Consider the system shown below of two masses of mass m, coupled together between two fixed walls via springs with varying spring constants.
Let x(t) and y(t) be the horizontal displacements of the two masses as a function of time.
(a)Write down a system of differential equations for x'' and y''.
(b)For the case when m = 1, k1 = 1 and k2 = 2, calculate the eigenvalues associated with this system, which are associated with the characteristic frequencies of vibration.
(c)Plot the eigenvalues as a function of k2 over the range 0 < k2 < 3. Discuss the physical interpretation of the changes to the eigenvalues as k2 is increased.
k2=0
(a) To write down a system of differential equations for x'' and y'', we need to apply Newton's second law to each mass.
For the first mass (mass m connected to the left wall):
By Newton's second law, the net force acting on the first mass is given by:
m*x'' = -k1*x + k2*(y-x)
For the second mass (mass m connected to the right wall):
By Newton's second law, the net force acting on the second mass is given by:
m*y'' = -k2*(y-x)
So, the system of differential equations for x'' and y'' is:
m*x'' = -k1*x + k2*(y-x)
m*y'' = -k2*(y-x)
(b) To calculate the eigenvalues associated with the system, we can rewrite the equations in matrix form.
Let X = [x, y] be a column vector, then the system can be written as:
m*X'' = A*X
where A is a 2x2 matrix given by:
A =
[ -k1+k2 k2 ]
[ k2 -k2 ]
Now, we need to find the eigenvalues λ of the matrix A by solving the characteristic equation:
det(A - λI) = 0
where I is the identity matrix.
For matrix A, the characteristic equation becomes:
|-k1+k2-λ k2 | = 0
| k2 -k2-λ |
Expanding the determinant:
(-k1+k2-λ)*(-k2-λ) - k2*k2 = 0
Simplifying the equation:
k1k2 + λ*(k1 - 2k2) + λ^2 = 0
Now, we can solve this quadratic equation to find the eigenvalues.
(c) To plot the eigenvalues as a function of k2 over the range 0 < k2 < 3, we need to plug different values of k2 into the quadratic equation obtained in part (b) and solve for the eigenvalues λ.
For each value of k2, calculate the eigenvalues using the quadratic formula:
λ = (-b ± √(b^2 - 4ac)) / 2a
where a = 1, b = k1 - 2k2, and c = k1k2.
Plot the eigenvalues (λ) on the y-axis and k2 on the x-axis. By varying k2 over the given range, observe how the eigenvalues change. Discuss the physical interpretation of the changes in the eigenvalues as k2 is increased.
The physical interpretation of the changes in the eigenvalues could be related to the characteristic frequencies of vibration of the system. The eigenvalues represent the natural frequencies of oscillation of the masses coupled with the springs. As k2 increases, it affects the stiffness of the spring connecting the two masses, and therefore, it would affect the natural frequencies of vibration. By analyzing the plot of eigenvalues as a function of k2, you can determine how changes in k2 impact the characteristic frequencies of the system.