The Research On The Statics And Dynamics Character Of The Beam With Transverse Load

Geometrical nonlinearity model of Euler-Bernoulli beam with transverse load can be established by regards to its axial distortion according to energy conservation. Constant coefficient differential equations about nonlinearity free vibration can be acquired by Hamilton variable theory and Kantorovich time average methods. The numerical solution to the bending distortion and free vibration can be achieved by shooting method, the yield of common principle are as following.1. Nonlinearity free vibration partial differential equations are established in condition of transverse uniform load for its axial distortion. The mentioned model is transformed into dimensionless differential groups for movement control which mainly be used as the beam's elastic distortion with regard to the elimination of dynamic items.2. Using shooting method, cantilever beam, simply supported beam, and clamped beam's bending distortion and vibration numerical solution are presented on condition of various load, as well as characteristic curve about distortion and load. It proves the relation of beams' distortion and the slender ratio, in dynamic response, the beams' three-order natural frequency is obtained, the chart about natural frequency, the slender ratio and load shows that the cantilever beam's front three-order free vibration frequency decrease monotonously with increase of load.Simply supported beam and clamped beam, frequency of all orders increase with load. The cantilever beam's frequency decreases with increase of the slender ratio. The other beams'frequency of all orders increase with the increase of the slender ratio. The front three-order vibration curve of above mentioned beams are presented.3. The cantilever beam and the simply supported beam's natural frequency are obtained by assumption vibration model and lumped mass method, a comparison is made between the above mentioned results and numerical solution of shooting method.