Study on Nonlinear Random Vibrations of Structures Carrying Liquid Tanks

研究了1：2内共振附近的弹性结构与圆柱形储罐内液体晃动的非线性随机相互作用。该结构由一个质量-弹簧-阻尼器系统模拟，该系统承载一个竖直的圆柱形液体储罐，在垂直方向上受到窄带随机激励。激励是由受高斯白噪声影响的线性成形滤波器的响应产生的。利用Galerkin方法，建立了考虑(1，1)、(0，1)和(2，1)三种晃动模式的分析模型。此外，节点直径与原振型(1，1)和(2，1)垂直的振型也被考虑在内。通过实验验证了该分析模型在正弦激励下的有效性。在随机激励的情况下，利用蒙特卡罗模拟方法对系统的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。结果表明，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非规则的能量交换。在接近这一条件下，由于同频率的正交模的非线性耦合，液体发生旋转(涡流)运动。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)、部分发展运动和完全发展的随机运动。当液面高度偏离正态分布时，结构响应概率密度函数近似为正态分布。发生液体运动的不稳定区域随着激发强度的增加或带宽的减小而变宽。此外，还发现如果只考虑一种晃动模式，则可以得到结构位移的统计量。

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.