phy293_l04.page2

# phy293_l04.page2 - This is the general equation of motion...

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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 = 0 Free SHO, γ = 0 undamped SHO, etc.) We now have a second-order, linear, non-homogeneous, 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. un-driven 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 steady-state part of the solution Since our forcing function has cos( ωt ) time dependence “try” a solution like: X ( t ) = G cos( ωt ) + H sin( ωt )
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