Abstract
This paper deals with the geometrically nonlinear free and forced vibration analysis of a multi-span Euler Bernoulli beam resting on arbitrary number N of flexible supports, denoted as BNIFS, with general end conditions. The generality of the approach is based on use of translational and rotational springs at both ends, allowing examination of all possible combinations of classical beam end conditions, as well as elastic restraints. First, the linear case is examined to obtain the mode shapes used as trial functions in the nonlinear analysis. The beam bending vibration equation is first written in each span. Then, the continuity requirements at each elastic support are stated, in addition to the beam end conditions. This leads to a homogeneous linear system whose determinant must vanish in order to allow nontrivial solutions to be obtained. Numerical results are given to illustrate the effects of the support stiffness and locations on the natural frequencies and mode shapes of the BNIFS. The nonlinear theory is then developed, based on the Hamilton’s principle and spectral analysis. The nonlinear beam transverse displacement function is defined as a linear combination of the linear modes calculated before. The problem is reduced to solution of a non-linear algebraic system using numerical or analytical methods. The nonlinear algebraic system is solved using an explicit method developed previously (second formulation) leading to the amplitude dependent nonlinear fundamental mode of the BNIFS.
Highlights
The purpose of this paper is to develop a model for the nonlinear free and forced transverse vibrations of beams supported by a finite number of flexible supports
In order to investigate the geometrically nonlinear dynamic behaviour of Euler-Bernoulli beams, with finite number of intermediate flexible supports located at arbitrary positions, in both free and forced cases, the Hamilton’s principle and spectral analysis are used, as in [1,2,3,4].First, linear case is examined to obtain the mode shapes in each span
By applying Hamilton’s principle with integration of the time functions over a period of vibration, or Lagrange’s equations [1] combined with the harmonic balance method (HBM), the nonlinear amplitude equation is obtained as a set of nonlinear algebraic system solved by injecting initial estimates for the basic function contribution coefficients in a multidimensional Newton-Raphson iteration algorithm
Summary
The purpose of this paper is to develop a model for the nonlinear free and forced transverse vibrations of beams supported by a finite number of flexible supports. In order to investigate the geometrically nonlinear dynamic behaviour of Euler-Bernoulli beams, with finite number of intermediate flexible supports located at arbitrary positions, in both free and forced cases, the Hamilton’s principle and spectral analysis are used, as in [1,2,3,4].First, linear case is examined to obtain the mode shapes in each span. By applying Hamilton’s principle with integration of the time functions over a period of vibration, or Lagrange’s equations [1] combined with the harmonic balance method (HBM), the nonlinear amplitude equation is obtained as a set of nonlinear algebraic system solved by injecting initial estimates for the basic function contribution coefficients in a multidimensional Newton-Raphson iteration algorithm. In the forced vibrations case, the single mode approach (SMA) is used the get a thirddegree equation solved by the classical Cardan method, in which the frequency ratio and the excitation level are the inputs and the real solution corresponds to the normalised mode shape contribution coefficient, the complex solutions being ignored
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