An aluminum rod of length L = 1m has mass density ρ = 2700 kgm3 andYoung’s modulus E = 70GPa. The rod is fixed at both ends. The exactnatural eigenfrequencies of the rod are ωexactn =πnLqEρfor n=1,2,3,. . . .1. What is the minimum number of linear elements necessary todetermine the fundamental frequency ω1 of the system? Discretizethe rod in that many elements of equal length, assemble the globalsystem of equations KU = ω2MU, and find the fundamentalfrequency ω1. Compute the relative error e1 = (ω1 − ωexact1)/ωexact1.Sketch the fundamental mode of vibration. An aluminum rod of length L = 1m has mass density p = 2700k andYoung's modulus E = 70 GPa. The rod is fixed at both ends. The exactπηnatural eigenfrequencies of the rod are exact = √ for n=1,2,3,..1. What is the minimum number of linear elements necessary todetermine the fundamental frequency w₁ of the system? Discretizethe rod in that many elements of equal length, assemble the globalsystem of equations KU = w²MU, and find the fundamentalfrequency w1. Compute the relative error e₁ = (w1 - wexact)/wexact.Sketch the fundamental mode of vibration.e1

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Answered: An aluminum rod of length L = 1m has…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

An aluminum rod of length L = 1m has mass density ρ = 2700 kg
m3 and
Young’s modulus E = 70GPa. The rod is fixed at both ends. The exact
natural eigenfrequencies of the rod are ω
exact
n =
πn
L
qE
ρ
for n=1,2,3,. . . .
1. What is the minimum number of linear elements necessary to
determine the fundamental frequency ω1 of the system? Discretize
the rod in that many elements of equal length, assemble the global
system of equations KU = ω
2MU, and find the fundamental
frequency ω1. Compute the relative error e1 = (ω1 − ω
exact
1
)/ωexact
1
.
Sketch the fundamental mode of vibration.