# A blueprint for truncation resonance placement in elastic periodic lattices
with unit cell asymmetry

Hasan B. Al Ba’ba’a  Department of Mechanical Engineering, Union College,
Schenectady, NY 12308  Department of Mechanical and Aerospace Engineering,
University at Buffalo (SUNY), Buffalo, NY 14260  Hosam Yousef  Department of
Mechanical and Aerospace Engineering, University at Buffalo (SUNY), Buffalo,
NY 14260  Mostafa Nouh  Department of Mechanical and Aerospace Engineering,
University at Buffalo (SUNY), Buffalo, NY 14260  Department of Civil,
Structural and Environmental Engineering, University at Buffalo (SUNY),
Buffalo, NY 14260

###### Abstract

Elastic periodic lattices act as mechanical filters of incident vibrational
loads. By and large, they forbid wave propagation within bandgaps and resonate
outside them. However, they often encounter ‘truncation resonances’ inside
bandgaps when certain conditions are met. This study provides an all-
encompassing blueprint for the design and selective placement of truncation
resonances in elastic periodic lattices, integrating all factors that play a
role in the onset of truncation resonances and shaping the finite lattice’s
response, from unit cell configuration, mass and stiffness contrasts, to types
and symmetry of boundary conditions.

At their very core, elastic periodic lattices are effective mechanical filters
of incident excitations. Lattices comprised of a sequence of two or more
alternating layers exhibit Bragg scattering preventing vibrations within
selected frequency ranges (bandgaps) from being transmitted through the
material Sigalas (1992); Sigalas and Economou (1995). They represent a class
of phononic crystals, owing to the similarity between the periodic array of
lumped spring-mass elements and the ordered arrangement of atoms in a
crystalline material Hussein, Leamy, and Ruzzene (2014); Vasileiadis  _et al._
(2021). The majority of structural resonances associated with such lattices
fall within its propagation frequencies or ‘pass bands’. Yet, finite lattices
are known to encounter natural frequencies which avert these pass bands and
end up inside bandgaps, forcing a unique combination of high-amplitude
deformations emblematic of resonances and an exponential spatial attenuation
of the incident wave towards the bulk of the medium Al Ba’Ba’A and Nouh
(2017). The result is a localized mode where a large amount of energy is
confined to one edge of the structure, opening up new avenues of research in
the field of wave focusing and guidance. Among other names, these unique
resonances have been referred to as truncation and bandgap resonances, given
their direct relation to the location at which an infinite array of unit cells
is truncated to form a finite one and their existence within bandgap regions.

Despite the widespread usage of truncation resonances in engineering
applications (topological insulation Roman and Sebastian (2015); Prodan  _et
al._ (2017); Rosa  _et al._ (2019), flow control Hussein  _et al._ (2015);
Balasubramanian  _et al._ (2022), and aerodynamic laminar-to-turbulent
transition delay Barnes  _et al._ (2021), to name a few), the study of their
existence criteria has been less prevalent. In 2011, Davis et al. examined
anomalous truncation resonances appearing in periodic flexural beams Davis
_et al._ (2011). Almost a decade later, Al Ba’ba’a et al. formalized the
theory of truncation resonance origination in solid continua, establishing the
role of symmetry in both unit cell choice and boundary conditions Al Ba’ba’a
_et al._ (2023). Rosa et al. further developed the topological origins
associated with some truncation resonances in continuous structures Rosa  _et
al._ (2023). Additional studies have shed light on different aspects of
truncation resonances, for example, those exhibited by locally resonant
metamaterials Sangiuliano  _et al._ (2020), bandgap resonances which accompany
defect modes Lv  _et al._ (2019); Katch  _et al._ (2023), and the use of such
resonances to manipulate Lamb Wu, Wu, and Hsu (2009) and surface acoustic
waves Oudich  _et al._ (2017). Despite the presence of some analogies, elastic
lattices of discrete atoms have been known to behave differently under similar
conditions and therefore merit their own framework. In 2017, Al Ba’ba’a et al.
derived the existence criteria of truncation resonances in elastic lattices
with two periodic atoms as well as uniform lattices with a periodic elastic
foundation Al Ba’ba’a, Nouh, and Singh (2017), and later generalized this
analysis to polyatomic Al Ba’ba’a, Nouh, and Singh (2019) and inertially
amplified lattices Al Ba’ba’a  _et al._ (2018). Bastawrous and Hussein
extended these criteria to general boundary conditions as well as lattices
with end masses Bastawrous and Hussein (2022). In this work, we present a
blueprint for the design and selective placement of truncation resonances in
elastic periodic lattices. The paradigm presented here absorbs all the
pertinent factors that play a role in the onset of truncation resonances and
shape the dynamic response of a finite lattice, spanning unit cell
configuration, mass and stiffness contrasts, as well as types and symmetry of
boundary conditions, and integrates them to delineate the theory of truncation
resonances in monatomic and diatomic lattices. In doing so, we also
demonstrate how the location at which deformations are confined in truncation
resonance edge modes are rooted in the lattice’s constitutive symmetry
parameters.

Monatomic lattice. Consider a monatomic lattice with stiffness $k$, mass $m$,
and $u_{i}$ denoting the displacement of the $i^{\text{th}}$ mass. Owing to
its periodicity, the building block of the infinite lattice (i.e., the unit
cell) can be defined in an infinite number of ways. Using a symmetry parameter
$\delta$, the lumped masses contained within a given unit cell can be
parameterized as $m_{\pm}=\frac{m}{2}(1\pm\delta)$, where $m_{\pm}$ represents
the masses connected to both ends of the spring $k$, as depicted in Fig. 1(a).
Consequently, the motion equations of an isolated unit cell simplify to

$\displaystyle(1-\delta)\ddot{u}_{i}+2\omega_{0}^{2}(u_{i}-u_{i+1})$
$\displaystyle=2f_{\text{L}}/m$ (1a)
$\displaystyle(1+\delta)\ddot{u}_{i+1}+2\omega_{0}^{2}(u_{i+1}-u_{i})$
$\displaystyle=2f_{\text{R}}/m$ (1b)

where $\ddot{(~{})}$ is an acceleration, $\omega_{0}=\sqrt{k/m}$, and
$f_{\text{L,R}}$ are the forces on both sides. Upon assuming a harmonic
solution and casting these equations in a transfer matrix format Hussein,
Leamy, and Ruzzene (2014), the state vectors at the $i^{\text{th}}$ and
${i+1}^{\text{th}}$ nodes of the lattice can be related via
$\begin{Bmatrix}u_{i+1}&\epsilon_{i+1}\end{Bmatrix}^{\text{T}}=\mathbf{T}\begin{Bmatrix}u_{i}&\epsilon_{i}\end{Bmatrix}^{\text{T}}$,
where $\epsilon_{i}=2f_{\text{L}}/k$, $\epsilon_{i+1}=-2f_{\text{R}}/k$, and
the transfer matrix $\mathbf{T}$ is given by

$\mathbf{T}=\begin{bmatrix}1-\frac{1}{2}(1-\delta)\Omega^{2}&-\frac{1}{2}\\\
\frac{1}{2}\Omega^{2}(4-(1-\delta^{2})\Omega^{2})&1-\frac{1}{2}(1+\delta)\Omega^{2}\end{bmatrix}$
(2)

where $\Omega=\omega/\omega_{0}$. The wave dispersion relation is defined as
$\text{tr}(\mathbf{T})=2\cos(q)$, which amounts to
$\Omega=\sqrt{2(1-\cos(q))}$ with $q$ being a nondimensional wavenumber. As
one might expect, we find the dispersion relation to be indifferent to the
unit cell choice, as evident by its independence of the symmetry parameter
$\delta$ Al Ba’ba’a  _et al._ (2023).

Once an infinite lattice is truncated to a finite size, $\delta$ plays an
instrumental role in the ensuing lattice behavior. The states at the two
lattice ends can be related via $\mathbf{T}^{n}$, where $n$ is the number of
cells in the finite lattice. An expression for $\mathbf{T}^{n}$ can be found
as follows Lin and McDaniel (1969)

$\mathbf{T}^{n}=\frac{\cos(nq)}{2\cos(q)}\left(\mathbf{T}+\mathbf{T}^{-1}\right)+\frac{\sin(nq)}{2\sin(q)}\left(\mathbf{T}-\mathbf{T}^{-1}\right)$
(3)

in which the four matrix elements are henceforth identified as $t_{11n}$ (top
left), $t_{12n}$ (top right), $t_{21n}$ (bottom left), and $t_{22n}$ (bottom
right), given by

$\displaystyle t_{11n}$
$\displaystyle=\cos(nq)+\frac{\sin(nq)}{2\sin(q)}\delta\Omega^{2}$ (4a)
$\displaystyle t_{12n}$ $\displaystyle=-\frac{\sin(nq)}{2\sin(q)}$ (4b)
$\displaystyle t_{21n}$
$\displaystyle=\Omega^{2}\left(4-(1-\delta^{2})\Omega^{2}\right)\frac{\sin(nq)}{2\sin(q)}$
(4c) $\displaystyle t_{22n}$
$\displaystyle=\cos(nq)-\frac{\sin(nq)}{2\sin(q)}\delta\Omega^{2}$ (4d)

Figure 1: Monatomic lattice. (a) Schematic diagram of a finite monatomic
lattice comprised of a series of identical masses $m$ connected via identical
springs $k$, subject to free and fixed boundary conditions. Owing to its
periodicity, the building block (unit cell) of the monatomic lattice can be
defined in an infinite number of ways using the single symmetry parameter
$\delta$, where $-1\leq\delta\leq 1$. While the choice of unit cell does not
affect the dispersion relation, a finite lattice shall generally have
different end masses whose values depend on the sign and value of $\delta$. A
positive $\delta$ will result in a heavier portion of the lumped mass ($>m/2$)
on the right end of the lattice, and vice versa. $\delta=0$ denotes a
perfectly symmetric unit cell where $m_{+}=m_{-}=m/2$. (b) Natural frequencies
of a free-free (left) and fixed-fixed (middle) finite monatomic lattice of
$n=10$ unit cells and $\delta=\pm 1/\sqrt{2}$, superimposed on the dispersion
diagram of the infinite lattice. Truncation resonances (TRs) are marked with a
circular marker of different color for distinction. One TR appears in the
free-free lattice at $\Omega=\sqrt{8}$ while none exist in the fixed-fixed
case. The rightmost plot tracks the location of the aforementioned TR as
$\delta$ varies, showing perfect mirror symmetry about $\delta=0$. The line
style depicts the lattice end associated with the TR’s edge mode localization
(solid: right-end edge mode, $\delta<0$; dashed: left-end edge mode,
$\delta>0$).

The resonances of an unconstrained (i.e., free-free) finite lattice are
captured by the characteristic equation $t_{21n}=0$, which gives three sets of
natural frequencies: (i) $\Omega=0$ (rigid body mode), (ii)
$\Omega=2(1-\delta^{2})^{-\frac{1}{2}}$, and (iii) a group of $n-1$
frequencies which can be computed by setting $q=r\pi/n$ in the lattice’s
dispersion relation, with $r=1,2,\dots,n-1$. Groups (i) and (iii) represent
$n$ natural frequencies that fall within the propagating zone of the monatomic
lattice and satisfy the dispersion relation. In other words, each of these
frequencies corresponds to a mode shape generating an $[\Omega,q]$ point which
lies on the lattice’s dispersion curve, as shown in Fig. 1(b). The natural
frequency given by (ii), however, is larger than the cut-off frequency of the
lattice at $\Omega=2$ for any non-zero values of $\delta\in(-1,1)$. It
coincides with the cut-off frequency at $\delta=0$ and rapidly approaches
infinity as $|\delta|\rightarrow 1$. This natural frequency indicates a unique
resonance which lies outside the lattice’s pass band and is therefore
unrecognizable by the dispersion analysis of the single unit cell. Instead, it
is perceived as a direct outcome of the finite lattice’s parameters, i.e.,
size and boundary conditions. Since these parameters culminate from trimming
an infinite lattice to form a finite one, this natural frequency is
customarily referred to as a ‘truncation resonance’ Al Ba’ba’a  _et al._
(2023). The leftmost panel of Fig. 1(b) shows the dispersion diagram of a
monatomic lattice with a unit cell configuration corresponding to $\delta=\pm
1/\sqrt{2}$. Since it consists of a repeated array of single degree-of-freedom
unit cells, the monatomic lattice has one wave mode described by a single
dispersive branch which ends at the edge of the irreducible Brillouin zone
($q=\pi$) and is followed by an unbounded stop band starting at the cut-off
frequency ($\Omega=2$). Shown in the same figure are the $n+1=11$ natural
frequencies of a finite free-free monatomic lattice constructed from $n=10$
unit cells of the same configuration. As can be seen, a truncation resonance
at $\Omega=\sqrt{8}$, as obtained from group (ii), can be spotted in the
lattice’s stop band. All natural frequencies are further confirmed via
conventional modal analysis of the lattice’s full mass and stiffness matrices.
The eleven eigenfrequencies (cross markers) perfectly match the analytical
natural frequencies (circular markers) obtained from all three groups.

The natural frequencies of a fixed-fixed finite lattice are effectively a
subset of the free-free resonances, and are therefore fairly straightforward.
They are obtained by setting $t_{12n}=0$, which gives a set of $n-1$
frequencies that are identical to those found in group (iii) of the free-free
system. Notably, the fixed-fixed lattice has two natural frequencies less than
its free-free counterpart due to the zero-displacement constraint on the
masses at both ends, effectively eliminating them from the dynamical model as
depicted in the schematic of Fig. 1(a). Physically speaking, one of these two
natural frequencies is the rigid body mode at $\Omega=0$ which no longer
exists once the lattice is anchored to fixed walls at both ends. The other one
is the truncation resonance which also no longer exists in the fixed-fixed
system since it is solely a function of $\delta$. The middle panel of Fig.
1(b) shows the dispersion diagram and natural frequencies obtained for the
same parameters used in the free-free case, namely $\delta=\pm 1/\sqrt{2}$ and
$n=10$. The results confirm the lack of a truncation resonance in the fixed-
fixed lattice. Since the dispersion curve is obtained from a unit cell
analysis, it is independent of the finite lattice’s boundary conditions as
evident by the identical dispersion diagram obtained for both systems.
Finally, it is important to note that the wave propagation profile associated
with a truncation resonance is an edge mode, where energy is confined to one
end of the lattice and is unable to infiltrate the elastic medium since the
frequency lies outside of its pass band Al Ba’ba’a  _et al._ (2023); Rosa  _et
al._ (2023). In the monatomic lattice, these localizations take place at the
edge with the ‘lighter’ mass. As a result, a negative $\delta$ results in an
energy localization at the right end of the lattice, while a positive $\delta$
induces localization at its left end. This symmetry effect is captured in the
rightmost panel of Fig. 1(b).

The two remaining scenarios, i.e., free-fixed and fixed-free lattices, are
indicative of asymmetric boundary conditions and their characteristic
equations are given by $t_{11n}=0$ and $t_{22n}=0$, respectively. The fact
that $t_{11n}(\delta)=t_{22n}(-\delta)$ shows that one of these cases can be
obtained from the other by inverting the sign of the symmetry parameter, and
makes it sufficient to study either one of them. Let’s start by examining a
couple of special cases. If $\delta=0$ (a perfectly symmetric unit cell), both
characteristic equations become identical and reduce to $\cos(nq)=0$,
resulting in $n$ pass-band natural frequencies satisfying
$q=\frac{2r-1}{2n}\pi$, with $r=1,2,\dots,n$. If $\delta=\pm 1$, the free-end
mass of the lattice is either $0$ or $m$, and the characteristic equation
$t_{22n}=0$ reduces to $\cos((2n\mp 1)q/2)=0$, once again revealing the
absence of truncation resonances. However, the pass-band natural frequencies
in this case satisfy $q=\frac{2r-1}{2(n\mp 1)}\pi$, with
$r=1,2,\dots,\varepsilon$, where $\varepsilon=n$ or $n-1$ depending on whether
the free-end mass exists or not, respectively. For a free-fixed or fixed-free
lattice corresponding to any other value of $\delta$ that is not equal to $0$
or $\pm 1$, a truncation resonance shall appear only if the free end contains
the ‘lighter’ mass, or in other words, a mass less than $m/2$. If such
condition is satisfied, the resulting truncation resonance will align
perfectly with that of the free-free lattice, assuming a sufficient number of
cells allowing this mode to materialize and the deformation profile to
exponentially decay from the free end into the bulk of the lattice,
effectively not seeing the other (fixed) end. As such, a free-fixed lattice
will only have a truncation resonance if $\delta>0$, while a fixed-free one
will only have a truncation resonance if $\delta<0$.

Figure 2: Diatomic lattice. (a) Schematic diagram of a finite diatomic lattice
comprised of a series of alternating masses $m_{a,b}$ connected via
alternating springs $k_{a,b}$, subject to free and fixed boundary conditions.
Owing to its periodicity, the building block (unit cell) of the diatomic
lattice can be defined in an infinite number of ways using the single symmetry
parameter $\delta$, where $-1\leq\delta\leq 1$, which dictates the values of
$m_{a+}$ and $m_{a-}$ (together forming the lumped mass $m_{a}$) which sit on
either side of the intermediate mass $m_{b}$. While the choice of unit cell
does not affect the dispersion relation, a finite lattice shall generally have
different end masses whose values depend on the sign and value of $\delta$. A
positive $\delta$ will result in a heavier portion of the lumped mass
($>m_{a}/2$) on the right end of the lattice, and vice versa. A zero value of
$\delta$ denotes a perfectly symmetric unit cell where
$m_{a+}=m_{a-}=m_{a}/2$. (b) Natural frequencies of a free-free (left) and
fixed-fixed (middle) finite diatomic lattice of $n=10$ unit cells,
$\delta=0.2$y, $\varrho=1/3$, and $\vartheta=-3/5$, superimposed on the
dispersion diagram of the infinite lattice. Truncation resonances (TRs) are
marked with a circular marker of different color for distinction. Two TRs
appear in the free-free lattice at $\Omega=0.742,1.796$ while one exists in
the fixed-fixed case at $\Omega=0.817$. The rightmost plot tracks the location
of the aforementioned TRs as $\delta$ varies, showing critical unit cell
configurations at which the localization edge shifts from one side of the
free-free diatomic lattice to the other, and confirms the indifference of the
fixed-fixed TR to the unit cell choice.

Diatomic lattice. Consider a diatomic lattice where the spring-mass systems
$[m_{a},k_{a}]$ and $[m_{b},k_{b}]$ alternate periodically, with $u_{i}$ and
$v_{i}$ denoting the displacements of the $i^{\text{th}}$ $m_{a}$ and $m_{b}$
masses, respectively. Similar to the monatomic lattice, the repeating unit
cell can be chosen based on $\delta$, as shown in Fig. 2(a). For any unit cell
choice, the motion equations of an isolated diatomic unit cell can be cast as
a matrix differential equation:
$\mathbf{M}\ddot{\mathbf{z}}+\mathbf{K}{\mathbf{z}}=\mathbf{f}$, where
$\mathbf{z}^{\text{T}}=\begin{Bmatrix}{v}_{i}&{u}_{i}&{u}_{i+1}\end{Bmatrix}$,
$\mathbf{f}^{\text{T}}=\begin{Bmatrix}0&{f}_{\text{L}}&{f}_{\text{R}}\end{Bmatrix}$,
$\mathbf{M}$ is a diagonal matrix of
$\begin{Bmatrix}m_{b}&m_{a_{-}}&m_{a_{+}}\end{Bmatrix}$ where
$m_{a_{\pm}}=m_{a}(1\pm\delta)/2$, and $\mathbf{K}$ is given by

$\mathbf{K}=\begin{bmatrix}k_{a}+k_{b}&-k_{a}&-k_{b}\\\ -k_{a}&k_{a}&0\\\
-k_{b}&0&k_{b}\\\ \end{bmatrix}$ (5)

To streamline the analysis, a normalizing frequency $\omega_{0}$, stiffness
contrast $\vartheta$, mass contrast $\varrho$, and a $\gamma$ parameter that
combines both contrasts are introduced as follows, respectively:

$\displaystyle\omega_{0}$
$\displaystyle=\sqrt{\frac{(k_{a}+k_{b})(m_{a}+m_{b})}{2m_{a}m_{b}}}$ (6a)
$\displaystyle\vartheta$ $\displaystyle=\frac{k_{b}-k_{a}}{k_{b}+k_{a}}$ (6b)
$\displaystyle\varrho$ $\displaystyle=\frac{m_{b}-m_{a}}{m_{b}+m_{a}}$ (6c)
$\displaystyle\gamma$ $\displaystyle=(1-\vartheta^{2})(1-\varrho^{2})$ (6d)

Both parameters $\vartheta$ and $\varrho$ can assume values between $-1$ and
$1$, resulting in $\gamma\in[0,1]$ for all values of $\varrho$ and
$\vartheta$. Upon assuming a harmonic solution and dissolving the internal
degree of freedom $v_{i}$ via Guyan reduction, a transfer matrix equation of
the form
$\begin{Bmatrix}u_{i+1}&\epsilon_{i+1}\end{Bmatrix}^{\text{T}}=\mathbf{T}\begin{Bmatrix}u_{i}&\epsilon_{i}\end{Bmatrix}^{\text{T}}$
is derived, where $\epsilon_{i}=f_{\text{L}}/(\omega_{0}^{2}m_{a})$ and
$\epsilon_{i+1}=-f_{\text{R}}/(\omega_{0}^{2}m_{a})$, with the four components
of $\mathbf{T}$ being

$\displaystyle t_{11}$
$\displaystyle=1-\frac{2}{\gamma}\Big{[}2-\delta(1-\varrho)-\vartheta(1+\varrho)\Big{]}\Omega^{2}+\frac{2}{\gamma}(1-\delta)\Omega^{4}$
(7a) $\displaystyle t_{12}$
$\displaystyle=\frac{4}{\gamma}\Big{[}\Omega^{2}-(1-\varrho)\Big{]}$ (7b)
$\displaystyle t_{21}$
$\displaystyle=\frac{1}{\gamma}\Omega^{2}\Big{[}2(1-\vartheta^{2})(1+\varrho)-\big{(}4-(1-\varrho)(1+\delta^{2})-2\delta\vartheta(1+\varrho)\big{)}\Omega^{2}+(1-\delta^{2})\Omega^{4}\Big{]}$
(7c) $\displaystyle t_{22}$
$\displaystyle=1-\frac{2}{\gamma}\Big{[}2+\delta(1-\varrho)+\vartheta(1+\varrho)\Big{]}\Omega^{2}+\frac{2}{\gamma}(1+\delta)\Omega^{4}$
(7d)

which results in the two following dispersion branches
$\Omega=\sqrt{1\pm\sqrt{\smash[b]{1-\gamma\sin^{2}(\frac{q}{2})}}}$, and a
Bragg bandgap sandwiched in between the two limits
$\Omega_{\pm}=\sqrt{1\pm\sqrt{1-\gamma}}$ with a width of
$\Delta\Omega=\sqrt{2-2\sqrt{\gamma}}$. As such, a bandgap opens up for all
values of $\gamma\neq 1$, with smaller values of $\gamma$ resulting in larger
bandgap widths. Finally, we note that the cut-off frequency of the diatomic
lattice remains constant at $\Omega=\sqrt{2}$ regardless of the magnitude of
$\gamma$.

We focus in the remainder of this discussion on the truncation resonances of
the finite diatomic lattice under different boundary conditions, with the
understanding that pass-band natural frequencies can always be obtained from
$\mathbf{T}^{n}$, following the same procedure outlined for the monatomic
lattice. As previously established for solid continua, the truncation
resonances of a bi-layered phononic crystal with symmetric boundary conditions
(e.g., free-free or fixed-fixed) coincide with the natural frequencies of the
isolated unit cell itself when subject to the same boundary conditions Al
Ba’ba’a  _et al._ (2023). As such, the truncation resonances of the free-free
diatomic lattice can be obtained from the roots of Eq. (7c), excluding
$\Omega=0$, resulting a biquadratic equation, from which the following
observations can be made:

  1. 1.

The equation has two positive roots culminating in two truncation resonances:
A lower frequency one inside the Bragg bandgap and a higher frequency one
above the lattice’s cut-off frequency (the latter lies exactly at the cut-off
frequency if $\delta=\vartheta$).

  2. 2.

The equation becomes quadratic when $|\delta|=1$, since the number of degrees
of freedom of the finite lattice reduces by one. The higher frequency
truncation resonance ceases to exists (mathematically, it moves to infinity
similar to the monatomic lattice case). The lower frequency truncation
resonance remains inside the bandgap unless $\varrho=0$ (zero mass contrast)
at which case it moves to one of the two bandgap limits, rendering the lattice
free of any bandgap resonances.

  3. 3.

A perfectly symmetric unit cell ($\delta=0$) combined with zero stiffness
contrast ($\vartheta=0$) moves the lower frequency truncation resonance to the
bandgap limit located at $\Omega=\sqrt{1+\varrho}$, which could be either the
lower or upper limit depending on the sign of $\varrho$.

The leftmost panel of Fig. 2(b) shows the dispersion diagram of a diatomic
lattice with a unit cell configuration corresponding to $\delta=0.2$, and mass
and stiffness contrasts of $\varrho=1/3$ and $\vartheta=-3/5$, respectively.
Since it consists of a repeated array of two degree-of-freedom unit cells, the
diatomic lattice has two wave modes described by an acoustic (lower) and an
optical (upper) branches which sit on both ends of the Bragg bandgap. Shown in
the same figure are the $2n+1=21$ natural frequencies of a finite free-free
diatomic lattice constructed from $n=10$ unit cells of the same configuration,
confirming the two truncation resonances stated in observation #1, one falling
inside the bandgap at $\Omega=0.742$ and the other taking place above the
optical branch at $\Omega=1.796$.

The fixed-fixed diatomic lattice has a single truncation resonance at
$\Omega=\sqrt{1-\varrho}$, representing the positive root of Eq. (7b), and
shows, once again, independence of the symmetry parameter $\delta$. The edge
mode profile accompanying this truncation resonance exhibits localized
displacement at the stiffer end of the lattice. In other words, the
localization takes place at the right end if the contrast $\vartheta$ is
positive and vice versa. Interestingly, in the case of $\varrho=0$, the
truncation resonance occurs at a constant frequency of $\Omega=1$ regardless
of the stiffness contrast $\vartheta$, a feature which has been exploited in
diatomic lattices with topological protection Chen, Nassar, and Huang (2018).
The middle panel of Fig. 2(b) displays the behavior of the finite fixed-fixed
diatomic lattice and confirms the presence of a sole truncation resonance
inside the bandgap at $\Omega=0.817$ for the same values of $\varrho$ and
$\vartheta$ used in the free-free example.

Finally, given the dependence of both truncation resonances of the free-free
diatomic lattice on $\delta$, it is important to highlight this effect of unit
cell symmetry on the edge mode profile associated with each of these two
resonances, which is succinctly captured by the rightmost plot Fig. 2(b). As
can be seen, there exists a critical value of $\delta$ for each truncation
resonance at which that particular resonance touches the lattice’s pass-band
regions, and at which the end of the lattice which exhibits the localized
displacement shifts from the one side to the other, a behavior which has been
similarly observed in recent studies Rosa  _et al._ (2019); Al Ba’ba’a  _et
al._ (2023). Consistent with the earlier hypothesis, the same plot shows a
flat line tracking the location of the fixed-fixed truncation resonance with a
varying $\delta$, confirming the lack of a symmetry effect for that system.

Acknowledgements. The authors acknowledge support of this work by the US
National Science Foundation through research awards no. 1847254 (CAREER) and
1904254, and the US Air Force Office of Scientific Research through award no.
FA9550-23-1-0564.

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