64. Linear quadratic drag feature

The identification tag for this tutorial is PDS-ABU. Pregenerated input files for this tutorial are found in the folder named PDS-ABU in the provided tutorial input files.

64.1. Tutorial overview

This tutorial covers:

Use of the RigidBodyLinearQuadraticDrag feature.

Fig. 64.1 Sphere with linear and quadratic damping

64.2. Linear and quadratic damping

Note

The RigidBody linear quadratic drag feature feature applies linear and quadratic damping to a RigidBody regardless of its level of submersion.
This feature could be used, for example, to apply roll damping to a floating body or model the drag behavior of an unmanned underwater vehicle (UUV) such as a towed body, AUV (autonomous underwater vehicle), or ROV (remotely operated vehicle).
A RigidBody obeys the Newton-Euler equations for RigidBody motion.
When a new RigidBody is created in ProteusDS, no drag, damping, buoyancy or control forces will apply loads to a RigidBody. To apply loads to a RigidBody, a hydrodynamic feature (e.g. spheroid, cuboid) could be added, or a connection made with a cable, or a controller connection made.
A simplified representation of the RigidBody dynamics can be written as:

$M\dot{v}+C(v)v+D(v)v+F_g=F_{env}+F_{cable}+F_{contact}+...$

In the above equation, $M$ is the combined added mass and inertia matrices, $C$ accounts for coriolis and centripetal effects, and $D$ is the total system damping. $F_g$ is gravitational and restoring effects, $F_{env}$ is the applied wind and wave loads, $F_{cable}$ are the forces and moments applied to the body due to the cable connections, and F contact are soil contact forces. Additional forces could be added and in ProteusDS may include, joint forces or controller forces. The velocity v is the combination of the ocean currents and the body relative velocity in terms of the RigidBody local coordinate frame.
The damping or drag on a RigidBody that is either deeply submerged or floating ($D(v)v$) is often conveniently represented by linear and nonlinear (or quadratic) components. Many sources of damping may exist for a hydrodynamic body including: radiation damping, skin friction, wave drift damping, lifting forces, pressure induced drag, vortex shedding damping. It is often difficult to separate these effects, so the damping on a RigidBody may be adjusted or fully represented using linear and quadratic drag matrices which might be identified through experiment, or other means (e.g. experience). Consider the relative velocity $v_r$, such that the damping of a RigidBody is represented as the combination linear and quadratic terms:

$D(v_r)=D_{linear} + D_{quadratic}(v_r)$

In the above equation, $D_{linear}$ and $D_{quadratic}$ are 6x6 damping matrices. Given that v is the body velocity in the local coordinate frame: $v=[u\ v\ w\ p\ q\ r]^T$ and likewise the relative velocity is $v_r=[u_r\ v_r\ w_r\ p_r\ q_r\ r_r]^T$, the linear drag (or damping) matrix takes the form:

$D_{linear} = -\ \underbrace{\begin{bmatrix}X_u & X_v & X_w & X_p & X_q & X_r \\Y_u & Y_v & Y_w & Y_p & Y_q & Y_r \\Z_u & Z_v & Z_w & Z_p & Z_q & Z_r \\K_u & K_v & K_w & K_p & K_q & K_r \\M_u & M_v & M_w & M_p & M_q & M_r \\N_u & N_v & N_w & N_p & N_q & N_r \end{bmatrix}}_{\$LinearDragCoefficients}$

The quadratic drag (or damping) matrix is:

$D_{quadratic}(v_r) = -\ \underbrace{\begin{bmatrix}X_{uu} & X_{vv} & X_{ww} & X_{pp} & X_{qq} & X_{rr} \\Y_{uu} & Y_{vv} & Y_{ww} & Y_{pp} & Y_{qq} & Y_{rr} \\Z_{uu} & Z_{vv} & Z_{ww} & Z_{pp} & Z_{qq} & Z_{rr} \\K_{uu} & K_{vv} & K_{ww} & K_{pp} & K_{qq} & K_{rr} \\M_{uu} & M_{vv} & M_{ww} & M_{pp} & M_{qq} & M_{rr} \\N_{uu} & N_{vv} & N_{ww} & N_{pp} & N_{qq} & N_{rr} \end{bmatrix}}_{\$QuadraticDragCoefficients} {\begin{bmatrix}|u_r| \\|v_r| \\|w_r| \\|p_r| \\|q_r| \\|r_r| \end{bmatrix} }$

In ProteusDS, $QuadraticDragCoefficients and $LinearDragCoefficients as shown in the above equations are specified in the RigidBodyLinearQuadraticDrag feature.

64.3. Sphere with roll, pitch and yaw damping

Create a new project in PST.
Add a new RigidBody.
Add an Ellipsoid feature to the RigidBody, creating a sphere hydrodynamic object of 1 m in diameter.
Set the $CDt property for the Ellipsoid to 0.001.
Set the mass of the RigidBody to 268 kg such that the body will float at the surface half submerged.
Set the $Ix, $Iy, and $Iz to 26.8 kgm², which is mass moment inertia of a sphere with a 1 m diameter.
Set an initial state with heading rate of 10 deg/s.
Run a simulation for 100 seconds and view the results in PostPDS.
Plot the yaw velocity of the sphere. Note that it takes approximately 100 seconds for the yaw velocity of the sphere to reach 0 deg/s.

Fig. 64.2 Yaw velocity of sphere

Note

Quadratic sources of drag are very weak when velocities become slow. This is illustrated in the exponential decay of the yaw velocity.
Increasing skin friction drag of the sphere CDt to a larger value (e.g. 0.1) is not very effective, and also may result in drag loads too conservatively high during extreme sea states.
An alternative is to add a source of linear drag to help reduce low speed conditions round and smooth hulls.

Add a new RigidBodyLinearQuadraticDrag feature to the Library in PST; this feature will allow specification of linear and rotational drag for each degree of freedom.
Add the newly created RigidBodyLinearQuadraticDrag feature to the RigidBody using the $LinearQuadraticDrag property in the RigidBody input file. Set the feature location to be at the origin, aligned with the RigidBody local coordinate frame.

// Mass properties
$Ix 26.8
$Iy 26.8
$Iz 26.8
$Ixy 0
$Ixz 0
$Iyz 0
$DefineInertiaAboutCG 0
$CGPosition 0 0 0
$Mass 268

// Numerical
$Kinematic 0

$Ellipsoid rigidBodyEllipsoid 0 0 0 0 0 0
$LinearQuadraticDrag rigidBodyLinearQuadraticDrag 0 0 0 0 0 0

Add quadratic roll, pitch, and yaw damping to the the RigidBodyLinearQuadraticDrag feature that was created: $K_{pp} = M_{qq} = N_{rr} = 100$.
Add a linear heave damping of 500 to reduce low amplitude heave oscillations: $Z_w = 500$.

// Fluid loading
$LinearDragCoefficients 0 0 0 0 0 0
$LinearDragCoefficients 0 0 0 0 0 0
$LinearDragCoefficients 0 0 500 0 0 0
$LinearDragCoefficients 0 0 0 0 0 0
$LinearDragCoefficients 0 0 0 0 0 0
$LinearDragCoefficients 0 0 0 0 0 0

$QuadraticDragCoefficients 0 0 0 0 0 0
$QuadraticDragCoefficients 0 0 0 0 0 0
$QuadraticDragCoefficients 0 0 0 0 0 0
$QuadraticDragCoefficients 0 0 0 100 0 0
$QuadraticDragCoefficients 0 0 0 0 100 0
$QuadraticDragCoefficients 0 0 0 0 0 100

Note

Fig. 64.3 and Fig. 64.4 show the heave motion and yaw velocity of the sphere. Note that now the heave motion of the sphere is damped, and the yaw velocity reduces to zero quicker with the increased quadratic drag.

Fig. 64.3 Heave motion of sphere with linear heave damping

Fig. 64.4 Yaw velocity of sphere with quadratic damping