Given that the eigenvalue for matrix [A] in equation (2) is A= -cos(Bn). i) Derive the characteristic equation of matrix [A] in equation (2). Show all your working when deriving the characteristic equation. Is it possible to solve for Bn analytically? If yes, solve it. If no, suggest another method to use to solve for Bn. ii)
Given that the eigenvalue for matrix [A] in equation (2) is A= -cos(Bn). i) Derive the characteristic equation of matrix [A] in equation (2). Show all your working when deriving the characteristic equation. Is it possible to solve for Bn analytically? If yes, solve it. If no, suggest another method to use to solve for Bn. ii)
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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![1. Given that the eigenvalue for matrix [A] in equation (2) is A= -cos(ßn).
i)
Derive the characteristic equation of matrix [A] in equation (2). Show all your
working when deriving the characteristic equation.
ii)
Is it possible to solve for Bn analytically? If yes, solve it. If no, suggest another
method to use to solve for Bn.
2. Based on the characteristic equation derived from Question 1,
i)
Suggest four methods you can use to solve for Bn numerically. Among the four
methods you have suggested, are there any methods that may be less favorable
to use? If yes, explain why.
Select a numerical method to solve for Bn and explain in detail and in your own
words, all the steps required to implement the chosen method for this problem.
ii)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57f33c8d-a162-41ea-8ce6-4d8ea1771ff0%2F35a91e11-c9db-4968-877c-06c422c09b58%2F07h2ugo_processed.png&w=3840&q=75)
Transcribed Image Text:1. Given that the eigenvalue for matrix [A] in equation (2) is A= -cos(ßn).
i)
Derive the characteristic equation of matrix [A] in equation (2). Show all your
working when deriving the characteristic equation.
ii)
Is it possible to solve for Bn analytically? If yes, solve it. If no, suggest another
method to use to solve for Bn.
2. Based on the characteristic equation derived from Question 1,
i)
Suggest four methods you can use to solve for Bn numerically. Among the four
methods you have suggested, are there any methods that may be less favorable
to use? If yes, explain why.
Select a numerical method to solve for Bn and explain in detail and in your own
words, all the steps required to implement the chosen method for this problem.
ii)
![The determination of the fundamental natural frequency for cantilever beams is crucial in many
engineering applications. Here, cantilever beam refers to beam that is fixed on one end and free
on the other end as shown in Figure 1 below.
Figure 1: Cantilever Beam.
Generally, a cantilever beam would have and an infinite number of natural frequencies
corresponding to its vibration modes. However, the fundamental natural frequency refers to the
natural frequency of the beam under its first vibration mode. For thin beams, the Euler-Bernoulli
beam theory can be used to determine this natural frequency.
The fundamental natural frequency of a cantilever beam is determined by the following equation:
Et2
| 12PL*
(1)
where wi is the fundamental natural frequency of the beam in Hz, E is the beam Young's
modulus, t is the thickness of the beam, p is the beam's density, L is the length of the beam and
ß1 is the fundamental eigenfrequency of the beam which can be determined as the smallest
absolute value from the characteristic equation of the following matrix in equation (2). In
equation (2), n correspond to the vibration mode number.
[A] = [
cosh(Bn)
sinh(Bn) + sin(Bn) ]
(2)
sinh(Bn) – sin(Bn)
cosh(Bn)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57f33c8d-a162-41ea-8ce6-4d8ea1771ff0%2F35a91e11-c9db-4968-877c-06c422c09b58%2Fhjwwrrq_processed.png&w=3840&q=75)
Transcribed Image Text:The determination of the fundamental natural frequency for cantilever beams is crucial in many
engineering applications. Here, cantilever beam refers to beam that is fixed on one end and free
on the other end as shown in Figure 1 below.
Figure 1: Cantilever Beam.
Generally, a cantilever beam would have and an infinite number of natural frequencies
corresponding to its vibration modes. However, the fundamental natural frequency refers to the
natural frequency of the beam under its first vibration mode. For thin beams, the Euler-Bernoulli
beam theory can be used to determine this natural frequency.
The fundamental natural frequency of a cantilever beam is determined by the following equation:
Et2
| 12PL*
(1)
where wi is the fundamental natural frequency of the beam in Hz, E is the beam Young's
modulus, t is the thickness of the beam, p is the beam's density, L is the length of the beam and
ß1 is the fundamental eigenfrequency of the beam which can be determined as the smallest
absolute value from the characteristic equation of the following matrix in equation (2). In
equation (2), n correspond to the vibration mode number.
[A] = [
cosh(Bn)
sinh(Bn) + sin(Bn) ]
(2)
sinh(Bn) – sin(Bn)
cosh(Bn)
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