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In this work, piezoelectric energy harvesting (PEH) performance via friction-induced vibration (FIV) is studied numerically. A nonlinear two-degree-of-freedom friction system (mass-on-belt) with piezoelectric elements, which simultaneously considers the stick-slip motion, model coupling instability, separation, and reattachment between the mass and belt, is proposed. Both complex eigenvalue analyses and transient dynamic analysis of this nonlinear system are carried out. Results show that it is feasible to convert FIV energy to electrical energy when the friction system is operating in the unstable vibration region. There exists a critical friction coefficient (_{c}) for the system to generate FIV and output visible voltage. The friction coefficient plays a significant role in affecting the dynamics and PEH performance of the friction system. The friction system is able to generate stronger vibration and higher voltage in the case that both the kinetic friction coefficient and static friction coefficient are larger than _{c}. Moreover, it is seen that the separation behavior between contact pair can result in overestimating or underestimating the vibration magnitude and output voltage amplitude, and the overestimate or underestimate phenomenon is determined by the located range of friction coefficient. Furthermore, it is confirmed that an appropriate value of external resistance is beneficial for the friction system to achieve the highest output voltage. The obtained results will be beneficial for the design of PEH device by means of FIV.

Recently, with the problem of energy shortage becoming more and more serious, energy harvesting from ambient sources has become an important research field and received lots of attention from academia and industry [

Generally, five main techniques can realize the conversion of vibration energy to electric energy, i.e., electromagnetic, electrostatic, piezoelectric, magnetostrictive, and triboelectric [

To solve this problem, a new approach is proposed: seeking a kind of vibration energy which is independent of the ambient frequency and can be used for harvesting and conversion, such as friction-induced vibration (FIV). FIV is a typical self-excited vibration, which does not require any exterior force excitation, but it can be triggered under specific working conditions [

Up to now, many works have pointed out that the most probable reasons for the occurrence of FIV are stick-slip motion and model coupling phenomenon [

In this work, the PEH performance by means of FIV is analyzed numerically. A nonlinear two-degree-of-freedom (DOFs) friction system (mass-on-belt) with piezoelectric elements, which simultaneously considers the stick-slip motion, model coupling, separation, and reattachment between the mass and belt, is proposed. Through introducing piezoelectric elements into the friction system, the energy harvesting behaviors along both the tangential and normal directions by means of FIV are verified. Moreover, the effects of friction coefficient and the separation behavior between contact pair on the dynamics and energy harvesting are studied. Finally, the dynamics behaviors of the friction system with different external electric resistances are analyzed, and the role of electric resistances in affecting PEH performance is revealed. Some conclusions on the piezoelectric energy harvesting related to FIV are presented.

A friction system with two DOFs is employed to study the PEH via FIV; this system is extended from the work of Tadokoro et al. [

A 2-DOF friction system.

The slider (_{1} and the damper _{1} connect with the slider along the tangential direction (_{2} connects with the slider along the normal direction (_{3} is linked to the slider at a 135° angle relative to the tangential direction, which provides coupling in the tangential and normal directions. Moreover, spring _{2} and _{nl}, which are characterized by linear and cubic spring, respectively, are used to simulate the contact stiffness between contact pair. The effects of impact at reattachment between the slider and belt are neglected [

If the slider is still in contact with the belt, the contact force _{N} between the contact pair is expressed as

The equations of friction motion for the slider yield

In the above equation, the parameters

If _{2} and _{nl} will not work on the contact stiffness. As a consequence, the situation where the slider separates with the belt can be depicted as shown in Figure

The situation where slider separates with the sliding belt.

Assuming that the belt surface is rigid, the condition of separation between the contact pair is determined by the normal displacement of slider, which is written as

When the normal displacement

It is meaningful to study the stability of the slider in the vicinity of the origin, which will be beneficial for detecting the critical friction coefficient

The equilibrium points (EPs) (_{e}, _{e}, _{xe}, and _{ye}) of the friction system are found by solving (

Define the transformation relationships

The Jacobian matrix can be derived via expanding the nonlinear equation (

The stability of the system can be evaluated by calculating the eigenvalues

The parameters set in this analysis are from the work performed by Tadokoro et al. [

The parameters of the friction system set in the eigenvalues analysis.

Mass (slider) | 0.08 kg | _{3} (coupling spring in two directions) | 600 N/m |

_{1} (including the stiffness of Piezo-1) | 890 N/m | _{1} = _{2} (including the damping values of piezo elements) | 0.077 N/(m/s) |

_{nl} (nonlinear contact stiffness) | 1200 N/m^{3} | 0.00036 N/V | |

_{2} (linear contact stiffness) | 500 N/m | 1.79 × 10^{−7} F | |

5 mm/s∼200 mm/s | 20 N | ||

_{1} | 10 kΩ | _{2} | 10 kΩ |

The real parts (a) and imaginary parts (b) of the friction system with the increase of friction coefficient.

In the transient dynamic analysis, the stick, slip, separation, and reattachment behaviors between the slider and belt are taken into consideration. Assume that the relative sliding velocity

Therefore, the friction system has two motion phases when the slider is sliding on the belt (^{−6}).

When

Here the parameter

The applied horizontal force is expressed as

and therefore, in the stick phase, the friction force

Then equation (

During sliding stage, the dynamical equation of the system is determined by equation (

Equation (

Visibly, it is a typical nonlinear, nonsmooth, and discontinuous friction system, which brings some difficulties in determining the moment that the motion of the slider transits from one stage to another stage. The flow chart of the numerical computation is shown in Figure

The flow chart of the transient dynamic analysis.

The critical friction coefficient value for the generation of self-excited vibration is determined in the above section, which is equal to 0.85. In this section, friction coefficient is set as 0.6, 0.85, and 1.2, respectively, and used in the transient dynamic analysis. These friction coefficients are lower, equal to, and larger than the critical friction coefficient, respectively. The belt velocity (

The tangential displacement (a), normal displacement (b), tangential output voltage (c), and normal output voltage (d) of the friction system with different friction coefficients.

Figure

The tangential displacement versus tangential velocity. (a) Normal displacement versus normal velocity. (b) Phase portraits of the friction system.

Expectedly, the large-amplitude periodic (limit-cycle) response on the high-energy orbit of the friction system with larger coefficient is also observed in the velocity versus voltage trajectory shown in Figure

The open-circuit voltage versus velocity phase trajectories.

It has been reported that the dynamics of friction system are very sensitive to the contact parameters [

According to the value of critical friction coefficient (0.85), the dynamic behaviors of the friction system with kinetic and static friction coefficients, denoted by _{e}, 0.1, _{e}, 0, 0, 0), with the tangential velocity of the slider equaling the velocity of the belt. The external electric resistance in this analysis is set at 10 KΩ.

The values of friction coefficients and the corresponding Eps in these three cases.

Case | Eps (_{e}, _{e}) | |||
---|---|---|---|---|

1 | 0.4 | 0.6 | 0.85 | (−0.00216, −0.0258) |

2 | 0.7 | 0.9 | 0.85 | (0.00104, −0.0246) |

3 | 1 | 1.2 | 0.85 | (0.00397, −0.0235) |

Figure

The tangential displacement (a), normal displacement (b), tangential output voltage (c), and normal output voltage (d) of the friction system with three different cases of friction coefficients.

Figure

The displacement verus velocity trajectory of the friction system in tangential direction (a) and normal direction (b). The velocity versus voltage trajectory of the friction system in tangential direction (c) and normal direction (d).

The velocity versus voltage trajectory of the friction system in three cases of friction coefficient is plotted. In the tangential direction, it is visible that the orbit of friction system with Case 3 possesses biggest limit cycle, which suggests that the friction system can generate higher amplitude voltage than the other two cases. Meanwhile the output voltage amplitude in Case 1 shows the smallest limit cycle. Indeed, both trajectories from Case 1 and Case 2 exhibit very complicated orbits. In the normal direction, as expected, the orbit of friction system with Case 3 exhibits the largest amplitude of limit cycle, which indicates that a sustained and high voltage is outputted in this direction during the sliding process. Additionally, the orbit of friction system with Case 1 forms a smallest limit cycle, which reflects that relative lower voltage is generated in this situation. This phenomenon can well reflect the time domain signal of voltage shown in Figure

To sum up, the dynamic and output voltages are strongly related to the relationship among the kinetic friction coefficient

In most previous studies of FIV of small models, it is assumed that the slider is still in contact with belt during vibration process [

The flow chart of transient dynamic analysis when the friction system ignores the separation between slider and belt is shown in Figure

The flow chart of transient dynamic analysis when the friction system ignores the separation between belt and slider.

In the previous study, it is found that the separation can occur in Case 2 and Case 3; thus, it is worth noting that comparison between ignoring separation and considering separation will be performed in these two cases.

Numerical results with _{k} = 0.7, _{s} = 0.9, and initial condition (_{e}, 0.1, _{e}, 0, 0, 0) are shown in Figure

The tangential displacement (a), normal displacement (b), tangential voltage (c), normal voltage (d), and contact force (e) of the friction system in the time domain when _{k} = 0.7 and _{s} = 0.9.

The contact forces of these two situations are plotted in Figure

The displacement verus velocity phase trajectories of the friction system in both directions are shown in Figures

The displacement versus velocity phase trajectory of the friction system in tangential direction (a) and normal direction (b). The velocity versus voltage trajectory of the friction system in tangential direction (c) and normal direction (d) when _{k} = 0.7 and _{s} = 0.9.

Furthermore, the velocity versus voltage trajectory is plotted in Figures

Figure _{k} = 1 and _{s} = 1.2, and the initial condition of system is set at (_{e}, 0.1, _{e}, 0, 0, 0). It can be seen that the dynamics and output voltage behaviors show significant difference between these two cases (ignoring and considering separation). For the tangential direction, the displacement signal shows larger amplitude compared with the situation of considering separation. However, the displacement amplitude in the normal direction shows totally opposite results (Figure

The tangential displacement (a), normal displacement (b), tangential voltage (c), normal voltage (d), and contact force (e) of the friction system in the time domain when _{k} = 1 and _{s} = 1.2.

Expectedly, when separation is ignored, the displacement versus velocity trajectory shows larger limit cycle compared to the case of considering separation in the tangential direction, and thus the corresponding velocity versus voltage trajectory when ignoring separation possesses larger limit cycle in this direction, as shown in Figures

The displacement versus velocity phase trajectory of the friction system in tangential direction (a) and normal direction (b). The velocity versus voltage trajectory of the friction system in tangential direction (c) and normal direction (d) when _{k} = 1 and _{s} = 1.2.

Combining the puzzling results shown in these two situations, a possible physical explanation is provided. When ignoring the separation behavior of the friction interface, the normal spring is stretched and provides continuous resistance to the slider, which accordingly prevents the slider from continuously moving upward, while the friction force still contributes to the system vibration in the tangential direction. In contrast, when separation is considered in this system, the normal contact stiffness will lose its effect when separation occurs; at this time, the friction force disappears, and the normal contact stiffness will not provide resistance force to the normal vibration. Since the tangential and normal vibration is coupled, the vibration of this system when separation is considered is actually more complex. The friction coefficient also plays a significant role in affecting the dynamics and energy harvesting performance of the friction system; visibly, the different friction coefficient will cause complete opposite results. These are very interesting findings and they are useful for designing PEH devices via FIV.

In this section, the effect of

The displacement and output voltage of friction system when

Additionally, the output voltage amplitudes and average generated electric power versus

The output voltage amplitude and electric power of friction system with different

For the curve of output voltage amplitude (red line), it increases gradually at first and then remains stable basically with the increase of

A 2-DOF friction system (slider-on-belt model) with piezoelectric elements, which accounts for the stick, slip, separation, and reattachment between contact pairs, is employed to study the energy harvesting performance via friction-induced vibration (FIV). Both the eigenvalues analysis and transient dynamic analysis are carried out to study the dynamics and energy harvesting behavior of this friction system, and the main conclusions are summarized as follows:

Eigenvalue analysis results show that friction coefficient value (

The dynamic and output voltages are crucially dependent on the relationship among the kinetic friction coefficient

If the separation between contact pair is ignored, it is likely to overestimate or underestimate the vibration magnitude and output voltage amplitude. The overestimation or underestimation phenomenon is mainly determined by the located range of friction coefficient. These are very interesting findings and can be exploited in design of energy harvester by means of FIV.

The external resistance

The data used to support the findings of this study are included within the article.

The authors declare no conflicts of interest with respect to the research, authorship, and/or publication of this article.

This research received financial support from the National Natural Science Foundation of China (no. 51505396).