Section 21–5 Forced oscillations

(Steady-state response / Transient response / Resonance response)

In this section, Feynman discusses the steady-state response, transient response, and resonance response of an oscillator. However, it could be better reorganized as transient response, steady-state response, and resonance response.

1. Transient response:

“…we have found is the solution only if things are started just right, for otherwise there is a part which usually dies out after a while. This other part is called the transient response to F(t) (Feynman et al., 1963, section 21–5 Forced oscillations).”

Feynman discusses the forced harmonic oscillator that is expressed by the equation md²x/dt² = −kx + F(t). In this section, he focuses on the solution that is called a steady-state response instead of the transient response. Mathematically speaking, the general solution of the above equation is the sum of a particular solution (steady-state response) and the “homogeneous” solution (transient response). In chapter 24 on Transients, Feynman elaborates on the transient response with more details. In section 24-3 Electrical transients, he mentions that there are “dying exponentials,” in which one may have a much faster “dying rate” than the other. Thus, Feynman may agree with using the term “dying response” that could be better than “transient response.”

Feynman simply says that the other part which usually dies out after a while is called the transient response to F(t). However, this is potentially misleading because the transient response is a solution to F(t) = 0 and there should be resistive forces (e.g., air resistance such as bv). In essence, the transient response is a dying response because the driving force F(t) = 0 means it is not a periodic force that can continuously influence the oscillatory motion. Furthermore, the effects of the initial conditions will die out after some time because of the resistive forces. On the other hand, there are three types of dying responses to F(t) = 0: underdamped, overdamped, and critically damped. These different responses to F(t) = 0 are dependent on the relative magnitude of the resistive force and the natural frequency ω₀.

2. Steady-state response:

“Of course, the solution we have found is the solution only if things are started just right… while x = Ccos ωt and C = F₀/m(ω₀²−ω²) are called the steady-state response (Feynman et al., 1963, section 21–5 Forced oscillations).”

Feynman shows that the particular solution (steady-state response) of md²x/dt² = −kx + F(t) is x = Ccos ωt in which C = F₀/m(ω₀²−ω²). That is, an object with mass m oscillates at the driving frequency (ω), but its amplitude depends on the magnitude of the force (F₀) and the difference between the natural frequency (ω₀) of the oscillator and the driving frequency of F(t). In chapter 25, Feynman renames the steady-state response as the “forced” solution that is dependent on the “forcing frequency” (ω) of the driving force F(t). The steady-state response is sinusoidal in shape that remains unchanged while the ‘transient response” dies out. We should notice that there are no arbitrary constants in the “forced” solution that are related to the initial position or initial velocity.

According to Feynman, if ω is very small compared with ω₀, the displacement and the force are in the same direction. One may clarify that the equation md²x/dt² = −kx + F(t) can be reduced to kx = F(t) for very low frequency when the object moves slowly in the same direction as the force (md²x/dt² ® 0). If ω is very high, Feynman uses C = F₀/m(ω₀²−ω²) to explain that C is negative and the denominator becomes very large, and thus, the amplitude is small. Alternatively, one may reduce the equation md²x/dt² = −kx + F(t) to md²x/dt² = F(t) because the term kx is smaller relative to md²x/dt² as higher frequencies implies larger acceleration. By integrating the equation twice, we have x = -(F₀/mω²) sin ωt that has a negative sign and it means that the displacement is “out of phase” to the driving force.

3. Resonance response:

“If we happen to get the right timing, then the swing goes very high, but if we have the wrong timing, then sometimes we may be pushing when we should be pulling, and so on, and it does not work (Feynman et al., 1963, section 21–5 Forced oscillations).”

Theoretically, the swing should oscillate at an infinite amplitude if ω is exactly equal to ω₀. Feynman explains that this is impossible because the equation is wrong by excluding some frictional terms, and other forces that occur in the real world. Some may prefer a more realistic model that includes resistive forces that could be represented as kv. Essentially, this resonance response is an idealization that is applicable to an unrealistic undamped forced oscillator that has no resistive forces. However, this ideal model may be used to explain the response of a bound electron to an electromagnetic field such as the scattering of light in the classical theory (Kleppner & Kolenkow, 1973).

Feynman provides another reason why the amplitude of oscillation does not reach infinity. In short, it is likely that the spring breaks! This ending may seem abrupt but one may emphasize that the spring does not definitely obey Hooke’s law within the elastic limit. As a suggestion, we may use Feynman’s (1988) investigation of the Space Shuttle Challenger disaster to conclude the chapter. In general, the elastic constant of an object is dependent on its temperature. Importantly, Feynman demonstrated that the O-ring made of rubber doesn’t spring back quickly at the temperature of ice water. The O-ring of the space shuttle Challenger, having lost its ability to seal, caused the disaster.

Questions for discussion:

1. Does an undamped forced oscillator have a transient response?

2. How would you explain the steady-state response of a forced oscillator if the driving frequency is very low or very high?

3. What should be the response of an oscillator when the driving frequency is equal to the natural frequency?

The moral of the lesson: an idealized undamped forced oscillator that has no resistive forces should have a steady-state response or resonance response.

References:

1. Feynman, R. P. (1988). What Do You Care What Other People Think? New York: W W Norton.

2. 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.

3. Kleppner, D., & Kolenkow, R. (1973). An Introduction to Mechanics. Singapore: McGraw-Hill.