Band Gap Design And Calculation For One-Dimensional Artificial Periodic Structure

Artificial periodic structure,which can prohibit effectively the elastic wave propagation in the band gap frequency range,has broad application prospects in the vibration and noise reduction areas,and has increasingly attracted more and more attention.How to design artificial periodic structure with the specific band gap frequency range has become an important issue for phononic crystals and acoustic metamaterials.The study on this field has very important meanings in theory and engineering application.The band gap design of artificial periodic structure is mainly based on two theories: Bragg scattering and local resonance.In this paper,both perturbation method and singularity analysis theory were utilized to analyze the dispersion curves and determine the starting and cutoff frequencies of the band gap based on Bloch’s theorem.Thus the influences of mass ratio,stiffness ratio and other structure parameters on the band gap were discussed,and then the band gap design and nonlinearity influence were explored.(1)Dispersion curve equations of one-dimensional periodic structure with three oscillators were investigated and analyzed with singularity theory.The transition set calculation result was empty set,which indicated that the topology of the dispersion curves did not change with mass,stiffness and other parameters.Then the starting and cutoff frequencies of the band gap could be determined,and the band gap design method was proposed and verified subsequently.(2)For the one-dimensional periodic rod with two components,lumped mass method was used to discretize the rod into one-dimensional spring oscillator periodic structure.The periodic structures of different numbers of oscillators obtained from the discretization were discussed respectively,the relationship curves were obtained among 21r(two component density ration),21e(elastic modulus ratio)and f(the starting and cutoff frequencies for first band gap),and then a design method for the first band gap is proposed and verified by three oscillators as an example with FEM.and the results were in accordance with the design requirements.(3)The exact expression of the band gap starting and cutoff frequencies was deduced for the linear local resonance periodic structure with two oscillators,and the influences of mass ratio and stiffness ratio of two oscillators were analyzed.The research suggested that the band gap starting frequency was lower than the internal oscillator resonant frequency,was monotonically increasing with stiffness ratio,and decreasing with mass ratio.And the band gap cutoff frequency was only monotonically decreasing with mass ratio.In the band gap range,besides the effective negative mass region as reported in the literature,there was also effective positive mass region.(4)The acoustic metamaterial containing Duffing oscillator with positive stiffness was studied,and the first-order perturbation solutions of acoustic and optical branches were obtained.The nonlinear parts of the dispersion curves were related to the coefficients before the nonlinear item,steady-state amplitude of Duffing oscillator and mass ratio.It was found that the soft/hard characteristics of Duffing oscillators could lead to the lower/upper movement of the starting and cutoff frequencies of the band gap.The stronger the nonlinearity is,the more obvious the trend of the movement is.The translation effect of the external excitation amplitude on the band gap range and the zero mass phenomenon at the band gap cutoff frequency were discussed.(5)The acoustic metamaterial containing Duffing oscillator with negative stiffness was studied,and the first-order perturbation solutions of acoustic and optical branches were derived.The influences of nonlinearity and other structure parameters on the band gap starting and cutoff frequencies were similar to the research results of the acoustic metamaterial containing Duffing oscillator with positive stiffness.However,the band gap width was expanded to 2 times.The influence of the external excitation amplitude increase on the band gap width reduction and the zero mass phenomenon at the band gap cutoff frequency were investigated.(6)The low frequency and broadband acoustic metamaterial containing Duffing oscialltor was studied,and the second-order perturbation solution of dispersion curve was derived.The influence of nonlinearity on the cutoff frequency was analysized.Numerical simulation was utilized to verify the influence of the external excitation amplitude on the band gap,the effective negative mass in bandgap,the effective positive mass and effective zero mass phenomenon at the band gap cutoff frequency.