The dynamic stability of a simply supported beam with additional discrete elements

Abstract

The dynamic stability of a simply supported beam with additional discrete elements was investigated in the paper. Those elements were an elastic spring, a concentrated mass and an undamped harmonic oscillator connected to the beam. All the discrete elements could be mounted at any chosen position along the beam length. The beam was axially loaded by a harmonic force. The problem of dynamic stability was solved by applying the mode summation method. The obtained Mathieu equation allowed the influence of additional elements on the position of solutions on a stability chart to be analysed. The analysis relied on testing the influence of individual discrete elements on the value of coefficient b in the Mathieu equation. The research carried out showed that both the concentrated mass and oscillator mass had a destabilising effect (maximum in the middle position) on the investigated system. The rigidity of the support and the oscillator had an influence on an increase in the stability of the investigated system. An increase in the loading force, independently of the relation between the mass and rigidity of discrete elements, had an influence on the increase in coefficient b in the Mathieu equation (the less stable system). The considered beam is treated as a Bernoulli–Euler beam in accordance with the small bending theory.