Elastic pendulum

[[File:Spring pendulum.gif|thumb|300px|2 DOF elastic pendulum with polar coordinate plots.

{{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}}

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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.

The spring has the rest length $l\_0$ and can be stretched by a length $x$. The angle of oscillation of the pendulum is $\backslash theta$.

The Lagrangian $L$ is:

:$L\; =\; T\; -\; V$

where $T$ is the kinetic energy and $V$ is the potential energy.

Hooke's law is the potential energy of the spring itself:

:$V\_k=\backslash frac\{1\}\{2\}kx^2$

where $k$ 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:

:$V\_g=-gm(l\_0+x)\backslash cos\; \backslash theta$

where $g$ is the gravitational acceleration.

The kinetic energy is given by:

:$T=\backslash frac\{1\}\{2\}mv^2$

where $v$ is the velocity of the mass. To relate $v$ to the other variables, the velocity is written as a combination of a movement along and perpendicular to the spring:

:$T=\backslash frac\{1\}\{2\}m(\backslash dot\; x^2+(l\_0+x)^2\backslash dot\; \backslash theta^2)$

So the Lagrangian becomes:

:$L\; =\; T\; -V\_k\; -\; V\_g$

:$L[x,\backslash dot\; x,\backslash theta,\; \backslash dot\; \backslash theta]\; =\; \backslash frac\{1\}\{2\}m(\backslash dot\; x^2+(l\_0+x)^2\backslash dot\; \backslash theta^2)\; -\backslash frac\{1\}\{2\}kx^2\; +\; gm(l\_0+x)\backslash cos\; \backslash theta$

With two degrees of freedom, for $x$ and $\backslash theta$, the equations of motion can be found using two Euler-Lagrange equations:

:$\{\backslash partial\; L\backslash over\backslash partial\; x\}-\{\backslash operatorname\; d\; \backslash over\; \backslash operatorname\; dt\}\; \{\backslash partial\; L\backslash over\backslash partial\; \backslash dot\; x\}=0$

:$\{\backslash partial\; L\backslash over\backslash partial\; \backslash theta\}-\{\backslash operatorname\; d\; \backslash over\; \backslash operatorname\; dt\}\; \{\backslash partial\; L\backslash over\backslash partial\; \backslash dot\; \backslash theta\}=0$

For $x$:

:$m(l\_0+x)\backslash dot\; \backslash theta^2\; -kx\; +\; gm\backslash cos\; \backslash theta-m\; \backslash ddot\; x=0$

$\backslash ddot\; x$ isolated:

:$\backslash ddot\; x\; =(l\_0+x)\backslash dot\; \backslash theta^2\; -\backslash frac\{k\}\{m\}x\; +\; g\backslash cos\; \backslash theta$

And for $\backslash theta$:

:$-gm(l\_0+x)\backslash sin\; \backslash theta\; -\; m(l\_0+x)^2\backslash ddot\; \backslash theta-\; 2m(l\_0+x)\backslash dot\; x\; \backslash dot\; \backslash theta=0$

$\backslash ddot\; \backslash theta$ isolated:

:$\backslash ddot\; \backslash theta=-\backslash frac\{g\}\{l\_0+x\}\backslash sin\; \backslash theta-\backslash frac\{2\backslash dot\; x\}\{l\_0+x\}\backslash dot\; \backslash theta$

These can be further simplified by scaling length $S=\backslash frac\{x\}\{l\_0\}$ and time $T\; =\; t\backslash sqrt\{\backslash frac\{g\}\{l\_0\}\}$. Expressing the system in terms of $S$ and $T$ results in nondimensional equations of motion. The one remaining dimensionless parameter $\backslash Omega^2\; =\; \backslash frac\{kl\_0\}\{mg\}$ characterizes the system.

:$\backslash frac\{\backslash mathrm\{d\}^2\; S\}\{\backslash mathrm\{d\}T^2\}\; =\; \backslash left(S+1\backslash right)\; \backslash left(\backslash frac\{\backslash mathrm\{d\}\backslash theta\}\{\backslash mathrm\{d\}T\}\backslash right)^2\; -\; \backslash Omega^2\; S\; +\; \backslash cos\backslash theta$

:$\backslash frac\{\backslash mathrm\{d\}^2\backslash theta\}\{\backslash mathrm\{d\}T^2\}\; =\; -\backslash frac\{\backslash sin\backslash theta\}\{S+1\}\; -\; \backslash frac\{2\}\{1+S\}\backslash left(\backslash frac\{\backslash mathrm\{d\}S\}\{\backslash mathrm\{d\}T\}\backslash right)\backslash left(\backslash frac\{\backslash mathrm\{d\}\backslash theta\}\{\backslash mathrm\{d\}T\}\backslash right)$

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{{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|pages = 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}} in this system for various values of the parameter $\backslash Omega^2$ and initial conditions $S$ and $\backslash theta$.

There is also a second example : Double Elastic Pendulum . See {{Cite journal |last=Haque |first=Shihabul |last2=Sasmal |first2=Nilanjan |last3=Bhattacharjee |first3=Jayanta K. |date=2024 |editor-last=Lacarbonara |editor-first=Walter |title=An Extensible Double Pendulum and Multiple Parametric Resonances |url=https://link.springer.com/chapter/10.1007/978-3-031-50631-4_12 |journal=Advances in Nonlinear Dynamics, Volume I |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|url-access=subscription }}

• Double pendulum
• Duffing oscillator
• Pendulum (mathematics)
• Spring-mass system

{{reflist}}

{{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

| doi = 10.1155/2009/104547

| url = http://dml.cz/bitstream/handle/10338.dmlcz/141703/MathBohem_136-2011-4_10.pdf

| doi-access = free

}}

{{refend}}

• 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

{{Chaos theory}}

Category:Chaotic maps

Category:Dynamical systems

Category:Mathematical physics

Category:Pendulums