Kelvin Viscoelasticity and Lagrange Multipliers Applied to the Simulation of Nonlinear Structural Vibration Control

Abstract This study proposes a new pure numerical way to model mass / spring / damper devices to control the vibration of truss structures developing large displacements. It avoids the solution of local differential equations present in traditional convolution approaches to solve viscoelasticity. The structure is modeled by the geometrically exact Finite Element Method based on positions. The introduction of the device's mass is made by means of Lagrange multipliers that imposes its movement along the straight line of a finite element. A pure numerical Kelvin/Voigt like rheological model capable of nonlinear large deformations is originally proposed here. It is numerically solved along time to accomplish the damping parameters of the device. Examples are solved to validate the formulation and to show the practical possibilities of the proposed technique摘要 本研究提出一种全新的纯数值建模方法，用于对承受大位移的桁架结构的振动控制用质量-弹簧-阻尼器（mass/spring/damper）装置进行建模。该方法规避了传统粘弹性求解卷积方法中所涉及的局部微分方程求解步骤。研究采用基于位置的几何精确有限元法（Finite Element Method）对目标结构进行建模。通过拉格朗日乘子（Lagrange multipliers）引入该装置的质量，该乘子可约束装置沿有限元的直线方向运动。本文首次提出一种可适配非线性大变形的纯数值型开尔文-沃伊特（Kelvin/Voigt）类流变模型，并通过时域数值求解获取该装置的阻尼参数。最后通过若干算例验证了所提公式的正确性，并展示了该技术的实际应用潜力。