Band gap analysis of a three-mass locally resonant structure

Abstract

A mass in a mass locally resonant system has been studied using a numerical and analytical method. This study is performed to compute the band gap and transmission coefficient of a mass–spring locally resonant system. A locally resonant structure is a periodic structure which exhibits negative effective properties in a certain frequency band and reveals band gaps below Bragg’s frequency. In this work, two substructures are attached with main mass so that the system will act as two masses in a mass system. It is found that the presented structure shows two band gaps below 500 Hz with negative effective properties. Addition of a third substructure with the main mass provides an additional band gap at low frequency. The position and width of band gaps can be tuned by changing the values of masses and stiffness.

Keywords

Local resonant mass-in-mass system negative effective property band gap transmission coefficient

Introduction

From the last two decades, the periodic structure has been widely used for noise attenuation in a band gap centered on Bragg’s frequency.¹ In the periodic structure, the Bloch function is used to represent the wave vector.² There is a certain range of frequency in which Bloch waves cancel out each other due to destructive interference generated in the periodic structure, and the wave cannot propagate in that particular frequency range. To get the noise attenuation below Bragg’s frequency, application of the periodic structure can be further improved and made more compact by using the concept of local resonance in scatterers. A locally resonant structure is a periodic structure which exhibits negative effective properties in a certain frequency band and reveals band gaps below Bragg’s frequency. Two types of locally resonant structures (LRSCs) are studied in the literature. One is an LRSC with C-shaped scatterers, and the other is an LRSC with a mass in a mass structure.^3–7 In C-shaped scatterers, a rectangular slit is cut from the cylindrical scatterers so that it acts as a Helmholtz resonator. In a mass in a mass unit cell, a second substructure is attached to the main structure of the unit cell. In LRSC and meta-materials, the locally resonant band gap always coincides with the negative effective property region which is the function of frequency.⁸ Wang et al.⁹ presented a model which was formed using a periodically attached spring–mass system. They calculated the dispersion relationship for the system using the finite element method. It is found that the spring–mass system showed broad band gaps at low frequencies. Krushynska et al.¹⁰ proposed a single- and two-phase phononic meta-material and studied the band gap formation theoretically. The result showed the evidence of a wider band gap than the conventional Bragg’s band gap.

Oudich et al.⁷ presented an experimental study of a locally resonant sonic crystal composed of silicon rubber stubs periodically arranged on an aluminum plate. A finite element method was implemented to compute the band structure and transmission loss through the structure, and results were verified by the experimental results. The result showed the existence of a local resonant band gap. The center frequency of this band gap is significantly less and very far from Bragg’s frequency. So, the locally resonant structure violates Bragg’s criteria.

Qian et al.¹¹ made a 2-D periodic arrangement of mass–spring resonators which were connected with the upper and lower plates by springs. The finite element method was used to calculate the band gap and transmission loss. The results were verified with the extended plane wave expansion method. It was found that varying the density of the resonator or the stiffness of the spring affects the position of the band gap, whereas varying the stiffness of the spring affects the bandwidth of the locally resonant band gap. Liu et al.⁴ fabricated a periodic structure based on the idea of local resonance. They used lead balls covered with silicone rubber and arranged them in a honeycomb lattice in epoxy. Lattice constant was taken two orders smaller than the relative wavelength of the wave propagating in the medium. The result showed the formation of two band gaps below 1000 Hz.

Nouh et al.¹² presented the vibration characteristics of locally resonant meta-materials. They used a viscoelastic membrane to support a small mass. Frequency response and band gap analyses were done by the finite element method. The result showed that band gap formation started at a low frequency, and vibrations are significantly reduced in the band gap region. Sainidou et al.¹³ used layer-multiple-scattering and finite-difference time-domain methods for computation of acoustic properties of two- and three-dimensional periodic structures. They presented a local resonant phononic crystal with cylindrical and spherical scatterers. The result showed that the position of resonance could be controlled by different parameters like size of cavity and nature of material inside and outside the scatterer.

Three-mass locally resonant structure

A mass in a mass locally resonant system exhibits some hypothetical properties such as negative effective mass and negative effective stiffness. The negative effective properties lie in the frequency region in which masses of the system go out of phase with each other. In a single mass in a mass locally resonant periodic structure, a second mass is attached to the main mass of the unit cell. In a particular region of frequency, both masses move in opposite directions which create local resonance at a lower frequency than the single-mass system.

In this work, two masses are connected to the main mass of the unit cell by two springs of different stiffness so that it acts as two masses in a mass system as shown in Figure 1. In this locally resonant structure, a third substructure is provided to create an additional band gap below the natural frequency of the single-mass system.

Analytical method

In this section, an analytical method is used to compute the dispersion relationship and transmission coefficient of the locally resonant structure.¹⁴ For the dispersion relationship, only one unit cell is analyzed to find the relationship between angular frequency and the wave vector of the propagating wave. Let m₁ be the main mass. m₂ and m₃ are the masses of substructures which are attached to m₁ by springs of stiffness k₂ and k₃. There are a total of ten unit cells of the three-mass system which are connected by a spring of stiffness k₁ with each other as shown in Figure 1. Let u₁, u₂, and u₃ be the displacements of masses m₁, m₂, and m₃, respectively. L is the length of the unit cell. j is a natural number and represents the number of the unit cell.

Applying force balance condition at each mass of the jth unit cell

m_{1} \frac{d^{2} u_{1}^{j}}{d t^{2}} = k_{1} (u_{1}^{j + 1} - u_{1}^{j}) - k_{1} (u_{1}^{j} - u_{1}^{j - 1}) + k_{2} (u_{2}^{j} - u_{1}^{j}) + k_{3} (u_{3}^{j} - u_{1}^{j})

(1)

m_{2} \frac{d^{2} u_{2}^{j}}{d t^{2}} = - k_{2} (u_{2}^{j} - u_{1}^{j})

(2)

m_{3} \frac{d^{2} u_{3}^{j}}{d t^{2}} = - k_{3} (u_{3}^{j} - u_{1}^{j})

(3)

Equations (1)–(3) are the equations of motion of mass m₁, m₂, and m₃. To solve these equations, let us consider that u₁, u₂, and u₃ are the functions of time and distance which are represented by equations (4)–(6)

u_{1^{j}} = U_{1} e^{i (jql - ω t)}

(4)

u_{2^{j}} = U_{2} e^{i (jql - ω t)}

(5)

u_{3^{j}} = U_{3} e^{i (jql - ω t)}

(6)

U₁, U₂, and U₃ are constants. q is the wave number and can be computed as 2pi/λ, λ is the wavelength, L is the lattice constant, and ω is the natural frequency.

Differentiating equations (4)–(6) two times

\frac{d^{2} u_{1}^{j}}{d t^{2}} = - ω^{2} U_{1} e^{i (j q L - ω t)}

(7)

\frac{d^{2} u_{2}^{j}}{d t^{2}} = - ω^{2} U_{2} e^{i (j q L - ω t)}

(8)

\frac{d^{2} u_{3}^{j}}{d t^{2}} = - ω^{2} U_{3} e^{i (j q L - ω t)}

(9)

Combining equations (1)–(3) and (7)–(9) and writing in a matrix form

[\begin{matrix} m 1 ω^{2} + 2 k 1 (\cos q L - 1) - k 2 - k 3 & k 2 & k 3 \\ k 2 & m 2 ω^{2} - k 2 & 0 \\ k 3 & 0 & m 3 ω^{2} - k 3 \end{matrix}] [\begin{matrix} U 1 \\ U 2 \\ U 3 \end{matrix}] = 0

(10)

In order to obtain nontrivial solution, the determinant of this matrix must be zero.

Equating the determinant of the matrix to zero

(m_{1} - \frac{k_{2} m_{2}}{m_{2}^{2} - k_{2}} - \frac{k_{3} m_{3}}{m_{3}^{2} - k_{3}}) ω^{2} = 2 k_{1} (1 - cos q L)

(11)

Equation (11) represents the dispersion relationship between angular frequency ω and wave number q for a three-mass system.

If m₂ = m₃ = 0

ω^{2} = \frac{2 k_{1}}{m_{1}} [1 - \cos q L]

(12)

where

ω^{2} = 4 k_{1} / m_{1} si n^{2} (q L / 2)

and

ω = \sqrt{4 k_{1} / m_{1}} \sin (q L / 2)

When $ω = \sqrt{4 k_{1} / m_{1}}$ , $\sin (q L / 2) = 1$ and $q = 2 π / 2 L$ .

Equation (12) represents the dispersion relation for a single-mass system. √(4k₁/m₁) is the stop frequency of the system. The single-mass system possesses the wavelength (λ) equal to 2L at this frequency. So, group velocity $(d ω / d q)$ of the system becomes zero at this frequency. It means that the wave cannot propagate through the single-mass structure above this frequency √(4k₁/m₁).

To represent a three-mass system in a single-mass system, the effective mass of the three-mass system is calculated by equating the dispersion relations equation of both systems. Equating this equation with equations (11) and (12)

m_{e f f} = m_{1} - \frac{k_{2} m_{2}}{m_{2} ω^{2} - k_{2}} - \frac{k_{3} m_{3}}{m_{3} ω^{2} - k_{3}}

Figure 2 represents an equivalent system to the three-mass system which is shown in Figure 1. X_j represents the displacement of the jth unit cell.

Applying the force balance condition at the jth unit cell

m_{eff} \frac{d^{2} X_{j}}{d t^{2}} = k_{1} (X_{j + 1} + X_{j - 1} - 2 X_{j})

(13)

Let $X_{j} = x_{j} e^{i (j q L - ω t)}$

So . \frac{d^{2} x_{j}}{d t^{2}} = - ω^{2} x_{j} e^{i (j q L - ω t)}

(14)

Combining equations (13) and (14)

(2 k_{1} - m_{e f f} ω^{2}) x_{j} = k_{1} (x_{j + 1} + x_{j - 1})

(15)

where j = 1, 2 3,…, N −1, and N is the total number of unit cells.

For the last unit cell when j=N, the equation of motion will be

(2 k 1 - m_{e f f} ω^{2}) x_{j} = k_{1} x_{j - 1}

(16)

Using equations (15) and (16), the ratio of the transmitted component of displacement to the incident displacement has been calculated

For j = N; \frac{x_{N}}{x_{N - 1}} = \frac{k_{1}}{(2 k 1 - m_{e f f} ω^{2})}

(17)

\begin{array}{l} For j = N - 1; (2 k_{1} - m_{e f f} ω^{2}) x_{N - 1} = k_{1} (x_{N} + x_{N - 2}) \\ \Rightarrow (2 k_{1} - m_{e f f} ω^{2}) x_{N - 1} = k_{1} (\frac{k_{1}}{(2 k 1 - m_{e f f} ω^{2})} x_{N - 1} + x_{N - 2}) \\ \Rightarrow (2 k_{1} - m_{e f f} ω^{2} - \frac{k_{1}^{2}}{(2 k 1 - m_{e f f} ω^{2})}) x_{N - 1} = k_{1} x_{N - 2} \end{array}

(18)

Combining equations (17) and (18)

\frac{x_{N}}{x_{N - 2}} = \frac{k_{1}}{(2 k 1 - m_{e f f} ω^{2})} \frac{k_{1}}{(2 k_{1} - m_{e f f} ω^{2} - k_{1}^{2} / (2 k 1 - m_{e f f} ω^{2}))}

The similar process has been followed to obtain x_N as a function of x₀

Transmittance (T) = \log | \frac{x_{N}}{x_{0}} |

(19)

Transmittance has been calculated on the log scale as represented by equation (19). The dispersion relationship and transmission coefficient have been calculated for a locally resonant structure which is having two additional substructures. This result has been calculated for some chosen values of masses and stiffness.

Let us consider m₁ = 1 kg, m₂ = 0.5 kg, m₃ = 1.2 kg, k₁ = 10⁷ N/m, k₂ = 3 × 10⁶ N/m, and k₃ = 2 × 10⁵ N/m. Figure 3 shows the normalized properties with respect to ω₂. It is found that negative effective mass presents only in the band gap region. Two band gaps are found below 500 Hz with the present values of masses and stiffness as shown in Figure 4(b). Addition of a third substructure with the main mass provides an additional band gap at very low frequency. In this case, two band gaps are found ranging from 63 Hz to 88 Hz and 365 Hz to 480 Hz. In the band gap region, the structure shows negative transmittance as shown in Figure 4(a).

The band gaps can be tuned by changing the masses and stiffness. When the value of third mass and stiffness change to 2 kg and 1 × 10⁶ N/m, respectively, a different band structure appears as shown in Figure 5(a). It shows two band gaps below 500 Hz with a wider bandwidth than the previous case. In this case, the band gap ranges from 108 Hz to 168 Hz and 370 Hz to 489 Hz. When the third substructure is removed, the system will act as a single mass in a mass system, and it shows only one band gap as shown in Figure 5(b). The second band gap is vanished with vanishing of the third mass in the system. The objective of this study is to show that a periodic two mass in a mass system shows two band gaps in the low-frequency region (less than 500 Hz in this case), and effective mass in this band gap region always turns negatives. If one mass is removed, it will act as a mass in a mass system. In this case, the structure shows only one band gap. Location of band gaps in these types of structures is always less than the natural frequency of the main mass (m₁). For m₁ = 1 kg, m₂ = 0.5 kg, m₃ = 2 kg, k₁ = 10⁷ N/m, k₂ = 3 × 10⁶ N/m, and k₃ = 1 × 10⁶ N/m, natural frequency (√(k/m)) of the main mass is 503 Hz, and the location of band gaps in this case ranges from 108 Hz to 168 Hz and from 370 Hz to 489 Hz as shown in Figure 5.

Numerical modeling

Numerical modeling is performed to verify the analytical results computed in Section 2.1. ANSYS software is used to compute the displacement as a function of time. Ten unit cells are considered in two mass in a mass locally resonant structure. The parameters are considered as m₁ = 1 kg, m₂ = 0.5 kg, m₃ = 1.2 kg, k₁ = 10⁷ N/m, k₂ = 3 × 10⁶ N/m, and k₃ = 2 × 10⁵ N/m. Lattice constant (L) is taken sufficiently smaller than the propagating wavelength.

Mass 21 and combin 14 are used to describe the mass and spring as an element in ANSYS. All masses are represented by nodes which are shown in Figure 6. As the main mass (m₁) is connected to two different substructures, two imaginary nodes are created to represent mass m₁. These nodes are coupled together in such a way that these nodes have the same degree of freedom as gained by the main mass. The transient study is performed for ten unit cells of the structure with a total simulation time of 1 s. The step size of time is 0.0001 s. Four dampers are attached in series to the tenth unit cell to avoid wave reflections from the end. A harmonic displacement of particular frequencies is given to the main mass of the first unit cell. The amplitude of the given displacement is 1. The displacement is computed at the main mass of the 10th unit cell.

Using analytical method, it is found that for the given values of masses and stiffness, the negative effective mass region lies between 63 Hz to 88 Hz and 365 Hz to 480 Hz. To verify the analytical results, a few frequencies of input excitation have been taken. Figure 7(a) and (b) represents the displacement of the 10th unit cell when excitation frequency of displacement at the first unit cell is 200 Hz and 400 Hz. Figure 7(a) shows no displacement attenuation at the 10th unit cell at frequency 200 Hz. At 400 Hz, the wave cannot propagate in the structure as shown in Figure 7(b) because 400 Hz is present in the band gap region of the given locally resonant structure.

Some more simulations have been performed with different frequency of input excitation. It is found that the wave signals, which belong to the band gap region, are reduced significantly and approach to zero as shown in Figure 8. The numerical simulation is performed to verify the analytical results. The simulations results are in good agreement with the analytical results. Figure 9 shows the displacement of the main mass at the 9th and 10th unit cells when excitation frequency is 400 Hz (present in the band gap region). It is clearly seen that displacement of both unit cells is of smaller amplitude than the given excitation and both signals are out of phase. Displacement amplitude of the 10th unit cell is less than the displacement at the 9th unit cell.

Three-mass LRSC are useful to create an additional band gap at low frequencies. It can be used to treat low-frequency vibrations at certain places. Parameters can be tuned according to the requirements.

Conclusion

A three-mass periodic structure has been presented which acts as a locally resonant structure. The structure shows negative effective properties in a certain frequency range below Bragg’s frequency. The locally resonant structure has been studied analytically as well as numerically, and both results are in good agreement. It is found that adding a third substructure in a mass-in-mass system leads to create an additional band gap below Bragg’s frequency, and the band gap region shows negative transmission coefficients on log scale. Bandwidth and its location can be tuned by changing the masses and stiffness of springs in the system. The numerical and analytical results have not been compared together on one graph. The location of the band gap has been computed using the analytical method. By using numerical simulation, it is verified that the wave of frequencies lying in the band gap region could not pass through the structure. The wave of frequencies, which are out of the band gap region, passes from the 10th unit cell with somewhat higher amplitude than the input excitation at the first unit cell. The presented model shows two band gaps in the low-frequency region ranging from 63 Hz to 88 Hz and 365 H to 480 Hz. This type of a mass-in-mass locally resonant structure can be used for low-frequency wave attenuation or filtering. The work presented here can be extended to beams and plates having mass-in-mass periodic units.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to acknowledge the facility of the Acoustics and Vibration Lab provided by the Indian Institute of Technology, Mandi, for carrying out this work. The authors would also like to acknowledge the assistance provided by SERB through the DST project YSS/2015/001245.

ORCID iD

Preeti Gulia

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