Monday, 27 May 2019

homework and exercises - Modeling a 2-dimensional mass spring system


First of all, I am unfortunately not an expert in physics, so please be indulge with me. I am trying to model a 2-dimensional mass-spring system with 1 mass and 3 springs to solve a dynamics problem in frequency domain. I've been looking for a solution for a similar problem but I couldn't find anything useful. Are these classical newton equation of motion mass-spring systems limited to 1D?


The mass m is connected to 3 springs k1,k2,k3, which are fixed at their endpoints, rotations are possible. The springs are assumed linear and can be simplified k1=k2=k3. In the equilibrium state, the angle between the springs is 120.


2d mass spring system



Answer



The simplest setup is for small displacements. Suppose the spring rest lengths are L1,L2,L3, the mass has mass m, the springs have constants k1,k2,k3, the angle is 120 degrees between attachments, and the attachment points are set up so that at rest, the springs are all unstretched.


The potential becomes U(r)=3j=1kj2(|ruj|Lj)2

where uj={Ljcos(2πj/3),Ljsin(2πj/3)}.
It's easy to verify that U(0)=0
which means that the system is at equilibrium when the mass sits at the origin.



Defining H=U(0)=(14(k1+k2+4k3)143(k2k1)143(k2k1)34(k1+k2))

we obtain eigenvalues λ±=12(k1+k2+k3±k21k2k1k3k1+k22+k23k2k3)
and the ordinary vibrating frequencies become ω±=12πλ±2m.
Notice that the lengths are irrelevant, and that in the case k1=k2=k3 the frequencies become identical, thus becoming like a 2D spherical oscillator.


When you add more masses to the system, things get interesting.


What manner of driving force are you planning on applying?


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