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2280 12:55pm Lectures Week 7 S2018

Last Modified: February 23, 2018, 12:38 MST.    Today: March 24, 2018, 23:04 MDT.
  Sections 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.

     Pure 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
PDF: 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.
PDF: Beats y=sin(10x)sin(x/2) (68.9 K, pdf, 18 Mar 2013) Theory of Practical Resonance
Slides: 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. The practical resonance frequency is the number omega defined by the above square root expression. Circuits EPbvp3.7 and Electrical resonance Derivation 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 Experiment The 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
Slides: 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)
References from week 6

 Chapter 3 references. Sections 3.4, 3.5, 3.6. Forced/Unforced oscillations.
Slides: 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)
Slide: Drawing for Exercise 3.4-34 (26.0 K, pdf, 23 Feb 2018)

Resonance Videos

Projection: glass-breaking video. Wine glass experiment. Tacoma narrows.
Video: 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)