On Asymptotic Approximations to Beam Model Shapes

1984 ◽  
Vol 51 (2) ◽  
pp. 439-439 ◽  
Author(s):  
E. H. Dowell
Keyword(s):  
Boundary Conditions ◽  
Mode Shapes ◽  
Asymptotic Approximations ◽  
General Interest ◽  
Beam Model ◽  
Asymptotic Character

Recently the author had occasion to investigate inter alia the asymptotic character of the mode shapes of uniform beams. These results do not seem to have been presented before in the literature and so are given here as they appear to be of general interest. Clamped-free and free-free beams are considered, although other beam boundary conditions may be treated in a similar manner.

scholarly journals Stability of a flexible structure with destabilizing boundary conditions

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160109 ◽  
Author(s):  
M. Shubov ◽  
V. Shubov
Keyword(s):  
Boundary Conditions ◽  
Control Parameter ◽  
The Other ◽  
Mode Shapes ◽  
Beam Model ◽  
Control Parameters ◽  
The Stability ◽  
Dissipative Boundary ◽  
The Right ◽  
Observation Vector

The Euler–Bernoulli beam model with non-dissipative boundary conditions of feedback control type is investigated. Components of the two-dimensional input vector are shear and moment at the right end, and components of the observation vector are time derivatives of displacement and slope at the right end. The codiagonal matrix depending on two control parameters relates input and observation. The paper contains five results. First, asymptotic approximation for eigenmodes is derived. Second, ‘the main identity’ is established. It provides a relation between mode shapes of two systems: one with non-zero control parameters and the other one with zero control parameters. Third, when one control parameter is positive and the other one is zero, ‘the main identity’ yields stability of all eigenmodes (though the system is non-dissipative). Fourth, the stability of eigenmodes is extended to the case when one control parameter is positive, and the other one is sufficiently small. Finally, existence and properties of ‘deadbeat’ modes are investigated.

scholarly journals A Cyclosymmetric Beam Model and a Spring-Supported Annular Plate Model for Automotive Disc Brake Vibration

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Dennis Boennen ◽  
Stephen James Walsh
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Plate Theory ◽  
Forced Response ◽  
Analytical Models ◽  
Mode Shapes ◽  
Brake Disc ◽  
Disc Brake ◽  
Beam Model ◽  
Static Test

This paper discusses two simplified analytical models for automotive disc brake vibration which can be used to complement more complex finite element methods. The first model approximates the brake disc as a simple beam structure with cyclosymmetric boundary conditions. Since the beam model is a one-dimensional approach, modelling of the inner boundary condition of the brake disc, at the interface of the brake rotor and the central hat, is not possible. The second model, which is established based upon Kirchhoff’s thin plate theory, is presented in this paper in order to incorporate the vibrational deformation at the hat-disc interface. The mode shapes, natural frequencies, and forced response of a static disc are calculated using different inner boundary conditions. Among others, the spring-supported boundary condition is proposed and applied in this paper to make appropriate predictions. The predicted results are compared with measurements of the vibration characteristics of a solid brake disc mounted upon a static test rig. These comparisons demonstrate that the most appropriate model for the inner boundary condition of the measured brake disc is the proposed spring-supported inner boundary condition.

An experimental study of cable tension identification based on measurements of mode shapes

2021 ◽  
Author(s):  
Zhan-hang Liu ◽  
Lin Chen ◽  
Li-Min Sun ◽  
Yi-qing Zou
Keyword(s):  
Experimental Study ◽  
Boundary Conditions ◽  
Mode Shapes ◽  
Beam Model ◽  
Vibration Measurements ◽  
Cable Tension ◽  
Sensor Arrangement ◽  
Axial Tension ◽  
Higher Order Modes ◽  
Unknown Length

<p>Cable tension identification based on mode shapes extracted from vibration measurements is a relatively new method. In this method, the cable is equivalent to a beam model hinged at its ends and under axial tension with an unknown length to eliminate the effects of boundary conditions. This study focuses on the influences of sensor arrangement in the measurements on the accuracy of the tension identification. For this purpose, full-scale cable experiments have been carried out, where a number of sensors were attached to the cable to record cable acceleration during artificial excitation. The eigenvalue realization algorithm (ERA) has then been applied to identify the mode shapes and frequencies of the cables from the multiple acceleration measurements. The effects of different sensor arrangement schemes and cable tension identification method based on higher- order modes are compared and discussed.</p>

Vibration of a Rotating Stepped Beam by the Distributed Transfer Function Method

1993 ◽  
Author(s):  
W. Kuang ◽  
C. A. Tan
Keyword(s):  
Transfer Function ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Beam Model ◽  
Simply Supported ◽  
Rayleigh Beam ◽  
Stepped Beam ◽  
Distributed Transfer Function ◽  
Forced Responses

Abstract In this paper, the distributed transfer function formulation is used to investigate the dynamic characteristics of a rotating stepped beam. The Rayleigh beam model with general boundary conditions is considered. Exact closed-form solutions are obtained for the free and forced responses. Results for the natural frequencies and mode shapes of the stepped beam are presented. For the purpose of illustration, three sets of boundary conditions are considered: simply-simply supported, clamped-clamped, and clamped-simply supported. Effects of system parameters such as the length ratio and diameter ratio of the beam on the free and forced responses are examined and discussed. It is shown that for different combinations of boundary conditions, the pair of natural frequencies for the forward and backward precession modes are associated with different mode shapes except for the simply-simply supported beam. Moreover, the difference between the forward and backward mode shapes increases with the rotation speed of the beam.

Free Bending Vibrations of a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) With a Gap in Mindlin Plates or Panels

2007 ◽  
Author(s):  
U. Yuceoglu ◽  
O. Gu¨vendik ◽  
V. O¨zerciyes
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Shear Stresses ◽  
Lap Joint ◽  
Bending Vibrations ◽  
Joint System ◽  
Adhesive Layers ◽  
Bonded Joint ◽  
Free Bending

In this present study, the “Free Bending Vibrations of a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) with a Gap in Mindlin Plates or Panels” are theoretically analyzed and are numerically solved in some detail. The “plate adherends” and the upper and lower “doubler plates” of the “Bonded Joint” system are considered as dissimilar, orthotropic “Mindlin Plates” joined through the dissimilar upper and lower very thin adhesive layers. There is a symmetrically and centrally located “Gap” between the “plate adherends” of the joint system. In the “adherends” and the “doublers” of the “Bonded Joint” assembly, the transverse shear deformations and the transverse and rotary moments of inertia are included in the analysis. The relatively very thin adhesive layers are assumed to be linearly elastic continua with transverse normal and shear stresses. The “damping effects” in the entire “Bonded Joint” system are neglected. The sets of the dynamic “Mindlin Plate” equations of the “plate adherends”, the “double doubler plates” and the thin adhesive layers are combined together with the orthotropic stress resultant-displacement expressions in a “special form”. This system of equations, after some further manipulations, is eventually reduced to a set of the “Governing System of the First Order Ordinary Differential Equations” in terms of the “state vectors” of the problem. Hence, the final set of the aforementioned “Governing Systems of Equations” together with the “Continuity Conditions” and the “Boundary conditions” facilitate the present solution procedure. This is the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials). The present theoretical formulation and the method of solution are applied to a typical “Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) with a Gap”. The effects of the relatively stiff (or “hard”) and the relatively flexible (or “soft”) adhesive properties, on the natural frequencies and mode shapes are considered in detail. The very interesting mode shapes with their dimensionless natural frequencies are presented for various sets of boundary conditions. Also, several parametric studies of the dimensionless natural frequencies of the entire system are graphically presented. From the numerical results obtained, some important conclusions are drawn for the “Bonded Joint System” studied here.

Flexural Vibration of Prestressed Concrete Bridges with Corrugated Steel Webs

2017 ◽  
Vol 17 (02) ◽  
pp. 1750023 ◽  
Author(s):  
Xia-Chun Chen ◽  
Zhen-Hu Li ◽  
Francis T. K. Au ◽  
Rui-Juan Jiang
Keyword(s):  
Finite Element ◽  
Shear Deformation ◽  
Prestressed Concrete ◽  
Natural Frequencies ◽  
Sandwich Beam ◽  
Flexural Vibration ◽  
Concrete Bridges ◽  
Mode Shapes ◽  
Beam Model ◽  
Prestressed Concrete Bridges

Prestressed concrete bridges with corrugated steel webs have emerged as a new form of steel-concrete composite bridges with remarkable advantages compared with the traditional ones. However, the assumption that plane sections remain plane may no longer be valid for such bridges due to the different behavior of the constituents. The sandwich beam theory is extended to predict the flexural vibration behavior of this type of bridges considering the presence of diaphragms, external prestressing tendons and interaction between the web shear deformation and flange local bending. To this end, a [Formula: see text] beam finite element is formulated. The proposed theory and finite element model are verified both numerically and experimentally. A comparison between the analyses based on the sandwich beam model and on the classical Euler–Bernoulli and Timoshenko models reveals the following findings. First of all, the extended sandwich beam model is applicable to the flexural vibration analysis of the bridges considered. By letting [Formula: see text] denote the square root of the ratio of equivalent shear rigidity to the flange local flexural rigidity, and L the span length, the combined parameter [Formula: see text] appears to be more suitable for considering the diaphragm effect and the interaction between the shear deformation and flange local bending. The diaphragms have significant effect on the flexural natural frequencies and mode shapes only when the [Formula: see text] value of the bridge falls below a certain limit. For a bridge with an [Formula: see text] value over a certain limit, the flexural natural frequencies and mode shapes obtained from the sandwich beam model and the classical Euler–Bernoulli and Timoshenko models tend to be the same. In such cases, either of the classical beam theories may be used.

scholarly journals Detailed Analysis of the Acoustic Mode Shapes of an Annular Combustion Chamber

1999 ◽  
Author(s):  
Günther Walz ◽  
Werner Krebs ◽  
Stefan Hoffmann ◽  
Hans Judith
Keyword(s):  
Boundary Conditions ◽  
Acoustic Mode ◽  
Mode Shapes ◽  
Maximum Deviation ◽  
Minor Importance ◽  
Velocity Of Sound ◽  
Fe Method ◽  
Shift Frequency ◽  
Space Dependence ◽  
Eigenfrequency Spectrum

To get a better understanding of the formation of thermoacoustic oscillations in an annular gasturbine combustor, an analysis of the acoustic eigenmodes has been conducted using the Finite Element (FE) method. The influence of different boundary conditions and a space dependent velocity of sound has been investigated. The boundary conditions actually define the eigenfrequency spectrum. Hence, it is crucial to know e.g. the burner impedance. In case of the combustion system without significant mixing air addition considered in this paper, the space dependence of the velocity of sound is of minor importance for the eigenfrequency spectrum leading to a maximum deviation of only 5% in the eigenvalues. It is demonstrated that the efficiency of the numerical eigenvalue analysis can be improved by making use of symmetry, by splitting the problem into several steps with alternate boundaries conditions, and by choosing the shift frequency ωs in the range of frequencies one is interested in.

Approximate Formulas for Cylindrical Shell Free Vibration Based on Vlasov’s and Enhanced Vlasov’s Semi-Momentless Theory

2018 ◽  
Author(s):  
Igor Orynyak ◽  
Yaroslav Dubyk
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Middle Surface ◽  
Axial Direction ◽  
Mode Shapes ◽  
Natural Mode ◽  
Transverse Deflection ◽  
Approximate Formulas

Simple approximate formulas for the natural frequencies of circular cylindrical shells are presented for modes in which transverse deflection dominates. Based on the Donnell-Mushtari thin shell theory the equations of motion of the circular cylindrical shell are introduced, using Vlasov assumptions and Fourier series for the circumferential direction, an exact solution in the axial direction is obtained. To improve the results assumptions of Vlasov’s semimomentless theory are enhanced, i.e. we have used only the hypothesis of middle surface inextensibility to obtain a solution in axial direction. Nonlinear characteristic equations and natural mode shapes, are derived for all type of boundary conditions. Good agreement with experimental data and FEM is shown and advantage over the existing formulas for a variety of boundary conditions is presented.

scholarly journals Confinement of Vibrations in Variable-Geometry Nonlinear Flexible Beam

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
W. Gafsi ◽  
F. Najar ◽  
S. Choura ◽  
S. El-Borgi
Keyword(s):  
Passive Control ◽  
Inverse Eigenvalue Problem ◽  
Beam Dynamics ◽  
Mode Shapes ◽  
Physical Parameters ◽  
Geometrical Parameters ◽  
Flexible Beam ◽  
Beam Model ◽  
Nonlinear Frequency ◽  
Nonlinear Beam

In this paper, we propose a novel strategy for controlling a flexible nonlinear beam with the confinement of vibrations. We focus principally on design issues related to the passive control of the beam by proper selection of its geometrical and physical parameters. Due to large deflections within the regions where the vibrations are to be confined, we admit a nonlinear model that describes with precision the beam dynamics. In order to design a set of physical and geometrical parameters of the beam, we first formulate an inverse eigenvalue problem. To this end, we linearize the beam model and determine the linearly assumed modes that guarantee vibration confinement in selected spatial zones and satisfy the boundary conditions of the beam to be controlled. The approximation of the physical and geometrical parameters is based on the orthogonality of the assumed linear mode shapes. To validate the strategy, we input the resulting parameters into the nonlinear integral-partial differential equation that describes the beam dynamics. The nonlinear frequency response curves of the beam are approximated using the differential quadrature method and the finite difference method. We confirm that using the linear model, the strategy of vibration confinement remains valid for the nonlinear beam.

scholarly journals Vibration Characteristics of Piezoelectric Microbeams Based on the Modified Couple Stress Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
R. Ansari ◽  
M. A. Ashrafi ◽  
S. Hosseinzadeh
Keyword(s):  
Boundary Conditions ◽  
Couple Stress ◽  
Natural Frequencies ◽  
Couple Stress Theory ◽  
Equations Of Motion ◽  
Modified Couple Stress Theory ◽  
Beam Model ◽  
Governing Equations ◽  
Stress Theory ◽  
Modified Couple Stress

The vibration behavior of piezoelectric microbeams is studied on the basis of the modified couple stress theory. The governing equations of motion and boundary conditions for the Euler-Bernoulli and Timoshenko beam models are derived using Hamilton’s principle. By the exact solution of the governing equations, an expression for natural frequencies of microbeams with simply supported boundary conditions is obtained. Numerical results for both beam models are presented and the effects of piezoelectricity and length scale parameter are illustrated. It is found that the influences of piezoelectricity and size effects are more prominent when the length of microbeams decreases. A comparison between two beam models also reveals that the Euler-Bernoulli beam model tends to overestimate the natural frequencies of microbeams as compared to its Timoshenko counterpart.

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