scholarly journals Finite Dynamic Elements and Modal Analysis

1993 ◽  
Vol 1 (2) ◽  
pp. 171-176
Author(s):  
N.J. Fergusson ◽  
W.D. Pilkey
Keyword(s):  
Modal Analysis ◽  
Stiffness Matrix ◽  
Correction Term ◽  
Dynamic Stiffness ◽  
Mass Matrix ◽  
Power Series Expansion ◽  
Forced Response ◽  
Dynamic Correction ◽  
Analysis Scheme ◽  
Dynamic Element

A general modal analysis scheme is derived for forced response that makes use of high accuracy modes computed by the dynamic element method. The new procedure differs from the usual modal analysis in that the modes are obtained from a power series expansion for the dynamic stiffness matrix that includes an extra dynamic correction term in addition to the static stiffness matrix and the consistent mass matrix based on static displacement. A cantilevered beam example is used to demonstrate the relative accuracies of the dynamic element and the traditional finite element methods.

scholarly journals Modal Analysis of Blades With Densely Distributed Small Holes

1994 ◽  
Author(s):  
Jin Zhang ◽  
Zhen-Hua Wang ◽  
Ben-Hua Dong
Keyword(s):  
Modal Analysis ◽  
Stiffness Matrix ◽  
Mass Matrix ◽  
Turbine Rotor ◽  
Rotor Blades ◽  
Turbine Engines ◽  
Gas Turbine Engines ◽  
Element Analysis ◽  
Good Agreement ◽  
Small Holes

At present, when numerical methods are used to analyze the dynamic characteristics of blades with densely distributed small holes, it is difficult to consider the influence of small holes accurately. A new method is presented regarding air-cooled turbine rotor blades of gas turbine engines. To investigate the topic, a series of experimental rules are simulated effectively by theoretical formulae. The method is developed by combining the experimental results and finite element analysis. To include the effects of small holes on dynamic behaviors of blades, the equivalent concept is employed. In a generalized eigen-equation, the modified stiffness matrix and mass matrix are used to perform an accurate modal analysis of the blades. When making comparison between the analytical results obtained by using the method of this paper and the experimental data, a good agreement is seen. Accuracy, reliability, and practicality of the method presented in this paper are verified.

Frequency-Dependent Element Mass Matrices

1992 ◽  
Vol 59 (1) ◽  
pp. 136-139 ◽  
Author(s):  
N. J. Fergusson ◽  
W. D. Pilkey
Keyword(s):  
Stiffness Matrix ◽  
Shape Function ◽  
Mass Matrix ◽  
Power Series Expansion ◽  
Shape Functions ◽  
Frequency Dependent ◽  
Mass Matrices ◽  
The Matrix ◽  
Matrix Expansion ◽  
Consistent Mass Matrix

This paper considers some of the theoretical aspects of the formulation of frequency-dependent structural matrices. Two types of mass matrices are examined, the consistent mass matrix found by integrating frequency-dependent shape functions, and the mixed mass matrix found by integrating a frequency-dependent shape function against a static shape function. The coefficients in the power series expansion for the consistent mass matrix are found to be determinable from those in the expansion for the mixed mass matrix by multiplication by the appropriate constant. Both of the mass matrices are related in a similar manner to the coefficients in the frequency-dependent stiffness matrix expansion. A formulation is derived which allows one to calculate, using a shape function truncated at a given order, the mass matrix expansion truncated at twice that order. That is the terms for either of the two mass matrix expansions of order 2n are shown to be expressible using shape functions terms of order n. Finally, the terms in the matrix expansions are given by formulas which depend only on the values of the shape function terms at the boundary.

A Taylor Series Expansion Approach for Nonlinear Blade Forced Response Prediction Considering Variable Rotational Speed

2016 ◽  
Author(s):  
Torsten Heinze ◽  
Lars Panning-von Scheidt ◽  
Jörg Wallaschek ◽  
Andreas Hartung
Keyword(s):  
Series Expansion ◽  
Taylor Series ◽  
Rotational Speed ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Taylor Series Expansion ◽  
Dynamic Compliance ◽  
Forced Response ◽  
Compliance Matrix ◽  
Nonlinear Contact

In the field of turbomachinery, great efforts are made to enhance computational tools to obtain reliable predictions of the vibrational behavior of friction-damped bladed disks. As a trade-off between computational burden and level of simplification, numerous methods were developed to reduce the nonlinear systems dimension. Using component mode synthesis methods (CMS), one is capable to describe the systems motion by interface and modal coordinates. Subsequently or alternatively, the dynamic compliance matrix can be evaluated efficiently by means of modal superposition to avoid the inversion of the dynamic stiffness matrix. Only the equations corresponding to the degrees of freedom (DOF) subject to localized nonlinear contact forces need to be solved simultaneously, whereas the solution of the linear DOF is obtained by exploiting the algebraic character of the set of equations. In this paper an approach is presented to account for rotational speed-dependent stiffness in the subset of nonlinear DOF without the need to re-evaluate the associated eigenvalue problem (EVP) when rotational speed is changed. This is done by means of a Taylor series expansion of the eigenvalues and eigenvectors used for the modal superposition to re-construct the dynamic compliance matrix. In the context of forced response predictions of friction-damped blisks the expansion is performed up to different order for a simplified blisk model with nonlinear contact interfaces. The results are compared to the solution obtained by direct evaluation of the EVP at selected rotational speeds and the solution when dynamic compliance matrix is built up by direct inversion of the dynamic stiffness matrix. Finally, the proposed methods computational performance is analyzed.

A Taylor Series Expansion Approach for Nonlinear Blade Forced Response Prediction Considering Variable Rotational Speed

2017 ◽  
Vol 139 (6) ◽  
Author(s):  
Torsten Heinze ◽  
Lars Panning-von Scheidt ◽  
Jörg Wallaschek ◽  
Andreas Hartung
Keyword(s):  
Series Expansion ◽  
Taylor Series ◽  
Rotational Speed ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Taylor Series Expansion ◽  
Dynamic Compliance ◽  
Forced Response ◽  
Compliance Matrix ◽  
Nonlinear Contact

In the field of turbomachinery, great efforts are made to enhance computational tools to obtain reliable predictions of the vibrational behavior of friction-damped bladed disks. As a trade-off between computational burden and level of simplification, numerous methods were developed to reduce the nonlinear systems dimension. Using component mode synthesis methods (CMS), one is capable to describe the systems motion by interface and modal coordinates. Subsequently or alternatively, the dynamic compliance matrix can be evaluated efficiently by means of modal superposition to avoid the inversion of the dynamic stiffness matrix. Only the equations corresponding to the degrees-of-freedom (DOF) subject to localized nonlinear contact forces need to be solved simultaneously, whereas the solution of the linear DOF is obtained by exploiting the algebraic character of the set of equations. In this paper, an approach is presented to account for rotational speed-dependent stiffness in the subset of nonlinear DOF without the need to re-evaluate the associated eigenvalue problem (EVP) when rotational speed is changed. This is done by means of a Taylor series expansion of the eigenvalues and eigenvectors used for the modal superposition to reconstruct the dynamic compliance matrix. In the context of forced response predictions of friction-damped blisks, the expansion is performed up to different order for a simplified blisk model with nonlinear contact interfaces. The results are compared to the solution obtained by direct evaluation of the EVP at selected rotational speeds and the solution when dynamic compliance matrix is built up by direct inversion of the dynamic stiffness matrix. Finally, the proposed methods computational performance is analyzed.

Exact solution by dynamic stiffness method for the natural vibration of porous functionally graded plate considering neutral surface

2021 ◽  
pp. 146442072098817
Author(s):  
Md. Imran Ali ◽  
Mohammad Sikandar Azam
Keyword(s):  
Power Law ◽  
Functionally Graded Material ◽  
Stiffness Matrix ◽  
Natural Vibration ◽  
Dynamic Stiffness ◽  
Functionally Graded ◽  
Neutral Surface ◽  
Functionally Graded Plate ◽  
Dynamic Stiffness Matrix ◽  
Graded Material

This paper presents the formulation of dynamic stiffness matrix for the natural vibration analysis of porous power-law functionally graded Levy-type plate. In the process of formulating the dynamic stiffness matrix, Kirchhoff-Love plate theory in tandem with the notion of neutral surface has been taken on board. The developed dynamic stiffness matrix, a transcendental function of frequency, has been solved through the Wittrick–Williams algorithm. Hamilton’s principle is used to obtain the equation of motion and associated natural boundary conditions of porous power-law functionally graded plate. The variation across the thickness of the functionally graded plate’s material properties follows the power-law function. During the fabrication process, the microvoids and pores develop in functionally graded material plates. Three types of porosity distributions are considered in this article: even, uneven, and logarithmic. The eigenvalues computed by the dynamic stiffness matrix using Wittrick–Williams algorithm for isotropic, power-law functionally graded, and porous power-law functionally graded plate are juxtaposed with previously referred results, and good agreement is found. The significance of various parameters of plate vis-à-vis aspect ratio ( L/b), boundary conditions, volume fraction index ( p), porosity parameter ( e), and porosity distribution on the eigenvalues of the porous power-law functionally graded plate is examined. The effect of material density ratio and Young’s modulus ratio on the natural vibration of porous power-law functionally graded plate is also explained in this article. The results also prove that the method provided in the present work is highly accurate and computationally efficient and could be confidently used as a reference for further study of porous functionally graded material plate.

scholarly journals Reduced Mass-Weighted Proper Decomposition for Modal Analysis

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
Venkata K. Yadalam ◽  
B. F. Feeny
Keyword(s):  
Modal Analysis ◽  
Mass Distribution ◽  
Mass Matrix ◽  
Linear Interpolation ◽  
Modal Identification ◽  
Distributed Parameter ◽  
Arbitrary Mass ◽  
Spring System ◽  
Mass Spring ◽  
Proper Orthogonal

A method of modal analysis by a mass-weighted proper orthogonal decomposition for multi-degree-of-freedom and distributed-parameter systems of arbitrary mass distribution is outlined. The method involves reduced-order modeling of the system mass distribution so that the discretized mass matrix dimension matches the number of sensed quantities, and hence the dimension of the response ensemble and correlation matrix. In this case, the linear interpolation of unsensed displacements is used to reduce the size of the mass matrix. The idea is applied to the modal identification of a mass-spring system and an exponential rod.

Measured Static and Rotordynamic Coefficient Results for a Rocker-Pivot, Tilting-Pad Bearing With 50 and 60% Offsets

2012 ◽  
Vol 134 (5) ◽  
Author(s):  
Chris D. Kulhanek ◽  
Dara W. Childs
Keyword(s):  
Dynamic Stiffness ◽  
Mass Matrix ◽  
Excitation Frequency ◽  
Quadratic Function ◽  
Added Mass ◽  
Damping Coefficients ◽  
Stiffness Coefficients ◽  
Tilting Pad ◽  
Frequency Independent ◽  
Direct Stiffness

Static and rotordynamic coefficients are measured for a rocker-pivot, tilting-pad journal bearing (TPJB) with 50 and 60% offset pads in a load-between-pad (LBP) configuration. The bearing uses leading-edge-groove direct lubrication and has the following characteristics: 5-pads, 101.6 mm (4.0 in) nominal diameter,0.0814 -0.0837 mm (0.0032–0.0033 in) radial bearing clearance, 0.25 to 0.27 preload, and 60.325 mm (2.375 in) axial pad length. Tests were performed on a floating bearing test rig with unit loads from 0 to 3101 kPa (450 psi) and speeds from 7 to 16 krpm. Dynamic tests were conducted over a range of frequencies (20 to 320 Hz) to obtain complex dynamic stiffness coefficients as functions of excitation frequency. For most test conditions, the real dynamic stiffness functions were well fitted with a quadratic function with respect to frequency. This curve fit allowed for the stiffness frequency dependency to be captured by including an added mass matrix [M] to a conventional [K][C] model, yielding a frequency independent [K][C][M] model. The imaginary dynamic stiffness coefficients increased linearly with frequency, producing frequency-independent direct damping coefficients. Direct stiffness coefficients were larger for the 60% offset bearing at light unit loads. At high loads, the 50% offset configuration had a larger stiffness in the loaded direction, while the unloaded direct stiffness was approximately the same for both pivot offsets. Cross-coupled stiffness coefficients were positive and significantly smaller than direct stiffness coefficients. Negative direct added-mass coefficients were obtained for both offsets, especially in the unloaded direction. Cross-coupled added-mass coefficients are generally positive and of the same sign. Direct damping coefficients were mostly independent of load and speed, showing no appreciable difference between pivot offsets. Cross-coupled damping coefficients had the same sign and were much smaller than direct coefficients. Measured static eccentricities suggested cross coupling stiffness exists for both pivot offsets, agreeing with dynamic measurements. Static stiffness measurements showed good agreement with the loaded, direct dynamic stiffness coefficients.

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.

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