## Constraints: Calculated DOFs |

Calculated DOF constraints can be either **Cartesian** or **curvilinear**. The Cartesian type will be sufficient in the vast majority of modelling applications, but many advanced capabilities become available when using curvilinear constraints, such as the ability to model the motion of the out-frame along arbitrary curves and surfaces, mixed calculated and imposed motion, and dynamic release of the out-frame based upon user-specified criteria.

Determines which degrees of freedom (DOFs) are calculated. In each case, if **free** is checked then the DOF is calculated, otherwise it is fixed. The translational DOFs are **x**, **y**, **z**, and the rotational DOFs are **Rx**, **Ry**, **Rz**. These degrees of freedom are relative to the in-frame axes.

An **initial value** is required for each free DOF for the initial iteration of the static analysis. Often this can be left at the default of zero, but in some cases static convergence can only be achieved if this value is specified appropriately. The use calculated positions function can be used to capture this value following a successful statics analysis. If the constraint has a double-sided connection, then the initial DOF values are not available because they are superseded by the out-frame's initial position and orientation data.

Note: | The degrees of freedom page is only relevant to Cartesian constraints. It is not available when the model type is curvilinear. |

The out-frame can be disconnected from the in-frame at the start of any stage of the simulation, to release it from the in-frame. It will then move completely independently of the in-frame, as if all six of its DOFs are free. Constraint stiffness and damping forces will be ignored, but any applied loads will still have an effect. Any child objects of the constraint will remain connected to the out-frame after it has been released. For no disconnection, set this item to '~'.

Note: | If a constraint has no children connected to it, other than other constraint objects, then the dynamic equations associated with its free DOFs become singular because the constraint is effectively a massless object with no physical properties of its own. In this case, all free DOFs are ignored and the out-frame is assumed to remain rigidly attached to the in-frame in the position it was in when the last of its children (ignoring other constraints) was released. |

Multiple applied loads can be defined, with respect to either global axes or local out-frame axes. An applied load is defined by specifying its point of application, relative to the out-frame, and its components (which may be constant, vary with simulation time, or be given by an external function). An applied load acts as if its point of application were rigidly attached to the out-frame.

For some models it may be desirable to explicitly set a characteristic length and characteristic force for the constraint. These characteristic scales directly affect the convergence criteria of the iterative solvers employed in the analysis. The data does not appear on the constraint data form but can be found on the all objects data form.

Note: | These data appeared on the constraint data form prior to version 11.1; however, now that all objects with free degrees of freedom now have associated characteristic scales, the data have been moved to a single location on the all objects data form. |

The constraint stiffness represents translational and rotational springs which resist displacement of the out-frame relative to the in-frame. The damping represents translational and rotational dampers which resist velocity of the out-frame relative to the in-frame.

Stiffness and damping may be linear or nonlinear. Linear data are specified by a constant value that is interpreted as follows:

- Linear stiffness is multiplied by relative displacement to produce the stiffness load.
- Linear damping is multiplied by relative velocity to produce the damping load.

Nonlinear data are specified as variable data:

- Nonlinear stiffness is specified by tabulating stiffness load against relative displacement.
- Nonlinear damping is specified by tabulating damping load against relative velocity.

Nonlinear stiffness data can optionally be treated as hysteretic.

The stiffness and damping are *isotropic*, which means that each is the same in all directions, e.g. motion along the $x$-axis has the same stiffness as motion along the $y$-axis. *Non-isotropic* models can be constructed by connecting two or more constraint objects together in a chain, with different stiffness and damping data in each. For instance, the first constraint could specify a translational stiffness of 1kN/m with the $x$ DOF free, and the second constraint could specify a translational stiffness of 2kN/m with the $y$ DOF free.

Linear stiffness is defined by a single value, which applies to both positive and negative displacements.

Nonlinear stiffness, for a Cartesian constraint *with a single free DOF*, is *double-sided*. That is, the stiffness-displacement table should contain values of both positive and negative displacement. The values may be symmetrical about zero displacement, but do not have to be so (e.g. positive and negative displacements along the $x$-axis can give rise to different stiffnesses). This applies to the translational DOFs and the rotational DOFs separately and independently.

Nonlinear stiffness should otherwise be *single-sided* (with just positive displacements), for Cartesian constraints with more than one free translational DOF or more than one free rotational DOF, and for *all* curvilinear constraints. Why? Well because, for Cartesian constraints, it is not possible to unambiguously assign a sign to a displacement in more than one dimension without introducing discontinuities; for curvilinear constraints, there can be ambiguity even for one-dimensional motion (e.g. if the permitted motion of the out-frame does not pass through the origin of the in-frame).

This choice is available for directly solved Cartesian constraints with a single free rotational DOF. If rotational winding is enabled, then the angle of rotation, $\theta$, between the out-frame and the in-frame, is treated as unbounded (i.e. it can take any real value). This means that the stiffness moment continues to increase as the out-frame performs a full 360° rotation around the in-frame, and increases still further over each subsequent winding (e.g. $\theta$=720°, $\theta$=1080°, etc.). If rotational winding is not enabled, then $\theta$ is confined to lie within the primary range of -180° < $\theta \leq$ 180°. This means that a rotation of $\theta$ = 181° is treated as if it were a rotation of $\theta$ = -179° for purposes of the stiffness calculation.

Note: | Owing to technical considerations, rotational winding is not permitted if the constraint solution method is indirect, or if the constraint is not Cartesian, or if there is more than one free rotational DOF. |

Nonlinear stiffness variable data can optionally be treated as hysteretic. This is used to model non-elastic stiffness effects that depend upon the past history of the motion. The model used is directly analogous to the one used for line hysteretic bending. The independent variable in the line hysteretic bending model is curvature, whereas for constraints the independent variable is displacement or angular displacement; similarly, the dependent variable in the line model is bend moment, whereas for constraints it is stiffness force or moment. Hysteretic nonlinear stiffness data must always be single-sided, even in the case of a single free DOF, because the hysteresis model is not compatible with double-sided data.

Warning: | For constraints with two or three rotational degrees of freedom, the hysteretic rotational stiffness model is strictly only valid for small angular displacements. It may not be appropriate in cases where the out-frame rotates significantly relative to the in-frame. As a rough guide, angular displacements of greater than 30° should be treated with caution, but this will vary on a case-to-case basis. |

Linear damping is defined by a single value, which applies to both positive and negative velocity. Positive velocity corresponds to increasing displacement between the in-frame and the out-frame.

Nonlinear *translational* damping data are two-sided: they can be specified for both positive and negative velocity. The translational damping here is analogous to that of spring-damper link objects. Imagine an invisible link connecting the in-frame to the out-frame. This link applies a stiffness force between the two frames in the usual way. However, the damping force does not resist the instantaneous relative velocity of the out-frame relative to the in-frame, but rather resists the rate of change of the magnitude of the displacement between the two frames (i.e. the rate of change of the length of the link) and acts along the line between the two frames. Motions of the out-frame that do not change the magnitude of the displacement between the two frames (i.e. perpendicular to the link) are therefore undamped. This assumption is made to provide consistency with OrcaFlex links and to avoid introducing additional moments between the in-frame and the out-frame.

Nonlinear *rotational* damping data are one-sided, with only positive values of relative angular velocity. The damping moment applied is proportional to the magnitude of the instantaneous angular velocity between the out-frame and the in-frame, in the direction opposing that angular velocity.

To be well-defined, stiffness and damping data should usually specify zero load at zero displacement (stiffness) and zero relative velocity (damping). The only exception to this rule is the case of Cartesian constraints with no more than a single translational DOF and no more than a single rotational DOF, for which (as above) it is possible to unambiguously define a non-zero stiffness force at zero displacement and angular displacement. However, the damping force must always be zero at zero relative velocity in all cases, without exception.

Note: | The same stiffness and damping model is applied to both Cartesian and curvilinear constraints. In the case of curvilinear constraints, this means that the stiffness and damping forces depend only upon the position and velocity of the out-frame relative to the in-frame, and not upon the particular choice of curvilinear coordinates that have been defined. The resultant loads will be resolved into components acting tangentially and normal to the constraint manifold in order to compute their effects upon the out-frame's motion and connection loads respectively. |