Here the subscript ‘a’ indicates the centroid of attachment, and ‘c’ indicates the center of joint. Ra is the rotation matrix representing the rotation of the attachment, and
u represents the displacements.
When using weak constraints, the joint constraints are applied in a weak form, and the values of the Lagrange multipliers give the joint forces (Fw) and moments (
Mw). This works for both flexible and rigid components. The use of Lagrange multipliers can, however, have an effect on the structure of the equation system, which limits the solvers that can be used.
When using penalty method, the joint constraints are applied using a penalty formulation, and the values of the penalty forces (Fp) and moments (
Mp) give the joint forces and moments. This works for both flexible and rigid components. Moreover, this also works for overconstrained rigid-body systems in which the weak constraints method might not work due to an insufficient number of equations for the Lagrange multipliers.
where cu and
cθ are the translational and rotational violations of the joint constraints, respectively. The penalty factor
pj is the input having a default value,
0.1dEequ. In this expression,
Eequ is an effective Young’s modulus, and
d is the geometric diagonal of the bounding box of the geometry.