Examining Non-Linear Transverse Vibrations of Clamped Beams Carrying N Concentrated Masses at Various Locations Using Discrete Model

The discrete model used is an N-Degree of Freedom system made of N masses placed at the ends of solid bars connected by springs, presenting the beam flexural rigidity. The large transverse displacements of the bar ends induce a variation in their lengths giving rise to axial forces modeled by longitudinal springs causing nonlinearity. Nonlinear vibrations of clamped beam carrying n masses at various locations are examined in a unified manner. A method based on Hamilton's principle and spectral analysis has been applied recently to nonlinear transverse vibrations of discrete clamped beam, leading to calculation of the nonlinear frequencies. After solution of the corresponding linear problem and determination of the linear eigen vectors and eigen values, a change of basis, from the initial basis, i.e. the displacement basis (DB) to the modal basis (MB), has been performed using the classical matrix transformation. The nonlinear algebraic system has then been solved in the modal basis using an explicit method and leading to nonlinear frequency response function in the neighborhood of the first mode. If the masses are placed where the amplitudes are maximized, stretching in the bars becomes significant causing increased nonlinearity.