\"font-size:small;\">Coupled Oscillators . \"font-size:9pt;font-family:Verdana;\
ID: 2985047 • Letter: #
Question
"font-size:small;">Coupled Oscillators
.
"font-size:9pt;font-family:Verdana;">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 m
1 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 m
1 = m2 = m and k1 = k2 = k3 = k.
c) Find the general solution of this system and implement the initial conditions to determine the arbitrary 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 for this system.
f) Find the eigenvalues and eigenvectors of this system in the special case that m
1 = m2 = m and k1 = k2 = k3 = k.
g) Find the general solution of this system and implement the initial conditions to determine the arbitrary constants.
h) Graph the displacement functions for the two blocks and comment on the behavior of the system
Explanation / Answer
A fbd of m2 with a downward positive and T is the tension in the cord T=m2*(g-a) and the sliding block T=m1*a the a is the same for both in magnitude combine and solve for a a=m2*g/(m1+m2)
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