# Vibration Analysis of Piezoelectric Cantilever Beams with Bimodular Functionally-Graded Properties
## 1 Introduction  
The piezoelectric effect is among the most exploited transduction mechanisms for multiscale electromechanical applications, such as sensors, actuators and energy conversion devices, in which piezoelectric vibrational energy harvesting is attractive. For piezoelectric materials, there is a wide spectrum, from piezoelectric ceramics, perovskite structured lead zirconate titanate (PZT), to piezoelectric polymer films, polyvinylidene fluoride (PVDF). Among the piezoelectric materials, piezoelectric ceramics (PZT) is currently regarded as the most promising material system of piezoelectric vibrational energy harvesting devices since it can produce large output power, effective electromechanical coupling and high mechanical strain under an applied electric field [1, 2]. Usually, piezoelectric sensors are a laminated original made of ceramic slice; thus, it is easy to result in stress concentration and also to develop interfacial microcracks. For overcoming this difficulty, functionally-graded piezoelectric materials (FGPM), whose properties of materials continuously change along certain direction, are developed. In FGPM, the obvious interface is disappeared, thus effectively avoiding the damage caused by the stress concentration at the interface. Studies concerning FGPM and corresponding structures made of FGPM have attracted the interests of scholars from all over the world [3, 4, 5, 6, 7, 8]. In the existing works, the vibration problems of piezoelectric structures with functionally-graded properties have been extensively studied, and some valuable results were obtained. Based on the first-order shear theory, [PERSON] et al. [9] studied the free vibration of a rotating circular nanoplate composed of two directional functionally-graded piezo materials (two directional FGPM). The steady-state forced vibration of functionally-graded piezoelectric beams was investigated by [PERSON] and [PERSON] [10]. [PERSON] and [PERSON] [11] made static and dynamic analysis of thick functionally-graded plates with piezoelectric layers based on the layerwise finite element model. [PERSON] [12] investigated, analytically, the vibrations and dynamic response of functionally-graded plate, which is integrated with piezoelectric layers in thermal environment. Using a numerical method, [PERSON] et al. [13] investigated the transient response and natural vibration of FGPM curved beam. [PERSON] et al. [14] studied the free vibration of FGM beams with surface-bonded piezoelectric layers, which is statically thermal post-buckled and subjected to both voltage and temperature rise. [PERSON] and [PERSON] [15] investigated the dynamic response and vibration of FGM plates with piezoelectric actuators in thermal environments. [PERSON] et al. [16] made a nonlinear analysis of free vibration, dynamic stability and buckling for the FGPM beams in thermal environment. [PERSON] and [PERSON] [17] studied the free vibration of a FGPM beam by using state-space based differential quadrature. Considering that there have been many studies in this field, it is not necessary to review them in detail here. Compared with the FGPM, the bimodular effect of materials is relatively less known. However, many investigations have indicated that some materials [18; 19], such as graphite, plastics, ceramics, concrete, steel, polymeric materials, powder metallurgy materials and some composites, will perform different elastic properties when they are in tension and compression; that is, they have different moduli when tensioned and compressed and thus are named bimodular materials [20] (see Figure 1), in which \(\sigma\) is the stress, \(\varepsilon\) is the strain and \(E^{+}\) and \(E^{-}\) represents the tensile modulus of elasticity and compressive one, respectively. In 1982, the Elasticity Theory of Different Moduli, by [PERSON] [21], was published, in which the constitutive model for bimodular materials and the corresponding application in structural analysis were introduced systematically.