Abstract
This work looks at the effectiveness of constant weight metastructures for vibration suppression. A metastructure is a structure with distributed vibration absorbers. The metastructures are compared to a baseline structure of equal mass. The equal mass constraint shows that any increase in performance is due to the addition of the vibration absorbers and not due to adding additional mass to the structure. In this paper, two different metastructure designs are compared. These structures are designed to suppress longitudinal vibrations traveling along the length of the bar. The metastructures have ten vibration absorbers distributed on the length of the bar and the ratio of mass of the absorbers to mass of the host structure is 0.26. One metastructure has all the absorbers tuned to the natural frequency of the host structure and the other metastructure uses absorbers that are tuned to frequencies that have linearly varying natural frequencies. These structures were modeled using two different methods, a one-dimensional (1D) finite element method with lumped mass vibration absorbers and a fully three-dimensional (3D) finite element model. The results show that the metastructure with linearly varying natural frequencies outperforms that metastructure with vibration absorbers tuned to a single natural frequency.
Keywords
16.1 Introduction
Metastructures are a metamaterial inspired concept. Metamaterial research began with the investigation of electromagnetic metamaterials which exhibit a negative permittivity and or permeability [1, 2]. Inspired by the electromagnetic metamaterials, the concepts were extended to acoustic metamaterials [3]. 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 [4]. Liu et al. created the first physical metastructure using local vibration absorbers. 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 [5]. The research reported here emulates the concept of repeated elements from metamaterials’ research to devise a structure with repeated inserts consisting of vibration absorbers. A metastructure is defined as a structure with distributed vibration absorbers. Previous research often considers two structures with equal stiffness and compares the response of the structure [6, 7]. The addition of the distributed vibration absorbers leads to an increase in mass of the structure. For aerospace applications, the additional mass is undesirable. This work takes the alternative approach and keeps the mass constant between the baseline structure and the metastructure by redistributing the mass.
In this paper, three different structures are compared. These structures are designed to examine longitudinal vibrations traveling along the length of the bar. All three structures have the same weight. The baseline structure has no vibration absorbers and is used as a baseline to see the improved performance of the metastructures. The other two structures are both metastructures with ten absorbers distributed along the length of the bar. One structure has all the absorbers tuned to the natural frequency of the host structure and the other metastructure uses absorbers that are tuned to frequencies that have linearly varying natural frequencies. These structures were modeled using two different methods, a one-dimensional (1D) finite element method with lumped mass vibration absorbers and a fully three-dimensional (3D) finite element model.
16.2 Metastructure Design
The metastructures are designed to suppression vibrations in the longitudinal direction. The metastructures are composed of the host mass and the distributed absorbers. The host structure is the part of the structure not consisting of the vibration absorbers. The aim is to minimize the vibrations in the host structure. The host structure is a hollow square bar as seen in Fig. 16.1 with w = 50 mm and t = 5 mm. The length of the bar is 45 cm. There are ten vibration absorbers with in the hollow section distributed throughout the length of the bar. The vibration absorbs consist of a bar and a tip mass. The cantilever configuration is arranged such that the bending motion of the beam and tip mass suppress vibrations along the length of the bar. The specific geometry of the absorbers is modified to tune the natural frequency of the absorber. The mass ratio, μ of the metastructure is defined as the ratio of mass in the absorbers over the mass in the host structure. For this configuration, the mass ratio utilized is μ = 0.26. The host structure has a mass of 473 g leading to the mass of the absorbers to be 123 g. This gives the structure a total mass of 600 g. These parameters are used for both metastructures analyzed in this paper.
The baseline structure is used as a comparison to quantify the improved performance of the metastructure. The baseline structure has no vibration absorbers and consists of just the hollow square cross-section. The dimensions of the baseline bar are w = 50 mm and t = 6.5 mm. The length is also 45 cm. Since the baseline structure and the metastructure are the same weight, 600 g, the baseline structure has a slightly larger thickness to the account for the lack of vibration absorbers. 3D CAD models of the metastructure and the baseline structure are shown in Fig. 16.2. The material properties for all structures utilized are those from the Objet Connex 3D printer by Stratasys, specifically the DM8430 digital material. The Young’s modulus and density of the material are 1.97 GPa and 1168 kg/m3 respectively. The 3D printer will be used for the fabrication of an experimental prototype in future work.
16.3 Modeling
Two different modeling methods were utilized. A 1D model and a fully 3D finite element model. The 1D finite element was created in MATLAB. Because of the simplicity of the 1D model, the simulation runs quickly and can be optimized relatively easily in order the find the idea range of natural frequencies for the absorbers. The 3D finite element model was created in Abaqus using 3D tetrahedral elements. The 3D modeling allows us to capture the 3D effects present in all systems leading to a more accurate model. For both modeling procedures, we are interested in the frequency response function (FRF) of the structures. The frequency range of interest for these structures is 0–1500 Hz. As a performance measure, the area under the FRF within this frequency range is utilized. A smaller area corresponds to lower vibration response from the structure. Three different structures are analyzed, a baseline structure and two metastructures. The first metastructure has all the vibration absorbers tuned to the same natural frequency while the second one has the absorbers tuned to a range of frequencies. Pervious work has shown that varying these frequencies is beneficial [8] and this work seeks to confirm this using a 3D finite element model.
16.3.1 1D Finite Element Model
This 1D finite element model is depicted in Fig. 16.3. The host structure is discretized into 100 elements along the length of the bar and a simple bar 2-noded bar element is utilized. The vibration absorbers are modeled as lumped masses and springs and are distributed evenly through-out the length of the bar. The mass of the vibrations absorbers is chosen such that the mass ratio, μ = 0.26 is achieved and the stiffness of the absorbers varies such that specific natural frequencies can be achieved. The baseline structure is modeled using 100 bar elements. The finite element model allows us to determine the mass and stiffness matrices for this model. Using these matrices, the system can be transformed into state space and the frequency response function of the structure is calculated.
The simplicity of this model allows an optimization procedure to easily be implemented. Varying the stiffness of the vibration absorbers results in different natural frequencies of the absorbers leading to a different response in the structure. All other parameters are kept constant and the stiffness values are varied. The stiffness values are constrained to vary in a linear fashion thus the minimum and maximum values characterize these values. The objective function of the optimization is the area under frequency response function as described above. The optimization is done in MATLAB using a constrained non-linear interior point algorithm (fmincon). The results of the optimization show that the natural frequencies of the absorbers should vary from 450 Hz to 1100 Hz in order the achieve the best performance. The natural frequency of the host structure is 721 Hz.
16.3.2 3D Finite Element Model
The 3D finite element modeling is done in Abaqus. This allows us to capture the 3D effects of the structure which do not appear in a 1D model. Also, the 3D modeling allows for a realistic vibration absorber geometry to be determined which has a natural frequency that matches the desired frequency. The vibration absorbers utilized consist of a cantilever beam with a large tip mass. This design is the same design utilized by other researchers [9]. To vary the natural frequency of the absorber, the cross-sectional area of the beam is varied. A larger thickness of the beam leads to a stiffer absorber which raises the natural frequency. To keep the mass of all the vibration absorbers constant, the cross-sectional area of the beam is restricted to be constant throughout the entire structure. Thus, an increased thickness of the beam will result in a decrease in the width of the beam. To tune the natural frequencies of the vibration absorber, a 3D finite element model was necessary. The stiffness of the host structure contributes significantly to the natural frequency of the absorber thus must be accounted for in the modeling. To achieve this each absorber was modeled by itself attached to the host structure as seen in Fig. 16.4 and a modal analysis is performed. This allows the stiffness of the host structure to be accounted for while eliminating the interactions between the various vibration absorbers. The results of this modeling are shown in Table 16.1. The entire range of frequencies from the optimization procedure described above was not able to be achieved. The optimization called for frequencies ranging from 450 Hz to 1100 Hz. In order to reach frequencies above 940 Hz, the thickness of the beam needed to be greater than 21 mm. The absorber mass is 21 mm in length on each side, thus the dimensions of the beam must be less than 21 mm. The dimensions shown in Table 16.1 are used for the final design of the metastructure with varying frequencies. The metastructure with vibration absorbers having a single frequency used absorber seven for all ten of the vibration absorbers which has a natural frequency closest to that of the host structure, 721 Hz.
16.4 Results
For each of the three designs, (1) the baseline structure, (2) the metastructure with linearly varying natural frequencies and (3) the metastructure with vibrations absorbers with a single frequency both models were used to create the resulting FRF. All three of these structures have equal mass so any increase in performance is due to the addition of the vibration absorbers and not due the addition of mass to the structure. The results for the various structures and models are shown in Fig. 16.5. The FRF plotted is the response of the tip mass due to a loading at the tip mass. All the FRFs plotted are normalized with respect to the static response of the baseline structure.
As mentioned previously, the performance of the structure is characterized by the area under the FRF from 0 Hz to 1500 Hz and is compared to the response of the baseline structure. These results are shown Table 16.2. In both cases the metastructure with linearly varying natural frequencies performs better that the baseline structure but that is not the case for the metastructure with all the absorbers tuned to the same natural frequency. The 3D finite element model shows better performance but the 1D model does not. This discrepancy is likely due to damping of the structure not being modeled effectively. Even though the results for the two models do not match up very well, both models show the same trends. For the metastructure with a single vibration absorber, there are clearly two peaks on either side of the natural frequency of the baseline structure. The metastructure with multiple frequencies, many small peaks can be seen throughout the range of frequencies of interest. Both models show that the metastructure with absorbers tuned to multiple natural frequencies has better performance that the metastructure with the absorbers tuned to a single frequency. When using a model, it is hard to capture the true effects of damping. This shows the necessity of conducting experimental tests to better quantify how the damping contributes to the results.
References
Engheta, N., Ziolkowski, R.W.: Metamaterials: Physics and Engineering Explorations, IEEE (2006)
Laszlo, S., Shamonina, E.: A historical review. In: Waves in Metamaterials, pp. 315–324 (2009)
Cummer, S.A., Christensen, J., Alù, A.: Controlling sound with acoustic metamaterials. Nat. Rev. Mater. 1(3), 16001 (2016)
Milton, G.W., Willis, J.R.: On modifications of Newton’s second law and linear continuum elastodynamics. Proc. R. Soc. A Math. Phys. Eng. Sci. 463(August 2006), 855–880 (2007)
Liu, Z., Zhang, X., Mao, Y., Zhu, Y.Y., Yang, Z., Chan, C.T., Sheng, P.: Locally resonant sonic materials. Science. 289(5485), 1734–1736 (2000)
Baravelli, E., Ruzzene, M.: Internally resonating lattices for bandgap generation and low-frequency vibration control. J. Sound Vib. 332(25), 6562–6579 (2013)
Yu, T., Lesieutre, G.A.: Damping of sandwich panels via acoustic metamaterials. In: AIAA SciTech, 57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference (2016)
Reichl, K.K., Inman, D.J.: Modelling of low-frequency broadband vibration mitigation for a bar experiencing longitudinal vibrations using distributed vibration absorbers. In: 20th International Conference on Composite Materials (2015)
Hobeck, J.D., Laurant, C.M.V, Inman, D.J.: 3D printing of metastructures for passive broadband vibration suppression. In: 20th International Conference on Composite Materials (2015)
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.
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Reichl, K.K., Inman, D.J. (2017). Constant Mass Metastructure with Vibration Absorbers of Linearly Varying Natural Frequencies. In: Mains, M., Blough, J. (eds) Topics in Modal Analysis & Testing, Volume 10. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-54810-4_16
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