Coupled Vibration of Cracked Shafts

1992 ◽  
Vol 114 (4) ◽  
pp. 461-467 ◽  
Author(s):  
C. A. Papadopoulos ◽  
A. D. Dimarogonas
Keyword(s):  
Circular Cross Section ◽  
Vibration Modes ◽  
Coupled Vibration ◽  
Frequency Range ◽  
Bending Vibrations ◽  
Flexibility Matrix ◽  
Local Flexibility ◽  
Modes Of Vibration ◽  
Small Cracks ◽  
Cracked Shafts

The coupling of vibration modes of vibration of a clamped-free circular cross-section Timoshenko beam with a transverse crack is investigated in this paper. A 6 × 6 local flexibility matrix is used to simulate the crack. The nondiagonal terms of this matrix cause coupling between the longitudinal, torsional, and bending vibrations. Coupling is apparent in all spectra obtained with a harmonic sweeping excitation throughout the frequency range. The method is very sensitive even for small cracks.

Stability of Cracked Rotors in the Coupled Vibration Mode

1988 ◽  
Vol 110 (3) ◽  
pp. 356-359 ◽  
Author(s):  
C. A. Papadopoulos ◽  
A. D. Dimarogonas
Keyword(s):  
Harmonic Functions ◽  
Vibration Mode ◽  
Coupling Effect ◽  
Equations Of Motion ◽  
Vibration Modes ◽  
Coupled Vibration ◽  
Compliance Matrix ◽  
Stiffness Coefficients ◽  
Local Flexibility

A transverse surface crack is known to add to a shaft a local flexibility due to the stress-strain singularity in the vicinity of the crack tip. This flexibility can be represented, in the general case by way of a 6 × 6 compliance matrix describing the local flexibility in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling along the directions which are indicated by the off-diagonal terms. In addition, when the shaft rotates the crack opens and closes. Then the differential equations of motion have periodically varying stiffness coefficients and the solution can be expressed as a sum of harmonic functions of time. A method for the determination of the intervals of instability of the first and of second kind is developed. The results have been presented in stability charts in the frequency vs. depth of the crack domain. The coupling effect due to the crack leads to very interesting results such as new frequencies and vibration modes.

Coupled Longitudinal and Bending Vibrations of a Cracked Shaft

1988 ◽  
Vol 110 (1) ◽  
pp. 1-8 ◽  
Author(s):  
C. A. Papadopoulos ◽  
A. D. Dimarogonas
Keyword(s):  
Longitudinal Vibration ◽  
Slenderness Ratio ◽  
Crack Depth ◽  
Crack Identification ◽  
Open Crack ◽  
Vibration Modes ◽  
Bending Vibrations ◽  
Local Flexibility ◽  
Cracked Shaft ◽  
Identification Tool

This paper describes the coupling of bending and longitudinal vibration of a stationary cracked shaft with an open crack. The crack is modeled by way of a 2×2 local flexibility matrix with coupling terms. The elements of this matrix are obtained analytically. One of the elements compares well with experimental data of other investigators. The free vibration of the shaft, and the influence of the crack on the vibrational behavior of the shaft is studied. The relation of the eigenvalues of the system and the crack depth as functions of the slenderness ratio are presented. The forced vibration of the shaft is also studied and the coupling of the vibration modes is verified analytically and experimentally. The applicability of the method as a crack identification tool is demonstrated.

scholarly journals Design and Performance Evaluation of a Single-Phase Driven Ultrasonic Motor Using Bending-Bending Vibrations

Micromachines ◽  
2021 ◽  
Vol 12 (8) ◽  
pp. 853
Author(s):  
Dongmei Xu ◽  
Wenzhong Yang ◽  
Xuhui Zhang ◽  
Simiao Yu
Keyword(s):  
Single Phase ◽  
Oblique Line ◽  
Ultrasonic Motor ◽  
Vibration Modes ◽  
Bending Vibrations ◽  
Bending Vibration ◽  
Output Performance ◽  
Resonance Frequencies ◽  
And Performance ◽  
Line Trajectory

An ultrasonic motor as a kind of smart material drive actuator has potential in robots, aerocraft, medical operations, etc. The size of the ultrasonic motor and complex circuit limits the further application of ultrasonic motors. In this paper, a single-phase driven ultrasonic motor using Bending-Bending vibrations is proposed, which has advantages in structure miniaturization and circuit simplification. Hybrid bending vibration modes were used, which were excited by only single-phase voltage. The working principle based on an oblique line trajectory is illustrated. The working bending vibration modes and resonance frequencies of the bending vibration modes were calculated by the finite element method to verify the feasibility of the proposed ultrasonic motor. Additionally, the output performance was evaluated by experiment. This paper provides a single-phase driven ultrasonic motor using Bending-Bending vibrations, which has advantages in structure miniaturization and circuit simplification.

Bending Vibrations of Variable Section Beams

1956 ◽  
Vol 23 (1) ◽  
pp. 103-108
Author(s):  
E. T. Cranch ◽  
Alfred A. Adler
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Circular Cross Section ◽  
Beam Theory ◽  
Cantilever Beams ◽  
Constant Depth ◽  
Bending Vibrations ◽  
Variable Section

Abstract Using simple beam theory, solutions are given for the vibration of beams having rectangular cross section with (a) linear depth and any power width variation, (b) quadratic depth and any power width variation, (c) cubic depth and any power width variation, and (d) constant depth and exponential width variation. Beams of elliptical and circular cross section are also investigated. Several cases of cantilever beams are given in detail. The vibration of compound beams is investigated. Several cases of free double wedges with various width variations are discussed.

scholarly journals MATHEMATICAL MODELLING OF VIBRATIONS OF NON-UNIFORM RING-SHAPED ULTRASONIC WAVEGUIDES

2021 ◽  
Vol 3 (56) ◽  
pp. 90-96
Author(s):  
Dmitry A. STEPANENKO ◽  
◽  
Ksenija A. BUNCHUK ◽  
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Central Line ◽  
Balance Method ◽  
Vibration Modes ◽  
Constant Sign ◽  
Algebraic Equations ◽  
Bending Vibrations ◽  
Singular Matrices ◽  
Uniform Ring

The article describes technique for modelling of ultrasonic vibrations amplifiers, which are implemented in the form of non-uniform ring-shaped waveguides, based on application of harmonic balance method. Bending vibrations of the waveguide are described by means of non-uniform integral and differential equations equivalent to Euler–Bernoulli equations in order to simplify calculation of amplitude-frequency characteristics of vibrations, particularly, to exclude the need of working with singular matrices. Using harmonic balance method, equations of vibrations are reduced to overdetermined non-uniform linear system of algebraic equations, which least-squares solution is determined by means of pseudo-inverse matrix. On the basis of analysis of numerical example possibility of existence of variable-sign and constant-sign vibration modes of the waveguide is shown and it is determined that for realization of amplifying function it is necessary to use waveguide at constant-sign vibration mode. The constant-sign vibration modes are combinations of bending defor-mation and extensional deformation of central line of the waveguide and they are detected due to accounting extensibility of the central line in equations of vibrations. Validity of the obtained results is confirmed by comparing them to the results of modelling by means of finite element method.

scholarly journals A Medium-Sized Nanoclusters Au38: A Numerical Finite-Difference Method with DFTB Approach

2020 ◽  
Author(s):  
K. Vishwanathan
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Vibrational Spectrum ◽  
Difference Method ◽  
Group Symmetry ◽  
Frequency Range ◽  
Bending Vibrations ◽  
Geometrical Structures ◽  
Coordination Numbers ◽  
Standard Orientation

Abstract We present a vibrational spectrum analysis ofAu38cluster, with energy(-4.95 eV/atom) having a group symmetry C1. We have carried out DFTB calculation by using the numerical finite-difference method and extracted the required force constants. Our calculations have confirmed with the experimental results, that is, a thermodynamically very stable structure cannot be crystalline, but having a high probability of amorphus. The lowest energy geometrical structures are being confirmed by comparison (of the nuclei coordination numbers) of the different axis of rotations at a standard orientation of crystal shape. Moreover, we have accurately predicted the vibrational frequency range from 1.62 to 298.53 cm−1at ∆E = 0. Significantly, the spectrum has shown 10 sets of double state degeneracy and the rest of the 88 spectrum’s are having independent single state degeneracy. Amazingly, at NVM 101,102 that gives a pair of degeneracy {238.54, 238.89} cm−1 that has occurred within the range of Mid Infrared MIR, IR-C 3330-200 cm−1. Nevertheless, our investigation has revealed that the vibrational spectrum strongly depends upon the size, shape, and structure, as well as, stretching and bending vibrations of the atoms.

scholarly journals Comparison study of CC and CH vibration frequencies and eelectronic properties for mono, Di, Tri, and tetra-rings layer of arm chair (SWCNTs)

2019 ◽  
Vol 11 (20) ◽  
pp. 85-99
Author(s):  
Huda N. Al-Ani
Keyword(s):  
Point Group ◽  
Ir Absorption ◽  
Vibration Frequencies ◽  
Comparison Study ◽  
Symmetry Point Group ◽  
Bending Vibrations ◽  
Modes Of Vibration ◽  
Program Comparison ◽  
Semi Empirical ◽  
Equilibrium Geometries

Semi-empirical methods were applied for calculating the vibration frequencies and IR absorption intensities for normal coordinates of the {mono (C56H28), di (C84H28), tri (C112H28) and tetra (C140H28)} -rings layer for (7,7) armchair single wall carbon nanotube at their equilibrium geometries which were all found to have D7d symmetry point group. Assignment of the modes of vibration (3N-6) was done depending on the pictures of their modes by applying (Gaussian 03) program. Comparison of the vibration frequencies of (mono, di, tri and tetra) rings layer which are active in IR, and inactive in Ramman spectra. For C-H stretching vibrations, the results showed that vibration frequencies value increased with increased of length nano tube (rings layer SWCNT). The results include the relation for axial bonds, which are the vertical C-C bonds (annular bonds) in the rings and for circumferential bonds which are the outer ring bonds. Also include the assignment of puckering, breathing and clock-anticlockwise bending vibrations. They allow a comparative view of the charge density at the carbon atom too.

Unbalance Response of Rotors Supported on Hydrodynamic Bearings Placed Close to Nodal Points of Excited Vibration Modes

2002 ◽  
Author(s):  
Demetrio C. Zachariadis
Keyword(s):  
Tilt Angle ◽  
Rotor Dynamics ◽  
Journal Bearings ◽  
Vibration Modes ◽  
Time Varying ◽  
Rotor Systems ◽  
Unbalance Response ◽  
Nodal Points ◽  
Excited Vibration ◽  
Modes Of Vibration

The traditional 8-coefficient bearing model, used in linear rotor dynamics, is shown here to be inadequate for the unbalance response calculation of rotor systems supported on hydrodynamic journal bearings placed close to nodal points of excited modes of vibration. In such situations, one cannot neglect the time varying tilt angle between journals and bearings, whose consideration leads to the adoption of a 32-coefficient bearing model. Numerical results indicate that the differences between vibration amplitudes calculated using both bearing models can be greater than 100%, while discrepancies in the predicted stability thresholds are small. The conclusions of the study are coherent with previously published theoretical and experimental results.

Calculation of the Response of a Fighter Aircraft to Turbulent Air and a Comparison with Flight Measurements

1968 ◽  
Vol 72 (688) ◽  
pp. 303-309 ◽  
Author(s):  
C. G. B. Mitchell
Keyword(s):  
Structural Flexibility ◽  
Vibration Modes ◽  
Frequency Range ◽  
Aerodynamic Forces ◽  
Fighter Aircraft ◽  
Surface Theory ◽  
Short Period ◽  
Rigid Body Modes ◽  
Dutch Roll ◽  
Flight Measurements

Summary:— The response of a P1a fighter aircraft to vertical and lateral harmonic gusts is calculated, and the predicted spectra for response accelerations at many points on the structure compared with previous flight measurements in continuous turbulence. The frequency range for the comparison is 0.45 c/s to 30 c/s and the response includes the short period pitching and the dutch roll rigid–body modes as well as oscillations in five symmetric and four anti–symmetric flexible modes. Structural flexibility is represented by measured vibration modes and aerodynamic forces are calculated by an oscillatory lifting surface theory.

Coupled vibration modes

1975 ◽  
Vol 38 (1) ◽  
pp. 27-37 ◽  
Author(s):  
J.G.A. Croll
Keyword(s):  
Vibration Modes ◽  
Coupled Vibration
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