1. Reference problem#

1.1. Geometry#

_images/100000000000020F000000CCDA4B2773532EEC71.jpg

Beam cross sections:

  • support frame

\(\mathrm{S1}\):

\(A=1.2061{10}^{-2}{m}^{2}\)

\({I}_{z}=2.3681{10}^{-5}{m}^{4}\)

\(\mathrm{S5}\):

\(A=1.4621{10}^{-2}{m}^{2}\)

\({I}_{z}=2.8709{10}^{-5}{m}^{4}\)

\(\mathrm{S9}\):

\(A=1.5530{10}^{-2}{m}^{2}\)

\({I}_{z}=3.0493{10}^{-5}{m}^{4}\)

  • insulating columns

\(\mathrm{S2}\):

\(A=3.1428{10}^{-2}{m}^{2}\)

\({I}_{z}=4.5070{10}^{-5}{m}^{4}\)

\(\mathrm{S6}\):

\(A=3.2592{10}^{-2}{m}^{2}\)

\({I}_{z}=4.6738{10}^{-5}{m}^{4}\)

\(\mathrm{S10}\):

\(A=3.3416{10}^{-2}{m}^{2}\)

\({I}_{z}=4.7927{10}^{-5}{m}^{4}\)

  • fittings

\(\mathrm{S3},\mathrm{S11}\):

\(A=3.1944{10}^{-2}{m}^{2}\)

\({I}_{z}=1.15{10}^{-5}{m}^{4}\)

\(\mathrm{S7}\):

\(A=4.2130{10}^{-2}{m}^{2}\)

\({I}_{z}=1.15{10}^{-5}{m}^{4}\)

  • drivers

\(\mathrm{S4},\mathrm{S8}\):

circular \(R=6.055{10}^{-2}m\)

\(e=6.2{10}^{-3}m\)

1.2. Material properties#

\(\mathrm{M1}\):

\(E=2.{10}^{11}\mathrm{Pa}\)

\(\rho =8000\mathrm{kg}/{m}^{3}\)

(support frame)

\(\mathrm{M2}\):

\(E=5.{10}^{10}\mathrm{Pa}\)

\(\rho =2500\mathrm{kg}/{m}^{3}\)

(insulating column)

\(\mathrm{M3}\):

\(E=7.{10}^{10}\mathrm{Pa}\)

\(\rho =2700\mathrm{kg}/{m}^{3}\)

(aluminum connector and conductor)

1.3. Boundary conditions and loads#

Points \(A\), \(E\), \(I\): embedding

Points \(D\), \(L\): non-continuity of \({u}_{x}\), \({\theta }_{y}\), \({\theta }_{z}\)

Laplace forces on drivers \(\mathrm{DH}\), \(\mathrm{HL}\);

  • two-phase current \(\phi =\omega =100m\)

  • infinite conductors separated from \(1m\)

\(I\mathrm{=}{I}_{\mathit{eff}}\sqrt{2}(\mathrm{cos}(\omega t+\phi )\mathrm{-}{e}^{\mathrm{-}t\mathrm{/}\tau }\mathrm{cos}\phi )\)

\({I}_{\mathrm{eff}}\) effective current strength

\(\tau\) time constant

  • two short circuits with reclosing

1.4. Initial conditions#

\(t=0\), zero speed and acceleration.