Natural frequency of large degree of freedom system

[dxdy]

I have a spring-mass system with 200 masses and 199 springs. All masses are 100 tonnes and stiffness 20 MN. The boundary conditions are fixed-free.

I have constructed a lumped mass matrix and stiffness matrix and calculated the lowest natural frequency. Including the boundary conditions I calculated this to be 0.0124 Hz.

However, when I calculate the wave speed, I can calculate the period of the wave starting at one end, reflecting at the free end and returning. This frequency (1/T) is more than double the natural frequency I calculated. The wave speed I calculated was 141.4 m/s assuming the distance between masses is 10 metres.

Am I using the wrong approach here for calculating the natural frequency both ways?

 

[nasu]

How did you calculate the speed of the wave in this system? And what kind of wave?

 

[xts]

masses are 100 tonnes and stiffness 20 MN.

What does it mean stiffness 20MN? You, maybe, meant 20 MN/m?
And how are they connected? In a simple chain mass-spring-mass-sping-mass...mass-spring-mass, or rather in some mesh?

 

[dxdy]

Yes I did mean 20 MN/m.
Solved.

 

[sardar]

I appreciate your thorough analysis and calculations of the natural frequency of your spring-mass system. It seems that you have correctly calculated the lowest natural frequency using the lumped mass and stiffness matrices. However, when considering the wave speed and period of the wave, it is important to note that they are not directly related to the natural frequency of a system.

The natural frequency of a system is determined by its mass and stiffness properties, while the wave speed and period of the wave are influenced by the boundary conditions and overall geometry of the system. In your case, the fixed-free boundary conditions and the distance between masses of 10 metres are affecting the wave speed and period of the wave.

Therefore, it is not surprising that the calculated wave speed and period are different from the natural frequency you calculated using the lumped mass and stiffness matrices. This does not necessarily mean that you are using the wrong approach, but rather that you are considering different aspects of the system.

In conclusion, your approach for calculating the natural frequency using the lumped mass and stiffness matrices is correct. However, to accurately calculate the wave speed and period of the wave, you may need to consider the boundary conditions and overall geometry of the system in your calculations.