非经典阻尼分布参数系统复振型叠加方法

附加减震装置的一维杆或剪切梁模型属于非连续的非经典阻尼分布参数系统。对于它的动力分析,通常是建立分段的运动方程,然后利用各段动力反应的实振型叠加形式和连续条件进行动力计算。这是一种实模态综合方法,尽管它可以求得近似的动力反应,但反映不出阻尼对整体系统动力特性的影响。为了考虑附加减震装置引起的阻尼和刚度非连续性,基于广义函数理论,建立了整体系统的无量纲化运动方程,利用Laplace变换推导了振型函数和特征值方程,并给出了振型函数的正交条件,最终导出了适用于非经典阻尼分布参数系统的复振型叠加方法。由于特征值方程为复杂的超越方程,为了同时求出多个自振频率,建议了一种基于柯西积分定理的等效多项式方法。这种方法将自振频率转变成了线性代数方程组的求解,更简便、实用。最后以基底隔震分布参数系统为例,展示了复振型叠加法的应用,同时对隔震结构设计得出了有益的结论。给出的复振型叠加法是传统的经典阻尼连续系统实振型叠加法的推广,具有一定理论意义和应用价值。

Complex mode superposition method for distributed-parameter systems with non-classical damping

One-dimensional bar or shear-type beam with additional energy dissipation devices is a distributed-parameter system with non-classical damping.For its dynamic analysis,the conventional way is to construct an equation of motion for each segment and obtain the dynamic response by using the real mode superposition method and the continuous condition.In essence,this method is a component mode synthesis based on undamped modes of substructure.Even though approximated dynamic responses can be estimated,it cannot consider the effect of damping on the dynamic behavior.To consider the discontinuity of damping and stiffness resulting from the additional damper,utilizing the generalized function theory,one non-dimensional equation of motion for the whole system is constructed in this paper.Then,using the Laplace integral transformation,the eigen function(complex mode)and eigenvalue equation are derived.Finally,the complex mode superposition method for distributed-parameter systems with non-classical damping is developed based on the derived orthogonality condition of eigen functions.In addition,the eigenvalue equation is a very complex transcendental equation,in order to get several natural frequencies,an equivalent polynomial method based on the Cauchy integral theorem is proposed,in which the eigenvalue equation is transformed into a set of linear equations such that their solutions can be obtained more easily.In the last section of this paper,the application of the proposed method is illustrated in a base-isolated shear-type beam and some useful information for the design of base-isolated structures is provided.To summarize,the complex mode superposition method is an extension of the conventional real mode superposition method for classically damped continuous and distributed-parameter systems,which is meaningful and valuable in the theory and application.