Problem 12: Wilberforce Pendulum

A Wilberforce pendulum consists of a mass hanging from a vertically oriented helical spring. The mass can both move up and down on the spring and rotate about its
vertical axis. Investigate the behaviour of such a pendulum and how it depends on relevant parameters.


I. Phenomenon Demonstration

II. Books, Encyclopedia, Discussion and Forum Posts

III. Research Papers

  • Berg, R. E., & Marshall, T. S. (1991). Wilberforce pendulum oscillations and normal modes. American Journal of Physics, 59(1), 32–38.
  • de Bustos, M. T., López, M. A., & Martínez, R. (2016). On the periodic orbits of the perturbed Wilberforce pendulum. Journal of Vibration and Control, 22(4), 932–939.
  • Miro, Plavčić & Zupanovic, Pasko & Bonačić Lošić, Željana. (2009). The resonance of the Wilberforce pendulum and the period of beats. Latin-American Journal of Physics Education.
  • Debowska, E., Jakubowicz, S., & Mazur, Z. (1999). Computer visualization of the beating of a Wilberforce pendulum. European Journal of Physics, 20(2), 89–95.
  • Greczylo, T., & Debowska, E. (2002). Using a digital video camera to examine coupled oscillations. European Journal of Physics, 23(4), 441–447.
  • Kos, B., Grodzicki, M., & Wasielewski, R. (2018). Electronic system for the complex measurement of a Wilberforce pendulum. European Journal of Physics, 39(3), 035804.
  • The Wilberforce pendulum: a complete analysis through RTL and modelling
  • Köpf, U. (1990). Wilberforce’s pendulum revisited. American Journal of Physics, 58(9), 833–837. doi:10.1119/1.16376
  • Hübner, M., & Kröger, J. (2018). Experimental verification of the adiabatic transfer in Wilberforce pendulum normal modes. American Journal of Physics, 86(11), 818–824. doi:10.1119/1.5051179
  • Wilhelm, T.S., Orndorff, J., & Baak, D. (2010). The Avoided Crossing in the Normal-Mode Frequencies of a Wilberforce Pendulum.
  • Devaux, P., Piau, V., Vignaud, O., Grosse, G., Olarte, R., & Nuttin, A. (2019). Cross-camera tracking and frequency analysis of a cheap Slinky Wilberforce pendulum. Emergent Scientist, 3, 1.
  • NAKAMICHI, Y., FUNADA, T., SHIMIZU, K., IWAMOTO, D., FUNATSU, Y., & OOBA, K. (2010). Fundamental analysis of a wilberforce pendulum. Memoirs of Numazu College of Technology, (44), 369-374.
  • Greczyło, T. The Use of Digital Techniques in an Investigation of Coupled Oscillations of a Wilberforce Pendulum.
  • Acosta-Humánez, P. B., Blázquez-Sanz, D., & Delgado, J. (2011). Non-integrability Criterium for Normal Variational Equations around an integrable Subsystem and an example: The Wilbeforce spring-pendulum. arXiv preprint arXiv:1104.0312.
  • Wilberforce, L. R. (1894). XLIV. On the vibrations of a loaded spiral spring. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 38(233), 386–392. doi:10.1080/14786449408620648

  • Moradi, S., Anderson, J., & Gürcan, O. D. (2015). Predator-prey model for the self-organization of stochastic oscillators in dual populations. Physical Review E, 92(6), 062930.

  • Iqbal, A. (2019). Applications of an Extended Kalman Filter in nonlinear mechanics (Doctoral dissertation, PhD Thesis, University of Management and Technology).

  • Kutsenko, L., Vanin, V., Shoman, O., Yablonskyi, P., Zapolskiy, L., Hrytsyna, N., … & Shevchenko, S. (2019). Modeling the resonance of a swinging spring based on the synthesis of a motion trajectory of its load. Восточно-Европейский журнал передовых технологий, (3 (7)), 53-64.

  • Caballero Flores, R., & Prida Pidal, V. M. D. L. (2020). Resolución geométrica del péndulo de Wilberforce. Revista Española de Física, 33 (4).

  • Caballero Flores, R., & Prida, V. M. (2019). Resolución analítica del péndulo de Wilberforce. Real Sociedad Española de Física, Revista española de Física.

  • Silva, O. H. M., de Mello Arruda, S., Laburú, C. E., & Bueno, E. A. S. (2013). Pêndulo de Wilberforce: uma proposta de montagem para ambientes educativos informais e laboratórios didáticos. Caderno brasileiro de ensino de fisica, 30(2), 409-426. 
  • Geballe, R. (1958). Statics and Dynamics of a Helical Spring. American Journal of Physics, 26(5), 287–290. doi:10.1119/1.1996131

  • Yang, C. J., Zhang, W. H., Ren, G. X., & Liu, X. Y. (2013). Modeling and dynamics analysis of helical spring under compression using a curved beam element with consideration on contact between its coils. Meccanica, 49(4), 907–917. doi:10.1007/s11012-013-9837-1

  • Rashidi, M., Budhabhatti, S. P., & Frater, J. L. (2004). Dynamics of a Coulomb Damped Helical Spring: A Finite Element Approach. Dynamic Systems and Control, Parts A and B. doi:10.1115/imece2004-59625

  • Nagaya, K. (1987). Stresses in a Helical Spring of Arbitrary Cross Section With Consideration of End Effects. Journal of Vibration Acoustics Stress and Reliability in Design, 109(3), 289. doi:10.1115/1.3269434

  • Bergou, M., Wardetzky, M., Robinson, S., Audoly, B., & Grinspun, E. (2008). Discrete elastic rods. ACM Transactions on Graphics, 27(3), 1. doi:10.1145/1360612.1360662

  • YILDIRIM, V. (1996). INVESTIGATION OF PARAMETERS AFFECTING FREE VIBRATION FREQUENCY OF HELICAL SPRINGS. International Journal for Numerical Methods in Engineering, 39(1), 99–114. doi:10.1002/(sici)1097-0207(19960115)39:1<99::aid-nme850>;2-m

  • Deul, C., Kugelstadt, T., Weiler, M., & Bender, J. (2018). Direct Position-Based Solver for Stiff Rods. Computer Graphics Forum, 37(6), 313–324. doi:10.1111/cgf.13326