Bending of a Viscoelastic Timoshenko Cracked Beam Based on Equivalent Viscoelastic Spring Models

Considering the transverse crack as a massless viscoelastic rotational spring, the equivalent stiffness of the viscoelastic cracked beam is derived by Laplace transform and the generalized Dirac delta function. Using the standard linear solid constitutive equation and the inverse Laplace transform, the analytical expressions of the deflection and rotation angle of the viscoelastic Timoshenko beam with an arbitrary number of open cracks are obtained in the time domain. By numerical examples, the bending results of the analytical expressions are verified with those of the FEM program. Additionally, the effects of the time, slenderness ratio, and crack depth on the bending deformations of the different cracked beam models are revealed.

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The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions

Fractal and Fractional ◽

10.3390/fractalfract5010018 ◽

2021 ◽

Vol 5 (1) ◽

pp. 18

Author(s):

Nicos Makris

Keyword(s):

Laplace Transform ◽

Fractional Derivative ◽

Memory Function ◽

Recurrence Formula ◽

Delta Function ◽

Dirac Delta Function ◽

Inverse Laplace Transform ◽

Complex Materials ◽

Dirac Delta ◽

Anomalous Relaxation

Motivated from studies on anomalous relaxation and diffusion, we show that the memory function M(t) of complex materials, that their creep compliance follows a power law, J(t)∼tq with q∈R+, is proportional to the fractional derivative of the Dirac delta function, dqδ(t−0)dtq with q∈R+. This leads to the finding that the inverse Laplace transform of sq for any q∈R+ is the fractional derivative of the Dirac delta function, dqδ(t−0)dtq. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of sqsα∓λ where α<q∈R+, which is the fractional derivative of order q of the Rabotnov function εα−1(±λ,t)=tα−1Eα,α(±λtα). The fractional derivative of order q∈R+ of the Rabotnov function, εα−1(±λ,t) produces singularities that are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of q in association with the recurrence formula of the two-parameter Mittag–Leffler function.

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Electromagnetic Waves Through Disordered Systems: Comparison of Intensity, Transmission and Conductance

MRS Proceedings ◽

10.1557/proc-694-k5.8 ◽

2001 ◽

Vol 694 ◽

Author(s):

Fredy R Zypman ◽

Gabriel Cwilich

Keyword(s):

Probability Distribution ◽

Disordered Systems ◽

Electromagnetic Waves ◽

Delta Function ◽

Rayleigh Distribution ◽

Dirac Delta Function ◽

Universal Form ◽

Dirac Delta ◽

Diffusive Regime ◽

Analytical Expressions

AbstractWe obtain the statistics of the intensity, transmission and conductance for scalar electromagnetic waves propagating through a disordered collection of scatterers. Our results show that the probability distribution for these quantities x, follow a universal form, YU(x) = xne−xμ. This family of functions includes the Rayleigh distribution (when α=0, μ=1) and the Dirac delta function (α →+ ∞), which are the expressions for intensity and transmission in the diffusive regime neglecting correlations. Finally, we find simple analytical expressions for the nth moment of the distributions and for to the ratio of the moments of the intensity and transmission, which generalizes the n! result valid in the previous case.

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Laplace Transform for Finite Element Analysis of Electromagnetic Interferences in Underground Metallic Structures

Applied Sciences ◽

10.3390/app12020872 ◽

2022 ◽

Vol 12 (2) ◽

pp. 872

Author(s):

Andrea Cristofolini ◽

Arturo Popoli ◽

Leonardo Sandrolini ◽

Giacomo Pierotti ◽

Mattia Simonazzi

Keyword(s):

Finite Element ◽

Laplace Transform ◽

Electromagnetic Interference ◽

Inverse Laplace Transform ◽

Element Analysis ◽

Metallic Structures ◽

Numerical Inverse Laplace Transform ◽

Transient Electromagnetic ◽

The Time Domain ◽

High Voltage Power Lines

A numerical methodology is proposed for the calculation of transient electromagnetic interference induced by overhead high-voltage power lines in metallic structures buried in soil—pipelines for oil or gas transportation. A series of 2D finite element simulations was employed to sample the harmonic response of a given geometry section. The numerical inverse Laplace transform of the results allowed obtaining the time domain evolution of the induced voltages and currents in the buried conductors, for any given condition of the power line.

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Effect of East Gyro Drift and Initial Azimuth Error on the Compass Azimuth Alignment Convergence Time

Mathematical Problems in Engineering ◽

10.1155/2020/9042197 ◽

2020 ◽

Vol 2020 ◽

pp. 1-10

Author(s):

Dongxu He ◽

Xinle Zang ◽

Lei Ge

Keyword(s):

Laplace Transform ◽

Time Domain ◽

Oscillation Period ◽

Inverse Laplace Transform ◽

Convergence Time ◽

Initial Error ◽

Error Band ◽

Alignment System ◽

The Time Domain ◽

Gyro Drift

The effect of gyro constant drift and initial azimuth error on the convergence time of compass azimuth is analyzed in this article. Using our designed compass azimuth alignment system, we obtain the responses of gyro constant drift and initial azimuth error in the frequency domain. The corresponding response function in the time domain is derived using the inverse Laplace transform, and its convergence time is then analyzed. The analysis results demonstrate that the convergence time of compass azimuth alignment is related to the second-order damping oscillation period, the gyro constant drift, and the initial azimuth error. In this study, the error band is set to 0.01° to determine convergence. When the gyro drift is less than 0.05°/h, compass azimuth alignment can converge within 0.9 damping oscillation periods. When the initial azimuth error is less than 5°, compass azimuth alignment can converge within 1.4 damping oscillation periods. When both conditions are met, the initial error plays a major role in convergence, while gyro drift has a smaller effect on convergence time. Finally, the validity of our method is verified using simulations.

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On the Fractional Derivative of Dirac Delta Function and Its Application

Advances in Mathematical Physics ◽

10.1155/2020/1842945 ◽

2020 ◽

Vol 2020 ◽

pp. 1-7

Author(s):

Zaiyong Feng ◽

Linghua Ye ◽

Yi Zhang

Keyword(s):

Integral Equation ◽

Laplace Transform ◽

Fractional Derivative ◽

Fractional Order ◽

Delta Function ◽

Dirac Delta Function ◽

Fractional Order System ◽

Order System ◽

Dirac Delta ◽

Integer Order

The Dirac delta function and its integer-order derivative are widely used to solve integer-order differential/integral equation and integer-order system in related fields. On the other hand, the fractional-order system gets more and more attention. This paper investigates the fractional derivative of the Dirac delta function and its Laplace transform to explore the solution for fractional-order system. The paper presents the Riemann-Liouville and the Caputo fractional derivative of the Dirac delta function, and their analytic expression. The Laplace transform of the fractional derivative of the Dirac delta function is given later. The proposed fractional derivative of the Dirac delta function and its Laplace transform are effectively used to solve fractional-order integral equation and fractional-order system, the correctness of each solution is also verified.

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Semi-analytical solution for consolidation of vertical drain to unsaturated soils considering well resistance and smear effect under time-varying loadings

E3S Web of Conferences ◽

10.1051/e3sconf/202019802033 ◽

2020 ◽

Vol 198 ◽

pp. 02033

Author(s):

Aifang Qin ◽

Lianghua Jiang ◽

Linzhong Li ◽

Xinhao Li

Keyword(s):

Analytical Solution ◽

Laplace Transform ◽

Unsaturated Soils ◽

Inverse Laplace Transform ◽

Loading Frequency ◽

Time Varying ◽

Numerical Inverse Laplace Transform ◽

Vertical Drain ◽

Special Cases ◽

The Time Domain

In this paper, based on equal-strain assumption a semi-analytical solution, considering well re-sistance, smear effect and time-varying loading, is deduced for radial consolidation aided by vertical drain (VD) to unsaturated soils. Firstly, by employing the general integration, Laplace transform, decoupling methods and numerical inverse Laplace transform, the semi-analytical solution in the time domain is ob-tained. Then, its validity is verified by the special cases of the proposed solution under instantaneous loading. Finally, the case analysis show that the dissipation of excess pore pressures is accelerated with the decrease of smear coefficients (αa or αw) or well resistance factors (Ga or Gw). In addition, when the well resistance factor is less than 1, the barrier of VD material to flow can be ignored. Furthermore, a smaller value of the loading frequency of cyclic loading, the bigger the amplitude, and the less fluctuation period in the dissipa-tion rates. Moreover, the current solution can analyse the consolidation characteristics of unsaturated soils with VDs under arbitrary time-varying loadings, including cyclic loadings.

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ON THE PHENOMENOLOGY OF ELECTRICAL RELAXATION IN ROCKS

Geophysics ◽

10.1190/1.1440331 ◽

1973 ◽

Vol 38 (1) ◽

pp. 37-48 ◽

Cited By ~ 35

Author(s):

R. T. Shuey ◽

M. Johnson

Keyword(s):

Laplace Transform ◽

Time Domain ◽

Decay Time ◽

Inverse Laplace Transform ◽

Current Response ◽

Initial Value ◽

Real Frequency ◽

Voltage Curve ◽

Decay Times ◽

The Time Domain

The work of Keller (1959) with decay‐time distributions is extended by incorporating ideas developed by Gross (1953) for viscoelasticity. In addition to the usual premises of linear systems theory, we introduce the postulate that the response is nonresonant. Mathematically, this postulate is that the frequency response, continued off the real frequency axis, is an analytic function of the complex variable ω, except, possibly, at purely imaginary frequencies. The singularities in the response at purely imaginary frequencies are identified with decay times. There are two decay‐time distribution functions, one associated with resistivity and one associated with conductivity. The former is the inverse Laplace transform of the voltage response to a current step, and the latter is the inverse Laplace transform of the current response to a voltage step. These two decay spectra are not the same; in the normal situation where resistivity decreases and conductivity increases with increasing (real) frequency, the conductivity decay times are shorter than the corresponding resistivity decay times. There are several relations between the resistivity decay spectrum and the usual IP measures. The initial value of the time‐domain voltage curve is proportional to the integral over the spectrum of resistivity decay times. The area under the time‐domain voltage curve, divided by its initial value, is the mean of the distributed resistivity decay times. Finally, PFE and phase shift are roughly proportional to the strength of the resistivity decay in the relevant spectral region.

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