TopicsSections 7.1 to 7.6 The textbook topics, definitions, examples and theorems

Edwards-Penney Ch 3, 7.1 to 7.5 (21.0 K, txt, 22 Feb 2015)

## Tuesday: Resonance, Section 3.6. Circuits, Section 3.7.

ApplicationsPure Resonance x''+x=cos(t), frequency matching Solution explosion, unbounded solution x=(1/2) t sin t. Practical Resonance: x'' + x = cos(omega t) with omega near 1 Large amplitude harmonic oscillations: Pure resonance y = x sin(x) (74.7 K, pdf, 18 Mar 2013) Resonance examples: Soldiers marching in cadence, Tacoma narrows bridge, Wine Glass Experiment. Theodore Von Karman and vortex shedding. Cable model of the Tacoma bridge, year 2000. Resonance explanations. Millenium Foot-Bridge London Beats x''+x=cos(2t) Graphics for beats [x=sin(10 t)sin(t/2)], slowly-oscillating envelope, rapidly oscillating harmonic with time-varying amplitude.: Beats y=sin(10x)sin(x/2) (68.9 K, pdf, 18 Mar 2013)Theory of Practical Resonance: Forced vibrations and resonance (253.0 K, pdf, 08 Mar 2014) The equation is mx''+cx'+kx=F_0 cos(omega t) THEOREM. The limit of x_h(t) is zero at t=infinity THEOREM. x_p(t) = C(omega) cos(omega t - phi) C(omega) = F_0/Z, Z^2 = A^2+B^2, A and B are the undetermined coefficient answers for trial solution x(t) = A cos(omega t) + B sin(omega t). THEOREM. The output x(t) = x_h(t) + x_p(t) is graphically just x_p(t) = C(omega) cos(omega t - phi) for large t. Therefore, x_p(t) is the OBSERVABLE output. THEOREM. The amplitude C(omega) is maximized over all possible input frequencies omega>0 by the single choice omega = sqrt(k/m - c^2/(2m^2)). DEFINITION. TheSlidespractical resonance frequencyis the number omega defined by the above square root expression.Circuits EPbvp3.7 and Electrical resonanceDerivation from mechanical problems 5.6. THEOREM: omega = 1/sqrt(LC). REVIEW Impedance, reactance. Steady-state current amplitude Transfer function. Input and output equation..Wine Glass ExperimentThe lab table setup Speaker. Frequency generator with adjustment knob. Amplifier with volume knob. Wine glass. x(t)=deflection from equilibrium of the radial component of the glass rim, represented in polar coordinates, orthogonal to the speaker front. mx'' + cx' + kx = F_0 cos(omega t) The model of the wine glass m,c,k are properties of the glass sample itself F_0 = volume knob adjustment omega = frequency generator knob adjustment: Basic undetermined coefficients (147.6 K, pdf, 14 Feb 2018)Slides: Variation of parameters (164.5 K, pdf, 03 Mar 2012)Slides: Forced vibrations and resonance, Millenium Bridge, Wine Glass, Tacoma Narrows (253.0 K, pdf, 08 Mar 2014)Slides: Resonance and undetermined coefficients, cafe door, pet door, phase-amplitude (178.0 K, pdf, 08 Mar 2014)Slides: Electrical circuits (112.8 K, pdf, 19 Feb 2016)SlidesReferences from week 6Chapter 3 references. Sections 3.4, 3.5, 3.6. Forced/Unforced oscillations.: Unforced vibrations 2008 (647.6 K, pdf, 27 Feb 2014)Slides: Forced damped vibrations (263.9 K, pdf, 10 Feb 2016)Slides: Forced undamped vibrations (214.2 K, pdf, 03 Mar 2012)Slides: phase-amplitude, cafe door, pet door, damping classification (136.0 K, pdf, 08 Mar 2014)Slides: Drawing for Exercise 3.4-34 (26.0 K, pdf, 23 Feb 2018)Slide

## Resonance Videos

Projection: glass-breaking video. Wine glass experiment. Tacoma narrows.: Wine glass breakage (avi) (260.5 K, avi, 18 Feb 2015) 2015 Video: Glass breakage in slow motion, MIT 2009 Video: Glass breakage in slow motion, MIT (same video)Video: Same 2009 Glass Breakage, local copy (12992.3 K, mp4, 16 Feb 2016)Video: Wine glass experiment (12mb mpg, 2min) (12493.8 K, mpg, 01 Apr 2008)Video: Tacoma Narrows Bridge Nov 7, 1940 (18mb mpg, 4min) (18185.8 K, mpg, 01 Apr 2008)Video