A new beam structure with periodically attached multioscillators is proposed based on the idea of locally resonant (LR) phononic crystals (PCs) to reduce flexural vibrations in the frequency-multiplication ranges. Wave band structures of the new beam are derived by using the transfer matrix method. The multiple band gaps in the beam are then verified by the frequency response function (FRF), which is calculated through the finite element method. In addition, simplified models are proposed, which contribute to the calculation of the edge frequencies of the band gaps and enhance the understanding of the LR mechanism of PCs. The accuracy of the simplified models is proven by comparing them with the results derived from the analytical model under different beam structure parameters. The results suggest that lower frequencies and ranges of frequency multiplications can be achieved in the band gaps which are obtained from the new beam structure with multioscillators in a unit cell. Therefore, the ideas presented in this paper have the potential to be used in developing new devices with frequency-multiplication characteristics for vibration isolation or noise control in aerospace and civil structures.

Methods to control the propagation of elastic waves, such as vibration reduction and noise isolation, are often the focus of engineering studies. Much research has been conducted over many years to suppress unwanted vibration or noise. A variety of vibration control technologies, including visco-elastic materials, springs, soft materials, hydraulic dampers, and pneumatic isolators, among others, were gradually developed and are widely used in engineering practice [

In the last decade, the emergence and development of phononic crystals (PCs) have inspired new ideas for wave control [

There has been a great deal of research on the mechanisms and properties of band gaps. The earlier investigations of PCs are commonly based on the Bragg scattering mechanism [

Beams are typical structural elements of many engineering constructions and equipments. The control of wave propagation in beams is of great importance in aerospace and civil structures because the unwanted transmission of waves can lead to safety issues or environmental consequences. Based on the concept of LR PCs, some research focuses on the existence of low-frequency resonance gaps in infinite systems and the validation of gap characteristics by calculating/measuring the frequency response functions (FRFs) of finite samples [

Recently, the coexistence of resonance-type and Bragg-type band gaps was found in LR beams [

The main purpose of this paper is to achieve more flexible resonance-type multiband gaps by proposing a new beam with periodically attached multioscillators. The lower initial frequency and band gaps in the frequency-multiplication ranges are expected to be obtained in the new beam, which can meet the demand of wave attenuation in multiple frequency ranges in engineering. In addition, simplified models for the corresponding edge frequencies of the band gaps are studied, which can contribute to further understanding of the LR mechanism of PCs and the realisation of composite structures with multiple band gaps. The paper is organised as follows. The exact dispersion relations for the propagation of flexural vibrations in infinite Timoshenko beams that are periodically connected with multioscillators are derived in Section

In this section, the transfer matrix method is used to derive the exact dispersion relations for the Timoshenko beams with periodically attached multioscillators, which allows for continuity conditions at the two surface boundaries of each unit cell through the use of matrices [

The analytical models of a Timoshenko beam with periodically attached multioscillators. (a) Model A: the oscillators are connected to each other on the same side. (b) Model B: the oscillators are distributed on different sides, a configuration which was put forward and discussed in [

For Model A, which is shown in Figure

The interactive force between the first oscillator and the beam,

For the case of Model B studied in [

The solution is

From (

According to the continuity of the displacement, slope, bending moment, and shear force at the interface between the

By extracting the arbitrary coefficients from (

Due to the periodicity of the structure, the Bloch theorem states that

The dispersion relation between the wave vector

Figure

The sketch of a Timoshenko beam with periodically attached multioscillators. (a) Model A, (b) Model B.

All of the material parameters used in the calculations are listed in the Table

Material parameters.

Material | Density ^{3}) |
Young’s Modulus |
Shear Modulus |
Poisson ratio |
---|---|---|---|---|

Rubber | 1300 | 7.7 × 10^{5} |
2.6 × 10^{5} |
0.48 |

Al | 2600 | 7.0 × 10^{10} |
2.7 × 10^{10} |
0.3 |

Cu | 8950 | 1.646 × 10^{11} |
7.53 × 10^{10} |
0.093 |

Steel | 7780 | 2.106 × 10^{11} |
8.1 × 10^{10} |
0.3 |

The radial stiffness of the rubber ring can be calculated using [

For comparison, the structure parameters for Model B are taken from [

The band structures of both models are shown in Figure

The band structure of the infinite Timoshenko beam with periodically attached multioscillators. (a) Real wave vector of Model A. (b) Real wave vector of Model B.

Figure

The existence of the band gaps calculated from the infinite system can be verified by the transmission property derived from a corresponding finite system because PCs with a sufficient number of unit cells can provide a large wave attenuation in the corresponding band gap range [

(a) The calculation model for the FE method. (b) The corresponding calculated FRF.

Note the two sharp drops below the 0 dB line (dashed line) in the Figure

In this section, the corresponding simplified models for the initial and terminal frequencies of the band gaps for Model A are studied. The simplified models for Model B have been discussed previously in [

The initial frequency of the first band gap in a typical LR PC is determined by the resonance frequency of the oscillator in the same direction. In this resonance mode, the oscillators vibrate in specific directions, and the phases of the oscillator vibrations in adjacent unit cells are reversed to keep the dynamic balance [

The simplified model for the initial frequencies of band gaps.

The equations of motion for the model are as follows:

The natural angular frequency

Thus,

All of the oscillation phases of the unit cells are in the same direction at the terminal frequency of the band gap. The dynamic balance is given by the antiphases between the LR structures and the matrix [

The simplified model for the terminal frequencies of the band gaps.

The components to the right of the static point (dashed box) can be observed as a single unit. The natural angular frequency

Because the resonances of the matrix and the connected oscillators are at the same frequency,

Thus, the relation between

and the terminal frequencies of the first two band gaps are

Figures

Variation of the band gaps as a function of the oscillators’ mass ratio and verification of the simplified models. (a) Model A, (b) Model B. (

Variation of the band gaps as a function of the oscillators’ stiffness ratio and verification of the simplified models. (a) Model A, (b) Model B. (

Variation of the band gaps as a function of the beam’s mass and verification of the simplified models. (a) Model A, (b) Model B. (

Figures

By comparing the (a) subfigure with the (b) subfigure in Figures

In this paper, a new Timoshenko beam structure with periodically attached multioscillators is proposed to obtain band gaps in the frequency-multiplication ranges based on the LR mechanism of PCs. Explicit matrix formulations are derived for the calculation of wave band structures of the new beam by using the transfer matrix method. The gap characteristics of the beam are confirmed by calculating the FRF of the corresponding finite structure. The numerical calculations of the band structures and the analysis of the model parameters demonstrate that the beams with periodically attached multioscillators have more abundant gap characteristics than those with only one oscillator in a unit cell. By using common materials and an uncomplicated beam structure, multiple resonance-type band gaps with large wave-attenuation and frequency-multiplication ranges, together with the wider and lower first band gap, are derived in the new beam; this result was not illustrated in any of the previous studies on LR PC beams. In addition, simplified models are proposed to deduce accurate estimation formulae for the initial and terminal frequencies of the band gaps in the new beam. The simplified models will also contribute to enhanced understanding of the LR mechanism of PCs and will facilitate the analysis of similar structures.

The research findings presented in this paper provide suggestions for future studies of small-size PCs with low frequencies and multiple resonance-type band gaps. Moreover, the results can be employed to create new devices that reduce vibration and mitigate noise in the frequency-multiplication ranges for aerospace and civil structures.

This work is supported by the National Nature Science Foundation of China under Grant nos. 51079127, 51179171, and 51279180.