Forced Vibrations of a Body on an Infinite Elastic Solid

1955 ◽  
Vol 22 (3) ◽  
pp. 391-400
Author(s):  
R. N. Arnold ◽  
G. N. Bycroft ◽  
G. B. Warburton
Keyword(s):  
Elastic Solid ◽  
Infinite Medium ◽  
Elastic Model ◽  
Constant Depth ◽  
Amplitude Response ◽  
Torsional Mode ◽  
Exciting Frequency ◽  
Modes Of Vibration ◽  
Circular Base ◽  
Good Agreement

Abstract The paper considers the forced vibration of a rigid body resting on a homogeneous elastic medium of infinite surface area and constant depth which may be finite or infinite. Four modes of vibration for a body with a circular base are investigated; (a) vertical translation, (b) torsion, (c) horizontal translation, (d) rocking. For a semi-infinite medium the amplitude response can be obtained for any mass in terms of known constants of the system and two fundamental functions f1 and f2, which depend only on the exciting frequency and the properties of the medium. Close approximations to these functions have been evaluated for each mode. Experiments on an elastic model are described, the results of which are in good agreement with theoretical prediction. The behavior of a stratum of infinite area is more complex since functions f1 and f2 also depend on stratum depth. These functions have been evaluated for the torsional mode only, experimental results being given for the other modes.

Forced Vibration of a Body on an Elastic Stratum

1957 ◽  
Vol 24 (1) ◽  
pp. 55-58
Author(s):  
G. B. Warburton
Keyword(s):  
Resonant Frequency ◽  
Surface Area ◽  
Forced Vibration ◽  
The Body ◽  
Constant Depth ◽  
Disturbing Force ◽  
Axis Of Symmetry ◽  
Experimental Values ◽  
Circular Base ◽  
Good Agreement

Abstract The paper considers the forced vibration of a body with a circular base resting on a homogeneous elastic stratum of infinite surface area and constant depth. The body is subjected to a vertical disturbing force along the axis of symmetry. After evaluating an integral given by Arnold, Bycroft, and Warburton, curves are given showing the dependence of the resonant frequency on three parameters. Good agreement is obtained between theoretical values of the resonant frequency and experimental values taken from the previous paper.

Some Integrals Relating to the Vibration of a Cantilever Beam and Approximation for the Effect of Taper on Overtone Frequencies

1956 ◽  
Vol 7 (2) ◽  
pp. 109-124 ◽  
Author(s):  
A. I. Martin
Keyword(s):  
Turbine Blade ◽  
Cantilever Beam ◽  
Second Order ◽  
Modes Of Vibration ◽  
Turbine Blading ◽  
Good Agreement

SummaryIn the investigation of approximate numerical values of overtone frequencies of a turbine blade it is desirable to know formulae for certain integrals of the modes of vibration of an ordinary cantilever beam. The first object of this paper is to obtain such formulae and to arrange them in tabulated form. By proceeding along the lines of the calculus of perturbations, these results may then be used to obtain new formulae which give second order approximations for the effects of uniform breadth and thickness tapers on the overtone frequencies. The theory gives good agreement with experiments for tapers which do not exceed about 0·5, which is present-day practice for turbine blading.

scholarly journals Voltage–Amplitude Response of Superharmonic Resonance of Second Order of Electrostatically Actuated MEMS Cantilever Resonators

2019 ◽  
Vol 14 (3) ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Martin A. Botello ◽  
Christian A. Reyes ◽  
Julio S. Beatriz
Keyword(s):  
Numerical Integration ◽  
Multiple Scales ◽  
Low Voltage ◽  
Second Order ◽  
Voltage Amplitude ◽  
Amplitude Response ◽  
Superharmonic Resonance ◽  
Bifurcation Points ◽  
Electrostatically Actuated ◽  
Modes Of Vibration

This paper investigates the voltage–amplitude response of superharmonic resonance of second order (order two) of alternating current (AC) electrostatically actuated microelectromechanical system (MEMS) cantilever resonators. The resonators consist of a cantilever parallel to a ground plate and under voltage that produces hard excitations. AC frequency is near one-fourth of the natural frequency of the cantilever. The electrostatic force includes fringe effect. Two kinds of models, namely reduced-order models (ROMs), and boundary value problem (BVP) model, are developed. Methods used to solve these models are (1) method of multiple scales (MMS) for ROM using one mode of vibration, (2) continuation and bifurcation analysis for ROMs with several modes of vibration, (3) numerical integration for ROM with several modes of vibration, and (4) numerical integration for BVP model. The voltage–amplitude response shows a softening effect and three saddle-node bifurcation points. The first two bifurcation points occur at low voltage and amplitudes of 0.2 and 0.56 of the gap. The third bifurcation point occurs at higher voltage, called pull-in voltage, and amplitude of 0.44 of the gap. Pull-in occurs, (1) for voltage larger than the pull-in voltage regardless of the initial amplitude and (2) for voltage values lower than the pull-in voltage and large initial amplitudes. Pull-in does not occur at relatively small voltages and small initial amplitudes. First two bifurcation points vanish as damping increases. All bifurcation points are shifted to lower voltages as fringe increases. Pull-in voltage is not affected by the damping or detuning frequency.

An Algorithm for Fluid–Solid Coupling Based on SPH Method and Its Preliminary Verification

2018 ◽  
Vol 16 (02) ◽  
pp. 1846008
Author(s):  
X. J. Ma ◽  
M. Geni ◽  
A. F. Jin
Keyword(s):  
Elastic Solid ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Sph Method ◽  
Particle Hydrodynamics ◽  
Sph Model ◽  
Smoothed Particle ◽  
Feasible Algorithm ◽  
Coupling Algorithm ◽  
Good Agreement

Based on the fundamental theory of smoothed-particle hydrodynamics (SPH), a feasible algorithm for fluid–solid coupling on interface is applied to describe the dynamic behavior of fluid and solid by utilizing continuum mechanics governing equations. Numerical simulation is conducted based on the proposed SPH model and the fluid–solid interface coupling algorithm, and good agreement is observed with the experiment results. It is shown in the results that the present SPH model is able to effectively and accurately simulate the free-surface flow of fluid, deformation of the elastic solid and the fluid–solid impacting.

Oscillations of a rigid sphere embedded in an infinite elastic solid

1967 ◽  
Vol 63 (4) ◽  
pp. 1189-1205 ◽  
Author(s):  
P. Chadwick ◽  
E. A. Trowbridge
Keyword(s):  
Elastic Solid ◽  
Forced Oscillations ◽  
Graphical Form ◽  
Rigid Sphere ◽  
Free Oscillations ◽  
Modes Of Vibration ◽  
Fixed Axis ◽  
Steady Oscillations ◽  
Angular Oscillations ◽  
Elastic Shear

AbstractA detailed study is made of angular oscillations of small amplitude about a fixed axis of a rigid sphere embedded in an infinite elastic solid. Three modes of vibration of the sphere are considered: steady oscillations arising from the application of a periodic torque; forced oscillations produced by an arbitrary time-dependent torque; and free oscillations excited by an impulsive torque. Due to the transfer of energy to the surrounding material by the radiation of an elastic shear wave, free oscillations of the sphere are damped, the principal parameter affecting the damping being the density contrast between the sphere and its surroundings. Illustrative numerical results, referring to steady and free oscillations of the sphere, are presented in graphical form.

scholarly journals A Supporting Design Method When Longwall FaceStrides across and Passes through a Roadway

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Weidong Pan ◽  
Xinyuan Li ◽  
YiWei Li ◽  
Xiaobin Li ◽  
Qing Qiao ◽  
...  
Keyword(s):  
Stress Condition ◽  
Underground Mining ◽  
Design Method ◽  
Elastic Model ◽  
Mechanical Behaviours ◽  
Normal Production ◽  
Maximum Subsidence ◽  
Seam Mining ◽  
Rock Strata ◽  
Good Agreement

In coal underground mining, situation of longwall face striding across (SAR) or passing through roadway (PTR) is very common, especially in an inclined coal seam mining. A roadway supporting design method, consisting of a model using to determine the minimal rock strata thickness and a segmental supporting scheme, is developed. In addition, to represent the mechanical behaviour of the caved material authentically, an elastic model was developed. The results showed that the elastic model has a good agreement with the caved material mechanical behaviours at a relatively lower stress condition. By using a FDEM method, a real case in Xutuan coal mine is studied. Compared with the process without backfill, the z-displacement of cross-cut roof decreased shapely after backfilling, with a maximum z-displacement, decreases from 0.76 m to 0.13 m and from 0.39 m to 0.064 m in PTR and SAR section, respectively. Therefore, the possibility of fall of ground (FOG) and crushing accidents can be reduced effectively with backfilling material of wood cribs. And the maximum subsidence (SAR section side) of face floor is 0.16 m, which is small enough to ensure normal production. The results of this study are likely to be useful as a reference for the safe and efficient mining of coal resources under similar conditions.

Bond of tube in semi-infinite elastic solid

1980 ◽  
Vol 15 (2) ◽  
pp. 53-62 ◽  
Author(s):  
J W Ivering
Keyword(s):  
Half Space ◽  
Three Dimensional ◽  
Elastic Solid ◽  
Elastic Half Space ◽  
Infinite Medium ◽  
Bond Stress ◽  
Elastic Analysis ◽  
One Dimensional ◽  
Stress Curve ◽  
Tube Segment

The analysis of the bond stress of a thick-walled tube embedded at the surface of an elastic, isotropic, semi-infinite medium is presented. The condition of three-dimensional compatibility between the tube and the anchorage medium is taken into account. An equilibrium equation for a segment of an embedded tube is derived, from which bond stresses acting on the tube can be computed. The general solution obtained relates to the vector function for a uniform lineal load acting perpendicularly to the surface of an elastic half-space. The solution is in agreement with equations derived independently for cases of one-dimensional (lineal) compatibility. The equation of equilibrium derived for a tube segment embedded at the surface of an elastic half-space is transformed to a form suitable for solving the bond stresses of a tube anchorage embedded at some distance from the surface. A numerical solution of bond stresses obtained by elastic analysis is compared to the bond stress curve along the anchorage obtained experimentally.

scholarly journals Modeling of Brain Shift Phenomenon for Different Craniotomies and Solid Models

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Alvaro Valencia ◽  
Benjamin Blas ◽  
Jaime H. Ortega
Keyword(s):  
Elastic Solid ◽  
Brain Parenchyma ◽  
Solid Model ◽  
Elastic Model ◽  
Brain Shift ◽  
Healthy Human ◽  
Brain Surface ◽  
Solid Models ◽  
The Difference ◽  
The Brain

This study investigates the effects of different solid models on predictions of brain shift for three craniotomies. We created a generic 3D brain model based on healthy human brain and modeled the brain parenchyma as single continuum and constrained by a practically rigid skull. We have used elastic model, hyperelastic 1st, 2nd, and 3rd Ogden models, and hyperelastic Mooney-Rivlin with 2- and 5-parameter models. A pressure on the brain surface at craniotomy region was applied to load the model. The models were solved with the finite elements package ANSYS. The predictions on stress and displacements were compared for three different craniotomies. The difference between the predictions of elastic solid model and a hyperelastic Ogden solid model of maximum brain displacement and maximum effective stress is relevant.

scholarly journals Dynamic testing of a modern concrete bridge

1979 ◽  
Vol 6 (3) ◽  
pp. 447-455 ◽  
Author(s):  
J. H. Rainer ◽  
G. Pernica
Keyword(s):  
Natural Frequencies ◽  
Dynamic Testing ◽  
Dynamic Properties ◽  
Mode Shapes ◽  
Total Width ◽  
Concrete Bridge ◽  
Reinforced Concrete Bridge ◽  
Natural Modes ◽  
Modes Of Vibration ◽  
Good Agreement

A posttensioned reinforced concrete bridge, slated for demolition, was tested to obtain its dynamic properties. The 10 year old bridge consisted of a continuous flat slab deck of variable thickness having a total width of 103 ft (31.39 m) and spans of 28 ft 6 in. (8.69 m), 71 ft 0 in. (21.64 m), and 42 ft 6 in. (12.95 m). The entire bridge was skewed 10°50′ and the deck was slightly curved in plan.The mode shapes, natural frequencies, and damping ratios for the lowest five natural modes of vibration were determined using sinusoidal forcing functions from an electrohydraulic shaker. These modes, located at 5.7, 6.4, 8.7, 12.0, and 17.4 Hz, were found to be highly dependent on the lateral properties of the bridge deck. Damping ratios were determined from the widths of resonance peaks. The modal properties from the steady state excitation were compared with those obtained from measurements of traffic-induced vibrations and good agreement was found between the two methods.

In-Vacuo Modal Dynamic Response of the Human Skull

1974 ◽  
Vol 96 (2) ◽  
pp. 490-494 ◽  
Author(s):  
R. E. Nickell ◽  
P. V. Marcal
Keyword(s):  
Shell Element ◽  
Element Model ◽  
General Purpose ◽  
Element Formulation ◽  
Human Skull ◽  
Natural Modes ◽  
Modes Of Vibration ◽  
Doubly Curved ◽  
Insight Into ◽  
Good Agreement

A finite element model of a human skull is analyzed in order to determine the lowest natural modes of vibration. A doubly curved, triangular, thin shell element formulation is used, within the framework of a general-purpose program, to investigate the effect of various support systems on the frequencies and modal shapes. The frequencies are found to be in good agreement with the results of other investigators and the modal shapes offer some insight into a modified theory of craniocerebral damage that includes both skull rotation and cavitation as pathogenic mechanisms.

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