Refined Dynamic Equations Of Thick Beam Bending And Stretching And Its Analysis

The classical beam theories are the Euler-Bernoulli beam theory, the Rayleigh beam theory, the shear beam and the Timoshenko beam theory. Differential equations of these beam theories are derived based on element balance under many assumptions.In fact, stress state of the solid and the structure is very complex, The stress of the beam is not as simple as assumed in the classical theory. So the error of the analysis of complex structure using the above beam theories is self-evident. Han, Benaroya and Wei compared the four beam models under different supporting conditions, Timoshenko beam theroy had the highest precision, at the same time, it needed the most complex computation. The accuracy of the Euler beam model was the worst, but its governing equation was the most simple. In the case of low frequence, results gived by classical beam models were acceptable.With the increase of the thickness of the beam, using the traditional mechanics calculation method will lead to big error, especially the dynamics. At present, many beam height span ratio have already exceeded the scope of the classical theory of structure. As some structures need to take higher order vibration mode into consideration, to provide more accurate vibration model is becoming more and more important. Therefore, more accurate governing equations of the deep beam are required, in order to analyze the dynamic strength, vibration response and vibration control of thick wall structures.Different from the classical beam theory, in this paper, based on the theory of refined dynamic equations of thick plates bending, applied the differential operator algebra and decomposition of operator spectra, the refined dynamic equation of the beam flexural motion is first obtained by using proper gauge conditions and satisfying the boundary conditions. The dispersion relations, which are from the given beam theory, theory of Euler-Bernoulli beam and Timoshenko beam, respectively, are compared. The refined equations of thick beams and applicable condition are investigated and discussed. Since derivation of the refined dynamic equation is conducted without any assumptions, so the proposed equation of thick beams bending is exact, that can be used to analyze vibration of thick beams at the high frequency and to evaluate the applicable condition of the engineering beam theory.Based on the theory of refined dynamic equations of thick plates bending with lateral load, applied the differential operator algebra and decomposition of operator spectra, the refined dynamic equation of the beam flexural motion with lateral load is first obtained by using proper gauge conditions and satisfying the boundary conditions. The refined equation of beams is a fourth-order equationTaken a cantilever beam as example in this paper, the first ten order natural frequency were obtained based on theory in this paper. The results gived by the Euler-Bernoulli beam and the Timoshenko beam theory, and beam theory in this paper were compared. Using theory in this paper, the analysis of the frequence were more accurate.In this paper, based on the theory of refined dynamic equations of thick plate stretching, applied decomposition of operator spectra, a refined dynamic equation of the thick beam stretching was firstly obtained by using proper gauge conditions and satisfying the free boundary conditions in both sides of panels. The refined equation of beams governs the stretching vibrations of beams. Since derivation of the dynamic equation of beams is conducted without the assumptions, so the equation of thick beam stretching is exact, that can be used to analyze the stretching vibration of beams including the higher-order modes and to evaluate the applicable condition of the different stretching theories of engineering structures.