Harmonic Rocking of a Rigid Rectangular Body on a Semi-Infinite Elastic Medium

1966 ◽  
Vol 33 (3) ◽  
pp. 547-552 ◽  
Author(s):  
A. O. Awojobi
Keyword(s):  
Elastic Medium ◽  
Dynamic Stress ◽  
Graphical Representation ◽  
Mixed Boundary ◽  
The Body ◽  
Stress Distributions ◽  
Dual Integral Equations ◽  
Novel Method ◽  
Theoretical Results ◽  
Rectangular Body

Dual integral equations representing the mixed boundary-value problem of the rocking of an infinitely long, rigid rectangular body on the free surface of a semi-infinite elastic medium are here solved. Dynamic stress distributions under the rigid body and resonance curves for motion of the body are derived from the solution of these equations. A graphical representation of the theoretical results provides a novel method for estimating Poisson’s ratio of soil from a knowledge of the rocking frequency of a building. Suitable published results of tests on actual buildings have been used to indicate the application of this method.

Vibration of rigid bodies on semi-infinite elastic media

1965 ◽  
Vol 287 (1408) ◽  
pp. 27-63 ◽  
Keyword(s):  
Integral Equations ◽  
Mixed Boundary ◽  
Rigid Bodies ◽  
Stress Distributions ◽  
Dual Integral Equations ◽  
Response Curves ◽  
Rectangular Body ◽  
Vertical Translation

This mixed boundary-value problem gives rise to a set of dual integral equations which have not hitherto been solved. Four cases are analysed: the vertical translation and rotation about an axis normal to the surface of a rigid circular body and the vertical translation and rocking of an infinitely long rigid rectangular body. The dynamic stress distributions under the rigid bodies are determined and are shown to reduce to the known static distributions for zero frequency factors. The dual integral equations are solved by a series of expansion procedures. The calculated response curves for translation and rotation of a rigid circular body are compared with the experimental results by Arnold, By croft & Warburton (1955) and are shown to be an improvement over other approximate theories. A suggestion is made for using the results of this analysis for the determination of the dynamic elastic properties of a soil in situ .

Approximate Solution of High-Frequency-Factor Vibrations of Rigid Bodies on Elastic Media

1971 ◽  
Vol 38 (1) ◽  
pp. 111-117 ◽  
Author(s):  
A. O. Awojobi
Keyword(s):  
Approximate Solution ◽  
Elastic Medium ◽  
Half Space ◽  
High Frequency ◽  
Low Frequency ◽  
Frequency Factor ◽  
Rigid Bodies ◽  
Elastic Media ◽  
Stress Distributions ◽  
Rectangular Body

The mixed boundary-value problems of the vibrations of rigid bodies on elastic media are generally considered in the low-frequency-factor range. It is first established that, quite apart from a consideration of resonance, the usual assumption that this range predominates in practice is erroneous. The present work, therefore, is concerned with vibrations at frequency factors which are much greater than unity. Five cases have been considered: torsional vibration of a rigid circular body on a semi-infinite elastic medium and on an infinitely wide elastic stratum on a rigid bed; vertical vibration of a rigid circular body and of an infinitely long rectangular body on a semi-infinite elastic medium; rocking of a long rectangular body on a semi-infinite elastic medium. An estimate of both the unknown dynamic stress distribution under the rigid bodies and their amplitude responses has been obtained by finding an approximate solution to the exact governing dual integral equations. It is shown that at high-frequency factors, stress distributions are approximately constant for vertical vibrations and vary linearly from the center for rotational vibrations as in a Winkler model of theoretical soil statics contrary to increasing stresses with infinite edge stresses for low-frequency and static stress distributions of rigid bodies on elastic half space. We also obtain the important conclusion for amplitude response that it is predominantly governed by the inertia of the bodies because the contribution due to the dispersion of waves in the elastic medium is generally of a lower order of frequency factor than the inertia term except for an incompressible medium which has been analyzed separately and found to be of the same order leading to expressions for equivalent inertia of the vibrating medium. The theoretical results are used to derive the “tails” of resonance curves for both half space and stratum cases where experimental results are available. The agreement is fair and improves with increasing frequency factor.

Dynamic Stress Concentration Around Two Coplanar Griffith Cracks in an Infinite Elastic Medium

1978 ◽  
Vol 45 (4) ◽  
pp. 803-806 ◽  
Author(s):  
S. Itou
Keyword(s):  
Stress Intensity ◽  
Stress Concentration ◽  
Elastic Medium ◽  
Stress Intensity Factors ◽  
Dynamic Stress ◽  
Expansion Method ◽  
Intensity Factors ◽  
Dual Integral Equations ◽  
Dynamic Stress Intensity Factors ◽  
Series Expansion Method

The dynamic problem presented here is for an infinite elastic medium weakened by two coplanar Griffith cracks in which a self-equilibrated system of pressure is varied harmonically with time. The problem is reduced to dual integral equations and solved by a series-expansion method. The dynamic stress-intensity factors are computed numerically.

Forced vibrations of a rigid circular plate on a semi-infinite elastic space and on an elastic stratum

1956 ◽  
Vol 248 (948) ◽  
pp. 327-368 ◽  
Keyword(s):  
Free Surface ◽  
Circular Plate ◽  
Degrees Of Freedom ◽  
Mixed Boundary ◽  
Forced Vibrations ◽  
Reciprocal Theorem ◽  
Elastic Space ◽  
Dual Integral Equations ◽  
Theoretical Results ◽  
Finite Integrals

The impedance of a rigid circular plate attached to the free surface of a semi-infinite elastic space or an elastic stratum is determined for its four degrees of freedom. The solution of the dual integral equations arising from this mixed boundary-value problem is avoided by reference to Rayleigh’s reciprocal theorem. This enables the functions of frequency, which determine the in-phase and out-of-phase components of displacement of the plate, to be located between two close bounds and lying much closer to one than to the other. These bounds appear as infinite integrals involving branch functions and are reduced to tractable finite integrals by integration in the complex plane. Dissipation of waves to infinity produces an effective damping, and the added effect of the inclusion of true damping in the medium is discussed. It is to be expected, of course, that the unloaded rigid plate attached to the free surface of a semi-infinite elastic space does not resonate. The change of impedance of the plate with frequency is found to be similar for the two translations and also similar for the two rotations. Resonance occurs in the case of vertical and horizontal translation of the plate attached to the surface of an elastic stratum. However, it does not exist for rotations of the plate on the stratum. Instead, a maximum in the response appears, this maximum being more defined the greater the ratio of plate diameter to stratum depth. The addition of small true damping in the medium changes the characteristics very little. Experimental work substantiating these theoretical results, together with a general discussion of the results and their applications in geophysics and engineering, is being published shortly.

scholarly journals Study on the nonlinear vibration of embedded carbon nanotube via the Hamiltonian-based method

2021 ◽  
pp. 146134842110327
Author(s):  
Kang-Jia Wang ◽  
Guo-Dong Wang
Keyword(s):  
Carbon Nanotubes ◽  
Carbon Nanotube ◽  
Nonlinear Vibration ◽  
Elastic Medium ◽  
Numerical Results ◽  
Frequency Property ◽  
New Novel ◽  
Novel Method

This article mainly studies the vibration of the carbon nanotubes embedded in elastic medium. A new novel method called the Hamiltonian-based method is applied to determine the frequency property of the nonlinear vibration. Finally, the effectiveness and reliability of the proposed method is verified through the numerical results. The obtained results in this work are expected to be helpful for the study of the nonlinear vibration.

A dynamic model for far-field acceleration

1982 ◽  
Vol 72 (4) ◽  
pp. 1049-1068
Author(s):  
John Boatwright
Keyword(s):  
Stress Drop ◽  
High Frequency ◽  
Dynamic Stress ◽  
The Body ◽  
Body Waves ◽  
Far Field ◽  
Rupture Velocity ◽  
Frequency Level ◽  
Average Change ◽  
Dynamic Stress Drop

abstract A model for the far-field acceleration radiated by an incoherent rupture is constructed by combining Madariaga's (1977) theory for the high-frequency radiation from crack models of faulting with a simple statistical source model. By extending Madariaga's results to acceleration pulses with finite durations, the peak acceleration of a pulse radiated by a single stop or start of a crack tip is shown to depend on the dynamic stress drop of the subevent, the total change in rupture velocity, and the ratio of the subevent radius to the acceleration pulse width. An incoherent rupture is approximated by a sample from a self-similar distribution of coherent subevents. Assuming the subevents fit together without overlapping, the high-frequency level of the acceleration spectra depends linearly on the rms dynamic stress drop, the average change in rupture velocity, and the square root of the overall rupture area. The high-frequency level is independent, to first order, of the rupture complexity. Following Hanks (1979), simple approximations are derived for the relation between the rms dynamic stress drop and the rms acceleration, averaged over the pulse duration. This relation necessarily depends on the shape of the body-wave spectra. The body waves radiated by 10 small earthquakes near Monticello Dam, South Carolina, are analyzed to test these results. The average change of rupture velocity of Δv = 0.8β associated with the radiation of the acceleration pulses is estimated by comparing the rms acceleration contained in the P waves to that in the S waves. The rms dynamic stress drops of the 10 events, estimated from the rms accelerations, range from 0.4 to 1.9 bars and are strongly correlated with estimates of the apparent stress.

scholarly journals A mixed boundary value problem of a cracked elastic medium under torsion

2021 ◽  
pp. 10-10
Author(s):  
Belkacem Kebli ◽  
Fateh Madani
Keyword(s):  
Boundary Value Problem ◽  
Elastic Medium ◽  
Integral Equations ◽  
Integral Transformation ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Crack Problem ◽  
Fredholm Integral Equations ◽  
Mixed Boundary Value Problem ◽  
Penny Shaped Crack

The present work aims to investigate a penny-shaped crack problem in the interior of a homogeneous elastic material under axisymmetric torsion by a circular rigid inclusion embedded in the elastic medium. With the use of the Hankel integral transformation method, the mixed boundary value problem is reduced to a system of dual integral equations. The latter is converted into a regular system of Fredholm integral equations of the second kind which is then solved by quadrature rule. Numerical results for the displacement, stress and stress intensity factor are presented graphically in some particular cases of the problem.

Principles of minimum potential energy and complementary energy

2020 ◽  
pp. 228-248
Author(s):  
Lallit Anand ◽  
Sanjay Govindjee
Keyword(s):  
Boundary Condition ◽  
Potential Energy ◽  
Displacement Field ◽  
Mixed Boundary ◽  
Energy Functional ◽  
The Body ◽  
Minimum Potential Energy ◽  
Complementary Energy ◽  
Minimum Potential ◽  
Potential Energy Functional

With the displacement field taken as the only fundamental unknown field in a mixed-boundary-value problem for linear elastostatics, the principle of minimum potential energy asserts that a potential energy functional, which is defined as the difference between the free energy of the body and the work done by the prescribed surface tractions and the body forces --- assumes a smaller value for the actual solution of the mixed problem than for any other kinematically admissible displacement field which satisfies the displacement boundary condition. This principle provides a weak or variational method for solving mixed boundary-value-problems of elastostatics. In particular, instead of solving the governing Navier form of the partial differential equations of equilibrium, one can search for a displacement field such that the first variation of the potential energy functional vanishes. A similar principle of minimum complementary energy, which is phrased in terms of statically admissible stress fields which satisfy the equilibrium equation and the traction boundary condition, is also discussed. The principles of minimum potential energy and minimum complementary energy can also be applied to derive specialized principles which are particularly well-suited to solving structural problems; in this context the celebrated theorems of Castigliano are discussed.

scholarly journals Full-Span Flying Wing Wind Tunnel Test: A Body Freedom Flutter Study

Fluids ◽  
2020 ◽  
Vol 5 (1) ◽  
pp. 34
Author(s):  
Pengtao Shi ◽  
Jihai Liu ◽  
Yingsong Gu ◽  
Zhichun Yang ◽  
Pier Marzocca
Keyword(s):  
Wind Tunnel ◽  
Theoretical Analysis ◽  
Mass Balance ◽  
Degrees Of Freedom ◽  
Wind Tunnel Test ◽  
Suspension System ◽  
The Body ◽  
Structural Dynamic ◽  
Flutter Speed ◽  
Theoretical Results

Aiming at the experimental test of the body freedom flutter for modern high aspect ratio flexible flying wing, this paper conducts a body freedom flutter wind tunnel test on a full-span flying wing flutter model. The research content is summarized as follows: (1) The full-span finite element model and aeroelastic model of an unmanned aerial vehicle for body freedom flutter wind tunnel test are established, and the structural dynamics and flutter characteristics of this vehicle are obtained through theoretical analysis. (2) Based on the preliminary theoretical analysis results, the design and manufacturing of this vehicle are completed, and the structural dynamic characteristics of the vehicle are identified through ground vibration test. Finally, the theoretical analysis model is updated and the corresponding flutter characteristics are obtained. (3) A novel quasi-free flying suspension system capable of releasing pitch, plunge and yaw degrees of freedom is designed and implemented in the wind tunnel flutter test. The influence of the nose mass balance on the flutter results is explored. The study shows that: (1) The test vehicle can exhibit body freedom flutter at low airspeeds, and the obtained flutter speed and damping characteristics are favorable for conducting the body freedom flutter wind tunnel test. (2) The designed suspension system can effectively release the degrees of freedom of pitch, plunge, and yaw. The flutter speed measured in the wind tunnel test is 9.72 m/s, and the flutter frequency is 2.18 Hz, which agree well with the theoretical results (with flutter speed of 9.49 m/s and flutter frequency of 2.03 Hz). (3) With the increasing of the mass balance at the nose, critical speed of body freedom flutter rises up and the flutter frequency gradually decreases, which also agree well with corresponding theoretical results.

Power Transformer Constants Affecting its Impulse Voltage Distribution

1977 ◽  
Vol 14 (1) ◽  
pp. 53-63
Author(s):  
A. M. El-Arabaty ◽  
Ezzat A. A. Mansour ◽  
Osama A. M. Said
Keyword(s):  
Power Transformer ◽  
Power Transformers ◽  
Stress Distributions ◽  
Voltage Distribution ◽  
Impulse Voltage ◽  
The Difference ◽  
Theoretical Results

This work deals with the modification of the known calculating formulae for power transformer constants used for impulse voltage distribution, presents the effect of transformer constants and their modifications on impulse voltage and stress distributions in power transformers, and compares experimental and theoretical results considering measures taken to minimize the difference between them.

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