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.

Free Bending Vibrations of Adhesively Bonded Orthotropic Plates With a Single Lap Joint

1996 ◽  
Vol 118 (1) ◽  
pp. 122-134 ◽  
Author(s):  
U. Yuceoglu ◽  
F. Toghi ◽  
O. Tekinalp
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Mode Shapes ◽  
Mindlin Plate ◽  
Lap Joint ◽  
Orthotropic Plates ◽  
Adhesively Bonded ◽  
Bending Vibrations ◽  
Free Bending

This study is concerned with the free bending vibrations of two rectangular, orthotropic plates connected by an adhesively bonded lap joint. The influence of shear deformation and rotatory inertia in plates are taken into account in the equations according to the Mindlin plate theory. The effects of both thickness and shear deformations in the thin adhesive layer are included in the formulation. Plates are assumed to have simply supported boundary conditions at two opposite edges. However, any boundary conditions can be prescribed at the other two edges. First, equations of motion at the overlap region are derived. Then, a Levy-type solution for displacements and stress resultants are used to formulate the problem in terms of a system of first order ordinary differential equations. A revised version of the Transfer Matrix Method together with the boundary and continuity conditions are used to obtain the frequency equation of the system. The natural frequencies and corresponding mode shapes are obtained for identical and dissimilar adherends with different boundary conditions. The effects of some parameters on the natural frequencies are studied and plotted.

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 The Effect of Various Fiber Orientations and Boundary Conditions on Natural Frequencies of Laminated Composite Beam

2018 ◽  
Vol 7 (3.11) ◽  
pp. 67 ◽  
Author(s):  
M Arif Mat Norman ◽  
M Amiruddin Zainuddin ◽  
Jamaluddin Mahmud
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Aspect Ratio ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Composite Beam ◽  
Natural Frequencies ◽  
Laminated Composite ◽  
Mode Shapes ◽  
Laminated Composite Beam

This paper investigates the free vibration characteristics of laminate composite beam for various lamination schemes and under various boundary conditions. A beam model with the aspect ratio (length to thickness) of 25 to 150 made of carbon/ epoxy laminates under free vibration were constructed using a commercially available finite element software (ANSYS). The varied parameters are the lamination schemes (cross ply, angle ply and unidirectional ply) and boundary conditions (Clamp-Free (C-F), Clamp-Clamp (C-C), Clamp-Hanger (C-H), Free-Free (F-F) and Hanger-Hanger (H-H) ). For each case, finite element simulations were performed and the natural frequencies were determined. Mode shapes were also analyzed to observe the beam’s deformation behavior. Results showed that increasing aspect ratio will decrease natural frequencies for the first seven mode shapes. In terms of lamination scheme, the unidirectional ply produced the highest frequency (34.26 Hz), followed by cross ply (34.05 Hz) and angle ply (13.60 Hz) at the aspect ratio of 25. In terms of boundary conditions, the Hanger-Hanger boundary condition produced the highest natural frequency (2272.52  Hz) at the aspect ratio of 25, while Clamped-Free boundary condition produced the lowest frequency (2.28 Hz) at the aspect ratio of 150. In general, it can be concluded that the current study is useful and has contributed significant knowledge to better understand of effect of various fiber orientations and boundary conditions on the natural frequencies of laminated composite beam. 

scholarly journals Comparison Between Calculated and Measured Free-Free Modes for a Flexible Rotor

1998 ◽  
Author(s):  
José A. Vázquez ◽  
Lloyd E. Barrett
Keyword(s):  
Natural Frequencies ◽  
Forced Response ◽  
Modal Testing ◽  
Analytical Models ◽  
Mode Shapes ◽  
Reduced Order Models ◽  
Response Functions ◽  
Flexible Rotor ◽  
Reduced Order ◽  
Response Characteristics

Many industrial machines nowadays are sold based on analysis performed on mathematical models of the rotors, bearings, substructures, and other components. The validity of the analysts therefore depends on the accuracy of the models themselves. When the rotor is available, modal testing may be used to validate the model of the rotor by comparing the calculated and measured free-free natural frequencies and mode shapes. This work presents additional tools for the verification of analytical models against experimental data. These tools use models of the rotor constructed from the measured data and the analytical model. A comparison of the first six calculated and measured free-free natural frequencies and mode shapes for a multi-mass flexible rotor is presented. The natural frequencies compare within 1.8%. The calculated and measured mode shapes were used to construct independent reduced order models of the rotor. These models were used to perform forced response and stability analyses. Forced response functions are presented comparing the forced response characteristics obtained from the two models. This provides a comparison between the measured and calculated forced response functions for the same number of modes. For the stability analysis, identical bearing models were added to both reduced order models. The eigenvalues were calculated using both models for a range of bearing stiffness and damping coefficients and were plotted for comparison.

A Study of Constrained Layer Damping Models Under Clamped Boundary Conditions

1999 ◽  
Author(s):  
Peter Y. H. Huang ◽  
Per G. Reinhall ◽  
I. Y. Shen
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Frequency Response ◽  
Basic Assumption ◽  
Beam Model ◽  
Constrained Layer Damping ◽  
Response Functions ◽  
Frequency Response Functions ◽  
Modified Model ◽  
Cantilevered Beam

Abstract The most commonly used beam model for constrained layer damping was developed by Mead and Markus in 1969. Although three displacement variables were used in the model, only two of them were independent. As a result, boundary conditions that are allowed in the Mead-Markus formulation may sometimes be limited. For example, a simple lab setup often consists of a cantilevered base beam with free-free constraining layer. In this case, the axial displacements of the beam and the constrained layer are independent at the cantilevered end. This boundary condition violates the basic assumption of the Mead-Markus model and cannot be described under the Mead-Markus formulation. In this paper, we investigate a modified model that is able to incorporate such boundary conditions by using three independent displacement variables. The modified model is demonstrated on a cantilevered beam with a free-free constrained layer treatment. The frequency response functions were obtained both experimentally and analytically. Our results show that the modified model is able to accurately predict vibration response. An investigation into the frequency response functions of the Mead-Markus model under similar boundary conditions is also reported.

Analysis of Structure Dynamic Characteristics of a Metal Plate

2012 ◽  
Vol 268-270 ◽  
pp. 1075-1079
Author(s):  
Chen Zhang ◽  
Zhi Gang Yang ◽  
Yin Zhi He
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Modal Analysis ◽  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Modern Method ◽  
Metal Plate ◽  
Mode Shapes ◽  
Structure Dynamic ◽  
Fixed Boundary Condition

Modal analysis is a modern method to study structure dynamic characteristics. In this paper, computational modal analysis with Finite Element Method is applied to simulate an aluminum plate with the dimension of 160mm*240mm*1.5mm under different boundary conditions (Including free boundary condition and fixed boundary condition). The results of structure natural frequencies and mode shapes of this plate show obvious difference between the two boundary conditions.

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>

Vibrations of an Incompressible Linearly Elastic Plate Using Discontinuous Finite Element Basis Functions for Pressure

2019 ◽  
Vol 141 (5) ◽  
Author(s):  
Lisha Yuan ◽  
Romesh C. Batra
Keyword(s):  
Finite Element ◽  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Plate Theory ◽  
Mass Matrix ◽  
Basis Functions ◽  
Mode Shapes ◽  
Normal Strain ◽  
Simply Supported ◽  
Transverse Normal Strain

Abstract We numerically analyze, with the finite element method, free vibrations of incompressible rectangular plates under different boundary conditions with a third-order shear and normal deformable theory (TSNDT) derived by Batra. The displacements are taken as unknowns at the nodes of a 9-node quadrilateral element and the hydrostatic pressure at four interior nodes. The plate theory satisfies the incompressibility condition, and the basis functions satisfy the Babuska-Brezzi condition. Because of the singular mass matrix, Moler's QZ algorithm (also known as the generalized Schur decomposition) is used to solve the resulting eigenvalue problem. Computed results for simply supported, clamped, and clamped-free rectangular isotropic plates agree well with the corresponding analytical frequencies of simply supported plates and with those found using the commercial software, abaqus, for other edge conditions. In-plane modes of vibrations are clearly discerned from mode shapes of square plates of aspect ratio 1/8 for all three boundary conditions. The magnitude of the transverse normal strain at a point is found to equal the sum of the two axial strains implying that higher-order plate theories that assume null transverse normal strain will very likely not provide good solutions for plates made of rubberlike materials that are generally taken to be incompressible. We have also compared the presently computed through-the-thickness distributions of stresses and the hydrostatic pressure with those found using abaqus.

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.

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