Improved complex modal superposition method with inclusion of overdamped modes

To address the involvement of overdamped complex modes that appear in a nonproportionally damped system, an improved complex mode superposition (ICMS) theory is proposed for forced vibration analysis, which suggests the use of exact modes in pairs for CMS-based dynamic analysis of whether the modes are overdamped or not. A typical nonproportionally damped system, namely, a cantilever beam with attached multiple arbitrarily placed external dampers, is considered an example because the first mode of the system is likely to be overdamped with the increase in damping. First, the relationship between the response in a complex modal space and the actual dynamic response is elucidated, based on which complete theoretical ICMS approaches for attaining time-domain response, transfer functions, and variances are expounded in detail. By decomposing the governing equation into real and imaginary parts, the original equation of motion in the complex domain is represented by an augmented state-space equation with real-valued matrices, which considerably reduces the difficulties observed in computing the time-varying response using complex-valued matrices. Additionally, for external excitations that can be regarded as filtered white noise, an efficient method for evaluating the variance response is proposed, which effectively reduces the computational cost. The results from the application of the proposed CMS-based methods are compared with those obtained by an assumed-mode (AM) method and finite element analysis (FEA) . It can be found that the current results are closer to those obtained by FEA than those by the AM method. Finally, the optimal damping and optimal position of the dampers are investigated using an enumeration method, which reveals that the use of multiple dampers with small damping demonstrates a better effect than that of a single damper with large damping.