A Note on Computational Rotor Dynamics

1998 ◽  
Vol 120 (1) ◽  
pp. 228-233 ◽  
Author(s):  
W. J. Chen
Keyword(s):  
Computational Efficiency ◽  
Equations Of Motion ◽  
Rotor Dynamics ◽  
Memory Storage ◽  
Rotor Systems ◽  
Real Domain ◽  
The Matrix ◽  
System Equations ◽  
Ordering Algorithms ◽  
Matrix Bandwidth

Concise equations for improvements in computational efficiency on dynamics of rotor systems are presented. Two coordinate ordering methods are introduced in the element equations of motion. One is in the real domain and the other is in the complex domain. The two coordinate ordering algorithms lead to compact element matrices. A station numbering technique is also proposed for the system equations during the assembly process. The proposed numbering technique can minimize the matrix bandwidth, the memory storage and can increase the computational efficiency. Numerical examples are presented to demonstrate the benefit of the proposed algorithms.

Concise Equations for Rotor Dynamics Analysis

1995 ◽  
Author(s):  
W. J. Chen
Keyword(s):  
Equations Of Motion ◽  
Rotor Dynamics ◽  
Memory Storage ◽  
The Other ◽  
Dynamics Analysis ◽  
Real Domain ◽  
The Matrix ◽  
System Equations ◽  
Ordering Algorithms ◽  
Matrix Bandwidth

Abstract Concise equations for rotor dynamics analysis are presented. Two coordinate ordering methods are introduced in the element equations of motion. One is in the real domain and the other is in the complex domain. The two proposed ordering algorithms lead to more compact element matrices. A station numbering technique is also proposed for the system equations during the assembly process. This numbering technique can minimize the matrix bandwidth, the memory storage and can increase the computational efficiency.

A Variational-Vector Calculus Approach to Machine Dynamics

1986 ◽  
Vol 108 (1) ◽  
pp. 25-30 ◽  
Author(s):  
E. J. Haug ◽  
M. K. McCullough
Keyword(s):  
Mass Matrix ◽  
Equations Of Motion ◽  
Linear Structure ◽  
Rigid Bodies ◽  
Machine Dynamics ◽  
System A ◽  
Vector Calculus ◽  
The Matrix ◽  
Generalized Force ◽  
System Equations

A variational-vector calculus approach is presented to define virtual displacements and rotations and position, velocity, and acceleration of individual components of a multibody mechanical system. A two-body subsystem with both Cartesian and relative coordinates is used to illustrate a systematic method of exploiting the linear structure of both vector and differential calculus, in conjunction with a variational formulation of the equations of motion of rigid bodies, to derive the matrix structure of governing multibody system equations of motion. A pattern for construction of the system mass matrix and generalized force terms is developed and applied to derivation of the equations of motion of a vehicle system. The development demonstrates an approach to multibody machine dynamics that closely parallels methods used in finite-element structural analysis.

MESHFREE CELL-BASED SMOOTHED POINT INTERPOLATION METHOD USING ISOPARAMETRIC PIM SHAPE FUNCTIONS AND CONDENSED RPIM SHAPE FUNCTIONS

2011 ◽  
Vol 08 (04) ◽  
pp. 705-730 ◽  
Author(s):  
G. Y. ZHANG ◽  
G. R. LIU
Keyword(s):  
Computational Efficiency ◽  
Interpolation Method ◽  
Shape Functions ◽  
Generalized Gradient ◽  
Singularity Problem ◽  
Numerical Examples ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
System Equations ◽  
Gradient Smoothing

This paper presents two novel and effective cell-based smoothed point interpolation methods (CS-PIM) using isoparametric PIM (PIM-Iso) shape functions and condensed radial PIM (RPIM-Cd) shape functions respectively. These two types of PIM shape functions can successfully overcome the singularity problem occurred in the process of creating PIM shape functions and make the constructed CS-PIM models work well with the three-node triangular meshes. Smoothed strains are obtained by performing the generalized gradient smoothing operation over each triangular background cells, because the nodal PIM shape functions can be discontinuous. The generalized smoothed Galerkin (GS-Galerkin) weakform is used to create the discretized system equations. Some numerical examples are studied to examine various properties of the present methods in terms of accuracy, convergence, and computational efficiency.

Time Integration Incorporated Sensitivity Analysis With Generalized-α Method for Multibody Dynamics Systems

2011 ◽  
Author(s):  
Guang Dong ◽  
Zheng-Dong Ma ◽  
Gregory Hulbert ◽  
Noboru Kikuchi
Keyword(s):  
Sensitivity Analysis ◽  
Topology Optimization ◽  
Dynamic Loading ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Time Integration ◽  
Optimization Method ◽  
Sensitivity Analysis Method ◽  
System Equations ◽  
Consecutive Time

The topology optimization method is extended for the optimization of geometrically nonlinear, time-dependent multibody dynamics systems undergoing nonlinear responses. In particular, this paper focuses on sensitivity analysis methods for topology optimization of general multibody dynamics systems, which include large displacements and rotations and dynamic loading. The generalized-α method is employed to solve the multibody dynamics system equations of motion. The developed time integration incorporated sensitivity analysis method is based on a linear approximation of two consecutive time steps, such that the generalized-α method is only applied once in the time integration of the equations of motion. This approach significantly reduces the computational costs associated with sensitivity analysis. To show the effectiveness of the developed procedures, topology optimization of a ground structure embedded in a planar multibody dynamics system under dynamic loading is presented.

Optimal Design of Vibration Absorber Systems Supported by Elastic Base

1991 ◽  
Author(s):  
Hashem Ashrafiuon
Keyword(s):  
Dynamic Models ◽  
Damping Ratio ◽  
Equations Of Motion ◽  
Elastic Base ◽  
Design Parameters ◽  
Primary System ◽  
Vibration Absorbers ◽  
Equivalent Damping ◽  
System Equations ◽  
Different Levels

Abstract This paper presents the effect of foundation flexibility on the optimum design of vibration absorbers. Flexibility of the base is incorporated into the absorber system equations of motion through an equivalent damping ratio and stiffness value in the direction of motion at the connection point. The optimum values of the uncoupled natural frequency and damping ratio of the absorber are determined over a range of excitation frequencies and the primary system damping ratio. The design parameters are computed and compared for the rigid, static, and dynamic models of the base as well as different levels of base flexibility.

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.

Coupling of Rotor-Gear-Casing Vibrations During Extreme Operating Events

1992 ◽  
Vol 114 (4) ◽  
pp. 464-471 ◽  
Author(s):  
F. K. Choy ◽  
J. Padovan ◽  
Y. F. Ruan
Keyword(s):  
Element Model ◽  
Operating Conditions ◽  
Transient Vibration ◽  
Rotor Systems ◽  
Power Plant Equipment ◽  
Modal Synthesis ◽  
Operating Environments ◽  
Rotor Bearing ◽  
The Matrix ◽  
Plant Equipment

During extreme operating environments (i.e., seismic events, base motion-induced vibrations, etc.), the coupled vibrations developed between the rotors, bearings, gears and enclosing structure of gear-driven rotating equipment can be quite substantial. Generally, such large vibrational amplitudes may lead to failures in both the rotor-gearing system and/or the casing structure. This paper simulates the dynamic behavior of rotor-bearing-gear system resulting from motion of the enclosed structure. The modal synthesis approach is used in this study to synthesize the dynamics of the rotor systems with the vibrations of their casing structure in modal coordinates. Modal characteristics of the rotor-bearing-gear systems are evaluated using the matrix transfer technique, while the modal parameters for the casing structure are developed through a finite element model using NASTRAN. The modal accelerations calculated are integrated through a numerical algorithm to generate modal transient vibration analysis. Vibration results are examined in both time and frequency domains to develop representations for the coupled dynamics generated during extreme operating conditions. Typical three-rotor bull gear-driven power plant equipment (compressors, pumps, etc.) is used as an example to demonstrate the procedure developed.

Rolling Element Bearing Non-Linearity Effects

2000 ◽  
Author(s):  
N. S. Feng ◽  
E. J. Hahn
Keyword(s):  
Equations Of Motion ◽  
Rolling Element Bearing ◽  
Radial Clearance ◽  
Rotor Systems ◽  
Bearing Life ◽  
Rolling Element ◽  
Bearing Load ◽  
Rolling Elements ◽  
Element Bearing ◽  
Negative Clearance

Non-linearity effects in rolling element bearings arise from two sources, viz. the Hertzian force deformation relationship and the presence of clearance between the rolling elements and the bearing races. Assuming that centrifugal effects may be neglected and that the presence of axial preload is appropriately reflected in a corresponding change in the radial clearance, this paper analyses a simple test rig to illustrate that non-linear phenomena such as synchronous multistable and nonsynchronous motions are possible in simple rigid and flexible rotor systems subjected to unbalance excitation. The equations of motion of the rotor bearing system were solved by transient analysis using fourth order Runge Kutta. Of particular interest is the effect of clearance, governed in practice by bearing specification and the amount of preload, on the vibration behaviour of rotors supported by ball bearings and on the bearing load. It is shown that in the presence of positive clearance, there exists an unbalance excitation range during which the bearing is momentarily not transmitting force owing to contact loss, resulting in rolling element raceway impact with potentially relatively high bearing forces; and indicating that for long bearing life, operation with positive clearance should be avoided in the presence of such unbalance loading. Once the unbalance excitation is high enough to avoid such contact loss, it is the bearings with zero or negative clearance which produce maximum bearing forces.

The Effects of Harmonic Drive Gears on Robot Dynamics

1991 ◽  
Author(s):  
Tsung-Chieh Lin ◽  
K. Harold Yae
Keyword(s):  
Mathematical Models ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Robot Dynamics ◽  
Harmonic Drive ◽  
Dynamic Coupling ◽  
Manipulator Dynamics ◽  
Dynamic Formulation ◽  
System Equations ◽  
Modelling Assumptions

Abstract Mathematical models of the harmonic drive have been developed, and their effects on manipulator dynamics have been examined. The harmonic drive is modelled as a flexible gear with a high gear reduction ratio. The recursive Newton-Euler dynamic formulation is applied to deriving the system equations of motion that include the effects of the geared actuation. The equations include not only the nonlinear dynamic coupling between rotors and links but the gyroscopic effect due to the spinning rotors. Different modelling assumptions creates four models and their time responses are compared. As an example, a seven degree of freedom robot was chosen to make comparisons in time responses.

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