Section 23–2 The forced oscillator with damping

(Complex factor R / Phase angle θ / Quality factor Q)

In this section, Feynman discusses a complex factor R, phase angle θ, and quality factor Q of a forced oscillation.

1. Complex factor R:

“There is no technical name for this factor, no particular letter for it, but we may call it R for discussion purposes: R = 1/m(ω₀²−ω²+iγω)… (Feynman et al., 1963, section 23–2 The forced oscillator with damping).”

Feynman says that there is no technical name for the factor R = 1/m(ω₀²−ω²+iγω), but we may call it R for discussion purposes. By using this factor R, the amplitude of the response becomes F^R in which F^ is a complex force. However, in a mechanics textbook, Barger and Olsson (1973) call R a complex factor and clarify that it is a time-independent factor. In the first edition of this textbook, the equation of motion is dz²/dt² + 2g(dz/dt) + w₀²z = (F₀/m)e^iwt and they let R = 1/m(w₀²-w²+2igw) = re^-iq. This allows them to have z = RF₀e^iwt that is similar to Feynman’s approach. In the second edition, they let R = w₀²-w²+2igw and this allows them to have z = (f/R)e^iwt in which f = F₀/m. In a sense, this revised approach is better because the R factor is simpler and it also helps to understand the phase angle better (to be discussed later).

Feynman defines R = re^iq and calls r the magnification factor because it represents the size of the response. Furthermore, ρ² = 1/m²[(ω²−ω₀²)²+γ²ω²] is proportional to the square of the amplitude or the energy of the oscillator. Initially, Barger and Olsson (1973) defines R = re^-iq and thus, the magnification factor r² = |R|² = 1/m²[(w₀² - w²)² + 4g²w²]. In the second edition, Barger and Olsson (1995) defines R = re^iq and thus, r² = |R|² = (w₀²-w²)²+4g²w². The magnification factor r depends mainly on the relative magnitude of the natural frequency w₀ and the driving frequency w. However, it is not absolutely necessary to use the symbol r or r, but it is a shorthand that saves time, instead of writing (w₀² - w²)²+g²w² or (w₀²-w²)²+4g²w² again and again.

2. Phase angle θ:

“Therefore ρ and θ represent the size of the response and the phase shift of the response (Feynman et al., 1963, section 23–2 The forced oscillator with damping).”

According to Feynman, if we write 1/R = 1/ρe^iθ = (1/ρ)e^−iθ = m(ω₀²−ω²+iγω), then tan θ = −γω/(ω₀²−ω²). However, it should be easier to understand the derivation of phase angle θ if we choose ρe^iθ = m(ω₀²−ω²+iγω). We may explain that ρe^iθ = ρ(cos θ + isin θ) = m(ω₀²−ω²+iγω) or equivalently, ρ(cos θ) = m(ω₀²−ω²) and ρ(isin θ) = imγω by comparing the real and imaginary part. This allows us to easily deduce that tan θ = mγω/m(ω₀²−ω²) = γω/(ω₀²−ω²). Note that some textbook authors may prefer tan θ = 2γω/(ω₀²−ω²) because their resistive force is 2γv instead of γv. In addition, due to different definitions of θ, we should carefully distinguish tan θ = γω/(ω₀²−ω²) and tan θ = γω/(ω²−ω₀²) that involves a difference in sign.

Feynman explains that the phase angle θ is minus because tan (−θ) = −tan θ. He adds that a negative value for θ results for all ω and this corresponds to the displacement x lagging the force F. However, one may change tan θ = −γω/(ω₀²−ω²) into tan θ = γω/(ω²−ω₀²) without mentioning the minus sign and still explain that the displacement is lagging the force. Whether the phase angle is positive or negative is also dependent on your choice, e.g., let x = x₀e^i(ωt+θ) or x = x₀e^i(ωt-θ). More important, one has a choice to either say system A leads system B by q or system A lags system B by (2p - q). We choose to say that the force leads the displacement by the phase angle θ because we consider the force is the cause and the displacement is the effect that is lagging. The “delay” of the displacement is dependent on the magnitude of the resistive force.

3. Quality factor Q:

“As another measure of the width, some people use a quantity Q which is defined as Q = ω₀/γ (Feynman et al., 1963, section 23–2 The forced oscillator with damping).”

Feynman mentions that some people use a quantity Q which is defined as a measure of the frequency width (ω₀/γ). This quantity Q is commonly known as the “quality factor,” but Johnson who originates the term, says that it did not stand for quality (Green, 1955). He proposes the symbol Q to represent the ratio of reactance to effective resistance in a coil or anything else because the other letters of the alphabet had been used. In chapter 24, Feynman adds that many have tried to define Q in the simplest and most useful way such that various definitions differ a bit from one another, but they are in agreement if Q is very large. Interestingly, the term Q can be used to describe a characteristic of many physical systems such as a resonant circuit, a spectral line, a mechanical vibrator, and a bouncing ball.

Feynman explains that the narrower the resonance, the higher the Q, e.g., Q = 1000 means a resonance whose width is only 1000th of the frequency scale. In chapter 24, he provides another definition of Q: the mean stored energy is multiplied by 2π and is divided by the work done per cycle. One may clarify that we can define the Q factor in terms of energy of instead of frequency and this is why it is also known as the “storage factor.” This factor is dependent on the presence of resistive forces that affect the energy stored during an oscillation. Alternatively, Kleppner and Kolenkow (1973) explain that the degree of damping of an oscillator is specified by the quality factor and defines Q = (energy stored in the oscillator)/(energy dissipated per radian).

Questions for discussion:

1. Would you prefer to define R = 1/m(ω₀²−ω²+iγω) or simply R = (ω₀²−ω²+iγω)?

2. Would you prefer to have tan θ = γω/(ω₀²−ω²), tan θ = γω/(ω²−ω₀²), or tan θ = 2γω/(ω₀²−ω²)? 

3. Would you describe Q as a quality factor or storage factor?

The moral of the lesson: we would really appreciate the magic of the “complex” method that can obtain the response x to a given force F by comparing with the working needed if we use the old straightforward way, the “trignometric” method.

References:

1. Barger, V. D., Olsson, M. G. (1973). Classical mechanics: a modern perspective. New York: McGraw-Hill.

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. Green, E. I. (1955). The story of Q. American Scientist, 43(4), 584-594.

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