The energetics of self-oscillators

Alejandro Jenkins U. de Costa Rica &

Academia Nacional de Ciencias

International Conference on Advances in Vibrations U. do Porto, Portugal

30 March 2015

References• Self-oscillators maintain regular,

periodic motion, at expense of power source with no corresponding periodicity

• Positive feedback between oscillation and power modulation

• AJ, “Self-Oscillation”, Phys. Rep. 525, 167 (2013)

• AJ, The Physical Theory of Self-Oscillators, (in preparation)

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Control theory“A distinguishing feature of this new science is the total absence of considerations of energy, heat, and efficiency, which are so important in other natural sciences.”

- Qian Xuesen, Engineering Cybernetics (1954)

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Qian Xuesen (1911 – 2009)

Rayleigh

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q ↵q + q3/3 + !2q = 0

Combines linear anti-damping α

2nd ed. of Theory of Sound (1894-6) models “maintained oscillations”, including wind musical instruments, by

Ploss

= qFdamp

= mq4/3

Pgain = qFantidamp = ↵mq2

with non-linear damping β

Van der Pol

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V ↵ V 2

V + !2V = 0

Van der Pol (1920) uses eq. equivalent to Rayleigh’s:

V0 = 2r

↵

Steady amplitude:

Directly implementable as electric circuit:

Vin VoutR

L C

I0

V0

Idiode

V

I

I0

V0

Vin

= g · Vout

Limit cyclesx ↵

1 x

2x+ x = 0

x = ↵

y + x x

3/3

y = x/↵

Liénard transformation:

-2 -1 1 2 x

-2

-1

1

2y

↵ = 0.2

10 20 30 40 50 60 t

-2

-1

1

2Vx

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-2 -1 1 2 x

-1.0

-0.5

0.5

1.0

y

5 10 15 20 25 30 t

-2

-1

1

2

Vx

↵ = 5

- 3 -2 -1 1 2 3 4 x

- 3

-2

-1

1

2

y

↵ = 0.2 10 20 30 40 50 60 t

- 3

-2

-1

1

2

3

4Vx

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Positive feedback• Vout amplified & fed

back to Vin

• Resistance effectively negative

• Exponential growth limited by amplifier’s saturation (nonlinearity)

• All clocks work on this principle

Vin

= g · Vout

Vout

+1 g

RCV

out

+1

LCV

out

= 0

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Vin VoutR

L C

Millennium Footbridge (London)

http://www.youtube.com/watch?v=eAXVa__XWZ8

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Kelvin-Helmholtz

exp [ik(x vt)] v = !/k

Galileo: @

@t

! @

@t

+ V

@

@x

= i! + V · ik

= i!

1 V

v

See: Zel’dovich, JETP Lett. 14, 180 (1971)

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Air

Water v = wave velocity, seen by the water

V = water velocity, seen by the air

2 /k

Relaxation oscillation• Pearson-Anson flasher

(1922)

• Period not associated to resonance

• Non-linear switching at thresholds

• Theory by Van der Pol & Friedländer (1926)

• e.g., heart & neurons

VoutR

CV0neonlamp

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VoutVon

tVoff

Synchronization• Huygens (1665) noticed that two adjacent pendulum

clocks, mounted on wooden partition, ended up moving in anti-phase

• Locking of frequency, mode, or phase (synchronization) possible because non-linear oscillator’s frequency depends on its amplitude

• Amplitude may vary until phase relative to forcing motion makes power input match dissipation

• See discussion of “Duffing problem” in Sargent, Scully & Lamb, Laser Physics (1974), sec. 3-2

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Forcing• Relaxation oscillators

are particularly easy to entrain

• Also show:

A. demultiplication

B. quasi-periodicity

C. chaos

x 3

1 x

2x + x = 5 cos(1.788t)

10 20 30 40 50 60 t

-2

-1

1

2

x

x(0) = 0.1 , x(0) = 0

x(0) = 0.1 , x(0) = 0.01

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Chandler wobble• Extinctions, followed by phase

jumps, in 1850s, 1920s & 2000s

• Not associated with obvious geophysical events

• Work in progress: wobble as self-oscillation, powered by fluid circulations

• Turned on & off by stochastic perturbations (Hopf bifurcation)

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Malkin & Miller, Earth Planets Space 62, 943 (2010)

Singular spectrum analysis (SAS) filtered:

944 Z. MALKIN AND N. MILLER: CHANDLER WOBBLE: TWO MORE LARGE PHASE JUMPS REVEALED

Fig. 1. Original and filtered PM series used for our analysis, and corresponding spectra. One can see that both types of digital filtering allows us toeffectively suppress the annual signal. The CW signal looks similar in both filtered series. However, some differences can be seen near the ends ofthe interval.

for digital filtering of the PM series.

Singular spectrum analysis (SSA). This method allows usto investigate the time series structure in more detailthan other digital filters. As shown in previous studies,it can be effectively used in investigations of the Earthrotation (see, e.g., Vorotkov et al., 2002; Miller, 2008).

Fourier filtering. We used the bandpass Fourier transform(FT) filter with the window 1.19±0.1 cpy. Such a widefilter band was used to preserve the complicated CW

structure. In the filtered PM series, the amplitude ofthe remaining annual signal is about 0.5 mas, i.e. 0.5%of the original value.

Hereafter we will refer to filtered PM time series as CWseries. Analyzed PM and CW time series and their spec-tra are shown in Fig. 1. We can see two main spectralpeaks of about equal amplitude near the central period ofabout 1.19 yr, and several less intensive peaks in the CWfrequency band. Discussion on its origin, and even reality,

Classical engine• Two heat baths, working

fluid, piston

• Piston modulates power from working fluid, via fly-wheel & valves

• Positive feedback between piston & valve action allows net work extraction

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Andronov, Vitt & Khaikin, Theory of Oscillators (Dover, 1987 [1966]), ch. VIII, sec. 10

ħΩ

Quantum engine• Recent work by Alicki et al. on

quantum engines

• Solar cell: baths at phonon (room) temperature + incident photons at high effective temperature (~1000 K)

• Plasma oscillation at p-n junction may act as piston

• Ω/2π ~ 1 THz

• Maintains cyclic DC current

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Alicki, Gelbwaser-Klimovsky & Szczygielski, arXiv:1501.00701 [cond-mat.stat-mech]

Summary• Self-oscillation usually studied within mathematical

theory of non-linear dynamical systems

• Energetics dispels needless obscurities

• Intermittent self-oscillation (Hopf bifurcation) may account for some heavy-tailed distributions & other complex phenomena

• Picture of motors as self-oscillators useful to thermodynamics

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