scholarly journals Free and Forced Vibrations of Elastically Connected Structures

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
S. Graham Kelly
Keyword(s):  
Natural Frequencies ◽  
Iterative Solution ◽  
Forced Response ◽  
Inner Product ◽  
Mode Shapes ◽  
Square Roots ◽  
Parallel Structures ◽  
Special Cases ◽  
Definition Of ◽  
The Individual

A general theory for the free and forced responses of elastically connected parallel structures is developed. It is shown that if the stiffness operator for an individual structure is self-adjoint with respect to an inner product defined for , then the stiffness operator for the set of elastically connected structures is self-adjoint with respect to an inner product defined on . This leads to the definition of energy inner products defined on . When a normal mode solution is used to develop the free response, it is shown that the natural frequencies are the square roots of the eigenvalues of an operator that is self-adjoint with respect to the energy inner product. The completeness of the eigenvectors in is used to develop a forced response. Special cases are considered. When the individual stiffness operators are proportional, the problem for the natural frequencies and mode shapes reduces to a matrix eigenvalue problem, and it is shown that for each spatial mode there is a set of intramodal mode shapes. When the structures are identical, uniform, or nonuniform, the differential equations are uncoupled through diagonalization of a coupling stiffness matrix. The most general case requires an iterative solution.

scholarly journals Identification of Mistuning Characteristics of Bladed Disks From Free Response Data

1998 ◽  
Author(s):  
Marc P. Mignolet ◽  
Alejandro Rivas-Guerra
Keyword(s):  
Dynamic Models ◽  
Natural Frequencies ◽  
Dynamic Properties ◽  
Structural Parameters ◽  
Identification Problem ◽  
Forced Response ◽  
Mode Shapes ◽  
Likelihood Principle ◽  
Bladed Disks ◽  
Block Test

The focus of the present investigation is on the estimation of the dynamic properties, i.e. masses, stiffnesses, natural frequencies, mode shapes and their statistical distributions, of turbomachine blades to be used in the accurate prediction of the forced response of mistuned bladed disks. As input to this process, it is assumed that the lowest natural frequencies of the blades alone have been experimentally measured, for example in a broach block test. Since the number of measurements is always less than the number of unknowns, this problem is indeterminate in nature. Two distinct approaches will be investigated to resolve the shortfall of data. The first one relies on the imposition of as many constraints as needed to insure a unique solution to this identification problem. Specifically, the mode shapes and modal masses of the blades are set to their design/tuned counterparts while the modal stiffnesses are varied from blade-to-blade to match the measured natural frequencies. The second approach, based on the maximum likelihood principle, yields estimates of all the structural parameters of the blades through the minimization of a specified “cost function”. The accuracy of these two techniques in predicting the forced response of mistuned bladed disks will be assessed on simple dynamic models of the blades.

Accuracies of Reduced Order Models of a Bladed Rotor With Geometric Mistuning

2013 ◽  
Vol 136 (7) ◽  
Author(s):  
Yasharth Bhartiya ◽  
Alok Sinha
Keyword(s):  
Natural Frequencies ◽  
Domain Analysis ◽  
Forced Response ◽  
Reduced Order Model ◽  
Mode Shapes ◽  
Reduced Order Models ◽  
Order Model ◽  
Reduced Order ◽  
Model Based ◽  
Bladed Rotor

The results from a reduced order model based on frequency mistuning are compared with those from recently developed modified modal domain analysis (MMDA). For the academic bladed rotor considered in this paper, the frequency mistuning analysis is unable to capture the effects of geometric mistuning, whereas MMDA provides accurate estimates of natural frequencies, mode shapes, and forced response.

Modified Modal Domain Analysis of a Bladed Rotor Using Coordinate Measurement Machine Data on Geometric Mistuning

2015 ◽  
Vol 137 (4) ◽  
Author(s):  
Vinod Vishwakarma ◽  
Alok Sinha ◽  
Yasharth Bhartiya ◽  
Jeffery M. Brown
Keyword(s):  
Natural Frequencies ◽  
Domain Analysis ◽  
Reduced Order Modeling ◽  
Forced Response ◽  
Mode Shapes ◽  
Reduced Order ◽  
Coordinate Measurement ◽  
Coordinate Measurement Machine ◽  
Bladed Rotor ◽  
Proper Orthogonal

Modified modal domain analysis (MMDA), a reduced order modeling technique, is applied to a geometrically mistuned integrally bladed rotor to obtain its natural frequencies, mode shapes, and forced response. The geometric mistuning of blades is described in terms of proper orthogonal decomposition (POD) of the coordinate measurement machine (CMM) data. Results from MMDA are compared to those from the full (360 deg) rotor Ansys model. It is found that the MMDA can accurately predict natural frequencies, mode shapes, and forced response. The effects of the number of POD features and the number of tuned modes used as bases for model reduction are examined. Results from frequency mistuning approaches, fundamental mistuning model (FMM) and subset of nominal modes (SNM), are also generated and compared to those from full (360 deg) rotor Ansys model. It is clearly seen that FMM and SNM are unable to yield accurate results whereas MMDA yields highly accurate results.

Identification of Mistuning Characteristics of Bladed Disks From Free Response Data — Part II

1999 ◽  
Author(s):  
Marc P. Mignolet ◽  
Jason P. Delor ◽  
Alejandro Rivas-Guerra
Keyword(s):  
Dynamic Models ◽  
Natural Frequencies ◽  
Dynamic Properties ◽  
Structural Parameters ◽  
Identification Problem ◽  
Forced Response ◽  
Mode Shapes ◽  
Likelihood Principle ◽  
Bladed Disks ◽  
Block Test

The focus of the present investigation is on the estimation of the dynamic properties, i.e. masses, stiffnesses, natural frequencies, mode shapes and their statistical distributions, of turbomachine blades to be used in the accurate prediction of the forced response of mistuned bladed disks. As input to this process, it is assumed that the lowest natural frequencies of the blades alone have been experimentally measured, for example in a broach block test. Since the number of measurements is always less than the number of unknowns, this problem is indeterminate in nature. Three distinct approaches will be investigated to resolve the shortfall of data. The first one relies on the imposition of as many constraints as needed to insure a unique solution to this identification problem. Specifically, the mode shapes and modal masses of the blades are set to their design/tuned counterparts while the modal stiffnesses are varied from blade-to-blade to match the measured natural frequencies. The second approach, based on the maximum likelihood principle, yields estimates of all the structural parameters of the blades through the minimization of a specified “cost function”. Finally, the third approach provides a bridge between the first two methods being based on the second but yielding a mistuning model similar to that of the first approach. The accuracy of these three techniques in predicting the forced response of mistuned bladed disks will be assessed on simple dynamic models of the blades.

A generalization of the concept of ω-consistency

1954 ◽  
Vol 19 (3) ◽  
pp. 183-196 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Functional Calculus ◽  
Free Variable ◽  
Natural Numbers ◽  
Deductive Systems ◽  
Individual Variable ◽  
Special Cases ◽  
Definition Of ◽  
The Individual ◽  
Formal Properties ◽  
Use Of Models

In this paper we consider certain formal properties of deductive systems which, in special cases, reduce to the property of ω-consistency; and we then seek to understand the significance of these properties by relating them to the use of models in providing interpretations of the deductive systems.The notion of ω-consistency arises in connection with deductive systems of arithmetic. For definiteness, let us suppose that the system is a functional calculus whose domain of individuals is construed as the set of natural numbers, and that the system possesses individual constants ν0, ν1, ν2, … such that νi functions as a name for the number i. Such a system is called ω-consistent, if there is no well-formed formula A(x) (in which x is the only free variable) such that A(ν0), A(ν1), A(ν2), … and ∼(x)A(x) are all formal theorems of the system, where A(νi) is the formula resulting from A(x) by substituting the constant νi for each free occurrence of the individual variable x.Now consider an arbitrary applied functional calculus F, and let Γ be any non-empty set of its individual constants. In imitation of the definition of ω-consistency, we may say that the system F is Γ-consistent, if it contains no formula A(x) (in which x is the only free variable) such that ⊦ A (α) for every constant α in Γ, and also ⊦ ∼(x)A(x) (where an occurrence of “⊦” indicates that the formula which it precedes is a formal theorem). We easily see that the condition of Γ-consistency is equivalent to the condition that the system F contain no formula B(x) such that ⊦ ∼ B(α) for each α in Γ, and also ⊦ (∃x)B(x).

Modal Analysis of Nonviscously Damped Beams

2006 ◽  
Vol 74 (5) ◽  
pp. 1026-1030 ◽  
Author(s):  
S. Adhikari ◽  
M. I. Friswell ◽  
Y. Lei
Keyword(s):  
Natural Frequencies ◽  
Past History ◽  
Kernel Functions ◽  
Mode Shapes ◽  
Damping Force ◽  
The Past ◽  
Convolution Integrals ◽  
Special Cases ◽  
History Of ◽  
Damping Model

Linear dynamics of Euler–Bernoulli beams with nonviscous nonlocal damping is considered. It is assumed that the damping force at a given point in the beam depends on the past history of velocities at different points via convolution integrals over exponentially decaying kernel functions. Conventional viscous and viscoelastic damping models can be obtained as special cases of this general damping model. The equation of motion of the beam with such a general damping model results in a linear partial integro-differential equation. Exact closed-form equations of the natural frequencies and mode shapes of the beam are derived. Numerical examples are provided to illustrate the new results.

Free Oscillations of Edge-Connected Simply Supported Plate Systems

1961 ◽  
Vol 83 (4) ◽  
pp. 434-439 ◽  
Author(s):  
Eric E. Ungar
Keyword(s):  
Individual Component ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Free Oscillations ◽  
Common Edge ◽  
Simply Supported ◽  
The Individual ◽  
Plate System

A simple semigraphical method for calculating the natural frequencies of two-plate systems is developed, a two-plate system being one made up of two rectangular plates simply supported at all edges and joined at a common edge. Charts for easy determination of the afore-mentioned natural frequencies are developed. One of these gives, as a by-product, the natural frequencies of rectangular plates (of any dimensions) having one edge clamped, the remaining three simply supported. It is demonstrated that the higher natural frequencies of two-plate systems are very nearly equal to those of the individual component plates. Equations for the mode shapes are also given.

Dynamic Characteristics of Resonant Beams With Passive Thermal Stress Regulators

2018 ◽  
Author(s):  
Tyler Kellar ◽  
Pezhman Hassanpour
Keyword(s):  
Boundary Condition ◽  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Separation Of Variables ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Mode Shapes ◽  
Elastic Boundary ◽  
Special Cases ◽  
Angular Displacements

This paper addresses the dynamic characteristics of a beam with a particular elastic boundary condition. In this elastic boundary condition, the lateral and angular displacements of the beam are coupled through the elastic constraints. The dynamic characteristic, namely natural frequencies and mode shapes of vibrations are frequently encountered in the design and modeling of resonant micro-structures. The governing equations of motion of the beam is derived using Euler-Bernoulli beam theory considering the elastic coupling between the transverse and rotational displacements of the beam’s end. The characteristic equation for the natural frequencies and mode shapes of vibration is derived by applying the method of separation of variables to the governing partial differential equation of motion. The natural frequencies and mode shapes of the system are derived for various combinations of compliance values of the elastic support and are compared with those of several special cases, namely clamped-free, clamped-guided, clamped-pinned and clamped-clamped beams.

Linear Dynamics of a Translating String on an Elastic Foundation

1990 ◽  
Vol 112 (1) ◽  
pp. 2-7 ◽  
Author(s):  
N. C. Perkins
Keyword(s):  
Free Vibration ◽  
Elastic Foundation ◽  
Linear Response ◽  
Natural Frequencies ◽  
Cutoff Frequency ◽  
Forced Response ◽  
Mode Shapes ◽  
Translation Speed ◽  
Linear Dynamics ◽  
Distinct Solution

This paper examines the free and forced linear response of a string which is translating across an elastic foundation. Exact solutions are derived for the free vibration of the string which translates between fixed eyelets and across elastic foundations represented by (1) a single interior spring and (2) a uniform step foundation. Results illustrate the dependence of the string natural frequencies and mode shapes on the foundation stiffness, the foundation geometry, and the string translation speed. The forced response of the string to harmonic end excitation is computed in closed form for the case of a complete uniform foundation. A cutoff frequency separates three distinct solution forms. For excitation frequencies below the cutoff frequency, the response amplitude decays exponentially with distance from the driven end.

Identification of Mistuning Characteristics of Bladed Disks From Free Response Data—Part I

1999 ◽  
Vol 123 (2) ◽  
pp. 395-403 ◽  
Author(s):  
M. P. Mignolet ◽  
A. J. Rivas-Guerra ◽  
J. P. Delor
Keyword(s):  
Dynamic Models ◽  
Natural Frequencies ◽  
Dynamic Properties ◽  
Structural Parameters ◽  
Identification Problem ◽  
Forced Response ◽  
Mode Shapes ◽  
Likelihood Principle ◽  
Bladed Disks ◽  
Block Test

The focus of the present two-part investigation is on the estimation of the dynamic properties, i.e., masses, stiffnesses, natural frequencies, mode shapes and their statistical distributions, of turbomachine blades to be used in the accurate prediction of the forced response of mistuned bladed disks. As input to this process, it is assumed that the lowest natural frequencies of the blades alone have been experimentally measured, for example, in a broach block test. Since the number of measurements is always less than the number of unknowns, this problem is indeterminate in nature. In this first part of the investigation, two distinct approaches will be investigated to resolve the shortfall of data. The first one relies on the imposition of as many constraints as needed to ensure a unique solution to this identification problem. Specifically, the mode shapes and modal masses of the blades are set to their design/tuned counterparts while the modal stiffnesses are varied from blade to blade to match the measured natural frequencies. The second approach, based on the maximum likelihood principle, yields estimates of all the structural parameters of the blades through the minimization of a specified “cost function.” The accuracy of these two techniques in predicting the forced response of mistuned bladed disks will be assessed on simple dynamic models of the blades.

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