Study On The Band Gap Inverse Problems Of Phononic Crystal Structure Based On The Lumped Mass Method

Phononic crystals,a synthetic composite periodic structure with elastic wave band gaps,offer numerous applications in the realm of vibration and noise reduction.In practical engineering applications,a specific model must have bandgap characteristics for a specific frequency or frequency band.After this,the study of the band gap inverse problem is used to find the material or structural parameters that best suit the needs.The majority of current phononic crystal research relies on theoretical derivation and simulation to derive the band gap law of elastic wave propagation under the assumption of predetermined material and structural parameters.The engineering use of phononic crystals is severely constrained by the lack of progress in inverse problem research.The band gap inverse problems of phononic crystal rods,shafts,and beams are studied in this thesis using the lumped mass approach,giving a theoretical foundation for the practical engineering uses of phononic crystal rods,shafts,and beams.Firstly,the band gap inverse problems of phononic crystal rods with longitudinal wave band gap characteristics is studied.The discrete model of phononic crystal rod is analyzed and the longitudinal vibration equation based on the lumped mass method is derived theoretically.Combined with the theory of computation of phononic crystal band gap,the band gap characteristics of phononic crystal rod are explored,and the dispersion polynomial characterizing the band gap is calculated.The relationship between dispersion polynomial curve,band gap structure and attenuation factor curve are summarized to establish the existence criterion of band gap inverse problems.The feasible region of the band gap inverse problems is obtained through programming calculation,and the feasibility of the criterion for the longitudinal vibration of phononic crystal rod band gap inverse problems is verified through numerical calculation and finite element simulation.Secondly,the band gap inverse problems of the phononic crystal shafts with band gap characteristics of torsional waves is studied.Based on the lumped mass method,the torsional vibration equation of multi-element phononic crystal shafts with variable cross section are derived,and the band gap characteristics of the phononic crystal shafts are investigated by combining the Bloch theorem.Then,the dispersion polynomial of phononic crystal shafts is calculated,and the dispersion polynomial curve is used as a new way to characterize the band gap.Based on the characteristics of the dispersion polynomial,the existence criterion of the band gap of the phononic crystal shafts for the frequency point and frequency segment is established,and the parameter feasible region of the band gap inverse problems is obtained by solving an example.By means of numerical calculation and finite element simulation,the parameters of the feasible domain are analyzed to verify the feasibility of the existence criterion of phononic crystal axial band gap.Finally,based on the lumped mass method,the transverse vibrating phononic crystal beam was discretized into a discrete model consisting of a spring,a mass point and a massless rigid rod.The transverse vibration characteristics and band gap characteristics of phononic crystal beams are investigated.The dispersion polynomial of phononic crystal beam is derived theoretically,and the corresponding relationship between the dispersion polynomial curve and the band gap structure is studied.It is found that the dispersion polynomial curve representing the odd order band gap and its attenuation has a minimum solution independent of the band gap,and the functional relation of this minimum solution is deduced.The band gap existence criterion for inverse problems solving is established by combining the dispersion polynomial and band gap characteristics.A phononic crystal beam model composed of feasible domain parameters and non-feasible domain parameters is constructed.The band gap,transmissibility and stress distribution are compared and analyzed by numerical calculation and finite element simulation to verify the feasibility of the band gap existence criterion of phononic crystal beam inverse problem.Based on the criteria for the existence of band gap in the three basic components of phononic crystal rods,shafts,and beams,parameter feasible region analysis and finite element simulation verification of finite period models were conducted.The results show that the proposed theory of band gap inverse problems has certain feasibility.