Elastic pendulum explained

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. For specific energy values, the system demonstrates all the hallmarks of chaotic behavior and is sensitive to initial conditions. 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.

Analysis and interpretation

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

and can be stretched by a length . The angle of oscillation of the pendulum is . is:$$L\; =\; T\; -\; V$$where is the kinetic energy and is the potential energy.

Hooke's law is the potential energy of the spring itself:$$V\_k=\backslash frackx^2$$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:$$V\_g=-gm(\backslash ell\_0+x)\backslash cos\; \backslash theta$$where

is the gravitational acceleration.

The kinetic energy is given by:$$T=\backslash fracmv^2$$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:$$T\; =\; \backslash frac\; m\; \backslash left(\backslash dot\; x^2\; +\; \backslash left(\backslash ell\_0+x\backslash right)^2\; \backslash dot\; \backslash theta^2\backslash right)$$

So the Lagrangian becomes:^[1] $$L\; =\; T\; -V\_k\; -\; V\_g$$$$L[x,\backslash dot\; x,\backslash theta,\; \backslash dot\; \backslash theta]\; =\; \backslash fracm\; \backslash left(\backslash dot\; x^2\; +\; \backslash left(\backslash ell\_0+x\backslash right)^2\; \backslash dot\; \backslash theta^2\backslash right)\; -\backslash frackx^2\; +\; gm\; \backslash left(\backslash ell\_0+x\backslash right)\backslash cos\; \backslash theta$$

Equations of motion

With two degrees of freedom, for

and , the equations of motion can be found using two Euler-Lagrange equations:

For

:^[1] $$m\backslash left(\backslash ell\_0+x\backslash right)\; \backslash dot\; \backslash theta^2\; -kx\; +\; gm\backslash cos\; \backslash theta-m\; \backslash ddot\; x=0$$ isolated:$$\backslash ddot\; x\; =(\backslash ell\_0+x)\backslash dot\; \backslash theta^2\; -\backslash fracx\; +\; g\backslash cos\; \backslash theta$$

And for

:^[1] $$-gm\; \backslash left(\backslash ell\_0+x\backslash right)\; \backslash sin\; \backslash theta\; -\; m\; \backslash left(\backslash ell\_0+x\backslash right)^2\; \backslash ddot\; \backslash theta\; -\; 2m\; \backslash left(\backslash ell\_0+x\backslash right)\; \backslash dot\; x\; \backslash dot\; \backslash theta=0$$ isolated:$$\backslash ddot\; \backslash theta=-\backslash frac\backslash sin\; \backslash theta-\backslash frac\backslash dot\; \backslash theta$$

These can be further simplified by scaling length $s\; =\; /$ and time $\backslash tau\; =\; t\backslash sqrt$. Expressing the system in terms of

and results in nondimensional equations of motion. The one remaining dimensionless parameter characterizes the system.$$\backslash frac\; =\; \backslash left(s\; +\; 1\backslash right)\; \backslash left(\backslash frac\backslash right)^2\; -\; \backslash Omega^2\; s\; +\; \backslash cos\backslash theta$$$$\backslash frac\; =\; -\backslash frac\; -\; \backslash frac\; \backslash frac\; \backslash frac$$

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^[6] in this system for various values of the parameter

and initial conditions and .

There is also a second example : Double Elastic Pendulum . See ^[7]

See also

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

Further reading

• Pokorny . Pavel . Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum . Regular and Chaotic Dynamics . 13 . 3 . 155–165 . 2008 . 10.1134/S1560354708030027 . 2008RCD....13..155P. 56090968 .
• Pokorny . Pavel . Continuation of Periodic Solutions of Dissipative and Conservative Systems: Application to Elastic Pendulum . Mathematical Problems in Engineering . 2009 . 1–15 . 2009 . 104547 . 10.1155/2009/104547 . free .

External links

• 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

Notes and References

1. Web site: Dynamics of the Elastic Pendulum. Xiao. Qisong. et al..
2. Pokorny. Pavel. Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum. Regular and Chaotic Dynamics. 13. 3. 155–165. 2008. 10.1134/S1560354708030027. 2008RCD....13..155P. 56090968 .
3. Web site: Spring Pendulum. Sivasrinivas. Kolukula.
4. Web site: The spring pendulum. 19 July 2017. Hill. Christian.
5. Book: Leah, Ganis . The Swinging Spring: Regular and Chaotic Motion.
6. Understanding the order-chaos-order transition in the planar elastic pendulum. Physica D. 2020. 132256. 402. Anurag. Anurag. Mondal. Basudeb. Jayanta Kumar. Bhattacharjee. Sagar. Chakraborty. 10.1016/j.physd.2019.132256. 2020PhyD..40232256A . 209905775 . subscription.
7. Book: Haque . Shihabul . Sasmal . Nilanjan . Bhattacharjee . Jayanta K. . 2024 . Lacarbonara . Walter . An Extensible Double Pendulum and Multiple Parametric Resonances . NODYCON Conference Proceedings Series . Walter Lacarbonara . Advances in Nonlinear Dynamics, Volume I . https://link.springer.com/chapter/10.1007/978-3-031-50631-4_12 . en . Cham . Springer Nature Switzerland . 135–145 . 10.1007/978-3-031-50631-4_12 . 978-3-031-50631-4. subscription .