# 3.2.1: Driven Springs Simulation

- Page ID
- 26147

The following simulation shows five different masses, each attached to a spring of the same stiffness. The springs are mounted on a mechanical device that shakes the springs and attached masses. You can adjust the driving frequency, \(f\) in \(\text{Hz}\), of the shaking mechanism, the amplitude of the driving force, \(F_{o}\), in Newtons, the amount of friction, \(b\) in \(\text{Ns/m}\), and the stiffness of the springs, \(κ\), measured in \(\text{N/m}\).

## Simulation Questions:

- Start the simulation. Do any of the masses have a very large amplitude?
- Increase the amplitude of the driving force, \(F_{o}\). Now do any of the masses have a very large amplitude?
- Reset the simulation and change the driving frequency, \(\text{f}\), to \(0.5\text{ Hz}\). Wait a few seconds. What do you see now?
- Why is the oscillation of mass number five much larger than the other ones now?
- Reset the simulation and change the driving frequency to \(2.0\text{ Hz}\). What do you notice now?
- See if you can determine the resonance frequency of the center mass by trial and error. What is its resonance frequency?
- In the previous simulation we saw that the natural frequency, written as \(f_{o}\) is given by the stiffness of the spring, \(κ\), and the mass; \(f_{o} = (κ/m)^{1/2}/(2π)\). In this simulation the large mass is \(10\text{ kg}\) and the spring constant is initially \(100\text{ N/m}\) so \(f_{o}= 0.5\text{ Hz}\). This is why it has a large oscillation when driven at \(0.5\text{ Hz}\); it will resonate with a driving frequency equal to its natural frequency. The center mass (number three) is \(2.5\text{ kg}\) so the natural frequency is \(1.0\text{ Hz}\). Did you find a resonance frequency of \(1.0\text{ Hz}\) for this mass?
- Mass number two is \(1.25\text{ kg}\). Calculate its natural frequency. Verify your result by trying it out in the simulation.
- Reset the simulation so that none of the masses are in resonance. Why doesn't the driving amplitude have much of an effect on the oscillation of the masses?
- Change the spring stiffness, \(κ\) to \(150\text{ N/m}\). (This changes the stiffness of all the springs.) Do the masses have the same resonance frequencies? Explain.
- Use the formula for the natural frequency to calculate the natural frequency of the \(10\text{ kg}\) mass (mass number five) with a spring constant of \(150\text{ N/m}\). Verify your result using the simulation.