A smooth polynomial shaped command with an adjustable command time length is proposed for eliminating the residual vibrations of a multi-mode system. The ability of eliminating jerks and vibrational modes, regardless of their number, offers the most advantage of the proposed command. A numerical simulation is conducted to test the command’s effectiveness by eliminating the residual sloshing oscillations of a liquid-filled container conveyed by an overhead crane in a rest-to-rest manoeuvre. The governing equations of the liquid free-surface level are derived by modelling the sloshing dynamics by a series of mass–spring–damper harmonics. The proposed model accounts for the coupling between the pendulum dynamics and the sloshing equivalent mechanical model. The command’s robustness to the system parameters’ uncertainties, liquid depth and cable length, are investigated as well. The ability of adjusting the command length and retaining higher sloshing modes in command-designing are also outlined.

中文翻译：

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用于抑制悬浮液体容器晃动的平滑多项式命令

为了消除多模系统的残余振动，提出了一种具有可调命令时间长度的平滑多项式命令。消除急动和振动模式的能力，无论它们的数量如何，都提供了建议命令的最大优势。通过消除由桥式起重机运送的充满液体的容器在静止到静止动作中的残余晃动振荡，进行数值模拟以测试命令的有效性。液体自由表面水平的控制方程是通过一系列质量-弹簧-阻尼器谐波对晃动动力学建模得出的。所提出的模型考虑了摆动力学和晃动等效力学模型之间的耦合。命令对系统参数不确定性的鲁棒性，还研究了液体深度和电缆长度。还概述了在指令设计中调整指令长度和保持较高晃动模式的能力。