An Open Crack Model for Vibration Simulation of Beam Structures by Transfer Matrix Method

2007 ◽  
Vol 348-349 ◽  
pp. 893-896
Author(s):  
Ke Ming Wang ◽  
Song Xiang
Keyword(s):  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Crack Detection ◽  
Vibration Monitoring ◽  
Crack Model ◽  
Open Crack ◽  
Element Analysis ◽  
Beam Structures ◽  
Stiffness Reduction

Crack detection of critical beam structures such as bridges and aircraft wings by vibration monitoring is based on understanding how a crack affects the vibration characteristics of a beam structure. Transfer matrix method is a convenient, effective, and hence widely used approach to beam vibration analysis, but a crack in the beam makes this method ineffective. This paper proposes an open crack model that simulates the local stiffness reduction effect of a transverse crack by a rectangular slot to make the transfer matrix method able to analyze vibrations of a cracked beam. The depth of the slot is identical to the depth of the crack, and the equivalent width of the slot is obtained by comparison of stiffness reductions of finite element analysis results and the counterpart transfer matrix method results. Different dimensions of rectangular beams, different crack positions and loading conditions are considered and statistic method is used to improve the generality and accuracy of the model. A calculation example of a cracked cantilever beam is given and the validity of the proposed model is verified with available results of existing models.

A Finite-Width Crack Model for Rotor Dynamic Analysis by Transfer Matrix Method

2006 ◽  
Vol 324-325 ◽  
pp. 1003-1006
Author(s):  
Ke Ming Wang ◽  
Song Xiang
Keyword(s):  
Dynamic Analysis ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Transverse Crack ◽  
Rotor System ◽  
Crack Model ◽  
Finite Width ◽  
Stiffness Reduction ◽  
Rotor Dynamic

Understanding dynamic behavior of a rotor system with a transverse crack is of great significance for operation reliability of rotating machinery. The transfer matrix method is widely used for rotor dynamic analysis, but it encounters difficulties modeling a crack. This study proposes a finite-width crack model that simulates the local stiffness reduction effect of a gaping crack in transfer matrix method. This model is obtained by comparing finite element analysis results of a shaft with a zero-width crack and its counterpart transfer matrix calculation results with a trial equivalent finite-width slot. Different shaft geometry and loading modes are considered to improve the generality of the model. An application example is given that uses the proposed model to calculate critical speeds of a multi-disk rotor system with a transverse crack at different positions.

Crack Detection in a Rotor Dynamic System by Vibration Monitoring: Analysis and Experimental Results

2012 ◽  
Author(s):  
Philip Varney ◽  
Itzhak Green
Keyword(s):  
Transfer Matrix ◽  
Crack Detection ◽  
Early Stage ◽  
Weight Ratio ◽  
Fatigue Cracking ◽  
Vibration Monitoring ◽  
Crack Model ◽  
Test Rig ◽  
Response Component ◽  
Harmonic Response

As the power to weight ratio demand on rotordynamic systems increases, susceptibility to transverse fatigue cracking of the shaft increases as well. The ability to detect cracks in an early stage of progression is imperative for minimizing off-line repair time. A vibration monitoring system proposed prior is developed herein, employing the 2X harmonic response component of the rotor tilt as a signature indicating a transverse shaft crack. To effectively capture the 2X response, the crack model must include the local nature of the crack, the depth of the crack, and the stiffness asymmetry inducing the gravity-forced 2X harmonic response. The transfer matrix technique is well-suited to incorporate these crack attributes due to its modular nature. Two transfer matrix models are proposed to predict the 2X harmonic response. The first model applies local crack flexibility coefficients determined using the strain energy release rate, while the second incorporates the crack as a rectangular notch to emulate a manufactured crack used in the experiments. Analytic results are then compared to experimental measurement of the rotor tilt gleaned from an overhung rotor test rig originally designed to test seal face dynamics. The test rig is discussed, and experimental 2X harmonic amplitudes of the rotor tilt are provided for shafts containing manufactured cracks of depths between zero and 40 percent.

scholarly journals THE TRANSFER MATRIX METHOD FOR MODAL ANALYSIS OF CRACKED MULTISTEP BEAM

2017 ◽  
Vol 55 (5) ◽  
pp. 598
Author(s):  
Nguyen Tien Khiem ◽  
Vu Thi An Ninh ◽  
Tran Thanh Hai
Keyword(s):  
Modal Analysis ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Arbitrary Number ◽  
Matrix Method ◽  
Crack Detection ◽  
Natural Frequencies ◽  
Modal Testing ◽  
Testing Technique

The present study addresses the modal analysis of multistep beam with arbitrary number of cracks by using the transfer matrix method and modal testing technique. First, there is conducted general solution of free vibration problem for uniform beam element with arbitrary number of cracks that allows one to simplify the transfer matrix for cracked multistep beam. The transferring beam state needs to undertake only at the steps of beam but not through crack positions. Such simplified the transfer matrix method makes straightforward to investigate effect of cracks mutually with cross-section step in beam on natural frequencies. It is revealed that step-down and step-up in beam could modify notably sensitivity of natural frequencies to crack so that the analysis provides useful indication for crack detection in multistep beam. The proposed theory was validated by an experimental case study

Eigensolution of piezoelectric energy harvesters with geometric discontinuities: Analytical modeling and validation

2012 ◽  
Vol 24 (6) ◽  
pp. 729-744 ◽  
Author(s):  
Adam M Wickenheiser
Keyword(s):  
Finite Element ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Mode Shapes ◽  
Transfer Matrices ◽  
Coupling Effects ◽  
Beam Structures ◽  
Energy Harvesters ◽  
Piezoelectric Energy

Although cantilevered beams are the most prolific design for resonant piezoelectric energy harvesters, other topologies have been studied for their compactness or conformability to their host structures’ geometry. These more complex structures have been analyzed using custom analytical models developed from the first principles or finite-element methods to compute their eigensolutions and piezoelectric coupling effects. This article discusses the use of the transfer matrix method to derive analytical solutions to beam structures with pointwise discontinuities, bends, or lumped inertias between members or at the tip. Euler–Bernoulli beam theory is used to derive transfer matrices for the uniform beam segments, and point transfer matrices are derived to handle discontinuities in the structure between beam segments. The eigensolution of the transfer matrix is shown to produce the natural frequencies and mode shapes for these structures. Subsequently, the electromechanical coupling effects are incorporated, and the base excitation problem is considered. Parametric case studies are provided for beam structures with varying piezoelectric layer coverage and angle between members. Finally, these results are compared to finite-element solutions using COMSOL, and the modeling discrepancies are discussed. Based on the favorable comparison between these two methods, the utility and accuracy of the transfer matrix method are proven.

Generalized Eigensolution to Piecewise Continuous Distributed-Parameter Models of Piezoelectric Energy Harvesters Using the Transfer Matrix Method

2011 ◽  
Author(s):  
Adam M. Wickenheiser ◽  
Timothy Reissman
Keyword(s):  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Mode Shapes ◽  
Transfer Matrices ◽  
Coupling Effects ◽  
Beam Structures ◽  
Energy Harvesters ◽  
Piezoelectric Energy ◽  
Wide Range

Many multi-beam energy harvesters appearing in the literature require custom analytical or finite-element models to compute their eigensolutions and piezoelectric coupling effects. This paper discusses the use of the transfer matrix method to derive analytical solutions to beam structures with point-wise discontinuities or with lumped inertias between members or at the tip. In this method, transfer matrices are developed for the beam’s states (deflection, slope, shear force, and bending moment) analogously to the state transition matrix of a linear system. Euler-Bernoulli beam theory is used to derive transfer matrices for the uniform beam segments, and point transfer matrices are derived to handle discontinuities in the structure. The transfer matrix method is shown to be advantageous for analyzing complex structures because the size of the transfer matrix does not grow with increasing number of components in the structure. Furthermore, the same formulation can be used for a wide range of geometries, including arbitrary combinations of beam segments — single- or multi-layered — and lumped inertias. The eigensolution of the transfer matrix is shown to produce the natural frequencies and mode shapes for these structures. Subsequently, the electromechanical coupling effects are incorporated and the base excitation problem is considered. The electromechanical equations of motion are decoupled by mode and shown to be a generalization of existing analytical models. Parametric case studies are provided for beam structures with varying piezoelectric layer coverage.

Sign in / Sign up

Export Citation Format

Share Document