Frequency dependent mass and stiffness matrices of bar and beam elements and their equivalency with the dynamic stiffness matrix

2021 ◽  
Vol 254 ◽  
pp. 106616
Author(s):  
J.R. Banerjee
Keyword(s):  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Frequency Dependent ◽  
Dynamic Stiffness Matrix ◽  
Beam Elements ◽  
Stiffness Matrices ◽  
Dependent Mass

Steady-State Unbalance Response of Continuous Rotors on Anisotropic Supports

1993 ◽  
Author(s):  
Graziano Curti ◽  
Francesco A. Raffa ◽  
Furio Vatta
Keyword(s):  
Steady State ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Analytical Investigation ◽  
Dynamic Stiffness Matrix ◽  
Rotor Systems ◽  
Beam Elements ◽  
Unbalance Response ◽  
Constant Coefficients ◽  
Linearized Model

Abstract An analytical investigation of the steady-state unbalance response of axisymmetric rotor systems with anisotropic, flexible and damped bearings is presented. According to the exact approach of the dynamic stiffness method, the rotor is modelled by means of continuous beam elements. In this work, the expression of the 8 × 8 dynamic stiffness matrix of a rotating Timoshenko beam is derived and it is shown that it is related, by means of a simple law, to the previously published 4 × 4 dynamic stiffness matrix, which holds for the isotropic bearings case. The effects of concentrated disks and bearings are included into the formulation; in particular, each bearing is described by eight constant coefficients, according to the well-known linearized model of the bearing forces. The unbalance response of a typical rotor system taken from the literature is analyzed. A comparison is presented with the finite element results reported by other authors.

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.

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.

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