On the axial vibrations of rotating bars

One of the important developments in rotary wing aeroelasticity in the recent past has been the growing awareness and acceptance of the fact that the problem is inherently non-linear and that correct treatment of aeroelastic problems requires the development of a consistent mathematical model [l]. This has led to a number of studies devoted to the derivation of a consistent set of “second order” non-linear equations, for example, those of Hodges and Dowel1 [2], of Rosen and Friedmann [3], and of Kvaternik, White and Kaza [4], each of which differs from the others on the question of the inclusion of certain terms in the equations of motion. The final form of the equations depends first upon the ordering scheme used for characterizing the displacements and upon the consistency with which this is applied in omitting terms of lower order. The ideal way of achieving this would be to derive the equations of motion with all the terms first included regardless of their relative orders of magnitude and then to apply the ordering scheme.

Minimum error solutions of the Boltzmann equation for shock structure

‘Best’ solutions for the shock-structure problem are obtained by solving the Boltzmann equation for a rigid sphere gas by applying minimum error criteria on the Mott-Smith ansatz. The use of two such criteria minimizing respectively the local and total errors, as well as independent computations of the remaining error, establish the high accuracy of the solutions, although it is shown that the Mott-Smith distribution is not an exact solution of the Boltzmann equation even at infinite Mach number. The minimum local error method is found to be particularly simple and efficient. Adopting the present solutions as the standard of comparison, it is found that the widely used v2x-moment solutions can be as much as a third in error, but that results based on Rosen's method provide good approximations. Finally, it is shown that if the Maxwell mean free path on the hot side of the shock is chosen as the scaling length, the value of the density-slope shock thickness is relatively insensitive to the intermolecular potential. A comparison is made on this basis of present results with experiment, and very satisfactory quantitative agreement is obtained.