Study on Nonlinear Random Vibrations of Structures Carrying Liquid Tanks

在1：2内共振附近，研究了圆柱形贮箱中液体晃动与弹性结构的非线性随机相互作用。该结构由一个带有直立圆柱形贮液箱的质量-弹簧-阻尼器系统模拟，在竖直方向上受到窄带随机激励。激励是从经受高斯白色噪声的线性成形滤波器的响应产生的。采用Galerkin方法，建立了考虑（1，1）、（0，1）和（2，1）三种晃动模态的分析模型。此外，还考虑了节径与原模（1，1）和（2，1）节径垂直的模。通过实验验证了正弦激励下分析模型的有效性。在随机激励的情况下，系统的响应统计数字估计使用Monte Carlo模拟。分析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。接近于该条件，由于相同频率的正交模式的非线性耦合，液体旋转（涡旋）运动发生。取决于激励功率谱密度，液体自由表面经历零运动、不确定运动（不确定性）、部分发展的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高斯分布。液体运动发生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，研究发现，如果只考虑一种晃动模态，结构位移的统计特性可以得到。

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.