66 copyright c 2000 siam buy online from siam

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ially displaced from its equilibrium position by a small vertical distance—say bead k is displaced by an amount ck at t = 0. The beads are then released so that they can vibrate freely. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 560 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 m m L Equilibrium Position A Typical Initial Position It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Figure 7.6.1 Problem: For small vibrations, determine the position of each bead at time t > 0 for any given initial configuration. D E Solution: The small vibration hypothesis validates the following assumptions. • The tension T remains constant for all time. • There is only vertical motion (the horizontal forces cancel each other). • Only small angles are involved, so the approximation sin θ ≈ tan θ is valid. T H Let yk (t) = yk be the vertical distance of the k th bead from equilibrium at time t, and set y0 = 0 = yn+1 . IG y k +1 yk θk R Y θk–1 yk–1 k–1 P k θ k +1 k+1 Figure 7.6.2 If θk is the angle depicted in Figure 7.6.2, the diagram above, then the upward force on the k th bead at time t is Fu = T sin θk , while the downward force is Fd = T sin θk−1 , so the total force on the k th bead at time t is O C F = Fu − Fd = T (sin θk − sin θk−1 ) ≈ T (tan θk − tan θk−1 ) =T yk+1 − yk yk − yk−1 − L L = T (yk−1 − 2yk + yk+1 ). L Newton’s second law says force = mass × acceleration, so we set myk = T T (yk−1 − 2yk + yk+1 ) =⇒ yk + (−yk−1 + 2yk − yk+1 ) = 0 (7.6.1) L mL together with yk (0) = ck and yk (0) = 0 to model the motion of the k th bead. Altogether, equations (7.6.1) represent a system of n second-order linear differential equations, and each is coupled to its neighbors so that no single Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot7...
View Full Document

Ask a homework question - tutors are online