Nonlinear Forced Oscillations of a Shallow Spherical Shell

Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry

“…When N =1, formula (18) reduces to that obtained when keeping a single-mode in the truncation, thus recovering earlier results presented with this assumption [13][14][15][16]21]. Modal truncation will be studied by increasing N until convergence.…”

“…Evensen and Evan-Iwanowski [12] found a softening behaviour with the harmonic balance method, without studying the transition from hardening to softening behaviour. Many investigators used a single-mode approach to study the effect of geometry on the non-linear behaviour: Grossman et al [13] investigated different type of boundary conditions and mentioned the transition from hardening to softening behaviour as the riseto-thickness ratio increases, Yasuda and Kushida [14] found the first mode to be softening for a little curvature whereas the second stays of the hardening type. Singh et al [15] and Sathyamoorthy [16] studied the influence of transverse shear deformation and rotatory inertia in the case of a moderately thick shell.…”

Non-linear vibrations of free-edge thin spherical shells: Experiments on a 1:1:2 internal resonance

“…Amabili et al [9,10] experimentally investigated nonlinear vibrations of cylindrical shells, coupled with fluid or not, and reported other experimental works. On spherical shells, experimental results on snap-through behavior are exposed in [11] and a few qualitative experiments on the special case of a one-to-two (1:2) internal resonance between two axisymmetric modes are reported in [12]. To the knowledge of the authors, no experiments on multi-mode asymmetric nonlinear response of spherical shells have been proposed yet.…”

Chaotic Behavior Resulting in Transient and Steady State Instabilities of Pressure-Loaded Shallow Spherical Shells

“…These functions are the linear vibration modes of an unloaded clamped shallow spherical shell [11]. In equations (6a) and (6b) J 0 and J 1 are Bessel functions [31], and I 0 , and I 1 modified Bessel functions of the first kind, and K n (0 5 K 1 5 K 2 5 Á Á Á ) are the roots of the equation…”

“…Moreover, most of the studies on the axisymmetric dynamic buckling of shallow spherical caps have focused their attention on the snap-through buckling under step loading of infinite or finite duration [4][5][6][7][8][9]. Relatively little work has been done on the dynamic behavior of spherical caps under harmonic loading [10][11][12][13][14]. Studies have shown that such shells may display, due to their inherently non-linear nature, subharmonic and superharmonic oscillations, period-multiplying bifurcations, multiple solutions, chaotic motions and jumps due to the presence of competing potential wells and non-linear resonance curves within each well.…”