Unformatted text preview: • This is the general equation of motion for a forced simple harmonic oscillator • This is a complicated as we’ll make it in this course. All other cases we considered up to now were simplifications ( A = 0 Free SHO, γ = 0 undamped SHO, etc.) • We now have a secondorder, linear, nonhomogeneous, ordinary, differential equation • From the general theory of O.D.E. soultions (eg. Boyce & Diprima secs. 3.6 and 3.9) ◦ Look for a solution of the form: X ( t ) = C 1 x 1 ( t ) + C 2 x 2 ( t ) + X ( t ) ◦ The first two terms are the complementary solutions of the homogeneous equation (ie. undriven SHO studied up to now) ◦ These solutions are transient (oscillations will die away as energy is dissipated) • The third term is more interesting. It describes the forced response of the system • After the transient solutions have died away (like e = γt/ 2 ) this will be the only steadystate part of the solution • Since our forcing function has cos( ωt ) time dependence “try” a solution like:...
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This note was uploaded on 07/10/2011 for the course PHY 293 taught by Professor Pierresavaria during the Fall '07 term at University of Toronto.
 Fall '07
 PierreSavaria
 mechanics, Force, Statistical Mechanics

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