{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}


lecture_6_revised - Lecture 6 Harmonically forced...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 6

lecture_6_revised - Lecture 6 Harmonically forced...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online