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Unformatted text preview: AU OpenCourseWare Connors, Martin, Reading Notes on Vibrations and Waves. Athabasca, AB: Athabasca University, 2010. Please use the following citation format: Connors, Martin, Reading Notes on Vibrations and Waves. (Athabasca University: AU OpenCourseWare). (accessed [Date]). License: Creative Commons Attribution-NonCommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: PHYS 302 : Unit 4 Reading Notes – Athabasca University 1 PHYS 302: Vibrations and Waves – Unit 4 Reading Notes French, A. P. Vibrations and Waves . New York: W.W. Norton & Company, 1971. • Review Chapter 4: Forced Vibrations and Resonance (pp. 77–92). • Chapter 4: Forced Vibrations and Resonance (pp. 92–112). Chapter 4: Forced Vibrations and Resonance Transient Phenomena (p. 92) The important concept here (p. 92) is an initial transient stage, which is not represented in the steady state solutions we have sought so far. In the video lecture, Dr. Lewin refers to the solution as not having enough constants. In this section we will try to figure out what happened to those constants and to the transient solution. Eq. 4-16 takes Eq. 4-1 a step further by dividing by m and introducing m k = 2 ω . Recall that Eq. 4-1 arose from Newton’s second law (in the form ma=F ) applied to the driven mass on a spring: t F kx dt x d m ω cos 2 2 + − = . In the lecture, Dr. Lewin took damping into consideration, so the result for forced motion was more complex than Eq. 4-17. You should verify that if you put γ =0 into the solution ) cos( δ ω + = t A x used in the video lecture, you get back Eq. 4- 17 for undamped driven motion. It is easy to see that this form has an infinite response at ω=ω , where ω is the driving frequency and ω the natural frequency. Less obvious is the phase shift ongoing through resonance, where x is in phase with the driving below resonance, where 2 2 > − ω ω , and in antiphase above resonance, where 2 2 < − ω ω . The next paragraph discusses the missing constants and the non-physical results that are clear if Eq. 4-17 is assumed to work at time t=0. The resolution of this problem comes in introducing the transient solution. At the bottom page 93, you may recognize the homogeneous and inhomogeneous forms of the governing differential equation; the lecture briefly alludes to this. On page 94, Eq. 4-18 gives the general equation for the undamped forced mass on a spring. The text points out that the adjustable constants for initial conditions are now present and discusses what these constants are forced to be. In the undamped case, it is a bit misleading to refer to a transient since this “transient” never actually goes away....
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This note was uploaded on 02/18/2011 for the course PHYS 320 taught by Professor Martinconner during the Spring '10 term at Open Uni..

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