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.Is it possible to solve for Bn analytically? If yes, solve it. If no, suggest anothermethod to use to solve for Bn.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)

(1) (i) Given that, Where, is fundamental natural frequency is density is fundamental eigen…

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)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.4: Plane Curves And Parametric Equations

Problem 43E

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