Motion of Moving Concentrated Loads on a Simply-Supported Non-Uniform Rayleigh Beam with Non-Classical Boundary Conditions

Abstract

This paper investigates the transverse response of simply-supported non-uniform Rayleigh beams resting on a constant elastic foundation. The beams properties: moment of inertial I(x) and mass per unit length of the beam miu vary along the span L of the beam. The Mindlin and Goodman’s
technique is used to transform the governing non-homogeneous forth order partial differential equations with non-homogeneous boundary conditions into non-homogeneous forth order partial differential equations with homogeneous boundary conditions. The resultant transformed equation is then further treated using the versatile Generalized Galerkins’s method with the series representation of the Dirac Delta function, a modification of Struble’s asymptotic methods and the integral  transformation techniques in conjunction with the convolution theory. Analytical solution was obtained for the transverse displacement response of the non-uniform Rayleigh beam. Analytical and Numerical results reveal that the deflection profile of the non-uniform Rayleigh beam decreases as the value of the foundation stiffness K increases. It is also found that the increase of the foundation stiffness K causes increase in the critical velocity of the dynamical system, thereby reducing the risk of resonance.

Keywords: , Rayleigh beam, non-uniform, axial force, non-classical boundary, rotatory-inertia, Foundation-modulli, simply supported.