超材料混凝土单胞骨料的有阻尼自由振动特性研究

超材料概念的提出激发了功能混凝土的研究热潮，超材料混凝土的振动研究也成为热点之一。基于one-dimensional spring-mass system模型，建立了单胞骨料有阻尼自由振动的三维振动模型，模型中软涂层和砂浆均用弹簧等效代替，并计算了等效长度、等效刚度和等效阻尼。以单胞骨料为研究对象，用达朗贝尔原理导出有阻尼自由振动的非线性控制方程，获得系统的固有频率和振动解。用MATLAB软件编程进行算例分析，讨论初始条件、骨料内部尺寸及软涂层弹性模量等参数对固有频率和振动特性的影响。算例分析表明：非线性振动与线性振动中的无阻尼固有频率和有阻尼固有频率间的关系并不一致；阻尼对非线性振动固有频率的影响比线性振动的大；初速度对初相位的影响(非线性)比初始位移的大(“类线性”),初始条件既影响振幅，又影响固有频率，也会影响超材料混凝土抗冲击能力。减幅系数η随初始速度、软涂层弹性模量的增大而减小，随软涂层厚度的增大而增大，这可通过减小初始速度、增大软涂层弹性模量、增加重核的大小等方式降低减幅系数对超材料混凝土结构的影响。这对超材料混凝土抵抗高频荷载和冲击荷载的研究有工程指导价值。

A study of damped free vibration characteristics on a metaconcrete unit cell

The research boom of functional concrete is aroused by the concept of metamaterials. In the paper, based on the one-dimensional spring-mass system model, a three-dimensional vibration model of a unit cell that was taken as the study object was established under linearelastic deformation. The soft coating and mortar were assumed as spring that have the equivalent stiffness, the equivalent length and the equivalent damping in the model. Analyzed the force of the system, the equation of damped free vibration was deduced using the D’Alembert law. The system of vibration solution and natural frequency was obtained. The influences of the initial conditions, aggregate geometry data, the elastic modulus of the soft coating on the natural frequency and the vibration characteristics were analyzed by using the MATLAB. Results show that: the relationship between the undamped natural frequency and the damped natural frequency of the nonlinear vibration system is not consistent with the linear vibration system, and the influence of damping on the natural frequency of nonlinear vibration system is greater than that of linear vibration system. The initial velocity has greater influence on the initial phase than the initial displacement. The initial conditions affect both the amplitude and the natural frequency, as well as the impact resistance of metaconcrete. The reduction factor η decreases with the initial velocity and the elastic modulus of the soft coating. The reduction factor η increases with the thickness of the soft coating. The influence of the reduction factor on metaconcrete structure can be reduced by reducing the initial speed, selecting the soft coating with a larger elastic modulus and increasing the size of the heavy core.