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HW_5_090402

# HW_5_090402 - Adv EE II Introduction to Electronics Spring...

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Adv. EE II – Introduction to Electronics Course 300212 Spring term 2009 M. Bode Date: Apr. 02, 2009 Due: Apr. 16, 2009, in class Homework 5 This HW is about a simple analogue to atoms arranged in a crystal. The goal is to understand the basic mechanisms behind energy bands. As usual, do not forget to provide proper arguments for your conclusions. Task 1 (60) Band Structure (open ends) Consider a simple harmonic pendulum without damping: This describes the oscillating pendulum where x denotes the deviation of the pendulum from its equilibrium position as a function of time, and is its intrinsic or eigenfrequency. a) (10) Re-write the second order ode above as a pair of first order odes. To this end, introduce a new dynamical variable ( ) ( ) 2 t x t x 0 ω = ± ± 0 ω v ± = . Then find equations for and v in terms of x and v : b) (10) The equations you found should be linear ones. So you may write them in matrix notation: Find the corresponding matrix M and its eigenvalues x ± ± ( ) ( ) v x g v v x f x , , = = ± ± = v x M v x ± ± 1 λ and 2 λ . What is the unit of such an eigenvalue? How are the eigenvalues related to the eigenfrequency of the pendulum? ( ) ( ) t x t

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Adv. EE II – Introduction to Electronics Course 300212 Spring term 2009 M. Bode Date: Apr. 02, 2009 Due: Apr. 16, 2009, in class c) (20) Now, consider a pair of identically built pendulums that are coupled by means of an elastic springs: As above, introduce velocities v ( ) ( ) ( ) 2 2 ( ) ( ) ( ) 2 1 2 2 0 2 1 2 1 0 1 x x k t x t x x x k t x t x + = +
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