Keywords

1 Introduction

In recent years, the wireless sensor technology and intelligent monitoring technology has provided a new and effective way to replace the traditional wired underground environmental monitoring means in coal mines [1]. However, complex environment of coalmine means the difficulties of power supply of wired monitoring. The mechanical vibration produced in the working process of fully mechanized mining equipment in the coal mine is the most common kind of mechanical energy. Therefore, the construction of efficient vibration energy collection system is called upon to realize the self-power supply of wireless network nodes in coal mine [2, 3].

The piezoelectric vibration energy harvesting, owing to its superiority in structure and application, has been receiving more attention [4, 5]. The structural design of multi-field coupled piezoelectric energy harvester (MPEH) can effectively solve the problems of narrow resonance band of traditional linear piezoelectric harvester. Mechanical vibration is normally expressed as a broad-band random signal. Therefore, design and verification of MPEH should be studied not only under harmonic excitation, but also under random excitation.

2 Design and Modeling

The design of MPEH included four linear-arch beams, four flexible piezoelectric material polyvinylidene fluoride (PVDF), two permanent magnets and a mass block with permanent magnets in MPEH. The structure is illustrated in Fig. 1. Four PVDFs adhere to four metal beams, respectively. Four beams are connected to the same mass block, which has two permanent magnets on it. The distance between two adjustable magnets is changed by adjusting the screw thread of the casing.

Fig. 1
figure 1

Structural diagram of the MPEH

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Generalized Hamilton principle and piezoelectric theory are used [6] to obtain the model:

$$M\ddot{r}(t) + C\dot{r}(t) + Kr(t) - \theta v(t) - K_{1} r(t) - K_{2} r(t)^{3} = - H_{s} \ddot{z}(t)$$
(1)
$$\theta r(t) + C_{p} v(t) + q = 0$$
(2)

where \(r\left( t \right)\) and \(v\left( t \right)\) are the displacement mode function and voltage mode function, respectively. \(\ddot{z}\left( t \right)\) denotes the acceleration of the base excitation. And, it will be instead as Gaussian white noise \(W\left( \tau \right)\), which the spectral density is \(K_{n}\), when excitation is random.

The axial and radial vibration signals of rocker arm in no-load operation of coal mining machine are measured and analyzed, respectively. It is noted that the highest frequency range is between 15 and 25 Hz. Therefore, determining the structure size of the system to meet the requirements of excitation frequency is one of the important aspects of MPEH optimization design. The regularity of the influence of structural size is illustrated in Fig. 2.

Fig. 2
figure 2

Influence of structural parameters on resonance frequency of the MPEH

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3 Dynamic Response

In order to study the effect of magnet distance, noise spectral density and system damping on the response of the MPEH, the system response under random excitation is solved by the FPK equation method. The response results under longitudinal and horizontal excitation conditions are tested. In the following figures, the left are the longitudinal excitation condition, and the right are horizontal excitation (Figs. 3, 4, 5 and 6).

Fig. 3
figure 3

Stationary probability distribution under different magnetic distances

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Fig. 4
figure 4

Stationary probability distribution under different noise spectral densities

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Fig. 5
figure 5

Power spectral density under different magnetic distances

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Fig. 6
figure 6

Power spectral density under different damping ratios

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4 Conclusion

In order to adapt to the working frequency of coal mine machinery, structural optimization design of MPEH and the test under excitation conditions are carried out. Under longitudinal excitation, add magnetic attraction and the magnetic distance is reduced, the variance of vibration probability distribution of the system increases and the probability of large period vibration increases. However, the influence of the magnetic distance is the opposite under horizontal excitation. Reducing damping ratio or increasing noise spectral densities will increase the system output.