Theory for Joint Constraints

Constraints are used to restrict the relative motion between the two components sharing a joint. You can specify the upper and lower bound of the relative motion. Applying constraints on the relative motion can be interpreted as placing an imaginary stopper that restricts the motion of the destination attachment with respect to the source attachment in a prescribed direction. Translational Constraints, Rotational Constraints, Inclination Angle Constraints, and Axial Rotation Constraints are discussed in this section.

Translational constraints are available for the Prismatic Joint, Cylindrical Joint, Screw Joint, Planar Joint, Slot Joint, and Reduced Slot Joint features.

The maximum relative displacement (umax) and minimum relative displacement (umin) can be prescribed. The activation condition for the constraint is:

For Planar Joint with radial constraints, the activation condition for the constraint is:

where the constraint radius prescribes the maximum relative displacement in radial direction (umax). The relative displacement in radial direction (ur) is computed as the distance between the constraint center and the initial position of the center of the joint.

A penalty method is used to enforce the constraints.

The constraint on the relative displacement is enforced using a stiff spring between the components. The stiffness of the spring is defined by the penalty factor. When the constraint is activated in a dynamic analysis, the time steps are reduced in order to conserve the momentum in the system.

The default value of the penalty factor pu is evaluated as:

The numerator in Equation 3-1 is an assumed relative velocity between the components when the constraint is applied, and the denominator is the maximum allowable penetration. This ratio of the relative velocity and the maximum allowable penetration decides the required stiffness of the spring (the penalty factor). The default value of this ratio is 103 s−1. This could, for example, be a penetration of 1 mm with an impact speed of 1 m/s. You can change this ratio based on the dynamics of the system and the allowable constraint violation.

	For very high penalty factors, the solution can be inaccurate unless the tolerances are tightened.

The factor me in the penalty factor expression is the effective mass at the joint, defined as

The source and destination masses msrc and mdst are defined as

The following assumptions are made in the definition of the effective mass:

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The computation of msrc and mdest considers the domains that are directly adjacent to the source and destination attachment. These domains cannot be the only parts affecting the dynamics of an impact.

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The computation of msrc and mdest also do not account for the mass added using Added Mass features or Mass and Moment of Inertia features.

The default value for the maximum time step allowed in order to capture the event accurately is evaluated as:

where T is the time period for free vibration of a spring-mass system having a stiffness equal to the penalty factor and mass equal to the effective mass. As can be seen from the expressions above, the default maximum time step is approximately 0.1 ms.

The constraint is violated for half of the period (T/2). To capture that phenomenon accurately, approximately 30 time steps are used. There is a tradeoff between accuracy and computation time. A larger number of steps reduces the maximum allowable time step and increases the accuracy in conserving the momentum. However, this also increases the computation time.

To prevent the solver from taking time steps larger than the maximum allowable time step, the penalty factor actually used is:

This means that modified penalty factor pmu is equal to the given penalty factor as long as the time step taken by the solver is smaller than the maximum allowable time step. If the constraint is violated, and the time step taken by the solver is larger than the maximum allowable time step, the solver is forced to decrease the time step.

	The penalty factor is not modified for quasistatic and stationary-parametric types of analyses because the conservation of momentum is not relevant in these cases.

The force required to constrain the relative motion is

To enforce a constraint condition, a virtual work contribution is added when constraints are active:

In the case of a Planar Joint, the relative displacement is constrained along a user-defined axis (the constraint direction) oriented in a plane perpendicular to the joint axis. The relative displacement in the constraint direction, uc, is:

where ec is the normalized constraint axis given by the user.

Rotational constraints are available for the Hinge Joint, Cylindrical Joint, Screw Joint, Planar Joint, and Reduced Slot Joint features.

The theory for rotational constraints is similar to what is described above for Translational Constraints.

Here, the penalty factor should be considered as a torsional spring. The effective mass in the default penalty factor is replaced by an effective moment of inertia, having an analogous definition.

Inclination angle constraints are available for the Ball Joint and Slot Joint features.

Here, the relative rotation to which the constraint is applied is the inclination of the destination axis with respect to the reference axis (also called an inclination angle or polar angle). The reference axis can be chosen as the initial source axis, initial destination axis, or as a user-defined arbitrary axis. The reference axis is always following the source attachment.

The constraint is applied only on the upper limit of the inclination angle. The upper limit of the inclination angle should be less than 180°. The constraint on the inclination angle is independent of the azimuthal angle and cannot be a function of azimuthal angle.

You give the maximum relative inclination angle (θmax) as a limit on the inclination of the destination attachment with respect to the reference direction. The actual constraint is applied to the cosine of the inclination angle:

where eR is the reference axis and ed1 is the destination axis.

The activation condition for an inclination angle constraint is written as:

The default value for the penalty factor and maximum allowable time step is computed as for Rotational Constraints. The moments of inertia for the source and destination components are computed about an axis that is passing through the joint center and is perpendicular to both the source and destination axes.

In order to prevent the solver from taking time steps larger than the maximum allowable time step, a modified penalty factor is defined as:

The moment required to enforce the inclination angle constraint is made proportional to the difference in the cosines of the angles, rather than to the difference in the angles themselves. This avoids operating with inverse trigonometric functions.

To enforce the constraint condition, a virtual work contribution is added for the cosine of the inclination angle:

Axial rotation constraints are available for the Ball Joint and Slot Joint features.

The relative rotation, on which the constraint is applied, is the relative rotation of the destination attachment about its own axis.

The constraint can be applied as both the upper and lower limits on the relative rotation of the destination attachment about its own axis. This constraint is uniquely defined as long as the maximum relative rotation is less than 180 degrees.

You prescribe the maximum and minimum axial relative rotation of destination attachment Ψmax and Ψmin. The cosine of the relative axial rotation of the destination attachment is written as:

Here (ar, br, cr, dr) represents the relative quaternion, and er is the normalized relative rotation axis.

The activation conditions for the maximum and minimum relative axial rotation constraints are written as:

The default value computation for the penalty factor and maximum allowable time step is the same as for Rotational Constraints. The moment of inertia for the source and destination components are computed about an axis that is passing through the center of joint and is parallel to the destination axis.