On the dynamic stability of thin elastic shells filled with a liquid

Bublik, B. N.; Меркулов, В. И.

doi:10.1016/0021-8928(60)90123-4

Cited by 4 publications

(1 citation statement)

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“…Further, the symmetric components of displacement will be split into two parts. Thus, we will have Un WS2 V = Vn (8) Subscripts n denote nonsymmetric displacements, si denotes symmetric displacements induced by the nonsymmetric displacements, and s2 denotes symmetric displacements that occur because of a subsequent change in internal symmetric pressure loading. It is recognized that symmetric circumferential displacements are zero.…”

Section: Final-state Shell Equationsmentioning

confidence: 99%

Parametric oscillations of a longitudinally excited cylindrical shell containing liquid.

Craig

Kana

1968

Journal of Spacecraft and Rockets

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Parametric oscillations of a liquid-filled, circular cylindrical shell experiencing forced longitudinal excitation are investigated analytically, and numerical results are compared with experimental observations for several principal parametric modes. The linear axisymmetric forced response of the liquid-shell system forms an initial state in which the axisymmetric pressure exerts a parametric loading with respect to nonaxisymmetric coupled modes of the system. A perturbation technique is used to develop the equations governing the perturbed, or variational state. A linear analysis of the perturbed state leads to a Mathieu equation which predicts the dynamic stability of the system. Nonlinear shell displacements in the perturbed state lead to coupled nonlinear equations which are solved by methods of approximate analysis to predict the amplitude response of the nonlinear motion during the parametric resonance. Theoretical and experimental results show good agreement in describing the over-all vibrational behavior of the system. It is found that the parametric responses can be nonlinearly softening for some modes and hardening for others. Nomenclature a = mean radius of shell A m n = nonsymmetric shell amplitude in mnih mode B = material parameter (1 -v z )/Eh C e -equivalent sonic velocity CL = sonic velocity in pure liquid D = flexural rigidity [Eh*/I2(l -i> 2 )] E = elastic modulus of shell go = standard acceleration of gravity g x = excitational acceleration amplitude (^Oco 2 /gr 0 ) h = wall thickness of shell I n , J n = modified and ordinary Bessel functions of the first kind k m = shell displacement axial wavelength factor (m*/l) K = liquid compressibility I = shell length (also liquid depth) m = shell displacement axial wave number m mn = liquid apparent mass in wnth nonsymmetric mode m p = liquid apparent mass in pth symmetric mode ra c = mass per unit area of shell Me, M x e = moment resultants Downloaded by UNIVERSITY OF MICHIGAN on February 19, 2015 | http://arc.aiaa.org | SPACECRAFT n = shell displacement circumferential wave number Ne = membrane stress resultant Neo = symmetric membrane load in initial state Ne s = symmetric membrane load in variational state p = identification for pih axisymmetric mode q = variational pressure in liquid q 0 = initial-state symmetric pressure loading on shell q n = nonsymmetric pressure loading component q mn = nonsymmetric pressure loading in ranth mode q S) q s i } q s2 = symmetric pressure loading components Q = total pressure loading on shell Qe = cross-shear stress resultant r, 6, x = cylindrical coordinates for liquid t = time UQ, VQ, WQ = shell wall displacements in initial state u, v, w -shell wall displacements in variational state u n , v n , w n = nonsymmetric shell wall displacements in variational state U, V, W -shell wall displacements in final state Wmn -radial shell wall displacement in mnih nonsymmetric mode w s i -symmetric radial shell wall displacement induced by mnih nonsymmetric mode XQ = longitudinal excitational displacement AX, AF, AZ = redu...

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Section: Final-state Shell Equationsmentioning

confidence: 99%