CA-AA242B-Hw3

CA-AA242B-Hw3 - To exhibit this phenomenon, consider igure...

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AA 242B: Mechanical Vibrations (Spring 2010) Homework #3 Due May 11, 2010 1 Large rotation Consider a two-dimensional rectangular structure of arbitrary dimensions. Ro- tate this structure rigidly by an angle θ around an axis perpendicular to its plane and passing through the midpoint of one of its two short sides. 1. Compute the linear Green symmetric strain tensor at any point of this structure. What happens in this case if θ is: a) small, b) large? 2. Compute the “full” Green symmetric strain tensor at any point of this structure. What happens in this case if θ is: a) small, b) large? 2 Wave refections at interFaces The presence of boundary conditions or material discontinuities perturbs sim- ple waves and causes reFections or refractions. The three-dimensional nature of the problem also incudes another di±culty. Depending on the incident angle, a simple longitudinal or transverse wave which reFects on a free surface is trans- formed into a linear combination of a longitudinal wave and a transverse wave.
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Unformatted text preview: To exhibit this phenomenon, consider igure 1 where a harmonic plane wave ~u = u I i c L e i c L ( ~n ~x-c L t ) ~n ~n = sin ~ e 1 + cos ~ e 2 impinges on the free surface x 2 = 0 at an angle of incidence . 1. Derive the expression of the reFected wave in the form of a linear combi-nation of a longitudinal wave ~u L = u L i c L e i c L ( ~n L ~x-c L t ) ~n L propagating in the direction of ~n L and a transverse wave ~u = u T i c T e i c T ( ~n T ~x-c T t ) ~ e 3 ~n T propagating in the direction of ~n T , where ~n L = sin L ~ e 1-cos L ~ e 2 and ~n T = sin T ~ e 1-cos T ~ e 2 that is, nd L , T , u L and u T as functions of the given data. 1 Figure 1: Interface and incident wave. 2. Plot u L u I and u T u I as functions of for = 1 4 . 3. Explain why the change in type of the incident wave is a three-dimensional eect. 4. Report any other interesting observation. 2...
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This note was uploaded on 06/17/2010 for the course AA 242B taught by Professor Charbelfarhat during the Spring '10 term at Stanford.

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CA-AA242B-Hw3 - To exhibit this phenomenon, consider igure...

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