Emergence of Pseudo-Phononic Gaps in Periodically Architected Pendulums

Rejection of unmitigated vibrational disturbances represents an ongoing dilemma in complex linkage systems. In this work, we show evidence of an inherent and self-reliant vibration isolation mechanism in architected periodic chains of serially pivoted pendulums. Absorption of external excitations is achieved by virtue of Bragg-type band gaps which stem from the emergent chain dynamics. Owing to its coupled dynamics, the self-repeating cell of a Phononic Crystal (PC) is not trivial and cannot be readily identified from the periodic arrangement by inspection. As such, this work entails the extraction of a"pseudo"unit cell of an equivalent PC lattice from the derived motion equations of a finite chain, which departs from traditional wave dispersion methods. The model presented herein comprises a chain of linked pendulums with periodic variations of inertial/geometrical properties, carrying a payload at one end. We ultimately show evidence of forbidden wave propagation in prescribed frequency regimes, reminiscent of band gaps in PC lattices. The presented framework can be invaluable in applications that require vibration reduction in the delivery of payloads including gantry cranes, robotic arms and space tethers.


I. INTRODUCTION
Undesirable vibrations and mechanical disturbances are an ever-lasting concern in the vast majority of engineering applications. The effect of vibrations on a dynamical system ranges from shortening its longevity to sudden raptures, all of which impose serious operational challenges. The rational design of architected structures provides unique mechanisms by which unwanted arbitrary excitations can be rejected with minimal, and often non-existent, compromises in strength and resilience. Structural periodicity, whether in material composition [1], topology [2], boundary conditions [3], placement of local resonances [4] or via combinations of the previous [5], enable a myriad of unprecedented wave dispersion capabilities; tailored to address challenges in vibroacoustic mitigation [6,7]. Single and double negative materials [8,9], wave directivity [10], mechanical topological insulators [11], and one-way elastic diodes [12] are a few prime characteristics of engineered periodic systems.
This effort focuses on applications comprising mechanical pendulums and pendulum-like substructures suspended from a pivot and tasked with carrying and/or conveying a payload. Specifically, we address chains of coupled pendulums, the dynamics of which lie at the heart of several applications such as flexible robotic arms [13], unmanned aerial vehicles with payloads [14] and gantry (overhead) cranes [15]. In the latter, for instance, vibrations can be detrimental to the positioning accuracy of the terminal payload location. They elevate conveyance durations of payload and, consequently, create an operational bottleneck [16]. The pendular motion of the payload has been by far regulated using active control techniques, exploiting tools such as command shaping [15] and feedback control [17]. In its most common form, pay- * Corresponding author: mnouh@buffalo.edu load is often hoisted via a rope or a linkage chain and is transported via a cart/trolley (see left panel of Figure 1). The mechanism of the chain variant resembles that of a pendulum chain, where the tip of an individual pendulum (link) acts as a pivot to the subsequent one. Incorporating pendulums in novel metastructures has recently received increased attention. The most dominant example being 1D arrays of pendulums periodically coupled with springs which have been utilized to study a variety of intriguing phenomena pertaining to coupled nonlinear oscillators. These include solitons [18], breathers [19], energy transmission in band gaps [20] and most recently helical edge states in topological insulators [21]. To this end, however, exploiting the periodicity of pendulum chains as an inherent and self-reliant mechanism for vibration absorption remains uncharted territory. The aim of this work is to fill this gap by demonstrating phononiclike phenomena in the wave propagation profiles of such chains. Through a closed-form analysis of the dynamics of a periodic lumped pendulums chain with a payload (tip mass), as portrayed in Figure 1, we ultimately show evidence of emergent phononic band gaps. Such gaps, resulting from Bragg-scattering and interference effects, restrict the propagation of waves within certain (and ultimately favorable) frequency regimes.
Although the parameters of the chain of pendulums are periodic, the dynamics of each concentrated mass is strongly coupled with the whole chain as dictated by its motion equations [15]. As a result, the self-repeating unit cell of the chain arrangement (e.g. two masses m 1,2 and their links ℓ 1,2 in a diatomic configuration as shown in Figure 1) cannot be readily defined by the naked eye. As such, the most significant challenge unraveled here is the identification of a "pseudo" unit cell for Bloch-wave dispersion purposes and ultimately validating this cell choice from the actual pendulum response. Such identification process naturally departs from the conventional approach and instead relies on the dynamics of the finite chain as a reference point. We start by deriving the . . . equations of motion for a general chain of pendulums; the generalized model serves as the baseline for studying three distinct scenarios: (1) A uniform chain with and without a tip mass, (2) A chain with periodic variation of masses with a tip mass, and (3) A chain with periodic variation of the linkage lengths along with a tip mass. We mathematically show that the tip mass is crucial to obtaining a unit cell representation of an equivalent PC lattice which accurately captures the behavior of the chain under small vibrational amplitudes. The equivalent PC model is then compared with the actual linearized model of the periodic chain of pendulums in frequency and time domains. The latter is finally used to reconstruct the wave dispersion contours of the actual system, which validates the equivalent model for all three cases.

II. GENERALIZED EQUATIONS OF MOTION
For a chain of pendulums with masses m i connected by rigid connections with length ℓ i , the position vector r for the i th mass is given by and, hence, its velocity vector is where [˙] = d dt is the time derivative,î and are unit vectors in the x and y directions, respectively, and the angles φ j are labeled on the right panel of Figure 1. Following which, the potential energy V of the pendulums chain can be expressed as while the kinetic energy, T , is given by The which leads to the equation of motion of the i th mass of the system, given by Equation (8) Using Eq. (9) and setting the cart's acceleration, i.e. x, as an input, the equations of motion of the entire chain can be cast into a compact matrix form as where

A. Case I: Uniform chain
For a uniform chain with identical rigid links lengths and lumped masses, i.e. ℓ i = ℓ and m i = m, but without a tip mass, the linearized equation of motion of the i th mass given by Eq. (9) reduces to wheren i,j = n−max(i, j)+1, which leads to the following system of equations: The inverse of a matrix with the structure of M in Eq. (14a) has been shown to be a tridiagonal matrix [22,23]. For the uniform pendulums chain, the inverse of the matrix M in Eq. (14a) is derived as and the forcing term reduces tof T = 1 0 · · · 0 . By inspecting Eq. (17), it is evident that the structure of D is not periodic even though the physical chain has uniform properties. Consequently, a substructure consisting of one mass and its adjacent massless link can not be considered a self-repeating unit cell of the system for Bloch-wave analysis. Next, consider the same chain of masses with a tip mass m t at its end such that m t = ̺m. The linearized equation of motion of the i th mass now Casting the equations of motion into a matrix form similar to the previous analysis, it can be deduced that the inverse of the mass matrix in this case becomes Now, if the payload is much heavier than the rest of the masses chain, i.e. ̺ ≫n i,i , then the stiffness matrix can be approximated as K ≈ ̺I with I being the unit matrix; the equations of motion now reads: such that ω 0 = ̺g ℓ . Upon normalization using ω 0 , we arrive at the following reduced non-dimensional set of motion equations: where ( ′′ ) = d 2 dτ 2 and τ = ω 0 t. This system of equations identically resembles that of a monatomic lattice with free-free boundary conditions (since 2+̺ 1+̺ → 1 as ̺ → ∞), and a repeating unit cell of an infinite medium can now be defined. The equivalent monatomic lattice comprises a chain of lumped masses m connected via springs k = ω 2 0 m. As such, the medium's dispersion relation can be derived as [6] Ω = 2(1 − cosβ) (22) where Ω = ω ω0 andβ are the dimensionless temporal and spatial frequencies, respectively.
To verify these assumptions, we simulate the actual and equivalent systems in time and frequency domains. For the equivalent monatomic lattice, the end-to-end transfer function can be analytically derived as (see Al Ba'ba'a et al. [24] for details) where θ i depends on the boundary conditions of the system and is equal to i−1 n π in this case [25]. The roots of the denominator of Eq. (23) represent the system's poles (i.e. natural frequencies) and can be used to analytically determine the spectrum of the equivalent system.
It can be seen that the rigid body mode at s = 0 cancels perfectly with the s 2 term in the numerator; a result of the input being the acceleration of the cart, i.e.ẍ. Figures 2a and 2b show the frequency response for the system under consideration with ̺ = 10 2 and ̺ = 10 3 for both the equivalent and actual systems as well as the pole distribution on the complex s-domain. The results show excellent agreement between the equivalent and actual systems for ̺ = 10 3 while the frequency response and the pole distribution of the equivalent system slightly deviate from the actual one when ̺ = 10 2 .
To validate the dispersion analysis in Eq. (22), on the other hand, the system is excited with a wide band excitation and its dispersion profile is constructed from the time-transient response of the system using the spatiotemporal Fourier transform [26]. Figure 2c depicts the numerical dispersion contours for a chain of n = 31. The analytical dispersion relation of the equivalent system, i.e. Eq. (22), is displayed as a dashed line for comparison and shows a very decent agreement.

B. Case II: Periodic variation in masses
The previous results for the uniform chain of pendulums can be extended to a chain with a periodic arrangement of masses, as portrayed in Figure 3a. Assume that the odd numbered masses are m 1 = m while the even numbered are m 2 = µm, where µ = m2 m1 is the mass ratio. Making use of Eq. (9) and with few mathematical manipulations, the equations of motion simplify to Eq. (13) with the following new definitions for M, K and f : where ⌊·⌋ is the floor function and "mod" is the modulo operator. The mass matrix M in Eq. (24a) has the following inverse (which is shown for an even n): where ǫ = 1 + 1 µ+̺ for an even n while for an odd n, we get ǫ = 1 µ + 1 1+̺ . Note that ǫ reduces to either unity or 1 µ for an even or odd n, respectively, as ̺ → ∞ and µ = 1 recovers M −1 for the uniform chain of pendulums, i.e. Eq. (19). Since the payload is assumed to be much larger than the masses in the chain, the approximation K ≈ ̺I remains intact leading to the equations of motion in Eq. (21) for the equivalent system with M −1 defined in Eq. (25). The resultant equations of motion resemble that of a typical diatomic lattice (see Figure 3b) with identical masses, alternating springs, and free-free boundary conditions [3]. The unit cell can be now de-fined as two identical lumped masses m and two different springs k 1 = m µ ω 2 0 and k 2 = µk 1 , and, hence, the nondimensional dispersion relation for the equivalent system can be given by with an emergent Bragg-type frequency band gap whose bounds are √ 2 and 2/µ. A detailed derivation of the dispersion relation is provided in Appendix A. It is very intriguing to observe that the variation of the masses in the periodic pendulums influences the equivalent stiffness of the system and, as a result, the mass ratio µ now dictates the ratio between the stiffnesses in the equivalent system.
In line with the analysis performed for the uniform pendulums chain, the frequency and time responses of the equivalent and actual systems are computed to validate these predictions. Analytical transfer functions for a diatomic lattice have been derived in literature for any arbitrary combination of design parameters [3]. For brevity, the derivation details are omitted here and we only show the final forms of the end-to-end transfer functions. For an even n, we obtain while for an odd number n, the transfer function reads where θ i = 2iπ n in both cases. For n = 15, µ = 0.25, ̺ = 10 3 and ℓ = 1, Figure 3c shows the frequency responses of the equivalent and actual systems and their corresponding pole distributions. An excellent agreement between the two realizations is clearly observed. Unlike the case of the uniform pendulums, the natural frequency spectrum is split into two distinct groups of natural frequencies, before and after the band gap range, both of which have an essential role in the formation of the band gap itself as detailed in literature [3]. The band gap range in the frequency response spans the range √ 2 < Ω < 2 √ 2, which can be shown to match the band gap limits obtained from the dispersion relation in Eq. (26).
The steady state responses of the periodic pendulums chain at different frequencies is shown in Figure 3d. Within the band gap at Ω = 2.2, the angular displacement corresponding to the payload at the tip of the chain is nearly zero, in addition to most of the chain masses prior to the payload, which is indicative of a band gap. On the other hand, the responses of the chain for Ω outside the range of the band gap clearly show a wave-like motion as expected from pass band frequencies. Finally, the dispersion behavior in Eq. (26) is tested by simulating the system's transient response to a wave packet in the time domain, as illustrated earlier. Figure 3e shows the numerical dispersion contours for the same parameters used to generate Figure 3c but with n = 41. As depicted by the figure, the entirety of the pendulums' dispersion energy coincides with dispersion bands predicted from Eq. (26) for the finite system.

C. Case III: Periodic variation in linkage lengths
Consider the case of periodic pendulums with the lengths of odd and even numbered links being ℓ 1 = ℓ and ℓ 2 = κℓ, respectively, where κ is the length ratio of the periodic links; a visual illustration is provided in Figure 4a. As in the case of the periodically varying pendulums masses, the equations of motion simplify to Eq. (13) albeit with the following M, K and f definitions: and the inverse mass matrix in this case (shown for an even number n) is: For an even n, ε = 1 κ 2 (1 + 1 1+̺ ) while for an odd one, ε = 1 + 1 1+̺ . Unlike the case of periodic masses, the stiffness matrix in Eq. (29b) is approximated by a periodic diagonal matrix which is valid for a large ̺ value and, for an even n, is given by Since M −1 acts as the stiffness matrix of an equivalent PC lattice, the inverse of K in Eq. (31) can be treated as the equivalent mass matrix, and will ultimately yield the unit cell of the equivalent PC lattice. Observing the pattern of M −1 in Eq. (30) and the inverse of K in Eq. (31), it can be concluded that these matrices resemble the stiffness and mass matrices, respectively, for a diatomic lattice with periodic elastic foundations and varying masses, and with two boundary springs k η and k ξ . For visualization, this arrangement is depicted in Figure 4b. Based on Eq. (30), the PC chain has an equivalent uniform springs of k = m κ ω 2 0 and the stiffness of the periodic elastic foundations are k 1 = 2k (κ − 1) and k 2 = −k1 κ , which implies that either k 1 or k 2 has to be negative depending on the value of κ. The latter is interesting since a negative stiffness is usually induced by active elements, e.g. piezoelectric shunts [27]. In the present case, however, the negative stiffness is a direct consequence of the unique dynamics of the pendulums chain. Similarly, it can be shown that the spring constant k η = − k1 2 for all n, while k ξ = k η and k ξ = − k2 2 for odd and even values of n, respectively. In addition to the periodic elastic foundation, the variation in the length of the links creates a periodic variation in the masses for the equivalent lattice with values m 1 = m and m 2 = m 1 /κ. For these values of springs and masses, and for the unit cell marked on the schematic given by Figure 4b, the following dispersion relation can be derived: and the corresponding band gap bounds are √ 2 and 2/κ. Further details pertaining to this derivation are provided in Appendix B.
Given the boundary conditions of this PC model, a closed-form analytical solution for the system's poles remains elusive. As a result, the frequency response in this scenario is derived in semi-analytical form since the poles of the system are calculated numerically. Once the poles are computed, the following equation depicts the system's transfer function for all n (See Miu [28] for more details): In Eq. (33), Ω i is a normalized pole of the system. For κ = 1/5, n = 15, ℓ = 1 and ̺ = 10 3 , Figure 4c shows the end-to-end frequency response function for the actual and equivalent systems, where a very good match is observed. As expected, a band gap is created in the range √ 2 < Ω < √ 10 which coincides with the large attenuation region in the frequency response diagram and matches the range predicted from the dispersion relation in Eq. (32). The steady state responses provide a further confirmation of the occurrence of the band gap as can be seen from the displacement fields at frequencies inside (i.e. Ω = 2.4) and outside the band gap (i.e. Ω = 0.6, 1.25 and 3.3) as shown Figure 4d. It is worth noting that although a zero frequency band gap is traditionally onset in PC lattices with elastic foundations [3], the negative stiffness of the elastic foundation in this scenario neutralizes the effect of the grounded springs and effectively renders it a foundation-free lattice. Finally, the numerical dispersion contours for the same system parameters used in Figure 4c and for n = 41 is depicted in Figure 4e. As anticipated, the contours are again concentrated and centered around the dispersion bands predicted from Eq. (32).
On a final note, a succinct presentation of the three presented cases is provided in Table I. The latter summarizes the dual correlations between the actual pendulum chain and the hypothetical dispersion-equivalent system for cases I, II and III.

IV. SUMMARY
In this paper, the ability of a passive and nondissipative periodic pendulums chain to exhibit selfinduced vibration isolation capabilities was demonstrated. Given the non-trivial and coupled nature of the pendulum chain dynamics, an obvious definition of a unit cell -typically needed to conduct and predict Bragg band gaps in self-repeating structures-does not exist. Instead, a pseudo unit cell of an equivalent PC lattice was identified and extracted from the finite chain's response in a novel and innovative manner. The analysis showed that the payload carried by the pendulum chain becomes a key to the formation of the equivalent PC system and, in its presence, the system shows a similar dispersive behavior. As a result, periodic variations in the pendulums masses or links (with a tip mass) create a frequency band gap which has been verified in both frequency and time domains. As a very intriguing observation, we showed that: (1) The variation in masses resembles the effect of varying the stiffness in a conventional PC.
(2) On the other hand, a periodic change in the connecting links becomes reminiscent of a PC lattice with a periodic elastic foundations as well as a periodic variation in the masses. Both cases, however, act identically since the hypothetical elastic foundation comprises a negative stiffness which nullifies its own effect. This is particularly interesting given that negative stiffness systems are typically stimulated using actively controlled elements as opposed to the current case where it emerges solely from the pendulum's unique arrangement. The potential of the proposed pendulum chains can extend beyond industrial crane applications to ones which involve vibration control of payload deliveries including, but not limited to, robotic arms [29], descending payloads of unmanned aerial vehicles [14] as well as tethered space elevators [30,31].   Diatomic w/ elastic foundation • Varying stiffness k2/k1 = −1/κ • Varying masses with m2/m1 = 1/κ • k1 = 2m κ ω 2 0 (κ − 1) • k1 > 0 for κ > 1 and k1 < 0 for κ < 1 • k1k2 < 0, ∀κ • Fixed-fixed B.C. (finite structure)