Theory of the backward beam displacement on periodically corrugated surfaces and its relation to leaky Scholte-Stoneley waves
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A demonstration of the capability of the inhomogeneous wave theory to simulate backward displacement of ultrasonic-bounded beams [M. A. Breazeale and M. Torbett, Appl. Phys. Lett. 29, 456 (1976)] has been demonstrated recently [N. F. Declercq, J. Degrieck, R. Briers, and O. Leroy, Appl. Phys. Lett. 82, 2533 (2003)]. The current report applies the theory of the diffraction of inhomogeneous waves and shows how this theory is capable of simulating, explaining, and understanding the experiments mentioned above. The theory reveals the existence of leaky Scholte-Stoneley waves, a phenomenon suggested theoretically [N. F. Declercq, J. Degrieck, R. Briers, and O. Leroy, J. Acoust. Soc. Am. 112, 2414 (2002)] and observed experimentally [A. A. Teklu, M. A. Breazeale, J. Acoust. Soc. Am. 113, 2283 (2003)]. Moreover, the present paper shows that the classical Fourier decomposition of bounded beams is unable to simulate the backward beam displacement. This work also elucidates the nature of Wood anomalies in Diffraction spectra.Keywords:
Timoshenko beam theory
In this note, we prove that the displacement of a linearly damped string under bounded distributed forces is bounded. The proof is achieved by showing that an energy-like (Lyapunov) function of the string is bounded.
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The use of the Google Scholar produces about 78,000 hits on the term “Timoshenko beam.” The question of priority is of great importance for this celebrated theory. For the first time in the world literature, this study is devoted to the question of priority. It is that Stephen Prokofievich Timoshenko had a co-author, Paul Ehrenfest. It so happened that the scientific work of Timoshenko dealing with the effect of rotary inertia and shear deformation does not carry the name of Ehrenfest as the co-author. In his 2002 book, Grigolyuk concluded that the theory belonged to both Timoshenko and Ehrenfest. This work confirms Grigolyuk’s discovery, in his little known biographic book about Timoshenko, and provides details, including the newly discovered letter of Timoshenko to Ehrenfest, which is published here for the first time over a century after it was sent. This paper establishes that the beam theory that incorporates both the rotary inertia and shear deformation as is known presently, with shear correction factor included, should be referred to as the Timoshenko-Ehrenfest beam theory.
Timoshenko beam theory
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Abstract In this paper, the theory of a Timoshenko-Ehrenfest beam is revisited and given a new perspective with particular emphasis on the relative significances of the parameters underlying the theory. The investigation is intended to broaden the scope and applicability of the theory. It has been shown that the two parameters that characterise the Timoshenko-Ehrenfest beam theory, namely the rotary inertia and the shear deformation, can be related and hence they can be combined into one parameter when predicting the beam’s free vibration behaviour. A theoretical proof is given that explains why the effect of the shear deformation on the free vibration behaviour of a Timoshenko-Ehrenfest beam for any boundary condition will be always more pronounced than that of the rotary inertia. The range of applicability of the Timoshenko-Ehrenfest beam theory for realistic problems is demonstrated by a set of new curves, which provide considerable insights into the theory.
Timoshenko beam theory
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When a sound beam is incident onto a periodically corrugated surface, diffraction of the incident sound will be generated. The major diffraction phenomenon, which can be well explained by the classical grating equation, can be easily observed and has been intensively studied. In this work, we report an observation of diffracted waves whose intensity is much weaker than the major diffraction, and who are not expected to appear in the diffraction field. This secondary diffraction can be experimentally observed in the general diffraction configuration as well as in the Bragg diffraction configuration. The analysis of the direction and frequency of the diffracted waves based on the classical grating equation suggests that this diffraction is originated from a propagating wave along the corrugated surface. Such a propagating wave is possibly the experimental evidence of the existence of surface acoustic wave on corrugated interface generated by diffraction.
Diffraction topography
Diffraction efficiency
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Abstract In this paper, the theory of a Timoshenko–Ehrenfest beam is revisited and given a new perspective with particular emphasis on the relative significances of the parameters underlying the theory. The investigation is intended to broaden the scope and applicability of the theory. It has been shown that the two parameters that characterize the Timoshenko–Ehrenfest beam theory, namely the rotary inertia and the shear deformation, can be related, and hence, they can be combined into one parameter when predicting the beam’s free vibration behavior. It is explained why the effect of the shear deformation on the free vibration behavior of a Timoshenko–Ehrenfest beam for any boundary condition will be always more pronounced than that of the rotary inertia. The range of applicability of the Timoshenko–Ehrenfest beam theory for realistic problems is demonstrated by a set of new curves, which provide considerable insights into the theory.
Timoshenko beam theory
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It is shown that the Fraunhofer approximation, used in the kinematical theory of X-ray diffraction, may fail for a wide variety of crystals with different perfection. The kinematical theory describing the diffraction pattern in the general case is developed. The case of spherical-wave diffraction by a plane parallel crystal is considered in detail. The intensity distribution and the diffraction line width are ascertained to be essentially dependent on the region of diffraction in which the observation plane is located. On the other hand, the diffraction pattern geometry is independent of the diffraction region and is determined only by the crystal structure and the optics of diffraction. The geometry of the diffraction pattern recorded by the divergent-beam method is analysed in detail.
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Crystal (programming language)
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In this paper, we extend some recent results of Guedj-Lu-Zeriahi about psh envelopes of bounded functions on bounded domains in $\mathbb{C}^n$. We also present a result on the regularity of psh envelopes.
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This paper considers the input-output stability of a control system that is composed of a linear time-invariant multivariable system interconnecting with multiple decoupled time-invariant memoryless nonlinearities.The objectives of the paper are twofold.First and foremost, we prove (under certain assumptions) that if the multivariable Popov criterion is satisfied, then the system outputs and the nonlinearity inputs are bounded for any exogeneous input having bounded magnitude and bounded slope, and for all the nonlinearities lying in given sector bounds.As a consequence of using the convolution algebra, the obtained result is valid for rational and nonrational transfer functions.Second, for the case in which the transfer functions associated with the Popov criterion are rational functions, we develop a useful inequality for stabilizing the system by numerical methods.This is achieved by means of the positive real lemma and known results on linear matrix inequalities.To illustrate the usefulness of the inequality, a numerical example is provided.
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Timoshenko beam theory
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We consider real valued functions f defined in a closed interval I (bounded or unbounded), with nth differences $$ \Delta _{h}^{n}f(x) = \sum\limits_{i} {{{{( - 1)}}^{{n - i}}}} \left( {\begin{array}{*{20}{c}} n \\ i \\ \end{array} } \right)f(x + ih) $$ bounded for some fixed n.
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