scholarly journals Theoretical Analysis of the Shaft

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Jajneswar Nanda ◽  
D. R. Parhi
Keyword(s):  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Angular Position ◽  
Longitudinal Direction ◽  
Mode Shapes ◽  
Transverse Cracks ◽  
Compliance Matrix ◽  
Inference System ◽  
Crack Location ◽  
Adaptive Neurofuzzy Inference System

This paper represents the dynamic response of a steel shaft which is fixed at both ends by bearing. The shaft is subjected to both axial and bending loads. The behavior of the shaft in the presence of two transverse cracks subjected to the same angular position along longitudinal direction is observed by taking basic parameters such as nondimensional depth (bi/D), nondimensional length (Li/L), and three relative natural frequencies with their relative mode shapes. The compliance matrix is calculated from the stress intensity factor for two degrees of freedom. The dynamic nature of the cracked shaft at two cracked locations at a different depth is observed. The compliance matrix is a function of crack parameters such as depth and location of crack from any one of the bearings. The three relative natural frequencies and their mode shapes at a different location and depth obtained analytical and experimental method. Multiple adaptive neurofuzzy inference system (MANFIS) methodology (an inverse technique) is used for locating the cracks at any depth and location. The input of the MANFIS is provided with the first three natural frequencies and the first three mode shapes obtained from analytical method. The predicted result of the MANFIS (relative crack location and depth) has been validated using the results from the developed experimental setup.

Analysis of Detecting Multi Crack with Location and Size in Simply Supported Shaft Using ANFIS

2014 ◽  
Vol 541-542 ◽  
pp. 649-657
Author(s):  
J. Nanda ◽  
L.D. Das ◽  
H.C. Das ◽  
A. Biswal ◽  
A. Tripathy
Keyword(s):  
Complex Structure ◽  
Angular Position ◽  
Theoretical Data ◽  
Strain Energy Release ◽  
Mode Shapes ◽  
Multiple Cracks ◽  
Inference System ◽  
Acceptable Error ◽  
Crack Location ◽  
Bending Load

A methodology to detect multi crack, with crack location and size due to transverse loading is presented in this paper taking the advantage of Adaptive neuro-fuzzy inference system (ANFIS) using modal analysis of cracked shaft. A simple supported shaft is subjected with axial and bending load for same angular position along with its longitudinal direction. The natural frequency of the shaft with its various crack positions and size are accurately determined analytically using strain energy release rate and stress intensity factor. The vibration parameters such as first three relative natural frequencies with their mode shapes at different locations and size are supplied to ANFIS to optimize the results. The crack location and size predicted from ANFIS model are verified with the theoretical data with acceptable error. The present method used in this paper is suitable and can be easily extended to complex structure with different orientation of multiple cracks.

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.

Modeling and Free Vibration of Flex Circuits in Hard Disk Drives

2003 ◽  
Author(s):  
Jonathan Wickert
Keyword(s):  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Equilibrium Shape ◽  
Local Equilibrium ◽  
Hard Disk Drives ◽  
Laser Interferometry ◽  
Hard Disk ◽  
System Level ◽  
Mode Shapes ◽  
Free Length

A flex circuit connects the stationary electronic components in a hard disk drive to the rotating arm that carries the read/write heads and positions them above data tracks on the disk. Flex circuits are conventionally formed as a laminate of polyimide substrate, adhesive, and copper conductors. Deformation of a flex circuit is discussed in the context of the following stages: the initial unstressed shape, configurations in which stresses set and relax in response to elevated temperature, equilibrium, and small amplitude vibration. The model involves displacements of the flex circuit in the directions tangent and normal to the local equilibrium shape, and those motions couple with the arm’s dynamics. Nonlinearity associated with finite curvature, partial elastic springback, and the arm’s geometry and inertia properties are incorporated within the vibration model to predict system-level natural frequencies, mode shapes, and coupling factors between the circuit and the arm. Laboratory measurements using noncontact laser interferometry validate the model with respect to the circuit’s shape, stiffness, restoring moment, and natural frequencies. The primary degrees of freedom for optimizing flex circuit design are the thicknesses of the individual layers within the circuit, free length, and the locations and slopes of the circuit’s attachment points to the arm and electronics block. The model’s predictions and trends developed from a case study in free length are discussed with a view toward reducing coupling between the circuit and arm in certain vibration modes.

Free Vibration Analysis of an Inflated Toroidal Shell

2002 ◽  
Vol 124 (3) ◽  
pp. 387-396 ◽  
Author(s):  
Akhilesh K. Jha ◽  
Daniel J. Inman ◽  
Raymond H. Plaut
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Free Vibration Analysis ◽  
Longitudinal Direction ◽  
Variable Coefficients ◽  
Mode Shapes ◽  
Toroidal Shell

Free vibration analysis of a free inflated torus of circular cross-section is presented. The shell theory of Sanders, including the effect of pressure, is used in formulating the governing equations. These partial differential equations are reduced to ordinary differential equations with variable coefficients using complete waves in the form of trigonometric functions in the longitudinal direction. The assumed mode shapes are divided into symmetric and antisymmetric groups, each given by a Fourier series in the meridional coordinate. The solutions (natural frequencies and mode shapes) are obtained using Galerkin’s method and verified with published results. The natural frequencies are also obtained for a circular cylinder with shear diaphragm boundary condition as a special case of the toroidal shell. Finally, the effects of aspect ratio, pressure, and thickness on the natural frequencies of the inflated torus are studied.

Natural Vibrations of a Clamped-Clamped Arch With an Open Transverse Crack

1997 ◽  
Vol 119 (2) ◽  
pp. 145-151 ◽  
Author(s):  
M. Krawczuk ◽  
W. Ostachowicz
Keyword(s):  
Finite Element ◽  
Stiffness Matrix ◽  
Edge Crack ◽  
Natural Frequencies ◽  
Element Model ◽  
Mode Shapes ◽  
Curved Beam ◽  
Natural Vibrations ◽  
Crack Location ◽  
Cracked Arch

The paper presents a finite element model of the arch with a transverse, one-edge crack. A part of the cracked arch is modelled by a curved beam finite element with the crack. Parts of the arch without the crack are modelled by noncracked curved beam finite elements. The crack occurring in the arch is nonpropagating and open. It is assumed that the crack changes only the stiffness of the arch, whereas the mass is unchanged. The method of the formation of the stiffness matrix of a curved beam finite element with the crack is presented. The effects of the crack location and its length on the changes of the in-plane natural frequencies and mode shapes of the clamped-clamped arch are studied.

Effect of a Circumferential Arc Crack on the Vibration Characteristics of a Flexible Spinning Disk

2007 ◽  
Author(s):  
Nikhit N. Nair ◽  
Hamid N. Hashemi ◽  
Grant M. Warner ◽  
M. Olia
Keyword(s):  
Natural Frequencies ◽  
Crack Opening ◽  
Equations Of Motion ◽  
Bending Moment ◽  
Rotating Disk ◽  
Vibration Characteristics ◽  
Coupled Systems ◽  
Mode Shapes ◽  
Crack Location ◽  
Inner Region

The vibration characteristics of a circumferentially cracked rotating disk are investigated. The disk is assumed to be axisymmetric, flexible and clamped at the center. The crack increases the local flexibility of the disk at the crack location and is modeled as linear and torsional springs, connecting the two segments of the disk. The spring constants are evaluated by considering crack opening displacements due to bending moment and shear force at the crack location. The equations of motion of two segments of the disk, for disk operating in vacuum as well as subjected to shear fluid flow are developed. Using the Finite Difference Technique, the coupled systems of equations are solved and the natural frequencies and mode shapes are obtained. The mode shapes are seen to be comparatively flattened in the inner region of the crack and heightened towards the periphery of the disk. Shear fluid loading reduces the natural frequencies and results in a quicker onset of instability. It is observed that the effect of the crack on the vibration characteristics of the disk is mainly a function of the crack location.

Effect of Multiple Location Defects on the Dynamics of Draft Gear used in Freight Railway Wagon

2015 ◽  
Vol 7 (3) ◽  
Author(s):  
Sachin S. Harak ◽  
Satish C. Sharma ◽  
Sanjay Shukla ◽  
Parinay Gupta ◽  
Sanjay Kumar ◽  
...  
Keyword(s):  
Natural Frequency ◽  
Base Plate ◽  
Longitudinal Direction ◽  
Mode Shapes ◽  
Damage Scenarios ◽  
Railway Wagon ◽  
Crack Location ◽  
Draft Gear ◽  
Element Approach ◽  
Crack Defect

The present work investigates the effect of crack location on the modal frequency of draft gear used in autocouplers of freight railway wagon for various orientations. First seven mode shapes of a healthy draft gear have been determined using finite element approach. Defect of semi-elliptical shape is modelled in the lateral as well as longitudinal direction of the draft pad which is a component of draft gear. Various damage scenarios have been simulated by considering multiple locations of the crack in the draft gear for different orientations. Effect of crack orientation and defective pads location on the natural frequency of draft gear is analysed. It is seen that for single defective pad as well as multiple defective pads, the natural frequency of draft gear is dependent on the dynamics of draft pad. It is also observed that defect in consecutive pads causes more change in frequency as compared to single defective pad. As far as the location of defective pad is concerned, it is seen that the draft gear frequency is more sensitive to defective pads located either near the housing base plate or top follower. This study provides a tool to diagnose crack defect in draft gear based on vibration characteristics.

Detection of Cracks in Stationary Rotors via the Modal Frequency Changes Induced by a Roving Disc

2014 ◽  
Author(s):  
Z. N. Haji ◽  
S. O. Oyadiji
Keyword(s):  
Natural Frequencies ◽  
Crack Depth ◽  
Mode Shapes ◽  
Crack Identification ◽  
Open Crack ◽  
Suggested Approach ◽  
Rotor Systems ◽  
Crack Location ◽  
High Crack ◽  
Frequency Changes

In this study, a crack identification approach based on a finite element cracked model is presented to identify the location and depth ratios of a crack in rotor systems. A Bernoulli-Euler rotor carrying an auxiliary roving disc has been used to model the cracked rotor, in which the effect of a transverse open crack is modelled as a time-varying stiffness matrix. In order to predict the crack location in the rotor-disc-bearing system, the suggested approach utilises the variation of the normalized natural frequency curves versus the non-dimensional location of a roving disc which traverses along the rotor span. The merit of the suggested approach is to identify the location and sizes of a crack in a rotor by determining only the natural frequencies of the stationary rotor system. The first four natural frequencies are employed for the identification and localisation of a crack in the stationary rotor. Furthermore, this approach is not only efficient and practicable for high crack depth ratios but also for small crack depth ratios and for a crack close to or at the node of mode shapes, where natural frequencies are unaffected.

Coupled Transverse and Axial Vibrations of Cracked Cantilever Beams With Roving Mass and Rotary Inertia

2010 ◽  
Author(s):  
Hurang Hu ◽  
Akindeji Ojetola ◽  
Hamid Hamidzadeh
Keyword(s):  
Cantilever Beam ◽  
Natural Frequencies ◽  
Coupling Effect ◽  
Rectangular Cross Section ◽  
Crack Depth ◽  
Rotary Inertia ◽  
Mode Shapes ◽  
Axial Deformation ◽  
Crack Location ◽  
Axial Vibrations

The vibration behavior of a cracked cantilever beam with a stationary roving mass and rotary inertia is investigated. The beam is modeled as an Euler-Bernoulli beam with rectangular cross section. The transverse deformation and axial deformation of the cracked beam are coupled through a stiffness matrix which is determined based on fracture mechanics principles. The analytical solutions are obtained for the natural frequencies and mode shapes of a cracked cantilever beam with a roving mass and rotary inertia. The effects of the location and depth of the crack, the location and the weight of the roving mass and rotary inertia on the natural frequencies and mode shapes of the beam are investigated. The numerical results show that the coupling between the transverse and axial vibrations for moderate values of crack depth and/or roving mass and rotary inertia is weak. Increasing the crack depth and the mass and rotary inertia will increase the coupling effect. Detection of the crack location using natural frequencies and mode shapes as parameters is also discussed.

scholarly journals An analytical formulation to evaluate natural frequencies and mode shapes of high-rise buildings

2021 ◽  
Vol 8 (1) ◽  
pp. 307-318
Author(s):  
Giuseppe Nitti ◽  
Giuseppe Lacidogna ◽  
Alberto Carpinteri
Keyword(s):  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Shear Walls ◽  
Vertical Axis ◽  
Mode Shapes ◽  
Analytical Formulation ◽  
High Rise ◽  
Open Section ◽  
Stiffness Matrices ◽  
Thin Walled

Abstract In this paper, an original analytical formulation to evaluate the natural frequencies and mode shapes of high-rise buildings is proposed. The methodology is intended to be used by engineers in the preliminary design phases as it allows the evaluation of the dynamic response of high-rise buildings consisting of thin-walled closed- or open-section shear walls, frames, framed tubes, and dia-grid systems. If thin-walled open-section shear walls are present, the stiffness matrix of the element is evaluated considering Vlasov’s theory. Using the procedure called General Algorithm, which allows to assemble the stiffness matrices of the individual vertical bracing elements, it is possible to model the structure as a single equivalent cantilever beam. Furthermore, the degrees of freedom of the structural system are reduced to only three per floor: two translations in the x and y directions and a rigid rotation of the floor around the vertical axis of the building. This results in a drastic reduction in calculation times compared to those necessary to carry out the same analysis using commercial software that implements Finite Element models. The potential of the proposed method is confirmed by a numerical example, which demonstrates the benefits of this procedure.

Sign in / Sign up

Export Citation Format

Share Document