Multibody Dynamics (MBD)
Joints, Constraints & Flexible Body Motion

MBD is the right tool here. Regular FEM handles deformation but assumes small overall rigid body motion. Your door swings through 90° of rotation — that's the opposite: large rigid body motion with (hopefully) small structural deformation. MBD models the door and body as rigid or semi-rigid bodies connected by a revolute hinge joint that constrains 5 of the 6 relative DOF, leaving 1 rotation free. You drive the analysis with a motion profile or an applied opening force. The output gives you hinge pin forces, door velocity, and if needed you can feed those forces back into a detailed FEM stress analysis of the hinge bracket.

Start with N bodies, each with 6 DOF (3 translational, 3 rotational) — that's 6N coordinates total. Each joint adds constraint equations that remove DOF. A revolute joint removes 5 DOF; a ball-and-socket (spherical) joint removes 3; a prismatic joint removes 5 (keeps only one translation). The net system DOF = 6N minus the number of constraint equations. The equations of motion are typically assembled using Lagrange's equations with multipliers:

Here \(q\) are generalized coordinates, \(\Phi_q\) is the constraint Jacobian, \(\lambda\) are the Lagrange multipliers (physically: the constraint reaction forces at joints), \(Q\) is the applied force vector, and \(\gamma\) is the kinematic acceleration term. This is a DAE — solving it requires specialised integrators (like Gear's method or the HHT-I3 method) that maintain constraint drift under control.

Both formulations are equivalent — they produce the same trajectories. The conceptual difference: Lagrange equations work in generalised (possibly minimal) coordinates and naturally eliminate constraint forces from the dynamics — you only solve for the free DOF. This is elegant and efficient for simple chain topologies (like a robot arm). Newton-Euler equations work directly with Cartesian positions and orientations plus explicit constraint force terms. They produce larger systems (including constraint forces as unknowns) but handle arbitrary topologies — closed loops, parallel mechanisms — more naturally. Modern MBD codes like MSC ADAMS use the Newton-Euler + Lagrange multiplier approach (the "augmented" formulation) because it handles closed kinematic loops and redundant constraints robustly without special treatment.

Rigid body MBD is accurate when structural deformations are small compared to the mechanism's range of motion. You need flexible multibody dynamics when deformation affects the motion trajectory — for example, a long robotic arm whose tip deflects under load, changing the end-effector position; or an automotive leaf spring whose bending compliance is essential to the suspension kinematics. The Craig-Bampton Component Mode Synthesis (CMS) method reduces the FEM body to a compact superelement: you retain a set of "fixed-interface normal modes" (vibration modes with joints fixed) plus "constraint modes" (static shapes due to unit displacement at each joint DOF). Together, these maybe 50–200 modes represent the flexible body in the MBD simulation with a tiny fraction of the full FEM DOF count. This is the standard approach in ADAMS Flex, SIMPACK, and Abaqus/Explicit flexible-body coupling.

The standard library covers most mechanisms. A revolute joint (hinge) constrains all DOF except one rotation — door hinges, crankshafts, wheel hubs. A prismatic joint allows only one translational DOF — hydraulic cylinders, drawer slides. A spherical joint (ball-and-socket) allows all three rotations but no translation — steering tie rods, universal joints. A cylindrical joint allows one rotation and one co-axial translation — a nut on a threaded rod. A planar joint constrains one translation and two rotations — a part sliding on a flat surface. Beyond standard joints, modern codes support bushing elements (springs and dampers in all 6 DOF), friction, and clearance joints — critical for realistic mechanism analysis. In automotive suspension, for example, rubber bushings with frequency-dependent stiffness are modelled as ADAMS bushing elements between subframe and body.