A 4 kg mass is suspended from a spring with stiffness $k = 400$ N/m. Determine the natural frequency and period of vibration.

A simple pendulum consists of a 2 kg bob attached to a 1.2 m long string. Calculate the period of small oscillations. If the length is doubled, what is the new period?

A mass of $m = 10 \text{ kg}$ is suspended by a spring with stiffness $k = 4000 \text{ N/m}$. A viscous dashpot is attached, providing a damping coefficient of $c = 120 \text{ N}\cdot\text{s/m}$. Determine if the system is underdamped, critically damped, or overdamped. Then, calculate the damped natural frequency and the logarithmic decrement.

The infamous 1940 collapse of the original Tacoma Narrows Bridge ("Galloping Gertie") is a stark reminder of the destructive power of vibrations in structural engineering. It failed dramatically in a relatively mild 40 mph wind. Analyze this event from the perspective of mechanical vibrations, specifically focusing on the concepts of natural frequency, forced vibration, and aerodynamic flutter (a form of self-excited vibration).

Tall buildings, such as the Taipei 101 skyscraper, are highly susceptible to wind-induced vibrations and seismic activity. To mitigate these dangerous sways and improve occupant comfort, engineers often install massive "Tuned Mass Dampers" (TMDs) near the top of the structure. Analyze how a TMD works using the principles of forced vibration and coupled dynamic systems.