Three-dimensional MEMS simulation using euler parameters

2004 ◽  
Author(s):  
G. Casinovi
Keyword(s):  
Three Dimensional ◽  
Euler Parameters

scholarly journals A Survey on the Computation of Quaternions From Rotation Matrices

2019 ◽  
Vol 11 (2) ◽  
Author(s):  
Soheil Sarabandi ◽  
Federico Thomas
Keyword(s):  
Three Dimensional ◽  
Rotation Matrix ◽  
Three Dimensions ◽  
Attitude Dynamics ◽  
Euler Parameters ◽  
Four Dimensions ◽  
Rotation Matrices ◽  
Spacecraft Attitude Dynamics ◽  
Applied Fields ◽  
Quaternion Representation

The parameterization of rotations is a central topic in many theoretical and applied fields such as rigid body mechanics, multibody dynamics, robotics, spacecraft attitude dynamics, navigation, three-dimensional image processing, and computer graphics. Nowadays, the main alternative to the use of rotation matrices, to represent rotations in ℝ3, is the use of Euler parameters arranged in quaternion form. Whereas the passage from a set of Euler parameters to the corresponding rotation matrix is unique and straightforward, the passage from a rotation matrix to its corresponding Euler parameters has been revealed to be somewhat tricky if numerical aspects are considered. Since the map from quaternions to 3 × 3 rotation matrices is a 2-to-1 covering map, this map cannot be smoothly inverted. As a consequence, it is erroneously assumed that all inversions should necessarily contain singularities that arise in the form of quotients where the divisor can be arbitrarily small. This misconception is herein clarified. This paper reviews the most representative methods available in the literature, including a comparative analysis of their computational costs and error performances. The presented analysis leads to the conclusion that Cayley's factorization, a little-known method used to compute the double quaternion representation of rotations in four dimensions from 4 × 4 rotation matrices, is the most robust method when particularized to three dimensions.

scholarly journals Hamilton’s Equations With Euler Parameters for Rigid Body Dynamics Modeling

2004 ◽  
Vol 126 (1) ◽  
pp. 124-130 ◽  
Author(s):  
Ravishankar Shivarama ◽  
Eric P. Fahrenthold
Keyword(s):  
Rigid Body ◽  
Three Dimensional ◽  
Nonlinear Problems ◽  
Potential Energy Function ◽  
Rigid Body Dynamics ◽  
Dynamics Modeling ◽  
Euler Parameters ◽  
Strongly Nonlinear ◽  
Body Dynamics ◽  
General Potential

A combination of Euler parameter kinematics and Hamiltonian mechanics provides a rigid body dynamics model well suited for use in strongly nonlinear problems involving arbitrarily large rotations. The model is unconstrained, free of singularities, includes a general potential energy function and a minimum set of momentum variables, and takes an explicit state space form convenient for numerical implementation. The general formulation may be specialized to address particular applications, as illustrated in several three dimensional example problems.

scholarly journals A Consistent Dynamic Finite Element Formulation For a Pipe Using Euler Parameters

1998 ◽  
Vol 5 (2) ◽  
pp. 111-117
Author(s):  
Ara Arabyan ◽  
Yaqun Jiang
Keyword(s):  
Mass Matrix ◽  
Three Dimensional ◽  
Element Formulation ◽  
Flexible Pipe ◽  
Euler Parameters ◽  
Rotation Rates ◽  
Large Rotations ◽  
Gyroscopic Effects ◽  
External Forces ◽  
Dynamic Loading Conditions

A pipe element developed earlier for the analysis of combined large bending and torsional deformations of blood vessels under static loading is extended to model behavior in the presence of large rotations and time-varying external forces. As in the case of the earlier element, the enhanced element supports ovalization and warping of its cross-section. The enhancements presented in this paper are comprised of a mass matrix and gyroscopic effects resulting from fast rotation rates and large deformations. The effectiveness of the element is demonstrated by two examples, which simulate the three-dimensional behavior of a highly flexible pipe under dynamic loading conditions.

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.

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.

The Algebraic Classification of the Image Curves of Spherical Four-Bar Motion

1991 ◽  
Vol 113 (3) ◽  
pp. 227-231 ◽  
Author(s):  
Q. Jeffrey Ge ◽  
J. M. McCarthy
Keyword(s):  
Three Dimensional ◽  
Eigenvalues And Eigenvectors ◽  
Euler Parameters ◽  
Rotational Displacement ◽  
Algebraic Classification ◽  
Generalized Eigenvalues ◽  
Pass Through ◽  
Dimensional Projective Space ◽  
Four Bar Linkage

A rotational displacement of the coupler of a spherical four-bar linkage can be mapped to a point with coordinates given by the Euler parameters of the rotation. The set of rotational movements available to the coupler defines a curve in this three-dimensional projective space (four homogeneous coordinates). In this paper, we determine the generalized eigenvalues and eigenvectors of the pencil of quadrics that pass through this curve and examine their properties. The result is an algebraic classification of the image curves that parallels the well-known classification of spherical four-linkages. In addition, we find that the characteristic polynomial of the system yields Grashof’s criterion for the rotatability of cranks.

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.

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.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

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