Fig. 1 A bimorph piezoelectric cantilever beam periodically plucked by a magnet attached to a rotating, circular plate: ( a ) schematic of the system and magnetization directions, ( b ) dimensions and orientation of the magnets, and ( c ) schematic of the piezoelectric bimorph More

Fig. 2 Open-circuit ( β = 0) voltage at steady-state: ( a ) time history of the voltage v between a pluck ( τ = 0) and the next ( τ = T ) for ω = 0.10478 and α = 0.1, where α = d / R and T = 2 π / ω . −− −: numerical integration; —: MMS response and ( b ) mean squared open-circuit ... More

Fig. 3 Harmonic components V n of open-circuit voltage versus ω for α = 0.1, where n denotes the harmonic number: ( a ) ω near saddle-node bifurcation and ( b ) ω near Hopf bifurcation. It is noted that | V n | is plotted in a log scale More

Fig. 4 Divergence of the open-circuit voltage starting from ( a ) the saddle-node bifurcation point for ω = 0.10422 and ( b ) the Hopf bifurcation point for ω = 0.10478, and for α = 0.1. —: slowly varying envelope predicted by MMS. —: numerical integration. ×: instant the beam and plate coll... More

Fig. 5 Mean squared open-circuit voltage | v | a v 2 at steady-state versus ω for three values of h , where | v | a v 2 = ( 1 / T ) ∫ 0 T | v | 2 d τ : ( a ) h = 12, ( b ) h = 11, and ( c ) h = 10. —: MMS stable response. −− −: MMS u... More

Fig. 6 Average power β | v | a v 2 at steady-state versus ω for four values of β , and for α = 0.095. —: MMS stable response; −− −: MMS unstable response; °: numerical integration; and △ : Hopf bifurcation ( Hb ). More

Fig. 7 Comparison with the finite element method: ( a ) mode shape of the first mode of the cantilever beam in abaqus/standard and ( b ) mean squared beam tip displacement | x p | a v 2 at steady-state versus ω for four damping scenarios, where | x p | a v 2... More

Fig. 1 Infinite double concentric large cylindrical shells with periodic bulkheads and cavities submerged in the water More

Fig. 2 Positive loading directions of the reactive forces at the inner and outer edges of the annular bulkhead More

Fig. 3 SPL comparisons of the present method, analytical model, and experimental measurements: ( a ) θ = 0 and ( b ) θ = π /4 More

Fig. 4 Partial and total sound pressure power spectra of inner cylindrical shell at 2kHz: ( a ) total sound pressure power spectra of inner cylindrical shell calculated by u ~ 3 ( i ) and u ~ 3 ( o ) , ( b ) partial sound pressure power spectra of inner... More

Fig. 5 Partial and total sound pressure power spectra of outer cylindrical shell at 2kHz: ( a ) sound pressure power spectra of outer cylindrical shell on the bottom cylindrical surface calculated by u ~ 3 ( i ) and u ~ 3 ( o ) , ( b ) partial sound pre... More

Fig. 6 Sound pressure power spectra of the cylindrical interface between the outer cylindrical shell and external free fluid at 4 kHz: ( a ) a 1 = 1.5 m, a 2 = 5 m and ( b ) a 1 = 4.4 m, a 2 = 5 m More

Fig. 7 Sound pressure power spectra of stiffened double cylindrical shells on the interface between the outer cylindrical shell and external free fluid at 2 kHz: ( a ) without annular fluid, ( b ) with annular fluid, and ( c ) with acoustic cavities More

Fig. 8 Sound pressure power spectra of stiffened double cylindrical shells with acoustic cavities on the two fluid–solid coupling interfaces of annular fluid at 2 kHz: ( a ) sound pressure power spectra of the inner cylindrical shell and ( b ) sound pressure power spectra of the outer cylindrical ... More

Fig. 9 Sound pressure power spectra of stiffened double cylindrical shells on the cylindrical interfaces between outer cylindrical shell and external free fluid at 4 kHz: ( a ) without annular fluid, ( b ) with annular fluid, and ( c ) with periodic cavities More

Fig. 10 Sound pressure power spectra of stiffened double cylindrical shells on the two fluid–solid interfaces among inner cylindrical shell, annular cavities, and outer cylindrical shell at 4 kHz: ( a ) sound pressure power spectra of inner cylindrical shell and ( b ) sound pressure power spectra ... More