2. Benchmark solution#

2.1. Calculation method used for the reference solution#

2.1.1. Undamped structure#

In this case, the reference solution can be obtained analytically:

\(m(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array})(\begin{array}{c}\ddot{{x}_{1}}\\ \ddot{{x}_{2}}\\ \ddot{{x}_{3}}\end{array})+k(\begin{array}{ccc}2& \mathrm{-}1& 0\\ \mathrm{-}1& 2& \mathrm{-}1\\ 0& \mathrm{-}1& 2\end{array})(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array})\mathrm{=}(\begin{array}{c}1\\ 0\\ 0\end{array})\)

The natural pulsations of the mass-spring system are equal to:

\({\omega }_{1}^{2}=(2-\sqrt{2})\frac{k}{m}\) \({\omega }_{2}^{2}\mathrm{=}2\frac{k}{m}\) \({\omega }_{3}^{2}\mathrm{=}(2+\sqrt{2})\frac{k}{m}\)

of respective modal deformations:

\({\phi }_{1}\mathrm{=}(\begin{array}{c}\sqrt{2}\\ 2\\ \sqrt{2}\end{array})\) \({\phi }_{2}=(\begin{array}{}1\\ 0\\ -1\end{array})\) \({\phi }_{3}\mathrm{=}(\begin{array}{c}\mathrm{-}\sqrt{2}\\ 2\\ \mathrm{-}\sqrt{2}\end{array})\)

Projected on the basis of eigenmodes, the transitory equation becomes \({\eta }_{i}\) with the following generalized coordinates:

\(m(\begin{array}{ccc}8& 0& 0\\ 0& 2& 0\\ 0& 0& 8\end{array})(\begin{array}{c}\ddot{{\eta }_{1}}\\ \ddot{{\eta }_{2}}\\ \ddot{{\eta }_{3}}\end{array})+\mathrm{4k}(\begin{array}{ccc}4\mathrm{-}\sqrt{2}& 0& 0\\ 0& 1& 0\\ 0& 0& 4+2\sqrt{2}\end{array})(\begin{array}{c}{\eta }_{1}\\ {\eta }_{2}\\ {\eta }_{3}\end{array})\mathrm{=}(\begin{array}{c}+\sqrt{2}\\ 2\\ \mathrm{-}\sqrt{2}\end{array})\)

The system can be solved analytically. We get:

\(\left\{\eta (t)\right\}\mathrm{=}\frac{1}{\mathrm{2m}}(\begin{array}{c}\frac{\sqrt{2}}{4{\omega }_{1}^{2}}(1\mathrm{-}\mathrm{cos}{\omega }_{1}t)\\ \frac{1}{{\omega }_{2}^{2}}(1\mathrm{-}\mathrm{cos}{\omega }_{2}t)\\ \frac{\sqrt{2}}{4{\omega }_{3}^{2}}(\mathrm{cos}{\omega }_{3}t\mathrm{-}1)\end{array})\)

The solution on a physical basis is obtained using the Ritz transformation:

\(x(t)\mathrm{=}(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array})\mathrm{=}\phi \eta \mathrm{=}(\begin{array}{ccc}\sqrt{2}& 1& \mathrm{-}\sqrt{2}\\ 2& 0& 2\\ \sqrt{2}& \mathrm{-}1& \sqrt{2}\end{array})(\begin{array}{c}{\eta }_{1}\\ {\eta }_{2}\\ {\eta }_{3}\end{array})\)

2.1.2. Damped structure#

Damping is applied to the natural modes of the projection bases of embedded substructures (reduced damping). In this case, we end up with the following transitory equation in generalized coordinates (bib [1]):

\(m(\begin{array}{ccc}8& 0& 0\\ 0& 2& 0\\ 0& 0& 8\end{array})(\begin{array}{c}\ddot{{\eta }_{1}}\\ \ddot{{\eta }_{2}}\\ \ddot{{\eta }_{3}}\end{array})+4\varepsilon \sqrt{\mathrm{2km}}(\begin{array}{ccc}3\mathrm{-}2\sqrt{2}& 0& \mathrm{-}1\\ 0& 1& 0\\ \mathrm{-}1& 0& 3+2\sqrt{2}\end{array})(\begin{array}{c}\dot{{\eta }_{1}}\\ \dot{{\eta }_{2}}\\ \dot{{\eta }_{3}}\end{array})+\mathrm{4k}(\begin{array}{ccc}4\mathrm{-}\sqrt{2}& 0& 0\\ 0& 1& 0\\ 0& 0& 4+2\sqrt{2}\end{array})(\begin{array}{c}{\eta }_{1}\\ {\eta }_{2}\\ {\eta }_{3}\end{array})\mathrm{=}(\begin{array}{c}+\sqrt{2}\\ 1\\ \mathrm{-}\sqrt{2}\end{array})\)

Since this system was not decoupled, it was solved using the Maple software. We got \((\varepsilon \mathrm{=}0.01)\):

\(\left\{\eta (t)\right\}\mathrm{=}\frac{1}{\mathrm{2m}}(\begin{array}{c}\frac{\sqrt{2}}{4{\omega }_{1}^{2}}(1\mathrm{-}{e}^{\mathrm{-}\frac{t}{{\tau }_{1}}}\mathrm{cos}{\omega }_{1}t)\\ \frac{1}{{\omega }_{2}^{2}}(1\mathrm{-}{e}^{\mathrm{-}\frac{t}{{\tau }_{2}}}\mathrm{cos}{\omega }_{2}t)\\ \frac{\sqrt{2}}{4{\omega }_{3}^{2}}({e}^{\mathrm{-}\frac{t}{{\tau }_{3}}}\mathrm{cos}{\omega }_{3}t\mathrm{-}1)\end{array})\)

with \({\tau }_{1}\mathrm{=}1.65{10}^{3}s\), \({\tau }_{2}\mathrm{=}\frac{1}{\varepsilon {\omega }_{2}}\mathrm{=}\frac{100}{\sqrt{2}}\), and \({\tau }_{3}\mathrm{=}4.85{10}^{1}s\)

A formulation similar to the non-amortized case is therefore obtained, but in which exponential terms that characterize damping come into play.

The solution on a physical basis is obtained using the Ritz transformation:

2.2. Benchmark results#

Non-amortized structure:

Movement, speed, and acceleration of node \({x}_{2}\) at time \(t\mathrm{=}\mathrm{80 }s\):

\(\begin{array}{ccc}{x}_{2}(80)& \mathrm{=}& 4.1700{10}^{\mathrm{-}1}m\\ \dot{{x}_{2}}(80)& \mathrm{=}& \mathrm{-}4.3011{10}^{\mathrm{-}1}{\mathit{m.s}}^{\mathrm{-}1}\\ \ddot{{x}_{2}}(80)& \mathrm{=}& 3.3749{10}^{\mathrm{-}1}{\mathit{m.s}}^{\mathrm{-}2}\end{array}\)

Damped structure:

Move node \({x}_{2}\) to instant \(t=\mathrm{80 }s\):

\({x}_{2}(80)\mathrm{=}4.9867{10}^{\mathrm{-}1}m\)

2.3. Uncertainty about the solution#

Non-amortized case: analytical solution.

Damped case: semi-analytical solution.

2.4. Bibliographical reference#

1. VARE - Report HP61 /95/025/A- « Implementation of non-linear transient computation by substructuring in Code_Aster ».