determination of the flexibility of the pipe bend Determination of the flexibility of the pipe bend

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

IPS NASU Determination of the flexibility of the pipe bend Determination of the flexibility of the pipe bend Solution for the cylindrical shell Salley L. and Pan J. A study of the modal characteristics of curved pipes // Applied Acoustics. – – V.63. – pp

IPS NASU Determination of the flexibility of the pipe bend Determination of the flexibility of the pipe bend Решение для гиба The sought for solution : : The resulting equations:

IPS NASU Determination of the flexibility of the pipe bend Determination of the flexibility of the pipe bend - The coefficient of flexibility at harmonical vibrations Assume: if then we obtain : Results:

IPS NASU L. Salley and J. Pan. A study of the modal characteristics of curved pipes // Applied Acoustics. – – V.63. – pp Е = 2.07∙106 МПа;  = 0.3;  = 8000 кг/м3; R = м; h = м; В = м l=0.2 м l l R h B Determination of the flexibility of the pipe bend Determination of the flexibility of the pipe bend

IPS NASU Е = 2∙10 6 МПа; G = 8∙10 5 МПа;  = 0.3;  = 8000 кг/м 3 ; l = 5 м; R = 0.1 м; h = м. 1. The graph of bending moment in the central point of supported-supported beam Restoration of the outer force from the known displacements in arbitrary point 1 Abilities of «3D PipeMaster» for vibrodiagnostics

IPS NASU Abilities of «3D PipeMaster» for vibrodiagnostics 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

IPS NASU Abilities of «3D PipeMaster» for vibrodiagnostics Е = ∙10 6 МПа; μ= 0.3;  = кг/м 3 ; l = м; Δl=0.3048м; R = м; t = м Гц, Гц, Гц. 1.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». 2.The system (beam) is loaded by «the real» displacements in a few (or one) points, the moments and displacements are calculated. 3.The calculated in 2 results are compared with «real data». The frequency of outer force is given but the point of its application is unknown. The gauges measure the displacements

IPS NASU Abilities of «3D PipeMaster» for vibrodiagnostics =21 Гц =8 Гц 2 points of measurements

IPS NASU Abilities of «3D PipeMaster» for vibrodiagnostics =100 Гц =80 Гц =60 Гц

IPS NASU Abilities of «3D PipeMaster» for vibrodiagnostics =140 Гц =60 Гц 2 points of measurements

IPS NASU Abilities of «3D PipeMaster» for vibrodiagnostics =100 Гц =60 Гц 4 points of measurements

IPS NASU Abilities of «3D PipeMaster» for vibrodiagnostics 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

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

IPS NASU Abilities of «3D PipeMaster» for vibrodiagnostics Examples of the piping configuration

IPS NASU Abilities of «3D PipeMaster» for vibrodiagnostics Determination of the maximal stresses based on the measurements of velocities Е = ∙10 6 МПа;  = 7834 кг/м 3 ; l = 18 м; R = 0.1 м; t = 0.01 м. J. C. Wachel, Scott J. Morton, Kenneth E. Atkins. Piping vibration analysis Theoretical value:

IPS NASU Abilities of «3D PipeMaster» for vibrodiagnostics Determination of the maximal stresses based on the measurements of velocities When the exciting frequency exceeds the first natural frequency the correlation between the vibrovelocity and maximal stresses is good Е = ∙10 6 МПа; ρ= кг/м 3 ; R = м; t = m For parameters Theoretical value hertz Theoretical value hertz ω, Гц E E E E E E+07 obtained value The results of calculation:

IPS NASU Conclusion 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