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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 208209 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 springmass 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.
 Fall '09
 EdrayGoins

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