Design of double-layered framed plate with equivalent impedance

. To get clear the vibration mechanism of sandwith shell using the experiment method, a double-layered plate is designed for the sandwich shell by matching the equivalent impedance. Numerical analysis is conducted, and In 10Hz-4000Hz, the differenceof impedance from double-layered plate and sandwich shell should be less than 10%. A new design is given for the double-layered plate, and which is quite simple for construction. Then, the double-layered plate could be applied to conduct series vibration tests to get clear the influence of sandwich core material on structural vibration, while it could also reduce the costs of money and construction time.


Introduction
Sandwich structures have wild application in engineering, double-layered plate or shell is one kind of sandwich structures, it consists of two plates, the up and down plate and the sandwich cores is frame. Many works have been done about the mechanic, vibration and dynamic performance of sandwich structures. Chen et.al. [1] Analyzed the free vibration of sandwich structures with different lattice type. Lu et.al. [2] Studied the influence of random damage on dynamic performance sandwich plate with metal core, with the application of finite element model. Wu et.al. [3] Conducted system work for mechanic performance of very light sandwich plate with frictional pyramid core. Besides, sandwich structure could be applied to reduce structural vibration and noise level by selecting reasonable core material [4][5]. For a sandwich shell, the final design is to be made for good vibration and noise reduction. The best way to select suitable sandwich core for the shell is to construct a series of shell model filled with different sandwich cores. But this will cost much money and time, due to the difficulty of construction for sandwich shell. Therefore, a double-layered plate is designed with equivalent impedance to the sandwich shell, so as to lay foundation for future work on sandwich cores selection. Since the time and money cost for doublelayered plate is relatively much smaller. In this work, to study the influence of sandwich core on vibration of sandwich cylindrical shell, a double-layered plate design is put forward by using the impedance equivalent method. In the first step, three different doublelayered plate designs are given for the sandwich cylindrical shell, and the structural impedance are calculated and compared with that of the sandwich cylindrical shell. With comparative analysis, a good double-layered plate design is selected, which will lay foundation for future work to study the influence of sandwich core on vibration of sandwich cylindrical shell.

Target model
The sandwich cylindrical shell is the target model here, and the geometry is shown in Fig.1. The shell is of length 690mm, thickness of the outside plate is 3.2mm, the frame is 2mm, and the inner plate is 2.7mm.

Material
The material of sandwich cylindrical shell is titanium alloy, elastic module is 108GPa, density is 4510kg/m 3 , the Poisson's ratio is 0.34. The material of double-layered plate is steel, elastic module is 210GPa, density is 7850kg/m 3 , the Poisson's ratio is 0.3.

Criterion
During the numerical simulation, the difference of impedance from double-layered plate and sandwich cylindrical shell should be less than 10% in frequency range, 10Hz-4000Hz.

Multi-plans for double-layered plate 1) Plate 1
Structural parameters o plan 1 is the same to those of sandwich cylindrical shell, so thick of the up plate is 3.2mm, the down plate is 2.7mm, and the frame and plates at the ends are 2mm.
2) Plate 2 Some structural parameters in plan 2 has changed for easy construction, thickness for the up and down plate is 3mm, the frame and plates at the ends are 2mm. The other parameters are the same with the sandwich shell.
3) Plate 3 Some structural parameters in plan 3 is different, thickness for the up plate is 2mm, the down plate is 3mm, the frame and plates at the ends are 2mm. The other parameters are the same with the sandwich shell.

Equivalent impedance analysis process
1) Construct the numerical model of sandwich shell and double-layered plate with the CATIA software, according to the parameters in plans.
2) Numerical analysis pre-process, such as meshing, boundary condition setting, using the Hypermesh software.
3) Harmonic response analysis, using the LMS virtual.Lab software. 4) Numerical results process, obtain the acceleration response and impedance of shell and plates. 5) Comparative analysis on impedance from different plans to the sandwich shell, discuss and get some conclusions.

numerical results discussion
Comparative analysis on impedance curves from plates and sandwich shell against excitation frequency at point 1 is plotted in Fig.2, and impedance curves at point 2 is plotted in Fig.3, and the difference in 10Hz-4000Hz is shown in table X. It's noted from Fig.2 that at point 1: impedance from different plates varies up and down around that from the sandwich shell; fluctuation from plate 1 is minimum, plate 3 is the second, and plate 2 is the maximum. At point 2: impedance from different plates also varies up and down around that from the sandwich shell; fluctuation from plate 2 is minimum, plate 1 is the second, and plate 3 is the maximum. For the impedance difference: At point 1, plate 1 is minimum, plate 3 is the second, and plate 2 is the maximum, and they are all less than 10%; At point 2, plate 2 is minimum, plate 1 is the second, they are both less than 10%, and plate 3 is the maximum, 28.5%, which is much bigger than 10%. In a word, plate 2 is selected as the equivalent plate for the sandwich shell.

Conclusions
In this work, a double-layered plate is designed for the sandwich shell using the equivalent impedance method. In 10Hz-4000Hz, the impedance of double-layered plate and sandwich shell is nearly the same with that of sandwich shell, and the difference is less than 10%. Then, the double-layered plate could be applied to conduct series vibration tests to get clear the influence of sandwich core material on structural vibration, while it could also reduce the costs of money and construction time. This work will lay foundation for the vibration tests of sandwich shell with different cores in the future.