resonance

resonance - MA366 Sathaye Notes on resonant frequency We...

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Unformatted text preview: MA366 Sathaye Notes on resonant frequency We explain the calculations of the phenomenon of resonance. Compare with the pages 208-209 in the book. The aim is to show all the steps that the book leaves out. We also follow a different strategy of calculations which is more efficient. 1. We consider a solution of the equation for a spring-mass system: ( mD 2 + γD + k ) u = F cos( wt ) . (1) A guess for the particular solution u p is u p = R cos( wt- δ ) with R,δ to be determined. 2. We substitute in the LHS of the equation (1) and note: ( mD 2 + γD + k ) R cos( wt- δ ) = (- mw 2 + γD + k ) R cos( wt- δ ) = ( k- mw 2 ) R cos( wt- δ )- γR sin( wt- δ ) . (2) Note that we have used the fact that D 2 cos( at + b ) =- a 2 cos( at + b ) which is well known - or easy to verify! 3. Thus the substitution in LHS of equation (1) can be simplified as: ( k- mw 2 ) R cos( wt- δ )- γwR sin( wt- δ ) = A cos( wt- δ + θ ) (3) where we use the usual change of polar coordinates, taking advantage of the minus sign in the...
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This note was uploaded on 01/18/2012 for the course MATH 366 taught by Professor Edraygoins during the Fall '09 term at Purdue.

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resonance - MA366 Sathaye Notes on resonant frequency We...

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