LATERAL-TORSIONAL FLUTTER OF A DEEP CANTILEVER LOADED BY A LATERAL FOLLOWER FORCE AT THE TIP

As pointed out by Bolotin [1, 2], the study of the stability of structures under follower force systems apparently started with work by Nikolai in the late 1920s. Now there are many papers and even a few books devoted to this "eld; see, for example, the work of references [1]}[7]. Since the beginning of this period, much of the analytical research has focused on the e!ects of various physical phenomena, such as damping and transverse shear deformation, on the stability of beams subjected to various types of follower forces. In spite of all the published work, there seems to be very little literature concerned with the lateral-torsional stability of deep cantilevered beams loaded by a transverse follower force at the tip. This problem has some practical applications, such as the e!ect of jet engine thrust on the aeroelastic #utter of a #exible wing. According to Bolotin [1], this type of system was "rst considered by himself in reference [8]. Although the analysis presented therein is applicable to the tip-loaded cantilever case, no results speci"c to that case were presented. Como [9] analyzed a cantilevered beam subjected to a lateral follower force at the tip. The distributed mass and inertia properties of the beam were neglected, although a concentrated mass and inertia at the tip were included. Without neglecting the distributed mass and inertia properties of the beam, Wohlhart [10] undertook an extensive study, and results for a wide variation of several parameters were presented in this truly excellent paper. The results of the present study agree with those of Wohlhart. Later work by Feldt and Herrmann [11] added the in#uence of #uid #owing past a wing, the structural properties of which are represented by a beam undergoing bending and torsion. Unfortunately, the results for cases in which the aerodynamic forces are neglected agree with neither those of reference [10] nor those of the present work. Other than these three papers, to the best of the author's knowledge, this problem appears to have received no further attention in the literature. It is the objective of the present paper to consider further this non-conservative elastic stability problem and present a few results and observations that go beyond those of reference [10]. We "rst develop a weak form of the partial di!erential equations of motion for a deep, symmetric beam under the action of a tip follower force acting in the plane of symmetry. Then an approximate solution using cantilever beam bending and torsional modes is obtained. The e!ects of three parameters are investigated: the ratio of the uncoupled fundamental bending and torsional frequencies and dimensionless parameters re#ecting the mass radius of gyration and the o!set from the elastic axis of the mass centroid.