Vibration Characteristics of Straight Blades of Asymmetrical Aerofoil Cross-Section

1969 ◽  
Vol 20 (2) ◽  
pp. 178-190 ◽  
Author(s):  
W. Carnegie ◽  
B. Dawson
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Limited Range ◽  
First Order ◽  
Special Cases ◽  
Theoretical Results ◽  
Theoretical Procedure ◽  
Good Agreement

SummaryTheoretical and experimental natural frequencies and modal shapes up to the fifth mode of vibration are given for a straight blade of asymmetrical aerofoil cross-section. The theoretical procedure consists essentially of transforming the differential equations of motion into a set of simultaneous first-order equations and solving them by a step-by-step finite difference procedure. The natural frequency values are compared with results obtained by an analytical solution and with standard solutions for certain special cases. Good agreement is shown to exist between the theoretical results for the various methods presented. The equations of motion are dependent upon the coordinates of the axis of the centre of flexure of the beam relative to the centroidal axis. The effect of variations of the centre of flexure coordinates upon the frequencies and modal shapes is shown for a limited range of coordinate values. Comparison is made between the theoretical natural frequencies and modal shapes and corresponding results obtained by experiment.

Vibration of a Rotating Asymmetric Shaft Supported in Asymmetric Bearings

1968 ◽  
Vol 10 (3) ◽  
pp. 252-261 ◽  
Author(s):  
H. F. Black ◽  
A. J. McTernan
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Forced Vibrations ◽  
Variation Method ◽  
Approximate Theory ◽  
Cross Sectional ◽  
Mass Unbalance ◽  
The Mean ◽  
Theoretical Results ◽  
Good Agreement

The parametrically excited vibrations of this system with assumed small asymmetry of the shaft cross-section are discussed in terms of the motion of a symmetric shaft having the mean cross-sectional flexibility, and the equations of motion are solved by the approximate perturbation-variation method of Hsu. Both features yield a more lucid appreciation of the motions expected than previous treatments: in particular, simpler explicit expressions for unstable bounds are given and forced vibrations due to mass unbalance are discussed with greater facility. The practically important case of nearly coincident natural frequencies is examined. The theoretical results are compared with analogue computation: good agreement with the approximate theory is found even for quite large shaft asymmetry.

Models for New Corrugated and Porous Solar Air Collectors under Transient Operation

2017 ◽  
Vol 42 (1) ◽  
Author(s):  
Qahtan Adnan Abed ◽  
Viorel Badescu ◽  
Adrian Ciocanea ◽  
Iuliana Soriga ◽  
Dorin Bureţea
Keyword(s):  
Climatic Conditions ◽  
Meteorological Conditions ◽  
Bias Error ◽  
First Order ◽  
Solar Air Collector ◽  
Section Surface ◽  
South Eastern Europe ◽  
Main Components ◽  
Theoretical Results ◽  
Good Agreement

AbstractMathematical models have been developed to evaluate the dynamic behavior of two solar air collectors: the first one is equipped with a V-porous absorber and the second one with a U-corrugated absorber. The collectors have the same geometry, cross-section surface area and are built from the same materials, the only difference between them being the absorbers. V-corrugated absorbers have been treated in literature but the V-porous absorbers modeled here have not been very often considered. The models are based on first-order differential equations which describe the heat exchange between the main components of the two types of solar air heaters. Both collectors were exposed to the sun in the same meteorological conditions, at identical tilt angle and they operated at the same air mass flow rate. The tests were carried out in the climatic conditions of Bucharest (Romania, South Eastern Europe). There is good agreement between the theoretical results and experiments. The average bias error was about 7.75 % and 10.55 % for the solar air collector with “V”-porous absorber and with “U”-corrugated absorber, respectively. The collector based on V-porous absorber has higher efficiency than the collector with U-corrugated absorber around the noon of clear days. Around sunrise and sunset, the collector with U-corrugated absorber is more effective.

scholarly journals New Results on the Motions of Asteroids in Resonances

1992 ◽  
Vol 152 ◽  
pp. 145-152 ◽  
Author(s):  
R. Dvorak
Keyword(s):  
Initial Conditions ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Close Approach ◽  
Integration Time ◽  
Time Interval ◽  
Time Intervals ◽  
3 Dimensional ◽  
Special Cases ◽  
Good Agreement

In this article we present a numerical study of the motion of asteroids in the 2:1 and 3:1 resonance with Jupiter. We integrated the equations of motion of the elliptic restricted 3-body problem for a great number of initial conditions within this 2 resonances for a time interval of 104 periods and for special cases even longer (which corresponds in the the Sun-Jupiter system to time intervals up to 106 years). We present our results in the form of 3-dimensional diagrams (initial a versus initial e, and in the z-axes the highest value of the eccentricity during the whole integration time). In the 3:1 resonance an eccentricity higher than 0.3 can lead to a close approach to Mars and hence to an escape from the resonance. Asteroids in the 2:1 resonance with Jupiter with eccentricities higher than 0.5 suffer from possible close approaches to Jupiter itself and then again this leads in general to an escape from the resonance. In both resonances we found possible regions of escape (chaotic regions), but only for initial eccentricities e > 0.15. The comparison with recent results show quite a good agreement for the structure of the 3:1 resonance. For motions in the 2:1 resonance our numeric results are in contradiction to others: high eccentric orbits are also found which may lead to escapes and consequently to a depletion of this resonant regions.

Natural frequencies of parabolic arches with a single crack on opposite cross-section sides

2019 ◽  
Vol 25 (7) ◽  
pp. 1313-1325 ◽  
Author(s):  
U Eroglu ◽  
G Ruta ◽  
E Tufekci
Keyword(s):  
Cross Section ◽  
Natural Vibration ◽  
Natural Frequencies ◽  
Damage Identification ◽  
Crack Opening ◽  
Curved Beams ◽  
Governing Equations ◽  
First Order ◽  
Crack Location ◽  
Original Contribution

We study natural vibration of elastic parabolic arches, modeled as plane curved beams susceptible to elongation, shear, and bending, exhibiting small concentrated cracks. The crack is simulated by springs between regular chunks, with stiffness evaluated following stress concentration in usual crack opening modes. We evaluate and compare the linear dynamic response of the undamaged and damaged arch in nondimensional form. The governing equations are turned into a system of first-order differential equations that are solved numerically by the so-called matricant. The original contribution of this study lies in highlighting the dependence of the variation of the first natural frequencies on the crack location not only along the axis but also on opposite sides of the cross-section. We obtain the relative variations of the first frequencies in terms of the two crack locations. The result of this direct problem provides information on the possibility to detect such locations, and gives indications on structural monitoring and damage identification.

scholarly journals A Semi-Analytical Solution for Elastic Analysis of Rotating Thick Cylindrical Shells with Variable Thickness Using Disk Form Multilayers

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Mohammad Zamani Nejad ◽  
Mehdi Jabbari ◽  
Mehdi Ghannad
Keyword(s):  
Cylindrical Shell ◽  
Analytical Solution ◽  
Differential Equations ◽  
Variable Thickness ◽  
Deformation Theory ◽  
Shear Deformation Theory ◽  
First Order ◽  
Thick Cylinder ◽  
Good Agreement

Using disk form multilayers, a semi-analytical solution has been derived for determination of displacements and stresses in a rotating cylindrical shell with variable thickness under uniform pressure. The thick cylinder is divided into disk form layers form with their thickness corresponding to the thickness of the cylinder. Due to the existence of shear stress in the thick cylindrical shell with variable thickness, the equations governing disk layers are obtained based on first-order shear deformation theory (FSDT). These equations are in the form of a set of general differential equations. Given that the cylinder is divided intondisks,nsets of differential equations are obtained. The solution of this set of equations, applying the boundary conditions and continuity conditions between the layers, yields displacements and stresses. A numerical solution using finite element method (FEM) is also presented and good agreement was found.

Dynamical Behaviour of a Materially Damped Flexible Towed Cable

1972 ◽  
Vol 23 (2) ◽  
pp. 109-120 ◽  
Author(s):  
T C Cannon ◽  
J Genin
Keyword(s):  
Closed Form ◽  
Natural Frequencies ◽  
Equilibrium Shape ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Dynamical Behaviour ◽  
Aerodynamic Loading ◽  
Towed Cable ◽  
Closed Form Approximation ◽  
Good Agreement

SummaryThe three-dimensional equations of motion of a flexible towed cable are developed. A closed-form approximation for the equilibrium shape of a cable subjected to arbitrary aerodynamic loading is developed and used in the study of a heavy, vibrating tow cable. Natural frequencies of vibration and cable shapes are computed for typical cables and are shown to be in good agreement with exact, numerically obtained values.

Effect of Slenderness Ratio on the Natural Frequencies of Pre-Twisted Cantilever Beams of Uniform Rectangular Cross-Section

1971 ◽  
Vol 13 (1) ◽  
pp. 51-59 ◽  
Author(s):  
B. Dawson ◽  
N. G. Ghosh ◽  
W. Carnegie
Keyword(s):  
Differential Equations ◽  
Shear Deformation ◽  
Cross Section ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Slenderness Ratio ◽  
Twist Angle ◽  
Equations Of Motion ◽  
Rotary Inertia ◽  
Cantilever Beams

This paper is concerned with the vibrational characteristics of pre-twisted cantilever beams of uniform rectangular cross-section allowing for shear deformation and rotary inertia. A method of solution of the differential equations of motion allowing for shear deformation and rotary inertia is presented which is an extension of the method introduced by Dawson (1)§ for the solution of the differential equations of motion of pre-twisted beams neglecting shear and rotary inertia effects. The natural frequencies for the first five modes of vibration are obtained for beams of various breadth to depth ratios and lengths ranging from 3 to 20 in and pre-twist angle in the range 0–90°. The results are compared with those obtained by an alternative method (2), where available, and also to experimental results.

Dynamic Analysis of the Rotating Composite Thin-Walled Cantilever Beams with Shear Deformation

2013 ◽  
Vol 690-693 ◽  
pp. 309-313
Author(s):  
Yong Sheng Ren ◽  
Qi Yi Dai
Keyword(s):  
Shear Deformation ◽  
Cross Section ◽  
Fiber Orientation ◽  
Model Comparison ◽  
Natural Frequencies ◽  
Theoretical Study ◽  
Equations Of Motion ◽  
Cantilever Beams ◽  
The Finite Element Method ◽  
Rotating Speed

This paper presents a theoretical study of the dynamic characteristics of rotating composite cantilever beams. Considering shear deformation and cross section warping, the equations of motion of the rotating cantilever beams are derived using Hamilton’s principle. The Galerkin’s method is used in order to analysis the free vibration behaviors of the model. Comparison of the theoretical solutions has been made with the results obtained from the finite element method, which prove the validity of the model presented in this paper. Natural frequencies are obtained for circular tubular composite beams. The effects of fiber orientation, rotating speed and structure parameters on modal frequencies are investigated.

The steady flow due to a rotating sphere at low and moderate Reynolds numbers

1980 ◽  
Vol 101 (2) ◽  
pp. 257-279 ◽  
Author(s):  
S. C. R. Dennis ◽  
S. N. Singh ◽  
D. B. Ingham
Keyword(s):  
Reynolds Number ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Reynolds Numbers ◽  
Low Reynolds Numbers ◽  
Finite Set ◽  
Flow Variables ◽  
Theoretical Results ◽  
Radial Variable ◽  
Good Agreement

The problem of determining the steady axially symmetrical motion induced by a sphere rotating with constant angular velocity about a diameter in an incompressible viscous fluid which is at rest at large distances from it is considered. The basic independent variables are the polar co-ordinates (r, θ) in a plane through the axis of rotation and with origin at the centre of the sphere. The equations of motion are reduced to three sets of nonlinear second-order ordinary differential equations in the radial variable by expanding the flow variables as series of orthogonal Gegenbauer functions with argument μ = cosθ. Numerical solutions of the finite set of equations obtained by truncating the series after a given number of terms are obtained. The calculations are carried out for Reynolds numbers in the range R = 1 to R = 100, and the results are compared with various other theoretical results and with experimental observations.The torque exerted by the fluid on the sphere is found to be in good agreement with theory at low Reynolds numbers and appears to tend towards the results of steady boundary-layer theory for increasing Reynolds number. There is excellent agreement with experimental results over the range considered. A region of inflow to the sphere near the poles is balanced by a region of outflow near the equator and as the Reynolds number increases the inflow region increases and the region of outflow becomes narrower. The radial velocity increases with Reynolds number at the equator, indicating the formation of a radial jet over the narrowing region of outflow. There is no evidence of any separation of the flow from the surface of the sphere near the equator over the range of Reynolds numbers considered.

A Comparison of Micro-PIV Experiments in a Mini-Channel to Numerical and Analytical Solutions

2003 ◽  
Author(s):  
D. Newport ◽  
D. Curtin ◽  
M. Davies
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Exact Analytical Solution ◽  
Measurement Tool ◽  
Channel System ◽  
Micro Piv ◽  
Software Packages ◽  
Micro Systems ◽  
Experimental Rig ◽  
Good Agreement

In this paper, measurements are presented of the velocity profile in a mini-channel at different locations. The channel is rectangular in cross-section, approximately 1.2mm wide, 1.4mm deep and 29mm long. A micro-PIV system was used to obtain the velocity profiles at the inlet, mid-length and exit of the channel. The raw image maps were processed using three different commercial PIV software packages, and compared to an exact analytical solution. The mini-channel system was also simulated using a commercial CFD code as a further check on the dataset, and the experimental rig itself. It was found that the different processing procedures had little influence on the micro-PIV data, and good agreement was found with theory, numerical prediction and experiment. This establishes confidence in micro-PIV as a measurement tool in micro-systems.

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