Unformatted text preview: m d 2 x dt 2 + kx = A cos( ω 1 t ) , where m is the mass, k is the spring’s constant (cfr Hooke’s law). The external force is periodic with amplitude A > 0 and frequency ω 1 / 2 π . If there is no external forcing ( A = 0), we get a harmonic oscillator with natural frequency ω / 2 π , where ω = r k/m . Show that the general solution y g ( t ) is bounded if ω 1 n = ω (a function f ( t ) is called bounded if there is some positive constant K so that  f ( t )  < K for all t .), but that y g ( t ) is not bounded if ω 1 = ω . This last case is a typical example of what is known as a resonance phenomenon . Such phenomena may occur when a frequency of an external forcing term matches a natural frequency of the unforced system. 5. Find the general solution to the system dx dt = x + y dy dt = 2 x + y * MAP 2302; Instructor: Patrick De Leenheer. 1...
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 Summer '06
 DeLeenheer
 Solar System, Mass, general solution, scaled version

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