DYNAMICS OF A SLENDER BEAM WITH AN ATTACHED MASS UNDER COMBINATION PARAMETRIC AND INTERNAL RESONANCES PART I: STEADY STATE RESPONSE

✍ Scribed by S.K Dwivedy; R.C. Kar

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 349 KB
• Volume: 221
• Category: Article
• ISSN: 0022-460X
• DOI: 10.1006/jsvi.1998.2035

✦ Synopsis

The non-linear behaviour of a slender beam with an attached mass at an arbitrary position under vertical base excitation is investigated with combination parametric and internal resonances. The governing equation which retains the cubic non-linearities of geometric and inertial type is discretized by using Galerkin's method and the resulting second order temporal dierential equation is then reduced by the method of multiple scales to a set of ®rst order non-linear dierential equations. Steady state response and its stability are obtained numerically from these reduced equations. Super-and sub-critical Hopf bifurcations in the trivial as well as non-trivial branches and the saddlenode or fold type bifurcations in the non-trivial branches of the response curves are found. The eect of damping, amplitude as well as frequency of base excitation, the mass ratio and the location of the concentrated mass on the non-linear response of the system having internal resonance of 3:1 is studied at length. Hysteresis, saturation and blue sky catastrophe phenomena with bistability interval in the response curves are observed for a wide range of bifurcating parameters.