Forced vibration and vibrational energy flow in cracked variable-thickness plates: A semi-analytical model

Variable-thickness plates are widely used in lightweight engineering structures, where thickness gradients improve structural efficiency but also complicate vibration transmission and crack-dependent wave scattering. When a surface crack is present, the local compliance introduced by the crack interacts with the non-uniform impedance caused by thickness variation, leading to forced vibration and energy transfer characteristics that are difficult to capture using conventional uniform-thickness or purely numerical models. This study develops a semi-analytical forward model for the forced vibration and vibrational energy flow of cracked variable-thickness plates under harmonic excitation. The proposed formulation combines an established symplectic wave description for a uniformly cracked segment with a segmented representation of the thickness gradient, so that the coupled effects of crack compliance and impedance variation can be captured within a unified framework. Based on the resulting wave amplitudes, analytical expressions are derived for the input energy flow, transmitted energy flow, kinetic energy, and strain energy. The proposed model is validated directly against finite element simulations for simply supported plates, while experiments under a free boundary test configuration are used to support the finite element reference model employed in the validation chain. The results reveal how crack compliance and thickness gradient jointly govern resonance shifts, wave scattering, energy transmission, and spatial energy redistribution. In particular, the high-frequency response is more sensitive to crack-induced stiffness loss, and the energy transfer characteristics differ markedly between cracks located near the principal transmission path and those close to the plate boundary. Finally, an illustrative contour-based parameter-estimation example using normalized input energy flow is presented to show the estimation potential of the forward model under idealized conditions. The present work provides an efficient forward framework for analysing vibration transmission and energy flow mechanisms in cracked variable-thickness thin plates and offers a useful basis for subsequent model-based studies.