A dynamic modeling and simulation analysis of hose-paradrogue aerial refueling systems is presented. A set of governing equations of motion is derived using a finite-segment approach that describes the dynamics of the hoseparadrogue assembly under a prescribed motion of the tanker. The hose is modeled by a series of ball-and-socketconnected rigid links subject to gravitational and aerodynamic loads that account for the effects of tanker wake, steady wind, and atmospheric turbulence. Numerical simulations show a good correlation of the model's steady-state characteristics with previously reported flight-test data. Also investigated are the dynamic characteristics of the paradrogue assembly resulting from atmospheric turbulence and a typical pitch doublet maneuver of the tanker. Finally, the dynamic motion resulting from an in-flight adjustment of the paradrogue drag associated with strutangle changes is studied. Nomenclature a K = acceleration of lumped mass K, ft=s 2 a 0 = acceleration of the system tow point (hose exit point from a pod), ft=s 2 B = vehicle mass center C drogue = drag coefficient of paradrogue C X N Z N = vertical plane in which the aircraft moves c n;K , c t;K = normal and tangential drag coefficient of the link K D K = aerodynamic force acting on link K, lb d K , d drogue = diameters of link K and paradrogue F K = frame fixed in link K with axes x K ; y K ; z K F N = inertial frame with axes X N ; Y N ; Z N F W = aircraft (tanker) mean air trajectory frame with axes X W ; Y W ; Z W c GS = normalized paradrogue gore spacing; ranges from ( 1-1) for gore spacing from 5-8:5 deg g = gravitational acceleration vector, ft=s 2 L H = straight-line distance from hose exit point to paradrogue coupling, ft ' K = length of link K, ft m drogue = mass of paradrogue, slug m K = mass of lumped mass K (one-half of the total mass of adjoining links), slug N = number of links n K1 , n K2 , n K3 = mutually perpendicular unit vectors fixed in link K n K1 = unit vector pointing from lumped mass K to lumped mass J (along link K O N = origin of the inertial frame p K = position vector of lumped mass K relative to J; components in F W , ft p K; Ki = @p K =@ Ki , ft Q K = external force vector acting on lumped mass K (one-half of the total force acting on the adjoining links), lb r K = position vector of lumped mass K relative to an inertial frame, ft c SA = normalized paradrogue canopy characteristic length, ranges from ( 1-1) for lengths from 3-5:5 in: (7:62-13:97 cm) S Ki , C Ki = sine and cosine of Ki t K = tension in link K, lb u K = local air velocity due to steady wind, tanker wake, and turbulence at lumped mass K, ft=s V D = altitude difference from hose exit point to paradrogue coupling, ft V 1 = tanker speed, ft=s B t = inertial velocity of the vehicle mass center, ft=s K = velocity of lumped mass K, ft=s K=air = velocity of lumped mass K relative to the local air velocity ( K u K ), ft=s K;n , K;t = vector components of K normal and tangent to link K, ft=s w 1 , w 2 , w 3 = mutually perpendicular unit vector...

A new automated procedure for obtaining and solving the governing equations of motion of constrained multibody systems is presented. The procedure is applicable when the constraints are either (a) geometrical (for example, “closed-loops”) or (b) kinematical (for example, specified motion). The procedure is based on a “zero eigenvalues theorem,” which provides an “orthogonal complement” array which in turn is used to contract the dynamical equations. This contraction, together with the constraint equations, forms a consistent set of governing equations. An advantage of this formulation is that constraining forces are automatically eliminated from the analysis. The method is applied with Kane’s equations—an especially convenient set of dynamical equations for multibody systems. Examples of a constrained hanging chain and a chain whose end has a prescribed motion are presented. Applications in robotics, cable dynamics, and biomechanics are suggested.

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