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 yourworking when deriving the characteristic equation.ii)Is it possible to solve for Bn analytically? If yes, solve it. If no, suggest anothermethod 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 fourmethods you have suggested, are there any methods that may be less favorableto use? If yes, explain why.Select a numerical method to solve for Bn and explain in detail and in your ownwords, 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 manyengineering applications. Here, cantilever beam refers to beam that is fixed on one end and freeon 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 frequenciescorresponding to its vibration modes. However, the fundamental natural frequency refers to thenatural frequency of the beam under its first vibration mode. For thin beams, the Euler-Bernoullibeam 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'smodulus, t is the thickness of the beam, p is the beam's density, L is the length of the beam andß1 is the fundamental eigenfrequency of the beam which can be determined as the smallestabsolute value from the characteristic equation of the following matrix in equation (2). Inequation (2), n correspond to the vibration mode number.[A] = [cosh(Bn)sinh(Bn) + sin(Bn) ](2)sinh(Bn) – sin(Bn)cosh(Bn)

“Since you have asked multiple questions, we will solve the first question for you. If you want any…

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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Question

100%

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)

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