Dynamic analysis of negative stiffness noise absorber with

magnet

Min Yang 1 , Weiming Xiao, Erjing Han, Junjuan Zhao, Wenjiang Wang, Yunan Liu Beijing Key Lab of Environmental Noise and Vibration, Institute of urban safety and environmental science, Beijing academy of science and technology No.55, Taoranting Road, Xicheng, Beijing, 100054, China

ABSTRACT In the paper, the negative stiffness membrane absorber with magnet has been taken as a nonlinear noise absorber. The dynamic characteristics of the nonlinear noise absorber have been studied by nonlinear dynamics theory and numerical simulation. The dynamic equations of the system were established under harmonic excitation. The slow flow equations of the system are derived by using complexification averaging method, and the nonlinear equations which describe the steady-state response are obtained. The amplitude frequency diagram and phase diagram are used to study the nonlinear response of structures. The resulting equations are verified by comparing the results which respectively obtained from complexification-averaging method and Runge-Kutta method. It is helpful to optimize the structural parameters and further improve the sound absorption performance to study the variation of the sound absorption performance of magnet negative stiffness membrane absorber system with its structural parameters. 1 INTRODUCTION In recent years, acoustic metamaterials, because of their exotic properties, have received significant attention from physicists as well as acoustical engineers[1-2]. Among these materials, membrane-type acoustic metamaterials (MAMs) are relatively simple and lightweight and possess the ability to realize a subwavelength scale and tunable absorption design [3]. These properties are particularly important for noise control. Ma et al.[4] reported a novel acoustic m etasurface with hybrid resonances that can achieve robust impedance matching and perfect absorption. However, this type of metasurface has an extremely narrow absorption bandwidth as a result of the causality constraint on the minimal structural thickness [5]. To implement a compact sound absorber and expand bandwidth as much as possible in the low frequency, Zhao et al. [6] propo sed a membrane sound absorber (MSA) with a magnet. The study mainly focused on the absorption of a single state, did not further consider the continuously tunable characteristics and frequency-tuning range of the structure, and, specifically, did not provide a precise theoretical model to quantitatively calculate and predict the

1 yangmin@bmilp.com

absorption properties. Zhao et al. [7] p resented a theoretical model based on a multi- mechanism coupling impedance method. The model predicts the absorption coefficients and resonant frequencies of the MSA at different tuning magnetic states for three cavity configurations. In this paper, the dynamic characteristics of the nonlinear noise absorber have been studied by nonlinear dynamics theory and numerical simulation under harmonic excitation. 2. THEORETICAL MODEL

om ega1=

Figure. 1 Schematic diagram of the magnetically tunable membrane-type acoustic metamaterial

Structurally, a membrane sound absorber consists of three parts: a rectangular membrane with a small cylindrical magnet A , and a cylindrical magnet B to generate a magnetic field, and a cylindrical cavity formed between the solid support frame and membrane; see Fig. 1.

This section presents the derivation of a theoretical model to calculate the absorption coefficients using the mode-superposition method. The methods of determining and analyzing the parameters of the theoretical mode superposition model are also described.

The equation of motion of the membrane is given by [8]

     3 3

2

2 f d x t dx t M C Kx t p t S dt d x t k t     (1)

,0 / i i M M   , 2 2 i i C M c S     and

where the dot denotes the time derivative, 2

the magnetic force between the proof mass and the external magnets is approximated

by 3 1 3 f k x k x   , 2 2 2

1 / i K M c s D k       .

Complexification averaging method can be used to solve the equation. By introducing a complex function

  



  

    

i t

i t x t i x t t e

(2)

  

 

x t i x t t e

where  t  denotes the complex conjugate of  t  ， i is an imaginary number sign ，

 is frequency. We can obtain from Eq.(2)

 



    

 

i t i t

e e x i e e x

2

 



 

i t i t

(3)

2

 

  

  

i t i t

x e i e i x

Introducing Eq.(3) into Eq.(1) ， Algebraic equations are obtained

3 3 1 3 2 2 2 8 2 k C K SP i i i M M M M            

2 3

(4)

We express

re im i      (5)

Introducing Eq.(5) into Eq.(4) ， Eq. (4) can be divided into its real and imaginary part

                     

3 1 2 2 2 8 2 3 1 2 2 2 8

k R SP M MC M M k R

 

     

2 2 3

NL re re im im re im im

(6)

 

    

2 2 3

NL im im re re re im re

M MC M

Considering the steady-state response of the system, we have the following equation

                 

3 1 0 4 3 1 0 4

k R SP M MC M M k R M MC M

 

      

2 2 3

NL re im im re im im

(7)

 

     

2 2 3

NL im re re re im re

Substituting equation Eq. (5) into Eq.(3), the expression of amplitude is

2 2 re im x   

  (8)

3. NUMERICAL SIMULATION In order to verify the correctness of the derivation process, the results obtained by complexification averaging method and Runge-Kutta method are compared, as shown in Figure 2, which is in good agreement. We choose M =0.0027, C ms = 0.2185mmN -1 , S =0.0032, K = 2.2143, k 3 = 5.5849*10 13 .The velocity and phase diagram of upward sweep and downwarp sweep are shown in Figure 3, Figure 4 and Figure 5.

-5

x 10

1.8

Complexification averaging method Runge Kutta method

1.6

1.4

1.2

Amplitude(m)

1

0.8

0.6

0.4

0.2

0 50 100 150 200 250 300 350 400 450 500 0

Frequency(Hz)

Figure 2. The comparison between results obtained by complexification averaging

method and Runge-Kutta method.

Figure 3. The comparison between velocity results obtained by upward sweep and

downwarp sweep.

Figure 2. The phase diagram obtained by upward sweep.

Figure 2. The phase diagram obtained by downwarp sweep.

4. CONCLUSION

In the paper, the negative stiffness membrane absorber with magnet has been taken as a nonlinear noise absorber. The dynamic characteristics of the nonlinear noise absorber have been studied by nonlinear dynamics theory and numerical simulation. The dynamic equations of the system were established under harmonic excitation. The slow flow equations of the system are derived by using complexification averaging method, and the nonlinear equations which describe the steady-state response are obtained. The amplitude frequency diagram and phase diagram are used to study the nonlinear response of structures. The resulting equations are verified by comparing the results

which respectively obtained from complexification-averaging method and Runge- Kutta method. It is helpful to optimize the structural parameters and further improve the sound absorption performance to study the variation of the sound absorption performance of magnet negative stiffness membrane absorber system with its structural parameters.

5. ACKNOWLEDGEMENTS

This work is supported by BJAST Innovation Cultivation Programes (No. 11000022T000000468173, 11000022T000000468161); the Beijing Natural Science Foundation (No. 1202008); BJAST Young Scholar Programs (B) (No. YS202101).

6. REFERENCES

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acoustic metamaterial with negative dynamic mass, Phys. Rev. Lett. 101, 204301.

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