Understanding Harmonic Oscillators: Overview and Applications
Explore the world of harmonic oscillators with an in-depth look at simple harmonic motion, driven oscillators, resonance, damping effects, and different cases such as underdamped, overdamped, and critically damped. Discover the interplay between spring and mass, general forms, and solutions for driven harmonic oscillators.
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Presentation Transcript
Harmonic Oscillators Will Bergman and Mike Ma
Overview Simple Harmonic Motion Driven Simple Harmonic Oscillators Spring and Mass Periodic Driving Force General Form Resonance Damped Simple Harmonic Oscillators Tacoma Bridge Example Conclusion and Further Applications Underdamped Case Overdamped Case Critically Damped Case 2
Spring and Mass ? = ?? ??2? ??2= ?? Guess: ? ? = ??? ? ?,?2= ? ? ? ?1= +? https://commons.wikimedia.org/wiki/File:Simple_ harmonic_oscillator.gif ? ? = ?1?1? + ?2?2? ? ? = ?1??1?+ ?2??1? (Taylor, 2003) 3
General Form ??2? ??2= ?? ? ? = ?1+ ?2??? ?? + ? ?1 ?2??? ?? ? ? = ?1??? ?? + ?2??? ?? (Taylor, 2003) 4
Damped Simple Harmonic Oscillators ?2? or ?2? ??2= ??? ??2+ ??? ?? ?02? ??+ ?02? B damping constant ?0 -natural frequency Relationship between B and ?0determine different cases of damping Solution form: ? ? = ??? 5
Underdamped Case (? < ?0) ?2 ?02= ? ?02 ?2= ??1 ? ? = ? ??(?1???1?+ ?2? ??1?) Amplitude of oscillations decrease exponentially (Taylor, 2003) 6
Overdamped Case (? > ?0) ?2 ?02?+ ?2? ?2 ?02?) ? ? = ? ??(?1? No Oscillations! (Taylor, 2003) 7
Critically Damped Case (? = ?0) Repeated Eigenvalues ? = ?0is a bifurcation value ? ? = ? ?? ?(?) = ?? ?? ? ? = ?1? ??+ ?2?? ?? (Taylor, 2003) (Blanchard et al., 2012) 8
Driven Simple Harmonic Oscillators ?2? ??2+ ??? ??+ ?02? = ?(?) Solution = general solution of homogeneous equation (unforced) + one particular solution to nonhomogeneous equation (forced) ? ? = ?1?1? + ?2?2? + ??(?) Resonance- the frequency of the driving force is equal to the natural frequency of the oscillating system http://www.acs.psu.edu/drussell/Demos/SHO/m ass-force.html 9
Conclusion and Further Applications of Theory https://www.youtube.com/watch?v=vPZuHFrawz4 https://www.youtube.com/watch?v=3mclp9QmCGs 10
References https://commons.wikimedia.org/wiki/File:Simple_harmonic_oscillator.gif Taylor, John R. "Chapter 5: Oscillations." Classical Mechanics. Sausalito, CA: U Science, 2005. 161-203. Print. Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. "Chapter 2.3: The Damped Harmonic Oscillator." Differential Equations. Boston, MA: Brooks/Cole, Cengage Learning, 2012. 183-88. Print. http://www.acs.psu.edu/drussell/Demos/SHO/mass-force.html 11
