Study on dynamic characteristic of closedcell aluminum foam
Qiang Lei^{1} , Junlong Ren^{2} , Hongyi Ren^{3} , Hongxiao Chao^{4} , Wenbin Du^{5}
^{1, 2, 3, 4, 5}Northwest Institute of Mechanical and Electrical Engineering, 712099, Xianyang, P. R. China
^{1}Corresponding author
Vibroengineering PROCEDIA, Vol. 28, 2019, p. 142147.
https://doi.org/10.21595/vp.2019.20998
Received 7 September 2019; accepted 19 September 2019; published 19 October 2019
JVE Conferences
Closedcell aluminum foam has been widely used in aerospace, rail transit and mechanical for its outstanding performance. But for a long time, the research on its vibration damping performance is only limited to the material damping test, there are relatively few studies on its dynamic characteristics. In this paper, we studied the relationship between dynamic characteristic and feature parameters. Modal assurance criterion and Finite element method were used to verify the accuracy of experimental model. It turned out that the average pore diameter of closedcell aluminum foam conforms to Gaussian distribution. The modal analysis method can be used in the research of dynamic characteristic of closedcell aluminum foam. Its damping ratio showed increasing trend with the increase of porosity, natural frequency and the decrease of mean pore size. Each order natural frequency increases along with the increase of porosity.
 Modal analysis method can effectively analyze the dynamic characteristics of closedcell aluminum foam.
 The natural frequency of closedcell aluminum foam is negatively correlated to porosity and it increases with the increase of sample height when the axial dimension smaller than the radial dimension.
 FEM can be used to study the dynamic characteristic of closedcell aluminum foam and compared with experiment it has some advantages.
Keywords: closedcell aluminum foam, feature parameters, dynamic characteristic, MAC, FEM.
1. Introduction
Closedcell aluminum foam is a new kind of multifunctional metal porous material, which has the characteristics of both metal and bubbles. It’s becoming one of the hot research topics in the current metal material field for its excellent mechanical and physical properties [1], such as low density, high specific strength, high specific stiffness, high energy absorption, high damping vibration, phonics, electromagnetic shielding and its multifunction compatible [2]. One of its main uses is as damping material; it can effectively reduce the vibration and noise that has a great significance to improve the accuracy and life of equipment as well as the working environment.
Now, some aspects of closedcell aluminum foam have been deeply researched by experiment and FEM, such as the production process [3], dynamic and static compression [4] and energy absorption [5]. For its dynamic characteristics, especially the vibration damping performance mainly concentrated in the damping test referenced the standard of ASTM E75605. Han et al. [6] and Liu et al. [7] have researched the factors influencing the damping property of closedcell aluminum foam and damping mechanism. Golovin and Sinning [8] have researched its mechanical damping in a wide range of deformation amplitude. L. Dahil et al. [9] have studied the relationship between density and damping ratio of foamed aluminum using modal analysis method.
In this article, first, describe the analysis of feature parameters, adopted a new and simple method to analyze the porosity, characteristics of pore. Then using the modal analysis method to research the relationship between dynamic characteristic and feature parameters. Modal assurance criterion (MAC) and coherence function were used to verify the correctness of the model. We also compared the results by Finite element method (FEM). Last, the conclusions.
2. Experimental procedures
2.1. Feature parameters
The samples were made by melt foaming method and incised by wireelectrode cutting method. The sample size, outside diameter is Ø440 mm, inside diameter is Ø280 mm and the height is unequal.
Porosity is an important parameter to describe the closedcell aluminum foam. The measurement method of porosity is divided into weighing method and microscopic method. For its simple and high precision, weighing method is commonly used. According to the dimension and weight of the sample, using the Eq. (1) to calculate the porosity:
In which, $m$ and $V$ are the mass and volume of the sample, respectively, ${\rho}_{s}$ is the density of matrix material.
The measurement of pore parameters mainly includes direct method and indirect method that using the software to analysis the surface topography of the sample [19]. In this article, we applied a new and simple method to measure the pore parameters. It has the advantages of low cost and high precision. The main equipment is digital camera and the main software is Photoshop, Matlab and ImagePro Plus. The basic procession is shown in Fig. 1.
Fig. 1. The basic procession
2.2. Dynamic characteristic test
The dynamic characteristic was performed by experimental modal analysis and FEM. As shown in Fig. 2, the whole testing system composed by four parts; suspension part, shaking part, testing part and software analysis part. The suspension part includes rigid bracket and elastic soft cord to simulate the free boundary condition. The shaking part is impact hammer which is commonly used in the singleinput singleoutput (SISO) modal analysis to produce pulse signal. The testing part is PCB accelerometer and highspeed data acquisition system. The software analysis part is Virtual lab/Modal Analysis software.
Fig. 2. The testing system
In the modeling process, the model is simplified to octagon with 48 nodes and ignores the mounting holes, as shown in Fig. 3. In which point 1:1 to 1:24 is the master node and the rest is the slave node. The PCB acceleration sensor is installed in the 1:3 points. During the experiment, using the force hammer which installed rigid head to beat the other master node, every node beats at least three times. The related experimental parameters as shown in Table 1.
Fig. 3. The geometric model
Table 1. Related experimental parameters
Force sensor sensitivity

PCB sensor sensitivity

Bandwidth

Resolution

Averages

Response windowing

4 PC/N

101.7 mv/g

4096 Hz

1 Hz

3

Exponential

MAC was used to test the linear independence of each order. MAC is a good tool to evaluate the modal vector space angle. It can be expressed as follows:
where ${\phi}_{i}$and ${\phi}_{j}$ are the corresponding freedom of the $i$ and $j$ order calculation mode, respectively. The smaller of the offdiagonal matrix is the better of the independence of each calculation mode.
In this article, we also used the ABAQUS software to give a modal analysis in order to better verify the testing results. The basic steps of modal analysis in ABAQUS include modeling, select the analysis type, set the corresponding parameters, applying the boundary conditions, solve and results post processing. The parameters used in the simulation are shown in Table 2.
Table 2. Parameters used in FEM
Height

Porosity

Density (kg/m^{3})

Young’s modulus × 10^{6}^{}(Pa)

Poisson’s ratio

50

85.8 %

380

428.61

0.28

50

86.4 %

370

401.14

0.28

80

81.9 %

490

603.07

0.28

80

85.0 %

400

484

0.28

120

83.9 %

430

524.36

0.28

120

85.2 %

400

454.67

0.28

150

84.6 %

420

502.18

0.28

150

86.0 %

380

451.43

0.28

3. Results and discussion
3.1. Feature parameters analysis
Table 3 shows the feature parameters of aluminum foam. It can be found that the porosity is among 81.9 % to 86.42 %, the relative density is among 0.136 to 0.181, the average diameter is among 2.42 to 3.23 mm.
Table 3. Parameters of aluminum foam
No.

Height / mm

Mass / g

Porosity

Diameter / mm

Relative density

1

50

1761

85.79 %

3.23

0.142

2

50

1665

86.42 %

2.65

0.136

3

80

2930

85.01 %

2.77

0.150

4

80

3540

81.90 %

2.42

0.181

5

120

4694

83.92 %

2.44

0.161

6

120

4334

85.21 %

2.72

0.148

7

150

5629

84.61 %

2.70

0.154

8

150

5132

85.95 %

3.09

0.140

3.2. Experimental modal analysis
3.2.1. Results and validation
In this article, we analyzed the first six orders. As shown in Fig. 4 is the MAC of each sample. Through the MAC it can be found that except the diagonal correlation is 100 %, the other offdiagonal correlation is fewer than 10 % and mostly fewer than 5 %. According the above criteria of MAC, it can be found that the node configuration is reasonable, and each order modal has higher orthogonality.
Fig. 4. Modal assurance criterion
a) Sample one
b) Sample two
3.2.2. Discussion of experimental modal results
(1) Relationship between damping ratio and porosity.
As shown in Fig. 5 is the result of damping ratio of each sample. It can be seen from Fig. 5 each order damping ratio increases with the increase of porosity, the damping ratio and porosity is positively correlated. But in some orders the damping ratio decreases with the increase of porosity for the manufacturing defect.
Fig. 5. Relationship between damping ratio and porosity
a) Sample one
b) Sample two
(2) Relationship between natural frequency and porosity.
Material’s vibration damping performance and damping ratio are closely related, but there is another important factor that affects the vibration damping performance is the natural frequency of the sample. As shown in Fig. 6, it can be found that the natural frequency of each order decreases with the increase of porosity and increases with the increase of height of the sample. In this experiment, the sample’s axial dimension is less than the radial dimension, the stiffness of axial increases greater than the increase of mass, its natural frequency is increased with the increase of height.
Fig. 6. Relationship between natural frequency and porosity
a) Sample one
b) Sample two
3.3. Comparison between experiment and FEM
By comparing the simulation results and experimental results can make a mutual authentication between the two methods. The results are shown in the Table 4. As it can be seen the simulation results can be good fit with the testing results. So, in the next study, we can use the FEM to study its dynamic characteristic.
4. Conclusions
First, modal analysis method can effectively analyze the dynamic characteristics of closedcell aluminum foam. Second, the natural frequency of closedcell aluminum foam is negatively correlated to porosity and it increases with the increase of sample height when the axial dimension smaller than the radial dimension.
Third, FEM can be used to study the dynamic characteristic of closedcell aluminum foam and compared with experiment it has some advantages; it can make a further optimization analysis of the sample shape by FEM.
Table 4. Comparison between FEM and experiment
Height

Porosity

Item

First order
frequency

Second order
frequency

Third order
frequency

Forth order
frequency

Fifth order
frequency

Sixth order
frequency

50

0.8579

Test value

412.521

467.224

1156.314

1569.274

1991.156

2275.407

FEM value

441.43

464.18

1172

1558.6

2041

2259.5


Error

0.07

–0.0065

0.014

–0.0068

0.025

–0.0069


50

0.8642

Test value

403.873

465.858

1147.42

1550.125

1924.597

2261.659

FEM value

432.93

445.16

1144.3

1572.7

1957.4

2166.9


Error

0.072

–0.044

–0.0027

0.015

0.017

–0.042


80

0.819

Test value

624.184

715.786

1652.535

2046.163

2352.014

3219.411

FEM value

627.88

712.81

1855.4

2084.4

2312.3

3259.3


Error

0.0059

–0.0042

0.123

0.019

–0.0017

0.012


80

0.8501

Test value

611.275

694.411

1614.625

2016.47

2312.069

3153.054

FEM value

635.52

649.99

1607.7

2109.8

2314.9

3123.4


Error

0.04

–0.064

–0.0043

0.046

0.0012

–0.0094


120

0.8392

Test value

771.856

871.648

1838.794

2316.829

2412.699

3698.126

FEM value

784.96

869.53

1871

2407.3

2423.7

3701.1


Error

0.017

–0.0024

0.018

0.039

0.0046

0.001


120

0.8521

Test value

745.5

839.099

1751.82

2244.576

2335.451

3574.857

FEM value

767.51

849.5

1806.4

2324.2

2340

3598.4


Error

0.03

0.013

0.03

0.035

0.002

0.007


150

0.8461

Test value

805.083

918.122

1790.455

2343.469

2972.636

3238.733

FEM value

789.37

914.77

1770

2298.6

3034.9

3203.2


Error

–0.02

–0.004

–0.01

0.019

0.021

–0.011


150

0.8595

Test value

794.044

895.565

1766.027

2315.208

2931.719

3118.055

FEM value

786.83

911.79

1764.3

2291.2

3025.1

3137.9


Error

–0.01

0.018

–0.001

–0.01

0.032

0.007

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