Abhigna Bhatt

Abhigna Bhatt

she/her

ANRF National Postdoctoral Fellow at IISc Bengaluru, working on elastic waves, metamaterials and vibration isolation.

Concept Demonstrations

Waves through a locally resonant metamaterial

A chain of twelve unit cells sits between two non-dispersive media. Choose a frequency and see whether vibration crosses the chain, and how fast its phase and energy travel.

Drag the frequency line on the plots, or use the slider. On the left plot, the slope of the line from the origin gives the phase velocity, and the slope of the tangent gives the group velocity (plot axes are not to equal scale). The motion is drawn vertically, and resonator motion is capped for display.

Formulation

This follows the negative mass (NM) monocoupled system and the method of Bhatt, Banerjee and Adhikari (2022), obtained from its generalised system by setting \(m_a=0\) and \(m_2=0\), with \(m_1\), \(k_1\), \(\theta_{r1}\), \(\eta_{r1}\) and \(\zeta_{r1}\) written as \(m\), \(k\), \(\theta_r\), \(\eta_r\) and \(\zeta_r\).

1. Unit cell and dynamic stiffness

Main-chain masses \(M\) at nodes 1 and 4 are joined by a spring \(K\). Each carries a resonator \(m\) (nodes 2 and 3) through a spring \(k\). The equations of motion are

Unit cell: masses M at nodes 1 and 4 joined by spring K, with resonators m at nodes 2 and 3 attached through springs k
\[ M\ddot{u}_1+K(u_1-u_4)+k(u_1-u_2)=0 \]
\[ m\ddot{u}_2+k(u_2-u_1)=0 \]
\[ m\ddot{u}_3+k(u_3-u_4)=0 \]
\[ M\ddot{u}_4+K(u_4-u_1)+k(u_4-u_3)=0 \]

With \(\ddot{(\bullet)}=-\omega^2(\bullet)\), condensing out nodes 2 and 3 gives the dynamic stiffness matrix, and its inverse gives the receptance matrix:

\[ \mathbf{D}_{yc}=\begin{bmatrix} D & -K\\ -K & D\end{bmatrix} \]
\[ D=-\omega^2M+K+k+\frac{k^2}{\omega^2m-k} \]
\[ \boldsymbol{\alpha}_R=\mathbf{D}_{yc}^{-1}, \qquad \alpha_{LL}=\frac{D}{|\mathbf{D}_{yc}|}, \qquad \alpha_{LR}=\frac{K}{|\mathbf{D}_{yc}|} \]

2. Dispersion relation

Bloch's theorem relates the left and right displacements and forces through the propagation constant \(\kappa=\alpha+i\beta\):

\[ d_R=e^{i\kappa}d_L, \qquad f_R=-e^{i\kappa}f_L, \qquad \cos\kappa=\frac{\alpha_{LL}}{\alpha_{LR}}=g(\omega^2)=x \]

The wave propagates with \(\alpha=\cos^{-1}x\) and \(\beta=0\) when \(x^2\lt 1\). When \(x^2\gt 1\) it attenuates, with \(\alpha=\pi\) for \(x\lt -1\), \(\alpha=0\) for \(x\gt 1\), and \(\beta=\cosh^{-1}|x|\).

3. Peak and bounds of the attenuation band

Using \(\eta=\omega/\sqrt{K/M}\), \(\theta_r=m/M\) and \(\eta_r=\sqrt{k/m}/\sqrt{K/M}\), the dispersion relation takes the rational form

\[ g(\eta^2)=1+\frac{R(\eta^2)}{Q(\eta^2)} \]
\[ R=\eta^4-\eta^2\eta_r^2(1+\theta_r), \qquad Q=\eta_r^2-\eta^2 \]

The attenuation peak is at \(Q=0\), and the band boundaries are at \(R=0\) (\(\kappa=0\)) and \(R+2Q=0\) (\(\kappa=\pi\), the Bragg condition):

\[ \xi_{NM}=\eta_r, \qquad \zeta_r=\eta_r\sqrt{1+\theta_r} \]
\[ \eta_{B}^2=\frac{1}{2}\left[b\mp\sqrt{b^2-8\eta_r^2}\right], \qquad b=2+\eta_r^2(1+\theta_r) \]

The demonstration uses \(\theta_r=0.5\) and \(\eta_r=1\). The Bragg boundaries are then \(\eta_B=0.848\) and \(1.668\), the peak is at \(\eta=1\), and \(\zeta_r=1.225\).

Velocity formulation

In a propagation band, with \(L\) the cell length, the velocities in units of \(L\sqrt{K/M}\) follow from \(\cos\alpha=g(\eta^2)\):

\[ c_p=\frac{\eta}{\alpha}, \qquad c_g=\frac{d\eta}{d\alpha}=-\frac{\sin\alpha}{dg/d\eta} \]

\(\sin\alpha\) vanishes at both \(\kappa=0\) and \(\kappa=\pi\), so \(c_g=0\) at every band boundary, while \(c_p\) stays finite at the Bragg boundary. Solving \(\cos\kappa=g(\eta^2)\) gives each band exactly, with the minus sign for the lower band and the plus sign for the upper band:

\[ \eta^2(\kappa)=\frac{1}{2}\left[B\mp\sqrt{B^2-4(1-\cos\kappa)\,\eta_r^2}\right] \]
\[ B=\eta_r^2(1+\theta_r)+1-\cos\kappa \]

The wave packet panel is an exact sum of Bloch waves with a Gaussian spectrum of width \(\sigma_\kappa\), so it includes dispersive spreading:

\[ u(x,t)=\mathrm{Re}\sum_j e^{-(\kappa_j-\alpha)^2/2\sigma_\kappa^2}\,e^{i\left[\kappa_jx-\eta(\kappa_j)t\right]} \]

Impedance formulation

From the receptance relation and Bloch's theorem, with \(D=K\cos\kappa\), the force at the left node of the cell is

\[ f_L=\left(D-Ke^{i\kappa}\right)d_L=-iK\sin\kappa\;d_L \]

Dividing force by velocity amplitude gives the Bloch impedance of a propagating wave:

\[ Z_B=\frac{K\sin\alpha}{\omega}, \qquad \frac{Z_B}{\sqrt{KM}}=\frac{\sin\alpha}{\eta} \]

This value applies when the cell boundary passes through the node masses, as in this unit cell. It depends on where the boundary is placed. In the long-wave limit, it gives the impedance chosen for the two media:

\[ Z_0=\lim_{\omega\to0}Z_B=\sqrt{2K(M+m)}, \qquad c_0=\frac{\omega}{\kappa_0}=L\sqrt{\frac{K}{2(M+m)}} \]

Finite chain between the two media

The media are semi-infinite, so their far ends do not reflect. For incident, reflected and transmitted amplitudes \(U_I\), \(U_R\) and \(U_T\), with \(\kappa_0\) the wavenumber per cell and \(n\) measured from each interface:

\[ u_{\mathrm{left}}=\left(U_Ie^{i\kappa_0n}+U_Re^{-i\kappa_0n}\right)e^{-i\omega t} \]
\[ u_{\mathrm{right}}=U_Te^{i\kappa_0n}e^{-i\omega t} \]

Assembling \(N=12\) cells gives a tridiagonal system for the node displacements \(d_0\) to \(d_N\):

\[ (D-i\omega Z_0)\,d_0-Kd_1=-2i\omega Z_0U_I \]
\[ -Kd_{n-1}+2Dd_n-Kd_{n+1}=0, \qquad 0\lt n\lt N \]
\[ -Kd_{N-1}+(D-i\omega Z_0)\,d_N=0 \]
\[ U_R=d_0-U_I, \qquad U_T=d_N, \qquad |U_R|^2+|U_T|^2=|U_I|^2 \]

In a propagation band this solves to the closed form

\[ \left|\frac{U_T}{U_I}\right|^2=\frac{1}{1+\frac{1}{4}\left(\frac{Z_0}{Z_B}-\frac{Z_B}{Z_0}\right)^2\sin^2(N\alpha)} \]

What happens at the band edges

\(Z_B\to0\) and \(c_g\to0\) at a band boundary. Taking this limit at \(\eta_B\) gives a transmission that is small but not zero, and falls as \(1/N\) instead of exponentially:

\[ \left|\frac{U_T}{U_I}\right|_{\eta_B}=\frac{1}{\sqrt{1+\left(N\eta_BZ_0/2\sqrt{KM}\right)^2}} \]

For this chain that is 11% at \(\eta=0.848\) and 6% at \(\eta=1.668\). The infinite-chain band edge is sharp, but a finite chain has only a gradual transition around it. The impedance mismatch also reflects most of the wave just inside the band, so a finite chain is not a sharp switch at this frequency.

References

  1. A. Bhatt, A. Banerjee and S. Adhikari, Closed-form solutions for attenuation peaks and band boundaries of general monocoupled systems, Journal of Sound and Vibration, 541, 117318, 2022.
  2. L. Brillouin, Wave Propagation in Periodic Structures, McGraw-Hill, 1946.
  3. M. I. Hussein, M. J. Leamy and M. Ruzzene, Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook, Applied Mechanics Reviews, 66, 040802, 2014.
  4. D. J. Colquitt, R. V. Craster and M. Makwana, High frequency homogenisation for elastic lattices, on behaviour close to band edges.