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Model check
The science
Curving, cant and the balanced speed
On a curve of radius R a vehicle at speed v needs the centripetal acceleration ac = v2/R. A track banked by the angle φ (cant h = s sin φ, rail-centre distance s = 1.5 m) provides g sin φ along the plane. What is left is the side acceleration in the track plane, ay = ac cos φ - g sin φ. It is zero at the balanced speed v0 = √(g R tan φ); the equilibrium cant is heq = s v2/(g R) and the cant deficiency is D = heq - h. A deficiency of about 150 mm gives roughly 1 m/s2, a commonly used comfort level for conventional trains.
Tilting trains
A tilting body rolls by an extra angle τ towards the inside, so the floor roll is Φ = φ + τ and passengers feel ap = ac cos Φ - g sin Φ along the floor and the apparent gravity gp = g cos Φ + ac sin Φ through it. The servo here asks for τ = gain (atan(ac/g) - φ), limited to the maximum tilt, with a 1.2 s lag, a 4 deg/s rate limit and a 0.8 s look-ahead, so the body rolls in as the easement starts. Tilting makes high speed through curves comfortable, but the track still carries the full lateral force and the CG shifts only a little.
The strap and the coffee
A strap of length l = 0.7 m hanging in the carriage obeys l β'' = -gp sin β + ap cos β - c l β' (RK4 in the page). It settles at tan β = ap/gp, and the coffee surface stays perpendicular to the same apparent gravity (check 2).
Wheel loads and overturning
Quasi-statically, the outer wheel carries the share f = 1/2 + (ay hcg - gn δ)/(gn s) where gn = g cos φ + ac sin φ and δ is the inward CG shift caused by body tilt about a roll centre 0.9 m above the rail. The inner wheel unloads completely at f = 1; the page also solves the three static equations (vertical force, lateral force, moment about the CG) and compares (check 3). Real vehicles are designed with large safety margins against this limit, and the page shows only the static value.
Geometry
The test line is a stadium: straight, easement (curvature and cant both increase linearly with distance), circular arc, easement, then the same again; the heading is integrated numerically and the loop closes within centimetres (check 1 also measures the curvature of three track points independently of the formula).
Honest simplifications
Vehicle masses, the bogie and tilt servo parameters, the strap length and damping, the 1.5 m rail-centre distance, the 0.9 m roll centre and the comfort thresholds are illustrative round values, not data of a real train or standard. There is no suspension dynamics, wind, rail irregularity or flange contact; the track is flat; the speed changes at a limited rate. Educational model, not for rail design or safety assessment.