Bifurcation theory has been used to study Ihe nonlinear dynamics of the F-14. An 8 degree-of-freedom model that does not include the control system present in operational F-14's has been analyzed. The aerodynamic model, supplied by NASA, includes nonlinearlties as functions of the angles of attack and sideslip, the rotation rate about the velocity vector, and the elevator deflection. A continuation method has been used to calculate the steady states of the F-14 as continuous functions of the elevator deflection. Bifurcations of these steady states have been used to predict the onset of wing rock, spiral divergence, and jump phenomena that cause the aircraft to enter a spin. A simple feedback control system was designed to eliminate the wing rock and spiral divergence instabilities. The predictions were verified with numerical simulations.

The flow field inside a cylindrical container induced by the rotation of the top and bottom end walls with a fixed sidewall is described. For this problem, this paper shows that stagnation points occur along the axis of rotation between the midplane of symmetry and the rotating end walls for appropriate values of the characteristic parameters, viz., the Reynolds number and the aspect ratio of the container. Aspect ratios of 0.5, 0.8, 1.0, and 1.5 were examined over a range of Reynolds numbers from 100 to 2000. As the Reynolds number increased beyond a critical value a recirculation zone surrounding a columnar vortex core in the meridional-plane flow pattern is predicted to occur near the midplane. This toroidal vortex is different from the type B vortex breakdown phenomenon that occurs in cylindrical containers with only one end wall rotating.

Two important approximations have been incorporated in much of the work with approximate analysis of unsteady motions in combustion chambers: truncation of the series expansion to a finite number of modes, and time averaging. A major purpose of the analysis reported in this paper has been to investigate the limitations of those approximations. In particular two fundamental problems of nonlinear behavior are discussed: the conditions under which stable limit cycles of a linearly unstable system may exist; and conditions under which bifurcations of the limit cycle may occur. A continuation method is used to determine the limit cycle behavior of the equations representing the time dependent amplitudes of the longitudinal acoustic modes in a cylindrical combustion chamber. The system includes all linear processes and second-order nonlinear gas dynamics. The results presented show that time averaging works well only when the system is, in some sense, only slightly unstable. In addition, the stability boundaries predicted by the two-mode approximation are shown to be artifacts of the truncation of the system. Systems of two, four, and six modes are analyzed and show that more modes are needed to analyze more unstable systems. For the six-mode approximation with an unstable second mode two bifurcations are found to exist. A pitchfork bifurcation causes a new branch of limit cycles to exist in which the odd acoustic modes are excited. This new branch of limit cycles then undergoes a torus bifurcation that causes the system to exhibit stable quasi-periodic motions.

Two important approximations have been incorporated in much of the work with approximate analysis of unsteady motions in combustion chambers: truncation of the series expansion to a finite number of modes, and time averaging. A major purpose of the analysis reported in this paper has been to investigate the limitations of those approximations. In particular two fundamental problems of nonlinear behavior are discussed: the conditions under which stable limit cycles of a linearly unstable system may exist; and conditions under which bifurcations of the limit cycle may occur. A continuation method is used to determine the limit cycle behavior of the equations representing the time dependent amplitudes of the longitudinal acoustic modes in a cylindrical combustion chamber. The system includes all linear processes and second-order nonlinear gas dynamics. The results presented show that time averaging works well only when the system is, in some sense, only slightly unstable. In addition, the stability boundaries predicted by the two-mode approximation are shown to be artifacts of the truncation of the system. Systems of two, four, and six modes are analyzed and show that more modes are needed to analyze more unstable systems. For the six-mode approximation with an unstable second mode two bifurcations are found to exist. A pitchfork bifurcation causes a new branch of limit cycles to exist in which the odd acoustic modes are excited. This new branch of limit cycles then undergoes a torus bifurcation that causes the system to exhibit stable quasi-periodic motions.

The flow field induced inside a cylindrical container by the rotation of the two end walls is described. It is shown that stagnation points leading to separation bubbles occur on the axis of rotation and/or the bottom end wall for certain ranges of the characteristic parameters; the Reynolds number, the aspect ratio of the container, and the ratio of the rotation rates of the end walls. Flow fields in a container of aspect ratio 2.0 are examined for Reynolds numbers from 100 to 3000 and ratios of the rotation rates of the top and bottom end walls from −0.10 to 1.0. For a range of ratios of the rotation rates of the top and bottom end walls and Reynolds numbers it is shown that ring vortices surrounding a columnar vortex core exist.