Generation of Equations of Motion of Flying Vehicles as Multibody Systems - General Remarks

The methods of deriving of equations of motion discussed above are, in the author's opinion, a convenient and simple formalism of modelling of motion dynamics of aircrafts, treated as multibody mechanical systems. They are a universal tool, which can be used for holonomic and non-holonomic systems. The analysis can be performed with both generalised and quasi-velocities. The matrix record kept and the slightly modified formulations of these methods shall allow making certain comparisons between them and arriving at more general conclusions regarding the rules of choice of a method for specific types of tasks of flying vehicles dynamics. − group of methods using the rules of general mechanics, − group of methods using the methods of analytic mechanics. In the group of methods using the rules of general mechanics at least two approaches to deriving equations of motion can be identified. The first one consists in the use of the rule of alteration of momentum and angular momentum separately for all bodies into which the aircraft has been divided (e.g. helicopter fuselage, hub and the blades of the main rotor, hub and the blades of the tail rotor). This way one gets more equations than the number of degrees of freedom of the discussed system. Equations derived this way contain internal reactions which are not always interesting for solving the problem. The second approach consists in the use of the rule of change of momentum and angular momentum for an aircraft treated as an integral whole (non-deformable body). The rule of change of angular momentum can be used in relation to arbitrarily chosen pole (not necessarily in relation to the centre of mass). Within the group of methods of analytical mechanics also at least two approaches to derivation of the equations of motion of complex dynamic systems can be differentiated: a) approach based on the used of generalised inert coordinates and referring directly to the use of Hamilton's rule or Lagrange's equations. A certain inconvenience occurs here, since the generalised aerodynamic forces are usually defined as functions of quasi-velocity. Therefore, it is necessary to constantly make transformations of generalised velocities into quasi-velocities. It is undoubtedly a weak point of this approach. Its advantage is the avoidance of singularities which appear in the equations of kinematic relationships in certain flight conditions; b) approach consisting in using equations of analytical mechanics in quasi-coordinates (e.g. Boltzmann-Hamel, Apel-Gibs or Keyn equations). Equations of motion of mechanical systems notated in quasi-coordinates and quasi-velocities are usually more complex. And so in the case of using of Boltzmann-Hamel equations a necessity to define three-indicatory Boltzmann symbols occurs. It gives a large number of components of equations. The probability of making an error rises and problems with verification of the derived equations arise (physical sense of the transformations made is lost). The advantage of analytical methods is that basing on them one may create algorithms of automatic generation of equations of motion with the use of digital machines. II. Derivation of dynamic equations of motion on the basis of general rules of classic mechanics