Lumped mass model of a 1D metastructure for vibration suppression with no additional mass
Introduction
Metastructures are a metamaterial inspired concept. Metamaterial research began by investigating electromagnetic metamaterials which exhibited a negative permittivity and or permeability [1], [2]. Inspired by the electromagnetic metamaterials, the concepts were extended to acoustic metamaterials [3]. Traditional metamaterials utilize the theory of Bragg scattering. The lattices are created such that when the waves reflect off the structure, they destructively interfere with each other. In order for the Bragg scattering mechanism to work, the periodic length of the material must be of similar length as the wavelength. Thus, for low frequencies very large structures are required [4]. Metamaterials that rely on the Bragg scattering mechanism are commonly called phononic crystals. Phononic crystals are materials which exhibit some type of periodicity and are reviewed in a paper by Hussein et al. [5]. Milton and Willis were the first to conceive the idea of using local absorbers to create structures with negative effective mass that varies with frequency [6]. Liu et al. created the first physical metastructure that was able to create a bandgap at a frequency lower than that of the Bragg scattering mechanism. This structure is designed to suppress acoustic waves above 300 Hz. Their acoustic metamaterial contains lead spheres coated in a silicone rubber within an epoxy matrix. The lead balls in the rubber are referred to as local resonators. The local resonator mechanism is the same mechanism used for vibration suppression [7]. Since then locally resonant metamaterials have been studied extensively for both acoustic and vibration isolation applications. The work presented here deals exclusively with vibration mitigation applications. Structures or materials capable of suppressing vibrations using these local resonators are often referred to as elastic metamaterials. In a review paper by Zhu et al., the authors provide a review of various type of plate-type elastic metamaterials and discuss possible applications. They also provide an explanation of the negative mass density and negative bulk modulus [8]. Here the term metastructure is used to refer to structures with distributed vibration absorbers. These structures use conventional materials with absorbers integrated into the structure through geometry and material changes on the centimeter scale. The periodic-type nature of these structures was inspired from metamaterials but the larger scale modifications makes the term structure a more fitting term for this work. In the literature, these are also referred to as locally resonant phononic crystals or elastic metamaterials. The field of auxetics also has considerable overlap with metastructures. Auxetics are materials that exhibit a negative Poisson's ratio. These materials are realized by creating periodic lattice structures. Because of the periodic nature of auxetics, they affect how waves propagate through them and thus can be used for vibration suppression among other applications [9].
As Hussein et al. describes in their review paper, metastructures are at the cross roads of vibration and acoustics engineering, and condensed matter physics [5]. Thus, it is important that strengths from both fields are considered and reviewed for relevancy. Sun et al. and Pai looked into the working mechanism of metastructures for both bending and longitudinal motion. They were able to conclude that the working mechanism that leads to vibration suppression is based on the concept of mechanical vibration absorbers that do not need to be small or closely spaced [10], [11]. Therefore, it is also relevant to explore the literature regarding vibration absorbers. Vibration absorbers can also be called tuned mass dampers (TMDs) or dynamic vibration absorbers. TMDs typically consist of mass-spring-damper systems, while a vibration absorber does not use a damper to add significant localized damping. Although there is no localized damper added to the vibration absorber, there is still a small amount of material damping which is inherent in all structures. TMDs are studied widely in the field of earthquake engineering. Igusa and Xu were the first researchers to look at the effects of using multiple (TMDs) to suppress a single mode of a structure [12], [13]. Later, this was also studied by Yamaguchi and Harnpornchai [14]. Their work focuses on attaching multiple TMDs to a single degree of freedom system and shows that multiple TMDs can be more effective than a single TMD. These results can be leveraged in metastructure research.
Another important aspect of the TMD literature is how the optimal parameters for the TMDs were determined. Many methods have been used and applied to various systems. DenHartog developed the optimal parameters for a single TMD as an analytical expression [15] and this result has since been studied by many others as summarized in Sun et al.'s review paper [16]. The work presented here focuses on some of the numerical methods utilized by many TMD researchers specifically the norm. Parameters are chosen such that the norm is minimized. This performance metric describes the response of a structure excited across all frequencies [17], [18]. The norm provides different results than those obtained by suppressing a specific frequency range and tend to suppress the fundamental mode which typically has the largest magnitude response.
The model used in this paper is a one-dimensional lumped mass model, which was chosen for its simplicity, and allows the dynamics to be understood more thoroughly. Some of the most relevant work related to 1D metastructures is from Pai who models a longitudinal metastructure consisting of a hollow tube with many small mass-spring systems distributed throughout the bar. He suggests that the ideal design for a metastructure, involves absorbers with varying tuned frequencies [11]. Xiao et al. looks at a similar structure as Pai but considers multiple degree of freedom resonators. Their work focuses on modeling procedures and understanding the bandgap formation mechanisms [19]. The favorable dynamics response of these structures can also be described as having a negative effective mass which has been shown analytically and experimentally [6], [20], [21]. In addition, other researchers have conducted experiments on longitudinal metastructures. Zhu et al. looked at a thin plate with cantilever absorbers cut out of the plate. They were able to show the ability to accurately predict the band-gap and also compared various absorber designs [22]. Wang et al. tested a glass bar with cantilever absorbers made out of steel slices and a mass [23]. With the rise in additive manufacturing, 3D printing has become a good method to realize the complex geometry needed for these structures [24]. Hobeck et al. and Nobrega et al. both created longitudinal 3D metastructures and obtained experimental results [25], [26].
In this paper, the feasibility of adding distributed vibration absorbers to a structure without increasing the overall weight of the structure is presented. When creating these metastructures, much of the previous work looks at keeping the stiffness constant, which requires adding additional mass to the structure [27], [28], [29]. Taking an alternative approach, this paper keeps the mass constant by redistributing mass from the host structure to the distributed absorber system. Mass and suppression properties are inherently tied together which motivates this approach. Simply adding mass to a structure will result in increased suppression. It is important to isolate the effects of the dynamics of the vibration absorbers from the added mass. It must be shown that the additional mass is not causing the increased performance. Since a lumped mass model is used, the redistribution of mass does not affect the stiffness of the structure. In real structure where mass and stiffness are coupled, the stiffness would be affected. Other researchers have looked at constant stiffness structures but have not shown if the additional mass or the dynamics of the absorbers are causing the increased performance [27], [28], [29]. By using a constant mass constraint, this work shows that the vibration absorber dynamics, and not the additional mass, are increasing the suppression. Additionally, in both automotive and aerospace structures, it is critical to keep the total weight of the structure low. The strategy taken here is to model a discrete system undergoing axial vibrations. This is used as a starting point because the analysis for a unidirectional discrete problem is relatively straight-forward and enables understanding of the basic issues and phenomenon. The model consists of masses and springs connected in series with an absorber attached to each one of the masses. The mass and stiffness of these components are varied in order to get the desired dynamics. Material damping inherent in all structures is represented by a proportional damping model and is added to this model to keep the response bounded and to emulate material damping. The metastructure described here is compared to a baseline structure which has no vibration absorbers, which illustrates that the favorable damping comes from the addition of the distributed vibration absorbers and not from adding mass to the system. The focus of the vibration suppression is on the fundamental natural frequency of the structure as opposed to creating a band-gap at higher frequencies, which much of the metamaterial research examines. In engineering applications, it is important to suppress the frequencies near the fundamental mode of vibration, as these frequencies typically result in the highest magnitude response. A similar approach is taken when designing TMDs, explaining the motivation for using performance metrics from the TMD literature.
The work of Igusa and Xu is similar to the work presented here but there are some important differences. They are comparing the effectiveness of a single TMD and multiple TMDs whereas this work compares multiple vibration absorbers to no absorbers. Thus their structure with vibration suppression is heavier than their structure without suppression. In this work the suppression system does not add weight to the structure. Additionally, Igusa and Xu use TMDs so they can tune the mass, stiffness and damping of each absorber [13]. The work presented here does not add dampers with high levels of localized damping to the vibration absorbers, thus only mass and stiffness can be tuned.
The methodology used for this paper begins by introducing the model used and the parameters that characterize this model. The main parameters varied throughout this study are the number of absorbers, the mass ratio (ratio of absorber mass to mass of the rest of the structure), and the natural frequencies of the individual absorbers. Other variables in the model are calculated such that the mass of the structure is constant and the fundamental frequency of the entire structure stays relatively constant throughout the analysis. Both steady state and transient responses are examined. Next, the analysis of the model is described and details about the performance measures are provided. These are the and norms, which measure the total energy of the system and the maximum response respectively. An optimization procedure is set up to minimize the norm and shows the trade-offs between various parameters. All of the metastructure models created are compared to a baseline structure which has equal mass but no distributed vibration absorbers. The constant mass means any increase in performance can be attributed to the addition of the absorbers. Lastly, the simulation results will be compared to a 3D finite element model from previous work by the authors [30], to show that trends found from this longitudinal model match the trends found from in finite element models.
Section snippets
Lumped mass model
Consider the lumped mass model shown in Fig. 1(a) which represents a metastructure bar. This model consists of masses and springs connected in series and all the deformation occurs in the horizontal direction. The model contains the host structure with vibration absorbers distributed along the length of the bar. The larger masses and springs make up the host structure while the smaller masses and springs represent the vibration absorbers. Small deformation is desired in the host structure. To
Performance measures
This section describes the performance measures used to determine how effectively the structure reduces vibrations. Here the and the norms will be utilized, which are widely used in control literature to develop optimal control theory. The norm is related to the total energy in the system and the H∞ norm is related to the maximum energy. To begin, the system must be transformed into state space. The equations of motion for the structure can be converted into state space and then
Simulation results
Initially, a metastructure model with three main masses, two absorbers, and a mass ratio of is examined. The corresponding baseline structure has a fundamental natural frequency of 546 Hz. The two absorbers in the metastructure are tuned to that same frequency, 546 Hz. Using equations Eqs. (3), (4), and are calculated.
The results of this simulation are shown in Fig. 3. The FRF on the left clearly shows that the natural frequency peak of the baseline model gets split into two slightly
Optimization
Next, an optimized version of this model is examined using a similar procedure to that of Zuo and Nayfeh, who applied their methods to multiple degree of freedom tuned mass dampers [18]. Zuo and Nayfeh optimized their model by minimizing the norm. A similar approach is taken in this model, but will maximize the percent decrease in the norm from the baseline structure to the metastructure. The negative of this percentage is used as the objective function and is minimized. The optimization
Parameters to vary
A variety of parameters are examined, to see what the model is capable of and to help understand any basic phenomenon. This section explores the various parameters of the models to see how to get the best response in terms of the performance measures defined in the sections below.
Finite element verification
This section takes the optimal lumped mass model determined above with parameters and and performs a finite element verification on that model. The finite element model is based off the parameters from Table 1 and uses a design from previous work [30]. The 3D finite element model was created in Abaqus using 3D tetrahedral elements. The lumped mass model used in this section is modified slightly from the one described in the above sections. This section begins by describing the
Conclusions
The results of these simulations show that it is possible to use distributed vibration absorbers to reduce the response of a system without adding additional mass to the structure. These simulations found that the distributed absorbers should be designed such that their natural frequencies span a range of frequencies. For this specific structure, the results show that the mass ratio (mass of the absorbers over the mass of the host structure) should be around 0.30 and the number of absorbers
Acknowledgements
This work is supported in part by the US Air Force Office of Scientific Research under the Grant number FA9550-14-1-0246 “Electronic Damping in Multifunctional Material Systems’’ monitored by Dr. BL Lee and in part by the University of Michigan, through the “Kelly" Johnson Collegiate Professorship.
References (31)
- et al.
Vibration control using multiple tuned mass dampers
J. Sound Vib.
(1994) - et al.
Minimax optimization of multi-degree-of-freedom tuned-mass dampers
J. Sound Vib.
(2004) - et al.
On the negative effective mass density in acoustic metamaterials
Int. J. Eng. Sci.
(2009) - et al.
Experimental and numerical study of guided wave propagation in a thin metamaterial plate
Phys. Lett. A
(2011) - et al.
Vibration band gaps for elastic metamaterial rods using wave finite element method
Mech. Syst. Signal Process.
(2016) - et al.
Waves in Metamaterials
(2009) - et al.
Introduction, history, and selected topics in fundamental theories of metamaterials
- et al.
Controlling sound with acoustic metamaterials
Nat. Rev. Mater.
(2016) - et al.
Sound attenuation by sculpture
Nature
(1995) - et al.
Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook
Appl. Mech. Rev.
(2014)
On modifications of Newton's second law and linear continuum elastodynamics
Proc. R. Soc. A Math. Phys. Eng. Sci.
Locally resonant sonic
Mater. Sci.
Microstructural designs of plate-type elastic metamaterial and their potential applications: a review
Int. J. Smart Nano Mater.
Three decades of auxetics research − materials with negative Poisson's ratio: a review
Adv. Eng. Mater.
Theory of metamaterial beams for broadband vibration absorption
J. Intell. Mater. Syst. Struct.
Cited by (68)
Forced vibrations of a finite length metabeam with periodically arranged internal hinges and external supports
2024, European Journal of Mechanics, A/SolidsOn the influence of internal oscillators on the performance of metastructures: Modelling and tuning conditions
2023, Mechanical Systems and Signal ProcessingCentrifugal forces enable band gaps that self-adapt to synchronous vibrations in rotating elastic metamaterial
2023, Mechanical Systems and Signal ProcessingBroad bandgap active metamaterials with optimal time-delayed control
2023, International Journal of Mechanical SciencesTransfer matrix method for linear vibration analysis of flexible multibody systems
2023, Journal of Sound and VibrationAcoustic metamaterials with controllable bandgap gates based on magnetorheological elastomers
2023, International Journal of Mechanical SciencesCitation Excerpt :The bandgap is determined by the resonant frequencies of the resonators, a phenomenon which makes such media suitable for suppressing low-frequency waves [33–36]. Numerous applications are based on this property of “resonant” metamaterials: vibration suppression [37–42], stress wave amplitude reduction [25,43], and seismic isolation [44–47] to name a few. In turn, the successful solution of the problems of practical use involving acoustic metamaterials requires the development of methods for generating a controlled spectrum of propagating acoustic waves [17,48–52].