Study on Nonlinear Random Vibrations of Structures Carrying Liquid Tanks

研究了圆柱槽内液体晃动弹性结构在1:2内共振邻域下的非线性随机相互作用。该结构以承载直立圆柱形液罐的质量-弹簧-阻尼系统为模型，在垂直方向上受到窄带随机激励。该激励由高斯白噪声作用下的线性整形滤波器的响应产生。采用伽辽金方法，建立了考虑三种晃动模式的解析模型；（1,1）（0,1）和（2,1）。此外，还考虑了节点直径垂直于原模态（1,1）和（2,1）的模态。在正弦激励下，通过实验验证了分析模型的有效性。在随机激励情况下，采用蒙特卡罗模拟方法对系统响应统计量进行了数值估计。给出了随机激励的中心频率、带宽和液位对系统响应的影响。研究发现，当中心频率接近结构固有频率时，结构与液体自由表面运动之间存在不规则的能量交换。在此条件下，液体的旋转（旋流）运动是由同频率正交模态的非线性耦合引起的。根据激励功率谱密度的不同，液体自由表面经历零运动、不确定运动（间歇性）、部分展开运动和完全展开的随机运动。结构响应概率密度函数近似为高斯分布，而液体高度偏离正态分布。随着激发强度的增大或带宽的减小，液体运动发生的不稳定区域变宽。此外，还发现仅考虑一种晃动模态即可获得结构位移的统计量。

The nonlinear random interaction of an elastic structure with liquid sloshing in a cylindrical tank is investigated in the neighborhood of a 1 : 2 internal resonance. The structure, modeled by a mass-spring-dashpot system carrying an upright cylindrical liquid tank, is vertically subjected to a narrow-band random excitation. The excitation is generated from the response of a linear shaping filter subjected to a Gaussian white noise. By using the Galerkin method, the analytical model is constructed taking into account three sloshing modes ; (1,1), (0,1) and (2,1). In addition, the modes whose nodal diameters are perpendicular to those of the original modes (1,1) and (2,1) are also taken into account. The validity of the analytical model was certified by experiments when the excitation was sinusoidal. In the case of random excitation, the system response statistics are numerically estimated using Monte Carlo simulation. The influence of the center frequency and the bandwidth of the random excitation and the liquid level on the system responses are presented. It is found that there is an irregular energy exchange between the structure and the liquid free surface motion when the center frequency is close to the structure natural frequency. Close to this condition, the liquid rotational (swirl) motion takes place due to the nonlinear coupling of orthogonal modes of the same frequency. Depending on the excitation power spectral density, the liquid free surface experiences zero motion, uncertain motion (intermittency), partially developed motion, and fully developed random motion. The structure response probability density function is almost Gaussian, while the liquid elevation deviates from normality. The unstable region, where the liquid motion occurs, becomes wider as the excitation intensity increases or as the bandwidth decreases. In addition, it is found that the statistics of the structure displacement can be obtained if only one sloshing mode is considered.