The Finite Element Technique for Modelling of Pipe

Available online at www.sciencedirect.com

ScienceDirect Procedia Engineering 176 (2017) 681 – 688

Dynamics and Vibroacoustics of Machines (DVM2016)

The finite element technique for modelling of pipe system vibroacoustical characteristics Tatiana B. Mironova *, Andrey B. Prokofiev, Victor Y. Sverbilov Samara National Research University, Moskovskoe shosse 34, Samara 443086, Russian Federation

Abstract In this paper, we consider two finite element techniques developed by the authors earlier for the solving differential equation systems of the complex configuration pipelines dynamics and compare calculation results with experimental data. One of the techniquies uses the new type of seven node finite elements for modeling vibroacoustical interaction between solid structures and oscillating fluid, another - two node finite elements. The proposed finite element techniques allow calculation of pipe vibroacoustical characteristics in the space-time domain. Because these techniques use the new element types, the computational coast is 3 orders less than for available finite element techniques, which use different element types for a fluid and a solid structure. The developed techniques are designed for a pipeline diameter much smaller than an acoustic wavelength in a fluid. The experimental research was conducted for the pipe system of the hydraulic test bench for generating pressure pulsation. The results of calculation by the proposed finite element techniques were confirmed by experimental data. ©2017 2016The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license © Published by Elsevier Ltd. This Peer-review under responsibility of organizing committee of the Dynamics and Vibroacoustics of Machines (DVM2016). (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the international conference on Dynamics and Vibroacoustics of Machines Keywords: finite element techniques; vibroacoustical characteristics, pipe, experimental researche

1. Introduction It is well known that the pipeline pressure pulsation produced by the pumps and others sources are a course of noise and vibration. The pressure fluctuation is the reason of force vibrations of the pipe system components. There are may have a significant influence on the reliability of a given installation. It is highly desirable to be able to predict pressure pulsations at the design stage of an installation so that appropriate steps may be taken to minimize

* Corresponding author. Tel.: +7-906-345-5132; fax: +7-846-335-1905. E-mail address: [email protected]

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the international conference on Dynamics and Vibroacoustics of Machines

doi:10.1016/j.proeng.2017.02.313

682

Tatiana B. Mironova et al. / Procedia Engineering 176 (2017) 681 – 688

their levels and their influence. Thus, the special study is required to solve the problem of the pipeline vibration reduction. The dynamic processes in the complex configuration pipe system can be described as the non-linear differential equation systems [1-6]. The boundary conditions for the equation system are also represented by nonlinear differential equations, algebraic or integral equations [1, 2, 5, 7]. These equations describe unsteady motion of the fluid, fluid–structure interaction between fluid and elements of the mechanical subsystem [1-3, 6, 7] which interacted with different elements, units and environments of the hydraulic system. Computer simulation and numerical methods is the key to solution of the problem. The finite element method is widely used for calculating of vibroacoustical characteristics of pipe systems. However, it has limitations. It is the meshing complexity for complex configuration piping systems [8, 9] when the regular grid is not possible. In such cases, the order of the differential equation system is more than maximum order equations of the regular software. In this paper, we consider two finite element techniques developed earlier [1-3] for complex configuration piping systems dynamics. The developed techniques allow using the single type of element for modelling vibroacoustical characteristics of pipe systems under a force excitation by an oscillating fluid flow. The computational effort of this techniques decreases because a pipeline is considered like a beam [1-3, 7, 10] and a single type of the element is using for modelling vibroacoustical interaction between a solid structure and an oscillating fluid. Nomenclature w nondimensional working fluid flow velocity p nondimensional pressure  coordinate, measured along the centroidal line of the pipeline sections from zero point to the arbitrary cross-section  non-dimensional time e1  ,  unitary vector, binormal to the piping center line e2  ,  unitary vector, normal to the piping center line e3  ,  unitary vector, binormal to the piping center line ui vibration displacement in ei  ,   direction  nonplanar center line curvature vector friction coefficient H i the angle of rotation of the coordinate axis  angular frequency damping coefficient  mass coefficient n pipeline length l Q vector of internal forces in a pipeline q vector of external forces in a pipeline f vector of forces of interaction between solid and oscillating fluid in the pipeline M ci vector of point moment in a cross-section with coordinate si  nondimensional coordinate  vector of distributed forces from field of forces A pipeline stiffness Ni basis functions

2. Finite element techniques The techniques are based on the following mathematical model [1-3]:

683

Tatiana B. Mironova et al. / Procedia Engineering 176 (2017) 681 – 688

w  2 u Q  p   Bтр   q  f  1, ,    2  ~ ~   u     ,  e1  e10 , M  A   0   M 0 ,    m M  e1  Qб    M сi     i   0, f  f 1e1  f 2 e2  f 3 e3 ,  i 1

(1)



n(

 pe1   2u  2 u w  w   u   u    u   u    e1    e2    e3   w  e1    e2    e3 )   w  2  f 2                 2  3   2  3  

The equation system has 14 boundary conditions: 12 for the mechanical (solid) subsystem and 2 for the hydraulic subsystem (fluid). The boundary conditions for mechanical subsystem are defined by differential equations describing the piping end tailing types [1-3, 7, 10].The boundary conditions for hydraulic subsystem are defined for each harmonic oscillation component by the combination of two parameters from the next four: the complex pressure and velocity amplitudes at the pipe inlet and outlet, together with the load and input impedances (Zload and Zin). [1-3, 7, 10]. As an example, it is considered solving differential equation system for the pipe lying in one plane under force excitation by an oscillating fluid flow [1, 3, 7]

u2 

1 u1 ,  3 

 2 u1  4u  4u  2 u1 5u  6u1  4u  2 1 2  nw 3 1  nw 32  H 32 4 1    p  nw 2   2  32  41  2 6           2  u1   p  nw2    302  302  nw 302 w  n 302 w ,  2   2 1  u1 3    3u 1 ,  3  2

 302

(2)

For the equation system (2) two finite element techniques are implemented. One of them uses seven node finite elements [3], another - two node finite elements [1]. The solution algorithm of the techniques are based on the partial discretization and the weighted residual methods [11]. The new proposed space-time joint type elements based on Lagrange basis function (Fig.1) is used in the first technique. The linear function is used as a basis function in the second technique [1] (Fig.2).

Fig. 1. One-dimensional element and the Lagrange basis functions of 6-th order in the space domain.

684

Tatiana B. Mironova et al. / Procedia Engineering 176 (2017) 681 – 688

Fig. 2. One-dimensional elements and linear basis functions in the space domain [1].

Then the method of basis functions in the time domain is used. Finite elements are also used to represent the time domain of infinite extent [1-3]. The finite element techniques allow calculation of vibroacoustical characteristics of a pipe in the space-time domain so as natural and forced modes of vibration (Fig. 3).

Fig. 3. Time response of the nondimensional normal vibration displacement.

So as we use the new element types for modelling of vibroacoustical interaction between a solid structure and an oscillating fluid, the technique computational coast is 3 orders less than available finite element techniques, where the different element types are used for the both cases. The number of nodes for the pipeline model for Ansys is much more than for the finite element techniques (Fig. 4).

Fig. 4. The space-time joint element types and Ansys finite elements.

For equal number of elements the technique with seven node elements is more accurate than the two-node element technique. But the second technique computational coast is less than the first one [1-3, 11].

Tatiana B. Mironova et al. / Procedia Engineering 176 (2017) 681 – 688

3. Experimental research The tested pipeline of the hydraulic test bench for generating pressure pulsation is presented in Fig. 5. Fluid parameters: ρfluid=870 kg/m3, с=1300 m/s. Points of impulse excitations and registration of the signals are shown Fig. 5. For the pipe excitation the impact hammer PCB 086С03 was used (Fig. 6). Vibration measurements were made with the Portable Digital Vibrometer PDV-100 and mobile testing hardware LMS SCADAS Mobile (SCM05). For the analysis of experimental results LMS Test.Xpress software was used (Fig. 6).

Fig. 5. Measuring Schema: 1- supports; 2 – pressure pulsation sensor M101A06 PCB; 3 – focus point of laser vibrometr.

Fig. 6. Vibration measurements: 1 – pipeline; 2 - vibrometr PDV-100; 3 - 12- channel analyzer LMS SCADAS Mobile; 4 – laptop.

685

686

Tatiana B. Mironova et al. / Procedia Engineering 176 (2017) 681 – 688

The hydraulic test bench of pressure pulsation is presented in Fig. 7. For the recording of vibration signals the laser vibrometr PDV-100 was used, for the recording of pulsation signals - PCB pressure pulsation sensor M101A06 and 12 channel signal analyzer LMS SCADAS Mobile.

I

II

III

Fig. 7. Hydraulic test bench of pressure pulsation: I – fluid preparation unit; II - oscillation generator; III - test item; 1 - booster pump; 2, 10 filters; 3 - high-pressure pump; 4, 12, 13 - reservoirs; 5 - pipe; 6 - oscillation generator; 7 - pipe; 8, 9 - taps; 11 - tank.

The hydraulic test bench for generating pressure pulsation consists of fluid preparation unit and oscillation generator and is presented in Fig. 8. a

b

Fig. 8. (a) fluid preparation unit: 1 – tank; 2 – booster pump; 3 - filter; 4 - high-pressure pump; 5 – reservoirs; 6 –pipeline; 7– heat-exchange unit; (b) oscillation generator: 1 – pulsator; 2 – pulsator gearing; 3 – pressure pipeline; 4 - reservoirs; 5 – pipeline of supply pressure .

The hydraulic test bench allows measuring vibroacoustical characteristics of the hydraulic systems in the wide range of parameters: frequency range of pressure pulsation from 20 to 900 Hz; pressure pulsation amplitude up to 2 MPa; static pressure from 0.5 to 20 MPa. The independent control of amplitude, frequency and static parameters were provided. The pressure pulsation generator provides the frequency range of the first harmonic pressure oscillation from 30 to 700 Hz. Middle pressure in hydraulic system - 1.2 MPa. The pressure pulsation and vibration signals were recorded at inlet and outlet sections of the pipeline. LMS Test.Xpress were used for signal processing of test data. The sample rate was 51200 Hz. Time of signal recording was 1 sec. The analysis was carried out for the first four harmonics. It was allowed extending the frequency range to 2.5 kHz and defining the highest resonance frequencies. The experimental results are presented in Fig. 9.

Tatiana B. Mironova et al. / Procedia Engineering 176 (2017) 681 – 688

a

b

c

Fig. 9 Amplitude-frequency responses of the pipeline.

The amplitude-frequency responses of the pipe received from the experimental research and finite element techniques are presented in the Fig. 9. We took 270 finite element nodes in both techniques. c 1133,5 Resonance frequencies correspond to the quarter-wave resonance f    184 Hz , where c - reduced 4l 4  1.54 wave speed in the pipeline, including elasticity of pipe wall material, l – pipeline length. The quarter-wave resonance frequencies and resonance frequencies of the mechanical subsystem are presented in the Fig. 9. The frequencies 152, 440 and 770 Hz correspond to the 2, 5 и 9 oscillation mode. These frequencies correspond to the natural frequencies from experiments and simulation. The calculated curves received from the finite element technique using seven node elements and the finite element technique using two node finite elements are coincided.. 4. Conclusion The new types of finite elements are proposed for modeling vibroacoustical interaction between a solid structure and an oscillating fluid. The developed techniques allow calculation of complex configuration pipe system dynamics. The finite element techniques provide calculation of vibroacoustical characteristics of a pipe in the spacetime domain. The usage of the new element types provides a decrease in the computational coast by three orders compared to the available finite element techniques using different element types for solid and fluid structures. The number of nodes for finite element techniques is less than for Ansys. The results of calculation by the proposed finite element techniques were confirmed by experimental data.

687

688

Tatiana B. Mironova et al. / Procedia Engineering 176 (2017) 681 – 688

Acknowledgements This work was supported by the Ministry of Education and Science of the Russian Federation in the framework of the implementation of the Program “Research and development on priority directions of scientific-technological complex of Russia for 2014– 2020”. References [1] T.B. Mironova, A.B. Prokofiev, The finite element technique for modelling of pipe vibroacoustical characteristics based on space-time joint type two node elements, Proceedings of the 22th International Congress on Sound and Vibration, Florence, Italy, 12–16 July, (2015). [2] T.B. Mironova, A.B. Prokofiev, The finite element technique of the solving non-linear non-stationary differential equation system of the complex configuration piping system dynamics, Proceedings of the 23th International Congress on Sound and Vibration, Athens, Greece, 10– 14 July, (2016). [3] T.B. Mironova, A.B. Prokofiev, V.N. Iliukhin, The finite element technique for modelling of pipe vibroacoustical characteristics, Proceedings of the 23th International Congress on Sound and Vibration, Beijing, China, 13–17 July, (2014). [4] N. M.C. Martins, A. K. Soares, H. M. Ramos, Covas D. I.C., CFD modeling of transient flow in pres-surized pipes, Computers & Fluids, 1 (15), 129–140, (2016). [5] M. P. Paidoussis, Fluid-Structure Interactions (Second Edition) Volume 1: Slender Structures and Axi-al Flow, Elsevier Ltd., (2014) [6] M. Picasso, V. Prachittham, , An adaptive algorithm for the Crank–Nicolson scheme applied to a time-dependent convection–diffusion problem, Journal of Computational and Applied Mathematics, 4 (233), 1139–1154, (2009). [7] A.B.Prokofiev, Developing of the companion analysis method of the dynamic and strength of the pipe system with pressure pulsation damper, Doctor’s Degree thesis, Graduate Program in acoustics, Gradu-ate Program in dynamic and strength of the equipments and machines, Samara State Aerospace Univer-sity, (2008), (in Russian). [8] N.M.C. Martins, A.K. Soares, H.M. Ramos, D.I.C. Covas, CFD modeling of transient flow in pressurized pipes, Computers & Fluids, 1(15), 2016, pp. 129–140. [9] G.M. Makaryants, A.B. Prokofiev, E.V. Shakhmatov, Modeling of pipeline vibroacousticalcharacteristics by the finite element method, News of Samara scientific centre of the Russian Academy of Sciences, 4 (28), 2002, pp. 327-333. [10] I.M. Babakov, "Oscillation theory" , Moscow (2004), (In Russian). [11] O.C Zienkiewicz, Konechnie elementi i approksimacii (Finite elements and approximation), Mir, Moscow (1986). pp. 154–184.