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Spring element

The spring element may act on translations or rotations with a multi-linear stiffness. Such an element is associated to a diagram giving the corresponding effort as a functiun of the displacement or rotation.

Translation

Let \(L_o\) be the initial length of the spring and \(L\) the actual distance between both ends are \(X_1\) and \(X_2\). Then at a given iteration, \(\Delta x_k = L - Lo\) and the force generated by the spring is calculated as follows :

  • The modulus of the spring force is looked for in the diagram :

  • Then \(\vec{F_1}=F_k\frac {X_1\vec{X_2}} {L}\) and \(\vec{F_2}=-F_k\frac {X_1\vec{X_2}} {L}\)

The contribution of a spring element to the Virtual work principle writes :

\[ G^{spring}_{internal}=\vec{F_1}.\delta\vec{X_1}+\vec{F_2}.\delta\vec{X_2} \]

Rotation

For a flexjoint, the calculations are similar. The positions are replaced by the rotations of both ends and a moment is generated.

Its contribution to the Virtual work principle writes :

\[ G^{flexjoint}_{internal}=\vec{M_1}.\delta\vec{\Theta}_{x1}+\vec{M_2}.\delta\vec{\Theta}_{x2} \]

Note

Springs and flexjoints elements have no proper inertia.