ProductsAbaqus/StandardAbaqus/Explicit A universal joint is implemented in Abaqus/Standard as a multi-point constraint, defining the total rotation of the constrained (“secondary”) node (the first node of the MPC), C(ϕS) , as the total rotation of the “main node” (the second node of the MPC), C(ϕM) , followed by two relative rotations: ϕ1 about the first axis of the joint a1 , then ϕ2 about the second axis of the joint a2 (which is orthogonal to a1 ): C(ϕS)=C(ϕ2a2)⋅C(ϕ1a1)⋅C(ϕM).
The first joint axis, a1 , rotates with the rotation of the main node: a1=C(ϕM)⋅a1|0.
The second joint axis has this rotation plus the rotation about the first joint axis: a2=C(ϕ1a1)⋅C(ϕM)⋅a2|0.
The angular velocity of the secondary node is ˙ϕS=˙ϕM+˙ϕ1a1+˙ϕ2a2;
and the virtual variations of the rotations are, likewise, δϕS=δϕM+δϕ1a1+δϕ2a2.
Thus, the joint imposes three constraints (each component of the angular velocity of the secondary node is constrained) but introduces two additional degrees of freedom in the form of the relative rotations ϕ1 and ϕ2 . This means the joint provides a total of one constraint to the model if ϕ1 and ϕ2 are not prescribed or up to three constraints if they are. The virtual work contribution of the joint is MS⋅δϕS+MM⋅δϕM+M1δϕ1+M2δϕ2=0,
where MS is the total moment at node S, MM is the total moment at node M, and M1 and M2 are the moments in the joint hinges. Applying the constraints (Equation 1), this is (MS+MM)⋅δϕM+(MS⋅a1+M1)δϕ1+(MS⋅a2+M2)δϕ2=0.
If there are no further constraints associated with the nodes of the joint, δϕM, δϕ1 and δϕ2 are independent variations, so that the constrained virtual work equation implies that MS=-MM,
M1=-MS⋅a1=MM⋅a1
and M2=-MS⋅a2=MM⋅a2.
Because the universal joint is implemented in this manner, the relative rotations in the joint, ϕ1 and ϕ2, appear as degrees of freedom in the model (degree of freedom 6 at the third and fourth nodes of the MPC). Moments M1 and M2 can, therefore, be applied in the joint by specifying their values as concentrated loads; ϕ1 and ϕ2 can be given prescribed variations in time by specifying boundary conditions; or stiffness and/or damping can be associated with relative rotations of the joint by attaching springs and/or dashpots to ground to these degrees of freedom (springs or dashpots to ground are used because the variables are relative rotations). |