DAE index 3

DAE in one paragraph

For an unconstrained rigid body, Newton’s second law gives ODEs in position and velocity. Add a constraint — a pin joint, a slider, a gear mesh — and the equations of motion now contain both ODEs and algebraic relations that the configuration must satisfy at every instant. The combined system is a differential-algebraic equation. The index counts how many times the constraint equations must be differentiated to recover an ODE.

Mechanical multi-body systems with holonomic constraints (geometric constraints on positions only) are typically index-3:

• Differentiate once → constraint becomes a velocity-level relation.
• Differentiate twice → constraint becomes an acceleration-level relation.
• Differentiate three times → the system is recoverable as an ODE.

Why index-3 is hard

The natural form of the equations of motion for a constrained system is the augmented Lagrange formulation:

M q̈ + Cqᵀ λ = F
C(q)        = 0

where q is the generalised position, M is the mass matrix, C(q) are the holonomic constraints, Cq is the constraint Jacobian, λ is the Lagrange-multiplier vector, and F are applied forces. The constraint C(q) = 0 lives at the position level — three orders below the dynamics. Naive integration of the ODE part alone lets C(q) drift; constraint stabilisation is required to keep ‖C‖ bounded across long simulations.

Solver structure

Our dynamics layer integrates the augmented system directly:

[M     Cqᵀ] [q̈]   [F − γ]
[Cq    0  ] [λ ] = [γq]

where γ is a quadratic-velocity term and γq is the second derivative of the constraint. At every integration step:

1. Position-level Newton-Raphson solve enforces C(q) = 0.
2. Velocity-level projection enforces Cq q̇ = 0.
3. The augmented linear system above gives q̈ and λ simultaneously.
4. RK45 advances the (q, q̇) state.
5. Stabilisation keeps the position and velocity constraints satisfied to machine precision.

The reward for the bookkeeping: λ falls out of the solver as joint reaction forces in Newtons, directly usable for fatigue analysis and bearing sizing — no post-processing.

Why this matters

Most introductory MBSD treatments either:

• Reduce the index to 1 by analytical elimination of the constraints (only feasible for textbook examples), or
• Use penalty methods that add stiff springs at the constraint level (numerically badly conditioned, kills RK45 step-size).

Our solver goes through the augmented index-3 form directly because:

• The four classical filters in synthesis (Grashof, transmission angle, branch, order) have direct readings on the index-3 quantities.
• λ is a primary output, not a derived one. Joint forces in Newtons arrive without an additional model.
• The same solver runs the practica four-bar, the engine NVH chain, and the quarter-car suspension. One implementation, three benchmarks.

See also

• Lagrange multiplier — the λ in the augmented system, exposed as joint reaction force.
• Four-bar linkage — the canonical index-3 example.
• Practica case study — full solver pipeline end-to-end.