## 3 Analysis for Forced Vibration Excited by Piezoelectric Sheets
### Forced Vibration Analysis of a Touch Beam
Figure 6: 3 rd–6 th order forced response image of the cantilever touch beam. (**a**) Third-order forced response; (**b**) Fourth-order forced response; (**c**) Fifth-order forced response; (**d**) Sixth-order forced response. From Figure 7, the following points are worth noting. * We took the peak values of excitation voltage of 100 V, 150 V, and 200 V as calculation examples. As the value of the excitation voltage increases, the amplitude of the forced response gradually increases, and the larger the excitation frequency, the smaller the amplitude increase. It can be seen from the curves that when the excitation frequency is at the sixth-order natural frequency, the excitation voltage is increased from 100 V to 200 V, and the vibration amplitude is changed from \(1.44\times 10^{-6}\) m to \(2.62\times 10^{-6}\) m. More importantly, it can be speculated that increasing the excitation voltage can make the tactile change of the cantilever touch beam easier to perceive. * The effect of choosing piezoelectric sheets of different thicknesses on the forced response is verified by bringing in piezoelectric sheets with thicknesses of 0.3 mm, 0.4 mm, and 0.5 mm. As the thickness \(h_{p}\) decreases, the vibration amplitude of the cantilever touch beam gradually increases. The larger the excitation frequency, the smaller the amplitude increases. For increasing the displacement of forced vibration to improve the sensitivity of touch sensation, we can use the thinner piezoelectric sheets as an excitation element. * Investigating the influence of the material properties of the touch beam substrate on the vibration amplitude of the forced response, especially the elastic modulus \(E\), can provide a basis for selecting materials during design. We chose the elastic modulus of the touch beam as a research subject and analyze the effect of the three materials of brass, phosphor bronze, and spring steel, respectively, on the forced response. As the elastic modulus increases, the forced response of the cantilever touch beam gradually decreases. Therefore, the material of the touch beam substrate with the higher elastic modulus would reduce the forced response displacement to a greater extent. * In addition, the piezoelectric constant that measures the quality of the piezoelectric material is also an index that affects the forced response performance. The piezoelectric constant \(e_{31}\) was changed so that \(e_{31}\) takes values of 2.0 c/m\({}^{2}\), 2.5 c/m\({}^{2}\), and 3.0 c/m\({}^{2}\), respectively. As the piezoelectric constant of the piezoelectric element increases, the response amplitude of the cantilever touch beam gradually increases. It can be seen from the image that when the excitation frequency is at  
Figure 7: 6 th-order forced response amplitude of the touch beam. (**a**) Forced response changes of different voltage \(V_{3}\); (**b**) Forced response changes at different the thickness of piezoelectric sheet \(hp\); (**c**) Forced response changes of different elastic modulus \(E\); (**d**) Forced response changes of different piezoelectric constant \(e_{31}\). the sixth-order natural frequency, the piezoelectric constant is increased from 2.0 to 3.0, and the vibration amplitude increases from \(1.44\times 10^{-6}\) m to \(2.37\times 10^{-6}\) m, and the variation is also obvious. We recommend using a high-quality piezoelectric material with a large piezoelectric constant to enhance the perceptibility of the touch beam.