Compression performance and low-frequency damping characteristics of pyramidal lattice cylinder skeleton structure

The lattice structure has obvious advantages over traditional materials in terms of light weight, energy absorption, vibration and noise reduction, and is therefore widely used in many fields such as shipping and aerospace. In this work, a cylinder skeleton structure is designed based on pyramidal lattice structure by circumferential and axial array arrangement. The load-bearing capacity of the pyramidal lattice cylinder skeleton structure under quasi-static compression and its axial vibration characteristics are investigated by numerical method. The effects of key element geometric parameters such as diameter of metal wire, circumferential angle and axial angle on the compression and vibration damping performances of the pyramidal lattice cylinder skeleton structure were investigated. The results show that the developed structure has excellent compression and vibration damping capacity. It is able to provide a pronounced damping effect in the low-frequency range of 204-367 Hz, and the attenuation intensity of elastic wave in this frequency range can be up to 20dB.


Introduction
As a truss structure, the proposed pyramidal lattice cylinder skeleton structure can provide a new solution for the performance requirements such as lightweight, pressure-resistance, and vibrationreduction in pipeline systems.It can be an enhanced components such as pressure hull, expansion joint and noise attenuation muffler in the submarine and bridges engineering applications.These components are commonly used to mitigate or suppress the adverse effects of noise and vibrations, such as the forces generated by propellers, which produce radiated noise and induce vibrations in submarine hulls, negatively impacting stealth capabilities and overall safety [1] .Bridge expansion joints, which are exposed to intense vibrations over the long term, represent potential weak points.Vertical vibration is the primary cause of expansion joint failure, while horizontal vibration can affect the safety and service life of modular bridge expansion joint connections [2] .Traditional expansion chamber silencers have widespread use in pipeline systems.However, to eliminate low-frequency noise, these silencers require lengthy expansion chambers, and they are prone to periodic transmission losses [3] .Therefore, reasonable suppression of structural vibration can prolong the service life of machinery and increase the safety and comfort of structure.With the advancement of technology, researchers aspire to develop novel materials with exceptional properties, such as negative dielectric constants [4][5][6] , negative Poisson's ratios [7,8] , or vibration band gaps [9,10] .Lattice materials, as a new type of lightweight material, demonstrate remarkable specific strength and unparalleled design flexibility.Additionally, it demonstrated exceptional vibration suppression properties in the low frequency range.The definition of lattice structures associated with microstructure periodic arrangements was first introduced by Evans and Hutchinson, [11] .Since then, lattice structures have garnered considerable attention from a multitude of researchers due to the excellent mechanical performance.
In terms of compression and damping properties of lattice structures, Bai et al. [12] investigated the mechanical properties of gradient body columnar cubic (GBCC) lattice structures through experiments and finite element (FE) simulations.Kumar et al. [13] introduced a sea urchin (SU) lattice structure based on the bionic principle and investigated its compression properties experimentally.Huang et al. [14] studied energy absorption in compression by varying the cross-sectional shape and tilt angle of the pyramidal lattice structure.Zhao et al. [15] investigated the effect of local geometric features on the compressive properties of the structure.Li et al. [16] used theoretical and numerical methods to study the effect of different plate layers on the inherent frequencies of multilayer composite pyramidal sandwich plate structures.An et al. [17] proposed a truss lattice structure with a low-frequency band gap and verified the results by vibration transfer tests.
Most of the previous researches have focused on the static mechanical properties of lattice structures, with a lack of studies on vibration damping and noise reduction.Cylinders, being a versatile structural form, find extensive applications in various fields submarine and bridges engineering applications.Typically, these structures need to possess a certain level of compressive strength while also achieving the desired vibration isolation effects.Thus, this work is to design a novel lattice cylindrical framework based on a pyramid-shaped unit cell.This structure differs from the traditional lattice structures based on two-dimensional planes and three-dimensional cubes.It arranges the pyramidal units in the circumferential and axial directions.The static mechanical properties of the lattice structure are analyzed, as well as its vibration transmission characteristics.The cylindrical structure with skeletal configuration holds a significant advantage for achieving lowfrequency vibration reduction and noise attenuation.Through finite element simulation analysis, the compressive and vibration isolation performances of the structure are investigated with respect to three key geometric parameters: metal wire diameter, circumferential angle, and axial angle of the unit cell.Numerical simulation results indicate that by using appropriate geometric parameters, it is possible to widen the low-frequency band gap of the structure while maintaining a certain load-bearing capacity.The structure with the geometric configuration (i.e., 0.7 , 1 50 θ = ° and 2 60 θ = °) exhibits an exceptional performance, achieving significant vibration reduction in the low-frequency range of 204-367Hz.The vibration attenuation amplitude can reach up to 20dB.

Unit geometry design
In this work, pyramid-shaped unit was used as the basic units of the skeleton lattice structure.It consists of four cylindrical metal wire rod elements, and the pyramidal element and the overall structure are shown in Figure 1.The basic main parameters of the pyramidal unit include wire diameter ( d ), circumferential angle ( 1 θ ) and axial angle ( 2 θ ).The unit is combined into a lattice cylinder skeleton structure by circumferential array and axial array, and the number of circumferential array and axial array are m and n , respectively.The inner and outer diameters of the pyramidal lattice cylindrical skeleton structure are defined as i R and o R , respectively.The rod length can be expressed by the following equation: In this work, 304 stainless steel was selected as the basic material for the pyramidal lattice cylinder skeleton structure.Its static mechanical properties are shown in table 1.

Calculation of structural equivalent stiffness
The Euler-Bernoulli beam theory assumes that the deformed cross section remains perpendicular to the neutral axis.Therefore, the calculated deflection of the beam obtained from this theory is slightly lower than that of the actual beam and only applicable to slender beams.The flexural members in the pyramidal lattice units can be simplified to the planar model in Figure 2. Based on the Euler-Bernoulli beam theory, the differential equation for the deflection curve of the AB section of the bent rod is as follows: where , and E is the elastic modulus and I is the moment of inertia of the AB part.

Numerical simulation of quasi-static compression
The geometrical model is created by SOLIDWORKS and imported into Abaqus for finite element analysis.The numerical operation process is performed using a dynamic explicit algorithm.To reduce the computational cost, one-half of the symmetric model is used for the analysis.A tetrahedron element type was chosen for meshing.The main boundary conditions are set as follows: the lower

Numerical simulation of vibration response
The numerical operation process is performed using a dynamic steady-state step.A displacement excitation of amplitude δ is applied in the Y-direction of the structure, and the displacement response is extracted at the other end.The attenuation capability of the structure to elastic waves can be obtained visually through the response curves.The frequency response function (FRF) is defined as follows: where 1 u and 2 u are the excitation and response displacements.

Quasi-static compression simulation results
4.1.1.Comparison of theoretical and simulated equivalent stiffness.Figure 4 shows that the forcedisplacement curves can be divided into elastic phase, plastic phase and densification phase.Among them, the plastic phase can be divided into a plateau region with slowly changing loads and a spike region with fluctuating changes.The presence of the spike region can be attributed to the phenomenon that as further deformation requires higher loads, the deformation is transferred to different locations of the structure, leading to a redistribution of the loads.The comparison of equivalent stiffnesses derived between the theoretical and simulated results is depicted in table 2, as well as their deviation.It can be seen that there is still a significant deviation close to 30%.For the pyramidal lattice cylinder skeleton structure, the macroscopic configuration is a whole rather than combined by superposition of individual cell elements, thus the equivalent stiffness has deviations from the theoretical calculation.4(a)-(c) show the force-displacement curves for the three lattice cylinder skeleton structures.These structures are composed of triangular, quadrilateral, and Kagome units, with 15 units arranged radially and 3 units arranged axially.From the force-displacement curves, it can be observed that the force-displacement curve of the pyramidal lattice structure is the smoothest of the three structures, with a long plastic plateau region.Whereas the Kagome structure shows the best performance in terms of compressive capacity, the triangular structure is relatively poor in terms of compressive capacity.5(a), the effect of wire diameter on the force-displacement curves of the structures is investigated.The force-displacement curves of the three different wire diameter structures show a consistent trend.In the elastic phase, there is a positive correlation between the wire diameter and the peak load.The value of the platform load for the structure with a wire diameter of 0.5 mm hardly fluctuates, while the structures with wire diameters of 0.7 mm and 0.9 mm exhibits slightly increase in a platform load.The duration and magnitude of the spike region decrease with increasing wire diameter.As shown in Figure 5(b), the peak load is relatively great for a circumferential angle of 60°.Increasing the circumferential angle can significantly reduce the number and the amplitude of fluctuations in the spike region.Figure 5(c) shows that the peak load of the structure with an axial angle of 65° is the maximum.In the peak region, the fluctuation amplitude and duration increase with the increase of the axial angle.In summary, the wire diameter has the most significant effect on the platform load value during the plastic stage.Increasing the axial angle significantly reduces the amplitude of fluctuations in the peak region, while increasing the circumferential angle has a negative effect on this region.

Correctness verification of numerical simulation.
In order to assess of the correctness of the numerical model, the typical structure configuration with a wire diameter of 0.7 mm, circumferential angle 1 50 θ = °and axial angle 2 60 θ = ° is selected for compression analysis.As shown in Figure 6 the ratio of kinetic energy to internal energy reaches only 5.08%, which is within an acceptable value of 10%.Similarly, the ratio of strain energy to internal energy is also less than 5%.It should be noted that the numerical operation process does not exhibit the hourglass effect.Therefore, the correctness of numerical simulation for the quasi-static analysis is verified.The frequency response curves of the structures with different filament diameters are obtained by simulation.Figure 8(a) displays the frequency response curve of the structure with a wire diameter of 0.5mm, where the yellow region represents the bandgaps.The overall effective bandwidth of the bandgaps is up to 328Hz.In the frequency response curve, the first and second bandgaps exhibit a relatively wide range, while the third and fourth bandgaps are narrow and within a high frequency range.The first bandgap has the significant attenuation of elastic waves, reaching approximately 20dB. Figure 8(b) shows the frequency response curve for a wire diameter of 0.7mm, with a total effective bandwidth of 311Hz.Compared to Figure 8(a), the width of the first bandgap reduces and the attenuation of the second bandgap increases by about 15dB.The third and fourth band gaps are combined.Figure 8(c) illustrates the frequency response curve of the structure with a wire diameter of 0.9 mm and a total effective bandwidth of 313 Hz.The frequency response curve of this structure is similar to that of the wire diameter of 0.5 mm. Figure 9(a)-(c) illustrate the impact of the circumferential angle on the structure's frequency response curve.When the circumferential angle is 55º, the total effective bandwidth is up to 329Hz.When the circumferential angle is further increased to 60º, the band gap distribution and attenuation levels are similar to those of the structure with an azimuth angle of 50°, with the only significant enhancement being the attenuation level of the first band gap.At a frequency of approximately 100Hz, the attenuation can reach to 24dB, and the total effective bandwidth is 313Hz.When the axial angle is 65º, the total effective bandwidth is up to 330Hz, the band gap distribution and attenuation levels are similar to those of the structure with an axial angle of 60°, with the only significant enhancement being the attenuation level of the second band gap.At a frequency of approximately 247Hz, the attenuation can reach to 25dB.When the axial angle is further increased to 70º, the total effective bandwidth is 264 Hz, the attenuation levels of each bandgap decreases, where the second and fourth bandgaps split into two narrower bandgaps.

Vibration damping analysis.
In order to verify the vibration damping effect of the lattice structure at various frequencies, the typical structure configuration with a wire diameter of 0.7 mm, circumferential angle and axial angle is selected for vibration damping analysis, as depicted in Figure 11(a).The vibration distribution at two excitation frequencies of 12Hz and 300Hz can be observed from the vibration patterns displayed in Figure 11(b).It can be seen that 300 Hz is in the band gap range and the structure has an excellent damping performance at this frequency.

Conclusions
This work presents a pyramidal lattice cylinder skeleton structure and evaluates its compression and vibration damping performance using the finite element method.The work focuses on the influence of geometric parameters on the compression performance and vibration reduction effect of the structure.The main conclusions are drawn as follows: (1) The role of key geometric parameters of the pyramid element including wire diameter, circumferential angle, and axial angle play a considerable effect on mechanical performances associated with peak load, plateau load, and peak fluctuation region.(2) The pyramidal lattice cylinder skeleton structure exhibits an excellent axial vibration damping performance.The bandgap width and the strength attenuation during vibration can be improved by means of an alternative element structure configuration associated with wire diameter, circumferential angle, and axial angle.

Figure 1 .
Figure 1.(a) Front view of the pyramidal lattice cylinder skeleton structure, (b) top view of the structure, (c) schematic diagram of the pyramidal lattice unit.In this work, 304 stainless steel was selected as the basic material for the pyramidal lattice cylinder skeleton structure.Its static mechanical properties are shown in table1.

Figure 2 .
Figure 2. The force analysis of bent rod ABC.The displacement of the endpoint in the Y-axis direction is given by:

Figure 3 .
Figure 3.The finite element model of quasi-static compression.

Figure 4 .
Figure 4. Force-displacement curves: (a) pyramid unit, (b) triangular unit, (c) Kagome unit.As shown in Figure5(a), the effect of wire diameter on the force-displacement curves of the structures is investigated.The force-displacement curves of the three different wire diameter structures show a consistent trend.In the elastic phase, there is a positive correlation between the wire diameter and the peak load.The value of the platform load for the structure with a wire diameter of 0.5 mm hardly fluctuates, while the structures with wire diameters of 0.7 mm and 0.9 mm exhibits slightly increase in a platform load.The duration and magnitude of the spike region decrease with increasing wire diameter.As shown in Figure5(b), the peak load is relatively great for a circumferential angle of 60°.Increasing the circumferential angle can significantly reduce the number and the amplitude of fluctuations in the spike region.Figure5(c) shows that the peak load of the structure with an axial angle of 65° is the maximum.In the peak region, the fluctuation amplitude and duration increase with the increase of the axial angle.In summary, the wire diameter has the most significant effect on the platform load value during the plastic stage.Increasing the axial angle significantly reduces the amplitude of fluctuations in the peak region, while increasing the circumferential angle has a negative effect on this region.

Figure 6 .
Figure 6.Kinetic energy-internal energy ratio curve and strain energy-internal energy ratio curve.

4. 2 .
Vibration damping performance analysis 4.2.1.Vibration transmission characteristics curve.The vibration damping performance of three kinds of unit were analyzed, and the vibration transmission curves are shown in Figure.7. It shows that the pyramid structure has the largest range of low-frequency bandgap present in the 0-500 Hz range compared to the other two structures, and the attenuation is also the salient of the three structures.

Figure 7 .
Figure 7. Vibration transmission characteristics curve: (a) pyramid structure, (b) triangular structure, (c) Kagome structure.The frequency response curves of the structures with different filament diameters are obtained by simulation.Figure8(a) displays the frequency response curve of the structure with a wire diameter of 0.5mm, where the yellow region represents the bandgaps.The overall effective bandwidth of the bandgaps is up to 328Hz.In the frequency response curve, the first and second bandgaps exhibit a relatively wide range, while the third and fourth bandgaps are narrow and within a high frequency range.The first bandgap has the significant attenuation of elastic waves, reaching approximately 20dB.Figure8(b)shows the frequency response curve for a wire diameter of 0.7mm, with a total effective bandwidth of 311Hz.Compared to Figure8(a), the width of the first bandgap reduces and the attenuation of the second bandgap increases by about 15dB.The third and fourth band gaps are combined.Figure8(c) illustrates the frequency response curve of the structure with a wire diameter of 0.9 mm and a total effective bandwidth of 313 Hz.The frequency response curve of this structure is similar to that of the wire diameter of 0.5 mm.

Figure 8 .
Figure 8.The frequency response curves of structures with different wire diameters.Figure9(a)-(c) illustrate the impact of the circumferential angle on the structure's frequency response curve.When the circumferential angle is 55º, the total effective bandwidth is up to 329Hz.When the circumferential angle is further increased to 60º, the band gap distribution and attenuation levels are similar to those of the structure with an azimuth angle of 50°, with the only significant enhancement being the attenuation level of the first band gap.At a frequency of approximately 100Hz, the attenuation can reach to 24dB, and the total effective bandwidth is 313Hz.

Figure 9 .
Figure 9.The frequency response curves of structures with different circumferential angle.Figure 10(a)-(c) illustrate the impact of the axial angle on the structure's frequency response curve.When the axial angle is 65º, the total effective bandwidth is up to 330Hz, the band gap distribution and attenuation levels are similar to those of the structure with an axial angle of 60°, with the only significant enhancement being the attenuation level of the second band gap.At a frequency of approximately 247Hz, the attenuation can reach to 25dB.When the axial angle is further increased to 70º, the total effective bandwidth is 264 Hz, the attenuation levels of each bandgap decreases, where the second and fourth bandgaps split into two narrower bandgaps.

Figure 10 (
Figure 9.The frequency response curves of structures with different circumferential angle.Figure 10(a)-(c) illustrate the impact of the axial angle on the structure's frequency response curve.When the axial angle is 65º, the total effective bandwidth is up to 330Hz, the band gap distribution and attenuation levels are similar to those of the structure with an axial angle of 60°, with the only significant enhancement being the attenuation level of the second band gap.At a frequency of approximately 247Hz, the attenuation can reach to 25dB.When the axial angle is further increased to 70º, the total effective bandwidth is 264 Hz, the attenuation levels of each bandgap decreases, where the second and fourth bandgaps split into two narrower bandgaps.

Figure 10 .
Figure 10.The frequency response curves of structures with different axial angle.

Figure 11 .
Figure 11.(a) Schematic diagram of the structure, (b) vibration distribution of the structure.

Table 2 .
Comparison of equivalent stiffness and deviation.