Vibrational Analysis Of Laminated Composite Beam

Abstract

A composite beam consists of laminate consisting of more than one lamina bonded together through their thickness. Thicknesses of Lamina are of order 0.005 inch (0.125mm), implying that to take realistic loads several laminae will be required. For a typical unidirectional lamina the mechanical properties are severely limited in the transverse direction. Stacking several unidirectional layers, may lead to an optimum laminate for unidirectional loads. However, this would not be desirable for complex loading and stiffness requirements,. One can overcome this problem by making a laminate with layers stacked at different angles to withstand different loading and stiffness requirements. Usually more than one lamina are bonded together through their thickness to get real structure. Each layer can be identified, its material, its angle of orientation with respect to a reference axis and by location in the laminate.
Reduced stiffness matrix was obtained by using properties of composite material. These properties are Longitudinal elastic modulus , Transverse elastic modulus , Major poisons ratio and Shear modulus . Using these properties composite compliance matrix was obtained . Inverse of compliance matrix was taken and reduced stiffness matrix was obtained . Then reduced stiffness matrix for each and every layer was calculated . Reduced stiffness matrix for each and every layer was calculated taking in consideration angles of the fiber in lamina .
Mid plane symmetry was taken and position of each layer was calculated with respect to mid plane .D11 matrix was determine by formula using relative position of layer from mid plane and reduced stiffness matrix of all lamina ( effect of angle of fiber was included ) . Density of composite material was obtained by using densities of each material and there volume composition. Value of natural frequency in rad/sec and per sec was obtain by using formulation for finding frequency of composite material . Frequency was obtained for all the supports i.e simple –simple , simple – clamped , clamped –clamped and clamped –free and for first three mode of vibration . Glass/epoxy , graphite /epoxy and boron epoxy composites were used to obtain tabulation for natural frequency in hertz . Comparison of frequency for these composite , frequency of composite under different mode condition were done for these composites and required histogram was plotted . Taking beam as euler beam , equation of euler beam was considered and was solved for simple –simple case taking in consideration boundary condition of simple – simple support condition i.e displacement at support and bending moment at support is equal to zero .