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The equations of motion of three equal masses connected by springs of equal stiffness are i = -2x+y ÿ=x-2y+z ž=y-2z Show that for normal modes of oscillation x = Xcos cot, y = Y cos cot, z = Z cos cot to exist then the condition on λ = ² is 2-2 1 0 1 2-2 1 = 0 0 1 1 2-2 Find the three values of that satisfy this condition, and find the ratios X: Y: Z in each case.

The equations of motion of three equal masses connected by springs of equal stiffness are i = -2x+y ÿ=x-2y+z ž=y-2z Show that for normal modes of oscillation x = Xcos cot, y = Y cos cot, z = Z cos cot to exist then the condition on λ = ² is 2-2 1 0 1 2-2 1 = 0 0 1 1 2-2 Find the three values of that satisfy this condition, and find the ratios X: Y: Z in each case.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The equations of motion of three equal masses connected by springs of equal stiffness are *=-2x+y ÿ = x-2y+z ž=y - 2z Show that for normal modes of oscillation x = X cos@ot, y = Y cos cot, z = Zcos cot to exist then the condition on λ = @o² is 2-2 0 1 1 = 0 0 1 1 2-2 Find the three values of λ that satisfy this condition, and find the ratios X: Y: Z in each case. F λ-2
Transcribed Image Text:The equations of motion of three equal masses connected by springs of equal stiffness are *=-2x+y ÿ = x-2y+z ž=y - 2z Show that for normal modes of oscillation x = X cos@ot, y = Y cos cot, z = Zcos cot to exist then the condition on λ = @o² is 2-2 0 1 1 = 0 0 1 1 2-2 Find the three values of λ that satisfy this condition, and find the ratios X: Y: Z in each case. F λ-2
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