# Friction theory

OrcaFlex provides a simple friction model that can give an approximate representation of contact friction. This is commonly used to model seabed friction, friction on mid-water arches, guide tube friction etc.

### Seabed friction

In reality, seabed interaction is much more complicated than simple friction: it involves effects such as the soil being displaced by a line as it moves, accumulation of soil in front of the line, etc. Some of these effects can be captured by defining a general seabed tangential resistance profile; however, to model seabed interaction accurately would require much more information about the soil structure and would involve modelling the soil itself, which is beyond the scope of OrcaFlex.

### Overview of the OrcaFlex friction model

OrcaFlex models friction for contact with the seabed or elastic solids. Hereafter we use the term solids to refer to either the seabed or elastic solids.

 Notes: Friction for contact with elastic solids is only included during the dynamic simulation, and not in statics. Not all OrcaFlex model objects take into account contact with solids: lines, 3D buoys and 6D buoys are the only objects which do.

Friction is modelled as Coulomb friction in the solid plane. Most objects which are in contact with a solid keep track of a friction target position, and a friction force is applied which acts towards this target position. The exception is lines in contact with the seabed that have opted to use the decoupled lateral and axial friction model. This model is slightly different and will be explained below.

## Coulomb friction model

The standard Coulomb friction model states that a friction force of $\mu R$ is applied, where $\mu$ is the friction coefficient and $R$ is the contact reaction force. However, this model defines a discontinuous force / deflection relationship, which would be impossible for a program like OrcaFlex to solve. Instead we use a modified Coulomb model, as illustrated here:

 Figure: Standard Coulomb and modified Coulomb friction models

In the modified Coulomb friction model, the transition from a friction force of $-\mu R$ to $+\mu R$ takes place as a linear variation over the deflection range $-\Dcrit$ to $+\Dcrit$. We define $\Dcrit$ as $$\Dcrit = \frac{\mu R}{k\urm{s}a}$$ where

$k\urm{s}$ is the shear stiffness

$a$ is the contact area.

The friction force can be thought of as being ramped from 0 to a maximum value of $\mu R$ as the deflection increases. Higher values of $k\urm{s}$ lead to the ramping taking place over a shorter distance, lower values lead to a longer ramping distance.

### Calculation of friction coefficient

The friction coefficient $\mu$ is defined as follows:

• For contact between lines and the seabed the friction coefficient is defined by the line type data.
• For contact between 3D buoys and the seabed the friction coefficient is defined by the 3D buoy data.
• For contact between 6D buoys and the seabed the friction coefficient is defined by the 6D buoy data.
• For contact between lines, 3D buoys and 6D buoys and elastic solids, the friction coefficient is defined on the friction coefficients data form.

For line friction, both lateral and axial coefficients can be defined. If these values are different, OrcaFlex uses a single value of $\mu$ defined at each node as $$\mu = \lvert \mu_\mathrm{l} d_\mathrm{l} + \mu_\mathrm{a} d_\mathrm{a} \rvert$$ where, defining $\vec{d}$ as a unit vector in the plane of the solid, from the node towards the target position, $d_\mathrm{l}$ and $d_\mathrm{a}$ are the vector components of $\vec{d}$ in the node's lateral and axial directions respectively.

### Target positions in statics

Friction is only applied in statics for contact between lines and the seabed.

If the step 1 statics method is prescribed then the target positions are laid out along the prescribed shape.

For all other step 1 statics methods the target positions are laid along the seabed in the direction specified by the lay azimuth direction.

The distance between each target position is determined by the unstretched segment length and the As laid tension. If each node was to sit at its target position then the effective tension throughout the line would be the As laid tension.

### Target positions in dynamics

During the dynamic simulation, at the end of each time step, each friction target position is updated using the following rules:

• At the start of the dynamic simulation, for lines in contact with the seabed, the target position from statics is used. Otherwise the target position is set equal to the contact position.
• If an object is not in contact then the target position is not defined since no friction is applied.
• If an object has just come into contact then the target position is set equal to the contact position.
• If the target position is a distance greater than $\Dcrit$ from the contact position then the target position is placed at a distance exactly $\Dcrit$ from the contact position on the vector $\vec{d}$, as shown in the following figure. This can be thought of as dragging the target position towards the contact position.
• Otherwise the target position is not modified.

 Figure: Updating friction target positions during dynamics

### Shear damping

Objects moving in the solid plane may also experience a damping force. Let $d$ be the distance between the contact position and the target position. If $0 \leq d \leq \Dcrit$, then a damping force $f_\mathrm{D}$ is applied in the solid plane to the object with magnitude given by $$f_\mathrm{D} = \left(1-\frac{d}{\Dcrit}\right) \min\left\{ 2\lambda (mka)^{{1/2}} v_\mathrm{s},\ \mu R \right\}$$ where

$\lambda$ is percentage of critical contact damping / 100

$m$ is the mass of the object

$k$ is the solid normal stiffness

$a$ is the contact area

$v_\mathrm{s}$ is the component of velocity in the solid plane.

## Decoupled lateral and axial Coulomb friction model

This model is only used for lines in contact with the seabed, and only if the user has chosen this option on the line data form. This model does not average the lateral and axial friction coefficients. Instead, the lateral displacement (perpendicular to the axis of the line) and axial displacement (parallel to the axis of the line) are considered independently. Accordingly, there are separate $\Dcrit$ values, $\Dcritl$ and $\Dcrita$, for each direction:

$$\Dcritl = \frac{\mu_\mathrm{l} R}{k\urm{s}a} \qquad \Dcrita = \frac{\mu_\mathrm{a} R}{k\urm{s}a}$$

The lateral and axial displacements are accumulated by considering the displacement in each time-step and projecting this onto the lateral and axial directions at the the start of that static analysis/dynamics time step. This yields separate forces in the lateral and axial directions, $\vec{f}_l$ and $\vec{f}_a$, respectively, which are then summed vectorially to produce the total reaction force. This is the same procedure by which the lateral and axial contributions are decoupled for general tangential resistance profiles and more precise definitions can be found by referring to that help section.

As with the coupled friction model, there is the concept of a friction target position; however, rather than a single, two-dimensional, target position, there are instead two separate one-dimensional target positions, one lateral and one axial. The two one-dimensional target positions are updated in the analogous way to the their two-dimensional, coupled counterpart. Similarly, the concept of shear damping is also carried over from the two-dimensional model to the two one-dimensional models in the corresponding way.

 Note: Matters are slightly more complicated for a curved seabed. This is handled in the same way as for a general tangential resistance profile.

## Moments induced by friction

### Lines

If torsion is included on the line then typically the friction force is applied at the point of contact with the solid. For lines this is at the line outer edge, as defined by the outer contact diameter. In the case of seabed contact only, the apply seabed contact forces at centreline option can be selected to no longer consider this offset to the contact point.

Moments in OrcaFlex are reported at the centreline and so the moment arm effect of applying the friction force at the outer edge results in a moment being applied to the node. The applied moment is given by the vector cross-product $\vec{p}{\times}\vec{f}$, where $\vec{p}$ is the point of contact relative to the node origin and $\vec{f}$ the applied friction force.

This frictional moment effect can most easily be visualised by considering a pipe being dragged laterally across a surface with friction. It is intuitively obvious that the friction force will result in the pipe being twisted.

If torsion is not included or (for seabed contact only) the apply seabed contact forces at centreline option has been selected, then the contact forces are always applied at the line centreline.

### 3D and 6D buoys

Since 3D buoys do not have rotational degrees of freedom, no such moments are applied.

For 6D buoys, the friction force is applied at each vertex in contact with the solid. If the vertex if offset from the buoy origin then the appropriate moment, about the buoy origin, will be generated.