Friction Damping of Hollow Airfoils: Part I — Theoretical Development

1996 ◽  
Author(s):  
J. H. Griffin ◽  
W.-T. Wu ◽  
Y. El-Aini
Keyword(s):  
Analytical Model ◽  
Natural Frequencies ◽  
Time Integration ◽  
Hollow Structure ◽  
Companion Paper ◽  
Theoretical Development ◽  
Life Cycle Costs ◽  
Friction Dampers ◽  
Internal Cavities ◽  
Mass Test

The quest for higher performance engines in conjunction with the requirement for lower life cycle costs has resulted in stage configurations that are more susceptible to high cycle fatigue. One solution is the use of innovative approaches that introduce additional mechanical damping. The present paper describes an approach that may be used to assess the benefits of friction dampers located within internal cavities of a hollow structure. The friction dampers used in this application are often relatively thin devices that, if unconstrained, have natural frequencies in the same range as the natural frequencies of the hollow airfoil. Consequently, the analytical approach that is developed is distinct in that it has to take into account the dynamic response of the damper and how it changes as the amplitude of the vibration increases. In this paper, results from the analytical model are compared with independently generated results from a time integration solution of a three mass test problem. Results from the analytical model are compared with experimental data in a companion paper.

Friction Damping of Hollow Airfoils: Part I—Theoretical Development

1998 ◽  
Vol 120 (1) ◽  
pp. 120-125 ◽  
Author(s):  
J. H. Griffin ◽  
W.-T. Wu ◽  
Y. EL-Aini
Keyword(s):  
Analytical Model ◽  
Natural Frequencies ◽  
Time Integration ◽  
Hollow Structure ◽  
Companion Paper ◽  
Theoretical Development ◽  
Life Cycle Costs ◽  
Friction Dampers ◽  
Internal Cavities ◽  
Mass Test

The quest for higher performance engines in conjunction with the requirement for lower life cycle costs has resulted in stage configurations that are more susceptible to high cycle fatigue. One solution is the use of innovative approaches that introduce additional mechanical damping. The present paper describes an approach that may be used to assess the benefits of friction dampers located within internal cavities of a hollow structure. The friction dampers used in this application are often relatively thin devices that, if unconstrained, have natural frequencies in the same range as the natural frequencies of the hollow airfoil. Consequently, the analytical approach that is developed is distinct in that it has to take into account the dynamic response of the damper and how it changes as the amplitude of the vibration increases. In this paper, results from the analytical model are compared with independently generated results from a time integration solution of a three mass test problem. Results from the analytical model are compared with experimental data in a companion paper.

Updating of Non-Conservative Systems Using Inverse Methods

1997 ◽  
Author(s):  
Ladislav Starek ◽  
Milos Musil ◽  
Daniel J. Inman
Keyword(s):  
Analytical Model ◽  
Degrees Of Freedom ◽  
Ad Hoc ◽  
Natural Frequencies ◽  
Eigenvalue Problems ◽  
Model Updating ◽  
Analytical Models ◽  
Mode Shapes ◽  
Element Analysis ◽  
Modal Data

Abstract Several incompatibilities exist between analytical models and experimentally obtained data for many systems. In particular finite element analysis (FEA) modeling often produces analytical modal data that does not agree with measured modal data from experimental modal analysis (EMA). These two methods account for the majority of activity in vibration modeling used in industry. The existence of these discrepancies has spanned the discipline of model updating as summarized in the review articles by Inman (1990), Imregun (1991), and Friswell (1995). In this situation the analytical model is characterized by a large number of degrees of freedom (and hence modes), ad hoc damping mechanisms and real eigenvectors (mode shapes). The FEM model produces a mass, damping and stiffness matrix which is numerically solved for modal data consisting of natural frequencies, mode shapes and damping ratios. Common practice is to compare this analytically generated modal data with natural frequencies, mode shapes and damping ratios obtained from EMA. The EMA data is characterized by a small number of modes, incomplete and complex mode shapes and non proportional damping. It is very common in practice for this experimentally obtained modal data to be in minor disagreement with the analytically derived modal data. The point of view taken is that the analytical model is in error and must be refined or corrected based on experimented data. The approach proposed here is to use the results of inverse eigenvalue problems to develop methods for model updating for damped systems. The inverse problem has been addressed by Lancaster and Maroulas (1987), Starek and Inman (1992,1993,1994,1997) and is summarized for undamped systems in the text by Gladwell (1986). There are many sophisticated model updating methods available. The purpose of this paper is to introduce using inverse eigenvalues calculated as a possible approach to solving the model updating problem. The approach is new and as such many of the practical and important issues of noise, incomplete data, etc. are not yet resolved. Hence, the method introduced here is only useful for low order lumped parameter models of the type used for machines rather than structures. In particular, it will be assumed that the entries and geometry of the lumped components is also known.

Free Vibration of Bistable Clamped–Clamped Beams: Modeling and Experiment

2020 ◽  
Vol 143 (4) ◽  
Author(s):  
Xiaolei Song ◽  
Haijun Liu
Keyword(s):  
Analytical Model ◽  
Natural Frequencies ◽  
Classical Problem ◽  
Mode Shapes ◽  
Wide Range ◽  
Fundamental Understanding ◽  
Clamped Beams ◽  
Transverse Deflection ◽  
Symmetric And Antisymmetric Modes ◽  
Good Agreement

Abstract Bistable clamped–clamped beams have been used in a wide range of applications such as switches, resonators, energy harvesting, and vibration reduction. Most studies on this classic buckling problem focus on obtaining either the static configuration and the required critical axial load or the natural frequencies and mode shapes of postbuckling vibrations analytically. In this article, we present our study including analytical modeling and experimental method on bistable clamped–clamped beams, aiming to understand the detailed snap-through process and the ensuing vibration. In the analytical model, by decomposing the transverse deflection into static buckling configuration and linear vibration, we obtain the natural frequencies and mode shapes for the buckled beam and investigate the effects of static deflection on the symmetric and antisymmetric modes. An experimental design using noncontact methods is implemented to directly measure the response of the whole beam in the snap-through process and the sound generated by the vibrating beam. The measurements are characterized in both time and frequency domain and found to be in good agreement with the analytical model. The study presented in this article enhances the fundamental understanding of the classical problem of bistable clamped–clamped beams.

Analytical Dynamic Performance Modeling for Individual C-block Actuators

1999 ◽  
Vol 121 (2) ◽  
pp. 221-230 ◽  
Author(s):  
A. J. Moskalik ◽  
D. Brei
Keyword(s):  
Analytical Model ◽  
Performance Modeling ◽  
Natural Frequencies ◽  
Block Design ◽  
Dynamic Performance ◽  
Design Parameters ◽  
Good Match ◽  
Amplitude Response ◽  
Trade Offs ◽  
Analytical Dynamic

C-blocks are mid-range piezoelectric actuators that show promise for use in dynamic applications, such as noise and vibration control. This paper presents an analytical model of an individual C-block actuator, including the identification of the natural frequencies and the description of the amplitude response across the frequency spectrum. In addition, an experimental study with three case studies is presented investigating the accuracy of the model and the sensitivity of the overall dynamic performance to C-block design parameters. The experimental results showed a good match to the analytical model and outlined the trade-offs between displacement amplitude and bandwidth.

Methods of Preserving Symmetry in Model Updating

1995 ◽  
Vol 117 (3A) ◽  
pp. 349-354
Author(s):  
M. J. Lam ◽  
D. J. Inman
Keyword(s):  
Experimental Data ◽  
Analytical Model ◽  
Natural Frequencies ◽  
Model Updating ◽  
Mode Shapes ◽  
Modal Parameter ◽  
Modal Data ◽  
Model Correction ◽  
Damped Systems ◽  
Stiffness And Damping

This work examines the model updating technique for both conservative and nonproportionally damped systems. In model updating, also referred to as model correction, the analytical model is updated until it agrees with the experimental data available. In this paper it is assumed that the measured modal data, i.e., natural frequencies and in some instances mode shapes, disagrees in part with the modal parameter predicted by the analytical model. Many model updating schemes tend to produce nonsymmetric updated stiffness (and damping) matrices. The methods presented here focus on retaining the desired symmetry in the updated model

An Analytical Model for a Clamped Isotropic Beam Under Thermal Effects

2002 ◽  
Author(s):  
Rodrigo F. A. Marques ◽  
Daniel J. Inman
Keyword(s):  
Boundary Conditions ◽  
Analytical Model ◽  
Thermal Effects ◽  
Natural Frequencies ◽  
Dynamical Behavior ◽  
Beam Theory ◽  
Optimization Procedure ◽  
Temperature Changes ◽  
Thermally Expand ◽  
Spring Constants

Structures and industrial equipment often operate in environments where temperature variations take place. Although thermal effects may be negligible in some cases, they have caused the unexpected failure of mechanical systems many times. Whether or not temperature has significant effects on the dynamical behavior of such machines and structures depends upon several aspects, amongst which are geometry, material properties and boundary conditions. In this paper we investigate the dynamical behavior of a clamped beam under the influence of a uniform, quasi-statically varying temperature field. An analytical model was used, based on Euler-Bernoulli’s beam theory with the introduction of the proper boundary conditions. Temperature effects are included in terms of an axial force that shows up when the beam tends to thermally expand, but this expansion is restrained by the clamping. Preliminary results do not agree with experimental data, since perfect clamping is difficult to achieve in practice. Finally the model is updated with the inclusion of axial and torsional springs connecting the beam to the support. The spring constants were calculated through optimization procedure to minimize the differences between the natural frequencies obtained from the analytical model and the corresponding experimental ones. Agreement with experimental results is reasonable up to the 4th mode of the beam. In the future, this analytical model is to be used for design and simulation of an active controller that accounts for temperature changes in the structure.

Design a New Spherical Super Element to Static and Modal Analysis of Spherical Structures

2011 ◽  
Author(s):  
Hamid Faghani ◽  
M. T. Ahmadian
Keyword(s):  
Finite Element ◽  
Modal Analysis ◽  
Natural Frequencies ◽  
Accurate Result ◽  
Three Dimensional ◽  
Hollow Structure ◽  
Pressure Vessels ◽  
Spherical Structure ◽  
Spherical Structures ◽  
Spherical Pressure

Hollow Spherical Structures such as spherical pressure vessels are widely used in industries. Using faster numerical methods to solve these spherical problems has been one of the research interests in the last decade. Super elements are elements capable of producing the same accurate result in comparison with several conventional elements. In this paper, a new spherical super element is designed and implemented to study the static deformation and modal analysis of a spherical structure. In the static analysis mean of the inner radial displacements of the hollow structure under the internal pressure is evaluated, also performing the modal analysis, natural frequencies of the structure under the specified boundary condition are obtained. Findings are compared with those of conventional finite element methods and very good agreements are achieved. Using two spherical super element result the same displacement in conventional finite element with 2164 three dimensional brick elements.

Fluid Flow Through a Class of Highly-Deformable Porous Media. Part I: Experiments With Air

1981 ◽  
Vol 103 (3) ◽  
pp. 432-438 ◽  
Author(s):  
G. S. Beavers ◽  
A. Hajji ◽  
E. M. Sparrow
Keyword(s):  
Porous Media ◽  
Fluid Flow ◽  
Analytical Model ◽  
Mass Flow ◽  
Applied Pressure ◽  
Companion Paper ◽  
Working Fluid ◽  
Internal Stresses ◽  
Deformable Porous Media ◽  
Flow Through

This paper, together with a companion paper which follows, describes a many-faceted experimental investigation aimed at determining basic characteristics of fluid flow through deformable porous media. A major focus of the work is to establish the validity and the range of applicability of a simple analytical model for the fluid flow. The present paper describes experiments with a gas (air) as the working fluid, while the companion paper (Part II) deals with liquid-flow (i.e., water-flow) experiments. The experiments encompassed three distinct phases. In the first phase, the stress-deformation characteristics were measured (without fluid flow). In the second, flow-related material properties that are relevant to the analytical model (e.g., permeability, Forchheimer coefficient) were determined. The third phase consisted of measurements of mass flow rate as a function of applied pressure differential. The results of the first two phases were used as input to the analytical model, which yielded predictions of mass flow versus applied pressure. These predictions were shown to be in very good agreement with the experimental results, for those conditions where the model is applicable. Two unusual features of the participating deformable materials (polyurethane foams) were encountered, namely, a decrease of cross-sectional area with increasing compression and a slow relaxation of the internal stresses at a fixed compression.

On Vibrations of a System With an Eigenfrequency Identical to That of One of Its Subsystems, Part 2

1996 ◽  
Vol 118 (3) ◽  
pp. 414-416
Author(s):  
A. V. Pesterev ◽  
L. A. Bergman
Keyword(s):  
Complex System ◽  
Free Vibration ◽  
Natural Frequency ◽  
Natural Frequencies ◽  
Companion Paper ◽  
Synthesis Methods ◽  
Interaction Forces ◽  
Modal Synthesis ◽  
Green’S Operator ◽  
Simple Systems

The problem of free vibration of a complex system with a natural frequency identical to that of one of its subsystems is further discussed. Such eigenvibrations need special consideration in many modal synthesis methods, as the Green’s operator of the resonating subsystem does not exist at subsystem natural frequencies. A general treatment of this problem has been given by the authors in a companion paper. In this supplement, the previous work is extended to include the case of interaction forces applied to the resonating subsystem at points where the corresponding eigenfunction of the subsystem has maxima. Examples of such eigenvibrations are presented for two simple systems. The differences between these examples and those of the previous paper are noted and discussed.

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