# Dynamic Finite Element Model Based on Timoshenko Beam Theory for Simulating High-Speed Nonlinear Helical Springs

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## Abstract

**:**

## 1. Introduction

## 2. Timoshenko Beam Theory for Curved Beams

## 3. Dynamic Finite Element Analysis Based on Beam Elements

## 4. Results and Discussions

^{−4}s. This also shows that the significant spring forces generated at the 256-degree cam angle in Figure 7 and Figure 8 are caused by the coil clash between the second and third coils. These findings also coincide with the conclusions drawn from the previous research [1] that developed an FE spring model using solid elements to simulate the dynamic responses of helical springs at high engine speeds.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Gu, Z.; Hou, X.; Keating, E.; Ye, J. Non-linear finite element model for dynamic analysis of high-speed valve train and coil collisions. Int. J. Mech. Sci.
**2020**, 173, 105476. [Google Scholar] - Adeodato, A.; Duarte, B.T.; Monteiro, L.L.S.; Pacheco, P.M.C.; Savi, M.A. Synergistic use of piezoelectric and shape memory alloy elements for vibration-based energy harvesting. Int. J. Mech. Sci.
**2021**, 194, 106206. [Google Scholar] - Sutrisno, A.; Braun, D.J. How to run 50% faster without external energy. Sci. Adv.
**2020**, 6, eaay1950. [Google Scholar] - Wahl, A.M. Mechanical Springs; Penton Publishing Company: Dublin, Ireland, 1944. [Google Scholar]
- Renno, J.M.; Mace, B.R. Vibration modelling of helical springs with non-uniform ends. J. Sound Vib.
**2012**, 331, 2809–2823. [Google Scholar] - Qiu, D.; Paredes, M.; Seguy, S. Variable pitch spring for nonlinear energy sink: Application to passive vibration control. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2019**, 233, 611–622. [Google Scholar] - Nazir, A.; Ali, M.; Hsieh, C.-H.; Jeng, J.-Y. Investigation of stiffness and energy absorption of variable dimension helical springs fabricated using multijet fusion technology. Int. J. Adv. Manuf. Technol.
**2020**, 110, 2591–2602. [Google Scholar] - Gu, Z.; Hou, X.; Ye, J. Advanced static and dynamic analysis method for helical springs of non-linear geometries. J. Sound Vib.
**2021**, 513, 116414. [Google Scholar] - Lee, J.; Thompson, D. Dynamic stiffness formulation, free vibration and wave motion of helical springs. J. Sound Vib.
**2001**, 239, 297–320. [Google Scholar] [CrossRef] - Flenker, C.; Uphoff, U. Efficient valve-spring modelling with MBS valve-train design. MTZ Worldw.
**2005**, 66, 6–8. [Google Scholar] [CrossRef] - Liu, H.; Kim, D. Effects of end coils on the natural frequency of automotive engine valve springs. Int. J. Automot. Technol.
**2009**, 10, 413–420. [Google Scholar] - Kushwaha, M.; Rahnejat, H.; Jin, Z. Valve-train dynamics: A simplified tribo-elasto-multi-body analysis. Proc. Inst. Mech. Eng. Part K J. Multi-Body Dyn.
**2000**, 214, 95–110. [Google Scholar] [CrossRef] [Green Version] - Mclaughlin, S.; Haque, I. Development of a multi-body simulation model of a Winston Cup valvetrain to study valve bounce. Proc. Inst. Mech. Eng. Part K J. Multi-Body Dyn.
**2002**, 216, 237–248. [Google Scholar] - Liu, H.; Kim, D. Estimation of valve spring surge amplitude using the variable natural frequency and the damping ratio. Int. J. Automot. Technol.
**2011**, 12, 631. [Google Scholar] [CrossRef] - Hsu, W.; Pisano, A. Modeling of a finger-follower cam system with verification in contact forces. J. Mech. Des.
**1996**, 118, 132–137. [Google Scholar] - Huber, R.; Clauberg, J.; Ulbrich, H. An efficient spring model based on a curved beam with non-smooth contact mechanics for valve train simulations. SAE Int. J. Engines
**2010**, 3, 28–34. [Google Scholar] - Kim, D.; David, J.W. A Combined Model for High Speed Valve Train Dynamics (Partly Linear and Partly Nonlinear); Technical Report for SAE; SAE: Warrendale, PA, USA, 1990. [Google Scholar]
- Wittrick, W.H. On elastic wave propagation in helical springs. Int. J. Mech. Sci.
**1966**, 8, 25–47. [Google Scholar] [CrossRef] - Frikha, A.; Treyssede, F.; Cartraud, P. Effect of axial load on the propagation of elastic waves in helical beams. Wave Motion
**2011**, 48, 83–92. [Google Scholar] - Pearson, D.; Wittrick, W. An exact solution for the vibration of helical springs using a Bernoulli-Euler model. Int. J. Mech. Sci.
**1986**, 28, 83–96. [Google Scholar] - Zhang, J.; Qi, Z.; Wang, G.; Guo, S. High-efficiency dynamic modeling of a helical spring element based on the geometrically exact beam theory. Shock Vib.
**2020**, 2020, 8254606. [Google Scholar] - Yang, C.; Zhang, W.; Ren, G.; Liu, X. Modeling and dynamics analysis of helical spring under compression using a curved beam element with consideration on contact between its coils. Meccanica
**2014**, 49, 907–917. [Google Scholar] - Zhang, Z.; Qi, Z.; Wu, Z.; Fang, H. A spatial Euler-Bernoulli beam element for rigid-flexible coupling dynamic analysis of flexible structures. Shock Vib.
**2015**, 2015, 208127. [Google Scholar] - Sun, W.; Thompson, D.; Zhou, J.; Gong, D. Analysis of dynamic stiffness effect of primary suspension helical springs on railway vehicle vibration. J. Phys. Conf. Ser.
**2016**, 744, 012149. [Google Scholar] - Michalczyk, K. Analysis of lateral vibrations of the axially loaded helical spring. J. Theor. Appl. Mech.
**2015**, 53, 745–755. [Google Scholar] [CrossRef] - Čakmak, D.; Wolf, H.; Božić, Ž.; Jokić, M. Optimization of an inerter-based vibration isolation system and helical spring fatigue life assessment. Arch. Appl. Mech.
**2019**, 89, 859–872. [Google Scholar] - Yousefi, A.; Rastgoo, A. Free vibration of functionally graded spatial curved beams. Compos. Struct.
**2011**, 93, 3048–3056. [Google Scholar] - Meier, C.; Popp, A.; Wall, W.A. An objective 3D large deformation finite element formulation for geometrically exact curved Kirchhoff rods. Comput. Methods Appl. Mech. Eng.
**2014**, 278, 445–478. [Google Scholar] - Love, A.E.H. The Propagation of Waves of Elastic Displacement Along a Helical Wire. Trans. Camb. Philos. Soc.
**1900**, 18, 364–374. [Google Scholar] - Becker, L.; Chassie, G.; Cleghorn, W. On the natural frequencies of helical compression springs. Int. J. Mech. Sci.
**2002**, 44, 825–841. [Google Scholar] - Becker, L.E.; Cleghorn, W. On the buckling of helical compression springs. Int. J. Mech. Sci.
**1992**, 34, 275–282. [Google Scholar] - Chassie, G.G.; Becker, L.; Cleghorn, W. On the buckling of helical springs under combined compression and torsion. Int. J. Mech. Sci.
**1997**, 39, 697–704. [Google Scholar] [CrossRef] - Beck, A.T.; Da Silva, C.R., Jr. Timoshenko versus Euler beam theory: Pitfalls of a deterministic approach. Struct. Saf.
**2011**, 33, 19–25. [Google Scholar]

**Figure 1.**(

**a**) One spring coil in the Serret-Frenet coordinates. (

**b**) Unit rod element of a spring coil.

**Figure 2.**(

**a**) The beehive spring sample. (

**b**) The FE model of the beehive spring using the Timoshenko beam element.

**Figure 3.**The FE models of beehive spring with (

**a**) 2 mm and (

**b**) 0.4 mm element sizes and the convergency study of the size of elements at (

**c**) 7 mm, (

**d**) 5 mm and (

**e**) 3 mm compressions.

**Figure 4.**(

**a**) The valve train system of a V8 sports car engine. (

**b**) The FE spring model with a 0.4 mm element size and its (

**c**) static and (

**d**) dynamic status.

**Figure 5.**Comparison of the spring reaction forces at 5600 RPM engine speed of the dynamic FE results using beam elements and solid elements and the results of the engine head test.

**Figure 6.**Comparison between the spring reaction forces at 5600 RPM engine speeds of the dynamic FE results using beam elements and solid elements and the results of the engine head test (zoom-in area between the 220-degree and 280-degree cam angles).

**Figure 7.**Comparison between the spring reaction forces at 8000 RPM engine speed of the dynamic FE results using beam elements and solid elements and the results of the engine head test.

**Figure 8.**Comparison between the spring reaction forces at 8000 RPM engine speeds of the dynamic FE results using beam elements and solid elements and the results of the engine head test (zoom-in area between the 220-degree and 280-degree cam angles).

**Figure 9.**The motion status of the FE spring model at (

**a**) 255-degree, (

**b**) 256-degree and (

**c**) 257-degree cam angles and the (

**d**) acceleration and (

**e**) velocity of the node on the third coil from the lower spring end.

Coil Revolution | Helix Height (mm) | Spring Pitch (mm) | Coil Diameter (mm) |
---|---|---|---|

1 | 3.865 | 5.73 | 22.25 |

2 | 9.132 | 4.804 | 22.25 |

3 | 17.77 | 12.47 | 22.25 |

4 | 26.57 | 5.134 | 22.25 |

5 | 35.21 | 12.14 | 21.405 |

6 | 43.03 | 3.484 | 10.017 |

7 | 47.35 | 5.152 | 18.35 |

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**MDPI and ACS Style**

Zhao, J.; Gu, Z.; Yang, Q.; Shao, J.; Hou, X.
Dynamic Finite Element Model Based on Timoshenko Beam Theory for Simulating High-Speed Nonlinear Helical Springs. *Sensors* **2023**, *23*, 3737.
https://doi.org/10.3390/s23073737

**AMA Style**

Zhao J, Gu Z, Yang Q, Shao J, Hou X.
Dynamic Finite Element Model Based on Timoshenko Beam Theory for Simulating High-Speed Nonlinear Helical Springs. *Sensors*. 2023; 23(7):3737.
https://doi.org/10.3390/s23073737

**Chicago/Turabian Style**

Zhao, Jianwei, Zewen Gu, Quan Yang, Jian Shao, and Xiaonan Hou.
2023. "Dynamic Finite Element Model Based on Timoshenko Beam Theory for Simulating High-Speed Nonlinear Helical Springs" *Sensors* 23, no. 7: 3737.
https://doi.org/10.3390/s23073737