The system is much more complex than a simple pendulum, as the properties of the spring add an extra dimension of freedom to the system. For example, when the spring compresses, the shorter radius causes the spring to move faster due to the conservation of angular momentum. It is also possible that the spring has a range that is overtaken by the motion of the pendulum, making it practically neutral to the motion of the pendulum.
Lagrangian
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The spring has the rest length and can be stretched by a length . The angle of oscillation of the pendulum is .
Hooke's law is the potential energy of the spring itself:
where is the spring constant.
The potential energy from gravity, on the other hand, is determined by the height of the mass. For a given angle and displacement, the potential energy is:
where is the velocity of the mass. To relate to the other variables, the velocity is written as a combination of a movement along and perpendicular to the spring:
The elastic pendulum is now described with two coupled ordinary differential equations. These can be solved numerically. Furthermore, one can use analytical methods to study the intriguing phenomenon of order-chaos-order[7] in this system.
^ abcdXiao, Qisong; et al. "Dynamics of the Elastic Pendulum" (PDF).
^ abcPokorny, Pavel (2008). "Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum" (PDF). Regular and Chaotic Dynamics. 13 (3): 155–165. Bibcode:2008RCD....13..155P. doi:10.1134/S1560354708030027. S2CID 56090968.
^Hill, Christian (19 July 2017). "The spring pendulum".
^Leah, Ganis. The Swinging Spring: Regular and Chaotic Motion.
^Simionescu, P.A. (2014). Computer Aided Graphing and Simulation Tools for AutoCAD Users (1st ed.). Boca Raton, Florida: CRC Press. ISBN 978-1-4822-5290-3.
^Anurag, Anurag; Basudeb, Mondal; Bhattacharjee, Jayanta Kumar; Chakraborty, Sagar (2020). "Understanding the order-chaos-order transition in the planar elastic pendulum". Physica D. 402: 132256. Bibcode:2020PhyD..40232256A. doi:10.1016/j.physd.2019.132256. S2CID 209905775.
Further reading
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Pokorny, Pavel (2008). "Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum" (PDF). Regular and Chaotic Dynamics. 13 (3): 155–165. Bibcode:2008RCD....13..155P. doi:10.1134/S1560354708030027. S2CID 56090968.
Pokorny, Pavel (2009). "Continuation of Periodic Solutions of Dissipative and Conservative Systems: Application to Elastic Pendulum" (PDF). Mathematical Problems in Engineering. 2009: 1–15. doi:10.1155/2009/104547.
External links
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Holovatsky V., Holovatska Y. (2019) "Oscillations of an elastic pendulum" (interactive animation), Wolfram Demonstrations Project, published February 19, 2019.
Holovatsky V., Holovatskyi I., Holovatska Ya., Struk Ya. Oscillations of the resonant elastic pendulum. Physics and Educational Technology, 2023, 1, 10–17, https://doi.org/10.32782/pet-2023-1-2 http://journals.vnu.volyn.ua/index.php/physics/article/view/1093