The Variational Principle Of Extending Cantilever Beam

In this paper, variational integral method is used to study the extending cantilever-beam variational principle. The general step of variational integral method is that the basic equations of dynamics are multiplied by corresponding virtual quantities, integrated and then added algebraically, according to the corresponding relations between generalized forces and generalized displacements, then we will get various corresponding variational principle and generalized variational principle. It makes of organic unity partial analysis and the overall analysis.In this paper, Euler coordinates is used to describe the dynamics of the extending cantilever structure. First, in the case of small deformation, the dynamics equations of Euler-Bernoulli cantilever-beam and Timoshenko cantilever-beam are established, then the various variational principle and generalized variational principle of both cantilever-beam dynamics models will be established by the variational integral method. Specific details are as follows:Firstly, in the case of small deformation, I establish the dynamics equations of general Euler-Bernoulli cantilever-beam and general Timoshenko cantilever-beam. Then through the variational integral method, the various variational principle and generalized variational principle of both general cantilever-beam dynamics models can be established.Secondly, in the case of small deformation, considering the coupling between cantilever-beam extending movement and its horizontal vibration, I set up the dynamics equations of extending Euler-Bernoulli cantilever-beam and extending Timoshenko cantilever-beam. And both of extending cantilever-beam dynamics models' various variational principle and generalized variational principle can be established by the variational integral method. Thirdly, in the case of small deformation, considering the coupling between cantilever-beam rotating movement and its horizontal vibration, I set up the dynamics equations of rotating Euler-Bernoulli cantilever-beam and rotating Timoshenko cantilever-beam. Then the various variational principle and generalized variational principle of both rotating cantilever-beam dynamics models can be established by the variational integral method.Then, in the case of small deformation, considering the coupling among the extending movement and the rotating movement of cantilever-beam and its horizontal vibration, I establish the extension and rotation of Euler-Bernoulli cantilever-beam's and the extension and rotation of Timoshenko cantilever-beam's dynamics equations. And the various variational principle and generalized variational principle of the two extending- rotating cantilever-beam dynamics models can be established by the variational integral method.Finally, considering the geometric nonlinear deformation, I establish the extension and rotation of Euler-Bernoulli cantilever-beam's and the extension and rotation of Timoshenko cantilever-beam's dynamics models in the case of large deformation.