3.2.1: Driven Springs Simulation
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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 Hz, of the shaking mechanism, the amplitude of the driving force, Fo, in Newtons, the amount of friction, b in 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.