# Elastic pendulum

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Concept in physics and mathematics

"Spring pendulum" redirects here. Not to be confused with the one-dimensional vertical [spring-mass system](/source/Spring-mass_system) with gravity, cf. also [Simple harmonic motion#Mass on a spring](/source/Simple_harmonic_motion#Mass_on_a_spring).

Motion of an elastic pendulum - you can see the effect of overlapping vibrations of different frequencies (a composite of the vibrations of a simple pendulum and a spring pendulum)

This article is missing information about the characteristics of chaotic motion in the system, cf. Double pendulum#Chaotic motion. Please expand the article to include this information. Further details may exist on the talk page. (October 2019)

In [physics](/source/Physics) and [mathematics](/source/Mathematics), in the area of [dynamical systems](/source/Dynamical_systems), an **elastic pendulum**[1][2] (also called **spring pendulum**[3][4] or **swinging spring**) is a [physical system](/source/Physical_system) where a piece of mass is connected to a [spring](/source/Spring_(device)) so that the resulting motion contains elements of both a [simple pendulum](/source/Pendulum_(mathematics)) and a [one-dimensional](/source/One-dimensional) [spring-mass system](/source/Spring-mass_system).[2] For specific energy values, the system demonstrates all the hallmarks of [chaotic behavior](/source/Chaos_theory) and is [sensitive](/source/Butterfly_effect) to [initial conditions](/source/Initial_conditions).[2] At very low and very high [energy](/source/Energy), there also appears to be regular motion.[5] The motion of an elastic pendulum is governed by a set of coupled [ordinary differential equations](/source/Ordinary_differential_equation). This behavior suggests a complex interplay between energy states and [system dynamics](/source/System_dynamics).

## Analysis and interpretation

2 DOF elastic pendulum with polar coordinate plots.[6]

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](/source/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

The spring has the rest length ℓ 0 {\displaystyle \ell _{0}} and can be stretched by a length x {\displaystyle x} . The angle of oscillation of the pendulum is θ {\displaystyle \theta } .

The [Lagrangian](/source/Lagrangian_(field_theory)) L {\displaystyle L} is: L = T − V {\displaystyle L=T-V} where T {\displaystyle T} is the [kinetic energy](/source/Kinetic_energy) and V {\displaystyle V} is the [potential energy](/source/Potential_energy).

[Hooke's law](/source/Hooke's_law) is the potential energy of the spring itself: V k = 1 2 k x 2 {\displaystyle V_{k}={\frac {1}{2}}kx^{2}} where k {\displaystyle k} is the spring constant.

The potential energy from [gravity](/source/Gravity), on the other hand, is determined by the height of the mass. For a given angle and displacement, the potential energy is: V g = − g m ( ℓ 0 + x ) cos ⁡ θ {\displaystyle V_{g}=-gm(\ell _{0}+x)\cos \theta } where g {\displaystyle g} is the [gravitational acceleration](/source/Gravitational_acceleration).

The kinetic energy is given by: T = 1 2 m v 2 {\displaystyle T={\frac {1}{2}}mv^{2}} where v {\displaystyle v} is the [velocity](/source/Velocity) of the mass. To relate v {\displaystyle v} to the other variables, the velocity is written as a combination of a movement along and perpendicular to the spring: T = 1 2 m ( x ˙ 2 + ( ℓ 0 + x ) 2 θ ˙ 2 ) {\displaystyle T={\frac {1}{2}}m\left({\dot {x}}^{2}+\left(\ell _{0}+x\right)^{2}{\dot {\theta }}^{2}\right)}

So the Lagrangian becomes:[1] L = T − V k − V g {\displaystyle L=T-V_{k}-V_{g}} L [ x , x ˙ , θ , θ ˙ ] = 1 2 m ( x ˙ 2 + ( ℓ 0 + x ) 2 θ ˙ 2 ) − 1 2 k x 2 + g m ( ℓ 0 + x ) cos ⁡ θ {\displaystyle L[x,{\dot {x}},\theta ,{\dot {\theta }}]={\frac {1}{2}}m\left({\dot {x}}^{2}+\left(\ell _{0}+x\right)^{2}{\dot {\theta }}^{2}\right)-{\frac {1}{2}}kx^{2}+gm\left(\ell _{0}+x\right)\cos \theta }

### Equations of motion

With two [degrees of freedom](/source/Degrees_of_freedom), for x {\displaystyle x} and θ {\displaystyle \theta } , the equations of motion can be found using two [Euler-Lagrange equations](/source/Euler-Lagrange_equation): ∂ L ∂ x − d d t ∂ L ∂ x ˙ = 0 ∂ L ∂ θ − d d t ∂ L ∂ θ ˙ = 0 {\displaystyle {\begin{aligned}{\frac {\partial L}{\partial x}}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {x}}}}&=0\\[1ex]{\frac {\partial L}{\partial \theta }}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\theta }}}}&=0\end{aligned}}}

For x {\displaystyle x} :[1] m ( ℓ 0 + x ) θ ˙ 2 − k x + g m cos ⁡ θ − m x ¨ = 0 {\displaystyle m\left(\ell _{0}+x\right){\dot {\theta }}^{2}-kx+gm\cos \theta -m{\ddot {x}}=0} x ¨ {\displaystyle {\ddot {x}}} isolated: x ¨ = ( ℓ 0 + x ) θ ˙ 2 − k m x + g cos ⁡ θ {\displaystyle {\ddot {x}}=(\ell _{0}+x){\dot {\theta }}^{2}-{\frac {k}{m}}x+g\cos \theta }

And for θ {\displaystyle \theta } :[1] − g m ( ℓ 0 + x ) sin ⁡ θ − m ( ℓ 0 + x ) 2 θ ¨ − 2 m ( ℓ 0 + x ) x ˙ θ ˙ = 0 {\displaystyle -gm\left(\ell _{0}+x\right)\sin \theta -m\left(\ell _{0}+x\right)^{2}{\ddot {\theta }}-2m\left(\ell _{0}+x\right){\dot {x}}{\dot {\theta }}=0} θ ¨ {\displaystyle {\ddot {\theta }}} isolated: θ ¨ = − g ℓ 0 + x sin ⁡ θ − 2 x ˙ ℓ 0 + x θ ˙ {\displaystyle {\ddot {\theta }}=-{\frac {g}{\ell _{0}+x}}\sin \theta -{\frac {2{\dot {x}}}{\ell _{0}+x}}{\dot {\theta }}}

These can be further simplified by scaling length s = x / ℓ 0 {\textstyle s={x}/{\ell _{0}}} and time τ = t g / ℓ 0 {\textstyle \tau =t{\sqrt {{g}/{\ell _{0}}}}} . Expressing the system in terms of s {\displaystyle s} and τ {\displaystyle \tau } results in [nondimensional](/source/Nondimensionalization) equations of motion. The one remaining dimensionless parameter Ω 2 = k ℓ 0 m g {\displaystyle \Omega ^{2}={\frac {k\ell _{0}}{mg}}} characterizes the system. d 2 s d τ 2 = ( s + 1 ) ( d θ d τ ) 2 − Ω 2 s + cos ⁡ θ {\displaystyle {\frac {d^{2}s}{d\tau ^{2}}}=\left(s+1\right)\left({\frac {d\theta }{d\tau }}\right)^{2}-\Omega ^{2}s+\cos \theta } d 2 θ d τ 2 = − sin ⁡ θ s + 1 − 2 1 + s d s d τ d θ d τ {\displaystyle {\frac {d^{2}\theta }{d\tau ^{2}}}=-{\frac {\sin \theta }{s+1}}-{\frac {2}{1+s}}{\frac {ds}{d\tau }}{\frac {d\theta }{d\tau }}}

The elastic pendulum is now described with two coupled [ordinary differential equations](/source/Ordinary_differential_equations). These can be solved [numerically](/source/Numerical_analysis). Furthermore, one can use analytical methods to study the intriguing phenomenon of order-chaos-order[7] in this system for various values of the parameter Ω 2 {\displaystyle \Omega ^{2}} and initial conditions s {\displaystyle s} and θ {\displaystyle \theta } .

There is also a second example : Double Elastic Pendulum . See [8]

## See also

- [Double pendulum](/source/Double_pendulum)

- [Duffing oscillator](/source/Duffing_oscillator)

- [Pendulum (mathematics)](/source/Pendulum_(mathematics))

- [Spring-mass system](/source/Spring-mass_system)

## References

1. ^ [***a***](#cite_ref-Xiao_et_al_1-0) [***b***](#cite_ref-Xiao_et_al_1-1) [***c***](#cite_ref-Xiao_et_al_1-2) [***d***](#cite_ref-Xiao_et_al_1-3) Xiao, Qisong; et al. ["Dynamics of the Elastic Pendulum"](https://www.math.arizona.edu/~gabitov/teaching/141/math_485/Midterm_Presentations/Elastic_Pedulum.pdf) (PDF).

1. ^ [***a***](#cite_ref-Pokorny_2008_2-0) [***b***](#cite_ref-Pokorny_2008_2-1) [***c***](#cite_ref-Pokorny_2008_2-2) Pokorny, Pavel (2008). ["Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum"](http://old.vscht.cz/mat/Pavel.Pokorny/rcd/RCD155-color.pdf) (PDF). *Regular and Chaotic Dynamics*. **13** (3): 155–165. [Bibcode](/source/Bibcode_(identifier)):[2008RCD....13..155P](https://ui.adsabs.harvard.edu/abs/2008RCD....13..155P). [doi](/source/Doi_(identifier)):[10.1134/S1560354708030027](https://doi.org/10.1134%2FS1560354708030027). [S2CID](/source/S2CID_(identifier)) [56090968](https://api.semanticscholar.org/CorpusID:56090968).

1. **[^](#cite_ref-sivasrinivas_3-0)** Sivasrinivas, Kolukula. ["Spring Pendulum"](https://sites.google.com/site/kolukulasivasrinivas/mechanics/spring-pendulum).

1. **[^](#cite_ref-hill_2017_4-0)** Hill, Christian (19 July 2017). ["The spring pendulum"](https://scipython.com/blog/the-spring-pendulum/).

1. **[^](#cite_ref-5)** Leah, Ganis. *The Swinging Spring: Regular and Chaotic Motion*.

1. **[^](#cite_ref-Simionescu_2014_6-0)** Simionescu, P.A. (2014). *Computer Aided Graphing and Simulation Tools for AutoCAD Users* (1st ed.). Boca Raton, Florida: CRC Press. [ISBN](/source/ISBN_(identifier)) [978-1-4822-5290-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4822-5290-3).

1. **[^](#cite_ref-7)** Anurag, Anurag; Basudeb, Mondal; Bhattacharjee, Jayanta Kumar; Chakraborty, Sagar (2020). ["Understanding the order-chaos-order transition in the planar elastic pendulum"](https://www.sciencedirect.com/science/article/pii/S0167278919300119). *Physica D*. **402** 132256. [Bibcode](/source/Bibcode_(identifier)):[2020PhyD..40232256A](https://ui.adsabs.harvard.edu/abs/2020PhyD..40232256A). [doi](/source/Doi_(identifier)):[10.1016/j.physd.2019.132256](https://doi.org/10.1016%2Fj.physd.2019.132256). [S2CID](/source/S2CID_(identifier)) [209905775](https://api.semanticscholar.org/CorpusID:209905775).

1. **[^](#cite_ref-8)** Haque, Shihabul; Sasmal, Nilanjan; Bhattacharjee, Jayanta K. (2024). ["An Extensible Double Pendulum and Multiple Parametric Resonances"](https://link.springer.com/chapter/10.1007/978-3-031-50631-4_12). In [Lacarbonara, Walter](/source/Walter_Lacarbonara) (ed.). *Advances in Nonlinear Dynamics, Volume I*. NODYCON Conference Proceedings Series. Cham: Springer Nature Switzerland. pp. 135–145. [doi](/source/Doi_(identifier)):[10.1007/978-3-031-50631-4_12](https://doi.org/10.1007%2F978-3-031-50631-4_12). [ISBN](/source/ISBN_(identifier)) [978-3-031-50631-4](https://en.wikipedia.org/wiki/Special:BookSources/978-3-031-50631-4).

## Further reading

- Pokorny, Pavel (2008). ["Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum"](http://www.nhn.ou.edu/~johnson/Education/Juniorlab/Pendula/2008-Pokorny-Chaos-ElasticPendulum.pdf) (PDF). *Regular and Chaotic Dynamics*. **13** (3): 155–165. [Bibcode](/source/Bibcode_(identifier)):[2008RCD....13..155P](https://ui.adsabs.harvard.edu/abs/2008RCD....13..155P). [doi](/source/Doi_(identifier)):[10.1134/S1560354708030027](https://doi.org/10.1134%2FS1560354708030027). [S2CID](/source/S2CID_(identifier)) [56090968](https://api.semanticscholar.org/CorpusID:56090968).

- Pokorny, Pavel (2009). ["Continuation of Periodic Solutions of Dissipative and Conservative Systems: Application to Elastic Pendulum"](http://dml.cz/bitstream/handle/10338.dmlcz/141703/MathBohem_136-2011-4_10.pdf) (PDF). *Mathematical Problems in Engineering*. **2009** 104547: 1–15. [doi](/source/Doi_(identifier)):[10.1155/2009/104547](https://doi.org/10.1155%2F2009%2F104547).

## External links

- Holovatsky V., Holovatska Y. (2019) ["Oscillations of an elastic pendulum"](http://demonstrations.wolfram.com/OscillationsOfAnElasticPendulum/) (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](https://doi.org/10.32782/pet-2023-1-2) [http://journals.vnu.volyn.ua/index.php/physics/article/view/1093](http://journals.vnu.volyn.ua/index.php/physics/article/view/1093)

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Adapted from the Wikipedia article [Elastic pendulum](https://en.wikipedia.org/wiki/Elastic_pendulum) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Elastic_pendulum?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.