Elastic 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 and mathematics, in the area of dynamical systems, an elastic pendulum^[1][2] (also called spring pendulum^[3][4] 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.^[2] For specific energy values, the system demonstrates all the hallmarks of chaotic behavior and is sensitive to initial conditions.^[2]At very low and very high 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.This behavior suggests a complex interplay between energy states and system dynamics.

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 ${\displaystyle l_{0}}$ and can be stretched by a length ${\displaystyle x}$. The angle of oscillation of the pendulum is ${\displaystyle \theta }$.

The Lagrangian ${\displaystyle L}$ is:

${\displaystyle L=T-V}$

where ${\displaystyle T}$ is the kinetic energy and ${\displaystyle V}$ is the potential energy.

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

${\displaystyle V_{k}={\frac {1}{2}}kx^{2}}$

where ${\displaystyle k}$ 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:

${\displaystyle V_{g}=-gm(l_{0}+x)\cos \theta }$

where ${\displaystyle g}$ is the gravitational acceleration.

The kinetic energy is given by:

${\displaystyle T={\frac {1}{2}}mv^{2}}$

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

${\displaystyle T={\frac {1}{2}}m({\dot {x}}^{2}+(l_{0}+x)^{2}{\dot {\theta }}^{2})}$

So the Lagrangian becomes:^[1]

${\displaystyle L=T-V_{k}-V_{g}}$

${\displaystyle L[x,{\dot {x}},\theta ,{\dot {\theta }}]={\frac {1}{2}}m({\dot {x}}^{2}+(l_{0}+x)^{2}{\dot {\theta }}^{2})-{\frac {1}{2}}kx^{2}+gm(l_{0}+x)\cos \theta }$

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

${\displaystyle {\partial L \over \partial x}-{\operatorname {d} \over \operatorname {d} t}{\partial L \over \partial {\dot {x}}}=0}$

${\displaystyle {\partial L \over \partial \theta }-{\operatorname {d} \over \operatorname {d} t}{\partial L \over \partial {\dot {\theta }}}=0}$

For ${\displaystyle x}$:^[1]

${\displaystyle m(l_{0}+x){\dot {\theta }}^{2}-kx+gm\cos \theta -m{\ddot {x}}=0}$

${\displaystyle {\ddot {x}}}$ isolated:

${\displaystyle {\ddot {x}}=(l_{0}+x){\dot {\theta }}^{2}-{\frac {k}{m}}x+g\cos \theta }$

And for ${\displaystyle \theta }$:^[1]

${\displaystyle -gm(l_{0}+x)\sin \theta -m(l_{0}+x)^{2}{\ddot {\theta }}-2m(l_{0}+x){\dot {x}}{\dot {\theta }}=0}$

${\displaystyle {\ddot {\theta }}}$ isolated:

${\displaystyle {\ddot {\theta }}=-{\frac {g}{l_{0}+x}}\sin \theta -{\frac {2{\dot {x}}}{l_{0}+x}}{\dot {\theta }}}$

These can be further simplified by scaling length ${\displaystyle S={\frac {x}{l_{0}}}}$ and time ${\displaystyle T=t{\sqrt {\frac {g}{l_{0}}}}}$. Expressing the system in terms of ${\displaystyle S}$ and ${\displaystyle T}$ results in nondimensional equations of motion. The one remaining dimensionless parameter ${\displaystyle \Omega ^{2}={\frac {kl_{0}}{mg}}}$ characterizes the system.

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

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

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^[7] in this system for various values of the parameter ${\displaystyle \Omega ^{2}}$ and initial conditions ${\displaystyle S}$ and ${\displaystyle \theta }$.

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

• 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