Experimental Study on Wave Propagation in One-Dimensional Viscoelastic Metamaterial

Abstract

A locally resonant viscoelastic mass-spring cell is experimentally realized by a unit cell design fabricated by 3D printing. The standard linear solid model is introduced for the viscoelastic metamaterial. The complex band structures of both viscoelastic unit cell and elastic cases are presented to show the effect of viscoelasticity. Both the harmonic excitation and stochastic excitation are conducted on the finite viscoelastic metamaterial in experiments. Distinct wave attenuation is found in bandgap via sweep frequency response analysis under harmonic excitation. The experiments of the metamaterial under narrow-band noise excitation demonstrate good performance of wave attenuation in bandgap. Finally, the obtained bandgaps via numerical calculation are well consistent with the frequency ranges of wave attenuation from experiments, which confirm the effectiveness of the proposed viscoelastic model.

Appendix. Numerical Investigation of Finite Viscoelastic Metamaterial

The detailed procedures to get the responses of the finite system are described in this appendix. For the finite viscoelastic metamaterial composed of N cells shown in Fig. 5d, the dynamic equations are given in Eq. (A.1),

$$\begin{aligned}&\left\{ {{\begin{array}{*{20}l} {M\ddot{{u}}_{n} +k_{3} (2u_{n} -u_{n+\text{1 }} -u_{n-\text{1 }} )+c_{2} (\dot{{u}}_{n} -\dot{{q}}_{n} )+k_{1} (u_{n} -v_{n} )+k_{2} (u_{n} -p_{n} )+k_{4} (u_{n} -q_{n+\text{1 }} )=0,} \\ {c_{2} (\dot{{q}}_{n} -\dot{{u}}_{n} )+k_{4} (q_{n} -u_{n-\text{1 }} )=0,} \\ {m\ddot{{v}}_{n} +c_{1} (\dot{{v}}_{n} -\dot{{p}}_{n} )+k_{1} (v_{n} -u_{n} )=0,} \\ {c_{1} (\dot{{p}}_{n} -\dot{{v}}_{n} )+k_{2} (p_{n} -u_{n} )=0 \quad \text{( }n=1,\ldots ,N-1)} \\ \end{array} }} \right. \end{aligned}$$

(A.1a)

$$\begin{aligned}&\left\{ {{\begin{array}{*{20}l} {M\ddot{{u}}_{N} +k_{3} (u_{N} -u_{N-\text{1 }} )+c_{2} (\dot{{u}}_{N} -\dot{{q}}_{N} )+k_{1} (u_{N} -v_{N} )+k_{2} (u_{N} -p_{N} )=0} \\ {c_{2} (\dot{{q}}_{N} -\dot{{u}}_{N} )+k_{4} (q_{N} -u_{N-\text{1 }} )=0} \\ {m\ddot{{v}}_{N} +c_{1} (\dot{{v}}_{N} -\dot{{p}}_{N} )+k_{1} (v_{N} -u_{N} )=0} \\ {c_{1} (\dot{{p}}_{N} -\dot{{v}}_{N} )+k_{2} (p_{N} -u_{N} )=0} \\ \end{array} }} \right. \end{aligned}$$

(A.1b)

where $u_{0} (t)$ in Eq. (A.1a) is the displacement excitation. The multiple variables of the n-th unit cell are defined as

$$\begin{aligned}&{{{\varvec{y}}}}_{n} (t_{j} )\equiv \left\{ {u_{n} (t_{j} )\dot{{u}}_{n} (t_{j} )v_{n} (t_{j} )\dot{{v}}_{n} (t_{j} )p_{n} (t_{j} )q_{n} (t_{j} )} \right\} ^{\mathrm{T}}\text{( }n=1,2,\ldots ,N) \end{aligned}$$

(A.2a)

$$\begin{aligned}&t_{j} =t_{0} +h\cdot j \text{( }j=0,1,\ldots ,Nt) \end{aligned}$$

(A.2b)

where $t_{0}$ is the initial moment, h is the time step, and Nt is the total calculation steps. The Runge–Kutta method of order 4 (RK4) is utilized to numerically solve the ordinary differential Eq. (A.1). The initial conditions ${{\varvec{y}}}_{n} (t_{0} ),n=1,2,\ldots ,N$ should be given before calculation. Based on the finite-difference time-domain (FDTD) method [33], the second-order differential equations can be rewritten as the algebraic equations in matrix form,

$$\begin{aligned} {{{\varvec{y}}}}_{n} (t_{j+1} )={{{\varvec{ y}}}}_{n} (t_{j} )+\frac{1}{6}(\mathrm{{{\varvec{K}}}}_{\!1} +2{{{\varvec{K}}}}_{\!2} +2{{{\varvec{K}}}}_{\!3} +{{{\varvec{K}}}}_{\!4} )\cdot h \end{aligned}$$

(A.3)

where

$$\begin{aligned} \left\{ {{\begin{array}{*{20}l} {{{{\varvec{K}}}}_{\!1} ={{{\varvec{f}}}}(t_{j} ,{{{\varvec{y}}}}_{n} (t_{j} ))} \\ {{{{\varvec{K}}}}_{\!2} ={{{\varvec{f}}}}(t_{j} +\frac{1}{2}h,{{{\varvec{y}}}}_{n} (t_{j} )+\frac{1}{\!2}{{{\varvec{K}}}}_{\!1} h)} \\ {{{{\varvec{K}}}}_{\!3} ={{{\varvec{f}}}}(t_{j} +\frac{1}{2}h,{{{\varvec{y}}}}_{n} (t_{j} )+\frac{1}{\!2}{{{\varvec{K}}}}_{\!2} h)} \\ {{{{\varvec{K}}}}_{\!4} ={{{\varvec{f}}}}(t_{j} +h,{{{\varvec{y}}}}_{n} (t_{j} )+\mathrm{{{\varvec{K}}}}_{\!3} h)} \\ \end{array} }} \right. \end{aligned}$$

(A.4)

The function ${{\varvec{f}}}$ is defined as

$$\begin{aligned} {{{\varvec{f}}}}(t,{{{\varvec{y}}}}_{n} )={{{\varvec{A}}}}_{3} \cdot {{{\varvec{y}}}}_{n} +{{{\varvec{B}}}}_{n} (t)\quad (n=1,2,\ldots ,N) \end{aligned}$$

(A.5)

where

$$\begin{aligned}&{{{\varvec{A}}}}_{3} =\left[ {{\begin{array}{*{20}c} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ {\frac{-(2k_{3} +k_{1} +k_{2} +k_{4} )}{M}} &{} 0 &{} {\frac{k_{1} }{M}} &{} 0 &{} {\frac{k_{2} }{M}} &{} {-\frac{k_{4} }{M}} \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ {\frac{k_{1} +k_{2} }{m}} &{} 0 &{} {-\frac{k_{1} }{m}} &{} 0 &{} {-\frac{k_{2} }{m}} &{} 0 \\ {\frac{k_{2} }{c_{1} }} &{} 0 &{} 0 &{} 1 &{} {-\frac{k_{2} }{c_{1} }} &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} {-\frac{k_{4} }{c_{2} }} \\ \end{array} }} \right] \end{aligned}$$

(A.6a)

$$\begin{aligned}&\left\{ {\begin{array}{l} {{{\varvec{B}}}}_{1} (t)=\left\{ {{\begin{array}{*{20}l} 0 &{} {\frac{(k_{3} +k_{4} )u_{0} (t)+k_{3} u_{2} +k_{4} q_{2} }{M}} &{} 0 &{} 0 &{} 0 &{} {\frac{k_{4} u_{0} (t)}{c_{2} }} \\ \end{array} }} \right\} ^{\mathrm{T}} \\ {{{\varvec{B}}}}_{n} (t)=\left\{ {{\begin{array}{*{20}c} 0 &{} {\frac{(k_{3} +k_{4} )u_{n-1} +k_{3} u_{n+1} +k_{4} q_{n+1} }{M}} &{} 0 &{} 0 &{} 0 &{} {\frac{k_{4} u_{n-1} }{c_{2} }} \\ \end{array} }} \right\} ^{\mathrm{T}}\text{( }n=2,3,\ldots ,N-1) \\ {{{\varvec{B}}}}_{N} (t)=\left\{ {{\begin{array}{*{20}c} 0 &{} {\frac{(k_{3} +k_{4} )(u_{N} +u_{N-1} )}{M}} &{} 0 &{} 0 &{} 0 &{} {\frac{k_{4} u_{N-1} }{c_{2} }} \\ \end{array} }} \right\} ^{\mathrm{T}} \\ \end{array}} \right. \end{aligned}$$

(A.6b)

The displacement excitations and boundary conditions are all taken into account in Eq. (A.5). By the iterative computation of Eq. (A.3) in each time step, the transient simulation of proposed structure in the time domain would be obtained.