# Non-Linear Piezoelectric Actuator with a Preloaded Cantilever Beam

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preloaded Piezoelectric Actuators

_{s}and pretensions determine the degree of nonlinearity. At a critical pretension, the elastic restoring force of the beam is overcome by the spring tension, and the cantilever beam is deflected into one of two potential wells. When the pretension is close to the critical value of bifurcation, the effect of the tip displacement of the beam is enhanced.

_{0}cos(ω

_{v}t + φ

_{v}) was considered, where the driving amplitude V

_{0}and frequency were allowed to vary. The electric field across the thickness of the laminate generates a stress along the piezoelectric components, which bends the beam, and the inertia of the cantilever counteracts the bending force. For the sake of completeness, we fully derive a set of ordinary differential equations for the nonlinear system in the next section.

## 3. Mathematical Model of Preloaded Piezoelectric Actuators

^{2}/∂x

^{2}. Then, the potential energy in the coordinate reference frame is:

_{p}

_{,1}is independent of the electric field E

_{z}, and the subscript z denotes the direction along the thickness. U

_{p}

_{,2}couples the electric field with the stress and energy at the position corresponding to U

_{p}

_{,3}:

_{p}

_{,s}consists of the strain energy resulting from the bending of the piezoelectric patch and is:

_{p}is the laminate thickness. The piezoelectric coupling coefficient can be written as shown in Equation (9) in terms of the more commonly specified coupling coefficient d

_{ij}as:

_{ij}refer to the direction of the applied field and the polarity, respectively. Then, Equation (7) can be written as:

_{b}is the stiffness of the loaded spring and Δl

_{b}(t) represents the distance changes in the loading spring.

_{b}

_{,0}= L − l

_{b}

_{,0}and l

_{b}

_{,0}is the free length of the loaded spring. When the elongation of the loaded spring (Δl

_{b}

_{,}

_{m}) changes as described, we assume that the length of the beam, s(t), is equal to the initial length of the loaded spring, $s(t)={\displaystyle {\int}_{0}^{l}\sqrt{1+{w}^{\prime}{(x,t)}^{2}}dx}$ The initial length is the length of the loaded spring before the beam is bent. In addition, the distance between the two tips of the beam, l

_{b}(t), is equal to the instantaneous length of the loaded spring, ${l}_{b}(t)=\sqrt{{L}^{2}+w{(L,t)}^{2}}$.

_{b}is the kinetic energy of the loaded spring. Because the mass of the loading spring is assumed to be negligible compared to the beam mass, T

_{b}= 0.

Parameter | Symbol | Value |
---|---|---|

Substrate properties | ||

Length | l_{a} | 120 mm |

Width | b | 20 mm |

Thickness | h_{s} | 0.2 mm |

Density | ρ_{s} | 8650 kg/m^{3} |

Young’s modulus | E_{s} | 112 GPa |

Damping ratio | ζ | 0.01 |

Piezoelectric laminate properties | ||

Thickness | h_{p} | 0.25 mm |

Density | ρ_{p} | 7700 kg/m^{3} |

Young’s modulus | E_{p} | 63 GPa |

Coupling coefficient | d_{31} | −630 × 10^{−12} C/N |

Laminate permittivity | ε^{s}_{xx} | 3200 ε_{0} |

Permittivity of free space | ε_{0} | 8.854 × 10^{−12} F/m |

Loading spring properties | ||

Stiffness | k_{b} | 10 kN/m |

Length | l_{b}_{,0} | 100 mm |

Pretension | Δl_{b} | 1 mm |

## 4. Numerical Investigation of the Actuator for Nonlinear Oscillations

#### 4.1. The Nonlinear Elastic Energy of the Actuator

_{b}is the pretension of the loading spring. From Figure 3, it can be observed that as the pretension is increased, the repelling energy will exceed the total elastic potential of the actuator at a critical value.

**Figure 4.**Static mechanical potential energy for a loading spring pretension of (

**a**) 0.01 mm, (

**b**) 1.25 mm and (

**c**) 2 mm. The quadratic restoring potential of the beam (thin solid line) and the nonlinear potential of the loading spring (dotted line) are added to give the total potential energy (dark solid line) as a function of the tip displacement.

_{b}= 2 mm, a hardening spring force develops before a pitchfork bifurcation generates two new equilibrium points. Figure 4c shows the post-bifurcation result at an initial elongating length of 2 mm, where a symmetric double-well potential can be observed. Because this study involves a novel actuator, it was necessary to investigate its harmonic displacement response.

**Figure 5.**The equilibrium position of the loading piezoelectric actuator (k

_{b}= 10 kN/m). The dashed line denotes unstable equilibrium positions.

#### 4.2. Pre-Buckled Response

_{A}was used, which is the ratio of the amplitude with loading to that without loading. As observed from Figure 6b, λ

_{A}increases with the pretension of the loading spring when the cantilever beam is pre-buckled. Apparently, the extreme value of the amplification can be obtained when the cantilever approaches the critical scenario.

_{A}; the simulation results show that λ

_{A}decreases with increasing applied voltage. Specifically, when Δl

_{b}= 0.4 mm (Figure 7a) and the applied voltage increased to 200 V from 10 V, the displacement amplification coefficient decreased to 2.1545 from 2.1887; when Δl

_{b}= 0.8 mm (Figure 7b), λ

_{A}decreased to 4.5903 from 4.7681; and when Δl

_{b}= 1.2 mm (Figure 7c), λ

_{A}decreased to 11.4578 from 16.5957. For large applied voltages, the decrease is obvious.

**Figure 6.**The simulation results for the (

**a**) amplitudes and (

**b**) amplified factors for a stiffness of 10 kN/m, applied voltage of 10 V, and pretension of 0.4, 0.8, and 1.2 mm over a range of frequencies.

**Figure 7.**The simulated results for a stiffness of 10 kN/m, applied voltage of 10 V (dash-dot), 50 V (dot), 100 V (dashed), 200 V (solid), and pretension of (

**a**) 0.4 mm, (

**b**) 0.8 mm and (

**c**) 1.2 mm over a range of frequencies.

#### 4.3. Post-Buckled Response

_{b}≠ 0. Figure 8b,c show that the displacement of the beam and its amplification coefficient were significantly increased and were greater than the pre-buckled values. λ

_{A}decreases with increasing pretension when the cantilever is buckled. For instance, λ

_{A}decreases to 4.3129 from 131.9766 when the pretension increases to 1.6 from 1.3 mm. An interesting result from Figure 8a is that the equilibrium position varied with the frequency of the applied voltage. The oscillations of the cantilever were on either side of the unstable zero displacement position when the frequency was approximately the natural frequency. Under the other condition, the beam oscillated around one of the equilibrium positions because the frequency was far from the natural frequency.

_{A}is significant; specifically, there is an appropriate driven voltage for each pretension of the cantilever that can produce the maximum λ

_{A}. The largest λ

_{A}of the three driven voltages were obtained for 10, 100, and 200 V with pretensions of 1.30, 1.34, and 1.37 mm, respectively. Based on these results, we observe that the appropriate voltage increases with increasing pretension. According to Figure 3, the potential well depths of the cantilever increase with increasing pretension. The relationship between potential energy and kinetic energy in Equation (1) indicates that a deeper potential well requires more input energy, which is provided by the electric energy through the piezoelectric effect. In other words, the required voltage increases with increased pretention, as observed earlier based on the results in Figure 9.

**Figure 8.**The simulation results for the (

**a**) balance positions, (

**b**) amplitudes of the tip displacement and (

**c**) amplified factors for pretensions of 0 mm (dash-dot), 1.2 mm (dot), 1.3 mm (dash), and 1.6 mm (solid).

**Figure 9.**The simulated results with applied voltages of (

**a**) 10 V, (

**b**) 100 V, and (

**c**) 200 V for pretensions of 1.30 mm (dot), 1.34 mm (dashed), and 1.37 mm (solid).

## 5. Experimental Results and Discussion

#### 5.1. Measurement and Instrumentation

**Figure 10.**(

**a**) The structure of the experimental pretension actuator; (

**b**) the remaining necessary experimental equipment.

#### 5.2. Comparisons of Experiment with Theory

_{A}increases with increasing pretension. However, the amplitudes of the tip displacements increased from 0.19 mm to 1.2 mm when the pretension increased from zero to 1.29 mm. When the pretension was 1.4 mm, the beam was post-buckled, and the amplitudes of the tip displacements decreased to 0.5 mm. The balance position was one of two equilibrium values of the potential. The resonant piezoelectric cantilever beam provides greater values for the extremes of the tip displacement output only when the pretension is at or very close to its critical bifurcation pretension (1.29 mm); then, the displacement output of the actuator can be six times larger than the actuator without pretension. Hence, a displacement output that can increase by an order of magnitude larger over a frequency range can be expected with this device.

**Figure 11.**The response measured via laser experiment (points) and simulation (lines) for a stiffness of 10 kN/m and pretensions of Δl

_{b}= 0 mm (asterisks and solid line), 0.4 mm (circles and dashed line), 1.29 mm (triangles and dotted line), and 1.4 mm (squares and dashed-dotted line) over a range of frequencies.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Wu, Y.; Dong, J.; Li, X.; Yang, Z.; Liu, Q.
Non-Linear Piezoelectric Actuator with a Preloaded Cantilever Beam. *Micromachines* **2015**, *6*, 1066-1081.
https://doi.org/10.3390/mi6081066

**AMA Style**

Wu Y, Dong J, Li X, Yang Z, Liu Q.
Non-Linear Piezoelectric Actuator with a Preloaded Cantilever Beam. *Micromachines*. 2015; 6(8):1066-1081.
https://doi.org/10.3390/mi6081066

**Chicago/Turabian Style**

Wu, Yue, Jingshi Dong, Xinbo Li, Zhigang Yang, and Qingping Liu.
2015. "Non-Linear Piezoelectric Actuator with a Preloaded Cantilever Beam" *Micromachines* 6, no. 8: 1066-1081.
https://doi.org/10.3390/mi6081066