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 edit

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]

${\displaystyle L=T-V_{k}-V_{g}}$ 

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

Equations of motion edit

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:

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

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.

• 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