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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 16 Apr 2024 15:39:56 +0000Wave Propagation in an Elastic Plate Resting on an Elastic Foundation
https://resolver.caltech.edu/CaltechTHESIS:08172011-153814513
Authors: {'items': [{'id': 'Lloyd-James-Reily', 'name': {'family': 'Lloyd', 'given': 'James Reily'}, 'show_email': 'NO'}]}
Year: 1962
DOI: 10.7907/D5NC-XR09
Presented is an analysis of wave propagation in an infinite elastic plate or beam on an elastic foundation. The results are presented in two parts:
1. The frequency spectra (frequency as a function of wave number) for the problem based on existing approximate bending theories are compared with the spectra based on the exact equations of motion from linear elasticity theory. The existence of complex wave numbers is established in each case. A distinct similarity is found between the
spectrum representing the more exact theory of bending (Timoshenko bending mechanism) and the exact Rayleigh-Lamb spectrum for symmetric waves in a free elastic plate. Good agreement between approximate theories and the exact equations is found for soft foundations under the usual restrictions of low frequency-long waves.
2. The transient response is considered for the exact theory and the more exact theory of bending. In both cases suddenly applied line loads are considered. In the latter case the related point load problem is also studied. Two distinct integral transform methods of solution are
presented and used in these problems. For one of these methods the contributions from the various modes, including the complex arms, are identified with certain integrals that are components of the solution. Results from numerical computation of these integrals are presented and
analyzed for the more exact theory of bending using two different foundation stiffnesses.
https://thesis.library.caltech.edu/id/eprint/6589Analysis of Long Compressional Elastic Waves in Rods of Arbitrary Cross Section and Elastic Wave Fronts in Plates and Circular Rods
https://resolver.caltech.edu/CaltechTHESIS:08192011-084654605
Authors: {'items': [{'id': 'Rosenfeld-Robert-Leopold', 'name': {'family': 'Rosenfeld', 'given': 'Robert Leopold'}, 'show_email': 'NO'}]}
Year: 1962
DOI: 10.7907/WVNP-Z097
Long waves in elastic rods of arbitrary cross section are studied by writing a general expansion of the exact solution for three dimensional linear elasticity. The solution holds for transient excitation of the end of
a semi-infinite cylinder and is in terms of the harmonic modes of wave propagation for the infinite elastic cylinder. The major contribution to the solution for large distances from the end of the rod is found by making
approximations to the infinitely long wave length part of the solution. This is aided by using a perturbation method for long wave length to study the modes of propagation. An approximate theory for rods of arbitrary cross section is developed and compared to the exact theory for harmonic
waves of infinitely long wave lengths.
The amplitudes and locations of all wave fronts caused by certain suddenly applied loads on elastic plates and circular rods are presented. Both end loads on the rod and plate as well as normal line and point forces on the plate are considered. The problems are solved by expanding
double transforms into a series of terms, each term representing the disturbance following a single wave front. Evaluation of the terms for the wave front behavior is accomplished by Cagniard's method and the saddle point method. Ray theory aids in the interpretation of the
results and also serves to verify most of the formulas. The solution by Cagniard's method is exact for the plane strain problems studied and is plotted and compared to experiments.
https://thesis.library.caltech.edu/id/eprint/6600Transient Wave Propagation in Elastic Plates with Cylindrical Boundaries, Studied with the Aid of Multi-integral Transforms
https://resolver.caltech.edu/CaltechETD:etd-10042002-130206
Authors: {'items': [{'id': 'Scott-Richard-Anthony', 'name': {'family': 'Scott', 'given': 'Richard Anthony'}, 'show_email': 'NO'}]}
Year: 1964
DOI: 10.7907/G7HA-F836
Some mixed time dependent boundary value problems for isotropic elastic plates with circular cylindrical boundaries are studied using the linear equations of elasticity. A multi-integral transform approach is employed, necessitating the introduction of extended Hankel transforms, and formal solutions are obtained with the aid of residue theory. Some properties of the Rayleigh-Lamb frequency equation, pertinent to the inversion processes, are derived. The problem of a free infinite plate with a circular cylindrical cavity subjected to a step normal displacement is studied in detail and numerical information for the far-field, showing the effect of the cavity radius on the displacements, is obtained using stationary phase techniques.
The generation of transient elastic waves in free isotropic infinite elastic plates by time dependent body forces is also treated and the results for a radial body force, with step time-dependence, are compared with the corresponding platecavity results. Good agreement between the two is found in the far-field.
Similar problems for a free, transversely isotropic, semi-infinite plate (slab) are also studied and some numerical information for the farfield is obtained using the head of the pulse method. Stationary phase solutions for an isotropic slab subjected to a step edge displacement are obtained and compared with the corresponding plate-cavity results. It is found that at a fixed station the plate cavity solutions approach those for the slab, as the cavity radius goes to zero. A comparison between the head of the pulse and stationary pulse results for the isotropic slab is also made and some discrepancies between the two are found.https://thesis.library.caltech.edu/id/eprint/3895Plane-strain diffraction of transient waves by a circular cavity
https://resolver.caltech.edu/CaltechETD:etd-01142004-144633
Authors: {'items': [{'id': 'Peck-J-C', 'name': {'family': 'Peck', 'given': 'Jerry Clifford'}, 'show_email': 'NO'}]}
Year: 1965
DOI: 10.7907/GDNE-E586
The plane-strain problem of the diffraction of a transient plane dilatation wave by a circular cavity in an elastic medium is treated. The method used determines the (total) solution only in the shadow zone, i.e., those points which cannot be connected to the source of disturbance by straight-line rays. Numerical results are obtained for the velocities and displacements on the "back" surface of the cavity caused by a step-stress incident wave.
The analysis is based on a method devised by Friedlander (see his book Sound Pulses, Cambridge, 1958) for the analogous acoustic diffraction problem. This method converges most rapidly at short time, in contrast to Fourier series methods. The Friedlander method essentially employs integral transforms on both time and [Theta], the circumferential coordinate. In the shadow zone, the [Theta]-inversion can be performed by residue theory, the residues resulting from poles at the roots of a "frequency equation." The roots are infinite in number, and may be regarded as forming a dispersion spectrum relating the frequencies and angular wave numbers of a series of circumferential propagation modes. The time-transform inversion is carried out by contour integration and subsequent numerical evaluation.
The transient response results are found to compare well with the Fourier-series solutions at moderate to long times, but at short time the differences are marked, as would be expected. The fact that the present technique yields good long-time results suggests it is even more powerful than might be expected. The major limitation of the numerical method is its restriction to the shadow zone.https://thesis.library.caltech.edu/id/eprint/166Propagation of Harmonic Waves in an Elastic Rod of Elliptical Cross-Section
https://resolver.caltech.edu/CaltechETD:etd-09232002-145548
Authors: {'items': [{'id': 'Wong-Po-Kee', 'name': {'family': 'Wong', 'given': 'Po Kee'}, 'show_email': 'NO'}]}
Year: 1966
DOI: 10.7907/CX6S-JQ42
Using the potential equations of motion of linear elasticity, the propagation of harmonic waves in an infinite rod of elliptical cross-section is investigated. The frequency equations for the propagation of flexural waves in rods with (i) zero surface displacements, and (ii) zero surface stresses are obtained in the form of infinite determinants, the elements of which involve Mathieu functions and their derivatives. It is shown that these determinants can be written in diagonal form when the eccentricity goes to zero and in the light of this possible numerical procedures are discussed for small values of the eccentricity.https://thesis.library.caltech.edu/id/eprint/3715Diffraction of Transient Elastic Waves by a Spherical Cavity
https://resolver.caltech.edu/CaltechETD:etd-10022002-110856
Authors: {'items': [{'id': 'Norwood-Frederick-Reyes', 'name': {'family': 'Norwood', 'given': 'Frederick Reyes'}, 'show_email': 'NO'}]}
Year: 1967
DOI: 10.7907/ps9n-9z78
<p>The diffraction of transient elastic waves by a spherical cavity is treated. Two cases are considered: (a) a suddenly applied normal point load, and (b) the impingement of a plane transient pulse on the cavity. The method used determines the solution only in the shadow zones; that is, those points which cannot be connected to the source of disturbance by straight-line rays. Analytical results are obtained and evaluated for the displacements at the cavity wall.</p>
<p>The analysis is based on the Laplace transform (on time) and the Watson transformation. This well-known transformation makes it possible to convert an infinite series involving a discrete real wave number into one involving a generalized wave number. This leads to transient solutions the components of which have a one-to-one correspondence with the modes of the underlying frequency equation. These solutions have a form convenient for numerical analysis and for obtaining approximate solutions.</p>
<p>The results given here are for the displacements evaluated at the cavity wall. It is found that the behavior of the diffracted wave fronts is similar to that associated with the simpler equations governing scalar diffraction problems (see Friedlander, "Sound Pulses" , Cambridge, 1958). In both problems the Rayleigh disturbance predominates for long time, being singular at its arrival time in the point load case and non-singular in the plane wave case.</p>
https://thesis.library.caltech.edu/id/eprint/3861Transient excitation of an elastic half-space by a point load traveling on the surface
https://resolver.caltech.edu/CaltechETD:etd-03022006-134535
Authors: {'items': [{'id': 'Gakenheimer-D-C', 'name': {'family': 'Gakenheimer', 'given': 'David Charles'}, 'show_email': 'NO'}]}
Year: 1969
DOI: 10.7907/5FEH-EG57
The propagation of transient waves in an elastic half-space excited by a traveling normal point load is investigated. The load is suddenly applied and then it moves rectilinearly at a constant speed along the free surface. The displacements are computed for all points of the half-space as well as for all load speeds.
The disturbance is analyzed by using multi-integral transforms and an inversion scheme based on the well-known Cagniard technique. This reduces the displacements to single integral and algebraic contributions, each of which is identified as the disturbance behind a specific wave front. The same solution is valid for all load speeds, even though the wave front geometry varies greatly, depending on the speed of the load relative to the body wave speeds. Moreover, the surface displacements are obtained from the interior ones, but only after the Rayleigh waves are computed by a separate calculation. Then, by taking advantage of the form of the exact solution, wave front expansions and Rayleigh wave approximations are computed for all load speeds.
Several other analytical results are obtained for restricted values of the load speed. In particular, when it exceeds both of the body wave speeds the steady-state displacement field is separated from the transient one and reduced to algebraic form. Also, for the limit case of zero load speed a new representation of the interior displacements for Lamb's point load problem is displayed in terms of single integrals.
https://thesis.library.caltech.edu/id/eprint/837Modal Representations for the High-Frequency Response of Elastic Plates
https://resolver.caltech.edu/CaltechTHESIS:06082017-101810031
Authors: {'items': [{'id': 'Randles-Philip-Wayne', 'name': {'family': 'Randles', 'given': 'Philip Wayne'}, 'show_email': 'NO'}]}
Year: 1969
DOI: 10.7907/F4BM-J980
<p>Representations for the high-frequency response of a
suddenly loaded infinite plate are obtained from the modal
form of the exact solution. The method of approach is presented
by treating a linearly elastic, homogeneous, isotropic
plate subjected to a normal impulsive line load on
one face.</p>
<p>An investigation of the branches of the governing
Rayleigh-Lamb frequency equation is given. These branches
are closely related to the modes of propagation, the sum
of which is the modal solution. The relationship between
the high-frequency portions of the underlying frequency
spectra and the high-frequency response is brought out.</p>
<p>Series representations for the branches are used to
facilitate a summation over the branch (or mode) numbers.
This results in convenient high-frequency representations,
which exhibit all of the expected singular wave fronts in
the plate.</p>
<p>The method appears to be applicable to a broader
class of problems than other methods which have been used
for the high-frequency response of a plate.</p>https://thesis.library.caltech.edu/id/eprint/10315Strong Motion Accelerograph Evaluation
https://resolver.caltech.edu/CaltechTHESIS:06082017-105754537
Authors: {'items': [{'id': 'Peters-Rex-Bredesen', 'name': {'family': 'Peters', 'given': 'Rex Bredesen'}}]}
Year: 1969
DOI: 10.7907/TEJJ-XX37
<p>A brief study is made of the effect of common instrument
errors on the accuracy of data obtained from strong motion earthquake
accelerographs. Error sources considered include zero drift,
tilts, nonlinearities, cross-axis sensitivity, lack of initial conditions,
noise, and time base errors. It is concluded that most data of current
engineering interest are not critically affected by the level of errors
found in existing accelerographs. Techniques are suggested for
reducing or eliminating many of these errors by instrument design
changes.</p>
<p>An experimental study is made of a new strong motion
accelerograph during its engineering development. This new accelerograph
is designed to record an FM analog of ground acceleration
on magnetic tape, providing a record which may be rapidly and
automatically converted to digital form. The accuracy limits of the
accelerograph are explored and the design reasons for these limits
investigated. The more significant findings may be briefly
summarized:</p>
<p>(1) Static accuracy. The sensitivity and linearity of the instrument
are found to depend critically on a series of interdependent adjustments.
Reasonable care will bring errors in both of these
quantities to within ±2% of 1/2 g full scale. Higher accuracies
are possible, but require much more time and care, primarily
due to the limiting effect of mechanical drift in the accelerometers.</p>
<p>(2) Zero point drift. Uncertainties in the accelerometer zero
point arise from both mechanical and electronic drifts. Long
term drifts may be related to temperature or relative humidity, or
may be entirely random. Short term drifts of up to 2% of full scale
may occur during the course of a typical record. The total
variation may be as much as ± 30% of full scale for a 100°F range
of temperatures. These variations require adjustment of the data
before processing, but are not sufficient to interfere with operation
of the accelerograph.</p>
<p>(3) Noise. Random noise in the system as tested amounted to
1.4% of full scale, RMS, and was mostly due to the tape recording
system. By comparison with optical accelerographs, this noise
figure is marginal, but acceptable, and can be improved by changes
to the compensation system.</p>
<p>(4) Timing. The advantages of an effectively continuous time base
over discrete time marks were discovered and means devised to
obtain such a base from the test accelerograph. This method of
timing is a qualitative improvement over the best system which is
practical on optical recorders.</p>
<p>The overall performance of the test accelerograph is
adequate to yield acceptably accurate acceleration vs. time records
and Fourier spectra within the range of frequencies which are of
current engineering interest. It is able to produce useful displacement
records only for periods shorter than several seconds. The
reasons for this latter limitation are sufficiently fundamental that
markedly superior instruments are not expected to be available
within the next ten years.</p>https://thesis.library.caltech.edu/id/eprint/10316On Nonmixed Symmetric End-Load Problems in Elastic Waveguides
https://resolver.caltech.edu/CaltechTHESIS:07102018-114156713
Authors: {'items': [{'id': 'Sinclair-Glenn-Bruce', 'name': {'family': 'Sinclair', 'given': 'Glenn Bruce'}}]}
Year: 1973
DOI: 10.7907/ZDWG-ZG20
This investigation deals with the response of the semi-infinite,
linear elastic, homogeneous, isotropic plate in plane strain, subject
to symmetric normal loads acting, in the absence of shear stress,
on its edge. A double Laplace transform technique is used to obtain
long-time information for two problems; a uniform load and a line-load.
Near- and far-field approximations are found, the far-field approximations
giving the integral of the Airy integral for both problems.https://thesis.library.caltech.edu/id/eprint/11104Wave front analysis in the scattering of a plane compressional pulse by a cylindrical elastic inclusion
https://resolver.caltech.edu/CaltechETD:etd-10132005-083854
Authors: {'items': [{'id': 'Griffin-J-H', 'name': {'family': 'Griffin', 'given': 'Jerry Howard'}, 'show_email': 'NO'}]}
Year: 1974
DOI: 10.7907/AKFT-PB89
The plane-strain problem of a stress pulse striking an elastic circular cylindrical inclusion embedded in an infinite elastic medium is treated. The method used determines dominant stress singularities that arise at wave fronts from the focusing of waves refracted into the interior. It is found that a necessary and sufficient condition for the existence of a propagating stress singularity is that the incident pulse has a step discontinuity at its front. The asymptotic wave front behavior of the first few P and SV waves to focus are determined explicitly and it is shown that the contribution from other waves are less important. In the exterior, it is found that in most composite materials the reflected waves have a singularity at their wave front which depends on the angle of reflection. Also the wave front behavior of the first few singular transmitted waves is given explicitly.
The analysis is based on the use of a Watson-type lemma, developed here, and Friedlander's method (see his book Sound Pulses, Cambridge, 1958). The lemma relates the asymptotic behavior of the solution at the wave front to the asymptotic behavior of its Fourier transform on time for large values of the transform parameter. Friedlander's method is used to represent the solution in terms of angularly propagating wave forms. This method employs integral transforms on both time and [theta], the circumferential coordinate. The [theta] inversion integral is asymptotically evaluated for large values of the time transform parameter by use of appropriate asymptotics for Bessel and Hankel functions and the method of stationary phase. The Watson-type lemma is then used to determine the behavior of the solution at singular wave fronts.
The Watson-type lemma is generally applicable to problems which involve singular loadings or focusing in which wave front behavior is important. It yields the behavior of singular wave fronts whether or not the singular wave is the first to arrive. This application extends Friedlander's method to an interior region and physically interprets the resulting representation in terms of ray theory.
https://thesis.library.caltech.edu/id/eprint/4063