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.
The maximum relative displacement (umax) and minimum relative displacement (
umin) can be prescribed. The activation condition for the constraint is:
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.
The factor me in the penalty factor expression is the effective mass at the joint, defined 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.
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.
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.
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 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.
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 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.