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Many engineering problems can be solved using a linear approximation. In the Finite Element Analysis (FEA) the set of equations, describing the structural behaviour is then linear K d = F (1.1) In this matrix equation, K is the stiffness matrix of the structure, d is the nodal displacements vector and F is the external nodal force vector. Characteristics of linear problems is that the displacements are proportional to the loads, the stiffness of the structure is independent on the value of the load level. Though behaviour of real structures is nonlinear, e.g. displacements are not proportional to the loads; nonlinearities are usually unimportant and may be neglected in most practical problems.
"The vibration of a highly flexible cantilever beam is investigated. The order three equations of motion, developed by Crespo da Silva and Glyn (1978), for the nonlinear flexural-flexural-torsional vibration of inextensional beams, are used to investigate the time response of the beam subjected to harmonic excitation at the base. The equation for the planar flexural vibration of the beam is solved using the finite element method. The finite element model developed in this work employs Galerkin's weighted residuals method, combined with the Newmark technique, and an iterative process. This finite element model is implemented in the program NLB1, which is used to calculate the steady state and transient responses of the beam. The steady state response obtained with NLB is compared to the experimental response obtained by Malatkar (2003). Some disagreement is observed between the numerical and experimental steady state responses, due to the presence of numerical error in the calculation of the nonlinear inertia term in the former. The transient response obtained with NLB reasonably agrees with the response calculated with ANSYS®."--Abstract.
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