How does friction affect the energy and motion of a damped spring-mass system?

[hasan_researc]

Young and Freedman book Chapter 13 Periodic Motion:
In the third papargraph, the author writes that since a damped oscillator naturally vibrates a frequency of omega-prime, then we expect that an application of a driving force with omega close to omega-prime will cause the amplitude to become larger than normal.



I actually don't see why this has to be the case.

 

[hikaru1221]

(Oops, I almost forgot that posting a complete answer was forbidden)

1 - If the two frequencies are close to each other, at what frequency does the system vibrate?
2 - From the relation between the frequency the system vibrates at and the forcing frequency, could you predict what will happen to the energy of the system? Notice that the force and its work done on the system have something to do with this.
3 - If the energy of the system changes in such way above, how will the amplitude change?

This one is optional:
4 - Consider the spring - mass system for simplicity. From the three factors determining the vibration: the spring (which provides the system the free vibration), the friction (which prevents the vibration), the driving force (which forces the vibration to continue), which one did we not take into account? From that, could you predict under what condition does the phenomenon happen (besides natural frequency = forcing frequency)?

 

[hasan_researc]

1. The system vibrates at the mean of the two frequencies (??) (If so, why?)

2. I know the driving force does work on the system and increases its energy content. BUt how can we work out this fact from the relation between the natural freq and forced freq? What is the relation between the natural freq and the forced freq anyway?

3. Amplitude increases if energy increases.

4. We ignored friction. I don't understand why the phenomenon happens when natural freq = forced freq. And what the other condition for which resonance occurs is.

Please help!

 

[hikaru1221]

1 - Though the frequency the system vibrates at is not their mean value, it should be in relation with the 2 frequencies, right? But notice that the 2 frequencies are close to each other; for example, one is 10 Hz, the other is 10.1 Hz. What would you say about the frequency the system vibrates at approximately?

2 - No, it's not about natural frequency $$f_n$$ and forcing frequency $$f_F$$. As I pointed out in the question, it's about the frequency the system vibrates at $$f_v$$ and forcing frequency $$f_F$$ (you may be confused if you know question 1's answer, but there is a meaning behind this).

Okay, another hint:
$$f_v$$ is somehow related to the rate of change in the motion of the system.
$$f_F$$ is somehow related to the rate of change in the direction of the driving force.

Notice one thing: The force can do either positive or negative work on the system, so the energy may increase or decrease. So how would the energy benefit from that relation $$f_v$$ and $$f_F$$?

3 - Correct.

4 - Oh yes, we ignored friction! But in reality, friction always appears, however big or small it is. So when would we ignore friction in practice?

 

[hasan_researc]

2. I thought that the forced freq is the freq of oscillation. The freq of damped motion (natural freq) is a different freq. I actually don't understand why the freq the systme vibrates at is any diff from the forced freq.

4. We would ignore friction when it is much smaller than the driving force. (??)

 

[hikaru1221]

I'm sorry; I forgot one important point. I'll change the question.

Consider a spring-mass system in the air, which you excite to vibrate by your hand.
1 - You have: the forcing freq = the vibrating freq. From your experience, what do you think about the relation between the direction of the motion of the mass and the direction of the force at any time? Any conclusion about energy?

2 - Actually, only when the system is in steady state, forcing freq = vibrating freq. Before that, the motion of the system can be approximately described as the sum of 2 vibrations: natural vibration and forced vibration (when you use your hand to pull back and push forth the mass, at first, you will see that the mass seems not to move as you want, right? That's because of the natural vibration).

Now if natural freq = forcing freq, how would the mass vibrate before the steady state? From question #1, how would you conclude about the energy? Is the energy built up? Does the driving force, in this case, act somehow as an energy pump? Why?

Optional:
3 - The steady state is when the amplitude no longer changes with time. Remember that the friction is proportional to the speed, so as energy increases, friction increases and the rate of energy lose increases. At first, when the speed is still small, we don't mention friction as we did above. But at some point, it can no longer be ignored.

So in the steady state, what is the relation between the input energy from the driving force and the lost energy due to friction? How about the energy of the system at that time? Will it increase or decrease or remain unchanged?

4 - The other condition you found is that friction must be small (or more exactly, frictional coefficient is small). Do you feel like it is easier to move the mass as you want (i.e. you pull it, it comes close to you right away; you push it, it gets away from you right away) in the air than in the water or oil? Then friction has something to do with the phase lag between the driving force and the vibration, right?

a/ The greater friction is, the bigger the phase lag is. What do you think about the relation between the directions, and thus, the energy in question #1, if friction is large? What is the consequence?

b/ The greater friction is, the greater the energy lose is. What do you think about the total energy the system gains before the steady state happens if friction is large? What is the consequence?

Sorry, it's quite long :'(