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)
Until now several studies of rotating structures like rings or shells have been done. To model such problems in a right way, geometrical nonlinearity has to be considered. Different methods can be used, to solve the corresponding eigenvalue problem. In this article the focus will be placed on the finite element method.
In contrast to flat elements curved ones can cause some numerical inconveniences. They include a tremendous loss of convergence on the one hand, on the other hand the appearance of zero energy modes. In addition, centrifugal and Coriolis effects have to be taken into account. For certain examples the results of different FE‐models are shown and compared with global approaches as well as measurement data.
HSC (High Speed Cutting) milling is one of the most common production processes in tool-making and mold-making industries. One aim of current investigations is a better understanding of the dynamics of a tool shank with high slenderness. Often the revolution speed is higher than the lowest eigenfrequencies of the tool shank, so that resonance can appear during run-up or run-out. The focus will be placed on the influence of an unbalance. Starting with the geometrical nonlinear kinematics for the strains, the Hamilton's principle is evaluated to obtain the variational formulation of the tool shank. The unbalance is implemented with a stochastic WEDIG-DIMENTBERG model. In the following-up RITZ approach is used for discretization and the eigenfrequencies are computed. Afterwards the results are discussed for certain examples.
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