Dimensional analysis can be used in problems other than áuid mechanics ones. The important variablesaffecting the period of a vibrating beam (usually designated as T and with dimensions of time) are the beamlength `, area moment of inertia I, modulus of elasticity E, material density , and Poissonís ratio , so thatT = f cn(`; I; E; ; )Recall that the modulus of elasticity has typical units of N/m2 and Poissonís ratio is dimensionless.(a) Find dimensionless version of the functional relationship.(b) If E and I must always appear together (meaning that EI is e§ectively a single variable), Önd a dimensionless version of the functional relationship. **Dimensional Analysis in Beam Vibrations**Dimensional analysis can be used in problems other than fluid mechanics ones. The important variables affecting the period of a vibrating beam (usually designated as \(T\) and with dimensions of time) are the beam length \(l\), area moment of inertia \(I\), modulus of elasticity \(E\), material density \(\rho\), and Poisson’s ratio \(\sigma\), so that\[ T = f(l, I, E, \rho, \sigma) \]Recall that the modulus of elasticity has typical units of \( \text{N/m}^2 \) and Poisson’s ratio is dimensionless.- **(a)** Find the dimensionless version of the functional relationship.- **(b)** If \(E\) and \(I\) must always appear together (meaning that \(EI\) is effectively a single variable), find a dimensionless version of the functional relationship.

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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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Dimensional analysis can be used in problems other than áuid mechanics ones. The important variables
affecting the period of a vibrating beam (usually designated as T and with dimensions of time) are the beam
length `, area moment of inertia I, modulus of elasticity E, material density , and Poissonís ratio , so that
T = f cn(`; I; E; ; )
Recall that the modulus of elasticity has typical units of N/m2 and Poissonís ratio is dimensionless.
(a) Find dimensionless version of the functional relationship.
(b) If E and I must always appear together (meaning that EI is e§ectively a single variable), Önd a dimensionless version of the functional relationship.

**Dimensional Analysis in Beam Vibrations**

Dimensional analysis can be used in problems other than fluid mechanics ones. The important variables affecting the period of a vibrating beam (usually designated as \(T\) and with dimensions of time) are the beam length \(l\), area moment of inertia \(I\), modulus of elasticity \(E\), material density \(\rho\), and Poisson’s ratio \(\sigma\), so that

\[ T = f(l, I, E, \rho, \sigma) \]

Recall that the modulus of elasticity has typical units of \( \text{N/m}^2 \) and Poisson’s ratio is dimensionless.

- **(a)** Find the dimensionless version of the functional relationship.
- **(b)** If \(E\) and \(I\) must always appear together (meaning that \(EI\) is effectively a single variable), find a dimensionless version of the functional relationship.

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