Effects of Geometrical and Material Properties on Vibrations of Double Beams at Different Boundary Conditions

The double beams are important engineering structures that have been used in recent years in many engineering applications such as building aircraft structures, bridges, ships, and others. In this study, the variability of four factors of the geometry and the properties of the materials for these double beams were studied, which is the ratio of the density of the upper layer to the density of the lower layer, the ratio of the thickness of the upper layer to the thickness of the lower layer, the modulus of elasticity of the upper layer to the modulus of elasticity of lower layer, and the stiffness of the middle layer; the effect of these four factors on the variation of natural synchronous and asynchronous frequencies. A mathematical model is built to find synchronous and asynchronous natural frequencies at identical and non-identical boundary conditions; Clamped to Clamped, Simply-support to Simply-support, Sliding to Sliding, Free to Free, Clamped to Free, Clamped to Simply-Support, Clamped to Sliding, Free to Sliding, Free to Simply-support, and Simply-Support to Sliding. It was found that the variation of the elastic modulus ratios of the upper layer to the lower layer and the ratio of the thickness of the upper layer to the thickness of the lower layer is influencing the naturally synchronized frequencies directly when the ratio is less than one but it has little effect when the ratio is greater than one. As for the natural asynchronous frequencies, the ratio variation is not effective when the ratio is less than one, but it is very effective when the ratio is more than one. The variation of the ratio of the density of the upper layer to the density of the lower layer has little effect on synchronous frequencies and a non-linear inverse of the asynchronous frequencies when the ratio is less than one. While it is a non-linear inverse on the synchronous frequencies and little influence on the asynchronous frequencies when the ratios are more than one. The value of the modulus of elasticity has little effect on concurrent frequencies, but it has a non-linear direct effect on asynchronous frequencies.