Extension of dynamic stiffness method to complicated damped structures

The dynamic stiffness method is an exact method for structural dynamic analysis. By separating the variable of the displacement function in frequency domain, the dynamic stiffness matrix and frequency equation of the structure are obtained, and the structural dynamic analysis can then be achieved by solving the transcendental frequency equation. For undamped systems, the frequency equation can be accurately solved by the Wittrick-Williams algorithm, however, the frequency equation of damped structures is a complex transcendental equation and many root-search techniques performing well in real field including the Wittrick-Williams algorithm are no longer applicable. Therefore, the application of dynamic stiffness in damped structures is a major challenge and has not been well resolved. In view of this, aiming at the classically damped system in the project, this paper has improved the dynamic stiffness method from two aspects, (1) The calculation principle. By performing the variable separation in Laplace domain instead of frequency domain, this paper established the relationship between damped frequency and undamped frequency by a proposed method for the calculation of the damping ratio, thus avoiding the solution of the hard-to-solve complex transcendental frequency equation; (2) The solution method. To make the method widely applicable, an improved Wittrick-Williams algorithm is given in this paper to solve the frequency equation of complicated systems. Finally, numerical examples are used to verify the accuracy and universality of the proposed method.