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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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