lecture_6_revised - Lecture 6 Harmonically forced...

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Lecture 6 Harmonically forced vibrations In this section we study one of the most important phenomena in sci- ence and technology: that or resonance. If it was not for resonance, you would not have stadium rock concerts (might be for the better, I prefer smaller venues myself). Resonance can also cause a lot of trou- ble, as you could see in the Millenium bridge video shown in class, and of course the Tacoma bridge disaster that I mentioned in the introduc- tory lecture. I will explain the physics of the Tacoma bridge disaster towards the end of this lecture. Bit first, let us derive the equation of motion for the system shown in figure (1). Figure 1: A mass-spring-damper system acted upon by a time-dependent force F ( t ). Let us consider the case when the external force is harmonic, with zero phase. Remark 1 There is no loss of generality when we do this: remember that harmonic force can be written as F ( t ) = F 0 sin( ωt + ϕ ) , for some constant phase φ and constant force amplitude F 0 . If we now let τ = t + ϕ/ω , then the harmonic force becomes F ( τ ) = F 0 sin( ωτ ) , and since = dt , we have dx/dt = dx/dτ and d 2 /dt 2 = d 2 /dτ 2 . What we have done with this transformation is equivalent to looking at the process at an initial time t 0 = ϕ/ω instead of t = 0 . 1
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Taking into account the spring, damping and external harmonic forc- ing, the equation of motion reads m ¨ x + c ˙ x + kx = F 0 sin( ωt ) (1) From lecture 4, we know the solution to the homogeneous part m ¨ x + c ˙ x + kx = 0 , is x h ( t ) = x ( t ) = e - c 2 m t sin( ω d t + φ ) , and that solution decays exponentially fast in time to equilibrium x h = 0.
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