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Message ID: 1990     Entry time: Tue Jul 28 16:32:38 2015
Author: Alessandra 
Type: Misc 
Category: Seismometer 
Subject: IP spring constant and resonant frequency 

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:

 

-k\theta +(l_1+l_2+2r_3)Fcos\theta=I\ddot{\theta}

 

Where I is the moment of inertia of the IP.

In steady-state we have:

 

-k\theta +(l_1+l_2+2r_3)Fcos\theta=0

 

Applying a small-angle approximation:

 

cos\theta\simeq1 ;

 

\theta\simeq\frac{x}{l_1+l_2+2r_3}

 

we obtain:

 

k=\frac{(l_1+l_2+2r_3)^2F}{x}

 

I modeled the IP in Comsol, calculated x and deduced k.

 

The IP is made in Steel AISI 4340.1 (density: 7850 kg/m^3) and the values I used are:

 

r_1=0.001 \hspace{0.2cm} m; \hspace{1 cm} l_1=0.03 \hspace{0.2cm} m

r_2=0.005 \hspace{0.2cm} m; \hspace{1 cm} l_2=0.32 \hspace{0.2cm} m

r_3=0.0312 \hspace{0.2cm} m

 

applying a force F=1\hspace{0.2cm}N we obtain a displacement x=0.005493 \hspace{0.2cm}m 

 

 

and k results:

 

k=30.96 \hspace{0.2 cm} Nm

 

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:

-k\theta+m_1g\frac{l_1}{2}sin\theta+m_2g(l_1+\frac{l_2}{2})sin\theta+m_3g(l_1+l_2+r_3)sin\theta=I\ddot \theta

using the small-angles approximation we obtain:

I\ddot \theta+[k-m_1g\frac{l_1}{2}-m_2g(l_1+\frac{l_2}{2})-m_3g(l_1+l_2+r_3)]\theta=0

And the resonant frequency is

f_{IP}=\frac{1}{2\pi}\sqrt{\frac{k}{I}-\frac{g}{I}[m_1\frac{l_1}{2}+m_2(l_1+\frac{l_2}{2})+m_3(l_1+l_2+r_3)]}

Where moment of inertia I is:

I=\frac{1}{4}m_1r_1^{2}+\frac{1}{12}m_1l_1^{2}+m_1(\frac{l_1}{2})^{2}+\frac{1}{4}m_2r_2^{2}+\frac{1}{12}m_2l_2^{2}+m_2(l_1+\frac{l_2}{2})^{2}+\frac{2}{5}m_3r_3^{2}+m_3(l_1+l_2+r_3)^{2}

By plotting the f_{IP} expression above we can see that f_{IP} decreases as m_3 increases, as expected. We can also see that the spherical mass on top of the inverted pendulum, m_3, 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 m_3 increase I made r_3 increase keeping the total length of the IP constant.

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