A time-mode approach to nonlinear vibrations of orthotropic thin shallow spherical shells (Q1317133)

The problem studied is that of nonlinear vibrations of a shallow spherical shell. The state variable is the deflection of the cap. One of the principal features of the problem comes from the fact that the stiffness of the spherical cap varies according to the direction of the motion, downwards or upwards. (Let us recall, for instance, the dynamic problem of snap-through.) To this very complicated vibration problem is given a so-called ``analytical'' solution, contrarily to numerical ones, which leads to possible asymmetrical solutions for large amplitude vibrations. The von Kármán equations are written in a variational form by use of the Hamilton principle, the state variables being the deflection and a stress function (as usual for axisymmetrical bodies). A central regularity condition guarantees no singularity of the transverse shear stress. An examination of the simple case of a single degree of freedom shows that in the phase plane the mid-point of the motion shifts away from the static center, as the amplitude increases; a shift called drift or steady-stream. This example suggests a solution of the form \(w(x,\tau)= W(x) (\xi+\cos \omega \tau)\), where \(\xi\) is a parameter which models the asymmetry, \(\omega\) the nondimensional natural frequency, and other notations are obvious. The final state variables are now \(\beta\) and \(W\), which leads to a set of nonlinear coupled equations and boundary conditions (B.C.). A new solution method is applied as a modified iteration method which starts from \(\xi=0\), and, neglecting the nonlinear terms, a set of linear coupled equations are solved by an iteration process. The resolution leads to an infinite algebraic series. Successive iterations are then carried out from the first one, such that at each iteration the set of matrix equations may be solved through the vanishing of a determinant according to the B.C. . The discussion of the results shows that the vibrations depend on the B.C., giving softening or hardening nonlinearities, and more or less symmetry; results which also depend on the shallowness of the cap. To conclude, this interesting problem, solved with a new iterative method, remains like the results rather complicated. It will not be surprising that many engineers and researchers will prefer more direct and classical numerical methods though perhaps less attractive or convincing, but applicable to more general cases.