Case studies, mechanism tools and glossary entries on multibody dynamics and mechanism design.https://multibodysystemsdynamics.com/https://multibodysystemsdynamics.com/case-studies/engine-nvh-multicylinder/https://multibodysystemsdynamics.com/case-studies/engine-nvh-multicylinder/A multi-cylinder engine modelled as a coupled bank of slider-cranks. The dynamics solver decomposes the resulting shaking force and shaking moment into harmonic components — first, second, fourth, sixth — then drives a counterweight + balance-shaft co-design that flattens the dominant 1× and 2× signatures simultaneously. Same engine that synthesised the practica four-bar; same constraint Jacobian, same RK45 augmented solver.Fri, 08 May 2026 00:00:00 GMThttps://multibodysystemsdynamics.com/case-studies/practica/https://multibodysystemsdynamics.com/case-studies/practica/A three-position rigid-body guidance problem solved end-to-end — from DNI-keyed input through Burmester closed-form synthesis, all four defect filters, animated kinematics, and dynamic validation with joint reaction forces. Final design hits all three precision points at 0.0000 cm trace error and clears a 26.6° transmission-angle floor.Fri, 01 May 2026 00:00:00 GMThttps://multibodysystemsdynamics.com/case-studies/quarter-car-suspension/https://multibodysystemsdynamics.com/case-studies/quarter-car-suspension/A quarter-car suspension model — sprung mass, unsprung mass, tyre stiffness, spring-damper — exercised across the same MBSD engine that runs four-bar synthesis and engine NVH. We sweep passive design space (spring rate, damper coefficient) on a ride-vs-handling Pareto front, then layer an active control loop on the same plant. One simulator, three tasks: design space exploration, road-profile response, and active-control tuning.Fri, 08 May 2026 00:00:00 GMThttps://multibodysystemsdynamics.com/tools/four-bar-linkage-calculator/https://multibodysystemsdynamics.com/tools/four-bar-linkage-calculator/Grashof classification, rocker swing, transmission angle and coupler curves for any four-bar, checked live as you drag.https://multibodysystemsdynamics.com/tools/slider-crank-kinematics/https://multibodysystemsdynamics.com/tools/slider-crank-kinematics/Stroke, slider velocity, rod angle and quick-return ratio for in-line and offset slider-cranks, with the equations behind them.https://multibodysystemsdynamics.com/glossary/burmester-theory/https://multibodysystemsdynamics.com/glossary/burmester-theory/Burmester theory is the closed-form geometric method for synthesising a four-bar linkage that carries a rigid coupler body through a prescribed set of precision positions. For three positions there is a one-parameter family of solutions; for four or five positions the solutions reduce to a finite set determined by the Burmester curves of pole points. We use it to hit precision targets in microseconds — without optimisation, without seeding, without stochastic search.https://multibodysystemsdynamics.com/glossary/coupler-curve/https://multibodysystemsdynamics.com/glossary/coupler-curve/The coupler curve of a four-bar linkage is the trajectory traced by a single point on the coupler link as the crank rotates through one full revolution. For a generic planar four-bar the curve is algebraic of degree six (a sextic) and can include cusps, double-points, and self-intersections. It is the source of the four-bar's expressive power and the design target of path-generation synthesis.https://multibodysystemsdynamics.com/glossary/dae-index-3/https://multibodysystemsdynamics.com/glossary/dae-index-3/A differential-algebraic equation (DAE) is a system that mixes ordinary differential equations with algebraic constraints. Constrained multi-body mechanical systems naturally produce index-3 DAEs — three differentiations of the constraint equations are required to recover an ordinary differential equation. We solve them via the augmented mass-Jacobian system [M, Cqᵀ; Cq, 0] integrated with RK45, exposing the Lagrange multipliers as joint reaction forces in Newtons.https://multibodysystemsdynamics.com/glossary/force-balancing/https://multibodysystemsdynamics.com/glossary/force-balancing/Force balancing is the design problem of choosing counterweight masses on a moving mechanism such that the total inertia force transmitted to the ground frame is reduced — ideally to zero. For a four-bar linkage it amounts to placing two counterweights (one on the crank, one on the rocker) whose mass-radius products satisfy two algebraic equations. Done correctly, the shaking force collapses to zero across all harmonics of the crank rotation; the bolts holding the frame to the bench stop loosening.https://multibodysystemsdynamics.com/glossary/four-bar-linkage/https://multibodysystemsdynamics.com/glossary/four-bar-linkage/A planar four-bar linkage is a closed kinematic chain of four rigid links connected by four revolute (pin) joints. It is the simplest mechanism with a single degree of freedom, the canonical object of mechanism synthesis, and the building block from which slider-cranks, scotch yokes, and most balance-shaft drives are derived. Three of the four classical defect filters (Grashof, transmission angle, branch) are stated directly on its geometry.https://multibodysystemsdynamics.com/glossary/freudenstein-equation/https://multibodysystemsdynamics.com/glossary/freudenstein-equation/The Freudenstein equation is the algebraic identity that ties the four link lengths of a planar four-bar to the input and output crank angles. Written compactly as K1·cos(θ4) − K2·cos(θ2) + K3 = cos(θ2 − θ4), it reduces the position-analysis problem of a four-bar to a single transcendental equation in the output angle. It is the closed-form workhorse of function-generation synthesis — solve for the link lengths so the four-bar realises a prescribed input-output angle relationship.https://multibodysystemsdynamics.com/glossary/grashof-condition/https://multibodysystemsdynamics.com/glossary/grashof-condition/The Grashof condition states that a four-bar linkage will have at least one link capable of full rotation if and only if the sum of the shortest and longest links is less than or equal to the sum of the other two links (s + l ≤ p + q). It is the first algebraic filter applied in our synthesis pipeline because it eliminates non-rotatable candidates without simulation.https://multibodysystemsdynamics.com/glossary/lagrange-multiplier/https://multibodysystemsdynamics.com/glossary/lagrange-multiplier/In constrained Lagrangian mechanics, a Lagrange multiplier λ is the scalar (one per constraint) that enforces a holonomic constraint on the system. In multi-body dynamics it has a direct physical meaning — the joint reaction force, in Newtons, transmitted across that constraint. Exposing λ as a primary output of the dynamics solver gives engineers fatigue-relevant joint forces without post-processing or additional modelling.https://multibodysystemsdynamics.com/glossary/pareto-front/https://multibodysystemsdynamics.com/glossary/pareto-front/The Pareto front of a multi-objective optimisation problem is the set of designs for which no objective can be improved without making at least one other objective worse. It is the natural output of co-design — an engineer should pick a point on the front rather than commit to a single hard threshold on each objective separately. We use it whenever geometry trades against balance, ride trades against handling, or mechanical advantage trades against coupler-curve fit.https://multibodysystemsdynamics.com/glossary/transmission-angle/https://multibodysystemsdynamics.com/glossary/transmission-angle/In a planar four-bar linkage the transmission angle μ is the angle between the coupler link and the rocker (output) link at their shared joint. It is a direct geometric proxy for mechanical advantage — sin(μ) appears in the input-to-output force ratio. Industrial design practice keeps the worst-case μ above a 40° floor; synthesis filters reject candidates that violate it.