Eigenfunctions for a Uniform Free Beam - Revision history

Adi Kurniawan: /* Anti-symmetric modes */

2009-12-08T16:12:25Z

Anti-symmetric modes

← Older revision		Revision as of 16:12, 8 December 2009
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	</center>		</center>

	By imposing boundary conditions at <math>x = l</math> :		By imposing boundary conditions at <math>x = L</math> :

	<center>		<center>

Adi Kurniawan: /* Modes for \lambda = 0 */

2009-12-08T16:11:43Z

Modes for \lambda = 0

← Older revision		Revision as of 16:11, 8 December 2009
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	== Modes for <math>\lambda = 0</math> ==		== Modes for <math>\lambda = 0</math> ==

	There are two modes for <math>\lambda = 0</math> which are the two rigid body motions they are given by		There are two modes for <math>\lambda = 0</math> which are the two rigid body motions; they are given by
	{{rigid modes for an elastic plate}}		{{rigid modes for an elastic plate}}

Meylan at 09:19, 20 October 2009

2009-10-20T09:19:58Z

← Older revision		Revision as of 09:19, 20 October 2009
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			{{complete pages}}

	== Introduction ==		== Introduction ==

Meylan: /* Modes for $\lambda = 0$ */

2009-09-17T22:02:27Z

Modes for <math>\lambda = 0</math>

← Older revision		Revision as of 22:02, 17 September 2009
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	There are two modes for <math>\lambda = 0</math> which are the two rigid body motions they are given by		There are two modes for <math>\lambda = 0</math> which are the two rigid body motions they are given by
	~~<center>~~		{{rigid modes for an elastic plate}}
	~~<math>~~
	~~X_0 = \frac~~{~~1}{\sqrt{2L}}~~
	~~</math>~~
	~~</center>~~
	~~and~~
	~~<center>~~
	~~<math>~~
	~~X_1 = \sqrt{\frac{3}~~{~~2L^3~~}} x
	~~</math>~~
	~~</center>~~

	== Symmetric modes ==		== Symmetric modes ==

Meylan: /* Natural Frequencies */

2009-04-08T10:47:56Z

Natural Frequencies

← Older revision		Revision as of 10:47, 8 April 2009
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	{{solution for a uniform beam in eigenfunctions}}		{{solution for a uniform beam in eigenfunctions}}

	~~== Natural Frequencies ==~~

	~~The equation in the time domain for a beam is~~
	~~<center>~~
	~~<math>~~
	~~m\partial_t^2 w + EI \partial_x^4 w = 0~~
	~~</math>~~
	~~</center>~~
	~~so that, having obtained eigenvalues <math>\lambda_n</math>, the natural frequency can be readily calculated :~~
	~~<center>~~
	~~<math>\omega_n = \lambda_n^2 \sqrt\frac{EI}{m}</math>~~
	~~</center>~~

	== Matlab Code ==		== Matlab Code ==

Meylan: /* Modes for $\lambda = 0$ */

2009-04-08T10:47:29Z

Modes for <math>\lambda = 0</math>

← Older revision		Revision as of 10:47, 8 April 2009
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	<center>		<center>
	<math>		<math>
	~~w_0~~ = \frac{1}{\sqrt{2L}}		X_0 = \frac{1}{\sqrt{2L}}
	</math>		</math>
	</center>		</center>
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	<center>		<center>
	<math>		<math>
	~~w_1~~ = \sqrt{\frac{3}{2L^3}} x		X_1 = \sqrt{\frac{3}{2L^3}} x
	</math>		</math>
	</center>		</center>

Meylan: /* Anti-symmetric modes */

2009-04-08T10:41:17Z

Anti-symmetric modes

← Older revision		Revision as of 10:41, 8 April 2009
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	C_3\\		C_3\\
	\end{bmatrix}		\end{bmatrix}

	=		=

	\begin{bmatrix}		\begin{bmatrix}
	0\\		0\\
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	Anti-symmetric natural modes can be written in normalized form as :		Anti-symmetric natural modes can be written in normalized form as :
	<center>		<center>
	<math>w_{2n+1}(x) = \frac{1}{\sqrt{2L}}\left( \frac{\sin(\lambda_{2n+1} x)}{\sin(\lambda_{2n+1} L)}+\frac{\sinh(\lambda_{2n+1} x)}{\sinh(\lambda_{2n+1} L)} \right )		<math>X_{2n+1}(x) = \frac{1}{\sqrt{2L}}\left( \frac{\sin(\lambda_{2n+1} x)}{\sin(\lambda_{2n+1} L)}+\frac{\sinh(\lambda_{2n+1} x)}{\sinh(\lambda_{2n+1} L)} \right )
	\,\,\,n\geq 1		\,\,\,n\geq 1
	</math>		</math>

Meylan: /* Symmetric modes */

2009-04-08T10:40:51Z

Symmetric modes

← Older revision		Revision as of 10:40, 8 April 2009
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	Symmetric natural modes can be written in normalized form as :		Symmetric natural modes can be written in normalized form as :
	<center>		<center>
	<math>w_{2n}(x) = \frac{1}{\sqrt{2L}}\left( \frac{\cos(\lambda_{2n} x)}{\cos(\lambda_{2n} L)}+\frac{\cosh(\lambda_{2n} x)}{\cosh(\lambda_{2n} L)} \right )		<math>X_{2n}(x) = \frac{1}{\sqrt{2L}}\left( \frac{\cos(\lambda_{2n} x)}{\cos(\lambda_{2n} L)}+\frac{\cosh(\lambda_{2n} x)}{\cosh(\lambda_{2n} L)} \right )
	\,\,\,n\geq 1		\,\,\,n\geq 1
	</math>		</math>

Meylan: /* Solution */

2009-04-08T10:40:30Z

Solution

← Older revision		Revision as of 10:40, 8 April 2009
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	General solution of the differential equation for <math>\lambda \neq 0</math> is		General solution of the differential equation for <math>\lambda \neq 0</math> is
	<center>		<center>
	<math>~~w_n~~(x) = C_1 \sin(\lambda_n x) + C_2 \cos(\lambda_n x) + C_3 \sinh(\lambda_n x) + C_4 \cosh(\lambda_n x)\,</math>		<math>X_n(x) = C_1 \sin(\lambda_n x) + C_2 \cos(\lambda_n x) + C_3 \sinh(\lambda_n x) + C_4 \cosh(\lambda_n x)\,</math>
	</center>		</center>
	Due to symmetry of the problem, dry natural vibrations of a free beam can be split into two different sets of modes, symmetric (even) modes and skew-symmetric (odd) modes.		Due to symmetry of the problem, dry natural vibrations of a free beam can be split into two different sets of modes, symmetric (even) modes and skew-symmetric (odd) modes.

Meylan: /* Solution for a uniform beam in eigenfunctions */

2009-04-07T08:22:24Z

Solution for a uniform beam in eigenfunctions

← Older revision		Revision as of 08:22, 7 April 2009
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	</center>		</center>
	where the eigenfunctions have been chosen so that their inner products equal the Kronecker delta.		where the eigenfunctions have been chosen so that their inner products equal the Kronecker delta.

			== Equations for a beam ==
			{{equations for a beam}}

	== Solution for a uniform beam in [[Eigenfunctions for a Uniform Free Beam\|eigenfunctions]] ==		== Solution for a uniform beam in [[Eigenfunctions for a Uniform Free Beam\|eigenfunctions]] ==

Eigenfunctions for a Uniform Free Beam - Revision history

Adi Kurniawan: /* Anti-symmetric modes */

Adi Kurniawan: /* Modes for \lambda = 0 */

Meylan at 09:19, 20 October 2009

Meylan: /* Modes for \lambda = 0 */

Meylan: /* Natural Frequencies */

Meylan: /* Modes for \lambda = 0 */

Meylan: /* Anti-symmetric modes */

Meylan: /* Symmetric modes */

Meylan: /* Solution */

Meylan: /* Solution for a uniform beam in eigenfunctions */

Meylan: /* Modes for $\lambda = 0$ */

Meylan: /* Modes for $\lambda = 0$ */