Geometrically Nonlinear Formulations of Beams in Flexible Multibody Dynamics

1995 ◽  
Vol 117 (4) ◽  
pp. 501-509 ◽  
Author(s):  
J. Mayo ◽  
J. Dominguez ◽  
A. A. Shabana
Keyword(s):  
Stiffness Matrix ◽  
Strain Energy ◽  
Equations Of Motion ◽  
Strain Energy Function ◽  
Flexible Multibody Dynamics ◽  
Second Order ◽  
Computationally Efficient ◽  
Constant Stiffness ◽  
Stiffness Matrices ◽  
Flexible Multibody

In this paper, the equations of motion of flexible multibody systems are derived using a nonlinear formulation which retains the second-order terms in the strain-displacement relationship. The strain energy function used in this investigation leads to the definition of three stiffness matrices and a vector of nonlinear elastic forces. The first matrix is the constant conventional stiffness matrix; the second one is the first-order geometric stiffness matrix; and the third is a second-order stiffness matrix. It is demonstrated in this investigation that accurate representation of the axial displacement due to the foreshortening effect requires the use of large number or special axial shape functions if the nonlinear stiffness matrices are used. An alternative solution to this problem, however, is to write the equations of motion in terms of the axial coordinate along the deformed (instead of undeformed) axis. The use of this representation yields a constant stiffness matrix even if higher order terms are retained in the strain energy expression. The numerical results presented in this paper demonstrate that the proposed new approach is nearly as computationally efficient as the linear formulation. Furthermore, the proposed formulation takes into consideration the effect of all the geometric elastic nonlinearities on the bending displacement without the need to include high frequency axial modes of vibration.

Flexible Multibody Dynamics Formulation by Hamilton’s Equations

2010 ◽  
Author(s):  
Martin M. Tong
Keyword(s):  
Multibody Dynamics ◽  
Multibody System ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Flexible Multibody Dynamics ◽  
Flexible Body ◽  
Flexible Multibody System ◽  
Generalized Coordinates ◽  
Flexible Multibody ◽  
The Matrix

Numerical solution of the dynamics equations of a flexible multibody system as represented by Hamilton’s canonical equations requires that its generalized velocities q˙ be solved from the generalized momenta p. The relation between them is p = J(q)q˙, where J is the system mass matrix and q is the generalized coordinates. This paper presents the dynamics equations for a generic flexible multibody system as represented by p˙ and gives emphasis to a systematic way of constructing the matrix J for solving q˙. The mass matrix is shown to be separable into four submatrices Jrr, Jrf, Jfr and Jff relating the joint momenta and flexible body mementa to the joint coordinate rates and the flexible body deformation coordinate rates. Explicit formulas are given for these submatrices. The equations of motion presented here lend insight to the structure of the flexible multibody dynamics equations. They are also a versatile alternative to the acceleration-based dynamics equations for modeling mechanical systems.

A New Finding on the Dynamic Stiffness Matrices of Asymmetric and Axisymmetric Shafts

2001 ◽  
Author(s):  
Francesco A. Raffa ◽  
Furio Vatta
Keyword(s):  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Rotational Inertia ◽  
Dynamic Stiffness Matrix ◽  
Rayleigh Beam ◽  
Principal Moments Of Inertia ◽  
Stiffness Matrices ◽  
New Finding

Abstract In this paper the dynamic stiffness method is developed to analyze a rotating asymmetric shaft, i.e. a shaft whose transverse section is characterized by dissimilar principal moments of inertia. The shaft is modeled according to the Rayleigh beam theory including the effects of both translational and rotational inertia, and gyroscopic moments. The mathematical description is carried out in a reference system rotating at the shaft speed and is based on the exact solution of the governing differential equations of motion. The exact expressions of the shaft displacements are utilized for deriving the 8 × 8 complex dynamic stiffness matrix of the shaft. A new relationship is obtained which links the dynamic stiffness matrix of the asymmetric shaft to the 4 × 4 real dynamic stiffness matrix of the axisymmetric shaft.

Flexible Multibody Dynamics Based on a Fully Cartesian System of Support Coordinates

1993 ◽  
Vol 115 (2) ◽  
pp. 294-299 ◽  
Author(s):  
N. Vukasovic ◽  
J. T. Celigu¨eta ◽  
J. Garci´a de Jalo´n ◽  
E. Bayo
Keyword(s):  
Multibody Dynamics ◽  
Reference Frames ◽  
Deformable Body ◽  
Equations Of Motion ◽  
Flexible Multibody Dynamics ◽  
Body Motion ◽  
Cartesian System ◽  
Flexible Multibody ◽  
Local Reference ◽  
Rigid Multibody

In this paper we present an extension to flexible multibody systems of a system of fully cartesian coordinates previously used in rigid multibody dynamics. This method is fully compatible with the previous one, keeping most of its advantages in kinematics and dynamics. The deformation in each deformable body is expressed as a linear combination of Ritz vectors with respect to a local frame whose motion is defined by a series of points and vectors that move according to the rigid body motion. Joint constraint equations are formulated through the points and vectors that define each link. These are chosen so that a minimum use of local reference frames is done. The resulting equations of motion are integrated using the trapezoidal rule combined with fixed point iteration. An illustrative example that corresponds to a satellite deployment is presented.

Finite Elastic Deformation of an Annular Rotating Disk

1976 ◽  
Vol 98 (4) ◽  
pp. 375-379 ◽  
Author(s):  
J. B. Haddow ◽  
M. G. Faulkner
Keyword(s):  
Elastic Deformation ◽  
Strain Energy ◽  
Equations Of Motion ◽  
Strain Energy Function ◽  
Rotating Disk ◽  
Force Balance ◽  
Annular Disk ◽  
Incompressible Materials ◽  
Plane Stress Problem ◽  
Governing Differential Equation

An accurate approximate method for the solution of the generalized plane stress problem of finite elastic deformation of a rotating annular disk is given. The method is applicable to both compressible and incompressible materials. Use of the nonlinear governing differential equation is avoided by considering the disk to be an aggregate of discrete coaxial rings, equations of motion, force balance and compatibility relations being formulated for the rings. Results are given for neo-Hookean disks and disks of a compressible material which has a strain energy function proposed by Blatz and Ko [7].

Finite Element Method of Flexible Multibody Dynamics Based on a New Kind of Floating Frame

1991 ◽  
Author(s):  
You-Fang Lu ◽  
Zhao-Hui Qi ◽  
Bin Wang ◽  
Guan-Min Feng
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Analysis ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Flexible Multibody Dynamics ◽  
Flexible Multibody ◽  
Floating Frame ◽  
Element Method

Abstract A new kind of floating frame whose parameters do not appear in equations of motion as additional unknowns is defined. Numerical analysis of flexible multibody dynamics is much facilitated by using finite-element iteration of the corresponding equations based on this concept.

On the use of convected coordinate systems in the mechanics of continuous media

1951 ◽  
Vol 47 (3) ◽  
pp. 575-584 ◽  
Author(s):  
A. S. Lodge
Keyword(s):  
Coordinate System ◽  
Strain Energy ◽  
Equations Of State ◽  
Equations Of Motion ◽  
Strain Energy Function ◽  
Elastic Solid ◽  
Moving Medium ◽  
Coordinate Systems ◽  
Invariance Properties ◽  
The Right

The use of a coordinate system convected with the moving medium for describing its mechanics, first proposed by Hencky (5), has since been extended by several authors, and has several advantages over the more conventional use of a coordinate system fixed in space; Brillouin(1) has shown that the relation between the strain-energy function for an ideally elastic solid and the stress tensor takes a very simple form when the latter is referred to a convected coordinate system; Oldroyd(8) has given a very general discussion of the formulation of rheological equations of state and has shown that the right invariance properties are most readily recognized when the equations are referred to a convected coordinate system; Green and Zerna (4) have similarly expressed the equations of motion and boundary conditions; and Gleyzal (2), and Green and Shield (3) have applied the formalism to certain problems in elasticity theory.

A New Finding on the Dynamic Stiffness Matrices of Asymmetric and Axisymmetric Shafts

2002 ◽  
Vol 124 (4) ◽  
pp. 649-653
Author(s):  
Francesco A. Raffa ◽  
Furio Vatta
Keyword(s):  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Rotational Inertia ◽  
Dynamic Stiffness Matrix ◽  
Rayleigh Beam ◽  
Principal Moments Of Inertia ◽  
Stiffness Matrices ◽  
New Finding

In this paper the dynamic stiffness method is developed to analyze a rotating asymmetric shaft, i.e., a shaft whose transverse section is characterized by dissimilar principal moments of inertia. The shaft is modelled according to the Rayleigh beam theory including the effects of both translational and rotational inertia, and gyroscopic moments. The mathematical description is carried out in a reference system rotating at the shaft speed and is based on the exact solution of the governing differential equations of motion. The exact expressions of the shaft displacements are utilized for deriving the 8×8 complex dynamic stiffness matrix of the shaft. A new relationship is obtained which links the dynamic stiffness matrix of the asymmetric shaft to the 4×4 real dynamic stiffness matrix of the axisymmetric shaft.

Verification of Absolute Nodal Coordinate Formulation in Flexible Multibody Dynamics via Physical Experiments of Large Deformation Problems

2005 ◽  
Vol 1 (1) ◽  
pp. 81-93 ◽  
Author(s):  
Wan-Suk Yoo ◽  
Su-Jin Park ◽  
Oleg N. Dmitrochenko ◽  
Dmitry Yu. Pogorelov
Keyword(s):  
Large Deformation ◽  
Multibody Dynamics ◽  
Absolute Nodal Coordinate Formulation ◽  
Equations Of Motion ◽  
Geometrical Nonlinearity ◽  
Flexible Multibody Dynamics ◽  
Absolute Nodal Coordinate ◽  
Spatial Beams ◽  
Flexible Multibody ◽  
Large Deformation Problems

A review of the current state of the absolute nodal coordinate formulation (ANCF) is proposed for large-displacement and large-deformation problems in flexible multibody dynamics. The review covers most of the known implementations of different kinds of finite elements including thin and thick planar and spatial beams and plates, their geometrical description inherited from FEM, and formulations of the most important elements of equations of motion. Much attention is also paid to simulation examples that show reasonableness and accuracy of the formulations applied to real physical problems and that are compared with experiments having significant geometrical nonlinearity. Current and further development directions of the ANCF are also briefly outlined.

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