A nonlinear Euler–Bernoulli model of slender piezoelectric beams is employed to investigate parametric resonance motions driven by a pulsating voltage with a DC component. The beam model is based on 3D electric charge conservation and 1D reduction of the momentum balance laws assuming as space coordinate the arclength along the beam base line as space coordinate. The 3D constitutive relationships for a piezoelectric material are specialized according to the Euler–Bernoulli ansatz of transverse unshearability. The nonlinear piezoelastic problem of the parametrically excited beam is directly attacked by the method of multiple scales up to the fifth nonlinear order overcoming modal projection drawbacks and severe restrictions on the oscillation amplitude range. The transition curves, separating regions of stable and unstable trivial solutions in the voltage frequency–amplitude plane, are obtained for various modes of a PVDF beam. The onset of parametric resonances and the post-critical large motion are investigated upon variations of meaningful parameters such as the prestress DC voltage amplitude. The analytical outcomes are compared with those provided by the finite element code ABAQUS which yields the numerical solution of the full 3D nonlinear piezoelectric problem. Moreover, the frequency–response curves to within fifth order are computed applying the pseudo-arclength method to the algebraic frequency–response equation. The frequency–response curves of the lowest few modes reveal new aspects of the parametric resonance motions which can be exploited in applications such as dynamic morphing of thin surfaces.

细长压电梁的非线性 Euler-Bernoulli 模型用于研究由具有直流分量的脉动电压驱动的参数共振运动。梁模型基于 3D 电荷守恒和动量平衡定律的 1D 归约，假设空间坐标为沿着梁基线的弧长作为空间坐标。压电材料的 3D 本构关系是根据横向不可剪切性的 Euler-Bernoulli ansatz 专门化的。参数激励梁的非线性压弹性问题直接通过多尺度到五阶非线性的方法来解决，克服了模态投影的缺点和对振荡幅度范围的严格限制。过渡曲线，对于 PVDF 梁的各种模式，获得了电压频率 - 幅度平面中稳定和不稳定平凡解的分离区域。在有意义的参数（如预应力直流电压幅度）变化时，研究了参数共振的开始和临界后大运动。将分析结果与有限元代码 ABAQUS 提供的结果进行比较，后者产生全 3D 非线性压电问题的数值解。此外，将伪弧长方法应用于代数频率响应方程，计算出五阶以内的频率响应曲线。最低几种模式的频率响应曲线揭示了参数共振运动的新方面，可以在薄表面的动态变形等应用中加以利用。