Publication of this report does noL constitute Air Force approval of the report's findings or conclusions, It is published only for the exchange and stimulation of ideas.Dennis R. Cochran, Lt., USAF Project OfficerThe behavior of a shallow arch and a thin ring ,der a dynamic pulse loading is studied for a wide range of g.ometric and load parameters. The nonlinear dynamic response and static load deflection characteristics of the systems are related and employed to define dynamic elastic snapping and dynamic elastic buckling.When such a relationship cannot be established, the mechanism of dynamic elastic-plastic buckling is introduced. Critical dynamic load criteria are specified, and critical dynamic load data are developed as a function of structural geometry and load duration.Finally, a classification scheme for dynamic load problems is suggested.
The two-dimensional flutter of two-bay clamped-edge and simply supported panels at supersonic Mach numbers is studied to determine the influence of structural and aerodynamic coupling. It is found that the effect of structural coupling between the panel bays is unimportant both for the "single-degree-of-freedom" flutter modes that are critical at the lower supersonic Mach numbers and for the critical coupled flutter modes that develop at higher supersonic Mach numbers. The effect of aerodynamic coupling is found to be of importance for the single-degree-of-freedom flutter modes only. Furthermore, the severity of this effect is highly dependent upon the panel boundary conditions. NomenclatureE = Young's modulus of the panel material g -structural damping coefficient h = plate thickness k = &L/2U, reduced frequency of flutter ki = uiL/U, stiffness parameter L = length of panel bay M = freestream Mach number N = number of panel bays p = aerodynamic pressure vector Pa(x, 0 = aerodynamic pressure induced by deflection w(x, t) acting upon upper surface of plate t = time variable U = freestream velocity w(Xj t) = vertical deflection of panel, positive in direction of z axis x = streamwise coordinate Y = panel displacement vector Y(x) = plate mode shape | 8 = (M 2 -1) 1/2 , compressibility parameter Ax = distance between collocation points v -Poisson's ratio of panel material Ai = p s h/pL, mass parameter p = freestream density p s = density of panel material o> = flutter frequency w = 2kM 2 /(M 2 -1), supersonic reduced frequency wi = fundamental frequency of free vibration = (7T
SYMBOLS b k M p(x, U X P CO = = = t) = = = = =panel semi-chord cob/ U, reduced frequency free-stream Mach number aerodynamic pressure induced by deflection F(#)e 1 free-stream velocity streamwise spatial coordinate free-stream density frequency of oscillation A PPLICATION of the quasi-steady supersonic-flow theory to the problem of the flutter of two-dimensional panels led to the prediction that all panels would flutter (regardless of st frness) if the flow Mach number was less than V2, whereas analyses that employed the complete linearized supersonic-flow theory indicated that sufficiently stiff panels would not flutter in this range. 1 On the other hand, at Mach numbers above \/2 there was good agreement between the predictions of flutter analyses employing both aerodynamic theories. The purpose of the present note is to clarify this situation.According to the two-dimensional linearized supersonic-flow theory the aerodynamic pressure acting upon a harmonically oscillating surface, Y(x)e mt , with a fixed leading edge, iswhere D denotes a complicated integral term. The approximate quasi-steady result, which may be obtained by expanding Eq. (1) in terms of the reduced frequency k, assuming k <3C 1 and neglecting squares and higher powers of the reduced frequency, is written as p(The first-order frequency term in Eq.(2) vanishes at Mach number V2. Difficulties may therefore be expected in the vicinity of this Mach number because the neglected frequency terms will dominate the first-order contribution. However, this does not explain the failure of the theory at supersonic Mach numbers well below \/2. The explanation can be found by an examination 2 of the flutter modes that were obtained in flutter analyses that employed Eq.(1). The modes obtained at the lower supersonic Mach numbers (M < s/2) show very little phase shift between the displacement at various points along the panel chord, 3 whereas the flutter modes at the higher Mach numbers (M > y/~2) exhibit an f The work described herein was undertaken at the California Institute of Technology in connection with AFOSR Contract AF 49 (638)-220. appreciable amount of phase shift. This feature of the flutter modes explains the disagreement in the theories at the lower supersonic Mach numbers and the good agreement for Mach numbers above y2.The energy exchange at flutter between the panel and the airstream arising from the first term in Eq. (1) is related to the change of phase in the displacement across the panel chord. If there is no phase shift in the panel motion there will be no energy exchange arising from this term. Due to the small phase shift present in the flutter modes at the lower supersonic Mach numbers this term does not play a dominant role in the energy exchange at flutter, and if the third term in Eq. (1) is neglected, as in the quasi-steady theory, there will be no factor to balance the influence of the second term, which is destabilizing at Mach numbers less than y/% The third term in Eq. (1) is therefore essential at these Mach numbers and its neg...
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