Abstract
In a recent work, a discrete model for geometrically nonlinear transverse free constrained vibrations of beams with various end conditions has been developed and validated via comparison with known results corresponding to nonlinear vibration of clamped beams carrying a concentrated mass. It is extended here to continuous beams carrying two or three concentrated masses at various locations and subjected to large vibration amplitudes. The discrete model used is an N -dof ( N -Degrees 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 modelled by longitudinal springs causing nonlinearity. The calculations made allowed application of the semi-analytical model developed previously for nonlinear structural vibration involving three tensors, namely the mass tensor m ij , the linear rigidity tensor k ij and the nonlinearity tensor b ijkl presenting the effect of the change in the bar lengths. The addition of three concentrated masses studied here induces a change in the mass matrix. By application of Hamilton’s principle and spectral analysis in the modal basis, the nonlinear vibration problem is reduced to a nonlinear algebraic system, using an explicit method, developed previously for non-linear structural vibration. This study shows that concentrated masses may be used for practical purposes to shift the resonant frequency; if the three masses locations are appropriately chosen.
Highlights
Application of the discrete model developed in the work [1, 2, 3] is made here to a Bernoulli beam carrying two or three concentrated masses at various locations and subject to geometrical nonlinear vibration due to large transverse displacements
This study shows that the developed model may be used to study successfully non-linear vibrations of beams carrying many concentrated masses by changing the mass matrix but without any change in the linear and nonlinear stiffness tensors, corresponding to a uniform beam defined in [1]
It is noted that when the masses are placed in the middle of the beam, the first natural frequency decreases; this is due to the shape of the first vibration mode which has a maximum amplitude in the middle of the beam, while the second eigenmode remains unchanged at this location since the form of this mode has a node in the middle of the beam
Summary
Application of the discrete model developed in the work [1, 2, 3] is made here to a Bernoulli beam carrying two or three concentrated masses at various locations and subject to geometrical nonlinear vibration due to large transverse displacements. This study shows that the developed model may be used to study successfully non-linear vibrations of beams carrying many concentrated masses by changing the mass matrix but without any change in the linear and nonlinear stiffness tensors, corresponding to a uniform beam defined in [1]. This discrete model may be very adapted to the study of beams with variable cross sections, with concentrated masses or stiffness, or with discontinuities, the stiffness or the material properties This discrete model may be very adapted to the study of beams with variable cross sections, with concentrated masses or stiffness, or with discontinuities in the section, the stiffness or the material properties
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