An Accurate Singularity-Free Formulation of a Three-Dimensional Curved Euler–Bernoulli Beam for Flexible Multibody Dynamic Analysis

2016 ◽  
Vol 138 (5) ◽  
Author(s):  
W. Fan ◽  
W. D. Zhu
Keyword(s):  
Dynamic Analysis ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
Bernoulli Beam ◽  
Euler Parameters ◽  
Generalized Coordinates ◽  
Current Formulation ◽  
Multibody Dynamic ◽  
Flexible Multibody ◽  
Euler Bernoulli Beam

An accurate singularity-free formulation of a three-dimensional curved Euler–Bernoulli beam with large deformations and large rotations is developed for flexible multibody dynamic analysis. Euler parameters are used to characterize orientations of cross sections of the beam, which can resolve the singularity problem caused by Euler angles. The position of the centroid line of the beam is integrated from its slope, and position vectors of nodes of beam elements are no longer used as generalized coordinates. Hence, the number of generalized coordinates for each node is minimized. Euler parameters instead of position vectors are interpolated in the current formulation, and a new C1-continuous interpolation function is developed, which can greatly reduce the number of elements. Governing equations of the beam and constraint equations are derived using Lagrange's equations for systems with constraints, which are solved by the generalized- α method for resulting differential-algebraic equations (DAEs). The current formulation can be used to calculate static and dynamic problems of straight and curved Euler–Bernoulli beams under arbitrary, concentrated and distributed forces. The stiffness matrix and generalized force in the current formulation are much simpler than those in the geometrically exact beam formulation (GEBF) and absolute node coordinate formulation (ANCF), which makes it more suitable for static equilibrium problems. Numerical simulations show that the current formulation can achieve the same accuracy as the GEBF and ANCF with much fewer elements and generalized coordinates.

An Accurate Singularity-Free Formulation of a Three-Dimensional Curved Euler-Bernoulli Beam for Flexible Multibody Dynamic Analysis

2016 ◽  
Author(s):  
W. Fan ◽  
W. D. Zhu
Keyword(s):  
Dynamic Analysis ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
Bernoulli Beam ◽  
Euler Parameters ◽  
Generalized Coordinates ◽  
Current Formulation ◽  
Multibody Dynamic ◽  
Flexible Multibody ◽  
Euler Bernoulli Beam

An accurate singularity-free formulation of a three-dimensional curved Euler-Bernoulli beam with large deformations and large rotations is developed for flexible multibody dynamic analysis. Euler parameters are used to characterize orientations of cross-sections of the beam, which can resolve the singularity problem caused by Euler angles. The position of the centroid line of the beam is integrated from its slope, and position vectors of nodes of beam elements are no longer used as generalized coordinates. Hence, the number of generalized coordinates for each node is minimized. Euler parameters instead of position vectors are interpolated in the current formulation, and a new C1-continuous interpolation function is developed, which can greatly reduce the number of elements. Governing equations of the beam and constraint equations are derived using Lagrange’s equations for systems with constraints, which are solved by the generalized-α method for resulting differential-algebraic equations. The current formulation can be used to calculate static and dynamic problems of straight and curved Euler-Bernoulli beams under arbitrary concentrated and distributed forces. The stiffness matrix and generalized force in the current formulation are much simpler than those in the geometrically exact beam formulation (GEBF) and absolute node coordinate formulation (ANCF), which makes it more suitable for static equilibrium problems. Numerical simulations show that the current formulation can achieve the same accuracy as the GEBF and ANCF with much fewer elements and generalized coordinates.

A New Singularity-Free Formulation of a Three-Dimensional Euler-Bernoulli Beam Using Euler Parameters

2015 ◽  
Author(s):  
W. Fan ◽  
W. D. Zhu ◽  
H. Ren
Keyword(s):  
Mass Matrix ◽  
Three Dimensional ◽  
Constraint Equation ◽  
Bernoulli Beam ◽  
Differential Algebraic Equation ◽  
Euler Parameters ◽  
Large Rotation ◽  
Current Formulation ◽  
Nodal Coordinates ◽  
Euler Bernoulli Beam

In this investigation, a new singularity-free formulation of a three-dimensional Euler-Bernoulli beam with large deformation and large rotation is developed. The position of the centroid line of the beam is integrated from its slope, which can be easily expressed by Euler parameters. The hyper-spherical interpolation function is used to guarantee that the normalization constraint equation of Euler parameters is always satisfied. Hence, each node of a beam element has only four nodal coordinates, which is significantly fewer than an absolute node coordinate formulation (ANCF) and the finite element method (FEM). Governing equations of the beam and constraint equations are derived using Lagrange’s equations for systems with constraints, which are solved by an available differential algebraic equation solver. The current formulation can be used to calculate the static equilibrium and dynamics of an Euler-Bernoulli beam under arbitrary concentrated and distributed forces. While the mass matrix is more complex than that in an absolute nodal coordinate formulation, the stiffness matrix and generalized forces are simpler, which is amenable for calculating the equilibrium of the beam. Several numerical examples are presented to demonstrate the performance of the current formulation. It is shown that the current formulation can achieve the same accuracy as the FEM and ANCF with a fewer number of coordinates.

A New Singularity-Free Formulation of a Three-Dimensional Euler–Bernoulli Beam Using Euler Parameters

2016 ◽  
Vol 11 (4) ◽  
Author(s):  
W. Fan ◽  
W. D. Zhu ◽  
H. Ren
Keyword(s):  
Mass Matrix ◽  
Three Dimensional ◽  
Constraint Equation ◽  
Bernoulli Beam ◽  
Differential Algebraic Equation ◽  
Euler Parameters ◽  
Current Formulation ◽  
Nodal Coordinates ◽  
Distributed Forces ◽  
Euler Bernoulli Beam

In this investigation, a new singularity-free formulation of a three-dimensional Euler–Bernoulli beam with large deformations and large rotations is developed. The position of the centroid line of the beam is integrated from its slope, which can be easily expressed by Euler parameters. The hyperspherical interpolation function is used to guarantee that the normalization constraint equation of Euler parameters is always satisfied. Each node of a beam element has only four nodal coordinates, which are significantly fewer than those in an absolute node coordinate formulation (ANCF) and the finite element method (FEM). Governing equations of the beam and constraint equations are derived using Lagrange's equations for systems with constraints, which are solved by a differential-algebraic equation (DAE) solver. The current formulation can be used to calculate the static equilibrium and linear and nonlinear dynamics of an Euler–Bernoulli beam under arbitrary, concentrated, and distributed forces. While the mass matrix is more complex than that in the ANCF, the stiffness matrix and generalized forces are simpler, which is amenable for calculating the equilibrium of the beam. Several numerical examples are presented to demonstrate the performance of the current formulation. It is shown that the current formulation can achieve the same accuracy as the ANCF and FEM with a fewer number of coordinates.

A New Three-Dimensional Moving Timoshenko Beam Element for Moving Load Problem Analysis

2020 ◽  
Vol 142 (3) ◽  
Author(s):  
Yan Xu ◽  
Weidong Zhu ◽  
Wei Fan ◽  
Caijing Yang ◽  
Weihua Zhang
Keyword(s):  
Dynamic Analysis ◽  
Coordinate System ◽  
Timoshenko Beam ◽  
Moving Load ◽  
Three Dimensional ◽  
Beam Element ◽  
Superposition Method ◽  
Reference Coordinate System ◽  
Current Formulation ◽  
Timoshenko Beam Element

Abstract A new three-dimensional moving Timoshenko beam element is developed for dynamic analysis of a moving load problem with a very long beam structure. The beam has small deformations and rotations, and bending, shear, and torsional deformations of the beam are considered. Since the dynamic responses of the beam are concentrated on a small region around the moving load and most of the long beam is at rest, owing to the damping effect, the beam is truncated with a finite length. A control volume that is attached to the moving load is introduced, which encloses the truncated beam, and a reference coordinate system is established on the left end of the truncated beam. The arbitrary Lagrangian–Euler method is used to describe the relationship of the position of a particle on the beam between the reference coordinate system and the global coordinate system. The truncated beam is spatially discretized using the current beam elements. Governing equations of a moving element are derived using Lagrange’s equations. While the whole beam needs to be discretized in the finite element method or modeled in the modal superposition method (MSM), only the truncated beam is discretized in the current formulation, which greatly reduces degrees-of-freedom and increases the efficiency. Furthermore, the efficiency of the present beam element is independent of the moving load speed, and the critical or supercritical speed range of the moving load can be analyzed through the present method. After the validation of the current formulation, a dynamic analysis of three-dimensional train–track interaction with a non-ballasted track is conducted. Results are in excellent agreement with those from the commercial software simpack where the MSM is used, and the calculation time of the current formulation is one-third of that of simpack. The current beam element is accurate and more efficient than the MSM for moving load problems of long three-dimensional beams. The derivation of the current beam element is straightforward, and the beam element can be easily extended for various other moving load problems.

An Exact Solution for the Natural Frequencies of Flexible Beams Undergoing Overall Motions

2003 ◽  
Vol 9 (11) ◽  
pp. 1221-1229 ◽  
Author(s):  
Ali H Nayfeh ◽  
S.A. Emam ◽  
Sergio Preidikman ◽  
D.T. Mook
Keyword(s):  
Exact Solution ◽  
Natural Frequencies ◽  
Three Dimensional ◽  
Beam Theory ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Bernoulli Beam ◽  
Flexible Beams ◽  
Euler Bernoulli Beam

We investigate the free vibrations of a flexible beam undergoing an overall two-dimensional motion. The beam is modeled using the Euler-Bernoulli beam theory. An exact solution for the natural frequencies and corresponding mode shapes of the beam is obtained. The model can be extended to beams undergoing three-dimensional motions.

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