Nonlinear Vibration Analysis of Metamaterial Honeycomb Sandwich Structures with Negative Poisson’s Ratio

Abstract

The research on existence, bifurcation, and number of periodic solutions is closely related to Hilbert’s 16th problem. The main goal of this chapter is to investigate the nonlinear dynamic response and periodic vibration characteristic of a simply supported concave hexagonal honeycomb sandwich plate with negative Poisson’s ratio. The plate is subjected to its in-plane and transverse excitation. The curvilinear coordinate frame, Poincaré map, and improved Melnikov function are proposed to detect the existence and number of the periodic solutions. The theoretical analyses indicate the existence of periodic orbits and can guaranteeat most four periodic orbits under certain conditions. Numerical simulations are performed to verify the theoretical results. The relative positons as well as the vibration characteristics can also be clearly found from the phase portraits. Theperiodic motion for the equation is closely related to the amplitude modulated periodic vibrations of the plate. The results will provide theoretical guidance to nonlinear vibration control for the metamaterial honeycomb sandwich structures.

Appendix

Assuming that H _{1, j} = F _j, H _{2, j} = G _j, c _{1, k} = a _k, c _{2, k} = b _k (j = 1, 2; k = 1, 2, ⋯, 17). Then F = (F ₁, F ₂)^T and G = (G ₁, G ₂)^T of Eq. (2) can be expressed as

$$ {\begin{array}{c}{H}_{i,1}={c}_{i,16}-{c}_{i,2}\left(\left(2-i\right){x}_2+\left(i-1\right){y}_2\right)+2{c}_{i,3}{y}_1{y}_2+{c}_{i,5}\left({x}_2{y}_1+{x}_1{y}_2\right)\\[2pt] {}+2{c}_{i,4}{x}_1{x}_2+\left({c}_{i,6}{y}_2+{c}_{i,7}{y}_1+{c}_{i,8}{x}_2+{c}_{i,9}{x}_1\right)\left({y}_1^2+{y}_2^2\right)\\[2pt] {}+\left({c}_{i,10}{y}_2+{c}_{i,11}{y}_1+{c}_{i,12}{x}_2+{c}_{i,13}{x}_1\right)\left({x}_1^2+{x}_2^2\right)\\[2pt] {}+\left({c}_{i,14}{y}_2+2{c}_{i,9}{y}_1+{c}_{i,15}{x}_2+2{c}_{i,11}{x}_1\right)\left({x}_1{y}_1+{x}_2{y}_2\right)\end{array}} $$

$$ {\begin{array}{c}{H}_{i,2}={c}_{i,17}-5{c}_{i,2}\left(\left(2-i\right){x}_1+\left(i-1\right){y}_1\right)+{c}_{i,3}\left({y}_1^2+3{y}_2^2\right)+{c}_{i,4}\left({x}_1^2+3{x}_2^2\right)\\[2pt] {}+{c}_{i,5}\left({x}_1{y}_1+3{x}_2{y}_2\right)+\left({c}_{i,7}{y}_2-{c}_{i,6}{y}_1+{c}_{i,9}{x}_2-{c}_{i,8}{x}_1\right)\left({y}_1^2+{y}_2^2\right)\\[2pt] {}+\left({c}_{i,11}{y}_2-{c}_{i,10}{y}_1+{c}_{i,13}{x}_2-{c}_{i,12}{x}_1\right)\left({x}_1^2+{x}_2^2\right)\\[2pt] {}+\left(2{c}_{i,9}{y}_2-{c}_{i,14}{y}_1+2{c}_{i,11}{x}_2-{c}_{i,15}{x}_1\right)\left({x}_1{y}_1+{x}_2{y}_2\right)\end{array}} $$

where the coefficients are respectively

${a}_2={\alpha}_2^2/24\vphantom{\frac{\int}{\int}}$	a 3 = (α 5f 1 + 3α 4f 2)/16	a 4 = (3α 3f 1 + α 6f 2)/16
a 5 = (α 6f 1 + α 5f 2)/8	a 6 = 3α 4(σ 1 + σ 2)/8	a 7 = 3μα 4(3β 1 − α 1)/8
a 8 = α 5(2σ 1 − σ 2)/4	a 9 = μα 5β 1/4	a 10 = α 6(3σ 2 − σ 1)/8
a 11 = μα 6(α 1 + β 1)/8	a 12 = 3α 3σ 1/4	a 13 = 3μα 1α 3/4
a 14 = α 5σ 2/2	a 15 = α 6(3σ 1 − σ 2)/4	a 16 = μα 1f 1/16
a 17 = σ 1f 1/16	${b}_2={\beta}_2^2/24\vphantom{\frac{\int}{\int}}$	b 3 = (β 6f 1 + 3β 3f 2)/16
b 4 = (3β 4f 1 + β 5f 2)/16	b 5 = (β 6f 2 + β 5f 1)/8	b 6 = 3β 3σ 2/4
b 7 = 3μβ 1β 3/4	b 8 = β 6(3σ 1 − σ 2)/8	b 9 = μβ 6(α 1 + β 1)/8
b 10 = β 5(−σ 1 + 2σ 2)/4	b 11 = μα 1β 5/4	b 12 = 3β 4(σ 1 + σ 2)/8
b 13 = 3μβ 4(3α 1 − β 1)/8	b 14 = β 6(−σ 1 + 3σ 2)/4	b 15 = β 5σ 1/2
b 16 = μβ 1f 2/16	b 17 = σ 2f 2/16