Doubochinski's pendulum

{{Short description|Experiment}}

{{Primary sources|date=April 2021}}

Doubochinski's pendulum is a classical oscillator interacting with a high-frequency field in such a way that the oscillator takes on a discrete set of stable regimes of oscillation, each at a frequency near to the proper frequency of the oscillator, but each with a distinct, "quantized" amplitude.

{{Cite journal

| author = L. A. Vainshtein

|author2=Ya. B. Doubochinski

| title = On the low-frequency oscillations under the influence of high-frequency force

| series = 48

| journal = Zh. Tekh. Fiz [Sov. Phys.-Tech. Phys]

| volume = [23]

| year = 1978

| pages = 1321 [745]

| url = http://www.kapitza.ras.ru/index.php?cont=journ&lang=ru&year=1978

| issue = 1494

}}

{{Cite journal

| author = D. B. Doubochinski

|author2=Ya. B. Duboshinsky

| journal = Zh. Tech. Fiz [Sov. Phys.-Tech. Phys]| series = 49

| title = Discrete modes of a system subject to an inhomogeneous high-frequency force

| volume = [24]

| year = 1979

| pages = 1160 [642]

|display-authors=etal

}}

{{Cite book |author=Landa |first=P. S. |url=http://www.gbv.de/dms/goettingen/193719193.pdf |title=Nonlinear Oscillations and Waves in Dynamical Systems |publisher=Kluwer Academic Publishers |year=2001 |pages=307 |author-link=Polina Landa}}

{{Cite journal

| author = Weldon J. Wilson

| journal = Professor of Engineering and Physics, Weldon Wilson's Home Page

| title = Amplitude Quantization as a Fundamental Property of Coupled Oscillator Systems

| url = http://cyberphysics.org/wwilson/Talks/CNLS%20Talk%20v3/CNLS%20Talk.ppt

| year = 2010

}}{{cite journal |last1=Luo |first1=Yao |last2=Fan |first2=Wenkai |last3=Feng |first3=Chenghao |last4=Wang |first4=Sihui |last5=Wang |first5=Yinlong |title=Subharmonic frequency response in a magnetic pendulum |journal=American Journal of Physics |date=2020 |volume=88 |issue=2 |pages=115–123 |doi=10.1119/10.0000038 |bibcode=2020AmJPh..88..115L |s2cid=213115412 |quote=We study the subharmonic frequency response of a generalized driven oscillator excited by a nonlinear periodic force. We take a magnetic pendulum called the Doubochinski pendulum as an example.}}{{cite journal |last1=Pedersen |first1=Henrik B |last2=Madsen |first2=Magnus Linnet |last3=Andersen |first3=John E V |last4=Nielsen |first4=Torsten G |title=Investigation of argumental oscillations of a physical pendulum |journal=European Journal of Physics |date=10 February 2021 |volume=42 |issue=2 |pages=025012 |doi=10.1088/1361-6404/abcee4 |bibcode=2021EJPh...42b5012P |s2cid=229406696 |quote=For one realization of argumental oscillations (Doubochinski's pendulum), we report experimental frequency response curves}} The phenomenon of amplitude quantization in this sort of coupled system was first discovered by the brothers Danil and Yakov Doubochinski in 1968–69.

A simple demonstration apparatus consists of a mechanical pendulum interacting with a magnetic field.

{{Cite journal

| author = D. B. Doubochinski

| author2 = Ya. B. Doubochinski

| title = Wave excitation of an oscillator having a discrete series of stable amplitudes

| journal = Soviet Physics Doklady

| volume = 27

| year = 1982

| pages = 564

|bibcode=1982SPhD...27..564D

}}

{{Cite book

| author = Martin Beech

| title = The Pendulum Paradigm: Variations on a Theme and the Measure of Heaven and Earth

| publisher = Universal Publishers

| year = 2014

| page = 232

| url = https://books.google.com/books?id=qumVBAAAQBAJ

| isbn = 9781612337302

}} The system is composed of two interacting oscillatory processes: a pendulum arm with a natural frequency on the order of 0.5–1 Hz, with a small permanent magnet fixed at its moving end; and a stationary electromagnet (solenoid) positioned under the equilibrium point of the pendulum's trajectory and supplied with alternating current of fixed frequency, typically in the range of 10–1000 Hz.

The mechanical pendulum arm and solenoid are configured in such a way, that the pendulum arm interacts with the oscillating magnetic field of the solenoid only over a limited portion of its trajectory – the so-called "zone of interaction" – outside of which the strength of the magnetic field drops off rapidly to zero. This spatial inhomogeneity of the interaction is key to the discretized oscillation amplitudes and other unusual properties of the system.