Pewne Zagadnienie Stateczności Dynamicznej Pręta Pryzmatycznego z Masą na Końcu - Piszczek, K. (Engineering Transactions, 1959)
The problem is considered of dynamic stability of straight prismate bar with a mass attached to the end and two types of boundary conditions: hinged support and clamping with the possibility of moving the end with attached mass. Shear forces and geometrical non-linearity are taken into consideration. It is assumed that the mass at the end is considerably greater than that of the beam itself. The Kirchhoff hypothesis concerning force is used in the expression for the potential energy, the longitudinal which also. makes possible the solution in the cases of clamped ends. Confining ourselves to the fundamental vibration mode, the possibility is demonstrated of the appearance of three resonance images and the so-called, additional resonance due to the shear forces. For steady vibration, the transversal amplitudes are found, giving for both types of boundary conditions the general equations (4.2) and (4.11) which may be interpreted as equations of a surface in the coordinates (ω2, ε, A2). In the last paragraph substitute parameters are given for coil springs. Assuming these parameters, the equations of the preceding paragraphs give a solution of the problem of dynamic stability of springs with mass attached to the end. Internal and external damping is disregarded.