Elastic pendulum

{{Short description|Concept in physics and mathematics}} {{Redirect-distinguish2|Spring pendulum|the one-dimensional vertical spring-mass system with gravity, cf. also Simple harmonic motion#Mass on a spring}} thumb|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)

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{{Missing information|the characteristics of chaotic motion in the system, cf. Double pendulum#Chaotic motion|date=October 2019}}

In physics and mathematics, in the area of dynamical systems, an '''elastic pendulum'''<ref name="Xiao et al">{{Cite web|title=Dynamics of the Elastic Pendulum|url=https://www.math.arizona.edu/~gabitov/teaching/141/math_485/Midterm_Presentations/Elastic_Pedulum.pdf|surname1=Xiao|given1=Qisong|display-authors=et al.}}</ref><ref name="Pokorny 2008">{{cite journal|last=Pokorny|first=Pavel|title=Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum|journal=Regular and Chaotic Dynamics|volume=13|issue=3|pages=155–165|date=2008|doi=10.1134/S1560354708030027|url=http://old.vscht.cz/mat/Pavel.Pokorny/rcd/RCD155-color.pdf|bibcode=2008RCD....13..155P|s2cid=56090968 }}</ref> (also called '''spring pendulum'''<ref name="sivasrinivas">{{cite web|url=https://sites.google.com/site/kolukulasivasrinivas/mechanics/spring-pendulum|title=Spring Pendulum|last=Sivasrinivas|first=Kolukula}}</ref><ref name="hill 2017">{{cite web|url=https://scipython.com/blog/the-spring-pendulum/|title=The spring pendulum|date=19 July 2017|last=Hill|first=Christian}}</ref> or '''swinging spring''') is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system.<ref name="Pokorny 2008" /> For specific energy values, the system demonstrates all the hallmarks of chaotic behavior and is sensitive to initial conditions.<ref name="Pokorny 2008" /> At very low and very high energy, there also appears to be regular motion.<ref>{{Cite book |last=Leah |first=Ganis |title=The Swinging Spring: Regular and Chaotic Motion}}</ref> The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations. This behavior suggests a complex interplay between energy states and system dynamics.

==Analysis and interpretation== thumb|300px|2 DOF elastic pendulum with polar coordinate plots.<ref name="Simionescu 2014">{{cite book|last=Simionescu|first=P.A.|title=Computer Aided Graphing and Simulation Tools for AutoCAD Users|year=2014|publisher=CRC Press|location=Boca Raton, Florida|isbn=978-1-4822-5290-3|edition=1st}}</ref>

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=== The spring has the rest length <math>\ell_0</math> and can be stretched by a length <math>x</math>. The angle of oscillation of the pendulum is <math>\theta</math>.

The Lagrangian <math>L</math> is: <math display="block">L = T - V</math> where <math>T</math> is the kinetic energy and <math>V</math> is the potential energy.

Hooke's law is the potential energy of the spring itself: <math display="block">V_k=\frac{1}{2}kx^2</math> where <math>k</math> 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: <math display="block">V_g=-gm(\ell_0+x)\cos \theta</math> where <math>g</math> is the gravitational acceleration.

The kinetic energy is given by: <math display="block">T=\frac{1}{2}mv^2</math> where <math>v</math> is the velocity of the mass. To relate <math>v</math> to the other variables, the velocity is written as a combination of a movement along and perpendicular to the spring: <math display="block">T = \frac{1}{2} m \left(\dot x^2 + \left(\ell_0+x\right)^2 \dot \theta^2\right)</math>

So the Lagrangian becomes:<ref name="Xiao et al"/> <math display="block">L = T -V_k - V_g</math> <math display="block">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 </math>

===Equations of motion=== With two degrees of freedom, for <math>x</math> and <math>\theta</math>, the equations of motion can be found using two Euler-Lagrange equations: <math display="block">\begin{align} \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{align}</math>

For <math>x</math>:<ref name="Xiao et al"/> <math display="block">m\left(\ell_0+x\right) \dot \theta^2 -kx + gm\cos \theta-m \ddot x=0</math> <math>\ddot x</math> isolated: <math display="block">\ddot x =(\ell_0+x)\dot \theta^2 -\frac{k}{m}x + g\cos \theta</math>

And for <math>\theta</math>:<ref name="Xiao et al"/> <math display="block">-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</math> <math>\ddot \theta</math> isolated: <math display="block">\ddot \theta=-\frac{g}{\ell_0+x}\sin \theta-\frac{2\dot x}{\ell_0+x}\dot \theta</math>

These can be further simplified by scaling length <math display="inline">s = {x}/{\ell_0}</math> and time <math display="inline">\tau = t\sqrt{{g}/{\ell_0}}</math>. Expressing the system in terms of <math>s</math> and <math>\tau</math> results in nondimensional equations of motion. The one remaining dimensionless parameter <math>\Omega^2 = \frac{k\ell_0}{mg}</math> characterizes the system. <math display="block">\frac{d^2 s}{d\tau^2} = \left(s + 1\right) \left(\frac{d\theta}{d\tau}\right)^2 - \Omega^2 s + \cos\theta</math> <math display="block">\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} </math>

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<ref>{{Cite journal|title = Understanding the order-chaos-order transition in the planar elastic pendulum|url = https://www.sciencedirect.com/science/article/pii/S0167278919300119|journal = Physica D|date = 2020|article-number = 132256|volume = 402|first1 = Anurag|last1 = Anurag|first2 = Mondal|last2 = Basudeb|first3 = Jayanta Kumar|last3 = Bhattacharjee|first4 = Sagar|last4 = Chakraborty|doi=10.1016/j.physd.2019.132256| bibcode=2020PhyD..40232256A | s2cid=209905775 |url-access = subscription}}</ref> in this system for various values of the parameter <math>\Omega^2</math> and initial conditions <math>s</math> and <math>\theta</math>.

There is also a second example : Double Elastic Pendulum . See <ref>{{Cite book |last1=Haque |first1=Shihabul |last2=Sasmal |first2=Nilanjan |last3=Bhattacharjee |first3=Jayanta K. |date=2024 |editor-last=Lacarbonara |editor-first=Walter |chapter=An Extensible Double Pendulum and Multiple Parametric Resonances |series=NODYCON Conference Proceedings Series |editor1-link=Walter Lacarbonara |title=Advances in Nonlinear Dynamics, Volume I |chapter-url=https://link.springer.com/chapter/10.1007/978-3-031-50631-4_12 |language=en |location=Cham |publisher=Springer Nature Switzerland |pages=135–145 |doi=10.1007/978-3-031-50631-4_12 |isbn=978-3-031-50631-4|chapter-url-access=subscription }}</ref>

== References == {{reflist}}

== Further reading == {{refbegin}} * {{cite journal | last = Pokorny | first = Pavel | title = Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum | journal = Regular and Chaotic Dynamics | volume = 13 | issue = 3 | pages = 155–165 | date = 2008 | doi = 10.1134/S1560354708030027 | url = http://www.nhn.ou.edu/~johnson/Education/Juniorlab/Pendula/2008-Pokorny-Chaos-ElasticPendulum.pdf | bibcode = 2008RCD....13..155P | s2cid = 56090968 }} * {{cite journal | last = Pokorny | first = Pavel | title = Continuation of Periodic Solutions of Dissipative and Conservative Systems: Application to Elastic Pendulum | journal = Mathematical Problems in Engineering | volume = 2009 | pages = 1–15 | date = 2009 | article-number = 104547 | doi = 10.1155/2009/104547 | url = http://dml.cz/bitstream/handle/10338.dmlcz/141703/MathBohem_136-2011-4_10.pdf | doi-access = free }} {{refend}}

==External links== * Holovatsky V., Holovatska Y. (2019) [http://demonstrations.wolfram.com/OscillationsOfAnElasticPendulum/ "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

Category:Chaotic maps Category:Dynamical systems Category:Mathematical physics Category:Pendulums