Energy flow, energy density of Timoshenko beam and wave mode incoherence

Abstract

Time-averaged energy flow and energy density are of significance in vibration analysis. The wave decomposition method is more fruitful and global in physical sense than the state variables depicted point by point. By wave approach, the Timoshenko beam vibration field is decomposed into two distinct modes: travelling and evanescent waves. Consequently, the power and energy functions defined on these waves’ amplitude and phase need to be established. However, such formulas on Timoshenko beam are hardly found in literatures. Furthermore, the incoherence between these two modes is of theoretical and practical significance. This characteristic guarantees that the resultant power or energy of a superposed wave field is equal to the sum of the power or energy that each wave mode would generate individually. Unlike Euler–Bernoulli beam, such incoherence in the Timoshenko beam case has not been theoretically proved so far. Initially, the power and energy formulas based on wave approach and the corresponding incoherence proof are achieved by present work, both in theoretical and numerical ways. Fortunately, the theoretical and numerical results show that the travelling and evanescent wave modes are incoherent with each other both on power and energy functions. Notably, the energy function is unconventional and self-defined in order to obtain the incoherence. Some remarkable power transmission characteristics of the evanescent wave are also illustrated meanwhile.