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Modal characteristics of micro-perforated sandwich beams with square honeycomb-corrugation hybrid cores: A mixed experimental-numerical study
Thin-Walled Structures 137 (2019) 185–196
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Full length article
Modal characteristics of micro-perforated sandwich beams with square honeycomb-corrugation hybrid cores: A mixed experimental-numerical study
T
Zhi-jia Zhanga,b,c,e, Qian-cheng Zhanga,b,c, , Fei-chen Lia,c, Jian-wen Yangd, Ji-wu Liud, ⁎ Zhan-yi Liud, Feng Jina,c, ⁎
a
State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an Jiaotong University, Xi’an 710049, PR China Key Laboratory of Intense Dynamic Loading and Effect, Xi'an 710024, China MOE Key Laboratory for Multifunctional Materials and Structures, Xi'an Jiaotong University, Xi'an 710049, PR China d Science and Technology on Liquid Rocket Engine Laboratory, Xi'an Aerospace Propulsion Institute, Xi'an 710100, PR China e Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA b c
ARTICLE INFO
ABSTRACT
Keywords: Sandwich structure Honeycomb-corrugation hybrid core Micro-perforation Vibration
Modal performance of micro-perforated sandwich beams with honeycomb-corrugation hybrid cores was investigated. Finite element methods and modal analysis techniques have been used to predict their vibration characteristics (i.e. their natural frequencies and mode shapes). It is shown that the natural frequencies of sandwich beam with micro-perforation are slightly lower than those of corresponding order of sandwich beam without micro-perforation based on the experimental and three-dimensional finite element calculated results. To reveal the effect of micro-perforation diameters on the natural frequencies of sandwich beams, a dimensionless frequency parameter was proposed. The results demonstrate that the frequency parameter of sandwich panels decrease near-linearly with the increase of micro-perforation ratio. In addition, the frequency parameter is sensitive not to such parameters as the face sheet thickness ratio, the slenderness ratio of corrugated member and the relative density of filling honeycomb, but to the configuration of micro-perforation.
1. Introduction Due to their multi-functional characteristic, periodic cellular metal sandwich structures have been widely used in such engineering application fields as aerospace, high speed transportation, submarine [1,2], and so forth. Various types of topological configurations of sandwich cores have been developed [3–7], however, there are still a large amount of technology-based breakthrough innovations for honeycombs, corrugations and lattice truss structures at millimeter scale. In order to improve their spatial and structural efficiency, many hybrid core structures have been constructed to improve mechanical properties of sandwich structures. The hybrid sandwich cores have been proposed, including aluminum foam-filled metallic tubes [8–11] and metal foam-filled metallic corrugation [12–14]. It is surprising that metal foam filling could greatly increase the strength and energy absorption of sandwich structures. The completely surprising results are attributed to an underlying mechanism: the lateral support to the core member not by weakly but strongly filled foam, altering the
⁎
deformation modes and considerably delaying core member buckling. However, the weight and space efficiency of new composite structures has not fully been exploited. The main reason is that filled foams in these structures have been enhanced next to nothing by such ordered cellular materials as metallic corrugation. In order to further improve structural efficiency in composite cellular structures, new idea about honeycomb-filled corrugated core has been proposed [15], which significantly increases the weight and space efficiency of sandwich structures by mutual enhancement of corrugated member and filled honeycomb, i.e. the mutual constraint of corrugated member and honeycomb cell walls against buckling changes the deformation modes of the corresponding structures. Zhang et al. [16] investigated the vibration performance of sandwich beams with honeycomb-corrugation hybrid cores. It is found that the filling honeycombs have an effect on weakening the anisotropy of the stiffness and suppressing the local mode shape for sandwich beam. Similarly, Yin et al. [17] proposed a hollow lattice truss reinforced honeycombs (LTRHs), which were designed to combine lattice topology with square honeycombs. The crush
Corresponding authors at: State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an Jiaotong University, Xi’an 710049, PR China. E-mail addresses: [email protected] (Q.-c. Zhang), [email protected] (F. Jin).
https://doi.org/10.1016/j.tws.2019.01.004 Received 21 August 2018; Received in revised form 30 November 2018; Accepted 6 January 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.
Thin-Walled Structures 137 (2019) 185–196
Z.-j. Zhang et al.
Fig. 1. Schematic of perforated honeycomb-corrugation hybrid core sandwich structure, which is composed of micro-perforated panels as facesheets and a honeycomb-corrugation hybrid as core. The corrugation is also perforated, just beneath the hole in top facesheet.
force efficiency, energy absorption and specific energy absorption capacity of LTRHs were found to be superior to both square honeycombs and hollow lattice structures. Besides the outstanding mechanical performance, sandwich structures with micro-perforations also are well-known devices used for the attenuation of acoustic levels. Perforated pores in the face plates of the sandwich panels can provide effective sound absorption as micro-perforated panel layers, while the backed plates and core structures can act as sound insulation barriers. Meng et al. [18] investigated the low frequency sound absorption coefficient and sound transmission loss of corrugated sandwich panels with different perforation configurations by numerical calculation and experiment. Through the comparisons between the classical corrugated sandwich panels (without perforations) and corrugated sandwich panels with face plate perforations, it is proved out that the face plate perforations are effective in improving the sound absorption coefficient (SAC) and sound transmission loss (STL) at low frequencies. Sakagami and co-authors [19] found that the honeycomb not only stiffened the micro-perforated panel, but also improved its low-frequency sound absorption performance. More recently, Tang et al. [20,21] found that honeycomb-corrugation hybrid cored sandwich panel with heterogeneously perforated face sheet and perforated corrugation as a new kind of sound absorber, as well as lightweight load-bearing structure, behaved better in low-frequency sound absorption. In addition, engineering design often requires structural discontinuities such as perforation or cutoffs for connection, joint, access and fluid flow. In general, the presence of perforations leads to considerable reduction in load bearing capacities [22,23]. Therefore, the influence and sensitivity of perforation on structural mechanical behaviors signify a critical issue in practice. Currently, no reference has been reported to study the dynamic characteristics of micro-perforated sandwich structures with hybrid cores. In this study, the effects of micro-perforation on the dynamic behaviors of microperforated sandwich beams with square honeycomb-corrugation hybrid cores are firstly studied to guide the relevant structural analysis and design of micro-perforated in actual applications with a mixed experimental-numerical study. The influence of key geometrical parameters on their vibration performance is also explored.
honeycomb-corrugation hybrid core of thickness h , as shown in Fig. 1. Periodically distributed micro-perforations were introduced in both facesheets and corrugation. The honeycomb core has a square crosssection with an inner side length lh and a wall thickness th . The corrugation micro-perforated at vertical direction has an inclination angle and a web thickness tc . The thicknesses and perforation diameters of top face sheet and down face sheet are t f , d1, and d2 (d1 0, d 2=0 or d1 0, d 2=d1 or d1 = 0, d2 = 0 ), respectively. 2.2. Fabrication of sandwich panel Consider a sandwich panel having metallic honeycomb-corrugation hybrid core, with both of its face sheets and core made of 304 stainless steel sheet. For reference, the sandwich panels with honeycomb core and empty corrugated core were also considered. The honeycombcorrugation hybrid core sandwich panels were fabricated in a three-step process summarized schematically in Fig. 2. All samples (the featured geometrical dimensions as shown in Table 1) were manufactured by folding method, electro-discharge machining (EDM) and brazing as described below. Firstly, cross-slots were cut by EDM and then the square-honeycomb was assembled as sketched in Fig. 2(a). The clearance of 5 µm between sheet and slot facilitated assembly while providing a sufficiently tight fit to assure stability. The braze of the assembled sheets kept in step with pre-filling alloy (Ni–Cr25-P10 (wt%)) powder slurry into the gap. For the brazing process, the structure was placed in a dry oven and held for 2 h at 60 oC to remove water from surface. The samples in vacuum furnace were heated at the rate of 10 o C /min to 350 oC , kept for 30 min to volatilize the polymer binder. The brazing temperature was set at 1020 oC and held for 20 min before being cooled inside the furnace to room temperature. Then trapezoidal honeycomb blocks precisely cut from a brazed square honeycombs using EDM. Secondly, the corrugated sheet was produced using the folding method as sketched in Fig. 2(b). Then the molded corrugated sheets were cut into several corrugated webs using EDM. Thirdly, after being cut down to size, the trapezoidal honeycomb blocks and corrugated web were assembled together with face sheets to create a sandwich beam with honeycomb-corrugation hybrid core, shown in Fig. 2(c). Meanwhile, the braze alloy was applied uniformly over contact regions between the trapezoidal honeycomb blocks and corrugated webs as well as face sheets and hybrid core. After assembly, the whole structure was welded together by the same brazing process above. The perforation process involved moving a laser beam to create the defined micro-perforations in the brazed sandwich beams, shown in Fig. 2(d).
2. Fabrication method and experimental measurements 2.1. Structure description The proposed perforated honeycomb-corrugation hybrid core sandwich structure consists of two face sheets of thickness t f and a 186
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Fig. 2. (a)-(d) Schematic of the manufacturing processes of sandwich beam with honeycomb-corrugation hybrid core; (e) sample beam without perforation; (f) sample beam with perforation.
Table 1 The featured geometrical dimensions of sandwich panels (mm). L
H
W
lh
th
lc
h
tc
300
19.8
55
5
0.2
20
17.8
0.5
2.3. Modal testing
o
tf
d1
d2
60
1
2.5
2.5
experimental equipment consisted of an impact hammer, a B&K type accelerometer, a charge amplifier, a clamping system and a LMS-TestLab modal analysis system as shown in Fig. 3. The experiments were carried out by multi-point excitation & single point measurement. Firstly, the specimens were fixed at one end by means of the clamping
Modal testing was performed on sandwich beams under clampedfree boundary condition. The first three natural frequencies and the corresponding mode shapes of sandwich beams were reported. The 187
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Fig. 3. Test setup of modal testing.
Fig. 4. FEM model of sandwich beam with honeycomb-corrugation hybrid core.
Fig. 5. The experimental and numerical frequency response functions (FRFs) of sandwich beams without micro-perforation and with micro-perforation.
Fig. 6. A comparison of first four frequency values between experimental datum and numerical datum for sandwich panels (a) without micro-perforation; (b) with micro-perforation. 188
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Fig. 7. A comparison of first four modal shapes between experimental datum and numerical datum for sandwich panels (a) without micro-perforation; (b) with micro-perforation.
Fig. 8. Effect of face sheet thickness on frequencies.
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Fig. 9. Effect of the face sheet thickness ratio of t f / h on frequency parameter.
system. Then twenty excitation points were evenly arranged on the specimens. The point-by-point excitation method was used to excite vibrations of the specimen by the impact hammer and the force history of the hammer impact is recorded by the force transducer connected with the hammer. Analogously, the vibration response of the specimen is detected by an acceleration transducer. Then the excitation and response signals are collected and processed by the dynamic signal analyzers. Therefore, the frequency response function (FRF) is obtained and the modal parameters including natural frequencies, modal shapes can be analyzed and induced by the post-processing with LMS-Test-Lab software. The average values of three tests were reported in the present study.
mesh convergence study was carried out to make the meshes fine enough to provide accurately numerical results. Total elements and nodes of the FEM models are about 361,680 elements and 183,280 nodes for sandwich beam without micro-perforation, and about 385,790 elements and 202,180 nodes for sandwich beam with micro-perforation. Linear perturbation analysis step called “Frequency” is conducted to obtain the natural frequencies and mode shapes based on the Lanczos eigensolver. In addition, dynamic responses of such sandwich beams were derived by using a mode superposition method. All parts are considered to be perfectly bonded together. The 304 stainless steel is considered in finite element model with Young's modulus E = 203 GPa, Poisson's ratio = 0.3, the density = 7900 kg/ m3 .
3. Finite element simulations
4. Results and discussion
The commercial finite element software ABAQUS 6.11 was used to predict the vibration characteristics of sandwich beam with honeycomb-corrugation hybrid core with the clamped-free boundary condition. The reduced integration triangular conventional shell (S3R) elements were employed in finite element model as shown in Fig. 4. A
4.1. Experimental modal analysis Fig. 5 reports the comparisons of frequency response functions (FRFs) of sandwich beams without micro-perforation and with microperforation in a range of first several predominant frequencies based on
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Fig. 10. First four mode shapes of micro-perforation sandwich beam with face sheet thickness th = 0.5 mm.
Fig. 11. Effect of the corrugated member thickness on frequencies.
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Fig. 12. Effect of the slenderness ratio tc / lc of corrugated member on frequencies.
the experimental test and numerical simulation. It can be found the agreement between each other was very good and minor difference. Compared with sandwich beam without micro-perforation, sandwich beam with micro-perforation has declined significantly in first several predominant frequencies. The reason is mainly that micro-perforation might reduce both the mass and structural stiffness of sandwich beam to a certain extent. Compared with the mass, the structural stiffness plays a dominant role on the frequency of sandwich beam with micro-perforation. As a result, the natural frequency of sandwich beams decrease.
with those of experimental tests as shown in Fig. 7. The first four mode shapes of specimen are plotted: (a) first bending, (b) second bending, (c) first torsion, (d) third bending. It is found that numerical investigation of its first third natural frequencies for sandwich beams is less 3% than those of experimental tests, and there is a little bit difference (an estimated maximum error of 6%) at its fourth natural frequency between numerical simulations and experimental tests.
4.2. Experimental-numerical model correlation
The effects of such geometry parameters as the face sheet thickness ratio of t f / h , the relative density of filling honeycomb h and the slenderness ratio, tc / lc , of corrugated member, on the first four natural frequency of sandwich beam are discussed further using the commercial finite element software ABAQUS. With material parameters fixed, the geometric parameters of the sandwich are varied. A dimensionless parameter, i = ip / iw , is proposed to better understand the effect of micro-perforation on natural frequencies of sandwich beams, where ip and iw are the natural frequency of the sandwich beam with micro-
4.3. Effect of geometric parameters
The first four natural frequencies of sandwich beam without microperforation and with micro-perforation have been obtained by numerical model and experimental tests in Fig. 6. The results obtained from the analysis of numerical model meet good agreement with ones from experimental tests. In this work, the bending and torsional modes (the coupling between the bending mode and the lateral mode) are considered. The first four mode shapes of numerical model are consistent
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Fig. 13. Effect of the relative density of filling honeycomb on frequencies.
perforation (d1 = d2 0 ) and the corresponding sandwich beam without micro-perforation. In order to keep the total mass used constant, perforated face sheets and corrugated webs in sandwich beams keep their mass constant respectively by varying the thicknesses of facesheets and corrugated webs in numerical simulation models (the heights of cores remain constant). For studying the effect of geometric parameters on frequencies, only the considered parameter is varied and others keep constant.
perforation diameter, which means that micro-perforation reduces the stiffness of sandwich structure, to a certain extent. Likewise, the first four frequency parameters decrease with the increase of the microperforation diameter as shown in Fig. 9. The effect of the face sheet thickness ratio on the third frequency parameter increases with increasing the micro-perforation diameter, and has a little sensitivity to the first and second and forth frequency parameters. In addition, the vibration responses on the first four mode shapes of sandwich beams (i.e.th = 0.5 mm) with various diameter of micro-perforation are presented in Fig. 10. It can be found that when the diameter of microperforation is less than or equal to 2.5 mm, the first, second and fourth mode shapes of sandwich beams correspond to bending mode and the third mode shape is a torsion mode. However, when the diameter of perforation is up to 3 mm, the first, third and fourth mode shapes of sandwich beam correspond to bending mode and the second mode becomes torsion mode (The effect of other geometry parameters on mode shapes of sandwich beam with varied perforation is similar, so we won't talk about this in other sections).
4.3.1. Effect of face sheet thickness on frequencies Fig. 8 and Fig. 9 show the effect of face sheet thickness and face sheet thickness ratio on the first four natural frequencies of sandwich beams with varied diameter of micro-perforation. As can be clearly observed from Fig. 8, the first four natural frequencies increase with increasing the face sheet thickness. This phenomenon can be attributed to the fact that face sheets play an important role in promoting the bending stiffness of sandwich beam [16]. The first four natural frequencies of sandwich beams decrease with the increase of the micro-
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Fig. 14. Effect of the relative density of filling honeycomb on frequency parameter
4.3.2. Effect of the thickness of corrugated member on frequencies Influence of the thickness of corrugated member upon first four natural frequencies of sandwich beams (the fixed core height is 17.8 mm) with varied diameter of micro-perforation is presented in Fig. 11. It can be concluded that the natural frequencies decrease as the thickness of corrugated member increases. As the thickness of corrugated member increase, both the mass and structural stiffness of sandwich beams increase. However, compared with the structural stiffness, the mass plays a more dominant role on the frequency parameters of sandwich beams [16]. As a result, the natural frequencies of sandwich beam decrease. It is also observed that the first four natural frequencies and frequency parameter of sandwich beams decrease with the increase of micro-perforation diameter. However, the slenderness ratio tc / lc of corrugated web has little impact on the frequency parameters as shown in Fig. 12. The reason is that the thickness of corrugated member has small contribution to structural stiffness, and the frequency parameter is independent on the mass of sandwich structure. As a result, the natural frequencies of sandwich beam are insensitive to the slenderness ratio of corrugated member.
.
4.3.3. Effect of the relative density of filling honeycomb on frequencies Fig. 13 shows the variation of natural frequencies with respect to the different relative densities of filling honeycombs. Contrary to the effect of the thickness of corrugated member on frequencies, the natural frequencies increase with the increase of the relative densities of filling honeycombs. Therefore, as the relative density of filling honeycomb increases, both the mass and structural stiffness of sandwich beam increase. Compared with the structural mass, the structural stiffness plays a dominant role on decreasing the frequency of sandwich beam. In addition, it is also observed that the first four frequency parameter, , of sandwich beams decreases with increasing the diameter of microperforation in Fig. 14. 4.4. The effect of the configuration of micro-perforation on frequencies Investigations have been carried out to determine the relationship of variation of frequencies with perforation diameters for various the configuration of micro-perforation in Fig. 15 and Fig. 16. As can be seen in Fig. 15, the values of the natural frequencies decrease with
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Fig. 15. Effect of the configuration of micro-perforation on frequencies. Note: d1 = 0, d2 = 0 , d1 0, d2 = 0 and d1 0, d1 = d2 refer to sandwich beams without micro-perforation, sandwich beams with perforation of the top face sheet and corrugated member, sandwich beams with perforation of the top face sheet, bottom face sheet and corrugated member, respectively.
increasing the diameter of micro-perforation for all modes of beams. In the corresponding order, it is noted that the frequencies of sandwich beam with d1 = 0, d2 = 0 is the maximum, and the frequencies of sandwich beam with d1 0, d1 = d2 is minimal. Moreover, when the micro-perforation ratio is less than 0.1, the frequency parameters are insensitive to the configuration of micro-perforation. Subsequently, the frequency parameters decrease with the increase of the micro-perforation ratio. Meanwhile, the frequency parameters of sandwich beams with d1 0, d2 = 0 are higher than those of sandwich beams with d1 0, d1 = d2 as shown in Fig. 16.
that the natural frequencies of sandwich beam with micro-perforation diameter of 2.5 mm decrease significantly. To systematically reveal the effect of micro-perforation on natural frequencies of sandwich beam, a dimensionless frequency parameter is proposed. It is found that the frequency parameter of sandwich panels decrease near-linearly with the increase of micro-perforation ratio. Furthermore, the configuration of micro-perforation has greater impact on the frequency parameter, the effect becomes more pronounced with the increase of micro-perforation diameter. However, the face sheet thickness ratio, the slenderness ratio of corrugated member and the relative density of filling honeycomb have little impact on the frequency parameter. Obtained results can be utilized to determine the changes in dynamic characteristics of sandwich beam with micro-perforation with specified diameters and configuration. This work is helpful for researchers in design for the practical application of the honeycomb-corrugation hybrid-cored sandwich structures.
5. Conclusions Modal characteristics of micro-perforated sandwich beams with honeycomb-corrugation hybrid cores have been studied by using modal test and finite element simulation. Experimental investigation showed
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Fig. 16. Effect of the configuration on frequency parameters.
Acknowledgements
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Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Full length article
Modal characteristics of micro-perforated sandwich beams with square honeycomb-corrugation hybrid cores: A mixed experimental-numerical study
T
Zhi-jia Zhanga,b,c,e, Qian-cheng Zhanga,b,c, , Fei-chen Lia,c, Jian-wen Yangd, Ji-wu Liud, ⁎ Zhan-yi Liud, Feng Jina,c, ⁎
a
State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an Jiaotong University, Xi’an 710049, PR China Key Laboratory of Intense Dynamic Loading and Effect, Xi'an 710024, China MOE Key Laboratory for Multifunctional Materials and Structures, Xi'an Jiaotong University, Xi'an 710049, PR China d Science and Technology on Liquid Rocket Engine Laboratory, Xi'an Aerospace Propulsion Institute, Xi'an 710100, PR China e Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA b c
ARTICLE INFO
ABSTRACT
Keywords: Sandwich structure Honeycomb-corrugation hybrid core Micro-perforation Vibration
Modal performance of micro-perforated sandwich beams with honeycomb-corrugation hybrid cores was investigated. Finite element methods and modal analysis techniques have been used to predict their vibration characteristics (i.e. their natural frequencies and mode shapes). It is shown that the natural frequencies of sandwich beam with micro-perforation are slightly lower than those of corresponding order of sandwich beam without micro-perforation based on the experimental and three-dimensional finite element calculated results. To reveal the effect of micro-perforation diameters on the natural frequencies of sandwich beams, a dimensionless frequency parameter was proposed. The results demonstrate that the frequency parameter of sandwich panels decrease near-linearly with the increase of micro-perforation ratio. In addition, the frequency parameter is sensitive not to such parameters as the face sheet thickness ratio, the slenderness ratio of corrugated member and the relative density of filling honeycomb, but to the configuration of micro-perforation.
1. Introduction Due to their multi-functional characteristic, periodic cellular metal sandwich structures have been widely used in such engineering application fields as aerospace, high speed transportation, submarine [1,2], and so forth. Various types of topological configurations of sandwich cores have been developed [3–7], however, there are still a large amount of technology-based breakthrough innovations for honeycombs, corrugations and lattice truss structures at millimeter scale. In order to improve their spatial and structural efficiency, many hybrid core structures have been constructed to improve mechanical properties of sandwich structures. The hybrid sandwich cores have been proposed, including aluminum foam-filled metallic tubes [8–11] and metal foam-filled metallic corrugation [12–14]. It is surprising that metal foam filling could greatly increase the strength and energy absorption of sandwich structures. The completely surprising results are attributed to an underlying mechanism: the lateral support to the core member not by weakly but strongly filled foam, altering the
⁎
deformation modes and considerably delaying core member buckling. However, the weight and space efficiency of new composite structures has not fully been exploited. The main reason is that filled foams in these structures have been enhanced next to nothing by such ordered cellular materials as metallic corrugation. In order to further improve structural efficiency in composite cellular structures, new idea about honeycomb-filled corrugated core has been proposed [15], which significantly increases the weight and space efficiency of sandwich structures by mutual enhancement of corrugated member and filled honeycomb, i.e. the mutual constraint of corrugated member and honeycomb cell walls against buckling changes the deformation modes of the corresponding structures. Zhang et al. [16] investigated the vibration performance of sandwich beams with honeycomb-corrugation hybrid cores. It is found that the filling honeycombs have an effect on weakening the anisotropy of the stiffness and suppressing the local mode shape for sandwich beam. Similarly, Yin et al. [17] proposed a hollow lattice truss reinforced honeycombs (LTRHs), which were designed to combine lattice topology with square honeycombs. The crush
Corresponding authors at: State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an Jiaotong University, Xi’an 710049, PR China. E-mail addresses: [email protected] (Q.-c. Zhang), [email protected] (F. Jin).
https://doi.org/10.1016/j.tws.2019.01.004 Received 21 August 2018; Received in revised form 30 November 2018; Accepted 6 January 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.
Thin-Walled Structures 137 (2019) 185–196
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Fig. 1. Schematic of perforated honeycomb-corrugation hybrid core sandwich structure, which is composed of micro-perforated panels as facesheets and a honeycomb-corrugation hybrid as core. The corrugation is also perforated, just beneath the hole in top facesheet.
force efficiency, energy absorption and specific energy absorption capacity of LTRHs were found to be superior to both square honeycombs and hollow lattice structures. Besides the outstanding mechanical performance, sandwich structures with micro-perforations also are well-known devices used for the attenuation of acoustic levels. Perforated pores in the face plates of the sandwich panels can provide effective sound absorption as micro-perforated panel layers, while the backed plates and core structures can act as sound insulation barriers. Meng et al. [18] investigated the low frequency sound absorption coefficient and sound transmission loss of corrugated sandwich panels with different perforation configurations by numerical calculation and experiment. Through the comparisons between the classical corrugated sandwich panels (without perforations) and corrugated sandwich panels with face plate perforations, it is proved out that the face plate perforations are effective in improving the sound absorption coefficient (SAC) and sound transmission loss (STL) at low frequencies. Sakagami and co-authors [19] found that the honeycomb not only stiffened the micro-perforated panel, but also improved its low-frequency sound absorption performance. More recently, Tang et al. [20,21] found that honeycomb-corrugation hybrid cored sandwich panel with heterogeneously perforated face sheet and perforated corrugation as a new kind of sound absorber, as well as lightweight load-bearing structure, behaved better in low-frequency sound absorption. In addition, engineering design often requires structural discontinuities such as perforation or cutoffs for connection, joint, access and fluid flow. In general, the presence of perforations leads to considerable reduction in load bearing capacities [22,23]. Therefore, the influence and sensitivity of perforation on structural mechanical behaviors signify a critical issue in practice. Currently, no reference has been reported to study the dynamic characteristics of micro-perforated sandwich structures with hybrid cores. In this study, the effects of micro-perforation on the dynamic behaviors of microperforated sandwich beams with square honeycomb-corrugation hybrid cores are firstly studied to guide the relevant structural analysis and design of micro-perforated in actual applications with a mixed experimental-numerical study. The influence of key geometrical parameters on their vibration performance is also explored.
honeycomb-corrugation hybrid core of thickness h , as shown in Fig. 1. Periodically distributed micro-perforations were introduced in both facesheets and corrugation. The honeycomb core has a square crosssection with an inner side length lh and a wall thickness th . The corrugation micro-perforated at vertical direction has an inclination angle and a web thickness tc . The thicknesses and perforation diameters of top face sheet and down face sheet are t f , d1, and d2 (d1 0, d 2=0 or d1 0, d 2=d1 or d1 = 0, d2 = 0 ), respectively. 2.2. Fabrication of sandwich panel Consider a sandwich panel having metallic honeycomb-corrugation hybrid core, with both of its face sheets and core made of 304 stainless steel sheet. For reference, the sandwich panels with honeycomb core and empty corrugated core were also considered. The honeycombcorrugation hybrid core sandwich panels were fabricated in a three-step process summarized schematically in Fig. 2. All samples (the featured geometrical dimensions as shown in Table 1) were manufactured by folding method, electro-discharge machining (EDM) and brazing as described below. Firstly, cross-slots were cut by EDM and then the square-honeycomb was assembled as sketched in Fig. 2(a). The clearance of 5 µm between sheet and slot facilitated assembly while providing a sufficiently tight fit to assure stability. The braze of the assembled sheets kept in step with pre-filling alloy (Ni–Cr25-P10 (wt%)) powder slurry into the gap. For the brazing process, the structure was placed in a dry oven and held for 2 h at 60 oC to remove water from surface. The samples in vacuum furnace were heated at the rate of 10 o C /min to 350 oC , kept for 30 min to volatilize the polymer binder. The brazing temperature was set at 1020 oC and held for 20 min before being cooled inside the furnace to room temperature. Then trapezoidal honeycomb blocks precisely cut from a brazed square honeycombs using EDM. Secondly, the corrugated sheet was produced using the folding method as sketched in Fig. 2(b). Then the molded corrugated sheets were cut into several corrugated webs using EDM. Thirdly, after being cut down to size, the trapezoidal honeycomb blocks and corrugated web were assembled together with face sheets to create a sandwich beam with honeycomb-corrugation hybrid core, shown in Fig. 2(c). Meanwhile, the braze alloy was applied uniformly over contact regions between the trapezoidal honeycomb blocks and corrugated webs as well as face sheets and hybrid core. After assembly, the whole structure was welded together by the same brazing process above. The perforation process involved moving a laser beam to create the defined micro-perforations in the brazed sandwich beams, shown in Fig. 2(d).
2. Fabrication method and experimental measurements 2.1. Structure description The proposed perforated honeycomb-corrugation hybrid core sandwich structure consists of two face sheets of thickness t f and a 186
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Fig. 2. (a)-(d) Schematic of the manufacturing processes of sandwich beam with honeycomb-corrugation hybrid core; (e) sample beam without perforation; (f) sample beam with perforation.
Table 1 The featured geometrical dimensions of sandwich panels (mm). L
H
W
lh
th
lc
h
tc
300
19.8
55
5
0.2
20
17.8
0.5
2.3. Modal testing
o
tf
d1
d2
60
1
2.5
2.5
experimental equipment consisted of an impact hammer, a B&K type accelerometer, a charge amplifier, a clamping system and a LMS-TestLab modal analysis system as shown in Fig. 3. The experiments were carried out by multi-point excitation & single point measurement. Firstly, the specimens were fixed at one end by means of the clamping
Modal testing was performed on sandwich beams under clampedfree boundary condition. The first three natural frequencies and the corresponding mode shapes of sandwich beams were reported. The 187
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Fig. 3. Test setup of modal testing.
Fig. 4. FEM model of sandwich beam with honeycomb-corrugation hybrid core.
Fig. 5. The experimental and numerical frequency response functions (FRFs) of sandwich beams without micro-perforation and with micro-perforation.
Fig. 6. A comparison of first four frequency values between experimental datum and numerical datum for sandwich panels (a) without micro-perforation; (b) with micro-perforation. 188
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Fig. 7. A comparison of first four modal shapes between experimental datum and numerical datum for sandwich panels (a) without micro-perforation; (b) with micro-perforation.
Fig. 8. Effect of face sheet thickness on frequencies.
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Fig. 9. Effect of the face sheet thickness ratio of t f / h on frequency parameter.
system. Then twenty excitation points were evenly arranged on the specimens. The point-by-point excitation method was used to excite vibrations of the specimen by the impact hammer and the force history of the hammer impact is recorded by the force transducer connected with the hammer. Analogously, the vibration response of the specimen is detected by an acceleration transducer. Then the excitation and response signals are collected and processed by the dynamic signal analyzers. Therefore, the frequency response function (FRF) is obtained and the modal parameters including natural frequencies, modal shapes can be analyzed and induced by the post-processing with LMS-Test-Lab software. The average values of three tests were reported in the present study.
mesh convergence study was carried out to make the meshes fine enough to provide accurately numerical results. Total elements and nodes of the FEM models are about 361,680 elements and 183,280 nodes for sandwich beam without micro-perforation, and about 385,790 elements and 202,180 nodes for sandwich beam with micro-perforation. Linear perturbation analysis step called “Frequency” is conducted to obtain the natural frequencies and mode shapes based on the Lanczos eigensolver. In addition, dynamic responses of such sandwich beams were derived by using a mode superposition method. All parts are considered to be perfectly bonded together. The 304 stainless steel is considered in finite element model with Young's modulus E = 203 GPa, Poisson's ratio = 0.3, the density = 7900 kg/ m3 .
3. Finite element simulations
4. Results and discussion
The commercial finite element software ABAQUS 6.11 was used to predict the vibration characteristics of sandwich beam with honeycomb-corrugation hybrid core with the clamped-free boundary condition. The reduced integration triangular conventional shell (S3R) elements were employed in finite element model as shown in Fig. 4. A
4.1. Experimental modal analysis Fig. 5 reports the comparisons of frequency response functions (FRFs) of sandwich beams without micro-perforation and with microperforation in a range of first several predominant frequencies based on
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Fig. 10. First four mode shapes of micro-perforation sandwich beam with face sheet thickness th = 0.5 mm.
Fig. 11. Effect of the corrugated member thickness on frequencies.
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Fig. 12. Effect of the slenderness ratio tc / lc of corrugated member on frequencies.
the experimental test and numerical simulation. It can be found the agreement between each other was very good and minor difference. Compared with sandwich beam without micro-perforation, sandwich beam with micro-perforation has declined significantly in first several predominant frequencies. The reason is mainly that micro-perforation might reduce both the mass and structural stiffness of sandwich beam to a certain extent. Compared with the mass, the structural stiffness plays a dominant role on the frequency of sandwich beam with micro-perforation. As a result, the natural frequency of sandwich beams decrease.
with those of experimental tests as shown in Fig. 7. The first four mode shapes of specimen are plotted: (a) first bending, (b) second bending, (c) first torsion, (d) third bending. It is found that numerical investigation of its first third natural frequencies for sandwich beams is less 3% than those of experimental tests, and there is a little bit difference (an estimated maximum error of 6%) at its fourth natural frequency between numerical simulations and experimental tests.
4.2. Experimental-numerical model correlation
The effects of such geometry parameters as the face sheet thickness ratio of t f / h , the relative density of filling honeycomb h and the slenderness ratio, tc / lc , of corrugated member, on the first four natural frequency of sandwich beam are discussed further using the commercial finite element software ABAQUS. With material parameters fixed, the geometric parameters of the sandwich are varied. A dimensionless parameter, i = ip / iw , is proposed to better understand the effect of micro-perforation on natural frequencies of sandwich beams, where ip and iw are the natural frequency of the sandwich beam with micro-
4.3. Effect of geometric parameters
The first four natural frequencies of sandwich beam without microperforation and with micro-perforation have been obtained by numerical model and experimental tests in Fig. 6. The results obtained from the analysis of numerical model meet good agreement with ones from experimental tests. In this work, the bending and torsional modes (the coupling between the bending mode and the lateral mode) are considered. The first four mode shapes of numerical model are consistent
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Fig. 13. Effect of the relative density of filling honeycomb on frequencies.
perforation (d1 = d2 0 ) and the corresponding sandwich beam without micro-perforation. In order to keep the total mass used constant, perforated face sheets and corrugated webs in sandwich beams keep their mass constant respectively by varying the thicknesses of facesheets and corrugated webs in numerical simulation models (the heights of cores remain constant). For studying the effect of geometric parameters on frequencies, only the considered parameter is varied and others keep constant.
perforation diameter, which means that micro-perforation reduces the stiffness of sandwich structure, to a certain extent. Likewise, the first four frequency parameters decrease with the increase of the microperforation diameter as shown in Fig. 9. The effect of the face sheet thickness ratio on the third frequency parameter increases with increasing the micro-perforation diameter, and has a little sensitivity to the first and second and forth frequency parameters. In addition, the vibration responses on the first four mode shapes of sandwich beams (i.e.th = 0.5 mm) with various diameter of micro-perforation are presented in Fig. 10. It can be found that when the diameter of microperforation is less than or equal to 2.5 mm, the first, second and fourth mode shapes of sandwich beams correspond to bending mode and the third mode shape is a torsion mode. However, when the diameter of perforation is up to 3 mm, the first, third and fourth mode shapes of sandwich beam correspond to bending mode and the second mode becomes torsion mode (The effect of other geometry parameters on mode shapes of sandwich beam with varied perforation is similar, so we won't talk about this in other sections).
4.3.1. Effect of face sheet thickness on frequencies Fig. 8 and Fig. 9 show the effect of face sheet thickness and face sheet thickness ratio on the first four natural frequencies of sandwich beams with varied diameter of micro-perforation. As can be clearly observed from Fig. 8, the first four natural frequencies increase with increasing the face sheet thickness. This phenomenon can be attributed to the fact that face sheets play an important role in promoting the bending stiffness of sandwich beam [16]. The first four natural frequencies of sandwich beams decrease with the increase of the micro-
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Fig. 14. Effect of the relative density of filling honeycomb on frequency parameter
4.3.2. Effect of the thickness of corrugated member on frequencies Influence of the thickness of corrugated member upon first four natural frequencies of sandwich beams (the fixed core height is 17.8 mm) with varied diameter of micro-perforation is presented in Fig. 11. It can be concluded that the natural frequencies decrease as the thickness of corrugated member increases. As the thickness of corrugated member increase, both the mass and structural stiffness of sandwich beams increase. However, compared with the structural stiffness, the mass plays a more dominant role on the frequency parameters of sandwich beams [16]. As a result, the natural frequencies of sandwich beam decrease. It is also observed that the first four natural frequencies and frequency parameter of sandwich beams decrease with the increase of micro-perforation diameter. However, the slenderness ratio tc / lc of corrugated web has little impact on the frequency parameters as shown in Fig. 12. The reason is that the thickness of corrugated member has small contribution to structural stiffness, and the frequency parameter is independent on the mass of sandwich structure. As a result, the natural frequencies of sandwich beam are insensitive to the slenderness ratio of corrugated member.
.
4.3.3. Effect of the relative density of filling honeycomb on frequencies Fig. 13 shows the variation of natural frequencies with respect to the different relative densities of filling honeycombs. Contrary to the effect of the thickness of corrugated member on frequencies, the natural frequencies increase with the increase of the relative densities of filling honeycombs. Therefore, as the relative density of filling honeycomb increases, both the mass and structural stiffness of sandwich beam increase. Compared with the structural mass, the structural stiffness plays a dominant role on decreasing the frequency of sandwich beam. In addition, it is also observed that the first four frequency parameter, , of sandwich beams decreases with increasing the diameter of microperforation in Fig. 14. 4.4. The effect of the configuration of micro-perforation on frequencies Investigations have been carried out to determine the relationship of variation of frequencies with perforation diameters for various the configuration of micro-perforation in Fig. 15 and Fig. 16. As can be seen in Fig. 15, the values of the natural frequencies decrease with
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Fig. 15. Effect of the configuration of micro-perforation on frequencies. Note: d1 = 0, d2 = 0 , d1 0, d2 = 0 and d1 0, d1 = d2 refer to sandwich beams without micro-perforation, sandwich beams with perforation of the top face sheet and corrugated member, sandwich beams with perforation of the top face sheet, bottom face sheet and corrugated member, respectively.
increasing the diameter of micro-perforation for all modes of beams. In the corresponding order, it is noted that the frequencies of sandwich beam with d1 = 0, d2 = 0 is the maximum, and the frequencies of sandwich beam with d1 0, d1 = d2 is minimal. Moreover, when the micro-perforation ratio is less than 0.1, the frequency parameters are insensitive to the configuration of micro-perforation. Subsequently, the frequency parameters decrease with the increase of the micro-perforation ratio. Meanwhile, the frequency parameters of sandwich beams with d1 0, d2 = 0 are higher than those of sandwich beams with d1 0, d1 = d2 as shown in Fig. 16.
that the natural frequencies of sandwich beam with micro-perforation diameter of 2.5 mm decrease significantly. To systematically reveal the effect of micro-perforation on natural frequencies of sandwich beam, a dimensionless frequency parameter is proposed. It is found that the frequency parameter of sandwich panels decrease near-linearly with the increase of micro-perforation ratio. Furthermore, the configuration of micro-perforation has greater impact on the frequency parameter, the effect becomes more pronounced with the increase of micro-perforation diameter. However, the face sheet thickness ratio, the slenderness ratio of corrugated member and the relative density of filling honeycomb have little impact on the frequency parameter. Obtained results can be utilized to determine the changes in dynamic characteristics of sandwich beam with micro-perforation with specified diameters and configuration. This work is helpful for researchers in design for the practical application of the honeycomb-corrugation hybrid-cored sandwich structures.
5. Conclusions Modal characteristics of micro-perforated sandwich beams with honeycomb-corrugation hybrid cores have been studied by using modal test and finite element simulation. Experimental investigation showed
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Fig. 16. Effect of the configuration on frequency parameters.
Acknowledgements
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