Parametric Resonance and Stochastic Dynamic Stability of Structures: Theory, Experiments, and Applications

土木工程中最重要的任务之一是评估结构在动态荷载作用下的安全性和可靠性(例如，地震作用下的建筑物、湍动风激励下的桥梁、岩爆作用下的地下矿柱)。这些载荷通常是随机力，并将显示为控制运动方程中的系数之一。这样的系统被认为是参数激发的，与之相关的不稳定性称为参数共振。随着高层建筑、大跨度桥梁和地下结构的不断建设，结构在随机激励下的动力稳定性研究日益重要。参数共振比普通共振更危险，因为它的特征是响应幅度的指数增长--即使在存在阻尼的情况下--而普通共振的特征是响应幅度的线性增长。参数共振发生在参数空间的一个区域内，激励频率可能高于或低于固有频率。此外，参数共振可能以意想不到的方式与普通共振相互作用。在过去的18年里，申请者和合作者在随机动力学的研究方面取得了重大进展。特别是，开发了近似算法并将其应用于线性单自由度系统，并在莱克黑德大学进行了初步的实验室实验。我的研究团队所获得的知识、专业知识和实验技能将有助于拟议的结构随机稳定NSERC计划。为了将所发展的方法应用于实际的结构系统，有必要将研究扩展到高维系统。该研究项目的目的是从理论和实验上研究随机载荷作用下分数阶粘弹性非线性结构的参数共振和动力稳定性，并将其应用于土木工程和采矿工程。我们将发展有效的解析和数值方法来确定各种具有强和弱非线性的系统的动态稳定性，例如具有可公度和不可公度频率的耦合系统、陀螺系统、时滞系统以及在乘性和加性载荷激励下的多自由度系统。拟议的研究将大大促进与结构动力稳定性相关的知识和技术的进步。通过调整结构参数或爆破变量，增强对随机参数共振的理解，将为方案和措施提供信息，以避免或减轻由地震、风浪或应力波载荷引起的动态灾害。该研究计划将提供在这些相关学科培训HQP的独特和前沿机会。因此，该计划将使加拿大的建筑和挖掘行业受益。

One of the most important tasks in civil engineering is assessing structural safety and reliability under dynamic loads (e.g., buildings during earthquake, bridges excited by turbulent winds, and underground mine pillars subjected to rock blasting). These loadings are often random forces and would appear as one of the coefficients in the governing equations of motion. Such systems are said to be parametrically excited and the associated instability is called parametric resonance. With the construction of more and more high-rise buildings, long bridges and underground structures, investigations of dynamic stability of structures under stochastic excitation are increasingly important. Parametric resonance is more dangerous than ordinary resonance because it is characterized by exponential growth of the response amplitude-even in the presence of damping-whereas ordinary resonance is characterized by linear growth of the response amplitude. Parametric resonance occurs over a region of parameter space and the excitation frequencies may be higher or lower than the natural frequency. Moreover, parametric resonance may interact with ordinary resonance in unexpected ways. Over the last 18 years, the applicant and collaborators have made significant progress in the study of stochastic dynamics. In particular, approximate algorithms were developed and applied to linear single degree-of-freedom systems, and preliminary lab experiments were conducted at Lakehead University. The knowledge, expertise, and experimental skills acquired by my research team will benefit the proposed NSERC program for stochastic stability of structures. In order to apply the developed methods to real structural systems, it is imperative to extend the study to higher-dimensional systems. The objective of the proposed research program is to theoretically and experimentally study parametric resonance and dynamic stability of fractional viscoelastic non-linear structures subjected to stochastic loads, with applications to civil and mining engineering. We will develop efficient analytical and numerical methods to determine the dynamic stability of various systems with both strong and weak non-linearity, such as coupled systems with commensurable and non-commensurable frequencies, gyroscopic systems, time-delay systems, and multiple degree-of-freedom systems excited by both multiplicative and additive loads. The proposed research will contribute significantly to the advancement of knowledge and technologies associated with dynamic stability of structures. Enhanced understanding of stochastic parametric resonance will inform protocols and measures, by adjusting structural parameters or blasting variables, to avoid or alleviate dynamic disasters due to seismic, wind, or stress wave loadings. This research program will provide unique and leading-edge opportunities to train HQP in these related subjects. Thus, this program will benefit the construction and excavation industries in Canada.