Various approaches are usually used in the dynamic analysis of beams vibrating transversally. For this, numerical methods allowing the solving of the general eigenvalue problem are utilized. The equilibrium equations, describing the movement, result from the solution of a fourth order differential equation. Our investigation is based on the finite element method. The findings of these investigations are the vibration frequencies, obtained by the Jacobi method. Two types of elementary mass matrix are considered, representing a uniform distribution of the mass along the element and concentrated ones located at fixed points whose number is increased progressively separated by equal distances at each evaluation stage. The studied beams have different boundary constraints representing several classical situations. Comparisons are made for beams where the distributed mass is replaced by n concentrated masses. As expected, the first calculus stage is to obtain the lowest number of the beam parts that gives a frequency comparable to that issued from the Rayleigh formula. The obtained values are then compared to theoretical results based on the assumptions of the Bernoulli-Euler theory. These steps are used after for the second type mass representation in the same manner.
Different continuous variations in the inertia are considered in achieving cantilever tapered beams to match functional design and resistance requirements. In this investigation, the expressions of linear and cubic variations in the inertia are associated to a linear mass distribution. An exact solution of the fourth order differential equation, with none constant coefficients governing the studied tapered beam element equilibrium, is obtained for each case. These displacements functions are normalized and introduced in the well known Rayleigh quotient formula for calculating the fundamental natural frequency. The accuracy of the approach over the use of the above approximation has been validated by beam references results, produced quite interesting results in numerical comparisons. Also, two curves corresponding to fixed-free and free-fixed boundary conditions, for a graphical evaluation of the natural frequency are given. They consider various degrees of taper corresponding to the two inertia variations.
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