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.

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.

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.

Vibration Modeling of Machine Tool Spindles: A Calibrated Dynamic Stiffness Matrix Method

2013 ◽  
Vol 651 ◽  
pp. 710-716 ◽  
Author(s):  
Omar Gaber ◽  
Seyed M. Hashemi
Keyword(s):  
Stiffness Matrix ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
Dynamic Stiffness Matrix ◽  
Spring Element ◽  
Linear Spring ◽  
Beam Bending ◽  
The Stability ◽  
Spinning Speed ◽  
Vibration Modeling

The effects of spindles vibrational behavior on the stability lobes and the Chatter behavior of machine tools have been established. The service life has been observed to reducethe system natural frequencies. An analytical model of a multi-segment spinning spindle, based on the Dynamic Stiffness Matrix (DSM) formulation, exact within the limits of the Euler-Bernoulli beam bending theory, is developed. The system exhibits coupled Bending-Bending (B-B) vibration and its natural frequencies are found to decrease with increasing spinning speed. The bearings were included in the model usingboth rigid, simply supported, frictionless pins and flexible linear spring elements. The linear spring element stiffness is then calibrated so that the fundamental frequency of the system matches the nominal value.

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.

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.

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.

Free vibration analysis of multistepped nonlocal Bernoulli–Euler beams using dynamic stiffness matrix method

2020 ◽  
pp. 107754632093347
Author(s):  
Moustafa S Taima ◽  
Tamer A El-Sayed ◽  
Said H Farghaly
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequency ◽  
Stiffness Matrix ◽  
Matrix Method ◽  
Dynamic Stiffness ◽  
Nonlocal Parameter ◽  
Free Vibration Analysis ◽  
Nonlocal Elasticity Theory ◽  
Dynamic Stiffness Matrix

The free vibration of multistepped nanobeams is studied using the dynamic stiffness matrix method. The beam analysis is based on the Bernoulli–Euler theory, and the nanoscale analysis is based on the Eringen’s nonlocal elasticity theory. The nanobeam is attached to linear and rotational elastic supports at the start, end, and intermediate boundary conditions. The effect of the nonlocal parameter, boundary conditions, and step ratios on the nanobeam natural frequency is investigated. The results of the dynamic stiffness matrix methods are validated by comparing selected cases with the literature, which give excellent agreement with those literatures. The results show that the dimensionless natural frequency parameter is inversely proportional to the nonlocal parameters except in the first mode for clamped-free boundary conditions. Also, the gap between every two consecutive modes decreases with the increasing of the nonlocal parameter.

The Effect of Lumped Parameters on Beam Frequencies

1963 ◽  
Vol 14 (3) ◽  
pp. 224-240 ◽  
Author(s):  
F. A. Leckie ◽  
G.M. Lindberg
Keyword(s):  
Boundary Conditions ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Fourth Power ◽  
Dynamic Stiffness Matrix ◽  
Lumped Parameter ◽  
Lumped Parameters

SummaryAn investigation has been made into the errors involved in using certain lumped parameter methods for the solution of beam frequencies. It is found that existing methods are not consistent for all boundary conditions. A new dynamic stiffness matrix has been formulated, which gives consistently good results for even a few elements. The error in the solution is always inversely proportional to the fourth power of the number of elements used.

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.

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