1. Reference problem#

1.1. Geometry#

This is a problem originally proposed in the reference [bib1] and included in [bib2].

_images/10002198000069D500003D7F40D866C729DE86F7.svg

  • beam \(\mathrm{AB}\): slender, massless beam with a length \(\mathrm{AB}\), \(l=\mathrm{10 }m\) and a moment of inertia \({I}_{Z}=\mathrm{0,3285}{m}^{4}\).

  • point mass in \(B\): \(m=\mathrm{43,8}{10}^{\mathrm{3 }}\mathrm{kg}\)

1.2. Material properties#

Young’s module:

\(E=4.{10}^{10}\mathrm{Pa}\)

Density:

\(\rho =0\mathrm{kg}/{m}^{3}\)

1.3. Boundary conditions and loads#

Boundary conditions:

The only authorized movements are translations according to axis \(x\).

Point \(A\) is embedded: \(\mathrm{dx}=\mathrm{dy}=\mathrm{dz}=\mathrm{drx}=\mathrm{dry}=\mathrm{drz}=0\).

Loads:

  • modeling A: transverse acceleration at point A: \(\gamma (t)\)

_images/1000096200002834000009B79EBACC0FC9A77866.svg

  • B modeling: transverse force at point \(B\): \(\mathrm{Fx}(t)\) with \(\mathrm{Fx}(t)=–\mathrm{m.}\gamma (t)\)

1.4. Initial conditions#

The system is at rest: at \(t=0\), \(\mathrm{dx}(0)=0\), \(\mathrm{dx}/\mathrm{dt}(0)=0\) at all points.