2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The calculation is based on the dynamic equilibrium and behavioral relationships [bib2] as set out below:

\(\frac{\partial {M}_{y}}{\partial x}+{V}_{z}=⟨\rho I⟩\frac{{\partial }^{2}{\theta }_{y}}{\partial {t}^{2}}\)

\(\frac{\partial {V}_{z}}{\partial x}=⟨\rho S⟩\frac{{\partial }^{2}{v}_{z}}{\partial {t}^{2}}\)

\({M}_{y}=⟨EI⟩\frac{\partial {\theta }_{y}}{\partial x}\)

\({V}_{z}=k⟨GS⟩\left(\frac{\partial {v}_{z}}{\partial x}-{\theta }_{y}\right)\)

These relationships make it possible to write the equation of transverse dynamic flexure motion \(v(x,t)\). The natural frequency equation is obtained after combining the boundary conditions.

The natural frequency equation is written as:

\(\mathrm{sin}({X}_{2})=0\) with \({X}_{2}={\left[{\stackrel{ˉ}{\omega }}^{2}\frac{(1+a)}{2}+\sqrt{{\stackrel{ˉ}{\omega }}^{2}({\stackrel{ˉ}{\omega }}^{2}{(\frac{1-a}{2})}^{2}+\frac{1}{{r}^{2}})}\right]}^{1/2}\)

and

\({\overline{\omega }}^{2}=\frac{⟨\rho I⟩{\omega }^{2}{L}^{2}}{⟨EI⟩}\); \({\overline{r}}^{2}=\frac{⟨\rho I⟩}{⟨\rho S⟩{L}^{2}}\); \(a=\frac{⟨\rho S⟩⟨EI⟩}{k⟨\rho I⟩⟨GS⟩}\)

The solutions to the natural frequency equation are then written as: \({X}_{2}=n\pi (n=\mathrm{1,2}\mathrm{,3},\mathrm{...})\)

2.2. Benchmark results#

5 first frequencies and associated natural modes of bending.

  • Mode 1 frequency:

\(64.476\mathrm{Hz}\)

 _images/Object_23.svg

  • Mode 2 frequency:

\(131.918\mathrm{Hz}\)

 _images/Object_24.svg

  • Mode 3 frequency:

\(198.734\mathrm{Hz}\)

 _images/Object_25.svg

  • Mode 4 frequency:

\(265.383\mathrm{Hz}\)

 _images/Object_26.svg

  • Mode 5 frequency:

\(331.963\mathrm{Hz}\)

 _images/Object_27.svg

2.3. Uncertainties about the solution#

The reference solution is calculated using the assumptions of girder theory [bib2]: \({\sigma }_{y}={\sigma }_{z}=0\).

2.4. Bibliographical references#

  1. VPCS: Software for calculating composite structures; Validation examples. Review of Composites and Advanced Materials, Volume 5 - special issue/ 1995 - Hermes edition.

  2. CIEAUX J.M.: Dynamic flexure of composite beams with orthotropic phases; Validity of the quasistatic domain. Thesis from Paul Sabatier University Toulouse III, 1988.