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Locally resonant phononic crystals (LRPCs) beam is characterized by the band gaps; some frequency ranges within which flexural waves cannot propagate freely. So, the LRPCs beam can be used for noise or vibration isolation. In this paper, a LRPCs beam with distributed oscillators is proposed, and the general formula of band gaps and transmission spectrum are derived by the transfer matrix method (TMM) and spectrum element method (SEM). Subsequently, the parameter effects on band gaps are investigated in detail. Finally, a rubber concrete beam is designed to demonstrate the application of distributed LRPCs beam in civil engineering. Results reveal that the distributed LRPCs beam has multifrequency band gaps and the number of the band gaps is equal to that of the oscillators. Compared with others, the distributed LRPCs beam can reduce the stress concentration when subjected to vibration. The oscillator interval has no effect on the band gaps, which makes it more convenient to design structures. Individual changes of oscillator mass or stiffness affect the band gap location and width. When the resonance frequency of oscillator is fixed, the starting frequency of the band gap remains constant, and increasing oscillator mass of high-frequency band gap widens the high-frequency band gap, while increasing oscillator mass of low-frequency gap widens both high-frequency and low-frequency band gaps. External loads, such as the common uniform spring force provided by foundation in civil engineering, are conducive to the band gap, and when the spring force increases, all the band gaps are widened. Taken together, a configuration of LRPCs rubber concrete beam is designed, and it shows good isolation on the vibration induced by the railway. By the presented design flow chart, the research can serve as a reference for vibration isolation of LRPCs beams in civil engineering.

The concept of phononic crystals (PCs) was first proposed by Kushwaha et al. [

Generally, there are two kinds of interpretations for the band gap mechanism of PCs, including Bragg scattering mechanism [

In engineering fields, the beams are the widely used structure types, in which the flexural waves lead mostly to the vibration and acoustic noise [

Meanwhile, various kinds of engineering materials have been developed to attenuate nuisance vibration of structures. In civil engineering, tire rubber concrete, a main recycling strategy of the waste tire rubber, has been adopted worldwide for civil engineering recovery [

In present study, a LRPCs beam model with multioscillators is proposed to conduct a comprehensive investigation on the band gap properties of the LRPCs beams. Considering the periodicity of the beam, the general formula of band structure and transmission spectrum for LRPCs beam with distributed multioscillators are derived by TMM and SEM, respectively. The derivation is verified by finite element method (FEM). Then, the effects of beam and oscillators on the band gap are investigated in detail. Considering the application in civil engineering, the external loads provided by foundation are included in the model, and the effects of foundation spring force on the band gaps are analyzed as well. Based on this, a LRPCs rubber concrete beam is proposed to demonstrate the application of distributed LRPCs beam with rubber concrete. And the vibration isolation effect of the LRPCs rubber concrete beam is demonstrated. The results can provide reference for the band gap design of LRPCs beam in civil engineering.

The theoretical model of LRPCs beam with distributed multioscillators (_{1}, _{2,} and _{m}. The infinite model is composed of the period multioscillators element, also known as primitive cell. The spacing of adjacent primitive cell is called lattice constant, which is expressed as

Theoretical model of LRPCs beam with distributed multioscillators (

To clearly demonstrate the band gap calculation of the multioscillator configuration, a special case (_{1}, _{2}, and _{3}, and springs _{1}, _{2}, and _{3}) are alternately attached to a three-component Euler beam. The intervals between adjacent oscillators are _{1}, _{2}, and _{3}, respectively. In this paper, the transfer matrix method (TMM) is used to calculate the complex band structure. The three-oscillator primitive cell can be regarded as a supercell composed of three single-oscillators. Therefore, the complex band structure of LRPCs beam can be solved by the introduction of continuity condition at the interface of the supercell and the periodic boundary condition at the boundary of the supercell. The governing equation for free flexural vibration in Euler-Bernoulli beam can be expressed as

The primitive cell of LRPCs beam and the force equilibrium (

For simple harmonic wave, the harmonic solution has the form

For oscillator at _{ni}, the linear displacement of the oscillator can be expressed as

Then, the oscillators satisfy the dynamic equilibrium equation_{i} is stiffness of the spring at _{ni}; _{i} is mass of the oscillator at _{ni};

Equation (

For periodic beam, the displacement _{1}, _{2} and

The transmission matrix of the LRPCs beam with distributed

Furthermore, with the introduction of the periodic boundary conditions, namely, Bloch-Floquet theorem [

By the solution of the eigenvalues of the matrix equation (

Equation (

In practice, the LRPCs beam structure is of finite length, and thus, the vibration transmission spectrum is calculated based on SEM in this paper. The finite length LRPCs beam is shown in Figure

Setting for the element and node of finite LRPCs beam.

The dynamic stiffness matrix of spectral element

The elements of the matrix are

The additional dynamic stiffness at the attachment point of each resonator is

The action of resonators on the beam can be expressed in additional dynamic stiffness matrix form:

Assembling the dynamic stiffness matrix of each spectral element with the additional dynamic stiffness matrix of each oscillator can obtain the dynamic stiffness matrix of each spectral element. By assembling the dynamic stiffness matrix of each spectral element, we can obtain the dynamic stiffness matrix of the whole LRPCs beam. The dynamic stiffness matrix of the LRPCs beam can be expressed as

By employing equation (

Applying load at the left end of LRPCs beam and picking up the displacement vibration response at both ends of LRPCs beam, we can obtain the vibration transmission spectrum of the LRPCs beam as follows:

Equation (

To verify the theoretical derivation, a special calculation model (_{1} _{2} = 0.02 m, height _{1} _{2} = 0.004 m, elasticity modulus _{1} ^{10} Pa, _{2} ^{10} Pa, density _{1} = 7780 kg/m^{3}, _{2} = 2730 kg/m^{3}. The oscillators are displaced at equal intervals, _{1} = _{2} = 0.015 m: oscillators, _{1} = 9.28 × 10^{−3} kg, _{1} = 3.29 × 10^{5} N/m, _{2} = 3.71 × 10^{−2} kg, _{2} = 3.29 × 10^{5} N/m. According to equation (

Imaginary band structure of LRPCs beam (

As shown in Figure

According to equation (

Finite element model of LRPCs beam with multioscillators (

Material parameters of FEM beam model.

Material | Density (kg/m^{3}) | Young modulus (Pa) | Shear elasticity (Pa) |
---|---|---|---|

Beam 1 | 7780 | 21.06 | 8.10 |

Beam 2 | 2730 | 7.76 | 2.87 |

Oscillator 1 | 8950 | 16.46 | 7.53 |

Oscillator 2 | 19500 | 8.5 | 2.99 |

Silicone rubber | 1300 | 1.0 | 3.4 |

Structural parameters of the FEM beam model.

_{1} (m) | _{2} (m) | _{0} (m) | _{1} (m) | _{2} (m) | _{3} (m) | ||
---|---|---|---|---|---|---|---|

0.02 | 0.00833 | 0.015 | 0.015 | 0.004 | 0.00621 | 0.01144 | 0.001 |

Apply the acceleration excitation at one end of the finite element structure and pick it up at the other end to calculate the transmission spectrum. The transmission spectrum of LRPCs beam calculated by theoretical and finite element simulation is shown in Figure

Transmittance spectrum of finite LRPCs beam (6 cells).

It can be seen from Figure

To demonstrate the vibration mechanism of LRPCs beam with multioscillators, taking

Though the locally resonant band gaps are of low frequencies, the local resonance of oscillators may cause vibration stress concentration at attachment point of the beam. In this case, this paper chooses the parameters in reference [

The vibration stress nephogram of different types of LRPCs beam.

As depicted in Figure

In order to investigate the influence law of the interval between distributed oscillators on the band gaps of LRPCs beam, we also take the LRPCs beam model with 2 oscillators (

The dimensionless stiffness of the spring is defined as the ratio of the spring stiffness to the bending stiffness of the corresponding beam segment:

The dimensionless frequency is defined as the ratio of lattice constant to half wavelength:

According to equations (_{1} = 0.497, _{2} = 1.987; _{1} = 0.395, _{2} = 0.395; Ω_{1} = 0.301, Ω_{2} = 0.213. Defining oscillator interval ratio, _{1}/

Effects of oscillator interval on band gap of LRPCs beam.

As shown in Figure

Analogously, we can obtain the effects of the oscillator parameters on the band gaps of LRPCs beam. For convenience, we take LRPCs beam model with 2 oscillators (

With the mass of oscillator 2 kept constant and only the dimensionless mass of oscillator 1 varying from 0 to 5.0, the mass effects of oscillator 1 on the band gap of LRPCs beam can be analyzed. The band gap property of LRPCs beam is shown in Figure

Mass effects of oscillator 1 on band gap of the LRPCs beam.

As shown in Figure

Similarly, with only the mass of oscillator 2 varying, the mass influence of oscillator 2 on the band gaps of LRPCs beam can be obtained. The diagram is shown in Figure

Mass effects of oscillator 2 on band gap of the LRPCs beam.

The above variation indicates that increasing the mass of oscillator in high-frequency band gap can lower the position of the high-frequency band gap and increase its width, but the low-frequency band gap will be compressed and thus narrows. On the other hand, increasing the mass of oscillator in low-frequency band gap can lower the position of low-frequency band gap and increase its width, while the high-frequency band gap remains nearly unaffected.

Remaining the stiffness of oscillator 2 constant and varying the stiffness of oscillator 1 from 0 to 0.5, the stiffness effects of oscillator 1 on the band gap of LRPCs beam can be analyzed. The band gap property of LRPCs beam is shown in Figure

Stiffness effects of oscillator 1 on band gap of the LRPCs beam.

As can be seen from Figure

Similarly, keeping the stiffness of oscillator 1 constant and only varying the stiffness of oscillator 2, the stiffness effects of oscillator 2 on band gap of LRPCs beam can be analyzed. The band gap property of LRPCs beam is shown in Figure

Stiffness effects of resonator 2 on band gap of the LRPCs beam.

As can be seen from Figure

By the resonance frequency of oscillators fixed, the effects of oscillator mass (or stiffness) on the band gap of LRPCs beam can be obtained. The band gap property of LRPCs beam is shown in Figures

Mass effects of oscillator 1 on band gap of the LRPCs beam.

Mass effects of oscillator 2 on band gap of the LRPCs beam.

As can be seen from Figure

As shown in Figure

The beams are usually subjected to external loads in practice. For example, in civil engineering, the existence of beam resting on the foundation is often encountered [_{0}, where _{0} = 10^{6} N/m^{3}. In this case, the governing equation of (

Analogically, taking the LRPCs beam model with 2 oscillators (

Effects of uniform spring force on band gaps.

As shown in Figure ^{6} N/m^{3}). Then, knowing the effects of external spring force on the band gap is conducive to the vibration attenuation design of the beam-foundation system.

In can be concluded that the oscillator parameters and external load can affect both band gap location and band gap width. Therefore, to fabricate an optimized LRPCs structure, which is of the goal-oriented band gaps according to the need of vibration reduction, it is better to design the structure by varying these parameters based on above influence law.

Generally, various kinds of vibration exist in our surroundings, for example, the ground vibration induced by railway, a kind of nuisance vibration related closely to our daily life. In civil engineering, concrete beam is often used as the foundation beam under building. Therefore, the theory of LRPCs can be introduced into the foundation beam to attenuate the ground vibration. Based on the above parameter analysis, a configuration of rubber concrete is presented to form LRPCs beam, and the LRPCs rubber concrete beam is shown in Figure ^{5} Pa), and it is so soft. The rubber shell can be regarded as the soft coating layer, and it acts as the spring for oscillators in LRPCs rubber concrete beam. Meanwhile, the hard concrete core inside the rubber shell can act as the mass in LRPCs concrete beam, namely, the soft rubber shell and the hard concrete core form the local resonator together. In this case, the LRPCs rubber concrete beam can be designed to attenuate the ground vibration induced by the railway. The design flow chart of LRPCs rubber beam is shown in Figure

Vibration waves induced by railway and the LRPCs rubber concrete foundation beam.

Design flow chart of the LRPCs beam.

As shown in Figure _{c} = 3.45 × 10^{10} Pa, density _{c} = 2450 kg/m^{3}, Poisson’s ratio _{c} = 0.3. And assume that the stiffness of ^{6} N/m^{3}. Thirdly, choose appropriate oscillators. For convenience, we only choose one type of oscillator. Fourthly, determine the parameters of the local resonators. The mass and the spring equivalent stiffness can be adjusted by the radius and thickness of rubber shell, namely, the filling rate of ground rubber. The higher the filling rate of the rubber is, the lower the equivalent stiffness of the spring would be. In this paper, the parameters of rubber shell are as follows: Young’s modulus _{r} = 1.175 × 10^{5} Pa, density _{r} = 1300 kg/m^{3}, Poisson’s ratio _{r} = 0.42, inner radius _{r} = 0.0255 m, thickness of rubber shell _{r} = 0.01 m, equivalent stiffness coefficient of rubber shell ^{5} N/m; mass of concrete core ^{−6} m^{4}. Figure

Acceleration record of railway.

Band gaps of the LRPCs rubber concrete beam laying on foundation.

FEM model of the LRPCs rubber concrete beam (6-cell).

Acceleration time domain analysis of the LRPCs rubber concrete beam.

The multifrequency LRPCs beam with distributed multioscillators is proposed, and the band gap is figured out based on transfer matrix method (TMM) and Bloch’s theorem. Besides, the parameter effects on the band gaps are investigated. Finally, a LRPCs rubber concrete beam is designed to show its application in isolating the railway vibration. Main conclusions can be drawn as follows:

The distributed LRPCs beam has multifrequency band gaps, and the number of the gaps is equal to that of the local oscillators. Compared with other types of LRPCs beam, the distributed one can reduce the stress concentration when subjected to vibration.

The oscillator interval has no effects on the band gaps, making it more convenient to design structures.

Individual changes of oscillator mass or stiffness affect the band gap location and width. When the resonance frequency of oscillator is fixed, the starting frequency of the gap keeps constant. And increasing the oscillator mass in high-frequency part widens the high-frequency band gap, while increasing the oscillator mass in low-frequency part widens both high-frequency and low-frequency band gaps.

The external loads, such as the uniform spring force encountered in civil engineering, are conducive to the generation of band gaps, and when the spring force increases, all the band gaps widen with the starting frequencies keeping constant.

Taken together, a LRPCs rubber beam is designed, and it shows the good vibration isolation of LRPCs beam in civil engineering. By the design flow chart, the research can provide a reference for vibration isolation of LRPCs beams in civil engineering fields.

Finally, it should be noted that the external load in this paper is mainly concerned with the common spring force provided by the

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This study was supported by the National Natural Science Foundation (Grant No. 51108252) and Shandong Transportation Science and Technology Project (Grant No. 2017B59).