Acoustic scattering from an infinitely long cylindrical shell with an internal mass attached by multiple axisymmetrically distributed stiffeners

摘要

A thin infinitely long elastic shell is stiffened by $J$ in number identicallengthwise ribs distributed uniformly around the circumference and joined to arod in the center. The 2D model of the substructure is a rigid central masssupported by $J$ axisymmetrically placed linear springs. The response of theshell-spring-mass system is quite different from a fluid filled shell or thatof a solid cylinder due to the discrete number of contact points which couplethe displacement of the shell at different locations. Exterior acousticscattering due to normal plane wave incidence is solved in closed form forarbitrary $J$. The scattering matrix associated with the normal mode solutiondisplays a simple structure, composed of distinct sub-matrices which decouplethe incident and scattered fields into $J$ families. The presence of asprings-mass substructure causes resonances which are shown to be related tothe subsonic shell flexural waves, and an approximate analytic expression isderived for the quasi-flexural resonance frequencies. Numerical simulationsindicate that the new solution for $J\ge 3$ springs results in a complicatedscattering response for plane wave incidence. As the number of springs becomeslarge enough, the total scattering cross-section is asymptotically zero at lowfrequencies and slightly increased compared to the empty shell at moderatefrequencies due to the added stiffness and mass. It is also observed that thesensitivity to the angle of incidence diminishes as the number of springs isincreased. This system can be tuned by selecting the shell thickness, springstiffness and added mass to yield desired quasi-static effective propertiesmaking it a candidate element for graded index sonic crystals.