The conditions for mode decoupling of a linear vibration system with diagonalizable stiffness matrices via screw theory

2010 ◽  
Vol 225 (1) ◽  
pp. 101-111 ◽  
Author(s):  
M B Hong ◽  
Y J Choi
Keyword(s):  
Stiffness Matrix ◽  
Screw Theory ◽  
Geometric Approach ◽  
Vibration Modes ◽  
Vibration System ◽  
Linear Vibration ◽  
Linearly Independent ◽  
Stiffness Matrices ◽  
Spatial Displacements ◽  
Necessary And Sufficient

In this article, a new geometric approach to the conditions for mode decoupling of an elastically suspended rigid body with a diagonalizable stiffness matrix is presented. The necessary and sufficient condition for diagonalization of a spatial stiffness matrix by a proper co-ordinate transformation is derived using the form of the stiffness matrix expressed by uniquely determined three orthogonal lines and three orthogonal torsional springs. Three orthogonal planes are defined by use of the axes of the line springs. Both the spatial displacements lying on or perpendicular to each of the planes and the spatial forces induced due to the displacements are decoupled into two linearly independent three-systems of screws. From this property and the geometric relations between those planes obtained from the stiffness and the principal planes of inertia, the conditions for mode decoupling of the vibration system are derived and the possible locations of the axes of the decoupled vibration modes are identified.

scholarly journals A Symbolic Formulation for Analytical Compliance Analysis and Synthesis of Flexure Mechanisms

2012 ◽  
Vol 134 (5) ◽  
Author(s):  
Hai-Jun Su ◽  
Hongliang Shi ◽  
JingJun Yu
Keyword(s):  
Sensitivity Analysis ◽  
Finite Element ◽  
Stiffness Matrix ◽  
Screw Theory ◽  
Element Model ◽  
Fe Model ◽  
Flexure Mechanism ◽  
Stiffness Matrices ◽  
Analysis And Synthesis ◽  
Flexure Joints

This paper presents a symbolic formulation for analytical compliance analysis and synthesis of flexure mechanisms with serial, parallel, or hybrid topologies. Our approach is based on the screw theory that characterizes flexure deformations with motion twists and loadings with force wrenches. In this work, we first derive a symbolic formulation of the compliance and stiffness matrices for commonly used flexure elements, flexure joints, and simple chains. Elements of these matrices are all explicit functions of flexure parameters. To analyze a general flexure mechanism, we subdivide it into multiple structural modules, which we identify as serial, parallel, or hybrid chains. We then analyze each module with the known flexure structures in the library. At last, we use a bottom-up approach to obtain the compliance/stiffness matrix for the overall mechanism. This is done by taking appropriate coordinate transformation of twists and wrenches in space. Four practical examples are provided to demonstrate the approach. A numerical example is employed to compare analytical compliance models against a finite element model. The results show that the errors are sufficiently small (2%, compared with finite element (FE) model), if the range of motion is limited to linear deformations. This work provides a systematical approach for compliance analysis and synthesis of general flexure mechanisms. The symbolic formulation enables subsequent design tasks, such as compliance synthesis or sensitivity analysis.

Decoupling of the Cartesian Stiffness Matrix: A Case Study on Accelerometer Design

2011 ◽  
Author(s):  
Ting Zou ◽  
Jorge Angeles
Keyword(s):  
Mechanical Properties ◽  
Finite Element Analysis ◽  
Stiffness Matrix ◽  
Screw Theory ◽  
Rotational Stiffness ◽  
Element Analysis ◽  
Cartesian Stiffness Matrix ◽  
Stiffness Matrices ◽  
Cartesian Stiffness

The 6 × 6 Cartesian stiffness matrix obtained through finite element analysis for structures designed with material and geometric symmetries may lead to unexpected coupling that stems from discretization error. Hence, decoupling of the Cartesian stiffness matrix becomes essential for design and analysis. This paper reports a numerical method for decoupling the Cartesian stiffness matrix, based on screw theory. With the aid of this method, the translational and rotational stiffness matrices can be analyzed independently. The mechanical properties of the decoupled stiffness submatrices are investigated via their associated eigenvalue analyses. The decoupling technique is applied to the design of two accelerometer layouts, uniaxial and biaxial, with what the authors term simplicial architectures. The decoupled stiffness matrices reveal acceptable compliance along the sensitive axes and high off-axis stiffness.

On Line Screw Systems and Their Application to Flexure Synthesis

2011 ◽  
Vol 3 (1) ◽  
Author(s):  
Hai-Jun Su ◽  
Hafez Tari
Keyword(s):  
Case Studies ◽  
Computational Algorithm ◽  
Screw Theory ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Constrained Motion ◽  
Motion Patterns ◽  
Linearly Independent ◽  
On Line ◽  
Necessary And Sufficient

Motivated by the problem of synthesizing a pattern of flexures that provide a desired constrained motion, this paper presents a new screw theory that deals with “line screws” and “line screw systems.” A line screw is a screw with a zero pitch. The set of all line screws within a screw system is called a line variety. A general screw system of rank m is a line screw system if the rank of its line variety equals m. This paper answers two questions: (1) how to calculate the rank of a line variety for a given screw system and (2) how to algorithmically find a set of linearly independent lines from a given screw system. It has been previously found that a wire or beam flexure is considered a line screw, or more specifically a pure force wrench. By following the reciprocity and definitions of line screws, we have derived the necessary and sufficient conditions of line screw systems. When applied to flexure synthesis, we show that not all motion patterns can be realized with wire flexures connected in parallel. A computational algorithm based on this line screw theory is developed to find a set of admissible line screws or force wrenches for a given motion space. Two flexure synthesis case studies are provided to demonstrate the theory and the algorithm.

Mass and Stiffness Modifications with Unmodified Eigenstructure Preserving

2013 ◽  
Vol 401-403 ◽  
pp. 475-478 ◽  
Author(s):  
Jun Yang ◽  
Xu Shi Lu ◽  
Jia Fan Zhang ◽  
Hua Jiang Ouyang
Keyword(s):  
Sufficient Condition ◽  
Vibration System ◽  
Necessary And Sufficient Condition ◽  
Stiffness Matrices ◽  
Necessary And Sufficient ◽  
Application Prospects

A necessary and sufficient condition is proposed for the incremental mass and stiffness matrices that modify some eigenpairs while keeping other eigenpairs unchanged, which requires the knowledge of only the few eigenpairs to be modified of the original undamped vibration system. The application prospects are proposed based on this formulation.

scholarly journals Diagonal scaling of stiffness matrices in the Galerkin boundary element method

2000 ◽  
Vol 42 (1) ◽  
pp. 141-150 ◽  
Author(s):  
Mark Ainsworth ◽  
Bill McLean ◽  
Thanh Tran
Keyword(s):  
Integral Equation ◽  
Stiffness Matrix ◽  
Numerical Experiments ◽  
Degrees Of Freedom ◽  
Boundary Integral ◽  
Piecewise Constant ◽  
Stiffness Matrices ◽  
Diagonal Scaling ◽  
Mesh Ratio ◽  
Galerkin Boundary Element Method

AbstractA boundary integral equation of the first kind is discretised using Galerkin's method with piecewise-constant trial functions. We show how the condition number of the stiffness matrix depends on the number of degrees of freedom and on the global mesh ratio. We also show that diagonal scaling eliminates the latter dependence. Numerical experiments confirm the theory, and demonstrate that in practical computations involving strong local mesh refinement, diagonal scaling dramatically improves the conditioning of the Galerkin equations.

Some Generalizations Of Burnsides Theorem

1957 ◽  
Vol 9 ◽  
pp. 336-346
Author(s):  
N. A. Wiegmann
Keyword(s):  
Division Ring ◽  
Sufficient Condition ◽  
Complex Field ◽  
Necessary And Sufficient Condition ◽  
Group Representations ◽  
Linearly Independent ◽  
Necessary And Sufficient

1. Introduction. Burnside's Theorem in the theory of group representations states that a necessary and sufficient condition that a semigroup of matrices of degree n over the complex field be irreducible is that the semigroup contain n2 linearly independent matrices. In the course of dealing with sets of matrices with coefficients in a division ring, Brauer (1) obtained a generalization of this theorem which concerned irreducible semigroups with elements in a division ring.

Elastoplastic Analysis of Layered Metal Matrix Composite Cylinders—Part I: Theory

1996 ◽  
Vol 118 (1) ◽  
pp. 13-20 ◽  
Author(s):  
R. S. Salzar ◽  
M.-J. Pindera ◽  
F. W. Barton
Keyword(s):  
Metal Matrix Composite ◽  
Matrix Composite ◽  
Stiffness Matrix ◽  
Metal Matrix ◽  
Plastic Response ◽  
Solution Strategy ◽  
Composite Tubes ◽  
Elastic Plastic ◽  
Stiffness Matrices ◽  
Local Stiffness

An exact elastic-plastic analytical solution for an arbitrarily laminated metal matrix composite tube subjected to axisymmetric thermo-mechanical and torsional loading is presented. First, exact solutions for transversely isotropic and monoclinic (off-axis) elastoplastic cylindrical shells are developed which are then reformulated in terms of the interfacial displacements as the fundamental unknowns by constructing a local stiffness matrix for the shell. Assembly of the local stiffness matrices into a global stiffness matrix in a particular manner ensures satisfaction of interfacial traction and displacement continuity conditions, as well as the external boundary conditions. Due to the lack of a general macroscopic constitutive theory for the elastic-plastic response of unidirectional metal matrix composites, the micromechanics method of cells model is employed to calculate the effective elastic-plastic properties of the individual layers used in determining the elements of the local and thus global stiffness matrices. The resulting system of equations is then solved using Mendelson’s iterative method of successive elastic solutions developed for elastoplastic boundary-value problems. Part I of the paper outlines the aforementioned solution strategy. In Part II (Salzar et al., 1996) this solution strategy is first validated by comparison with available closed-form solutions as well as with results obtained using the finite-element approach. Subsequently, examples are presented that illustrate the utility of the developed solution methodology in predicting the elastic-plastic response of arbitrarily laminated metal matrix composite tubes. In particular, optimization of the response of composite tubes under internal pressure is considered through the use of functionally graded architectures.

Vibration Analysis of Nonlinear Isolator’s Characteristics by ADAMS

2012 ◽  
Vol 152-154 ◽  
pp. 1077-1081 ◽  
Author(s):  
Zhao Qi He ◽  
Yu Chao Song ◽  
Hong Liang Yu
Keyword(s):  
Nonlinear Vibration ◽  
Vibration Isolation ◽  
Excitation Frequency ◽  
Vibration Isolator ◽  
Vibration System ◽  
Linear Vibration ◽  
External Excitation ◽  
Mass Model ◽  
Adams Software ◽  
Isolation Systems

A nonlinear spring-mass model is established to study the dynamic characteristics of nonlinear vibration isolator. By use of ADAMS software, the influence of stiffness, foundation displacement excitation and frequency of external excitation on the nonlinear vibration isolation systems are analyzed. Results indicate that the linear vibration system needs 4s to achieve stability, but the nonlinear vibration system only needs 0.1s. The response value increases with the increase of excitation frequency, the response pick value increases by 61.58% and 102.35% and each corresponding stable value increases by 159.35% and 309.87%.

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