Pretwisted nonuniform beams with time-dependent elastic boundary conditions

The governing differential equations with the nonhomogeneous time-dependent elastic boundary conditions for the coupled bending-bending vibration of a pretwisted nonuniform beam are derived by Hamilton's principle. By taking a general change of dependent variable with shifting functions, the original system is transformed to be a system composed of two nonhomogeneous governing differential equations and eight homogeneous boundary conditions. Consequently, the method of separation of variables can be used to solve the transformed problem. The physical meanings of these shifting functions are explored. The orthogonality condition for the eigenfunctions of a pretwisted nonuniform beam with elastic boundary conditions is also derived. The stiffness matrix for a nonuniform beam with arbitrary pretwist is derived. As the coefficients of the matrix can be integrated analytically, the exact stiffness matrix is, therefore, obtained. The relation between the shifting functions and the stiffness matrix is derived. The influence of the pretwist angle on the dynamic response of the beam is studied. The vibration control of a pretwist beam with boundary inputs is investigated.