Skip to main content

Advertisement

SpringerLink
Log in

Nonlinear dynamics of flexible slender structures moving in a limited space with application in nuclear reactors

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper deals with the modelling and dynamic analysis of thin or slender flexible bodies characterized by axial motion, contacts and friction interaction. Such mechanical systems can be found in cable applications, in petroleum drilling industry, in biomedical applications and in nuclear equipments. An original combination of the absolute nodal coordinate formulation and the classical finite element method is used for the modelling of flexible slender structures moving inside flexible tubes and channels considering the overall system vibration. In a nuclear power plant, it allows to analyse mainly the influence of guide thimbles vibration on the control rod drop. The novelty of the introduced methodology is also in the original usage of the absolute nodal coordinate formulation in the problems of nuclear engineering. All the interaction phenomena such as effect of fluid flow, contact forces, gravitation and buoyancy forces are considered in the mathematical model. The particular application of the presented methodology on the problem of the nuclear control rod drop through vibrating bowed guide thimble helps to formulate interesting conclusions on the rod drop time affected by vibrating guide thimbles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Data availability statement

The datasets generated during and analysed during the current study are available from the corresponding author on reasonable request.

Notes

  1. VVER, also referred to as WWER, stands for water-cooled and water-moderated energetic reactor, number 1000 refers to the power provided by the reactor.

Abbreviations

\(\alpha \) :

Rayleigh proportional damping parameter

\(\alpha _c\) :

Angle defining the contact position

\(\beta \) :

Rayleigh proportional damping parameter

\(\delta \) :

Contact penetration depth

\(\eta \) :

FE dimensionless parameter

\(\kappa \) :

Curvature of element centre line

\(\mathbf{B} \) :

Global damping matrix

\(\mathbf{B} _{e}\) :

Element damping matrix

\(\mathbf{e} \) :

Vector of ANCF nodal coordinates

\(\mathbf{e} _\mathrm{FEM}\) :

Vector of FE beam nodal coordinates

\(\mathbf{f} _\mathrm{cont}^\mathrm{(CR)}\) :

Global vector of contact forces for the CR

\(\mathbf{f} _\mathrm{cont}^\mathrm{(GT)}\) :

Global vector of contact forces for the GT

\(\mathbf{f} _\mathrm{drag}^\mathrm{(CR)}\) :

Global vector of drag forces for the CR

\(\mathbf{f} _{e, \mathrm {cont}}^\mathrm{FEM}\) :

Generalized contact force vector of e-th FEM element

\(\mathbf{F} _{e}^\mathrm{FEM}\) :

Contact forces vector in e-th FEM element

\(\mathbf{f} _{k, \mathrm {cont}}^\mathrm{ANCF}\) :

Generalized contact force vector of k-th ANCF element

\(\mathbf{f} _\mathrm{{kin}, { L}}^\mathrm{(GT)}\) :

Kinematical excitation of lower node

\(\mathbf{f} _\mathrm{{kin}, { U}}^\mathrm{(GT)}\) :

Kinematical excitation of upper node

\(\mathbf{F} _{p,k}^\mathrm{ANCF}\) :

Contact forces vector in p-th contact point of k-th ANCF element

\(\mathbf{f} _\mathrm{st}^\mathrm{(CR)}\) :

Global vector of gravitation and buoyancy for the CR

\(\mathbf{g} \) :

Gravitational acceleration vector

\(\mathbf{G} _e\) :

Generalized gravity force vector

\(\mathbf{I} \) :

Identity matrix

\(\mathbf{K} _{ C}\) :

Global coupling stiffness matrix of spacer grids

\(\mathbf{K} \) :

Global stiffness matrix

\(\mathbf{K} _e^\mathrm{tang}\) :

ANCF tangent stiffness matrix

\(\mathbf{M} \) :

Global mass matrix

\(\mathbf{M} _e\) :

Element mass matrix

\(\mathbf{N} \) :

Matrix of FEM shape functions

\(\mathbf{Q} \) :

Global vector of elastic forces

\(\mathbf{q} \) :

Global vector of nodal coordinates

\(\mathbf{Q} _e\) :

Vector of element elastic forces

\(\mathbf{q} _e\) :

Vector of arbitrary beam point displacements and rotation angles

\(\mathbf{q} _s\) :

Vector of spacer grids deformations

\(\mathbf{Q} _\mathrm{be}\) :

Vector of bending elastic forces

\(\mathbf{Q} _\mathrm{le}\) :

Vector of longitudinal elastic forces

\(\mathbf{r} \) :

Position vector of a beam point

\(\mathbf{r} ^j\) :

Position vector of node j

\(\mathbf{r} ^j_{,x}\) :

Gradient of position vector

\(\mathbf{r} _c\) :

Position vector between cross-sectional centre points

\(\mathbf{r} _c^\mathrm{CR}\) :

Global position vector of CR cross-sectional centre point

\(\mathbf{r} _c^\mathrm{GT}\) :

Global position vector of GT cross-sectional centre point

\(\mathbf{r} _D\) :

Position vector from GT centre to detected contact point

\(\mathbf{S} \) :

Matrix of ANCF shape functions

\(\omega _{b1}\) :

First bending eigenfrequency

\(\omega _{b2}\) :

Second bending eigenfrequency

\(\rho \) :

Material density

\(\rho _c\) :

Water coolant density

\(\rho _e\) :

Density of element e

\(\rho _\mathrm{CR}\) :

Considered density of the CR

\(\varepsilon _x\) :

Axial strain

\(\varphi _x,\ \varphi _z\) :

Beam FE rotational angles

\(\xi \) :

ANCF dimensionless parameter

\(A^\mathrm{CR}\) :

Control rod cross-sectional area

\(A_e\) :

Beam cross-sectional area

\(c_f\) :

Friction coefficient

\(C_v\) :

Drag coefficient

D :

Damping ratio

\(D_c\) :

Contact damping factor

\(D_e\) :

Outer diameter of GT

\(d_e\) :

Inner diameter of GT

\(D_\mathrm{CR}\) :

Outer diameter of the CR

\(D_{p, k}^e\) :

Detected contact point between k-th ANCF element and e-th FEM element

E :

Young’s modulus

e :

Index of FEM beam element

\(E_e\) :

Young’s modulus of element e

F :

Free FE beam nodes

f :

Range of excitation frequencies

\(F_b\) :

Buoyancy force

\(F_N\) :

Normal contact force

\(F_T\) :

Friction force

\(F_\mathrm{drag}\) :

Hydrodynamic drag force

g :

Index of spacer grid (\(g = 2i\) in FEM node i)

\(g_\mathrm{const}\) :

Gravitational acceleration

i :

Index of FEM beam node

\(I_e\) :

Second moment of area

j :

Index of ANCF beam node

K :

Contact stiffness

k :

Index of ANCF beam element

\(k_{g,x},\ k_{g,z}\) :

Spacer grid stiffness

L :

Lower FE beam node

\(l_e\) :

Element length

\(l_\mathrm{CR}\) :

Total length of the CR

M :

Total number of ANCF nodes

m :

Weight of body

N :

Total number of FE beam nodes

n :

Hertz contact exponent

\(N_b\) :

FE shape functions, \(b = 1,\ \dots ,\ 8\)

\(n_{C}\) :

Total number of contact points per element

p :

Index of ANCF beam element contact point

\(r_\mathrm{CR}\) :

CR radius

\(r_\mathrm{GT}\) :

GT radius

\(S_a\) :

ANCF shape functions, , \(a = 1,\ \dots ,\ 4\)

\(t_\mathrm{lim}\) :

Drop time limit value

U :

Upper FE beam node

\(U_e\) :

Element deformation energy

\(u_x,\ u_z\) :

Beam FE lateral displacements

\(V_e\) :

Element volume

\(v_r\) :

Velocity threshold value for friction

\(v_Y\) :

Vertical velocity in the contact point

\(v_\mathrm{CR}\) :

CR velocity

\(v_\mathrm{flow}\) :

Fluid flow velocity

x :

Local axial coordinate of ANCF element

\(X,\ Y,\ Z\) :

Global coordinate system

\(x_{C_p}\) :

Axial parameter of ANCF contact points

\(x_{g,st}\) :

Lateral static deformation of LBS in SG

\(x_\mathrm{max}\) :

GT maximum deformation

\(y_\mathrm{{loc}, D}\) :

FE beam axial coordinate of detected contact point

\(y_\mathrm{loc}\) :

local axial parameter of FE beam

\(z_{g, \mathrm {st}}\) :

Lateral static deformation of LBS in SG

References

  1. Andersson, T., Almberger, J., Björnkvist, L.: A decade of assembly bow management at ringhals. In: IAEA-TECDOC-1454, Structural Behaviour of Fuel Assemblies for Water Cooled Reactors, pp. 129–136. IAEA (2005)

  2. Andriambololona, H., Bosselut, D., Massin, P.: Methodology for a numerical simulation of an insertion or a drop of the rod cluster control assembly in a PWR. Nucl. Eng. Des. 237(6), 600–606 (2007)

    Article  Google Scholar 

  3. Aullo, M., Rabenstein, W.: European fuel group experience on control rod insertion and grid to rod fretting. In: IAEA-TECDOC-1454, Structural Behaviour of Fuel Assemblies for Water Cooled Reactors, pp. 147–163. IAEA (2005)

  4. Bauchau, O.A., Han, S., Mikkola, A., Matikainen, M.K.: Comparison of the absolute nodal coordinate and geometrically exact formulations for beams. Multibody System Dynamics 32, 67–85 (2014). https://doi.org/10.1007/s11044-013-9374-7

    Article  MathSciNet  Google Scholar 

  5. Bozorgmehri, B., Yu, X., Matikainen, M.K., Harish, A.B., Mikkola, A.: A study of contact methods in the application of large deformation dynamics in self-contact beam. Nonlinear Dynamics 103, 581–616 (2021). https://doi.org/10.1007/s11071-020-05984-x

    Article  Google Scholar 

  6. Bulín, R., Hajžman, M., Polach, P.: Nonlinear dynamics of a cable-pulley system using the absolute nodal coordinate formulation. Mech. Res. Commun. 82, 21–28 (2017)

    Article  Google Scholar 

  7. Bulín, R., Hajžman, M., Polach, P.: Investigation of falling control rods in deformed guiding tubes in nuclear reactors using multibody approaches. In: Proceedings of The 5th Joint International Conference on Multibody System Dynamics, pp. 1–8. Técnico Lisboa (2018)

  8. Dibold, M., Gerstmayr, J., Irschik, H.: A detailed comparison of the absolute nodal coordinate and the floating frame of reference formulation in deformable multibody systems. Journal of Computational and Nonlinear Dynamics 4, 1–10 (2009). https://doi.org/10.1115/1.3079825

    Article  Google Scholar 

  9. Donis, J., Goller, H.: A mathematical model of a control rod drop. Nuclear Engineering and Design 23(1), 107–120 (1972). https://doi.org/10.1016/0029-5493(72)90193-8

    Article  Google Scholar 

  10. Dou, Y., Yao, W., Jiang, N., Wang, Y.: A scram analysis under accident conditions for a reactor control rod system. In: Transactions of the 14th International Conference on Structural Mechanics in Reactor Technology (SMiRT 14), pp. 241–248. IASMiRT (1997)

  11. Dyk, Š, Zeman, V.: Impact vibrations of guide thimbles in nuclear fuel assembly. Archive of Applied Mechanics 87, 231–244 (2017). https://doi.org/10.1007/s00419-016-1189-6

    Article  Google Scholar 

  12. Esfandiar, H., Korayem, M.H., Haghpanahi, M.: Large deformation modeling of flexible manipulators to determine allowable load. Struct. Eng. Mech. 62, 619–629 (2017)

    Article  Google Scholar 

  13. Feireisl, E.: On the motion of rigid bodies in a viscous fluid. Applications of mathematics 47(6), 463–484 (2002). https://doi.org/10.1023/A:1023245704966

    Article  MathSciNet  MATH  Google Scholar 

  14. Gerstmayr, J., Shabana, A.A.: Analysis of thin beams and cables using absolute nodal co-ordinate formulation. Nonlinear Dynamics 45, 109–130 (2006). https://doi.org/10.1007/s11071-006-1856-1

    Article  MATH  Google Scholar 

  15. Gerstmayr, J., Sugiyama, H., Mikkola, A.: Review on the absolute nodal coordinate formulation for large deformation analysis of multibody systems. J. Comput. Nonlinear Dyn. 8, 031011–03101612 (2013). https://doi.org/10.1115/1.4023487

    Article  Google Scholar 

  16. Hong, D., Ren, G.: A modeling of sliding joint on one-dimensional flexible medium. Multibody System Dynamics 26(1), 91–106 (2011). doi: 10.1007/s11044-010-9242-7

    Article  MathSciNet  MATH  Google Scholar 

  17. Huang, H., Wang, Z., Xu, W., Liu, T., Yang, Y., Li, P.: Seismic analysis of PWR control rod drop with the CRDAC scram performance code. Ann. Nucl. Energy 114, 624–633 (2018)

    Article  Google Scholar 

  18. Lee, J.W., Kim, H.W., Ku, H.C., Yoo, W.S.: Comparison of external damping models in a large deformation problem. Journal of Sound and Vibration 325, 722–741 (2009). doi: 10.1016/j.jsv.2009.04.018

    Article  Google Scholar 

  19. Li, Y., Wang, C., Huang, W.: Rigid-flexible-thermal analysis of planar composite solar array with clearance joint considering torsional spring, latch mechanism and attitude controller. Nonlinear Dynamics 96, 2031–2053 (2019). doi: 10.1007/s11071-019-04903-z

    Article  Google Scholar 

  20. Lin, Z., Zhai, L., Zhu, L., Wang, X., Zhang, X., Cao, Y., Wang, N.: Control rod drop dynamic analysis in the TMSR-SF1 based on numerical simulation and experiment. Nuclear Engineering and Design 322, 131–137 (2017). doi: 10.1016/j.nucengdes.2017.06.031

    Article  Google Scholar 

  21. Lindström, S.B., Uesaka, T.: Simulation of the motion of flexible fibers in viscous fluid flow. Physics of Fluids 19(113307), 1–16 (2007). doi: 10.1063/1.2778937

    Article  MATH  Google Scholar 

  22. Liu, J.P., Cheng, Z.B., Ren, G.X.: An arbitrary lagrangian-eulerian formulation of a geometrically exact timoshenko beam running through a tube. Acta Mechanica 229, 3161–3188 (2018). doi: https://doi.org/10.1007/s00707-018-2161-z

    Article  MathSciNet  MATH  Google Scholar 

  23. Lu, D., Wang, Y., Xie, Q., Zhang, H., Ali, M.: R&d on a nonlinear dynamics analysis code for the drop time of the control rod. Science and Technology of Nuclear Installations 2017, 1–8 (2017). https://doi.org/10.1155/2017/1494385

    Article  Google Scholar 

  24. Lugrís, U., Escalona, J.L., Dopico, D., Cuadrado, J.: Efficient and accurate simulation of the rope-sheave interaction in weight-lifting machines. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 225(4), 331–343 (2011). https://doi.org/10.1177/1464419311403224

    Article  Google Scholar 

  25. Oh, S.H., Kim, J.Y., Yoon, K.H., Choi, C., Son, S.M.: Numerical analysis on the free fall motion of the control rod assembly for the sodium cooled fast reactor. In: Transactions of the Korean Nuclear Society Autumn Meeting, pp. 1–4. Korean Nuclear Society (2015)

  26. Pavlov, S.: In-service change in the flexural rigidity of the VVER-1000 fuel assemblies. Nucl. Energy Technol. 2(4), 251–255 (2016)

    Article  Google Scholar 

  27. Pennestrì, E., Rossi, V., Salvini, P.: Review and comparison of dry friction force models. Nonlinear Dynamics 83, 1785–1801 (2016). doi: 10.1007/s11071-015-2485-3

    Article  MATH  Google Scholar 

  28. Polach, P., Hajžman, M.: Research report VYZ 1236/09: Verification of the multibody model of the LKP-M/3 control assembly intended for the control assembly moving parts drop calculation at the seismic event on the basis of experimental measurement results. Škoda výzkum s.r.o, Pilsen (2009)

    Google Scholar 

  29. Qi, Z., Wang, J., Wang, G.: An efficient model for dynamic analysis and simulation of cable-pulley systems with time-varying cable lengths. Mech. Mach. Theory 116, 383–403 (2017)

    Article  Google Scholar 

  30. Recuero, A.M., Aceituno, J.F., Escalona, J.L., Shabana, A.A.: A nonlinear approach for modeling rail flexibility using the absolute nodal coordinate formulation. Nonlinear Dynamics 83, 463–481 (2016). doi: 10.1007/s11071-015-2341-5

    Article  MathSciNet  MATH  Google Scholar 

  31. Romero, I.: A comparison of finite elements for nonlinear beams: the absolute nodal coordinate and geometrically exact formulations. Multibody System Dynamics 20, 51–68 (2008). doi: 10.1007/s11044-008-9105-7

    Article  MathSciNet  MATH  Google Scholar 

  32. Rong, B., Rui, X., Tao, L., Wang, G.: Theoretical modeling and numerical solution methods for flexible multibody system dynamics. Nonlinear Dynamics 98, 1519–1553 (2019). doi: 10.1007/s11071-019-05191-3

    Article  Google Scholar 

  33. Sarrate, J., Huerta, A., Donea, J.: Arbitraty lagrangian-eulerian formulation for fluid-rigid body interaction. Computer methods in applied mechanics and engineering 190, 3171–3188 (2001). doi: https://doi.org/10.1016/S0045-7825(00)00387-X

    Article  MATH  Google Scholar 

  34. Shabana, A.A.: Flexible multibody dynamics: Review of past and recent developments. Multibody Sys.Dyn. 1, 189–222 (1997). https://doi.org/10.1023/A:1009773505418

    Article  MathSciNet  MATH  Google Scholar 

  35. Shabana, A.A.: Dynamics of multibody systems, 3rd edn. Cambridge University Press (2013)

    Book  Google Scholar 

  36. Shabana, A.A., Sany, J.R.: A survey of rail vehicle track simulations and flexible multibody dynamics. Nonlinear Dynamics 26, 179–212 (2001). doi: 10.1023/A:1012976302105

    Article  MATH  Google Scholar 

  37. Somnay, R., Ibrahim, R.: Nonlinear dynamics of a sliding beam on two supports under sinusoidal excitation. Sadhana 31, 383–397 (2006). doi: 10.1007/BF02716783

    Article  MathSciNet  MATH  Google Scholar 

  38. Tang, L., Liu, J.: Modeling and analysis of sliding joints with clearances in flexible multibody systems. Nonlinear Dynamics 94, 2423–2440 (2018). doi: 10.1007/s11071-018-4500-y

    Article  Google Scholar 

  39. Tang, L., Liu, J.: Frictional contact analysis of sliding joints with clearances between flexible beams and rigid holes in flexible multibody systems. Multibody System Dynamics 49, 155–179 (2020). doi: 10.1007/s11044-019-09717-w

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, Q., Tian, Q., Hu, H.: Dynamic simulation of frictional multi-zone contacts of thin beams. Nonlinear Dynamics 83, 1919–1937 (2016). doi: 10.1007/s11071-015-2456-8

    Article  Google Scholar 

  41. Wanninger, A., Seidl, M., Macián-Juan, R.: Mechanical analysis of the bow deformation of a row of fuel assemblies in a PWR core. Nucl. Eng. Technol. 50(2), 297–305 (2018)

    Article  Google Scholar 

  42. Yang, C.J., Hong, D., Ren, G., Zhao, Z.H.: Cable installation simulation by using a multibody dynamic model. Multibody Sys. Dyn. 30, 433–447 (2013). https://doi.org/10.1007/s11044-013-9364-9

    Article  MathSciNet  Google Scholar 

  43. Yang, Y.B., Chiou, H.T.: Rigid body motion test for nonlinear analysis with beam elements. Journal of Engineering Mechanics 113(9), 1404–1419 (1987). https://doi.org/10.1061/(ASCE)0733-9399(1987)113:9(1404)

    Article  Google Scholar 

Download references

Acknowledgements

The work was supported from European Regional Development Fund-Project “Research and Development of Intelligent Components of Advanced Technologies for the Pilsen Metropolitan Area (InteCom)” (No. CZ. 02.1.01/0.0/0.0/17_048/0007267) and by the Motivation system of the University of West Bohemia—Part Postdoc.

Author information

Authors and Affiliations

  1. NTIS, Faculty of Applied Sciences, University of West Bohemia, Technická 8, Pilsen, Czech Republic

    R. Bulín, Š. Dyk & M. Hajžman

Authors
  1. R. Bulín
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Š. Dyk
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Hajžman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Bulín.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bulín, R., Dyk, Š. & Hajžman, M. Nonlinear dynamics of flexible slender structures moving in a limited space with application in nuclear reactors. Nonlinear Dyn 104, 3561–3579 (2021). https://doi.org/10.1007/s11071-021-06582-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06582-1

Keywords

Search

Navigation