回传射线矩阵法的理论及其应用
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摘要
回传射线矩阵法(MRRM)自1998年公开提出以来,已被成功应用于平面框架结构的瞬态响应分析以及各向同性和横观各向同性层状介质的瞬态波传播分析。为了更深刻地理解MRRM,进一步拓展其求解范围,以期推广MRRM在科学研究和实际工程中的应用,本文研究MRRM的数学理论基础,并探讨其在复杂问题中的应用。
首先,从矩阵微分方程的解出发,给出MRRM列式的数学依据和统一步骤,阐述MRRM所能求解的动力问题。从数学列式和物理含义的角度,给出回传射线矩阵法与频谱单元法、动力刚度法、有限单元法、传递矩阵法和波传散射法的联系和区别。提出基于MRRM列式借助计算机数值求解复杂结构系统各种动力问题所需要的计算方法。
其次,考虑各种复杂因素(包括节点集中质量、弹簧及阻尼支承、杆件单元的材料阻尼、多重调谐质量阻尼器、荷载作用和位移激励),推导便于编制通用计算程序的平面和三维复杂框架结构的MRRM列式,结合提出的计算方法求解复杂框架结构的固有特性、稳态响应和瞬态响应。通过多个数值算例验证推导的正确性,并进而探讨各种复杂因素对框架结构动力性能的影响。
再次,基于MRRM列式,从连续模型的角度证明无阻尼复杂框架结构的固有模态具有正交性,并以此为基础构造模态叠加法来求解无阻尼复杂框架结构的瞬态响应。
最后,提出修正的层状介质回传射线矩阵法(MMRRM)。MMRRM基于状态空间列式,可用来求解由任何能够建立起状态方程的材料所形成的层状介质中的波传播问题,同时避免相位关系中出现大数和散射关系中出现矩阵求逆,实现在任何情形下的数值计算稳定性。将MMRRM应用于功能梯度弹性和压电层状介质和半空间的导波分析,探讨梯度特性对导波弥散的影响。
研究表明,MRRM具有严密的数学基础、明确的物理意义和统一的求解步骤。大量数值算例和结果表明,MRRM结合适当的计算方法可以求解各种复杂框架结构的固有特性、稳态响应和瞬态响应以及均匀和功能梯度层状介质的弥散曲线,具有计算精度高、求解代价小、结果解读性强等优点。
Since its first publication in 1998, the method of reverberation-ray matrix (MRRM) has been successfully applied to analyzing transient responses of planar framed structures as well as transient waves in isotropic and transversely isotropic layered media. To have a thorough understanding of MRRM and extend its applicability, the mathematical foundation of MRRM is explored and the applications to some complex dynamic problems are investigated.
Firstly, based on the solution of matrix differential equations, a uniform procedure for formulating MRRM is established in a strict mathematical sense along with the utility of physical essentials of the problem. The similarities and differences between MRRM and various other methods, including the spectral element method (SEM), the exact dynamic stiffness method (EDSM), the finite element method (FEM), the method of transfer matrix (MTM) and the traveling wave approach (TWA), are explained from the viewpoint of mathematical formulation as well as physical interpretation. Numerical algorithms are proposed when adopting MRRM to perform dynamic analysis of complex structures via PCs.
Secondly, uniform formulations of MRRM for complex planar and space framed structures are derived in a way suitable for programming. Dynamic analyses, including modal analysis, steady-state response analysis and transient response analysis, of complex structures, are then carried out by further combining with the appropriate numerical algorithms. Numerical examples are given to validate the derivation and study the influence of various parameters of the complex framed structures on its dynamic behavior.
Thirdly, based on the continuum model, the natural modes of undamped complex framed structures are proved to be orthogonal with each other in a generalized sense. This leads to an alternative method for transient response analysis of undamped framed structures via the expansion of normal modes.
Finally, the method of reverberation-ray matrix is further modified for layered media based on state space formulism. Exponentially growing functions are excluded from the phase relation and matrix inversion operation is avoided in the scattering relation, guaranteeing the numerical stability in all cases. As examples, the modified method of reverberation-ray matrix (MMRRM) is applied to study the propagation of guided waves in functionally graded anisotropic elastic and piezoelectric layered media. The effect of material gradient on the characteristic of the guided waves is also shown graphically.
It is indicated that MRRM (or MMRRM) bears a solid mathematical basis, a clear physical background and a uniform formulation procedure. Numerical examples show that MRRM (or MMRRM) has a high accuracy and less computational cost when applied to dynamics of complex framed structures and propagation of guided waves in FGM and FGPM layered media.
首先,从矩阵微分方程的解出发,给出MRRM列式的数学依据和统一步骤,阐述MRRM所能求解的动力问题。从数学列式和物理含义的角度,给出回传射线矩阵法与频谱单元法、动力刚度法、有限单元法、传递矩阵法和波传散射法的联系和区别。提出基于MRRM列式借助计算机数值求解复杂结构系统各种动力问题所需要的计算方法。
其次,考虑各种复杂因素(包括节点集中质量、弹簧及阻尼支承、杆件单元的材料阻尼、多重调谐质量阻尼器、荷载作用和位移激励),推导便于编制通用计算程序的平面和三维复杂框架结构的MRRM列式,结合提出的计算方法求解复杂框架结构的固有特性、稳态响应和瞬态响应。通过多个数值算例验证推导的正确性,并进而探讨各种复杂因素对框架结构动力性能的影响。
再次,基于MRRM列式,从连续模型的角度证明无阻尼复杂框架结构的固有模态具有正交性,并以此为基础构造模态叠加法来求解无阻尼复杂框架结构的瞬态响应。
最后,提出修正的层状介质回传射线矩阵法(MMRRM)。MMRRM基于状态空间列式,可用来求解由任何能够建立起状态方程的材料所形成的层状介质中的波传播问题,同时避免相位关系中出现大数和散射关系中出现矩阵求逆,实现在任何情形下的数值计算稳定性。将MMRRM应用于功能梯度弹性和压电层状介质和半空间的导波分析,探讨梯度特性对导波弥散的影响。
研究表明,MRRM具有严密的数学基础、明确的物理意义和统一的求解步骤。大量数值算例和结果表明,MRRM结合适当的计算方法可以求解各种复杂框架结构的固有特性、稳态响应和瞬态响应以及均匀和功能梯度层状介质的弥散曲线,具有计算精度高、求解代价小、结果解读性强等优点。
Since its first publication in 1998, the method of reverberation-ray matrix (MRRM) has been successfully applied to analyzing transient responses of planar framed structures as well as transient waves in isotropic and transversely isotropic layered media. To have a thorough understanding of MRRM and extend its applicability, the mathematical foundation of MRRM is explored and the applications to some complex dynamic problems are investigated.
Firstly, based on the solution of matrix differential equations, a uniform procedure for formulating MRRM is established in a strict mathematical sense along with the utility of physical essentials of the problem. The similarities and differences between MRRM and various other methods, including the spectral element method (SEM), the exact dynamic stiffness method (EDSM), the finite element method (FEM), the method of transfer matrix (MTM) and the traveling wave approach (TWA), are explained from the viewpoint of mathematical formulation as well as physical interpretation. Numerical algorithms are proposed when adopting MRRM to perform dynamic analysis of complex structures via PCs.
Secondly, uniform formulations of MRRM for complex planar and space framed structures are derived in a way suitable for programming. Dynamic analyses, including modal analysis, steady-state response analysis and transient response analysis, of complex structures, are then carried out by further combining with the appropriate numerical algorithms. Numerical examples are given to validate the derivation and study the influence of various parameters of the complex framed structures on its dynamic behavior.
Thirdly, based on the continuum model, the natural modes of undamped complex framed structures are proved to be orthogonal with each other in a generalized sense. This leads to an alternative method for transient response analysis of undamped framed structures via the expansion of normal modes.
Finally, the method of reverberation-ray matrix is further modified for layered media based on state space formulism. Exponentially growing functions are excluded from the phase relation and matrix inversion operation is avoided in the scattering relation, guaranteeing the numerical stability in all cases. As examples, the modified method of reverberation-ray matrix (MMRRM) is applied to study the propagation of guided waves in functionally graded anisotropic elastic and piezoelectric layered media. The effect of material gradient on the characteristic of the guided waves is also shown graphically.
It is indicated that MRRM (or MMRRM) bears a solid mathematical basis, a clear physical background and a uniform formulation procedure. Numerical examples show that MRRM (or MMRRM) has a high accuracy and less computational cost when applied to dynamics of complex framed structures and propagation of guided waves in FGM and FGPM layered media.
引文
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