Coupled Oscillators . Imagine two blocks of mass m1 and m2 on a frictionless hor
ID: 2984474 • Letter: C
Question
Coupled Oscillators. Imagine two blocks of mass m1 and m2 on a frictionless horizontal surface attached to each other and to two walls by springs with spring constants k1, k2 and k3(see figure). Let
x(t)and y(t) denote the displacement of the left and right block, respectively, from their equilibrium positions, with positive displacement to the right. At time t = 0, the left block is held at its equilibrium position ( x(0) = 0 ) and the right block is displaced two units to the right of its equilibrium position ( y(0) = 2 ).
a) The governing equations for this system (assuming no friction and no external forcing) is
m1 x"(t) = -k1 x + k2 (y - x)
m2 y"(t) = -k2 (y - x) - k3 y
Briefly explain the meaning of each term of these equations. Write the system as a set of four linear
ODEs in the variables x, x', y, y'. Determine the matrix for this system.
b) Find the eigenvalues and eigenvectors of this system in the special case that m1 = m2 = m and k1 =
k2 = k3 = k.
c) Find the general solution of this system and implement the initial conditions to determine thearbitrary constants.
d) Graph the displacement functions for the two blocks and comment on the behavior of the system.
e) Write the system as a set of four linear ODEs in the variables x, x', y, y'. Determine the matrix forthis system.
f) Find the eigenvalues and eigenvectors of this system in the special case that m1 = m2 = m and k1 =
k2 = k3 = k.
g) Find the general solution of this system and implement the initial conditions to determine thearbitrary constants.
h) Graph the displacement functions for the two blocks and comment on the behavior of the system.
Explanation / Answer
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