Two Kinds of Finite Element Variables Based on B-Spline Wavelet on Interval for Curved Beam

2019 ◽  
Vol 11 (02) ◽  
pp. 1950017 ◽  
Author(s):  
Yanfei He ◽  
Xingwu Zhang ◽  
Jia Geng ◽  
Xuefeng Chen ◽  
Zengguang Li
Keyword(s):  
Finite Element ◽  
Potential Energy ◽  
Free Vibration Analysis ◽  
Energy Functional ◽  
Curved Beam ◽  
Spline Wavelet ◽  
B Spline ◽  
Generalized Potential ◽  
Potential Energy Functional ◽  
Two Kinds Of Variables

Curved beam structure has been widely used in engineering, due to its good load-bearing and geometric characteristics. More common methods for analyzing and designing this structure are the finite element methods (FEMs), but these methods have many disadvantages. Fortunately, the multivariable wavelet FEMs can solve these drawbacks. However, the multivariable generalized potential energy functional of curved beam, used to construct this element, has not been given in previous literature. In this paper, the generalized potential energy functional for curved beam with two kinds of variables is derived initially. On this basis, the B-spline wavelet on the interval (BSWI) is used as the interpolation function to construct the wavelet curved beam element with two kinds of variables. In the end, several typical numerical examples of thin to thick curved beams are given, which show that the present element is more effective in static and free vibration analysis of curved beam structures.

scholarly journals The Analysis of Curved Beam Using B-Spline Wavelet on Interval Finite Element Method

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Zhibo Yang ◽  
Xuefeng Chen ◽  
Yumin He ◽  
Zhengjia He ◽  
Jie Zhang
Keyword(s):  
Finite Element ◽  
Polynomial Interpolation ◽  
Physical Space ◽  
Curved Beam ◽  
Approximation Properties ◽  
Curved Beams ◽  
Spline Wavelet ◽  
B Spline ◽  
Wavelet Space ◽  
Interval Finite Element Method

A B-spline wavelet on interval (BSWI) finite element is developed for curved beams, and the static and free vibration behaviors of curved beam (arch) are investigated in this paper. Instead of the traditional polynomial interpolation, scaling functions at a certain scale have been adopted to form the shape functions and construct wavelet-based elements. Different from the process of the direct wavelet addition in the other wavelet numerical methods, the element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space by aid of the corresponding transformation matrix. Furthermore, compared with the commonly used Daubechies wavelet, BSWI has explicit expressions and excellent approximation properties, which guarantee satisfactory results. Numerical examples are performed to demonstrate the accuracy and efficiency with respect to previously published formulations for curved beams.

Principles of minimum potential energy and complementary energy

2020 ◽  
pp. 228-248
Author(s):  
Lallit Anand ◽  
Sanjay Govindjee
Keyword(s):  
Boundary Condition ◽  
Potential Energy ◽  
Displacement Field ◽  
Mixed Boundary ◽  
Energy Functional ◽  
The Body ◽  
Minimum Potential Energy ◽  
Complementary Energy ◽  
Minimum Potential ◽  
Potential Energy Functional

With the displacement field taken as the only fundamental unknown field in a mixed-boundary-value problem for linear elastostatics, the principle of minimum potential energy asserts that a potential energy functional, which is defined as the difference between the free energy of the body and the work done by the prescribed surface tractions and the body forces --- assumes a smaller value for the actual solution of the mixed problem than for any other kinematically admissible displacement field which satisfies the displacement boundary condition. This principle provides a weak or variational method for solving mixed boundary-value-problems of elastostatics. In particular, instead of solving the governing Navier form of the partial differential equations of equilibrium, one can search for a displacement field such that the first variation of the potential energy functional vanishes. A similar principle of minimum complementary energy, which is phrased in terms of statically admissible stress fields which satisfy the equilibrium equation and the traction boundary condition, is also discussed. The principles of minimum potential energy and minimum complementary energy can also be applied to derive specialized principles which are particularly well-suited to solving structural problems; in this context the celebrated theorems of Castigliano are discussed.

QUANTITATIVE IDENTIFICATION OF ROTOR CRACKS BASED ON FINITE ELEMENT OF B-SPLINE WAVELET ON THE INTERVAL

2009 ◽  
Vol 07 (04) ◽  
pp. 443-457 ◽  
Author(s):  
XUEFENG CHEN ◽  
BING LI ◽  
JIAWEI XIANG ◽  
ZHENGJIA HE
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Beam Element ◽  
Rotational Inertia ◽  
Spline Wavelet ◽  
B Spline ◽  
Rotor Systems ◽  
Crack Location ◽  
Contour Lines ◽  
Quantitative Identification

Based on finite element of B-spline wavelet on the interval (BSWI), the quantitative identification method of transverse crack for rotor systems was studied. The new model of BSWI Rayleigh–Euler rotary beam element considering gyroscopic effect and rotational inertia was constructed to solve the first three natural frequencies of the cracked rotor with high precision, and the first three frequencies solution surfaces of normalized crack location and size were obtained by using surface-fitting technique. Then the first three metrical natural frequencies were employed as inputs of the solution curve surfaces. The intersection of the three frequencies contour lines predicted the normalized crack location and size. The numerical and experimental examples were given to verify the validity of the beam element for crack quantitative identification in rotor systems. The new method can be applied to prognosis and quantitative diagnosis of cracks in the rotor system.

scholarly journals Magnetic motion of spherical frictional charged particles on the unit sphere

2019 ◽  
Vol 65 (5 Sept-Oct) ◽  
pp. 496 ◽  
Author(s):  
Talat Körpınar ◽  
Ridvan Cem Demirkol
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Potential Energy ◽  
Frictional Force ◽  
Poynting Vector ◽  
Positive Curvature ◽  
Charged Particles ◽  
Energy Functional ◽  
Ordinary Space ◽  
Potential Energy Functional

Mathematically, the sphere unit S² is described to be a 2-sphere in an ordinary space with a positive curvature. In this study, we aim to present the manipulation of a spherical charged particle in a continuous motion with a magnetic field on the sphere S² while it is exposed to a frictional force. In other words, we effot to derive the exact geometric characterization for the spherical charged particle under the influence of a frictional force field on the unit 2-sphere. This approach also helps to discover some physical and kinematical characterizations belonging to the particle such as the magnetic motion, the torque, the potential energy functional, and the Poynting vector.

Free vibration analysis of large deformed curved beam incorporating rotary inertia and shear deformation effects and its experimental validation

2020 ◽  
Vol 234 (24) ◽  
pp. 4780-4800
Author(s):  
Sushanta Ghuku ◽  
Kashinath Saha
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Shear Deformation ◽  
Governing Equation ◽  
Free Vibration Analysis ◽  
Static Problem ◽  
Rotary Inertia ◽  
Curved Beam ◽  
Free Vibration Frequency ◽  
Different Thickness

The paper theoretically and experimentally analyzes free vibration characteristics of statically loaded moving boundary type curved beam considering rotary inertia and shear deformation effects. Effects of rotary inertia and shear deformation are observed for different thickness to span ratios of curved beam. The subject problem is decoupled into two interrelated problems: determining equilibrium configuration under static load and finding the corresponding free vibration frequency. The static problem is analyzed incrementally in body fitted curvilinear frame as it involves geometric nonlinearity due to generalized curvature, large deformation, and moving boundaries. Variational energy principle is employed to derive governing equation. The nonlinear governing equation associated with complicated boundary conditions is solved through iterative geometry updation. Once static problem is solved for current load step, governing equation for dynamic characteristics is derived using Hamilton’s principle. The governing equation gets linearized by using the static configuration, which finally yields a linear eigenvalue problem. Experiment is performed in a dedicated setup with two master leafs having different thickness to span ratios. The roller supported specimens are excited with an instrumented hammer and response signals are captured by accelerometers. The excitation and response signals are recorded using HBM-MX840B data acquisition system. Frequency response functions of the curved beam systems under different static loads are obtained from postprocessing of the dynamic signals in MATLAB®. First two natural frequencies of the specimens are noted from the experimental results and the corresponding theoretical results are generated. The specimens are also modeled in ABAQUS® CAE and finite element results are computed. Comparison between the theoretical, experimental, and finite element results validates the present model. The study also provides some meaningful observations on effects of rotary inertia and shear deformation. Based on the observations, more results are generated for different thickness to span ratios and findings are reported suitably.

A vectorizable potential energy functional for reactive scattering

1987 ◽  
Vol 72 (4) ◽  
pp. 253-264 ◽  
Author(s):  
Ernesto Garcia ◽  
Luigi Ciccarelli ◽  
Antonio Lagan�
Keyword(s):  
Potential Energy ◽  
Energy Functional ◽  
Reactive Scattering ◽  
Potential Energy Functional
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