非齐次弹性动力方程的回传射线分析及结构无损检测
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摘要
根据不同的理论计算模型,现有的结构动力方法可分为连续系统分析方法和离散系统分析方法两类。近年来,随着大跨度桥梁、高层框架结构以及大跨度轻型空间结构的发展,基于连续模型的精确结构动力分析变得越来越重要。于1998年正式提出的回传射线矩阵法以一维弹性动力学为基础,已被成功应用于二维、三维框架结构和层状介质的波动响应分析,体现出早期瞬态响应计算精确等优点,具有独特的优势。但到目前为止,回传射线矩阵法仅局限于处理节点集中荷载。本文将从方法的基本思路出发,将其分析范围拓展到任意分布荷载和移动荷载情形,研究任意连续梁或连续框架在这两种荷载类型下的瞬态响应并探讨回传射线矩阵法在结构无损检测中的应用,最后设计相关的波动实验验证本文的理论分析结果。
分别采用初等杆理论和Timoshenko梁理论表示任意动力荷载作用下多跨连续结构的轴向和竖向波动方程,其通解包括齐次解和非齐次解两部分,其中后者包含了跟外荷载有关的源函数。对于集中荷载或移动荷载,均可将之视为一种特殊的分布荷载,用广义脉冲函数表示。将波动方程关于时间变量和空间变量进行两次Fourier积分变换,求解出方程的非齐次解,并结合齐次解列出节点的边界条件。在回传射线矩阵法的对偶局部坐标系下,两点边界值问题将被简化为单点边界值问题,由此大大降低了通解中未知系数的求解难度。
将变换域内的通解进行两次Fourier逆变换以得到时域内的瞬态波动响应。具体为:首先,关于波数进行一次Fourier逆变换,得到分布荷载在单点频率下的稳态波动解,该解同时包含空间和频率两个变量,可表示为关于分布荷载的一个卷积积分。对于固定的分布荷载,该卷积可直接求解,最后结合数值积分逆变换即得到所关心的时域内的瞬态波动响应。而对于移动荷载,其积分顺序需做相应的改变,以此解决模态叠加法中临界移动速度所对应的数值奇异问题。
同时,本文还将回传射线矩阵法应用于集中荷载或移动荷载作用下带裂缝结构的动力响应分析,其中裂缝所在截面用弹性转动铰模拟。在求得结构的早期精确瞬态响应后,对其进行希尔伯特-黄变换,根据信号的变换谱大致判断裂缝的位置,以此提出基于弹性波的局部无损检测方法。
对移动荷载所激发的信号进行Fourier变换可提取出桥梁的自振频率,本文同样计算了移动荷载作用下连续梁和连续框架的瞬态响应,并根据希尔伯特-黄变换谱定位裂缝损伤,从理论上探讨桥梁整体快速无损检测方法。
最后,设计集中荷载作用下完整或损伤简支梁的波动实验。利用应变片等测量集中荷载下简支梁的宽频波动信号,并将实验信号及其希尔伯特-黄变换谱与理论预测结果进行对比,发现均符合良好,由此也验证了本文所采用的裂缝模型的合理性以及弹性波局部检测方法的可行性。
Dynamics of structures composed of a large number of structural members connected by joints has been analyzed traditionally by modeling all structural members as a continuous system of rods and beams or by a discrete system of many mass particles with weightless elastic connectors.Due to recent development of large span bridges,high-rise framed buildings and lightweight large span aerospace structures subject to variety of dynamic loadings,there is revival of interest in rational analysis of a structure as continuum.Based on one-dimensional elastic theory,the reverberation-ray matrix analysis(RRM) was developed in 1998,and has been successfully applied to analyze transient wave in planar trusses,layered media and three-dimensional framed structures.The newly developed method could be an alternative to finite element method(FEM) or other discrete system analysis,and is particularly effective to determine precisely early-time transient responses.So far all papers on this method are limited to steady-state and transient responses of concentrated loads applied at joints or at one or several points of a structural member.In this thesis,we extend the method of RRM to investigate the dynamics of a structure subject to distributive loads or moving forces on surface-chord members of the structure, and conduct detailed evaluations and experiments of transient wave responses in a multi-bay structure subject to both types of loading.
Axial and flexural wave motions of the surface-chord members of multi-bay framed structures or a multi-span continuous beam are governed respectively by standard wave equation and Timoshenko beam equation in coupled bending and shear wave modes with inhomogeneous term of loading functions.The complete solution for each equation is obtained by superposing complementary solution for the homogeneous part and particular solution for the inhomogeneous part of the equation.The latter includes a source function representing distributive load or moving force.A concentrated load is treated as a distributive load with a delta function in space and moving force is represented by a delta function with travelling phase in space x and time t.The double Fourier transform in x and t is applied to obtain the particular solution in transformed domains,which are then combined with complementary solutions to satisfy the boundary conditions at both joints.The method of RRM reduces a two points boundary value problem in single coordinate axis for each member to a one point boundary value problem in dual coordinate (two opposite axes),which simplifies greatly the determination of unknown coefficients in the general solution from complicated boundary conditions.
The transient solution is then obtained by double inverse transform with respect to the wave numberλand the frequency parameterω.Carrying out the inverse transform inλfirst,we obtain the steady-state wave solution of the structural member excited by the distributive loads oscillating at a single frequency.This solution contains amplitude factor in x andω,which is a convolution of distributive loading function shifted by a spatial parameter s.For the case of stationary distributive loading,the convolution integral in s is carried out first and the inverse Fourier transform inωis performed by numerical integration as reported in previous papers.For moving distributive loading,the order of integrations in s andωmust be revised so as to avoid the artificially introduced singularities.Such a modification circumvents the difficulty of having singularities at critical moving load speeds in transient responses obtained by expansion in normal modes.
In this thesis,we have applied afore mentioned matrix analysis and proper order of inverse Fourier transform to determine the transient response of a cracked structure under the action of stationary and moving concentrated load.The crack is modeled by discontinuity of rotational angle of the defective cross-section,characterized by several geometric parameters and a rotational spring constant.The precisely determined transient responses recorded at a point is then analyzed by the newly developed Hilbert-Huang transform(HHT).From the time spectrum of HHT,we can then identify the location of the crack,which could be developed into a nondestructive testing technique for detecting a crack locally.
It is known that the natural frequencies of a continuous beam could be determined from the Fourier frequency spectrum of a transient wave record generated by a moving force.We have also calculated the transient response of a continuous beam or framed structure subject to a moving force,and then determine the location of a crack from the time spectrum of HHT.It could be developed into another global nondestructive testing technique.
In the meantime,we have conducted experiments on a simply supported beam with and without the defect.Transient wave generated by a concentrated impact loading are recorded with strain gauges and broadband electronic instruments.The recorded signals and its time spectrum of HHT are compared with theoretical results with very good agreements.The experimental investigation substantiates the proper modeling of a crack in a beam and the effectiveness of time spectrum analysis of HHT for detecting the location of the defect.
分别采用初等杆理论和Timoshenko梁理论表示任意动力荷载作用下多跨连续结构的轴向和竖向波动方程,其通解包括齐次解和非齐次解两部分,其中后者包含了跟外荷载有关的源函数。对于集中荷载或移动荷载,均可将之视为一种特殊的分布荷载,用广义脉冲函数表示。将波动方程关于时间变量和空间变量进行两次Fourier积分变换,求解出方程的非齐次解,并结合齐次解列出节点的边界条件。在回传射线矩阵法的对偶局部坐标系下,两点边界值问题将被简化为单点边界值问题,由此大大降低了通解中未知系数的求解难度。
将变换域内的通解进行两次Fourier逆变换以得到时域内的瞬态波动响应。具体为:首先,关于波数进行一次Fourier逆变换,得到分布荷载在单点频率下的稳态波动解,该解同时包含空间和频率两个变量,可表示为关于分布荷载的一个卷积积分。对于固定的分布荷载,该卷积可直接求解,最后结合数值积分逆变换即得到所关心的时域内的瞬态波动响应。而对于移动荷载,其积分顺序需做相应的改变,以此解决模态叠加法中临界移动速度所对应的数值奇异问题。
同时,本文还将回传射线矩阵法应用于集中荷载或移动荷载作用下带裂缝结构的动力响应分析,其中裂缝所在截面用弹性转动铰模拟。在求得结构的早期精确瞬态响应后,对其进行希尔伯特-黄变换,根据信号的变换谱大致判断裂缝的位置,以此提出基于弹性波的局部无损检测方法。
对移动荷载所激发的信号进行Fourier变换可提取出桥梁的自振频率,本文同样计算了移动荷载作用下连续梁和连续框架的瞬态响应,并根据希尔伯特-黄变换谱定位裂缝损伤,从理论上探讨桥梁整体快速无损检测方法。
最后,设计集中荷载作用下完整或损伤简支梁的波动实验。利用应变片等测量集中荷载下简支梁的宽频波动信号,并将实验信号及其希尔伯特-黄变换谱与理论预测结果进行对比,发现均符合良好,由此也验证了本文所采用的裂缝模型的合理性以及弹性波局部检测方法的可行性。
Dynamics of structures composed of a large number of structural members connected by joints has been analyzed traditionally by modeling all structural members as a continuous system of rods and beams or by a discrete system of many mass particles with weightless elastic connectors.Due to recent development of large span bridges,high-rise framed buildings and lightweight large span aerospace structures subject to variety of dynamic loadings,there is revival of interest in rational analysis of a structure as continuum.Based on one-dimensional elastic theory,the reverberation-ray matrix analysis(RRM) was developed in 1998,and has been successfully applied to analyze transient wave in planar trusses,layered media and three-dimensional framed structures.The newly developed method could be an alternative to finite element method(FEM) or other discrete system analysis,and is particularly effective to determine precisely early-time transient responses.So far all papers on this method are limited to steady-state and transient responses of concentrated loads applied at joints or at one or several points of a structural member.In this thesis,we extend the method of RRM to investigate the dynamics of a structure subject to distributive loads or moving forces on surface-chord members of the structure, and conduct detailed evaluations and experiments of transient wave responses in a multi-bay structure subject to both types of loading.
Axial and flexural wave motions of the surface-chord members of multi-bay framed structures or a multi-span continuous beam are governed respectively by standard wave equation and Timoshenko beam equation in coupled bending and shear wave modes with inhomogeneous term of loading functions.The complete solution for each equation is obtained by superposing complementary solution for the homogeneous part and particular solution for the inhomogeneous part of the equation.The latter includes a source function representing distributive load or moving force.A concentrated load is treated as a distributive load with a delta function in space and moving force is represented by a delta function with travelling phase in space x and time t.The double Fourier transform in x and t is applied to obtain the particular solution in transformed domains,which are then combined with complementary solutions to satisfy the boundary conditions at both joints.The method of RRM reduces a two points boundary value problem in single coordinate axis for each member to a one point boundary value problem in dual coordinate (two opposite axes),which simplifies greatly the determination of unknown coefficients in the general solution from complicated boundary conditions.
The transient solution is then obtained by double inverse transform with respect to the wave numberλand the frequency parameterω.Carrying out the inverse transform inλfirst,we obtain the steady-state wave solution of the structural member excited by the distributive loads oscillating at a single frequency.This solution contains amplitude factor in x andω,which is a convolution of distributive loading function shifted by a spatial parameter s.For the case of stationary distributive loading,the convolution integral in s is carried out first and the inverse Fourier transform inωis performed by numerical integration as reported in previous papers.For moving distributive loading,the order of integrations in s andωmust be revised so as to avoid the artificially introduced singularities.Such a modification circumvents the difficulty of having singularities at critical moving load speeds in transient responses obtained by expansion in normal modes.
In this thesis,we have applied afore mentioned matrix analysis and proper order of inverse Fourier transform to determine the transient response of a cracked structure under the action of stationary and moving concentrated load.The crack is modeled by discontinuity of rotational angle of the defective cross-section,characterized by several geometric parameters and a rotational spring constant.The precisely determined transient responses recorded at a point is then analyzed by the newly developed Hilbert-Huang transform(HHT).From the time spectrum of HHT,we can then identify the location of the crack,which could be developed into a nondestructive testing technique for detecting a crack locally.
It is known that the natural frequencies of a continuous beam could be determined from the Fourier frequency spectrum of a transient wave record generated by a moving force.We have also calculated the transient response of a continuous beam or framed structure subject to a moving force,and then determine the location of a crack from the time spectrum of HHT.It could be developed into another global nondestructive testing technique.
In the meantime,we have conducted experiments on a simply supported beam with and without the defect.Transient wave generated by a concentrated impact loading are recorded with strain gauges and broadband electronic instruments.The recorded signals and its time spectrum of HHT are compared with theoretical results with very good agreements.The experimental investigation substantiates the proper modeling of a crack in a beam and the effectiveness of time spectrum analysis of HHT for detecting the location of the defect.
引文
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