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A FAMILY OF MODAL METHODS FOR COMPUTING EIGENVECTOR DERIVATIVES WITH REPEATED ROOTS 被引量:2
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作者 王文亮 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 1996年第2期158-168,共11页
A family of modal methods for computing eigenvector derivatives with repeated roots are directly derived from the constraint generalized inverse technique which is originally formulated by Wang and Hu. Extensions are ... A family of modal methods for computing eigenvector derivatives with repeated roots are directly derived from the constraint generalized inverse technique which is originally formulated by Wang and Hu. Extensions are made to Akgun's method to allow treatment of eigensensitivity with repeated roots for general nondefective systems, and Bernard and Bronowicki's modal expansion approach is expanded to a family of modal methods. 展开更多
关键词 a family of modal methods nondefective system eigenvector derivative eigensensitivity with repeated roots
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Study on tapered crossed subwavelength gratings by Fourier modal method
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作者 陈熙 钟源 +2 位作者 王青 张冶金 陈良惠 《Chinese Physics B》 SCIE EI CAS CSCD 2010年第10期258-264,共7页
Fourier modal method incorporating staircase approximation is used to study tapered crossed subwavelength gratings in this paper. Three intuitive formulations of eigenvalue functions originating from the prototype are... Fourier modal method incorporating staircase approximation is used to study tapered crossed subwavelength gratings in this paper. Three intuitive formulations of eigenvalue functions originating from the prototype are presented, and their convergences are compared through numerical calculation. One of them is found to be suitable in modeling the diffraction efficiency of the circular tapered crossed subwavelength gratings without high absorption, and staircase approximation is further proven valid for non-highly-absorptive tapered gratings. This approach is used to simulate the "moth-eye" antireflection surface on silicon, and the numerical result agrees well with the experimental one. 展开更多
关键词 tapered crossed subwavelength gratings Fourier modal method staircase approximation eigenvalue functions
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Sparse Modal Decomposition Method Addressing Underdetermined Vortex-Induced Vibration Reconstruction Problem for Marine Risers 被引量:1
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作者 DU Zun-feng ZHU Hai-ming YU Jian-xing 《China Ocean Engineering》 SCIE EI CSCD 2024年第2期285-296,共12页
When investigating the vortex-induced vibration(VIV)of marine risers,extrapolating the dynamic response on the entire length based on limited sensor measurements is a crucial step in both laboratory experiments and fa... When investigating the vortex-induced vibration(VIV)of marine risers,extrapolating the dynamic response on the entire length based on limited sensor measurements is a crucial step in both laboratory experiments and fatigue monitoring of real risers.The problem is conventionally solved using the modal decomposition method,based on the principle that the response can be approximated by a weighted sum of limited vibration modes.However,the method is not valid when the problem is underdetermined,i.e.,the number of unknown mode weights is more than the number of known measurements.This study proposed a sparse modal decomposition method based on the compressed sensing theory and the Compressive Sampling Matching Pursuit(Co Sa MP)algorithm,exploiting the sparsity of VIV in the modal space.In the validation study based on high-order VIV experiment data,the proposed method successfully reconstructed the response using only seven acceleration measurements when the conventional methods failed.A primary advantage of the proposed method is that it offers a completely data-driven approach for the underdetermined VIV reconstruction problem,which is more favorable than existing model-dependent solutions for many practical applications such as riser structural health monitoring. 展开更多
关键词 motion reconstruction vortex-induced vibration(VIV) marine riser modal decomposition method compressed sensing
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A method for static aeroelastic analysis based on the high-order panel method and modal method 被引量:9
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作者 YANG Chao ZHANG BoCheng +1 位作者 WAN ZhiQiang WANG YaoKun 《Science China(Technological Sciences)》 SCIE EI CAS 2011年第3期741-748,共8页
A method for static aeroelastic analysis based on the high-order panel method and modal method is presented. The static aeroelastic characteristics of flexible wings are investigated using this method. Three-dimension... A method for static aeroelastic analysis based on the high-order panel method and modal method is presented. The static aeroelastic characteristics of flexible wings are investigated using this method. Three-dimensional aerodynamic models of flexible wings are constructed based on the geometry of wing configuration, and the modal method is adopted to achieve the fluid-structure coupling. The static aeroelastic characteristics of the AGARD445.6 wing and a low-aspect-ratio wing are investigated in this study. The influences of elastic structural deformation on aerodynamic forces are studied with an emphasis analyzing the aerodynamic coefficients, wing root loads, structural deformation and pressure distribution of different sections, and results are compared with the results from wind-tunnel tests and the elastic results based on experimental aerodynamic forces. It is concluded that aerodynamic forces can be accurately calculated with the high-order panel method. The method presented in this study is feasible, credible and efficient. Comprehensive static aeroelastic characteristics can be provided by the method for early phases of aircraft design. 展开更多
关键词 aeroelasticity flight loads high-order panel method modal method experimental aerodynamic forces
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Modal Identification of Offshore Platforms Using Statistical Method Based on ERA 被引量:6
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作者 王树青 李华军 T.Takayama 《China Ocean Engineering》 SCIE EI 2005年第2期175-184,共10页
Identification of modal parameters of a linear structure with output-only measurements has received much attention over the past decades. In the paper, the Natural Excitation Technique (NExT) is used for acquisition o... Identification of modal parameters of a linear structure with output-only measurements has received much attention over the past decades. In the paper, the Natural Excitation Technique (NExT) is used for acquisition of the impulse signals from the structural responses. Then Eigensystem Realization Algorithm (ERA) is utilized for modal identification. For disregarding the fictitious ‘computational modes', a procedure, Statistically Averaging Modal Frequency Method (SAMFM), is developed to distinguish the true modes from noise modes, and to improve the precision of the identified modal frequencies of the structure. An offshore platform is modeled with the finite element method. The theoretical modal parameters are obtained for a comparison with the identified values. The dynamic responses of the platform under random wave loading are computed for providing the output signals used for identification with ERA. Results of simulation demonstrate that the proposed method can determine the system modal frequency with high precision. 展开更多
关键词 modal identification eigensystem realization algorithm statistically averaging modal frequency method offshore platform
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Physical Parameter Identification Method Based on Modal Analysis for Two-axis On-road Vehicles:Theory and Simulation 被引量:3
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作者 ZHENG Minyi ZHANG Bangji +1 位作者 ZHANG Jie ZHANG Nong 《Chinese Journal of Mechanical Engineering》 SCIE EI CAS CSCD 2016年第4期756-764,共9页
Physical parameters are very important for vehicle dynamic modeling and analysis.However,most of physical parameter identification methods are assuming some physical parameters of vehicle are known,and the other unkno... Physical parameters are very important for vehicle dynamic modeling and analysis.However,most of physical parameter identification methods are assuming some physical parameters of vehicle are known,and the other unknown parameters can be identified.In order to identify physical parameters of vehicle in the case that all physical parameters are unknown,a methodology based on the State Variable Method(SVM) for physical parameter identification of two-axis on-road vehicle is presented.The modal parameters of the vehicle are identified by the SVM,furthermore,the physical parameters of the vehicle are estimated by least squares method.In numerical simulations,physical parameters of Ford Granada are chosen as parameters of vehicle model,and half-sine bump function is chosen to simulate tire stimulated by impulse excitation.The first numerical simulation shows that the present method can identify all of the physical parameters and the largest absolute value of percentage error of the identified physical parameter is 0.205%;and the effect of the errors of additional mass,structural parameter and measurement noise are discussed in the following simulations,the results shows that when signal contains 30 d B noise,the largest absolute value of percentage error of the identification is 3.78%.These simulations verify that the presented method is effective and accurate for physical parameter identification of two-axis on-road vehicles.The proposed methodology can identify all physical parameters of 7-DOF vehicle model by using free-decay responses of vehicle without need to assume some physical parameters are known. 展开更多
关键词 Parameter identification free-decay response state variable method modal parameter physical parameter
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Improved modal truncation method for eigensensitivity analysis of asymmetric matrix with repeated eigenvalues
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作者 张振宇 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2014年第4期437-452,共16页
An improved modal truncation method with arbitrarily high order accuracy is developed for calculating the second- and third-order eigenvalue derivatives and the first- and second-order eigenvector derivatives of an as... An improved modal truncation method with arbitrarily high order accuracy is developed for calculating the second- and third-order eigenvalue derivatives and the first- and second-order eigenvector derivatives of an asymmetric and non-defective matrix with repeated eigenvalues. If the different eigenvalues λ1, λ2,……, λs of the matrix satisfy |λ1| ≤... ≤|λr| and |λs| 〈|〈s+1| (s ≤r-l), then associated with any eigenvalue λi (i≤ s), the errors of the eigenvalue and eigenvector derivatives obtained by the qth-order approximate method are proportional to |λi|/λs+1|q+l, where the approximate method only uses the eigenpairs corresponding to λ1, λ2,……,λs A numerical example shows the validity of the approximate method. The numerical example also shows that in order to get the approximate solutions with the same order accuracy, a higher order method should be used for higher order eigenvalue and eigenvector derivatives. 展开更多
关键词 modal truncation method eigenvector derivative asymmetric matrix repeated eigenvalue
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Free-Interface Dual-Compatibility Modal Synthesis Substructure Method in Large-Scale Structures
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作者 彭桂瀚 林伟 +1 位作者 陈尚鸿 余洁歆 《Transactions of Tianjin University》 EI CAS 2015年第4期347-355,共9页
Free-interface dual-compatibility modal synthesis method(compatibility of both force and displacement on interfaces)is introduced to large-scale civil engineering structure to enhance computation efficiency. The basic... Free-interface dual-compatibility modal synthesis method(compatibility of both force and displacement on interfaces)is introduced to large-scale civil engineering structure to enhance computation efficiency. The basic equations of the method are first set up, and then the mode cut-off principle and the dividing principle are proposed. MATLAB is used for simulation in different frame structures. The simulation results demonstrate the applicability of this substructure method to civil engineering structures and the correctness of the proposed mode cut-off principle. Studies are also conducted on how to divide the whole structure for better computation efficiency while maintaining better precision. It is observed that the geometry and material properties should be considered, and the synthesis results would be more precise when the inflection points of the mode shapes are taken into consideration. Furthermore, the simulation performed on a large-scale high-rise connected structure further proves the feasibility and efficiency of this modal synthesis method compared with the traditional global method. It is also concluded from the simulation results that the fewer number of DOFs in each substructure will result in better computation efficiency, but too many substructures will be time-consuming due to the tedious synthesis procedures. Moreover, the substructures with free interface will introduce errors and reduce the precision dramatically, which should be avoided. 展开更多
关键词 SUBSTRUCTURE method dynamic SUBSTRUCTURE analysis free-interface dual-compatibility modal synthesismethod large-scale CIVIL engineering structure mode CUT-OFF principle
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Exploration of Multi-modal Teaching Methods in Teaching of Agricultural Landscape Planning
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作者 Tian HU Yarui WU +1 位作者 Dong LIU Jing HU 《Asian Agricultural Research》 2017年第9期72-74,共3页
Taking the teaching practice of agricultural landscape planning for example,this paper uses the multi-modal teaching idea for teaching design based on traditional lecture-style teaching,including multi-modal teaching ... Taking the teaching practice of agricultural landscape planning for example,this paper uses the multi-modal teaching idea for teaching design based on traditional lecture-style teaching,including multi-modal teaching materials,multi-modal teaching methods and multi-modal teaching evaluation. The results show that this method can effectively improve students' interest in learning,reinforce the theoretical basis of agricultural landscape planning theory,and improve agricultural landscape planning practical skills. It is the active exploration of multi-modal teaching model and useful complement to traditional classroom teaching. 展开更多
关键词 Agricultural landscape planning Multi-modal teaching Teaching method
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Modal and Thermal Analysis of a Modified Connecting Rod of an Internal Combustion Engine Using Finite Element Method
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作者 Nkrumah Jacob Kwaku Baba Ziblim +1 位作者 Sulemana Yahaya Sherry Kwabla Amedorme 《Modeling and Numerical Simulation of Material Science》 2023年第3期29-49,共21页
The connecting rod is one of the most important moving components in an internal combustion engine. The present work determined the possibility of using aluminium alloy 7075 material to design and manufacture a connec... The connecting rod is one of the most important moving components in an internal combustion engine. The present work determined the possibility of using aluminium alloy 7075 material to design and manufacture a connecting rod for weight optimisation without losing the strength of the connecting rod. It considered modal and thermal analyses to investigate the suitability of the material for connecting rod design. The parameters that were considered under the modal analysis were: total deformation, and natural frequency, while the thermal analysis looked at the temperature distribution, total heat flux and directional heat flux of the four connecting rods made with titanium alloy, grey cast iron, structural steel and aluminium 7075 alloy respectively. The connecting rod was modelled using Autodesk inventor2017 software using the calculated parameters. The steady-state thermal analysis was used to determine the induced heat flux and directional heat flux. The study found that Aluminium 7075 alloy deformed more than the remaining three other materials but has superior qualities in terms of vibrational natural frequency, total heat flux and lightweight compared to structural steel, grey cast iron and titanium alloy. 展开更多
关键词 Connecting Rod Steady-State Thermal Analysis DEFORMATION Heat Flux Thermal and modal Finite Element method
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A MODEL IDENTIFICATION METHOD OF VIBRATING STRUCTURES FROM INCOMPLETE MODAL INFORMATION
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作者 郑小平 姚振汉 蘧时胜 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1995年第5期971-976,共6页
The accurate mathematical models for complicated structures are verydifficult to construct.The work presented here provides an identification method for estimating the mass, damping , and stiffness matrices of linear ... The accurate mathematical models for complicated structures are verydifficult to construct.The work presented here provides an identification method for estimating the mass, damping , and stiffness matrices of linear dynamical systems from incompleteexperimental data. The mass, stiffness, and damping matrices are assumed to be real,symmetric, and positive definite. The partial set of experimental complex eigenvalues and corresponding eigenvectors are given. In the proposed method the least squaresalgorithm is combined with the iteration technique to determine systems identified matrices and corresponding design parameters. several illustrative examples, are presented to demonstrate the reliability of the proposed method .It is emphasized thatthe mass, damping and stiffness martices can be identified simultaneously. 展开更多
关键词 vibrating structures model identification incompleteexperiemntal modal data the least squares method iteration technique
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A MODEL IDENTIFICATION METHOD OF VIBRATING STRUCTURES FROM INCOMPLETE MODAL INFORMATION
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作者 郑小平 姚振汉 蘧时胜 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1995年第10期971-976,共6页
The accurate mathematical models for complicated structures are very difficult to construct.The work presented here provides an identification method for estimating the mass.damping,and stiffness matrices of linear dy... The accurate mathematical models for complicated structures are very difficult to construct.The work presented here provides an identification method for estimating the mass.damping,and stiffness matrices of linear dynamical systems from incomplete experimental data.The mass,stiffness and damping matrices are assumed to be real,symmetric,and positive definite The partial set of experimental complex eigenvalues and corresponding eigenvectors are given.In the proposed method the least squares algorithm is combined with the iteration technique to determine systems identified matrices and corresponding design parameters.Seeveral illustative examples,are presented to demonstrate the reliability of the proposed method .It is emphasized that the mass,damping and stiffness matrices can be identified simultaneously. 展开更多
关键词 vibrating structures model identification incomplete experiemntal modal data the least squares method iteration technique
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空间大型桁架结构模型试验与动力相似研究
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作者 李清歌 骆海涛 +2 位作者 何凤霞 李玉新 李延斌 《制造技术与机床》 北大核心 2025年第1期33-40,共8页
静力学和模态测试是空间桁架结构设计中的一个重要环节。由于空间桁架结构尺寸较大,进行全尺寸原型试验受到试验场地和设备等因素的限制。因此,空间大型桁架结构缩比模型试验不仅可以降低试验成本和风险,而且可以对空间桁架结构的静力... 静力学和模态测试是空间桁架结构设计中的一个重要环节。由于空间桁架结构尺寸较大,进行全尺寸原型试验受到试验场地和设备等因素的限制。因此,空间大型桁架结构缩比模型试验不仅可以降低试验成本和风险,而且可以对空间桁架结构的静力学和振动特性分析提供新的手段。以空间大型桁架结构为研究对象,基于相似理论,运用方程分析法和量纲分析法推导了空间桁架原型和1/3缩比模型之间的静力学相似关系和模态响应的相似关系。最后,对原型桁架和1/3缩比模型桁架进行静力学和模态仿真分析,并进行了模态试验,验证模态相似关系的正确性。结果表明,通过相似理论与模型试验预测大型结构体原型的静力学和振动特性的可行性。 展开更多
关键词 空间桁架 缩比模型试验 模态 相似关系 量纲分析法
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Modal-FEM混合法对复杂终端进气道RCS的计算
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作者 李欣 吴昌英 +2 位作者 邱兆杰 宋涛 许家栋 《系统仿真学报》 CAS CSCD 北大核心 2009年第7期2106-2108,共3页
对带有复杂结构终端进气道的电磁散射问题进行了研究。给出一种高效的Modal-FEM分析方法,采用改进的模式法处理前端空腔部分,而复杂终端则只需有限元法进行一次求解;最后,在分界面上耦合得到整体的散射结果。与其他数值算法相比,计算结... 对带有复杂结构终端进气道的电磁散射问题进行了研究。给出一种高效的Modal-FEM分析方法,采用改进的模式法处理前端空腔部分,而复杂终端则只需有限元法进行一次求解;最后,在分界面上耦合得到整体的散射结果。与其他数值算法相比,计算结果表明:混合算法在保证精度的前提下减少了计算时间和所需内存,为计算带复杂终端进气道的RCS找到了一条可行的路径。 展开更多
关键词 复杂终端 有限元法 模式传输法 雷达散射截面
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Multidimensional modal analysis of liquid nonlinear sloshing in right circular cylindrical tank
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作者 余延生 马兴瑞 王本利 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2007年第8期1007-1018,共12页
The multidimensional modal theory proposed by Faltinsen, et al. (2000) is applied to solve liquid nonlinear free sloshing in right circular cylindrical tank for the first time. After selecting the leading modes and ... The multidimensional modal theory proposed by Faltinsen, et al. (2000) is applied to solve liquid nonlinear free sloshing in right circular cylindrical tank for the first time. After selecting the leading modes and fixing the order of magnitudes based on the Narimanov-Moiseev third order asymptotic hypothesis, the general infinite dimensional modal system is reduced to a five dimensional asymptotic modal system (the system of second order nonlinear ordinary differential equations coupling the generalized time dependent coordinates of free surface wave elevation). The numerical integrations of this modal system discover most important nonlinear phenomena, which agree well with both pervious analytic theories and experimental observations. The results indicate that the multidimensional modal method is a very good tool for solving liquid nonlinear sloshing dynamics and will be developed to investigate more complex sloshing problem in our following work. 展开更多
关键词 circular cylindrical tank nonlinear free sloshing multidimensional modal method asymptotic modal system dispersion effect
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基于改进IDW法的铣削机器人模态参数预测
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作者 段玉斌 赵一阳 +2 位作者 刘金玉 曹家彬 闫维新 《中国测试》 北大核心 2025年第1期159-165,180,共8页
铣削加工工业机器人模态参数识别是分析其加工时振动现象的基础,模态参数与机器人的姿态高度相关,因此存在加工过程中其随时间改变的现象。传统的有限元法以及试验模态分析法难以快速准确地预测铣削机器人加工空间内所有姿态下的模态参... 铣削加工工业机器人模态参数识别是分析其加工时振动现象的基础,模态参数与机器人的姿态高度相关,因此存在加工过程中其随时间改变的现象。传统的有限元法以及试验模态分析法难以快速准确地预测铣削机器人加工空间内所有姿态下的模态参数,对此文章提出一种改进的距离反比加权(IDW)频响曲线预测法,实现机器人任意姿态下的频响曲线预测,并采用正交多项式曲线拟合法准确辨识机器人的模态参数。首先对铣削加工系统进行有限元模态仿真,在此基础上确定传感器布放位置;采用力锤冲击试验得到加工平面内有限位姿点的频响曲线,之后利用改进IDW法对部分位姿点的频响曲线进行预测;最后利用正交多项式曲线拟合法基于预测频响曲线实现模态参数的辨识并与实测模态参数进行对比,分析结果表明:1)改进IDW法可有效提高频响曲线的预测精度;2)基于预测频响曲线的模态参数识别结果与基于实测频响曲线的参数识别结果接近,低阶固有频率的预测误差在8%以内。 展开更多
关键词 铣削机器人 频响函数预测 模态参数辨识 改进距离反比加权法
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A review of welding residual stress test methods 被引量:8
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作者 Gan Shiming Liu Huaying +1 位作者 Zhai Zhiping Han Yongquan 《China Welding》 CAS 2022年第2期45-55,共11页
Due to local uneven heating during the welding process,the residual stress of the structure after welding affects the reliability of it.In order to ensure the reliability,it is of great significance to test the residu... Due to local uneven heating during the welding process,the residual stress of the structure after welding affects the reliability of it.In order to ensure the reliability,it is of great significance to test the residual stress distribution of the welded joint.It has always been the focus to find a simple and feasible method for residual stress testing to quickly and accurately obtain the residual stress distribution of welded joints.The mechanical measurement method has high measurement accuracy,convenient and easy operation,but it will cause certain damage to the components.Physical measurement method can avoid damage to components,but its test cost is usually high,and its measurement accuracy can also be affected by the material microstructure characteristics of welded components.Based on the advantages and disadvantages of these two residual stress test methods,a modal test method is proposed.This method is a non-destructive measurement method.Based on the mathematical relationship between the residual stress of the welded structure and the natural frequency(mathematical model),the natural frequency is measured through the modal test to calculate the residual stress quickly.However,it is difficult to establish a mathematical model with this method,and it is not suitable for realization. 展开更多
关键词 welding residual stress mechanical measurement method physical measurement method modal test method
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Numerical-experimental method for elastic parameters identification of a composite panel 被引量:3
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作者 Dong Jiang Rui Ma +1 位作者 Shaoqing Wu Qingguo Fei 《Theoretical & Applied Mechanics Letters》 CAS 2014年第6期20-25,共6页
A hybrid numerical-experimental approach to identify elastic modulus of a textile composite panel using vibration test data is proposed and investi- gated. Homogenization method is adopted to predict the initial value... A hybrid numerical-experimental approach to identify elastic modulus of a textile composite panel using vibration test data is proposed and investi- gated. Homogenization method is adopted to predict the initial values of elastic parameters of the composite, and parameter identification is transformed to an optimization problem in which the objective function is the minimization of the discrepancies between the experimental and numerical modal data. Case study is conducted employing a woven fabric reinforced composite panel. Three parameters (Ell, E22, G12) with higher sensitivities are selected to be identified. It is shown that the elastic parameters can be accurately identified from experimental modal data. 展开更多
关键词 composite panel elastic modulus parameter identification modal data numerical-experimental method
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Dynamic Characteristics of a Curved Cable-stayed Bridge:Operational Modal Testing and FE Modeling 被引量:1
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作者 Bruno Briseghella Fulvio Busatta +2 位作者 Carmelo Gentile Enzo Siviero Tobia Zordan 《施工技术》 CAS 北大核心 2010年第5期111-117,共7页
This paper summarizes the analytical and experimental dynamic analyses carried out to assess the actual dynamic behaviour of a curved cable-stayed bridge,recently erected in the commercial harbour of Porto Marghera ( ... This paper summarizes the analytical and experimental dynamic analyses carried out to assess the actual dynamic behaviour of a curved cable-stayed bridge,recently erected in the commercial harbour of Porto Marghera ( Venice,Italy). Ambient vibration tests were carried out to determine the dynamic characteristics of the bridge and more than 20 modes were identified in the frequency range 0~10Hz. In the theoretical study,a 3D FE model of the bridge was developed using an integrated CAD-FEA approach; subsequently,the information obtained from the field tests,combined with simple manual tuning,provided a linear elastic model,accurately fitting the modal parameters of the bridge in its present condition. 展开更多
关键词 动力学 CAD技术 施工技术 3D模型
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Analysis on Modals and Semi-Modals
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作者 陈力 《海外英语》 2014年第8X期242-243,248,共3页
The modal is a type of auxiliary verb that is used to indicate modality.There are thirteen modal auxiliaries including some past tense forms.In this paper,the modals and semi-modals will be studied through analyzing f... The modal is a type of auxiliary verb that is used to indicate modality.There are thirteen modal auxiliaries including some past tense forms.In this paper,the modals and semi-modals will be studied through analyzing fifteen sample sentences and referring some grammar books.At last,some suggestions for ESL or ESL teacher to teach modals will be provided. 展开更多
关键词 modalS AUXILIARIES TEACHING methods ESL EFL
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