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

The Lagrangian L {\displaystyle L} is:

L = T − V {\displaystyle L=T-V}

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

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, 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(l_{0}+x)\cos \theta }

where g {\displaystyle g} is the 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 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({\dot {x}}^{2}+(l_{0}+x)^{2}{\dot {\theta }}^{2})}

So the Lagrangian becomes:⁸

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({\dot {x}}^{2}+(l_{0}+x)^{2}{\dot {\theta }}^{2})-{\frac {1}{2}}kx^{2}+gm(l_{0}+x)\cos \theta }

Equations of motion

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

∂ L ∂ x − d d ⁡ t ∂ L ∂ x ˙ = 0 {\displaystyle {\partial L \over \partial x}-{\operatorname {d} \over \operatorname {d} t}{\partial L \over \partial {\dot {x}}}=0} ∂ L ∂ θ − d d ⁡ t ∂ L ∂ θ ˙ = 0 {\displaystyle {\partial L \over \partial \theta }-{\operatorname {d} \over \operatorname {d} t}{\partial L \over \partial {\dot {\theta }}}=0}

For x {\displaystyle x} :⁹

m ( l 0 + x ) θ ˙ 2 − k x + g m cos ⁡ θ − m x ¨ = 0 {\displaystyle m(l_{0}+x){\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}}=(l_{0}+x){\dot {\theta }}^{2}-{\frac {k}{m}}x+g\cos \theta }

And for θ {\displaystyle \theta } :¹⁰

− g m ( l 0 + x ) sin ⁡ θ − m ( l 0 + x ) 2 θ ¨ − 2 m ( l 0 + x ) x ˙ θ ˙ = 0 {\displaystyle -gm(l_{0}+x)\sin \theta -m(l_{0}+x)^{2}{\ddot {\theta }}-2m(l_{0}+x){\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}{l_{0}+x}}\sin \theta -{\frac {2{\dot {x}}}{l_{0}+x}}{\dot {\theta }}}

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

d 2 S d T 2 = ( S + 1 ) ( d θ d T ) 2 − Ω 2 S + cos ⁡ θ {\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 } d 2 θ d T 2 = − sin ⁡ θ S + 1 − 2 1 + S ( d S d T ) ( d θ d T ) {\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-order¹¹ 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 ¹²

See also

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

Further reading

• Pokorny, Pavel (2008). "Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum" (PDF). Regular and Chaotic Dynamics. 13 (3): 155–165. Bibcode:2008RCD....13..155P. doi:10.1134/S1560354708030027. S2CID 56090968.
• Pokorny, Pavel (2009). "Continuation of Periodic Solutions of Dissipative and Conservative Systems: Application to Elastic Pendulum" (PDF). Mathematical Problems in Engineering. 2009: 1–15. doi:10.1155/2009/104547.

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

References

1. Xiao, Qisong; et al. "Dynamics of the Elastic Pendulum" (PDF). https://www.math.arizona.edu/~gabitov/teaching/141/math_485/Midterm_Presentations/Elastic_Pedulum.pdf ↩

2. Pokorny, Pavel (2008). "Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum" (PDF). Regular and Chaotic Dynamics. 13 (3): 155–165. Bibcode:2008RCD....13..155P. doi:10.1134/S1560354708030027. S2CID 56090968. http://old.vscht.cz/mat/Pavel.Pokorny/rcd/RCD155-color.pdf ↩

3. Sivasrinivas, Kolukula. "Spring Pendulum". https://sites.google.com/site/kolukulasivasrinivas/mechanics/spring-pendulum ↩

4. Hill, Christian (19 July 2017). "The spring pendulum". https://scipython.com/blog/the-spring-pendulum/ ↩

5. Pokorny, Pavel (2008). "Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum" (PDF). Regular and Chaotic Dynamics. 13 (3): 155–165. Bibcode:2008RCD....13..155P. doi:10.1134/S1560354708030027. S2CID 56090968. http://old.vscht.cz/mat/Pavel.Pokorny/rcd/RCD155-color.pdf ↩

6. Pokorny, Pavel (2008). "Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum" (PDF). Regular and Chaotic Dynamics. 13 (3): 155–165. Bibcode:2008RCD....13..155P. doi:10.1134/S1560354708030027. S2CID 56090968. http://old.vscht.cz/mat/Pavel.Pokorny/rcd/RCD155-color.pdf ↩

7. Leah, Ganis. The Swinging Spring: Regular and Chaotic Motion. ↩

8. Xiao, Qisong; et al. "Dynamics of the Elastic Pendulum" (PDF). https://www.math.arizona.edu/~gabitov/teaching/141/math_485/Midterm_Presentations/Elastic_Pedulum.pdf ↩

9. Xiao, Qisong; et al. "Dynamics of the Elastic Pendulum" (PDF). https://www.math.arizona.edu/~gabitov/teaching/141/math_485/Midterm_Presentations/Elastic_Pedulum.pdf ↩

10. Xiao, Qisong; et al. "Dynamics of the Elastic Pendulum" (PDF). https://www.math.arizona.edu/~gabitov/teaching/141/math_485/Midterm_Presentations/Elastic_Pedulum.pdf ↩

11. Anurag, Anurag; Basudeb, Mondal; Bhattacharjee, Jayanta Kumar; Chakraborty, Sagar (2020). "Understanding the order-chaos-order transition in the planar elastic pendulum". Physica D. 402: 132256. Bibcode:2020PhyD..40232256A. doi:10.1016/j.physd.2019.132256. S2CID 209905775. https://www.sciencedirect.com/science/article/pii/S0167278919300119 ↩

12. Haque, Shihabul; Sasmal, Nilanjan; Bhattacharjee, Jayanta K. (2024). Lacarbonara, Walter (ed.). "An Extensible Double Pendulum and Multiple Parametric Resonances". Advances in Nonlinear Dynamics, Volume I. Cham: Springer Nature Switzerland: 135–145. doi:10.1007/978-3-031-50631-4_12. ISBN 978-3-031-50631-4. 978-3-031-50631-4 ↩