We consider a system of two masses, m1= 1, m2= 4, coupled by three springs with

Question

We consider a system of two masses, m1= 1, m2= 4, coupled by three springs with k1 = 1, k2 = 4, &3 = 28, as in the illustration on page 442. Matrix-Eigenvalue-Eigenvector approach Find the eigenvalues and eigenvectors of matrix B Find the general solution u(t) Find the general solution x(t) Approach of reducing the system to a single equation (a) Return to the second order system of two equations Solve the second equation for x1 Substitute the expression you've got for for xi into the first equation, and simplify. You should arrive at a fourth order differential equation in terms of x2 only. Find general solution x2(t) of the fourth order differential equation you've obtained. Compare it to the general solution you've obtained using matrix approach.

Explanation / Answer

I'm not familiar with that particular book, but the differential equations of simple harmonic motion are interesting. You can make the system as complex as you like. I've studied SHM in both maths and physics. I simulated SH oscillators with operational amplifiers; in other words an analogue computer. First you simulate a frictionless pendulum, then you introduce friction so the oscillations die away, then you introduce forcing terms of various frequencies to see what happens; like pushing a child on a swing. If you're really clever, you can introduce hysteresis (delay), and extra degrees of freedom (flexible joints in the pendulum), and vary the Q of the system (how sharply you can tune it to the resonant frequency). It's about the most open-ended project I ever saw. I did it in second-year physics. At its simplest, it's just sixth form maths, but you can turn it into an impossibly complex exercise in chaotic systems.

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