# Spring test

## 1 Test description

This test uses a model composed of two nodes and one element. One node is attached to a linear spring (translational or rotational). A load (force or moment) is applied to that node and the node displacement is observed. This test is carried out both with the linear and nonlinear solvers (see Time domain simulation). Aerodynamic loads and gravity loads are disabled.

Several load cases are run, with springs and loads in all 6 degrees of freedom

The figure below illustrates the model with a spring in the x-direction. The red vector represents a force in the x-direction.

## 2 Analytical solution

For a degree of freedom
$$i$$
, the displacement is equal to
$$\delta_i=\frac{P_i}{K_i}$$

where
• $$P_i$$
is the load in the
$$i^{th}$$
degree of freedom, in
$$\text{m}$$
for forces and
$$\text{Nm}$$
for moments
• $$K_i$$
is the stiffness in the
$$i^{th}$$
degree of freedom, in
$$\text{N}\cdot\text{m}^{-1}$$
for translational springs and
$$\text{Nm}\cdot\text{rd}^{-1}$$
for rotational springs

Note: the rotational displacement of a node in Ashes is given in degrees

The table below summarises the load cases run in this test and the analytical solution.The parameters given in this table are
• the forces along the x, y and z axes,
$$F_x, F_y$$
and
$$F_z$$
, in
$$\text{kN}$$
• the moments around the x, y and z axes,
$$M_x, M_y$$
and
$$M_z$$
, in
$$MNm$$
• the stiffnesses along the x, y and z axes,
$$K_{11}, K_{22}$$
and
$$K_{33}$$
, in
$$\text{kN}\cdot\text{m}^{-1}$$
• the stiffnesses around the x, y and z axes,
$$K_{44}, K_{55}$$
and
$$K_{66}$$
, in
$$\text{MNm}\cdot\text{rd}^{-1}$$
• the displacements of the node on the spring along the x, y and z axes
$$u,v$$
and
$$w$$
, in
$$\text{m}$$
• the rotational displacements of the node on the spring along the x, y and z axes
$$ru, rv$$
and
$$rw$$
, in
$$\text{deg}$$

Note: the displacement is only compared to an analytical solution for the degrees of freedom for which a stiffness is provided and a load is applied. For other degrees of freedom, the displacement will not be in general 0, but is not used in this test.

 Load case name Load Stiffness Displacement $$F_x$$ $$F_y$$ $$F_z$$ $$M_x$$ $$M_y$$ $$M_z$$ $$K_1$$ $$K_2$$ $$K_3$$ $$K_4$$ $$K_5$$ $$K_6$$ $$u$$ $$v$$ $$w$$ $$ru$$ $$rv$$ $$rw$$ Translational x-direction 100 0 0 0 0 0 5 000 0 0 0 0 0 0.02 - - - - - Translational x-direction, negative force -200 0 0 0 0 0 5 000 0 0 0 0 0 -0.04 - - - - - Translational y-direction 0 100 0 0 0 0 0 6 000 0 0 0 0 - 0.0167 - - - - Translational y-direction, negative force 0 -200 0 0 0 0 0 6 000 0 0 0 0 - -0.0333 - - - - Translational z-direction 0 0 10 0 0 0 0 0 1 000 0 0 0 - - 0.01 - - - Translational z-direction, negative force 0 0 -25 0 0 0 0 0 1 000 0 0 0 - - -0.025 - - - Translational xyz-direction 200 100 -25 0 0 0 5 000 5 000 1 000 0 0 0 0.04 0.02 -0.025 - - - Rotational x-direction 0 0 0 10 0 0 0 0 0 50 000 0 0 - - - 0.01146 - - Rotational y-direction 0 0 0 0 10 0 0 0 0 0 50 000 0 - - - - 0.01146 - Rotational z-direction 0 0 0 0 0 10 0 0 0 0 0 50 000 - - - - - 0.01146

In addition to this 9 load cases, another set of 9 load cases with the same input is run with the linear solver instead of the nonlinear one. It is expected that the results will be the same.

## 3 Results

In Ashes, the loads are applied with a ramp-up, i.e. they start at 0 and are progressively increased until their value (see Analysis). Therefore, the output produced by Ashes will show a ramp-up period as well.

A test is considered passed when all values of the last second of the results produced by Ashes are within 0.01% of the analytical solution

The report for this test can be found on the following link: