Problem 1

Use Langrange's method to solve problem 1.21 in the book.

Problem 2

An undamped system vibrates with a frequency of 10 Hz and amplitude of 1 mm. Calculate the maximum amplitude of the system's velocity and acceleration.

Problem 3

Using Python inside a Jupyter notebook plot the solutions given by:

for the case \(k = 1000 \textrm{N/m}\) and \(m = 10 \textrm{kg}\) for two complete periods for each of the following sets of initial conditions: a) \(x_0 = 0\), \(v_0= 1 \textrm{m/s}\), b) \(x_0= 0.01\textrm{m}\), \(v_0 = 0\), and c) \(x_0 = 0.01 \textrm{m}\), \(v_0 = 1 \textrm{m/s}\). Plot each result in a sub plot of a single figure.

Problem 4

For a damped system, \(m\), \(c\), and \(k\) are known to be \(m = 1 \textrm{kg}\), \(c = 2 \textrm{kg/s}\), \(k = 10 \textrm{N/m}\). Calculate the values of \(\zeta\) and \(\omega_n\). Is the system overdamped, underdamped, or critically damped?

Problem 5

Using Python in a Jupyter notebook plot \(x(t)\) for a damped system of natural frequency \(\omega_n = 2 \textrm{rad/s}\) and initial conditions \(x_0 = 1 \textrm{mm}\), \(v_0 = 0\), for the following values of the damping ratio: \(\zeta = 0.01\), \(\zeta = 0.2\), \(\zeta = 0.6\), \(\zeta = 0.1\), \(\zeta = 0.4\), and \(\zeta = 0.8\). Plot each line in one plot and add a legend for the different values of \(\zeta\). In addition, use the interact function to create a slider that allows you to interactively adjust the values of \(\zeta\).