Increasing interest in wave propagation in phononic systems and metamaterials motivates the development of experimental designs, measurement techniques, and fabrication methods for use in basic research and classroom demonstrations. The simplest phononic system, the monatomic chain, exhibits rich physics such as dispersion and frequency-domain filtering. However, a limited number of experimental studies showcase monatomic chains for macroscale observation of phonons. Herein, we discuss the design, fabrication, and testing of monatomic lattices as enabled by three-dimensional (3D) printing. Using this widely available technology, we provide design guidelines for realization of a monatomic chain composed of 3D printed serpentine springs and press-fitted cylindrical masses. We also present measurement techniques that record propagating waves and algorithms for the experimental determination of dispersion behavior.

## I. INTRODUCTION

Wave propagation in phononic lattices and metamaterials exhibits rich physics of interest to the education and research communities. Recent studies have documented the performance of phononic lattices as filters,^{1,2} lenses,^{3,4} and diodes.^{5,6} In classroom settings, phononic lattices provide an accessible case study on dispersion, group velocity, and phase velocity.^{7–9} Perhaps the simplest phononic lattice, the monatomic chain, is composed of a single mass and spring that repeats periodically.^{10–14} However, these manually assembled designs can be tedious to realize, exhibit vulnerability to impurities in their periodicity, are susceptible to out-of-plane and torsional motions, experience significant dissipation at interfaces and where contact is made with constraining structures, and lack the technological sophistication that excites students in science and engineering. In addition, traditional designs typically capture standing waves, as opposed to travelling waves, in finite lattices.^{15–17} The inherent challenges associated with fabricating and assembling a large number of unit cells with few defects and low dissipation create a formidable barrier to experimental observation of travelling waves in mechanical lattices.

Additive manufacturing is an advanced but increasingly widespread fabrication method capable of rapidly producing materials with a high degree of uniformity. 3D printing is a common additive manufacturing technique in which parts are created one layer at a time. Because of its repeatable, high throughput nature, 3D printing can quickly fabricate easy-to-assemble periodic structures with high geometric complexity and low disorder. While prior studies have successfully implemented 3D printing to produce lattice materials, they considered structures with continuous, multi-modal elastic members.^{18–20} Embedded masses vibrate “locally” relative to the host medium instead of directly coupling to their nearest neighbor via a massless spring-like connection. Consequently, higher-order structural dynamics of the elastic members between masses can be excited, which would be undesirable for a discrete lattice experiment showcasing one-dimensional phonon propagation.

In this paper, we discuss the design, fabrication, and testing of precision monatomic lattices as enabled by additive manufacturing. In doing so, we detail an experiment demonstrating travelling wave packets in a monatomic lattice suitable for an undergraduate physics course using equipment that is becoming increasingly widespread, user-friendly, and affordable. Although we focus on a monatomic lattice for our demonstration, we discuss generic experimental design guidelines, which can be applied to more complicated systems such as diatomic chains and Su–Schrieffer–Heeger lattices. We seek to inform and empower future experiments with tunable, rapidly fabricated, precision lattices that support macroscopic phonon propagation.

We organize this paper as follows: First, we review phononic behavior in infinite and finite monatomic lattices. Next, we discuss the experimental design and parameter choices. We then present the design of 3D printed serpentine springs with integrated mass casings for tuning the stiffness of a macro-scale monatomic lattice. We borrow the serpentine spring design from the micro-electromechanical systems (MEMS)^{21–23} community and use the design to overcome many issues with traditional methods noted earlier. Finally, we present results obtained from the fabricated setup that captures wave propagation and dispersion on the macroscale, before concluding with final remarks.

## II. THEORY OF WAVE PROPAGATION IN A MONATOMIC LATTICE

A monatomic lattice consists of a unit cell, composed of a single mass, coupled to its neighbors by force laws that can be implemented as springs. This system is often used as a model for phonons. For further details on the theory of phonons, the reader is referred to monographs on solid state physics.^{7–9}

Consider an infinite monatomic lattice as depicted in Fig. 1. The direct lattice spacing, *a*, is the distance between two equivalent positions in neighbouring unit cells with mass *m*, and the spring constant coupling them is represented by *k*. Consequently, the equation of motion governing the *n ^{th}* unit cell is

where $un$ denotes the displacement of the *n ^{th}* mass. Equation (1) admits a plane-wave solution as

where *A* denotes the amplitude of displacement, $\omega =2\pi f$ denotes the angular frequency where *f* is the temporal frequency, and *κ* denotes the wavenumber. The displacements are given by the real or imaginary parts of *u _{n}*. It is customary to re-express the wavenumber in terms of a dimensionless propagation constant

where the propagation constant *μ* is a (generally) complex quantity whose real and imaginary parts denote the phase and attenuation constants, respectively. In the absence of attenuation, *μ* is purely real. Substituting Eq. (2) into Eq. (1) leads to the dispersion relationship

By rearranging Eq. (4), the propagation constant is expressed in terms of the angular frequency as

The wavelength *λ* in the discrete lattice represents the number of unit cells (or equivalently the number of masses) in one spatial period of the wave. This quantity relates to the propagation constant as

For a monatomic lattice, the smallest wavelength spans two unit cells and corresponds to a propagation constant $\mu =\pi $. For each $|\mu |>\pi $, there exists a $\mu \u2208(\u2212\pi ,\pi )$ such that the lattice state (displacement and velocities of the masses) is the same. Hence, all waves are uniquely determined by wavenumbers in the region $\mu \u2208(\u2212\pi ,\pi )$. Figure 2 displays the band structure (consisting of a single dispersion curve) governing wave propagation in a monatomic lattice with mass *m *=* *0.1 kg and stiffness *k *=* *6000 N/m. The physical values used to generate Fig. 2 correspond to the mass and stiffness used in the experimental apparatus described later. Note that the *x*-axis of Fig. 2 uses the normalized propagation constant, $\mu /\pi $.

The monatomic lattice features a cut-off frequency above which wave propagation is prohibited, determined by evaluating the dispersion relationship at $\mu =\pi $,

Above this frequency, the propagation constant is an imaginary quantity, which results in evanescent waveforms, or exponential decay of a wave packet's amplitude through the lattice. The dispersion relationship can also be used to calculate phase and group velocities

Monatomic lattices are dispersive media: Both phase and group velocity depend on frequency (or, equivalently, the propagation constant). Figure 3 depicts the phase and group velocity as functions of the propagation constant. Note that at $\mu =\pi $ the group velocity goes to zero, implying standing waves in which energy does not propagate through the lattice. In this mode, adjacent masses have opposite displacements and velocities.

A finite one-dimensional lattice containing *N* masses has *N* modes of vibration. Under open (or free) boundary conditions, the *N* natural frequencies are found on the dispersion curves of the infinite lattice's band structure at equally spaced propagation constants. A monatomic lattice with *m *=* *0.1 kg and *k *=* *6000 N/m consisting of ten masses possesses ten natural frequencies as identified by the vertical lines in Fig. 4. A similar procedure yields the natural frequencies for fixed-fixed boundary conditions.

## III. EXPERIMENTAL DESIGN

In the experiment, we are interested in observing traveling wave packets in a monatomic lattice. The design of the wave propagation experiment consists of identifying the lattice parameters, i.e., the mass and stiffness of the unit cell, the length of the lattice (number of unit cells), and the number of cycles of input provided to form the traveling wave packet. The theoretical model of the monatomic lattice considers a rigid mass and a massless spring thereby making the model discrete. However, practically, an experiment consists of continuous bodies having both mass and stiffness. In order to minimize the influence of elastodynamics of the individual components, it is necessary that the body used as the mass be extremely stiff compared to the spring while the spring should have negligible contribution to the mass of the unit cell. The chosen mass and spring stiffness determine the cut-off frequency of the lattice. To explore the lattice dynamics in the actuator frequency range of 0–100 Hz, we chose the masses in the lattice to be 0.1 kg and the spring stiffness to be 6000 N/m. Using Eq. (7), this combination of lattice parameters results in a cut-off frequency of 78 Hz.

Equations (5) and (6) reveal that $\lambda \u2192\u221e$ as $\omega \u21920$. This implies a longer lattice is required to study wave propagation at lower frequencies. The number of cycles of input at the desired frequency dictates the length of the wave packet travelling in the lattice and, thus, directly affects the choice of number of unit cells in the lattice. From the experimental design point of view, to study wave propagation within a frequency range of interest, we choose the number of cycles and the length of the lattice such that it is possible to separate the incident and reflected wave packets. Considering available lab space, we chose a 4 cycle input signal and a 70 unit cell lattice. In order to observe separate incident and reflected wave packets, the wave packet must, at most, span twice the number of masses between the mass of observation and the closest end of the lattice. For example, when observing the wave packet pass through the middle of the lattice, the lowest frequency at which incident and reflected wave packets are separable corresponds to the wavelength *λ* such that $4\lambda \u226470$. This is because the length of the wave pulse is $4\lambda $ and in order to separate the incident and reflected portions of the wave packet, the wave packet length must, at most, span twice the distance from the center to the nearest end (70 unit cells). Using Eqs. (4)–(6), this suggests a frequency lower bound of approximately 14 Hz. Thus, for inputs with center frequencies above the determined lower limit and below the cutoff frequency, the incident and reflected wave packets are clearly distinguishable.

## IV. 3D PRINTED SERPENTINE SPRING LATTICE

3D printing technology offers rapid prototyping, low production times, and batch printing capabilities, which are appealing traits for the fabrication of lattices composed of several unit cells. Furthermore, this method limits the disorder in the assembled lattice and produces lightweight springs capable of encasing heavy masses.

Due to their planar shape and compatibility with additive manufacturing, we employ serpentine springs to provide linear restoring forces on the masses in the monatomic chain. Serpentine springs derive their name from their slender, circuitous geometry and are commonly used in micro-electromechanical systems (MEMS).^{21–23} These springs have out-of-plane and torsional rigidity, a feature that traditional coil springs lack. Previous monatomic designs have been subject to unwanted whirling behavior if not constrained,^{16} or suffer from appreciable viscous and/or dry friction damping due to the addition of roller bearings or guides. The serpentine design lends itself to mass encasement using vacant receptacles into which heavy masses can be press fitted. Mass encasement minimizes joints and, thus, promotes crystalline uniformity. 3 D-printing many serpentine springs simultaneously as a single body considerably simplifies assembly while eliminating troublesome joints. Figure 5(a) displays the structure we use to build the lattice for our experiment. Note the design of mass encasements, or void spaces for neighboring masses to be coupled by the serpentine springs. Relevant dimensions of the serpentine springs are labeled in the inset. We arrive at the dimensions listed in Table I by first using an analytical model to approximate the stiffness to be close to the desired 6000 N/m and then tune the design using finite element (FE) software to obtain a more accurate stiffness value.

Spring parameters . | |
---|---|

Length (L) | 10.50 mm |

Width (W) _{L} | 1.000 mm |

Thickness (t) | 5.000 mm |

Elastic modulus (E) | 2.600 GPa |

End thickness (W) _{S} | 3.000 mm |

End height (middle d_{1}) | 3.500 mm |

End height (top/bottom d_{2}) | 3.000 mm |

Spring parameters . | |
---|---|

Length (L) | 10.50 mm |

Width (W) _{L} | 1.000 mm |

Thickness (t) | 5.000 mm |

Elastic modulus (E) | 2.600 GPa |

End thickness (W) _{S} | 3.000 mm |

End height (middle d_{1}) | 3.500 mm |

End height (top/bottom d_{2}) | 3.000 mm |

The serpentine spring is modelled as a series of guided cantilever beams,^{23} and a closed form expression for the overall spring constant *k _{y}*, of each spring unit of the structure displayed in Fig. 5 is

where *k*_{0} denotes the spring constant for a single guided cantilever beam of length *L*, width *W _{L}*, thickness

*t*, elastic modulus

*E*, and area moment of inertia $I=tWL3/12$. The number of beams on each side of the spring is denoted by

*n*(which is 4 in our design). The factor of two in Eq. (11) results from two sets of beams in parallel. Inserting the values from Table I, Eq. (11) predicts a spring stiffness of 5615 N/m. Note that a different set of dimensional values can be used to arrive at a similar estimate. For instance, increasing

_{b}*t*and

*n*by a factor 2 would also produce the same estimate. However, this would also increase the size of the unit cell. Consequently, selecting appropriate final dimensions for the spring is generally an iterative process.

_{b}The analytical expression in Eq. (11) readily provides a first approximation of the spring stiffness. A more accurate prediction is obtained through FE analysis using the Abaqus^{24} software package, where we simulate the deformation of the designed spring. Figure 5(b) displays the computational model of the serpentine spring. The bottom surface of the serpentine spring is fixed while the top surface is displaced by 2 mm. Figure 5(c) depicts the deformation of the spring unit together with a schematic of beams in series and parallel to visualize the equivalent stiffness calculation. Simulating small displacements provides a direct proportionality between the reaction force and the applied displacement with the ratio of the two being the spring's stiffness. For the dimensions considered, the FE analysis predicts a stiffness of 6104 N/m, which is within 2% of the desired value of 6000 N/m.

An added benefit of using FE software is that it allows for calculation of the natural frequencies of the serpentine spring. Since it is desired for the springs to operate quasi-statically, it is possible to confirm that the natural frequencies of the spring are not being excited during wave propagation. The first natural frequency of the spring reported by the FE software is 302 Hz, which is significantly higher than the lattice's 78 Hz cutoff frequency. Computer aided design (CAD) files for the final design of the serpentine springs can be found on the authors' SourceForge website.^{25}

The serpentine springs are additively manufactured to build a monatomic lattice for the wave propagation experiments. We use a Raise3D Pro2 (Irvine, CA) printer to produce unit cells consisting of springs and mass encasements. The printer size enables printing batches of five unit cells at once. To form the entire finite lattice, a thin layer of Loctite super glue joins these batches. We press fit cylindrical stainless steel masses of 0.1 kg into the encasements positioned between each spring. In our design, the encasements serve two purposes. First, they hold the masses in place and confine the spring forces to a single plane. Second, they reduce the number of joints in the system, which are often associated with greater defects, dissipation, and nonlinearity. A silhouette of the masses is printed, and steel masses are press fit into these encasements. For the printing filament material, we use polylactic acid (PLA) with an approximate density of 1250 kg/m^{3}. Using the dimensions of the final serpentine spring design, we calculate the mass of each spring to be 0.001 39 kg, which is less than 70 times the mass chosen for the lattice's unit cells. Thus, the springs' mass is negligible in comparison to the steel masses.

We seek to study the linear wave propagation phenomenon in the macroscale monatomic chain. Consequently, we reference analytical expressions (i.e., Eqs. (10) and (11)) that assume linear deformation of the serpentine springs. To confirm the linear behavior of our fabricated springs, we compress our serpentine structures to known displacements and measure the reaction force using a force gauge (Mark-10 model M2–5). Figure 6 displays force measurements at various displacements for one of our spring samples. The spring's restoring force exhibits linear behavior for deformations under 3 mm. Nonlinearity develops at deformations above 3 mm, which we attribute to neighbouring beams within the spring unit coming into contact. In the linear region, similar behaviour was observed for tensile loads. We repeated the test for 7 individual springs isolated from separate batches giving an average stiffness of 6588 N/m and a standard deviation of 122 N/m, which is within 10% of the desired stiffness while the variation among the springs is within 4% of the average value.

## V. EXPERIMENTAL SETUP

Figure 7 presents the physical setup while Fig. 8 provides an outline of the data acquisition in the experiment. We make use of widely available and user-friendly vibration testing equipment. We generate a signal corresponding to the desired wave packet using National Instruments (NI) LabVIEW Signal Express and output it using an analog voltage output module (NI DAQ 9263) to an amplifier (B&K Type 2718), which drives an electrodynamic shaker (B&K Type 4809). A stinger transmits the shaker motion to the first mass in the lattice. To capture the time evolution of the velocity of a selected mass, we focus a laser Doppler vibrometer (Polytec PDV-100) on a flag attached to the mass (please refer to the inset of Fig. 7). The vibrometer evaluates the velocity by measuring the Doppler shift in the laser beam reflected from the measurement point and outputs a corresponding voltage, which is acquired by an analog voltage input module (NI DAQ 9234). For additional details on laser Doppler vibrometry, the interested reader is referred to a recent review.^{26} Alternatively, accelerometers can be mounted on masses to measure their acceleration as a function of time. However, the accelerometer's mass should be significantly lower than that of the mass in the chain to avoid introducing a defect in the lattice's periodicity. Since all the masses are inline, we include clamps in the design of the mass encasements as a provision to place a flag above a mass whose velocity is being measured thereby ensuring there is no obstruction in the line of sight of the laser. We suspend the 3 D-printed lattice from a wooden frame and model it as a forced-free chain. These boundary conditions arise from one end of the lattice being connected to the shaker, which inputs the wave packet, while the other end remains unconstrained. The lattice is positioned approximately *l *=* *33 cm below the wooden frame using fishing line. The natural frequency of the masses acting as pendulums is negligibly small ($1/2\pi g/l=0.87$ Hz, where *g *=* *9.81 m/s^{2} is the acceleration due to gravity), thus circumventing the rotational nonlinear dynamics of large angle pendulum oscillation at the frequencies of interest. Furthermore, suspending the masses exhibits very little loss due to dry friction or viscous damping as a low cost alternative to other support mechanisms such as roller bearings or guide rails.

## VI. RESULTS AND DISCUSSION

We first use our experimental setup to calibrate the unit cell spring stiffness by comparing an experimental frequency response to the one computed analytically. The frequency response is a standard characterization method in vibration testing wherein the system is excited and the ratio of an output physical quantity to input physical quantity is computed or measured. Typically, the input-output quantities are selected among displacement, velocity, acceleration, and force. We compare the ratio of output velocity to input velocity as these quantities are readily obtained by the laser Doppler vibrometer measurements. Once the springs are calibrated, we conduct wave propagation studies by injecting wave packets at different frequencies into the lattice. We acquire the velocity data for consecutive masses by performing a series of experiments with the same input signal while moving the laser flag for each iteration. The collected velocity data are used to experimentally determine the dispersion behavior in the macroscale monatomic lattice.

To calibrate the spring stiffness of the unit cell in our fabricated lattice, we compare the resonance peaks of an experimentally measured frequency response to the theoretical response computed using the matrix equations of motion. We obtain the frequency response of the finite lattice by providing a continuous harmonic excitation to the first mass of the lattice and collecting velocity data at the first (i.e., driven) and fifth masses. The frequency of input excitation is swept through values associated with the lattice's theoretical passband. The experimental and analytical frequency responses of the lattice are obtained by evaluating the ratio of the velocity amplitude at the fifth mass to that at the first mass for each frequency in the sweep. For further details of the theoretical computation please refer to the supplementary material.^{27}

Figure 9 displays the frequency response of the finite lattice structure obtained experimentally and computed analytically. A small amount of proportional damping is applied to the analytical frequency response such that the resonant peaks are in close agreement with the experimental frequency response. Using a root mean square method of error calculation, we find the stiffness that results in the best alignment between experimental and analytical frequency response to be 6055 N/m with an error of 7.4% compared to the analytical model. Even spacing between the resonant frequencies confirms the lattice's periodicity as observed in Fig. 4. Furthermore, the narrow width of the resonance peaks in the experimentally obtained frequency response of Fig. 9 confirms the low damping of the serpentine springs as well as the string used to support them. The experimentally characterized spring stiffness (6055 N/m) is in close agreement with the stiffness calculated using the FE method (6104 N/m). Additionally, the experimentally calibrated stiffness is within 1% of the desired design stiffness (6000 N/m). Hereafter, we use 6055 N/m as the stiffness of the lattice unit cell as it is calibrated to the experimental setup.

For wave propagation studies, we inject a 4-cycle wave packet into the lattice and measure the ensuing wave motion. Figure 10 depicts the input to the signal amplifier for a waveform with a center frequency of 30 Hz. Using a laser Doppler vibrometer, we track the motion of a flag placed over a mass whose velocity is to be recorded. We repeat the experiment with the same input for measuring velocities of 30 consecutive masses starting at the 15th mass from the input. As a reference signal, during each iteration of the experiment, we measure the velocity of mass 1 (input mass) connected to the shaker. Figure 11 displays the normalized velocity measured at mass 19 when the lattice is excited with the input signal in Fig. 10. The incident and reflected pulses are clearly identified. The decay in amplitude observed in the reflected wave packet is attributed to the damping present in the lattice.

We measure the velocities as described previously at various input frequencies: 20, 30, 50, 70, and 90 Hz. All measured signals are truncated to begin when the reference pulse reaches maximum amplitude. We then use the signals acquired from 30 consecutive masses, approximately in the midspan of the lattice, to compute the dispersion. We convert the spatiotemporal velocity data of the masses, time-windowed to exclude the reflected pulse, to the frequency-wavenumber domain using a two-dimensional fast Fourier Transform (2D-FFT)^{28,29} implemented in matlab.^{30} Measured data at 8000 points in time and 30 points in space are zero-padded to the nearest power of 2 for an efficient implementation of the 2D-FFT algorithm (for more details, please refer to the supplementary material^{27}). The Fourier transform converts temporal information to the frequency domain and spatial information to the wavenumber domain. The 2D-FFT operates on spatio-temporal wave information and picks out the dominant frequency-wavenumber pairs associated with the travelling waves in the medium. The Fourier transform preserves the phase information and thus is able to separate forward and backward propagating waves. We measure the spatio-temporal information in our experiment at different input frequencies. Figure 12 presents the computed 2D-FFT linearly summed over various input frequencies. Alternately, one could input a broadband frequency input and obtain similar results. We use matlab's interpolation method to generate the surface plot. The color scale of the plot indicates the absolute value of the complex amplitude of waves of different propagation constants and frequencies. The spectrum for each tested frequency is normalized with respect to its maximum value. As expected, prominent amplitudes appear only for positive propagation constants since the time window for signals is truncated to include only the incident pulse. Also as expected, no appreciable amplitude is observed for 90 Hz, as it is above the cut-off frequency. An overlaid analytical dispersion curve computed for mass and stiffness used in the experiment exhibits close agreement with the experimental results. Near $\mu =\pi $, where the wave is more dispersive, we notice a spread of the amplitude as the transform does not pinpoint a specific wavenumber at these higher frequencies. The experimental data as well as the associated program files for producing the dispersion curves can be downloaded from our data repository.^{25}

## VII. CONCLUDING REMARKS

In summary, we presented a macroscale monatomic lattice fabricated using additive manufacturing through which phononic behavior is experimentally observed. Additive manufacturing, e.g., 3D printing, enables intricate and tunable spring designs to be realized. We print several unit cells together to rapidly assemble large lattices and to minimize the amount of bonding or fastening. Although we present an example monatomic system composed of serpentine structures as springs and metallic cylinders as masses, we provide generic guidelines, which could be extended and applied to the study of one dimensional lattices with higher degrees of freedom. We also demonstrate an experimentally obtained dispersion curve using the 2D Fourier transform of velocity data of the monatomic lattice. This study enables researchers and educators to investigate and demonstrate macroscale phonon propagation in lattices composed of a large number of unit cells using tunable, additively manufactured designs.

## ACKNOWLEDGMENTS

The authors would like to thank the National Science Foundation for supporting this research under Grant No. 1929849 and Sandia National Laboratories under the Laboratory Directed Research and Development (LDRD) Program.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.