A geometric nonlinear model for rotating cylindrical shells

Abstract

Shell structures are extensively used in engineering for constructions with light weight and high strength. Classical applications are covers in automotive industry, aero-and astronautics. For this purpose the dynamical behaviour is of interest for safety and comfort. The focus will be placed on rotating circumferential cylindrical shells. Starting with the geometrical nonlinear kinematics for the strains, the HAMILTON's principle is evaluated. In the followingup to the calculus of variation RITZ approach is used, to compute eigenfrequencies. Afterwards the results are discussed for certain examples and compared to FE-solutions. Mechanical modelStarting point is the mechanical model of a rotating cylindrical shell, how it is shown in figure 1. L is the length of the shell, R m the mean radius, h the wall thickness and Ω the constant angular velocity. The displacement of a arbitrary point on the middle surface P is given by the three functions u (X, φ, t), v (X, φ, t) and w (X, φ, t). Those functions depend on the LAGNRANGE parameters X, φ and the time t. For the description of the dynamical behaviour HAMILTON's principleis used. The kinetic energy is given by E kin and the potential energy by E pot . It is assumed, that there are no potential-free forces which are producing a additional virtual work (δW = 0). The kinetic energy of the rotating shell is given by(2)Fig. 1: Mechanical model of a rotating circumferential cylindrical shell.For the potential energy the stresses and strains of the shell are needed. HOOKE's law is assumed as well as the plain-stresstheory (h << R m , h << L), so that the non-zero strains are given byThe strain in thickness direction can be expressed as ZZ = −ν ( XX + φφ ) / (1 − ν). With the previous considerations the potential energy E potI = E 2 (1 − ν 2 ) L 0 2π 0 h/2 −h/2 2 XX + 2 ν XX φφ + 2 φφ (R m + Z) dZ dφ dX + E 1 + ν L 0 2π 0 h/2 −h/2 2 Xφ (R m + Z) dZ dφ dX (4)

A comparison of different nonlinear shell theories for rotating fluid‐filled structures

“…Additionally measurement data from [3] is taken into account. It can be observed, that double eigenvalues of the non-rotating shell bifurcate with increasing values of Ω (already shown in [4]). Because of this, two different eigenfrequencies of a forward and backward mode occur.…”

Nonlinear vibrations of rotating rings and shells under the usage of the finite element method

“…The here shown results can also be extrapolated for rotating cylindrical shells. Therefore, a global approach is presented in [3]. A speciality of multidimensional finite element formulations is the appearance of zero-energy modes.…”

Simulation of a tool shank for HSC‐milling under the influence of an unbalance

“…A comparison of the bifurcation in the shank fixed and in the non rotating inertial coordinate system is shown in [3]. If the shank is modelled as a shell, stress stiffening effects can be observed also induced from the angular velocity, which is shown in [2].…”