This work represents an investigation of the complex modes of continuous vibration systems with nonmodal damping. As an example, a cantilevered beam with damping at the free end is studied. Assumed modes are applied to discretize the eigenvalue problem in state-variable form and then to obtain estimates of the true complex normal modes and frequencies. The finite element method (FEM) is also used to get the mass, stiffness, and damping matrices and further to solve a state-variable eigenvalue problem. A comparison between the complex modes and eigenvalues obtained from the assumed-mode analysis and the finite element analysis shows that the methods produce consistent results. The convergence behavior when using different assumed mode functions is investigated. The assumed-mode method is then used to study the effects of the end-damping coefficient on the estimated normal modes and modal damping. Most modes remain underdamped regardless of the end-damping coefficient. There is an optimal end-damping coefficient for vibration decay, which correlates with the maximum modal nonsynchronicity.

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