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.

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.

scholarly journals Exact and Direct Modeling Technique for Rotor-Bearing Systems with Arbitrary Selected Degrees-of-Freedom

1994 ◽  
Vol 1 (6) ◽  
pp. 497-506 ◽  
Author(s):  
Shilin Chen ◽  
Michel Géradin
Keyword(s):  
Stiffness Matrix ◽  
Degrees Of Freedom ◽  
Dynamic Stiffness ◽  
Modeling Technique ◽  
Dynamic Stiffness Matrix ◽  
Global Dynamic ◽  
Combination Methods ◽  
Stiffness Matrices ◽  
Direct Modeling ◽  
Rotor Bearing

An exact and direct modeling technique is proposed for modeling of rotor-bearing systems with arbitrary selected degrees-of-freedom. This technique is based on the combination of the transfer and dynamic stiffness matrices. The technique differs from the usual combination methods in that the global dynamic stiffness matrix for the system or the subsystem is obtained directly by rearranging the corresponding global transfer matrix. Therefore, the dimension of the global dynamic stiffness matrix is independent of the number of the elements or the substructures. In order to show the simplicity and efficiency of the method, two numerical examples are given.

Free Vibration of Helicoidal Beams of Arbitrary Shape and Variable Cross Section

2002 ◽  
Vol 124 (3) ◽  
pp. 397-409 ◽  
Author(s):  
Wisam Busool ◽  
Moshe Eisenberger
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Stiffness Matrix ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
Free Vibration Analysis ◽  
Rotational Inertia ◽  
Variable Cross ◽  
Dynamic Stiffness Matrix ◽  
Helical Springs

In this study, the dynamic stiffness method is employed for the free vibration analysis of helical springs. This work gives the exact solutions for the natural frequencies of helical beams having arbitrary shapes, such as conical, hyperboloidal, and barrel. Both the cross-section dimensions and the shape of the beam can vary along the axis of the curved member as polynomial expressions. The problem is described by six differential equations. These are second order equations with variable coefficients, with six unknown displacements, three translations, and three rotations at every point along the member. The proposed solution is based on a new finite-element method for deriving the exact dynamic stiffness matrix for the member, including the effects of the axial and the shear deformations and the rotational inertia effects for any desired precision. The natural frequencies are found as the frequencies that cause the determinant of the dynamic stiffness matrix to become zero. Then the mode shape for every natural frequency is found. Examples are given for beams and helical springs with different shape, which can vary along the axis of the member. It is shown that the present numerical results agree well with previously published numerical and experimental results.

Dynamic Stiffness Analysis of a Beam Based on Trigonometric Shear Deformation Theory

2007 ◽  
Vol 130 (1) ◽  
Author(s):  
Jun Li ◽  
Hongxing Hua ◽  
Rongying Shen
Keyword(s):  
Shear Deformation ◽  
Stiffness Matrix ◽  
Deformation Theory ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Beam Element ◽  
Mode Shapes ◽  
Dynamic Stiffness Matrix ◽  
Stiffness Analysis

The dynamic stiffness matrix of a uniform isotropic beam element based on trigonometric shear deformation theory is developed in this paper. The theoretical expressions for the dynamic stiffness matrix elements are found directly, in an exact sense, by solving the governing differential equations of motion that describe the deformations of the beam element according to the trigonometric shear deformation theory, which include the sinusoidal variation of the axial displacement over the cross section of the beam. The application of the dynamic stiffness matrix to calculate the natural frequencies and normal mode shapes of two rectangular beams is discussed. The numerical results obtained are compared to the available solutions wherever possible and validate the accuracy and efficiency of the present approach.

Dynamic Stiffness Matrix Method for the Free Vibration Analysis of Rotating Uniform Shear Beams

2009 ◽  
Author(s):  
Dominic R. Jackson ◽  
S. Olutunde Oyadiji
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Power Series Solution ◽  
Dynamic Stiffness Matrix ◽  
Uniform Shear ◽  
Natural Boundary Conditions

The Dynamic Stiffness Method (DSM) is used to analyse the free vibration characteristics of a rotating uniform Shear beam. Starting from the kinetic and strain energy expressions, the Hamilton’s principle is used to obtain the governing differential equations of motion and the natural boundary conditions. The two equations are solved simultaneously and expressed each in terms of displacement and slope only. The Frobenius power series solution is applied to solve the equations and the resulting solutions are also expressed in terms of four independent solutions. Applying the appropriate boundary conditions, the Dynamic Stiffness Matrix is assembled. The natural frequencies of vibration using the DSM are computed by employing the in-built root finding algorithm in Mathematica as well as by implementing the Wittrick-Williams algorithm in a numerical routine in Mathematica. The results obtained using the DSM are presented in tabular and graphical forms and are compared with results obtained using the Timoshenko and the Bernoulli-Euler theories.

scholarly journals On the Free Vibration Modeling of Spindle Systems: A Calibrated Dynamic Stiffness Matrix

2021 ◽  
Author(s):  
Omar Gaber ◽  
Seyed M. Hashemi
Keyword(s):  
Boundary Conditions ◽  
Fundamental Frequency ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Dynamic Stiffness Matrix ◽  
Simply Supported ◽  
Multiple Components ◽  
Stiffness Matrices ◽  
Linear Spring ◽  
Vibration Modeling

The effect of bearings on the vibrational behavior of machine tool spindles is investigated. This is done through the development of a calibrated dynamic stiffness matrix (CDSM) method, where the bearings flexibility is represented by mass less linear spring elements with tuneable stiffness. A dedicated MAT LAB code is written to develop and to assemble the element stiffness matrices for the system’s multiple components and to apply the boundary conditions.The developed method is applied to an illustrative example of spindle system.When the spindle bearings are modeled as simply supported boundary conditions, the DSM model results in a fundamental frequency much higher than the system’s nominal value.The simply supported boundary conditions are then replaced by linear spring elements, and the spring constants are adjusted such that the resulting calibrated CDSM model leads to the nominal fundamental frequency of the spindle system.The spindle frequency results are also validated against the experimental data.The proposed method can be effectively applied to predict the vibration characteristics of spindle systems supported by bearings.

Free Vibration Analysis of Frames Using the Transfer Dynamic Stiffness Matrix Method

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
Tsung-Hsien Tu ◽  
Jen-Fang Yu ◽  
Hsin-Chung Lien ◽  
Go-Long Tsai ◽  
B. P. Wang
Keyword(s):  
Free Vibration ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
System Matrix ◽  
Dynamic Stiffness Matrix ◽  
3D Space ◽  
Euler Bernoulli Beam ◽  
Good Agreement

A method for free vibration of 3D space frame structures employing transfer dynamic stiffness matrix (TDSM) method based on Euler–Bernoulli beam theory is developed in this paper. The exact TDSM of each member is assembled to obtain the system matrix that is frequency dependent. All free vibration eigensolutions including coincident roots for the characteristic equation can be obtained to any desired accuracy using the algorithm developed by Wittrick and Williams (1971, “A General Algorithm for Computing Natural Frequencies of Elastic Structures,” Q. J. Mech. Appl. Math., 24, pp. 263–284). Exact eigenfunction of structures can then be computed using the dynamic shape function and the corresponding eigenvector. The results showed good agreement with those computed by finite element method.

Simple Modeling and Analysis for Crankshaft Three-Dimensional Vibrations, Part 2: Application to an Operating Engine Crankshaft

1995 ◽  
Vol 117 (1) ◽  
pp. 80-86 ◽  
Author(s):  
T. Morita ◽  
H. Okamura
Keyword(s):  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Inertia Force ◽  
Connecting Rod ◽  
Automobile Engine ◽  
Dynamic Stiffness Matrix ◽  
Modeling And Analysis ◽  
Engine Crankshaft

The modeling and analysis procedures with the dynamic stiffness matrix method described in Part 1 were applied to a crankshaft system, consisting of crankshaft, front pulley, flywheel, piston, and connecting rod, under firing conditions. For firing conditions, (7) one half of the reciprocating masses consisting of the piston, piston pin, and connecting rod small end, and (2) rotating masses of the connecting rod big end mass, were attached to the two ends of the crankpin, taking account of the rigidity of the connecting rod. The excitation forces were calculated from the gas force and the inertia force due to the reciprocating masses. By solving the equations of motion derived in the form of the dynamic stiffness matrix, we calculated the three-dimensional steady-state vibrations of the crankshaft system under firing conditions. A crankshaft system for a four-cylinder in-line automobile engine was used for the analysis. We calculated the influence of the mass and moments of inertia of the front pulley on the behavior of the crankshaft vibrations and the excitation induced at the crankjournal bearings. Calculated values were compared with experimental results.

scholarly journals On the Free Vibration Modeling of Spindle Systems: A Calibrated Dynamic Stiffness Matrix

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Omar Gaber ◽  
Seyed M. Hashemi
Keyword(s):  
Boundary Conditions ◽  
Fundamental Frequency ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Dynamic Stiffness Matrix ◽  
Spindle System ◽  
Simply Supported ◽  
Multiple Components ◽  
Stiffness Matrices ◽  
Linear Spring

The effect of bearings on the vibrational behavior of machine tool spindles is investigated. This is done through the development of a calibrated dynamic stiffness matrix (CDSM) method, where the bearings flexibility is represented by massless linear spring elements with tuneable stiffness. A dedicated MATLAB code is written to develop and to assemble the element stiffness matrices for the system’s multiple components and to apply the boundary conditions. The developed method is applied to an illustrative example of spindle system. When the spindle bearings are modeled as simply supported boundary conditions, the DSM model results in a fundamental frequency much higher than the system’s nominal value. The simply supported boundary conditions are then replaced by linear spring elements, and the spring constants are adjusted such that the resulting calibrated CDSM model leads to the nominal fundamental frequency of the spindle system. The spindle frequency results are also validated against the experimental data. The proposed method can be effectively applied to predict the vibration characteristics of spindle systems supported by bearings.

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