2. Reference solution#

2.1. Calculation method#

The solution is calculated in a mixed analytical and numerical manner.

The dynamic (complex) stiffness of the viscoelastic beam is calculated according to [1] by the equation:

\({\mathit{EI}}^{\text{*}}={E}_{1}{I}_{1}\left(1+{e}_{2}{h}_{2}^{3}+\frac{3{(1+{h}_{2})}^{2}({e}_{2}\times {h}_{2})}{1+{e}_{2}\times {h}_{2}}\right)\) (1)

The meaning of the terms in the formula is given in the list below:

\({E}^{\text{*}}\): Young’s modulus of the viscoelastic beam

\(I\): second moment of inertia of the cross section of the viscoelastic beam

\({I}_{1}\): second moment of inertia of the cross section of the material in layer 1

\({E}_{j}^{\text{*}}\): Young’s modulus of the material in layer No. j

\({e}_{2}\): module ratio \({E}_{2}^{\text{*}}/{E}_{1}\) (the damping of material No. 1 is considered to be negligible)

\({H}_{j}\): thickness of the material of diaper no.*j*

\({h}_{2}\): thickness ratio \({H}_{2}/{H}_{1}\)

Damping \(\eta\) for complex modes is then calculated as the ratio of the imaginary part of the dynamic stiffness to its real part:

\(\eta =\frac{\Im (\mathit{EI})}{\Re (\mathit{EI})}\) (2)

Finally, the natural frequencies \({f}_{i}\) are calculated according to [2] by the following equation:

\({f}_{i}=\frac{{\lambda }_{i}^{2}}{{\mathrm{2\pi L}}^{2}}\sqrt{\frac{\Re (\mathit{EI})}{m}}\) (3)

With \({\lambda }_{i}\): modal coefficient (given in [2]) associated with the natural frequency \({f}_{i}\)

\(L\): length of the viscoelastic beam

\(m\): linear mass of the viscoelastic beam

To take into account the frequency dependence of the mechanical properties of the viscoelastic material, the damping and natural frequencies are calculated by an iterative method.

2.2. Reference quantities and results#

The values of the natural frequencies and the damping of some complex modes of the viscoelastic beam are tested.

2.3. Uncertainty about the solution#

Digital solution.

2.4. Bibliographical references#

  1. A.D. Nashif, D.I.G. Jones, J.P. Jones, J.P. Henderson, J.P. Henderson, Vibration Damping. John Wiley and Sons, 1985.

  1. Robert D. BLEVINS PhD, Formulas for natural frequency and mode shape, §8.1.2 « Single-span beams. » Krieger Publishing Company, Malabar, 2001.