A novel tension estimation approach for elastic cables by elimination of complex boundary condition effects employing mode shape functions

2018 ◽  
Vol 166 ◽  
pp. 152-166 ◽  
Author(s):  
Chien-Chou Chen ◽  
Wen-Hwa Wu ◽  
Shin-Yi Chen ◽  
Gwolong Lai
Keyword(s):  
Boundary Condition ◽  
Mode Shape ◽  
Shape Functions ◽  
Complex Boundary Condition ◽  
Complex Boundary ◽  
Elastic Cables

A Spatial and Temporal Harmonic Balance Method for Obtaining Periodic Solutions of a Nonlinear Partial Differential Equation With a Linear Complex Boundary Condition

2020 ◽  
Vol 142 (3) ◽  
Author(s):  
Xuefeng Wang ◽  
Weidong Zhu
Keyword(s):  
Boundary Condition ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Jacobian Matrix ◽  
Additional Function ◽  
Complex Boundary Condition ◽  
Complex Boundary ◽  
Hill’S Method ◽  
Sine Functions ◽  
Nonlinear String

Abstract A spatial and temporal harmonic balance (STHB) method is demonstrated in this work by solving periodic solutions of a nonlinear string equation with a linear complex boundary condition, and stability analysis of the solutions is conducted by using the Hill’s method. In the STHB method, sine functions are used as basis functions in the space coordinate of the solutions, so that the spatial harmonic balance procedure can be implemented by the fast discrete sine transform. A trial function of a solution is formed by truncated sine functions and an additional function to satisfy the boundary conditions. In order to use sine functions as test functions, the method derives a relationship between the additional coordinate associated with the additional function and generalized coordinates associated with the sine functions. An analytical method to derive the Jacobian matrix of the harmonic balanced residual is also developed, and the matrix can be used in the Newton method to solve periodic solutions. The STHB procedures and analytical derivation of the Jacobian matrix make solutions of the nonlinear string equation with the linear spring boundary condition efficient and easy to be implemented by computer programs. The relationship between the Jacobian matrix and the system matrix of linearized ordinary differential equations (ODEs) that are associated with the governing partial differential equation is also developed, so that one can directly use the Hill’s method to analyze the stability of the periodic solutions without deriving the linearized ODEs. The frequency-response curve of the periodic solutions is obtained and their stability is examined.

scholarly journals The Spatial and Temporal Harmonic Balance Method for Obtaining Periodic Responses of a Nonlinear Partial Differential Equation With a Linear Complex Boundary Condition

2017 ◽  
Author(s):  
Xuefeng Wang ◽  
Weidong Zhu
Keyword(s):  
Boundary Condition ◽  
Harmonic Balance ◽  
Jacobian Matrix ◽  
Nonlinear Partial Differential Equation ◽  
System Matrix ◽  
Complex Boundary Condition ◽  
Complex Boundary ◽  
Toeplitz Form ◽  
Hill’S Method ◽  
Nonlinear String

The spatial and temporal harmonic balance (STHB) method is used to solve the periodic solution for a nonlinear partial differential equation (PDE) demonstrated by a nonlinear string equation with a linear complex boundary condition, and stablity analysis is conducted for the periodic solutions using Hill’s method. In order to avoid the integration procedure for discretizing the PDE to obtain the ordinary differential equations (ODEs), spatial and temporal harmonic balance procedures are conducted simultaneously, which can be efficiently achieved by the discrete sine transform and the fast Fourier transform. An additional coordinate associated with the generalized coordinates of the trial functions for the spatial discretization is introduced to make the solution satisfy all boundary conditions, and a relationship of the additional coordinate and the generalized coordinates is developed and used in the STHB method so that the test functions can be the same to the trial functions. Jacobian matrix of the harmonic balanced residual is obtained analytically, which can be used in Newton method for solving the periodic response. The STHB method and Jacobian matrix make the calculation of the periodic solution for the nonlinear string with a linear spring boundary condition efficient and easy to be implemented by computer programs. The relationship between the Jacobian matrix and the system matrix of the linearized ODEs are developed, so that one can directly obtain the Toeplitz form of the system matrix, and Hill’s method can be used to analyze the stability with the eigenvalues of the Toeplitz-form system matrix without the derivation of the ODEs. The frequency curve of the periodic solutions is obtained and their stability is indicated by the method in this work.

Boundary Condition Updating of Bridge Structure Based on Mode Parameters

2010 ◽  
Vol 163-167 ◽  
pp. 3749-3756
Author(s):  
Zhou Shi ◽  
Ren Da Zhao ◽  
Shi Qiang Qin ◽  
Yu Feng Gao
Keyword(s):  
Boundary Condition ◽  
Constrained Optimization ◽  
Least Square ◽  
Inversion Method ◽  
Bridge Structure ◽  
Matrix Perturbation ◽  
Matrix Perturbation Theory ◽  
Complex Boundary Condition ◽  
Complex Boundary ◽  
Structure Boundary

According to the complex boundary condition of real bridge structure, appendix restraint parameters of structure boundary condition were detailed analyzed. The appendix horizontal and torsional restraint parameters were selected as research object. The least square constrained optimization inversion method was used to establish the question of reverse of bridge structure boundary appendix parameters based on tested and calculated mode parameters. The first order perturbation of eigen-value and eigen-vector based on matrix perturbation theory were substituted into the constrained optimization inversion equation, so the solving efficiency was improved greatly, and the iteration method was used to improve the precision. The corresponding program was made, and the example of a maglev railway girder shows that fairly well precise bridge structure boundary appendix parameters can be reserved with only former several modes parameters through four to seven times of iteration calculation.

scholarly journals RBF-Based Meshless Method for Large Deflection of Elastic Thin Rectangular Plates with Boundary Conditions Involving Free Edges

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Mohammed M. Hussein Al-Tholaia ◽  
Husain Jubran Al-Gahtani
Keyword(s):  
Boundary Condition ◽  
Basis Function ◽  
Meshless Method ◽  
Iterative Procedure ◽  
Large Deflection ◽  
Free Edge ◽  
Thin Plates ◽  
Rectangular Plates ◽  
Numerical Examples ◽  
Complex Boundary

An RBF-based meshless method is presented for the analysis of thin plates undergoing large deflection. The method is based on collocation with the multiquadric radial basis function (MQ-RBF). In the proposed method, the resulting coupled nonlinear equations are solved using an incremental-iterative procedure. The accuracy and efficiency of the method are verified through several numerical examples. The inclusion of the free edge boundary condition proves that this method is accurate and efficient in handling such complex boundary value problems.

Experimental System Identification of the Vibro-Impact Dynamics Towards Structural Health Monitoring and Damage Detection

2013 ◽  
Author(s):  
Heng Chen ◽  
Young S. Lee ◽  
Mehmet Kurt ◽  
D. Michael McFarland ◽  
Lawrence A. Bergman ◽  
...  
Keyword(s):  
System Identification ◽  
Structural Health Monitoring ◽  
Damage Detection ◽  
Mode Shape ◽  
Health Monitoring ◽  
Correlation Spectroscopy ◽  
Impact Dynamics ◽  
Shape Functions ◽  
Modal Assurance Criterion ◽  
Structural Health

We perform nonlinear system identification (NSI) on the acceleration signals that were experimentally measured at ten, almost evenly spaced positions along a cantilever beam undergoing vibro-impacts between two rigid stops with clearances. The NSI methodology is based on the correspondence between analytical and empirical slow-flow dynamics, with the first step requiring empirical mode decomposition (EMD) analysis of the measured time series leading to sets of intrinsic modal oscillators (IMOs) governing the vibro-impact dynamics at different time scales. By comparing the spatiotemporal variations of the nonlinear modal interactions (and hence the IMOs), we examine how vibro-impacts influence the low- and high-frequency modes in global and local senses. In applications of the NSI results to structural health monitoring and damage detection (SHM / DD), we calculate typical measures such as the modal assurance criterion (MAC) and the coordinate modal assurance criterion (COMAC) by extracting information about the mode shape functions from the spatiotemporal IMO solutions. Whereas the MAC provides a global aspect of damage occurrence (i.e., which modes are more affected by induced defects), the COMAC can narrow down the damage locations (i.e., where in the structure defects exist that yield low correlation values in specific modes). Finally, we discuss the use of the 2-dimensional correlation spectroscopy technique to SHM / DD, which is frequently used in optical chemistry areas. With the spatiotemporal IMOs the 2-D correlation intensity for the linear beam is proportional to the product of the two mode shape functions at the respective positions; hence any deviations from that may indicate the occurrence and locations of damage in the structure.

Influence of the Finite Element Model on the Inverse Determination of the Heat Transfer Coefficient Distribution over the Hot Plate Cooled by the Laminar Water Jets / Wpływ Modelu Metody Elementów Skonczonych Na Współczynnika Wymiany Ciepła Wyznaczany Z Rozwiazania Odwrotnego Procesu Laminarnego Chłodzenia Płyty Metalowej

2013 ◽  
Vol 58 (1) ◽  
pp. 105-112 ◽  
Author(s):  
B. Hadała ◽  
Z. Malinowski ◽  
T. Telejko ◽  
A. Szajding
Keyword(s):  
Heat Transfer ◽  
Boundary Condition ◽  
Heat Transfer Coefficient ◽  
Heat Conduction ◽  
Transfer Coefficient ◽  
Shape Functions ◽  
Strip Surface ◽  
Coefficient Distribution ◽  
Laminar Jets ◽  
The Heat Transfer Coefficient

The industrial hot rolling mills are equipped with systems for controlled cooling of hot steel products. In the case of strip rolling mills the main cooling system is situated at run-out table to ensure the required strip temperature before coiling. One of the most important system is laminar jets cooling. In this system water is falling down on the upper strip surface. The proper cooling rate affects the final mechanical properties of steel which strongly dependent on microstructure evolution processes. Numerical simulations can be used to determine the water flux which should be applied in order to control strip temperature. The heat transfer boundary condition in case of laminar jets cooling is defined by the heat transfer coefficient, cooling water temperature and strip surface temperature. Due to the complex nature of the cooling process the existing heat transfer models are not accurate enough. The heat transfer coefficient cannot be measured directly and the boundary inverse heat conduction problem should be formulated in order to determine the heat transfer coefficient as a function of cooling parameters and strip surface temperature. In inverse algorithm various heat conduction models and boundary condition models can be implemented. In the present study two three dimensional finite element models based on linear and non-linear shape functions have been tested in the inverse algorithm. Further, two heat transfer boundary condition models have been employed in order to determine the heat transfer coefficient distribution at the hot plate cooled by laminar jets. In the first model heat transfer coefficient distribution over the cooled surface has been approximated by the witch of Agnesi type function with the expansion in time of the approximation parameters. In the second model heat transfer coefficient distribution over the cooled plate surface has been approximated by the surface elements serendipity family with parabolic shape functions. The heat transfer coefficient values at surface element nodes have been expanded in time by the cubic-spline functions. The numerical tests have shown that in the case of heat conduction model based on linear shape functions inverse solution differs significantly from the searched boundary condition. The dedicated finite element heat conduction model based on non-linear shape functions has been developed to ensure inverse determination of heat transfer coefficient distribution over the cooled surface in the time of cooling. The heat transfer coefficient model based on surface elements serendipity family is not limited to a particular form of the heat flux distribution. The solution has been achieved for measured temperatures of the steel plate cooled by 9 laminar jets.

scholarly journals Geometric aspects of high-order eigenvalue problems I. Structures on spaces of boundary conditions

2004 ◽  
Vol 2004 (13) ◽  
pp. 647-678 ◽  
Author(s):  
Xifang Cao ◽  
Hongyou Wu
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Complex Number ◽  
Geometric Structure ◽  
Eigenvalue Problems ◽  
Arbitrary Order ◽  
High Order ◽  
Manifold Structure ◽  
Complex Boundary ◽  
Quasidifferential Equation

We consider some geometric aspects of regular eigenvalue problems of an arbitrary order. First, we clarify a natural geometric structure on the space of boundary conditions. This structure is the base for studying the dependence of eigenvalues on the boundary condition involved, and reveals new properties of these eigenvalues. Then, we solve the selfadjointness condition explicitly and obtain a manifold structure on the space of selfadjoint boundary conditions and several other consequences. Moreover, we give complete characterizations of several subsets of boundary conditions such as the set of all complex boundary conditions having a given complex number as an eigenvalue, and describe some of them topologically. The shapes of some of these subsets are shown to be independent of the quasidifferential equation in question.

Formulating frequency of uniform beams with tip mass under various axial loads

2013 ◽  
Vol 228 (1) ◽  
pp. 67-76 ◽  
Author(s):  
Renfan Luo
Keyword(s):  
Mode Shape ◽  
Shape Functions ◽  
Axial Loads ◽  
Concentrated Force ◽  
Element Analysis ◽  
Axial Acceleration ◽  
Free Transverse Vibration ◽  
Tip Mass ◽  
Energy Conservation Principle ◽  
Good Agreement

The energy conservation principle has been applied to derive the formulation of the frequencies of free transverse vibration of beams for a given mode shape. For uniform beams with a tip mass and with either a clamped/free or a clamped/sliding or a hinged/sliding constraint, under various loads, such a centrifugal force, axial acceleration and concentrated force, the mode shape functions for the free uniform beams have been employed to develop empirical formulae, which are capable of predicting their frequencies. The predictions show a good agreement with those given by finite element analysis.

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