Drgania Swobodne Układu z Nieliniowa Bezwładnością - Gryboś, R. (Engineering Transactions, 1965)
The object of this analysis is the natural damped vibration of a system as represented in Fig. 2. The motion of this system is described by the same equation as the natural damped vibration of a bar with two hinges and a concentrated mass at the moving end, if the bar is assumed to be incompressible. It is known [1] that this equation has a expression characteristic for a nonlinear inertion. The present considerations are generalized to the case of the bar having an initial curvature and a nonlinear elastic characteristic of the Duffing type. A first integral of the relevant differential equation is obtained by quadratures, thus enabling the representation of the motion in the phase plane. For a study of the stability of vibration of a linearly elastic system with no initial deflection, the Lapunov function is obtained in an evident form. In the general case the stability is investigated on the basis of Lapunov’s first method. Numerical examples show the influence of a nonlinear inertion and an initial curvature on the (fundamental) natural frequency and the amplitude of lateral vibration of a bar having an additional mass at the end. The equations obtained are characterized by considerable simplicity which enables convenient application to engineering problems.