The Dynamic Stiffness Method for Linear Rotor-Bearing Systems

1996 ◽  
Vol 118 (3) ◽  
pp. 332-339 ◽  
Author(s):  
F. A. Raffa ◽  
F. Vatta
Keyword(s):  
Finite Element ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Matrix Analysis ◽  
Theoretical Formulation ◽  
Dynamic Stiffness Matrix ◽  
Rotor Systems ◽  
Stiffness Method ◽  
Dynamic Stiffness Method ◽  
Rotor Bearing

The behavior of linear rotor-bearing systems is investigated by using the exact approach of the dynamic stiffness method, which entails the use of continuous rather than lumped models. In particular, the theoretical formulation for rotor systems with anisotropic bearings is developed by utilizing the complex representation of all the involved variables. The proposed formulation eventually leads to the 8 × 8 complex dynamic stiffness matrix of the rotating Timoshenko beam; this matrix proves to be related, by a simple rule, to the 4 × 4 dynamic stiffness matrix, which describes rotor systems with isotropic bearings. The method is first applied to the critical speeds evaluation of a simple rotor system with rigid supports; for this case, the exact results of the dynamic stiffness approach are compared to the usual convergence procedure of the finite element method. Successively, the steady-state unbalance response of two rotor systems with anisotropic supports is analyzed; for these examples, the dynamic stiffness results compare favorably with the results of the finite element and the transfer matrix analysis performed by other authors.

EXACT VIBRATION FREQUENCIES AND MODES OF BEAMS WITH INTERNAL RELEASES

2002 ◽  
Vol 02 (01) ◽  
pp. 63-75 ◽  
Author(s):  
M. EISENBERGER
Keyword(s):  
Stiffness Matrix ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
Beam Element ◽  
Mode Shapes ◽  
Vibration Frequencies ◽  
Dynamic Stiffness Matrix ◽  
Continuous Beams ◽  
Stiffness Method ◽  
Dynamic Stiffness Method

The exact vibration frequencies of continuous beams with internal releases are found using the dynamic stiffness method. Two types of releases are considered: hinge and sliding discontinuities. First, the exact dynamic stiffness matrix for a beam element with a release is derived and then used in the assembly of the structure dynamic stiffness matrix. The natural frequencies are found as the values of frequency that make this matrix singular. Then the mode shapes are found exactly. Examples are given for continuous beams with different releases.

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.

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.

Modified Interval Perturbation Finite Element Method for a Structural-Acoustic System With Interval Parameters

2013 ◽  
Vol 80 (4) ◽  
Author(s):  
Baizhan Xia ◽  
Dejie Yu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stiffness Matrix ◽  
Dynamic Equilibrium ◽  
Dynamic Stiffness ◽  
Response Analysis ◽  
Acoustic System ◽  
Dynamic Stiffness Matrix ◽  
Interval Parameters ◽  
Element Method

For the frequency response analysis of the structural-acoustic system with interval parameters, a modified interval perturbation finite element method (MIPFEM) is proposed. In the proposed method, the interval dynamic equilibrium equation of the uncertain structural-acoustic system is established. The interval structural-acoustic dynamic stiffness matrix and the interval force vector are expanded by using the first-order Taylor series; the inversion of the invertible interval structural-acoustic dynamic stiffness matrix is approximated by employing a modified approximate interval-value Sherman–Morrison–Woodbury formula. The proposed method is implemented at an element-by-element level in the finite element framework. Numerical results on a shell structural-acoustic system with interval parameters verify the accuracy and efficiency of the proposed method.

Frequency domain-based analysis of floor vibrations using the dynamic stiffness matrix method

2018 ◽  
Vol 25 (4) ◽  
pp. 763-776 ◽  
Author(s):  
Tong Guo ◽  
Zhiliang Cao ◽  
Zhiqiang Zhang ◽  
Aiqun Li
Keyword(s):  
Finite Element ◽  
Frequency Domain ◽  
Stiffness Matrix ◽  
Road Traffic ◽  
Dynamic Stiffness ◽  
Design Stage ◽  
Element Analysis ◽  
Frequency Spectra ◽  
Dynamic Stiffness Matrix ◽  
Floor Vibrations

Buildings may experience excessive floor vibrations due to inner excitations such as walking people and running machines, or ground motion caused by the road traffic. Therefore, it is often necessary to evaluate the vibration level at the design stage. In this paper, a frequency domain-based model for predicting vertical vibrations of a building floor is provided, where the floor is simplified as a rectangular plate stiffened by beams in two orthogonal directions, while vertical motion and rotation of the slab–column joints are viewed as the unknown degrees of freedom. The dynamic stiffness matrix of the whole structure is obtained from those of the floor and column elements. To validate the proposed solution, a five-story building was analyzed, and frequency spectra were compared with those from the finite element method. Besides, a prototype building was analyzed and validated based on field measured data. It is found that the proposed solution could predict vibration responses with satisfactory accuracy, and is more computationally efficient than finite element analysis.

Dynamic Analysis of Rotating Beams with Nonuniform Cross Sections Using the Dynamic Stiffness Method

2001 ◽  
Vol 123 (4) ◽  
pp. 536-539 ◽  
Author(s):  
K. J. Huang ◽  
T. S. Liu
Keyword(s):  
Dynamic Analysis ◽  
Cross Sections ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Equation Of Motion ◽  
The Finite Element Method ◽  
Rotating Beam ◽  
Rotating Beams ◽  
Stiffness Method ◽  
Dynamic Stiffness Method

This study develops dynamic analysis based on the dynamic stiffness method for a rotating beam of nonuniform cross-section. To deal with nonuniform beams, coefficients related to material and geometric properties in the equation of motion are expressed in a polynomial form. A dynamic stiffness matrix is accordingly formulated in terms of power series. The dynamic response of the rotating beam is calculated by performing modal analysis. It is demonstrated that the present method provides an alternative to the finite element method in dealing with nonuniform rotating beams.

Spectral Finite Element Formulation for Nanorods via Nonlocal Continuum Mechanics

2011 ◽  
Vol 78 (6) ◽  
Author(s):  
S. Narendar ◽  
S. Gopalakrishnan
Keyword(s):  
Finite Element ◽  
Continuum Mechanics ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Small Scale ◽  
Element Formulation ◽  
Shape Functions ◽  
Dynamic Stiffness Matrix ◽  
Exact Shape ◽  
Spectral Finite Element

In this article, the Eringen’s nonlocal elasticity theory has been incorporated into classical/local Bernoulli-Euler rod model to capture unique properties of the nanorods under the umbrella of continuum mechanics theory. The spectral finite element (SFE) formulation of nanorods is performed. SFE formulation is carried out and the exact shape functions (frequency dependent) and dynamic stiffness matrix are obtained as function of nonlocal scale parameter. It has been found that the small scale affects the exact shape functions and the elements of the dynamic stiffness matrix. The results presented in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave dispersion properties of carbon nanotubes.

scholarly journals Riser Dynamic Analysis Using WKB-Based Dynamic Stiffness Method

2012 ◽  
Author(s):  
Yongming Cheng ◽  
J. Kim Vandiver
Keyword(s):  
Dynamic Analysis ◽  
Natural Frequencies ◽  
Shape Function ◽  
Dynamic Stiffness ◽  
Mode Shapes ◽  
Marine Riser ◽  
Theoretical Formulation ◽  
Eigen Value ◽  
Stiffness Method ◽  
Dynamic Stiffness Method

Risers are fluid conduits from subsea equipment to surface floating production platforms. The integrity of a riser system plays a very important role in deepwater developments. Riser dynamic analysis is an important part to the system design. This paper investigates riser dynamic analysis using the WKB-Based dynamic stiffness method. This paper first presents a theoretical formulation of the dynamic stiffness method. It then combines the dynamic stiffness method with the WKB theory, which assumes that the coefficients in the differential equation of motion are slowly varying. The WKB-based dynamic stiffness method is derived and a frequency dependent shape function is expressed implicitly. The Wittrick and Williams (W-W) algorithm is further extended to solve eigen value problem for a general non-uniform marine riser. Examples of non-uniform riser are analyzed and the results show the efficiency of this method. In addition, a pipe-in-pipe riser system is analyzed for natural frequencies and mode shapes using the WKB-based dynamic stiffness method with the W-W algorithm. The characteristic of the mode shapes is described for such a riser system.

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