The lumped mass model is derived from a one-mode Galerkin discretization with the Gauss–Lobatto quadrature applied to the non-linear swelling pressure term. Our reduced-order model of the problem is then analyzed to study the essential dynamics of an elastic cantilever Euler–Bernoulli beam subject to the swelling pressure described by Grob’s law. The solutions to the initial value problem for the resulting nonlinear ODE are proved to be always periodic. The numerical solution to the derived lumped mass model satisfactorily matches the finite difference solution of the dynamic beam problem. Including the effect of oscillations at the base of the beam, we show that the model exhibits resonances that may crucially influence its dynamical behavior.

集总质量模型是从单模Galerkin离散化获得的，高斯-洛巴托正交应用于非线性膨胀压力项。然后，我们对问题的降阶模型进行了分析，以研究弹性悬臂式Euler–Bernoulli梁在Grob定律描述的膨胀压力作用下的基本动力学。所得非线性ODE的初值问题的解证明总是周期性的。导出的集总质量模型的数值解令人满意地匹配了动态梁问题的有限差分解。包括梁底部振动的影响，我们表明模型显示的共振可能会严重影响其动力学行为。