Section 25–3 Oscillations in linear systems

(Idealized friction / Real friction / Resonance curves)

 

In this section, Feynman discusses the idealized friction (weak & proportional to the velocity) and real friction (strong & constant) experienced by an oscillator and its resonance curves.

 

1. Idealized friction:

“… a special kind of friction must be carefully invented for the very purpose of creating a friction that is directly proportional to the velocity … (Feynman et al., 1963, section 25–3 Oscillations in linear systems).”

 

According to Feynman, a friction is invented such that it is directly proportional to the velocity and it is weaker for small oscillations. Based on this idealization, the spring force is reduced, the inertial effects are lower, the accelerations are weaker, and the friction is lesser. In a similar sense, many problems of oscillators are modeled using the equation mdv/dt + bv + kx = 0 in which bv is idealized as a weak frictional force. This is a valid model if our linear problem is essentially small oscillations. In the case of a pendulum, we have also idealized the period to be independent of the amplitude and it is independent of the weight of the pendulum. (In the last paragraph of the previous section, Feynman explains that sin q is practically equal to q for a simple pendulum if q is small; this last paragraph could be shifted to the present section.)

 

Feynman elaborates that the sizes of the oscillations are reduced by the same fraction of themselves in every cycle because of the weaker frictional force. Thus, the amplitude (A) of the oscillation can be expressed by the equation A = A₀aⁿ in which A₀ is the initial amplitude, a is the ratio of the two amplitudes between two successive cycles, and n is the number of cycles traversed. Importantly, the fact that n is directly proportional to t (total time) is approximately true for small oscillations (i.e., the period is assumed to remain unchanged). Furthermore, one should recall that a solution of m(dv/dt) + bv + kx = 0 is in the form of e^−ctcos ω₀t. It may not be trivial to explain that e^−3ct = (e^−ct)(e^−ct)(e^−ct) = (0.9)³ if e^−ct = 0.9 and thus, we have a choice to use A = A₀aⁿ (or A = e^−ct).

 

2. Real friction:

“What happens if the friction is not so artificial; for example, ordinary rubbing on a table, so that the friction force is a certain constant amount … (Feynman et al., 1963, section 25–3 Oscillations in linear systems).”

 

Feynman says that the frictional force, for example, ordinary rubbing on a table is a certain constant amount and it is independent of the size of the oscillation that reverses its direction. It seems that he suggests the friction is a constant something like the kinetic friction. This is also an idealization in which kinetic friction (µ_kN) is simply equal to a kinetic coefficient of friction (µ_k) times the normal reaction (N). However, a better model for real friction can be represented by the equation F = μN + kA, where kA is dependent on the area of contact between two surfaces (Besson et al., 2007). Strictly speaking, the real friction measured may decrease as the velocity is increased and it is more problematic (Ludema & Tabor, 1966), and thus, it has to be solved by a numerical method.

 

Feynman explains that a system does not oscillate at all if there is too much friction. If the energy in the spring is unable to overcome the frictional force, it would slowly reach the equilibrium point. In the last chapter (section 24-3), Feynman has already discussed the strong friction that results in heavy damping. In a sense, Feynman contradicts himself because he mentions earlier that the system can oscillate one cycle (instead of “does not oscillate”). Perhaps some may argue whether the oscillatory motion should be defined as a to-and-fro motion or repeated motion. However, a pendulum suspended inside a bottle of honey may not move more than a quarter of a cycle and it needs a long time to reach the equilibrium point.

 

3. Resonance curves:

“Qualitatively, we understand the resonance curve; in order to get the exact shape of the curve it is probably just as well to do the mathematics (Feynman et al., 1963, section 25–3 Oscillations in linear systems).”

 

In Fig. 25–5, Feynman shows resonance curves with various amounts of friction present. He says that the curve goes toward infinity as ω approaches ω₀ (the natural frequency of the oscillator). On the contrary, Feynman explains that the amplitude does not reach infinity for some reason; it may be that the spring breaks in section 21-5. Notably, Landau and Lifshitz (1976) write that “the amplitude of oscillations in resonance increases linearly with the time (until the oscillations are no longer small and the whole theory given above becomes invalid) (p. 62).” On the other hand, the collapse of the Tacoma bridge is related to an aerodynamically induced condition of self-excitation or “negative damping” instead of forced resonance (Billah & Scanlan, 1991). In short, forced resonance is not a necessary condition to break a bridge.

 

Feynman mentions that the resonance curve is usually plotted so that the top of the curve is called one unit. He adds that if there is lesser friction, this curve can have a higher peak as well as the narrower width at half the maximum height. One should re-read his explanation on the equation ρ² ≈ 1/4m²ω₀²[(ω₀−ω)² + γ²/4] in chapter 23: “We shall leave it to the student to show the following: if we call the maximum height of the curve of ρ² vs. ω one unit, and we ask for the width Δω of the curve, at one half the maximum height, the full width at half the maximum height of the curve is Δω = γ, supposing that γ is small (Feynman et al., 1963, section 23–2 The forced oscillator with damping).” That is, the resonance curve is related to the Q-factor that is a measure of the width (Δω = γ), which is defined as Q = ω₀/Δω (or ω₀/γ).

 

Questions for discussion:

1. How would you explain the amplitude (A) of the oscillation can be modeled by the equation A = A₀aⁿ?

2. Is it correct to say that ordinary rubbing on a table is a certain constant amount?

3. How would you explain the resonance curve can have a higher peak as well as a narrower width at half the maximum height?

 

The moral of the lesson: if there is lesser friction, we will have a higher resonance curve and a narrower width at half its maximum height (or using Q = ω₀/Δω = ω₀/γ in which γ is dependent on the friction).

 

References:

1. Besson, U., Borghi, L., De Ambrosis, A., & Mascheretti, P. (2007). How to teach friction: Experiments and models. American Journal of Physics, 75(12), 1106-1113.

2. Billah, K. Y., & Scanlan, R. H. (1991). Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks. American Journal of Physics, 59(2), 118-124.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Landau, L. D., & Lifshitz, E. M. (1976). Mechanics (3rd ed.). Oxford: Pergamon Press.

5. Ludema, K. C., & Tabor, D. (1966). The friction and viscoelastic properties of polymeric solids. Wear, 9, 329-348.