Calculation of the inverted pendulum spring constant
To calculate the IP spring constant I used an analytical model and Comsol.
I assumed the inverted pendulum to have the following shape:
By applying a force F on the top of the sphere (as in picture) and measuring the displacement of the IP it is possible to deduce the spring constant:
The equation of motion (without gravity) is:
Where I is the moment of inertia of the IP.
In steady-state we have:
Applying a small-angle approximation:
I modeled the IP in Comsol, calculated and deduced .
The IP is made in Steel AISI 4340.1 (density: 7850 kg/m^3) and the values I used are:
applying a force we obtain a displacement
Calculation of the inverted pendulum resonant frequency
Knowing the IP spring constant we can calculate the IP resonant frequency using the following analytical model.
The equation of motion is:
using the small-angles approximation we obtain:
And the resonant frequency is
Where moment of inertia I is:
By plotting the expression above we can see that decreases as increases, as expected. We can also see that the spherical mass on top of the inverted pendulum, , should be about 8.99 kg to reach the desired resonant frequency of 40 mHz.
This value is not realistic, but in order to reach the 40 mHz resonant frequency we can also change the bottom cylinder parameters and make k smaller.
Note: to make increase I made increase keeping the total length of the IP constant.