The effect of crack geometry on mode shapes of a cracked cantilever beam

2020 ◽  
pp. 1-12
Author(s):  
V. Khalkar ◽  
K. Logesh
Keyword(s):  
Cantilever Beam ◽  
Mode Shapes ◽  
Crack Geometry

An Approach to Approximate the Full Strain Field of Turbofan Blades During Operation

2018 ◽  
Author(s):  
Gen Fu ◽  
Alexandrina Untaroiu ◽  
Walter O’Brien
Keyword(s):  
Numerical Model ◽  
Cantilever Beam ◽  
Strain Field ◽  
Measured Data ◽  
Mode Shapes ◽  
Beam Model ◽  
Reference Solution ◽  
Full Field ◽  
Field Response ◽  
Critical Locations

The measurement of the aeromechanical response of the fan blades is important to quantifying their integrity. The accurate knowledge of the response at critical locations of the structure is crucial when assessing the structural condition. A reliable and low cost measuring technique is necessary. Currently, sensors can only provide the measured data at several discrete points. A significant number of sensors may be required to fully characterize the aeromechanical response of the blades. However, the amount of instrumentation that can be placed on the structure is limited due to data acquisition system limitations, instrumentation accessibility, and the effect of the instrumentation on the measured response. From a practical stand point, it is not possible to place sensors at all the critical locations for different excitations. Therefore, development of an approach that derives the full strain field response based on a limited set of measured data is required. In this study, the traditional model reduction method is used to expand the full strain field response of the structure by using a set of discrete measured data. Two computational models are developed and used to verify the expansion approach. The solution of the numerical model is chosen as the reference solution. In addition, the numerical model also provides the mode shapes of the structure. In the expansion approach, this information is used to develop the algorithm. First, a cantilever beam model is created. The influences of the sensor location, number of sensors and the number of modes included are analyzed using this cantilever beam model. The expanded full field response data is compared with the reference solution to evaluate the expansion procedure. The rotor 67 blade model is then used to test the expansion method. The results show that the expanded full field data is in good agreement with the calculated data. The expansion algorithm can be used for the full field strain by using the limited sets of strain data.

Reliable Mode Shape Expansion Method for Error Localisation and Model Updating

1997 ◽  
Author(s):  
Philippe Collignon ◽  
Jean-Claude Golinval
Keyword(s):  
Cantilever Beam ◽  
Mode Shape ◽  
Constitutive Equations ◽  
Structural Model ◽  
Model Updating ◽  
Failure Detection ◽  
Expansion Method ◽  
Relative Performance ◽  
Mode Shapes ◽  
Error Localization

Abstract Failure detection and model updating using structural model are based on the comparison of an appropriate indicator of the discrepancy between experimental and analytical results. The reliability of the expansion of measured mode shapes is very important for the process of error localization and model updating. Two mode shape expansion techniques are examined in this paper : the well known dynamic expansion (DE) method and a method based on the minimisation of errors on constitutive equations (MECE). A new expansion method based on some improvements of the previous techniques is proposed to obtain results that are more reliable for error localisation and for model updating. The relative performance of the different expansion methods is demonstrated on the example of a cantilever beam.

Derivatives and Parameter Designs of Arbitrary Cross-Section Inhomogenous Beams’ Modes

2013 ◽  
Author(s):  
J. W. Xing ◽  
G. T. Zheng
Keyword(s):  
Cantilever Beam ◽  
Cross Section ◽  
Analytical Solutions ◽  
Structural Parameters ◽  
Arbitrary Cross Section ◽  
Mode Shapes ◽  
Design Parameters ◽  
Structural Parameter ◽  
Highly Sensitive ◽  
Sensitivity Analysis Method

As highly sensitive to structural parameter variations, it is necessary to study relations between derivatives of displacement modes and structural design parameters. This paper proposes an integral technique for obtaining the analytical solutions of slope and curvature modes of arbitrary cross-section inhomogeneous cantilever beam. The method is validated by comparing the computation results of modal frequencies and shapes with both numerical and analytical solutions. Furthermore, based on the presented method, we have established explicit expressions for the structural parameters sensitivity of the slope/curvature mode shapes. An example of parameter design is also presented for a cantilever beam with the proposed sensitivity analysis method.

Two degree of freedom vibration based electromagnetic energy harvester for bridge health monitoring system

2020 ◽  
pp. 1045389X2095945
Author(s):  
Muhammad Masood Ahmad ◽  
Farid Ullah Khan
Keyword(s):  
Cantilever Beam ◽  
Energy Harvester ◽  
Low Frequency ◽  
Open Circuit ◽  
Electromagnetic Energy ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Fixed End ◽  
Two Degree Of Freedom ◽  
Maximum Voltage

This paper presents an electromagnetic energy harvester to extract low frequency and low acceleration vibration energy available in a bridge environment. The developed harvester is a multi-mode oscillator with dual electromagnetic transduction mechanisms. The harvester consists of two cantilever beams. The first cantilever beam is split into two equal pieces along its length and the second beam placed in between them coming back to the fixed end and attached at outer end to the first beam. This way instead of a long conventional cantilever beam a compact harvester is fabricated. Two magnets as proof masses are attached to each free end of the beam making it a two degree of freedom system (2-DOF). The magnets are positioned to oscillate inside hand wound coils during operation. An analytical model was developed and COMSOL multiphysics was used to simulate the mode shapes of the harvester. The mathematical model was simulated for open circuit voltage in MATLAB and showed closely matching results with the experimental values. The harvester is characterized in lab for its performance under sinusoidal vibrations at low frequency (3 Hz–15 Hz) and low acceleration (0.01–0.09 g) levels. The 2-DOF harvester has two resonant frequencies of 4.4 Hz and 5.5 Hz and a volume of 333 cm3. It produces maximum voltage of 0.6 V at first resonance on coil-1 and maximum voltage of 1.2 V on coil-2 at second resonance at 0.09 g. At acceleration of 0.09 g the harvester produced 2.51 mW at first resonant frequency and 10.7 mW at second resonance. Moreover, the AC output voltage of the harvester is rectified to DC voltage by a three-stage Cockcroft-Walton multiplier type circuit. The DC power output at 0.05 g was 0.939 mW at first resonance and 0.956 mW at second resonance while maximum voltages of 5.4 V on coil-1 and 4 V on coil-2 were produced.

scholarly journals Vibration-Based Damage Detection in Beams by Cooperative Coevolutionary Genetic Algorithm

2014 ◽  
Vol 6 ◽  
pp. 624949 ◽  
Author(s):  
Kittipong Boonlong
Keyword(s):  
Genetic Algorithm ◽  
Damage Detection ◽  
Cantilever Beam ◽  
Cross Section ◽  
Random Noise ◽  
Vibration Characteristics ◽  
Mode Shapes ◽  
Test Problems ◽  
Uniform Cross Section ◽  
Coevolutionary Genetic Algorithm

Vibration-based damage detection, a nondestructive method, is based on the fact that vibration characteristics such as natural frequencies and mode shapes of structures are changed when the damage happens. This paper presents cooperative coevolutionary genetic algorithm (CCGA), which is capable for an optimization problem with a large number of decision variables, as the optimizer for the vibration-based damage detection in beams. In the CCGA, a minimized objective function is a numerical indicator of differences between vibration characteristics of the actual damage and those of the anticipated damage. The damage detection in a uniform cross-section cantilever beam, a uniform strength cantilever beam, and a uniform cross-section simply supported beam is used as the test problems. Random noise in the vibration characteristics is also considered in the damage detection. In the simulation analysis, the CCGA provides the superior solutions to those that use standard genetic algorithms presented in previous works, although it uses less numbers of the generated solutions in solution search. The simulation results reveal that the CCGA can efficiently identify the occurred damage in beams for all test problems including the damage detection in a beam with a large number of divided elements such as 300 elements.

A New Reactive Hybrid Membership Function in Fuzzy Approach for Identification of Inclined Edge Crack in Cantilever Beam Using Vibration Signatures

2014 ◽  
Vol 592-594 ◽  
pp. 1996-2000 ◽  
Author(s):  
K.B. Ranjan ◽  
Sasmita Sahu ◽  
R. Parhi Dayal
Keyword(s):  
Cantilever Beam ◽  
Edge Crack ◽  
Fuzzy Controller ◽  
Experimental Results ◽  
Mode Shapes ◽  
Crack Identification ◽  
Hybrid Technique ◽  
Membership Functions ◽  
Inclined Edge Crack ◽  
Vibration Signatures

In this paper, the crack identification using smart technique (by several hybrid membership functions in a fuzzy controller) has been developed for inverse analysis of the vibration signatures (like modal frequencies, mode shapes) and crack parameters (like crack depth, crack location and crack inclination) of an inclined edge crack cantilever beam. The modal frequencies are obtained from finite element (using ANSYS) and experimental analysis which are used as inputs to the hybrid fuzzy controller. The hybrid fuzzy system is designed by taking different types of membership functions (MF) to determine the crack parameters. The calculated first three modal frequencies are used to create number of fuzzy rules with the three output crack parameters. Finally, the proposed hybrid technique is validated by comparing the results obtained from trapezoidal and Gaussian fuzzy controllers, FEA and experimental results. The outcomes obtained from hybrid fuzzy controller are in good agreement with experimental results. Nomenclature

On Utilizing Approximate Modal Shapes to Estimate the Effective Nonlinearity of a Structural Mode

2012 ◽  
Author(s):  
Clodoaldo J. Silva ◽  
Mohammed F. Daqaq
Keyword(s):  
Cantilever Beam ◽  
Multiple Scales ◽  
Nonlinear Partial Differential Equation ◽  
Least Square ◽  
Mode Shapes ◽  
Accurate Estimation ◽  
Approximate Analytical Expression ◽  
Linear Mode ◽  
Linear Eigenvalue Problem ◽  
Structural Mode

This effort investigates the accuracy of estimating the effective nonlinearity of a given vibration mode using approximate modal shapes. As an example, the problem of approximating the modal effective nonlinearities of a linearly-tapered cantilever beam (along the width) is considered. This example was intentionally selected because the linear eigenvalue problem cannot be solved analytically for the exact eigenfrequencies and actual linear mode shapes of the structure, which permits investigating the influence of approximating the mode shapes on the effective nonlinearity. The nonlinear partial differential equation governing the beam’s motion is first discretized into an infinite set of nonlinear ordinary differential equations. The method of multiple scales is then utilized to obtain an approximate analytical expression for the effective nonlinearity which depends on the assumed mode shapes used in the series discretization. To approximate the mode shapes, three methods were utilized: i) a crude approach which directly utilizes the linear modes of a regular (untapered) cantilever beam to estimate the effective nonlinearity, ii) a finite element approach wherein the structural modes are obtained in ANSYS, then fitted into orthonormal polynomial curves while minimizing the least square error in calculating the eigenfrequencies, and iii) a Rayleigh-Ritz approach which utilizes a set of orthonormal trial basis functions to construct the structural mode shapes as a linear combination of the trial functions used. A comparison among the three methods for eight different taperings reveals that, while the modal frequencies are well-approximated yielding less than 2% deviation among the three methods, there is a huge discrepancy in approximating the nonlinear coefficients including the effective nonlinearity. This leads to the conclusion that convergence of the eigenfrequencies is not sufficient for accurate estimation of the nonlinear parameters.

scholarly journals Influence of Crack on Modal Parameters of Cantilever Beam Using Experimental Modal Analysis

2018 ◽  
Vol 1 (1) ◽  
pp. 16-23 ◽  
Author(s):  
Siva Sankara Babu Chinka ◽  
Balakrishna Adavi ◽  
Srinivasa Rao Putti
Keyword(s):  
Numerical Analysis ◽  
Modal Analysis ◽  
Natural Frequency ◽  
Cantilever Beam ◽  
Damping Ratio ◽  
Crack Depth ◽  
Experimental Modal Analysis ◽  
Mode Shapes ◽  
Modal Parameters ◽  
Crack Location

In this paper, the dynamic behavior of a cantilever beam without and with crack is observed. An elastic Aluminum cantilever beams having surface crack at various crack positions are considered to analyze dynamically. Crack depth, crack length and crack location are the foremost parameters for describing the health condition of beam in terms of modal parameters such as natural frequency, mode shape and damping ratio. It is crucial to study the influence of crack depth and crack location on modal parameters of the beam for the decent performance and its safety. Crack or damage of structure causes a reduction in stiffness, an intrinsic reduction in resonant frequencies, variation of damping ratios and mode shapes. The broad examination of cantilever beam without crack and with crack has been done using Numerical analysis (Ansys18.0) and experimental modal analysis. To observe the exact higher modes of beam, discretize the beam into small elements. An experimental set up was established for cantilever beam having crack and it was excited by an impact hammer and finally the response was obtained using PCB accelerometer with the help sound and vibration toolkit of NI Lab-view. After obtaining the Frequency response functions (FRFs), the natural frequencies of beam are estimated using peak search method. The effectiveness of experimental modal analysis in terms of natural frequency is validated with numerical analysis results. This paper contains the study of free vibration analysis under the influence of crack at different points along the length of the beam.

The Udwadia–Kalaba Trajectory Control Applied to a Cantilever Beam—Experimental Results

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Raphael Pereira Spada ◽  
Rodrigo Nicoletti
Keyword(s):  
Cantilever Beam ◽  
Tracking Control ◽  
Equations Of Motion ◽  
Constrained Systems ◽  
Mode Shapes ◽  
Laser Vibrometer ◽  
Successful Implementation ◽  
Out Of Plane ◽  
Control Forces

The Udwadia–Kalaba methodology is a possible way of explicitly obtaining the equations of motion of constrained systems. From these equations of motion, one can estimate the necessary forces in the constraint to keep the system in a given motion. Hence, the Udwadia–Kalaba methodology can also apply to active tracking control of subsystems or the control of points of a structure. In this work, one investigates experimentally the benefits and drawbacks of such control strategy by applying it to the control of out-of-plane vibrations of a cantilever beam. The beam is excited by a shaker mounted near the clamped end of the beam. A second shaker applies the control forces in the free end of the beam, where an accelerometer is used for feedback. The vibration behavior of the beam under excitation/control is measured by a laser vibrometer. Results show that the methodology changes the dynamic behavior of the structure by changing its boundary conditions at the point of control, thus shifting natural frequencies and mode shapes. Results also show that the successful implementation of the method experimentally is sensitive to the quality of modeling of the structure.

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