Study on Kinetic Energy for Saving Materials with Analysis of Lumped Mass Matrix of Variable Cross-Section Beam Element

2013 ◽  
Vol 675 ◽  
pp. 158-161
Author(s):  
Lv Zhou Ma ◽  
Jian Liu ◽  
Yu Qin Yan ◽  
Xun Lin Diao
Keyword(s):  
Kinetic Energy ◽  
Cross Section ◽  
Mass Matrix ◽  
Nonlinear Problems ◽  
Variable Cross Section ◽  
Beam Element ◽  
Variable Cross ◽  
Lumped Mass ◽  
Geometrical Nonlinear ◽  
Properties Of Materials

Based on positional finite element method (FEM), a new, simple and accurate lumped mass matrix to solve dynamic geometrical nonlinear problems of materials applied to variable cross-section beam element has been proposed. According to Hamilton theory and the concept of Kinetic energy, concentrate the beam element mass to the two nodes in certain proportion, the lumped mass matrix is deduced. The lumped mass matrix is diagonal matrix and its calculated quantity is less than using consistent mass matrix about properties of materials under the same calculation precision.

Study on Flexible Spin-Up Maneuver with Variable Cross-Section Based on Positional FEM

2014 ◽  
Vol 670-671 ◽  
pp. 834-837
Author(s):  
Lv Zhou Ma ◽  
Yu Qin Yan ◽  
Xun Lin Diao ◽  
Jian Liu
Keyword(s):  
Cross Section ◽  
Mass Matrix ◽  
Variable Cross Section ◽  
Beam Element ◽  
Variable Cross ◽  
Lumped Mass ◽  
Nonlinear Fem ◽  
Calculation Results ◽  
Properties Of Materials ◽  
Consistent Mass Matrix

Based on positional finite element method (FEM) and MATLAB platform, program VBEP (Variable cross-section Beam Element based on Positional FEM) is compiled. Flexible spin-up maneuver is calculated. The calculation results show that positional FEM uses fewer elements and gains higher calculation precision and efficiency when compared with traditional nonlinear FEM, and that calculated quantity using lumped mass matrix is less than using consistent mass matrix about properties of materials under the same calculation precision.

scholarly journals Static, Vibration, and Transient Dynamic Analyses by Beam Element with Adaptive Displacement Interpolation Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-25
Author(s):  
Shuang Li ◽  
Jinjun Hu ◽  
Changhai Zhai ◽  
Lili Xie
Keyword(s):  
Cross Section ◽  
Material Properties ◽  
Polynomial Interpolation ◽  
Mass Matrix ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Lumped Mass ◽  
Beam Elements ◽  
Dynamic Analyses ◽  
Interpolation Functions

An approach to analyzing structures by using beam elements is developed with adaptive displacement interpolation functions. First, the element stiffness matrix and equivalent nodal loads are derived on the basis of the equilibrium between nodal forces and section forces rather than the compatibility between nodal deformations and section deformations, which avoids discretization errors caused by the limitation of conventional polynomial interpolation functions. Then, six adaptive element displacement interpolation functions are derived and extended to include several cases, such as beams with variable cross-section, variable material properties, and many different steps in cross-section and/or material properties. To make the element usable in dynamic analyses, consistent mass matrix (CMM) and diagonally lumped mass matrix (LMM) are constructed using the presented adaptive displacement interpolation functions. All these features have made the element elegant, which is tested with a number of simple static, vibration, and dynamic examples to show its accuracy.

Nonlinear Stain of Variable Cross-Section Beam Element Based on Positional FEM

2012 ◽  
Vol 557-559 ◽  
pp. 2371-2374
Author(s):  
Lv Zhou Ma ◽  
Jian Liu ◽  
Xun Lin Diao ◽  
Xiao Dong Jia
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Cross Section ◽  
Beam Width ◽  
Variable Cross Section ◽  
Beam Element ◽  
Variable Cross ◽  
Nonlinear Fem ◽  
Beam Elements ◽  
Beam Depth

Based on positional finite element method, this paper attempts to find beam elements that can show the characteristics of the variable cross-section beam and can be practically applied. In this paper, the stain on a random point of the variable cross-section beam element is obtained when beam depth changes in a linear or quadratic parabolic way and beam width is fixed. The calculation is different and simpler than the classical nonlinear FEM.

A Higher-Order Variable Cross-Section Viscoelastic Beam Element Via ANCF for Kinematic and Dynamic Analyses of Two-Link Flexible Manipulators

2017 ◽  
Vol 09 (08) ◽  
pp. 1750116 ◽  
Author(s):  
Haidong Yu ◽  
Chunzhang Zhao ◽  
Hui Zheng
Keyword(s):  
Large Deformation ◽  
Cross Section ◽  
Variable Cross Section ◽  
Beam Element ◽  
Higher Order ◽  
Flexible Manipulator ◽  
Variable Cross ◽  
Viscoelastic Beam ◽  
Shear Locking ◽  
Large Deformation Problems

A new viscoelastic beam element with variable cross-sections is developed based on the absolute nodal coordinate formulation, in which the higher-order slope coordinates are used to describe the variable geometric boundaries and circumvent possible shear-locking problem. The mass and stiffness matrices of the new element are derived by considering the variable geometrical boundary in the integration functions. The modified Kelvin–Voigt viscoelastic constitutive model for large deformation problems is introduced into the stiffness matrix. The dynamic model of a typical two-link manipulator with variable cross-section links is established where the constraint equations of revolute joints are considered with Lagrange multipliers. The kinematic trajectories of the manipulator with various materials and geometrical parameters are numerically studied. It is shown that the new element could circumvent shear-locking problem and yield improved accuracy and convergence compared with the conventional beam elements for solving large deformation problems. Also, the viscosity of the structural material helps to reduce the deformation of the links and improve the kinematic precision of the manipulator, hence the trajectory of the flexible manipulator could be controlled by changing the geometrical shape of the cross-section of links under the constraint of same mass.

Numerical Examples of Variable Cross-Section Beam Elements

2012 ◽  
Vol 557-559 ◽  
pp. 822-825
Author(s):  
Lv Zhou Ma ◽  
Jian Liu ◽  
Xun Lin Diao ◽  
Xiao Dong Jia
Keyword(s):  
Exact Solution ◽  
Cantilever Beam ◽  
Cross Section ◽  
Beam Width ◽  
Variable Cross Section ◽  
Beam Element ◽  
Variable Cross ◽  
Numerical Examples ◽  
Beam Elements ◽  
Beam Depth

Based on MATLAB platform, program VCBEP (Variable Cross-section Beam Element based on Positional FEM) is compiled, and the cantilever beam with linear profile and the parabolic simple supported beam are calculated. The variable cross-section beam element is proposed to analyze rectangular beam whose beam depth changes in a linear or quadratic parabolic way and beam width is fixed and the exact solution can be obtained.

The Nonlinear Static Formulation of Variable Cross-Section Beam Element Based on Positional Description

2012 ◽  
Vol 557-559 ◽  
pp. 2367-2370
Author(s):  
Lv Zhou Ma ◽  
Jian Liu ◽  
Xun Lin Diao ◽  
Yu Qin Yan
Keyword(s):  
Cross Section ◽  
Large Deflection ◽  
Variable Cross Section ◽  
Beam Element ◽  
Solution Procedure ◽  
Variable Cross ◽  
Hyper Elastic ◽  
Nonlinear Static ◽  
Newton Raphson ◽  
Raphson Iteration

Based on positional FEM (finite element method), the nonlinear static formulation to treat large deflection of variable cross-section beam element is created by using the lowest potential energy theory. Adopting linear constitutive relation for hyper-elastic materials, the formulation and the solution procedure by Newton-Raphson iteration method are very simple.

Lumped Mass Matrix of Three-Node Beam Element

2012 ◽  
Vol 616-618 ◽  
pp. 1969-1973 ◽  
Author(s):  
Lv Zhou Ma ◽  
Jian Liu ◽  
Xiao Dong Jia ◽  
Yu Qin Yan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Kinetic Energy ◽  
Mass Matrix ◽  
Beam Element ◽  
Diagonal Matrix ◽  
Lumped Mass ◽  
Consistent Mass Matrix ◽  
Element Method

According to Hamilton theory and the concept of Kinetic energy, lumped mass matrix of three-node beam element based on positional finite element method (FEM) is deduced. By concentrating the mass at the three nodes of beam element, the lumped mass matrix is characterized. When dynamic analysing, lumped mass matrix is simpler and more common than consistent mass matrix because lumped mass matrix is diagonal matrix, and calculation work is less under the same calculation precision.

Boundary element method in numerical study of three-dimensional Stokes flows in channels of arbitrary shape

2012 ◽  
Vol 9 (1) ◽  
pp. 94-97
Author(s):  
Yu.A. Itkulova
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Cross Section ◽  
Numerical Study ◽  
Three Dimensional ◽  
Cylindrical Tube ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Periodic Flows ◽  
Element Method

In the present work creeping three-dimensional flows of a viscous liquid in a cylindrical tube and a channel of variable cross-section are studied. A qualitative triangulation of the surface of a cylindrical tube, a smoothed and experimental channel of a variable cross section is constructed. The problem is solved numerically using boundary element method in several modifications for a periodic and non-periodic flows. The obtained numerical results are compared with the analytical solution for the Poiseuille flow.

Longitudinal oscillation of a rod with a variable cross section

2019 ◽  
Vol 14 (2) ◽  
pp. 138-141
Author(s):  
I.M. Utyashev
Keyword(s):  
Cross Section ◽  
Natural Frequencies ◽  
Liouville Problem ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Longitudinal Vibrations ◽  
Cross Sectional ◽  
Independent Functions ◽  
The Cross ◽  
The Right

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in percussion mechanisms. The strength of such parts directly depends on the natural frequencies of longitudinal vibrations. The paper presents a method that allows numerically finding the natural frequencies of longitudinal vibrations of an elastic rod with a variable cross section. This method is based on representing the cross-sectional area as an exponential function of a polynomial of degree n. Based on this idea, it was possible to formulate the Sturm-Liouville problem with boundary conditions of the third kind. The linearly independent functions of the general solution have the form of a power series in the variables x and λ, as a result of which the order of the characteristic equation depends on the choice of the number of terms in the series. The presented approach differs from the works of other authors both in the formulation and in the solution method. In the work, a rod with a rigidly fixed left end is considered, fixing on the right end can be either free, or elastic or rigid. The first three natural frequencies for various cross-sectional profiles are given. From the analysis of the numerical results it follows that in a rigidly fixed rod with thinning in the middle part, the first natural frequency is noticeably higher than that of a conical rod. It is shown that with an increase in the rigidity of fixation at the right end, the natural frequencies increase for all cross section profiles. The results of the study can be used to solve inverse problems of restoring the cross-sectional profile from a finite set of natural frequencies.

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