Question

Vibrations

Two masses are connected via three springs to each other and to the wall at left and right. The equations of motion are

-k_{1} x_{1}-k\left(x_{1}-x_{2}\right)=m_{1} \ddot{x}_{1}

-k_{2} x_{2}-k\left(x_{2}-x_{1}\right)=m_{2} \ddot{x}_{2}

Suppose the force constants of the left and right springs are k1= k2 =2 and that of the center spring is k=1 let the masses be m1=m2=1 the two masses are given initial displacements at t =0 of x1 ==1 and x2=0 the initial velocity of each mass at t=0 is zero

Your job is to calculate expressions giving the position of each mass as a function of time using the diagonalization method discussed in class. This classic problem can be solved in a number of different ways, but I will not give you credit unless you solve using the diagonalization method from the course.


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