Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

The reference solution is obtained analytically for a Timoshenko beam, taking into account the shear force deformation and the rotary inertia of the sections.

The solution is developed in a series of clean modes. Theoretical aspects are developed in the reference given in :ref:`2.4 <RefNumPara__5069_280392679>`.


Modal basis
~~~~~~~~~~~

Define the following dimensionless quantities:

:math:`{\lambda }_{n}={k}_{n}L` wavelengths

:math:`{\Omega }_{n}=\frac{\rho A{L}^{4}}{\mathit{EI}}{\omega }_{n}^{2}` eigenvalues

:math:`j=\frac{I}{A{L}^{2}}` rotary inertia

:math:`g=\frac{\mathit{EI}}{k\text{'}AG{L}^{2}}` shear coefficient

Each specific mode of wave number :math:`{k}_{n}` is characterized by the following quantities:

Natural frequencies:

:math:`{\Omega }_{\mathrm{1,2}}=\frac{\rho A{L}^{4}}{\mathit{EI}}{\omega }_{\mathrm{1,2}}^{2}=\frac{(g+j){\lambda }_{n}^{2}+1\pm \sqrt{{(g-j)}^{2}{\lambda }_{n}^{4}+2(g+j){\lambda }_{n}^{2}+1}}{2gj}`

with

:math:`{\lambda }_{n}=n\pi`, :math:`n=\mathrm{1,}\mathrm{2,}\mathrm{3,}\mathrm{...}`,

*(the indices 1 and 2 correspond to the + and — signs in front of the root) .*

The generalized masses:

:math:`{\mu }_{\mathrm{1,2}}=\rho A={\left({k}_{n}-\frac{{\omega }_{\mathrm{1,2}}^{2}\rho }{{k}_{n}k\text{'}G}\right)}^{2}\rho I`.

Critical depreciation percentages:

:math:`{ϵ}_{\mathrm{1,2}}=\frac{1}{2}\left(\alpha {\omega }_{\mathrm{1,2}}+\frac{\beta }{{\omega }_{\mathrm{1,2}}}\right)`


Harmonic response
~~~~~~~~~~~~~~~~~~~~

The amplitude and the phase of the arrow W are given by

:math:`W(x)=\sum _{n=1}^{+\infty }{p}_{n}\left[\sum _{m=1}^{2}\frac{1}{{\mu }_{m}({\omega }_{m}^{2}-{\omega }^{2}+2i{ϵ}_{m}{\omega }_{m}\omega )}\right]\mathrm{sin}({k}_{n}x)`

with

:math:`{p}_{n}=\frac{2{p}_{0}}{n\pi }\left[1-{(-1)}^{n}\right]`

Benchmark results
----------------------


+----------------------+-----------------------------+----------------------+
|Position              |Arrow :math:`W`                                     |
+                      +-----------------------------+----------------------+
|                      |Amplitude (:math:`m`)        |Phase                 |
+----------------------+-----------------------------+----------------------+
|:math:`x=\frac{L}{4}` |:math:`2.136\times {10}^{-5}`|:math:`22.4\text{°}`  |
+----------------------+-----------------------------+----------------------+
|:math:`x=\frac{L}{2}` |:math:`1.342\times {10}^{-5}`|:math:`-121.5\text{°}`|
+----------------------+-----------------------------+----------------------+
|:math:`x=3\frac{L}{4}`|:math:`2.136\times {10}^{-5}`|:math:`22.4\text{°}`  |
+----------------------+-----------------------------+----------------------+

The quantities actually tested in the test case are the parts real and imaginary whose values are given below.


+----------------------+-------------------------------------------+-------------------------------------------+
|Position              |Arrow :math:`W`                                                                        |
+                      +-------------------------------------------+-------------------------------------------+
|                      |Real part (:math:`m`)                      |Imaginary part (:math:`m`)                 |
+----------------------+-------------------------------------------+-------------------------------------------+
|:math:`x=\frac{L}{4}` |:math:`-1.9599467159360556\times {10}^{-5}`|:math:`-8.4917893914738073\times {10}^{-6}`|
+----------------------+-------------------------------------------+-------------------------------------------+
|:math:`x=\frac{L}{2}` |:math:`-6.9993870268574731\times {10}^{-6}`|:math:`-1.1450108350939712\times {10}^{-5}`|
+----------------------+-------------------------------------------+-------------------------------------------+
|:math:`x=3\frac{L}{4}`|:math:`-1.9599467159360556\times {10}^{-5}`|:math:`-8.4917893914738073\times {10}^{-6}`|
+----------------------+-------------------------------------------+-------------------------------------------+


Uncertainty about the solution
---------------------------


* Analytical solution.


.. _RefNumPara__5069_280392679:


Bibliographical references
---------------------------


* ROBERT G., Analytical solutions in structural dynamics, Samtech Report No. 121, Liège, 1996.