Sinusoidally Forced Damped and Undamped Harmonic Oscillator, Complexification, Beat Phenomenon

Bill Kinney's Differential Equations and Linear Algebra Course, Lecture 28A.

(a.k.a. Differential Equations with Linear Algebra, Lecture 28A, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 28A).

#forcedharmonicmotion #harmonicmotion #differentialequations

(0:00) Lecture plan
(0:42) Sinusoidally forced damped harmonic oscillator (nonhomogeneous 2nd order linear differential equation)
(5:23) Associated complexified equation and a particular complex solution
(7:05) Solve for the undetermined complex coefficient
(16:30) Particular solution yp of original forced harmonic oscillator
(17:34) General solution of original forced harmonic oscillator
(18:24) Transient and steady state solution (unforced response and forced response)
(20:07) Graph of the steady state solution (forced response) is sinusoidal
(21:08) Amplitude
(22:33) Phase angle
(26:58) Undamped sinusoidally forced harmonic oscillator
(32:35) Mathematica
(35:26) The beat phenomenon (a.k.a. "beating phenomenon")
(37:56) Complex algebra approach
(42:12) Period and frequency of the sinusoidal factors near resonance (slow oscillations (“beats”) and fast oscillation)
(44:57) Mathematica

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[georgesadler]

Professor Kinney, thank you for an exceptional explanation of the Sinusoidally Forced Damped and Undamped Harmonic Oscillator, Complexification and Beat Phenomenon graphically and analytically. Drawing a picture with the aid of Mathematica. always help with understanding the material. I have seen this material in Engineering Physics and Systems/ Circuits I and II. The explanation and analysis are similar in all classes/lectures.