WikiWaves - User contributions [en]

Category:Linear Hydroelasticity

2008-11-14T01:45:36Z

Marko tomic:

Problems in Linear Water-Wave theory in which there is an elastic body.

[[Category:Linear Water-Wave Theory]]

Finite Element Method (FEM) can be used to analyze any general 3D elastic structure using linear hydroelastic theory.

<center>
local FE <math>\Rightarrow </math> global FE model
</center>

Dynamic equation of motion in matrix form can be expressed as:
<center>
<math>
\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}D\end{bmatrix}+
\begin{bmatrix}S\end{bmatrix}\begin{bmatrix}\dot D\end{bmatrix}+
\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}\ddot D\end{bmatrix}=
\begin{bmatrix}F(t)\end{bmatrix}</math>, where:</center>

<center><math>\begin{bmatrix}K\end{bmatrix}</math> is structural stiffness matrix,</center>
<center><math>\begin{bmatrix}S\end{bmatrix}</math> is structural damping matrix,</center>
<center><math>\begin{bmatrix}M\end{bmatrix}</math> is structural mass matrix,</center>
<center><math>\begin{bmatrix}D\end{bmatrix}</math> is generalized nodal displacements vector,</center>
<center><math>\begin{bmatrix}F\end{bmatrix}</math> is generalized force vector (fluid forces, gravity forces,...).</center>

Left-hand side of the global FEM matrix equation represents "dry" (in vacuuo) structure, while the right-hand side includes fluid forces (and coupling between the surrounding fluid and the structure).

The eigenvalue problem for the "dry" natural vibrations yields:

<center><math>\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}D\end{bmatrix}+\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}\ddot D\end{bmatrix}=\begin{bmatrix}0\end{bmatrix}</math></center>

If one assumes trial solution as <math>\begin{bmatrix}D\end{bmatrix}=\begin{bmatrix}w\end{bmatrix}\,e^{i\omega t}</math> then the eigenvalue problem reduces to <math>\left( \begin{bmatrix}K\end{bmatrix}-\omega^2 \begin{bmatrix}M\end{bmatrix} \right )\begin{bmatrix}w\end{bmatrix}=\begin{bmatrix}0\end{bmatrix}</math>. As a solution of the eigenvalue problem for each natural mode one obtains <math>\omega_n</math>, the n-th dry natural frequency and <math>\begin{bmatrix}w_n\end{bmatrix}</math>, the corresponding dry natural mode.

Generalized nodal displacements vector can be expressed using calculated "dry" structure natural modes:

<center><math>\begin{bmatrix}D\end{bmatrix}</math>=<math>\begin{bmatrix}W\end{bmatrix}\cdot\begin{bmatrix}\xi\end{bmatrix}</math></center>

<center><math>\begin{bmatrix}W\end{bmatrix}</math>=<math>\begin{bmatrix}\mathbf{w_1}\,\mathbf{w_2}\,\ldots \end{bmatrix}</math> is matrix of dry natural modes, with modes being sorted column-wise,</center>

<center><math>\begin{bmatrix}\xi\end{bmatrix}</math>is natural modes coefficients vector (modal amplitudes).</center>

<center><math>\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}K\end{bmatrix} \begin{bmatrix}W\end{bmatrix}
\begin{bmatrix}\xi\end{bmatrix}+\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}S\end{bmatrix} \begin{bmatrix}W\end{bmatrix}
\begin{bmatrix}\dot\xi\end{bmatrix}
+\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}M\end{bmatrix} \begin{bmatrix}W\end{bmatrix}
\begin{bmatrix}\ddot\xi\end{bmatrix}</math>=<math>\begin{bmatrix}W\end{bmatrix}^T\begin{bmatrix}F(t)\end{bmatrix}</math></center>

<center><math>\begin{bmatrix}k\end{bmatrix} \begin{bmatrix}\xi\end{bmatrix}+\begin{bmatrix}s\end{bmatrix} \begin{bmatrix}\dot\xi\end{bmatrix}+\begin{bmatrix}m\end{bmatrix} \begin{bmatrix}\ddot\xi\end{bmatrix}</math>=<math>\begin{bmatrix}f(t)\end{bmatrix}</math></center>

<center><math>\begin{bmatrix}k\end{bmatrix}=\begin{bmatrix}W^T\end{bmatrix}\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}W\end{bmatrix}</math>is the modal stiffness matrix,
<math>\begin{bmatrix}m\end{bmatrix}=\begin{bmatrix}W^T\end{bmatrix}\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}W\end{bmatrix}</math>is the modal mass matrix.</center>

Hydroelastic analysis of the general 3D structure is thus preformed using the modal superposition method.

Let us assume time-harmonic motion. Then the following is valid:

<center><math>\begin{bmatrix}\xi(t)\end{bmatrix}=\begin{bmatrix}\tilde{\xi}(\omega)\end{bmatrix}\cdot e^{i \omega t},
\; \begin{bmatrix}f(t)\end{bmatrix}=\begin{bmatrix}\tilde{f}(\omega)\end{bmatrix}\cdot e^{i \omega t}</math></center>

<center><math>\left ( \begin{bmatrix}k\end{bmatrix}+i\omega\begin{bmatrix}s\end{bmatrix}-\omega^2\begin{bmatrix}m\end{bmatrix} \right ) \begin{bmatrix}\tilde{\xi}\end{bmatrix}=\begin{bmatrix}\tilde{f}\end{bmatrix}</math></center>

Modal hydrodynamic forces are calculated by pressure work integration over the wetted surface:

<center><math>\tilde{f}^{hd}_i(t)=-i\omega\rho\iint_{S}\tilde{\phi}\,\mathbf{h_i}\mathbf{n}\,\mbox{d}S</math></center>

Total velocity potential can be decomposed as:

<center><math>\tilde{\phi}=\tilde{\phi}^I+\tilde{\phi}^D-i\omega\sum_{j=1}^N\tilde{\xi}_j\,\tilde{\phi}_j^R</math></center>

to be continued............

Category:Linear Hydroelasticity

2008-11-14T01:33:31Z

Marko tomic:

Category:Linear Hydroelasticity

2008-11-14T01:29:04Z

Marko tomic:

Problems in Linear Water-Wave theory in which there is an elastic body.

[[Category:Linear Water-Wave Theory]]

Finite Element Method (FEM) can be used to analyze any general 3D elastic structure using linear hydroelastic theory.

<center>
local FE <math>\Rightarrow </math> global FE model
</center>

Dynamic equation of motion in matrix form can be expressed as:
<center>
<math>
\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}D\end{bmatrix}+
\begin{bmatrix}S\end{bmatrix}\begin{bmatrix}\dot D\end{bmatrix}+
\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}\ddot D\end{bmatrix}=
\begin{bmatrix}F(t)\end{bmatrix}</math>, where:</center>

<center><math>\begin{bmatrix}K\end{bmatrix}</math> is structural stiffness matrix,</center>
<center><math>\begin{bmatrix}S\end{bmatrix}</math> is structural damping matrix,</center>
<center><math>\begin{bmatrix}M\end{bmatrix}</math> is structural mass matrix,</center>
<center><math>\begin{bmatrix}D\end{bmatrix}</math> is generalized nodal displacements vector,</center>
<center><math>\begin{bmatrix}F\end{bmatrix}</math> is generalized force vector (fluid forces, gravity forces,...).</center>

Left-hand side of the global FEM matrix equation represents "dry" (in vacuuo) structure, while the right-hand side includes fluid forces (and coupling between the surrounding fluid and the structure).

The eigenvalue problem for the "dry" natural vibrations yields:

<center><math>\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}D\end{bmatrix}+\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}\ddot D\end{bmatrix}=\begin{bmatrix}0\end{bmatrix}</math></center>

If one assumes trial solution as <math>\begin{bmatrix}D\end{bmatrix}=\begin{bmatrix}w\end{bmatrix}\,e^{i\omega t}</math> then the eigenvalue problem reduces to <math>\left( \begin{bmatrix}K\end{bmatrix}-\omega^2 \begin{bmatrix}M\end{bmatrix} \right )\begin{bmatrix}w\end{bmatrix}=\begin{bmatrix}0\end{bmatrix}</math>. As a solution of the eigenvalue problem for each natural mode one obtains <math>\omega_n</math>, the n-th dry natural frequency and <math>\begin{bmatrix}w_n\end{bmatrix}</math>, the corresponding dry natural mode.

Generalized nodal displacements vector can be expressed using calculated "dry" structure natural modes:

<center><math>\begin{bmatrix}D\end{bmatrix}</math>=<math>\begin{bmatrix}W\end{bmatrix}\cdot\begin{bmatrix}\xi\end{bmatrix}</math></center>

<center><math>\begin{bmatrix}W\end{bmatrix}</math>=<math>\begin{bmatrix}\mathbf{w_1}\,\mathbf{w_2}\,\ldots \end{bmatrix}</math> is matrix of dry natural modes, with modes being sorted column-wise,</center>

<center><math>\begin{bmatrix}\xi\end{bmatrix}</math>is natural modes coefficients vector (modal amplitudes).</center>

<center><math>\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}K\end{bmatrix} \begin{bmatrix}W\end{bmatrix}
\begin{bmatrix}\xi\end{bmatrix}+\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}S\end{bmatrix} \begin{bmatrix}W\end{bmatrix}
\begin{bmatrix}\dot\xi\end{bmatrix}
+\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}M\end{bmatrix} \begin{bmatrix}W\end{bmatrix}
\begin{bmatrix}\ddot\xi\end{bmatrix}</math>=<math>\begin{bmatrix}W\end{bmatrix}^T\begin{bmatrix}F(t)\end{bmatrix}</math></center>

<center><math>\begin{bmatrix}k\end{bmatrix} \begin{bmatrix}\xi\end{bmatrix}+\begin{bmatrix}s\end{bmatrix} \begin{bmatrix}\dot\xi\end{bmatrix}+\begin{bmatrix}m\end{bmatrix} \begin{bmatrix}\ddot\xi\end{bmatrix}</math>=<math>\begin{bmatrix}f(t)\end{bmatrix}</math></center>

<center><math>\begin{bmatrix}k\end{bmatrix}=\begin{bmatrix}W^T\end{bmatrix}\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}W\end{bmatrix}</math>is the modal stiffness matrix,
<math>\begin{bmatrix}m\end{bmatrix}=\begin{bmatrix}W^T\end{bmatrix}\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}W\end{bmatrix}</math>is the modal mass matrix.</center>

Hydroelastic analysis of the general 3D structure is thus preformed using the modal superposition method.

Let us assume time-harmonic motion. Then the following is valid:

<center><math>\begin{bmatrix}\xi(t)\end{bmatrix}=\begin{bmatrix}\tilde{\xi}(\omega)\end{bmatrix}\cdot e^{i \omega t},
\; \begin{bmatrix}f(t)\end{bmatrix}=\begin{bmatrix}\tilde{f}(\omega)\end{bmatrix}\cdot e^{i \omega t}</math></center>

<center><math>\left ( \begin{bmatrix}k\end{bmatrix}+i\omega\begin{bmatrix}s\end{bmatrix}-\omega^2\begin{bmatrix}m\end{bmatrix} \right ) \begin{bmatrix}\tilde{\xi}\end{bmatrix}=\begin{bmatrix}\tilde{f}\end{bmatrix}</math></center>

Modal hydrodynamic forces are calculated by pressure work integration over the wetted surface:

<center><math>\mathbf{F}^{hd}(t)=-\rho\iint_{S}\partial_n\Phi\,\mathbf{n}\,\mbox{d}S</math></center>

Category:Linear Hydroelasticity

2008-11-14T01:10:02Z

Marko tomic:

Category:Linear Hydroelasticity

2008-11-14T00:37:02Z

Marko tomic:

Category:Linear Hydroelasticity

2008-11-14T00:35:15Z

Marko tomic:

Problems in Linear Water-Wave theory in which there is an elastic body.

[[Category:Linear Water-Wave Theory]]

Finite Element Method (FEM) can be used to analyze any general 3D elastic structure using linear hydroelastic theory.

<center>
local FE <math>\Rightarrow </math> global FE model
</center>

Dynamic equation of motion in matrix form can be expressed as:
<center>
<math>
\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}D\end{bmatrix}+
\begin{bmatrix}S\end{bmatrix}\begin{bmatrix}\dot D\end{bmatrix}+
\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}\ddot D\end{bmatrix}=
\begin{bmatrix}F(t)\end{bmatrix}</math>, where:</center>

<center><math>\begin{bmatrix}K\end{bmatrix}</math> is structural stiffness matrix,</center>
<center><math>\begin{bmatrix}S\end{bmatrix}</math> is structural damping matrix,</center>
<center><math>\begin{bmatrix}M\end{bmatrix}</math> is structural mass matrix,</center>
<center><math>\begin{bmatrix}D\end{bmatrix}</math> is generalized nodal displacements vector,</center>
<center><math>\begin{bmatrix}F\end{bmatrix}</math> is generalized force vector (fluid forces, gravity forces,...).</center>

Left-hand side of the global FEM matrix equation represents "dry" (in vacuuo) structure, while the right-hand side includes fluid forces (and coupling between the surrounding fluid and the structure).

The eigenvalue problem for the "dry" natural vibrations yields:

<center><math>\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}D\end{bmatrix}+\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}\ddot D\end{bmatrix}=\begin{bmatrix}0\end{bmatrix}</math></center>

If one assumes trial solution as <math>\begin{bmatrix}D\end{bmatrix}=\begin{bmatrix}w\end{bmatrix}\,e^{i\omega t}</math> then the eigenvalue problem reduces to <math>\left( \begin{bmatrix}K\end{bmatrix}-\omega^2 \begin{bmatrix}M\end{bmatrix} \right )\begin{bmatrix}w\end{bmatrix}=\begin{bmatrix}0\end{bmatrix}</math>. As a solution of the eigenvalue problem for each natural mode one obtains <math>\omega_n</math>, the n-th dry natural frequency and <math>\begin{bmatrix}w_n\end{bmatrix}</math>, the corresponding dry natural mode.

Generalized nodal displacements vector can be expressed using calculated "dry" structure natural modes:

<center><math>\begin{bmatrix}D\end{bmatrix}</math>=<math>\begin{bmatrix}W\end{bmatrix}\cdot\begin{bmatrix}\xi\end{bmatrix}</math></center>

<center><math>\begin{bmatrix}W\end{bmatrix}</math>=<math>\begin{bmatrix}\mathbf{w_1}\,\mathbf{w_2}\,\ldots \end{bmatrix}</math> is matrix of dry natural modes, with modes being sorted column-wise,</center>

<center><math>\begin{bmatrix}\xi\end{bmatrix}</math>is natural modes coefficients vector.</center>

<center><math>\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}K\end{bmatrix} \begin{bmatrix}W\end{bmatrix}
\begin{bmatrix}\xi\end{bmatrix}+\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}S\end{bmatrix} \begin{bmatrix}W\end{bmatrix}
\begin{bmatrix}\dot\xi\end{bmatrix}
+\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}M\end{bmatrix} \begin{bmatrix}W\end{bmatrix}
\begin{bmatrix}\ddot\xi\end{bmatrix}</math>=<math>\begin{bmatrix}W\end{bmatrix}^T\begin{bmatrix}F(t)\end{bmatrix}</math></center>

<center><math>\begin{bmatrix}k\end{bmatrix} \begin{bmatrix}\xi\end{bmatrix}+\begin{bmatrix}s\end{bmatrix} \begin{bmatrix}\dot\xi\end{bmatrix}+\begin{bmatrix}m\end{bmatrix} \begin{bmatrix}\ddot\xi\end{bmatrix}</math>=<math>\begin{bmatrix}f(t)\end{bmatrix}</math></center>

<center><math>\begin{bmatrix}k\end{bmatrix}=\begin{bmatrix}W^T\end{bmatrix}\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}W\end{bmatrix}</math>is the modal stiffness matrix,
<math>\begin{bmatrix}m\end{bmatrix}=\begin{bmatrix}W^T\end{bmatrix}\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}W\end{bmatrix}</math>is the modal mass matrix.</center>

Category:Linear Hydroelasticity

2008-11-14T00:23:27Z

Marko tomic:

Category:Linear Hydroelasticity

2008-11-14T00:22:56Z

Marko tomic:

Category:Linear Hydroelasticity

2008-11-14T00:21:39Z

Marko tomic:

Category:Linear Hydroelasticity

2008-11-14T00:19:17Z

Marko tomic:

Category:Linear Hydroelasticity

2008-11-13T07:20:52Z

Marko tomic:

Category:Linear Hydroelasticity

2008-11-13T07:02:41Z

Marko tomic:

Category:Linear Hydroelasticity

2008-11-13T05:52:36Z

Marko tomic:

Category:Linear Hydroelasticity

2008-11-13T05:38:24Z

Marko tomic:

Problems in Linear Water-Wave theory in which there is an elastic body.

[[Category:Linear Water-Wave Theory]]

local FE \Rightarrow global FE model

Category:Linear Hydroelasticity

2008-11-13T05:38:02Z

Marko tomic:

Problems in Linear Water-Wave theory in which there is an elastic body.

[[Category:Linear Water-Wave Theory]]

local FE

Category:Linear Hydroelasticity

2008-11-13T05:15:40Z

Marko tomic:

Problems in Linear Water-Wave theory in which there is an elastic body.

[[Category:Linear Water-Wave Theory]]

Category:Linear Hydroelasticity

2008-11-13T01:41:02Z

Marko tomic:

Problems in Linear Water-Wave theory in which there is an elastic body.

[[Category:Linear Water-Wave Theory]]
dfdfhd

Integral Equation for the Finite Depth Green Function at Surface

2008-11-09T23:09:11Z

Marko tomic:

We want to solve
<center><math>
\phi(x) = \int_{-L}^{L}G(x,\xi)
\left(
\alpha\phi(\xi) - w(\xi)
\right)d \xi
</math></center>
where
<math>G(x,\xi)</math> is the [[Free-Surface Green Function]] for two-dimensional waves, with singularity at
the water surface. We break the surface into <math>N</math> evenly spaced point
<math>x_n = -L = hn</math> where <math>h=2L/N</math> and <math>n=0,1,\dots,N</math>

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
[http://www.math.auckland.ac.nz/~meylan/code/boundary_element/matrix_G_surface.m matrix_G_surface.m]

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}

[[Category:Linear Water-Wave Theory]]
[[Category:Pages with Matlab Code]]

Integral Equation for the Finite Depth Green Function at Surface

2008-11-09T23:08:53Z

Marko tomic:

We want to solve
<center><math>
\phi(x) = \int_{-L}^{L}G(x,\xi)
\left(
\alpha\phi(\xi) - w(\xi)
\right)d \xi
</math></center>
where
<math>G(x,\xi)<\math> is the [[Free-Surface Green Function]] for two-dimensional waves, with singularity at
the water surface. We break the surface into <math>N</math> evenly spaced point
<math>x_n = -L = hn</math> where <math>h=2L/N</math> and <math>n=0,1,\dots,N</math>

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
[http://www.math.auckland.ac.nz/~meylan/code/boundary_element/matrix_G_surface.m matrix_G_surface.m]

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}

[[Category:Linear Water-Wave Theory]]
[[Category:Pages with Matlab Code]]

Integral Equation for the Finite Depth Green Function at Surface

2008-11-09T23:08:18Z

Marko tomic:

We want to solve
<center><math>
\phi(x) = \int_{-L}^{L}G(x,\xi)
\left(
\alpha\phi(\xi) - w(\xi)
\right)d \xi
</math></center>
where
G(x,\xi) is the [[Free-Surface Green Function]] for two-dimensional waves, with singularity at
the water surface. We break the surface into <math>N</math> evenly spaced point
<math>x_n = -L = hn</math> where <math>h=2L/N</math> and <math>n=0,1,\dots,N</math>

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
[http://www.math.auckland.ac.nz/~meylan/code/boundary_element/matrix_G_surface.m matrix_G_surface.m]

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}

[[Category:Linear Water-Wave Theory]]
[[Category:Pages with Matlab Code]]

Eigenfunctions for a Uniform Free Beam

2008-11-06T23:10:25Z

Marko tomic:

We can find a the eigenfunction which satisfy
<center>
<math>\partial_x^4 w_n = \lambda_n^4 w_n</math>
</center>
plus the edge conditions.
<center><math>\begin{matrix}
\frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l,
\end{matrix}</math></center>
<center><math>\begin{matrix}
\frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l.
\end{matrix}</math></center>

General solution of the differential equation is :

<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>
</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.

== Symmetric modes ==

<center>
<math>C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \cos(\lambda_n l)&\cosh(\lambda_n l)\\
\sin(\lambda_n l)&\sinh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_2\\
C_4\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

The first three roots are :

<center>
<math>\lambda_0 l = 0, \lambda_2 l = 2.365, \lambda_4 l = 5.497\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Symmetric natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\cos(\lambda_n x)}{\cos(\lambda_n l)}+\frac{\cosh(\lambda_n x)}{\cosh(\lambda_n l)} \right )
</math>
</center>

== Skew-symmetric modes ==

<center>
<math>C_2 = C_4 = 0 \Rightarrow w_n(x) = C_1 \sin(\lambda_n x) + C_3 \sinh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \sin(\lambda_n l)&\sinh(\lambda_n l)\\
-\cos(\lambda_n l)&\cosh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_1\\
C_3\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>-\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

The first three roots are :

<center>
<math>\lambda_1 l = 0, \lambda_3 l = 3.925, \lambda_5 l = 7.068\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Skew-symmetric natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\sin(\lambda_n x)}{\sin(\lambda_n l)}+\frac{\sinh(\lambda_n x)}{\sinh(\lambda_n l)} \right )
</math>
</center>

Eigenfunctions for a Uniform Free Beam

2008-11-06T23:08:17Z

Marko tomic:

We can find a the eigenfunction which satisfy
<center>
<math>\partial_x^4 w_n = \lambda_n^4 w_n</math>
</center>
plus the edge conditions.
<center><math>\begin{matrix}
\frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l,
\end{matrix}</math></center>
<center><math>\begin{matrix}
\frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l.
\end{matrix}</math></center>

General solution of the differential equation is :

<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>
</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.

== Symmetric modes ==

<center>
<math>C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \cos(\lambda_n l)&\cosh(\lambda_n l)\\
\sin(\lambda_n l)&\sinh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_2\\
C_4\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

The first three roots are :

<center>
<math>\lambda_0 l = 0, \lambda_2 l = 2.365, \lambda_4 l = 5.497\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Symmetric natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\cos(\lambda_n x)}{\cos(\lambda_n l)}+\frac{\cosh(\lambda_n x)}{\cosh(\lambda_n l)} \right )
</math>
</center>

== Skew-symmetric modes ==

<center>
<math>C_2 = C_4 = 0 \Rightarrow w_n(x) = C_1 \sin(\lambda_n x) + C_3 \sinh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \sin(\lambda_n l)&\sinh(\lambda_n l)\\
-\cos(\lambda_n l)&\cosh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_1\\
C_3\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>-\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Skew-symmetric natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\sin(\lambda_n x)}{\sin(\lambda_n l)}+\frac{\sinh(\lambda_n x)}{\sinh(\lambda_n l)} \right )
</math>
</center>

Eigenfunctions for a Uniform Free Beam

2008-11-06T23:06:52Z

Marko tomic:

We can find a the eigenfunction which satisfy
<center>
<math>\partial_x^4 w_n = \lambda_n^4 w_n</math>
</center>
plus the edge conditions.
<center><math>\begin{matrix}
\frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l,
\end{matrix}</math></center>
<center><math>\begin{matrix}
\frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l.
\end{matrix}</math></center>

General solution of the differential equation is :

<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>
</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.

== Symmetric modes ==

<center>
<math>C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \cos(\lambda_n l)&\cosh(\lambda_n l)\\
\sin(\lambda_n l)&\sinh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_2\\
C_4\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

The first three roots are :

<center>
<math>\lambda_0 l = 0, </math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Symmetric natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\cos(\lambda_n x)}{\cos(\lambda_n l)}+\frac{\cosh(\lambda_n x)}{\cosh(\lambda_n l)} \right )
</math>
</center>

== Skew-symmetric modes ==

<center>
<math>C_2 = C_4 = 0 \Rightarrow w_n(x) = C_1 \sin(\lambda_n x) + C_3 \sinh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \sin(\lambda_n l)&\sinh(\lambda_n l)\\
-\cos(\lambda_n l)&\cosh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_1\\
C_3\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>-\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Skew-symmetric natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\sin(\lambda_n x)}{\sin(\lambda_n l)}+\frac{\sinh(\lambda_n x)}{\sinh(\lambda_n l)} \right )
</math>
</center>

Eigenfunctions for a Uniform Free Beam

2008-11-06T23:02:54Z

Marko tomic:

We can find a the eigenfunction which satisfy
<center>
<math>\partial_x^4 w_n = \lambda_n^4 w_n</math>
</center>
plus the edge conditions.
<center><math>\begin{matrix}
\frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l,
\end{matrix}</math></center>
<center><math>\begin{matrix}
\frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l.
\end{matrix}</math></center>

General solution of the differential equation is :

<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>
</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.

== Symmetric modes ==

<center>
<math>C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \cos(\lambda_n l)&\cosh(\lambda_n l)\\
\sin(\lambda_n l)&\sinh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_2\\
C_4\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Symmetric natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\cos(\lambda_n x)}{\cos(\lambda_n l)}+\frac{\cosh(\lambda_n x)}{\cosh(\lambda_n l)} \right )
</math>
</center>

== Skew-symmetric modes ==

<center>
<math>C_2 = C_4 = 0 \Rightarrow w_n(x) = C_1 \sin(\lambda_n x) + C_3 \sinh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \sin(\lambda_n l)&\sinh(\lambda_n l)\\
-\cos(\lambda_n l)&\cosh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_1\\
C_3\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>-\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Skew-symmetric natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\sin(\lambda_n x)}{\sin(\lambda_n l)}+\frac{\sinh(\lambda_n x)}{\sinh(\lambda_n l)} \right )
</math>
</center>

Eigenfunctions for a Uniform Free Beam

2008-11-06T23:02:17Z

Marko tomic:

We can find a the eigenfunction which satisfy
<center>
<math>\partial_x^4 w_n = \lambda_n^4 w_n</math>
</center>
plus the edge conditions.
<center><math>\begin{matrix}
\frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l,
\end{matrix}</math></center>
<center><math>\begin{matrix}
\frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l.
\end{matrix}</math></center>

General solution of the differential equation is :

<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>
</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.

== Symmetric modes ==

<center>
<math>C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \cos(\lambda_n l)&\cosh(\lambda_n l)\\
\sin(\lambda_n l)&\sinh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_2\\
C_4\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Symmetrical natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\cos(\lambda_n x)}{\cos(\lambda_n l)}+\frac{\cosh(\lambda_n x)}{\cosh(\lambda_n l)} \right )
</math>
</center>

== Skew-symmetric modes ==

<center>
<math>C_2 = C_4 = 0 \Rightarrow w_n(x) = C_1 \sin(\lambda_n x) + C_3 \sinh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \sin(\lambda_n l)&\sinh(\lambda_n l)\\
-\cos(\lambda_n l)&\cosh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_1\\
C_3\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>-\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Skew-symmetrical natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\sin(\lambda_n x)}{\sin(\lambda_n l)}+\frac{\sinh(\lambda_n x)}{\sinh(\lambda_n l)} \right )
</math>
</center>

Eigenfunctions for a Uniform Free Beam

2008-11-06T23:01:22Z

Marko tomic:

We can find a the eigenfunction which satisfy
<center>
<math>\partial_x^4 w_n = \lambda_n^4 w_n</math>
</center>
plus the edge conditions.
<center><math>\begin{matrix}
\frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l,
\end{matrix}</math></center>
<center><math>\begin{matrix}
\frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l.
\end{matrix}</math></center>

General solution of the differential equation is :

<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>
</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.

== Symmetric modes ==

<center>
<math>C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \cos(\lambda_n l)&\cosh(\lambda_n l)\\
\sin(\lambda_n l)&\sinh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_2\\
C_4\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Symmetrical natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\cos(\lambda_n x)}{\cos(\lambda_n l)}+\frac{\cosh(\lambda_n x)}{\cosh(\lambda_n l)} \right )
</math>
</center>

== Skew-symmetric modes ==

<center>
<math>C_2 = C_4 = 0 \Rightarrow w_n(x) = C_1 \sin(\lambda_n x) + C_3 \sinh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \sin(\lambda_n l)&\sinh(\lambda_n l)\\
-\cos(\lambda_n l)&\cosh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_1\\
C_3\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>-\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Symmetrical natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\sin(\lambda_n x)}{\sin(\lambda_n l)}+\frac{\sinh(\lambda_n x)}{\sinh(\lambda_n l)} \right )
</math>
</center>

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:58:10Z

Marko tomic:

We can find a the eigenfunction which satisfy
<center>
<math>\partial_x^4 w_n = \lambda_n^4 w_n</math>
</center>
plus the edge conditions.
<center><math>\begin{matrix}
\frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l,
\end{matrix}</math></center>
<center><math>\begin{matrix}
\frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm l.
\end{matrix}</math></center>

General solution of the differential equation is :

<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>
</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.

== Symmetric modes ==

<center>
<math>C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \cos(\lambda_n l)&\cosh(\lambda_n l)\\
\sin(\lambda_n l)&\sinh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_2\\
C_4\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Symmetrical natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\cos(\lambda_n x)}{\cos(\lambda_n l)}+\frac{\cosh(\lambda_n x)}{\cosh(\lambda_n l)} \right )
</math>
</center>

== Skew-symmetric modes ==

<center>
<math>C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \cos(\lambda_n l)&\cosh(\lambda_n l)\\
\sin(\lambda_n l)&\sinh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_2\\
C_4\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Symmetrical natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\cos(\lambda_n x)}{\cos(\lambda_n l)}+\frac{\cosh(\lambda_n x)}{\cosh(\lambda_n l)} \right )
</math>
</center>

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:57:39Z

Marko tomic:

We can find a the eigenfunction which satisfy
<center>
<math>\partial_x^4 w_n = \lambda_n^4 w_n</math>
</center>
plus the edge conditions.
<center><math>\begin{matrix}
\frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L,
\end{matrix}</math></center>
<center><math>\begin{matrix}
\frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L.
\end{matrix}</math></center>

General solution of the differential equation is :

<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>
</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.

== Symmetric modes ==

<center>
<math>C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \cos(\lambda_n l)&\cosh(\lambda_n l)\\
\sin(\lambda_n l)&\sinh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_2\\
C_4\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Symmetrical natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\cos(\lambda_n x)}{\cos(\lambda_n l)}+\frac{\cosh(\lambda_n x)}{\cosh(\lambda_n l)} \right )
</math>
</center>

== Skew-symmetric modes ==

<center>
<math>C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \cos(\lambda_n l)&\cosh(\lambda_n l)\\
\sin(\lambda_n l)&\sinh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_2\\
C_4\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Symmetrical natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\cos(\lambda_n x)}{\cos(\lambda_n l)}+\frac{\cosh(\lambda_n x)}{\cosh(\lambda_n l)} \right )
</math>
</center>

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:56:24Z

Marko tomic:

We can find a the eigenfunction which satisfy
<center>
<math>\partial_x^4 w_n = \lambda_n^4 w_n</math>
</center>
plus the edge conditions.
<center><math>\begin{matrix}
\frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L,
\end{matrix}</math></center>
<center><math>\begin{matrix}
\frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L.
\end{matrix}</math></center>
This solution is discussed further in [[Eigenfunctions for a Free Beam]].

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.

General solution of the above stated equation is :

<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>
</center>

== Symmetric modes ==

<center>
<math>C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \cos(\lambda_n l)&\cosh(\lambda_n l)\\
\sin(\lambda_n l)&\sinh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_2\\
C_4\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Symmetrical natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\cos(\lambda_n x)}{\cos(\lambda_n l)}+\frac{\cosh(\lambda_n x)}{\cosh(\lambda_n l)} \right )
</math>
</center>

== Skew-symmetric modes ==

<center>
<math>C_1 = C_3 = 0 \Rightarrow w_n(x) = C_2 \cos(\lambda_n x) + C_4 \cosh(\lambda_n x)</math>
</center>

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

<center>
<math>
\begin{bmatrix}
- \cos(\lambda_n l)&\cosh(\lambda_n l)\\
\sin(\lambda_n l)&\sinh(\lambda_n l)\\
\end{bmatrix}

\begin{bmatrix}
C_2\\
C_4\\
\end{bmatrix}

=

\begin{bmatrix}
0\\
0\\
\end{bmatrix}
</math>
</center>

For a nontrivial solution one gets:
<center>
<math>\tan(\lambda_n l)+\tanh(\lambda_n l)=0\,</math>
</center>

Having obtained eigenvalues <math>\lambda_n</math>, natural requency can be readily calculated :
<center>
<math>\omega_n = \frac{(\lambda_n l)^2}{l^2}\sqrt\frac{EI}{m}</math>
</center>

Symmetrical natural modes can be written in normalized form as :
<center>
<math>w_n(x) = \frac{1}{2}\left( \frac{\cos(\lambda_n x)}{\cos(\lambda_n l)}+\frac{\cosh(\lambda_n x)}{\cosh(\lambda_n l)} \right )
</math>
</center>

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:55:22Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:53:33Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:52:48Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:48:19Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:47:36Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:46:12Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:45:12Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:42:05Z

Marko tomic: /* Symmetric modes */

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:40:34Z

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Eigenfunctions for a Uniform Free Beam

2008-11-06T22:38:44Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:33:58Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:31:51Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:28:02Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:27:21Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:26:36Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:25:14Z

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Eigenfunctions for a Uniform Free Beam

2008-11-06T22:24:21Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:23:33Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:23:11Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:22:51Z

Marko tomic:

Eigenfunctions for a Uniform Free Beam

2008-11-06T22:22:04Z

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