scholarly journals Nonlinear Flexural Vibrations of Thin Circular Rings

1966 ◽  
Vol 33 (3) ◽  
pp. 553-560 ◽  
Author(s):  
D. A. Evensen
Keyword(s):  
Equations Of Motion ◽  
Mode Shapes ◽  
Test Results ◽  
State Response ◽  
Coupled Mode ◽  
Bending Mode ◽  
Circular Rings ◽  
Flexural Vibrations ◽  
Bending Modes ◽  
Good Agreement

The nonlinear flexural vibrations of thin circular rings are analyzed by assuming two vibration modes and then applying Galerkin’s procedure on the equations of motion. The results show that vibrations involving either a single bending mode or two coupled bending modes can occur. Theory and experiment both indicate a nonlinearity of the softening type and the existence of these coupled-mode vibrations. Test results for the steady-state response are in good agreement with the calculated values, and the deflection modes used in the analysis agree with the experimental mode shapes. The analytical and experimental results exhibit several features that are characteristic of nonliner vibrations of axisymmetric systems in general and of circular cylindrial shells in particular.

Investigating the vibrational behaviour of a rotating two-blade propeller by using a self-tracking method

2018 ◽  
Vol 233 (3) ◽  
pp. 835-847 ◽  
Author(s):  
Mohammad-Reza Ashory ◽  
Farhad Talebi ◽  
Heydar R Ghadikolaei ◽  
Morad Karimpour
Keyword(s):  
Natural Frequency ◽  
Measurement Errors ◽  
Mode Shapes ◽  
Model Parameters ◽  
Test Results ◽  
Modal Assurance Criterion ◽  
Tracking Method ◽  
Strong Resemblance ◽  
Test Scenarios ◽  
Good Agreement

This study investigated the vibrational behaviour of a rotating two-blade propeller at different rotational speeds by using self-tracking laser Doppler vibrometry. Given that a self-tracking method necessitates the accurate adjustment of test setups to reduce measurement errors, a test table with sufficient rigidity was designed and built to enable the adjustment and repair of test components. The results of the self-tracking test on the rotating propeller indicated an increase in natural frequency and a decrease in the amplitude of normalized mode shapes as rotational speed increases. To assess the test results, a numerical model created in ABAQUS was used. The model parameters were tuned in such a way that the natural frequency and associated mode shapes were in good agreement with those derived using a hammer test on a stationary propeller. The mode shapes obtained from the hammer test and the numerical (ABAQUS) modelling were compared using the modal assurance criterion. The examination indicated a strong resemblance between the hammer test results and the numerical findings. Hence, the model can be employed to determine the other mechanical properties of two-blade propellers in test scenarios.

Flexural Vibration of Symmetrical Multi-Layer Beams with Viscoelastic Damping

1968 ◽  
Vol 10 (3) ◽  
pp. 269-281 ◽  
Author(s):  
J. A. Agbasiere ◽  
P. Grootenhuis
Keyword(s):  
Equations Of Motion ◽  
Finite Difference Methods ◽  
Flexural Vibration ◽  
Order Effect ◽  
Simultaneous Equations ◽  
Mode Shapes ◽  
Central Layer ◽  
Method Of Solution ◽  
Flexural Vibrations ◽  
Frequency Phase

The flexural vibrations of beams can be reduced by introducing sandwiched layers of energy-dissipating materials. The equations of motion are derived for flexural vibrations of symmetrical, multi-layer sandwich beams. Two types of beams are considered, depending on whether the central layer is energy dissipating when the number of layers will be n = 4 i − 1, where i = 1, 2,…, or whether the material of the central layer is perfectly elastic when n = 4 i + 1. The total number of equations of motion will be i + 1 for each type. These equations will be non-linear when the properties of the energy-dissipating materials are strain dependent. A numerical method of solution has been introduced by transforming the equations of motion into sets of non-dimensional simultaneous equations and using finite difference methods. The simplest type of beam has three layers but even this leads to a set of four simultaneous equations of the twelfth order. It is shown as an example that the strain dependence of a typical viscoelastic material has only a second order effect upon the computed response. Experimentally determined values of frequency, phase angle and mode shapes under conditions of steady state are compared with computed data and the agreements are such that the response to excitation in flexural motion of symmetrical, multi-layer damped beams can now be calculated with confidence even for the lower modes.

scholarly journals Nonlinear interactions in deformable container cranes

2015 ◽  
Vol 230 (1) ◽  
pp. 5-20 ◽  
Author(s):  
Andrea Arena ◽  
Walter Lacarbonara ◽  
Matthew P Cartmell
Keyword(s):  
Rigid Body ◽  
Internal Resonance ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Higher Order ◽  
Mode Shapes ◽  
Lagrange Equations ◽  
Internal Resonances ◽  
Container Cranes ◽  
Bending Mode

Nonlinear dynamic interactions in harbour quayside cranes due to a two-to-one internal resonance between the lowest bending mode of the deformable boom and the in-plane pendular mode of the container are investigated. To this end, a three-dimensional model of container cranes accounting for the elastic interaction between the crane boom and the container dynamics is proposed. The container is modelled as a three-dimensional rigid body elastically suspended through hoisting cables from the trolley moving along the crane boom modelled as an Euler-Bernoulli beam. The reduced governing equations of motion are obtained through the Euler-Lagrange equations employing the boom kinetic and stored energies, derived via a Galerkin discretisation based on the mode shapes of the two-span crane boom used as trial functions, and the kinetic and stored energies of the rigid body container and the elastic hoisting cables. First, conditions for the onset of internal resonances between the boom and the container are found. A higher order perturbation treatment of the Taylor expanded equations of motion in the neighbourhood of a two-to-one internal resonance between the lowest boom bending mode and the lowest pendular mode of the container is carried out. Continuation of the fixed points of the modulation equations together with stability analysis yields a rich bifurcation behaviour, which features Hopf bifurcations. It is shown that consideration of higher order terms (cubic nonlinearities) beyond the quadratic geometric and inertia nonlinearities breaks the symmetry of the bifurcation equations, shifts the bifurcation points and the stability ranges, and leads to bifurcations not predicted by the low order analysis.

Flexural–Torsional Free Vibration Analysis of a Double-Cantilever Structure

2021 ◽  
Vol 144 (3) ◽  
Author(s):  
Anahita Zargarani ◽  
S. Nima Mahmoodi
Keyword(s):  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Experimental Investigations ◽  
Torsional Vibrations ◽  
Cantilever Beams ◽  
Torsional Mode ◽  
Flexural Mode ◽  
Cantilever Structure ◽  
Flexural Vibrations

Abstract This paper aims to investigate the free coupled flexural–torsional vibrations of a double-cantilever structure. The structure consists of two identical Euler–Bernoulli cantilever beams with a piezoelectric layer on top. The beams are connected by a rigid tip connection at their free ends. The double-cantilever structure in this study vibrates in two distinct modes: flexural mode or coupled flexural–torsional mode. The flexural mode refers to the in-phase flexural vibrations of the two cantilever beams resulting in translation of the tip connection, while the coupled flexural–torsional mode refers to the coupled flexural–torsional vibrations of the cantilever beams resulting in rotation of the tip connection. The latter is the main interest of this research. The governing equations of motion and boundary conditions are developed using Hamilton’s principle. Two uncoupled equations are realized for each beam: one corresponding to the flexural vibrations and the other one corresponding to the torsional vibrations. The characteristic equations for both the flexural and the coupled flexural–torsional vibration modes are derived and solved to find the natural frequencies corresponding to each mode of vibration. The orthogonality condition among the mode shapes is derived and utilized to determine the modal coefficients corresponding to each mode of vibration. Moreover, the analytical and experimental investigations show that the coupled flexural–torsional fundamental frequency of the structure is dependent on dimensional parameters including the length of the cantilever beams and the length of the tip connection.

Dynamic Response of a Geared Rotor-Bearing System Under Residual Shaft Bow Effect

2006 ◽  
Author(s):  
T. N. Shiau ◽  
E. K. Lee ◽  
Y. C. Chen ◽  
T. H. Young
Keyword(s):  
Steady State ◽  
Natural Frequencies ◽  
Coupling Effect ◽  
Equations Of Motion ◽  
Transmission Error ◽  
Mode Shapes ◽  
Steady State Response ◽  
State Response ◽  
Rotor Bearing ◽  
Bearing System

The paper presents the dynamic behaviors of a geared rotor-bearing system under the effects of the residual shaft bow, the gear eccentricity and excitation of gear’s transmission error. The coupling effect of lateral and torsional motions is considered in the dynamic analysis of the geared rotor-bearing system. The finite element method is used to model the system and Lagrangian approach is applied to derive the system equations of motion. The dynamic characteristics including system natural frequencies, mode shapes and steady-state response are investigated. The results show that the magnitude of the residual shaft bow, the phase angle between gear eccentricity and residual shaft bow will significantly affect system natural frequencies and steady-state response. When the spin speed closes to the second critical speed, the system steady state response will be dramatically increased by the residual shaft bow for the in-phase case. Moreover the zero response can be obtained when the system is set on special conditions.

Large Amplitude Oscillations of Thin Circular Rings

1987 ◽  
Vol 54 (2) ◽  
pp. 315-322 ◽  
Author(s):  
S. P. Maganty ◽  
W. B. Bickford
Keyword(s):  
Large Amplitude ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Circular Ring ◽  
Bending Mode ◽  
Out Of Plane ◽  
Circular Rings ◽  
Linear Problems ◽  
Plane Bending ◽  
Nonlinear Equations Of Motion

Using an intrinsic formulation, an accurate set of geometrically nonlinear equations of motion is derived for the large amplitude oscillations of a thin circular ring. Non-dimensionalization of the equations of motion and the compatibility conditions indicates clearly that certain terms involving the extensional deformation, the shear deformation, and the rotatory inertia are relatively small and can be discarded. The resulting equations of motion are analyzed by the method of multiple scales with a single bending mode approximation to the linear problems indicating a softening type of nonlinearity for both the in-plane and the out-of-plane problems with the out-of-plane flexural motion experiencing a greater degree of softening when compared to that of the in-plane flexural motion. The results for the nonresonant case indicate that the frequency of an out-of-plane bending mode is significantly reduced by the presence of a nonzero in-plane bending amplitude, whereas the results for the resonant case indicate the presence of unsteady oscillations with an exchange of energy between the in-plane and the out-of-plane modes.

In-Plane Vibration of Simply Supported-Simply Supported Circular Ring Segments

1991 ◽  
Author(s):  
S. Azimi
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Circular Ring ◽  
Mode Shapes ◽  
Simply Supported ◽  
Plane Vibration ◽  
Ring Segment ◽  
Linear Spring ◽  
Circular Rings ◽  
Elastic Constraints

Abstract The general in-plane vibration problem of circular ring segments having linear and torsional elastic constraints at the ends has been studied. One of the interesting cases of concern for which there exists an exact solution is the case where the linear and torsional spring stiffnesses in the circumferential direction become zero (Ku = 0, Kt = 0) and the linear spring stiffness in the radial direction becomes infinity (Kw = ∞). This case is the so-called simply supported-simply supported case. The general form of the equations of motion of circular rings with the proper boundary conditions at the ends have been employed to investigate the vibration of the ring segment. Results for natural frequencies and mode shapes for different angles of the segment have been presented.

Tire Vibrations

1977 ◽  
Vol 5 (4) ◽  
pp. 202-225 ◽  
Author(s):  
G. R. Potts ◽  
C. A. Bell ◽  
L. T. Charek ◽  
T. K. Roy
Keyword(s):  
Natural Frequencies ◽  
Standing Waves ◽  
Resonant Mode ◽  
Mode Shapes ◽  
Resonant Vibrations ◽  
Resonant Frequencies ◽  
Radial Tire ◽  
Analysis Techniques ◽  
Thin Ring ◽  
Good Agreement

Abstract Natural frequencies and vibrating motions are determined in terms of the material and geometric properties of a radial tire modeled as a thin ring on an elastic foundation. Experimental checks of resonant frequencies show good agreement. Forced vibration solutions obtained are shown to consist of a superposition of resonant vibrations, each rotating around the tire at a rate depending on the mode number and the tire rotational speed. Theoretical rolling speeds that are upper bounds at which standing waves occur are determined and checked experimentally. Digital Fourier transform, transfer function, and modal analysis techniques used to determine the resonant mode shapes of a radial tire reveal that antiresonances are the primary transmitters of vibration to the tire axle.

Dynamic model for high-speed rotors based on their experimental characterization

2017 ◽  
Vol 2 (4) ◽  
pp. 25
Author(s):  
L. A. Montoya ◽  
E. E. Rodríguez ◽  
H. J. Zúñiga ◽  
I. Mejía
Keyword(s):  
Dynamic Model ◽  
High Speed ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Experimental Characterization ◽  
Rotating Systems ◽  
Computing Performance ◽  
The Impact ◽  
Stiffness Distribution ◽  
Good Agreement

Rotating systems components such as rotors, have dynamic characteristics that are of great importance to understand because they may cause failure of turbomachinery. Therefore, it is required to study a dynamic model to predict some vibration characteristics, in this case, the natural frequencies and mode shapes (both of free vibration) of a centrifugal compressor shaft. The peculiarity of the dynamic model proposed is that using frequency and displacements values obtained experimentally, it is possible to calculate the mass and stiffness distribution of the shaft, and then use these values to estimate the theoretical modal parameters. The natural frequencies and mode shapes of the shaft were obtained with experimental modal analysis by using the impact test. The results predicted by the model are in good agreement with the experimental test. The model is also flexible with other geometries and has a great time and computing performance, which can be evaluated with respect to other commercial software in the future.

scholarly journals Spine Examination during COVID-19 Pandemic via Video Consultation

2021 ◽  
Author(s):  
Tom Jansen ◽  
Martin Gathen ◽  
Amadeo Touet ◽  
Hans Goost ◽  
Dieter Christian Wirtz ◽  
...  
Keyword(s):  
Pain Intensity ◽  
Range Of Motion ◽  
Test Results ◽  
Risk Of Infection ◽  
Face To Face ◽  
Reliable Tool ◽  
Provocation Tests ◽  
Active Range Of Motion ◽  
Video Consultation ◽  
Good Agreement

Abstract Introduction During the current COVID-19 pandemic video consultations are increasingly common in order to minimize the risk of infection for staff and patients. The aim of this study was to evaluate the feasibility of a spine examination via video. Methods A total of 43 patients were recruited. Each participant underwent a video-based (VB) and a conventional face-to-face (FTF) spine examination. Pain intensity, active range of motion, inspection, a neurophysiologic basic exam and provocations tests were evaluated using video-based and face-to-face methods. Results The intra-rater reliability (IRR) was measured between both examinations. Good to very good IRR values were obtained in inspection (Kappa between 0,752 und 0,944), active range of motion and basic neurophysiological examination (Kappa between 0,659 und 0,969). Only moderate matches were found in specific provocation tests (Kappa between 0,407 und 0,938). A video-based spine examination is a reliable tool for measuring pain intensity, active range of motion and a basic neurophysiologic exam. Conclusion A basic spine examination during a video consultation is possible. A good agreement of the test results between video-based and face-to-face examination could be found.

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