Laplace Transform Research Articles

Control analysis of an underactuated bio-inspired robot

Abstract This article is devoted to the control of bio-inspired robots that are underactuated. These robots are composed of tensegrity joints remotely actuated with cables, which mimic the musculoskeletal system of the bird neck. A computed torque control (CTC) is applied to these robots as well as an original control called pseudo computed torque control (PCTC). This new control uses the dynamics and the pseudo-inverse of the Jacobian matrix. The stability of the two proposed controls is then analyzed through linearization of the dynamic model and expression of the closed-loop transfer function in the Laplace domain. We show that, depending on the desired trajectory, the CTC can be unstable when the controlled variables are the end effector position and orientation. For a robot with many joints and a limited number of cables, the CTC is always unstable. Instead, the PCTC shows a large domain of stability. The analysis is complemented by experimental tests demonstrating that the CTC and PCTC exhibit similar performance when the CTC is stable. Furthermore, the PCTC maintains stability on trajectories where the CTC becomes unstable, showing robustness to perturbations as well.

A Note on Some Novel Laplace and Stieltjes Transforms Associated with the Relaxation Modulus of the Andrade Model

In the framework of linear viscoelasticity, the authors have previously calculated a novel inverse Laplace transform involving the Mittag–Leffler function in order to calculate the relaxation modulus in the Andrade model. Here, we generalize this result, calculating the inverse Laplace transform of a given function Fα,βs by using two different approaches: the Bromwich integral and the decomposition of Fα,βs in simple fractions. From both calculations, we obtain a set of novel Laplace and Stieltjes transforms.

Conformable Double Laplace Transform Method (CDLTM) and Homotopy Perturbation Method (HPM) for Solving Conformable Fractional Partial Differential Equations

In the present article, the method which was obtained from a combination of the conformable fractional double Laplace transform method (CFDLTM) and the homotopy perturbation method (HPM) was successfully applied to solve linear and nonlinear conformable fractional partial differential equations (CFPDEs). We included three examples to help our presented technique. Moreover, the results show that the proposed method is efficient, dependable, and easy to use for certain problems in PDEs compared with existing methods. The solution graphs show close contact between the exact and CFDLTM solutions. The outcome obtained by the conformable fractional double Laplace transform method is symmetrical to the gain using the double Laplace transform.

A novel approach for fractional pharmacokinetics modeling and integrating stochastic simulation techniques using Sibuya distribution

Fractional compartment models are widely used to simulate pharmacokinetics with power-law behavior. In this work, we study an approach, previously mainly used in Epidemiology for the age model, employing the Sibuya distribution, a discrete-time, non-Markov probability distribution, to effectively emulate Mittag-Leffler functions. We demonstrate the validity of our method by applying it to various pharmacokinetic datasets and comparing the results with traditional methods like the Numerical Inverse Laplace Transform (NILT) algorithm. Additionally, we propose two stochastic simulation algorithms, where the Monte Carlo SSA allows us to capture the randomness of data and the Gillespie-like stochastic numerical simulation technique can optimize computational resources in fractional analysis.

Ramping cells in the rodent medial prefrontal cortex encode time to past and future events via real Laplace transform

In interval reproduction tasks, animals must remember the event starting the interval and anticipate the time of the planned response to terminate the interval. The interval reproduction task thus allows for studying both memory for the past and anticipation of the future. We analyzed previously published recordings from the rodent medial prefrontal cortex [J. Henke et al., eLife10, e71612 (2021)] during an interval reproduction task and identified two cell groups by modeling their temporal receptive fields using hierarchical Bayesian models. The firing in the "past cells" group peaked at the start of the interval and relaxed exponentially back to baseline. The firing in the "future cells" group increased exponentially and peaked right before the planned action at the end of the interval. Contrary to the previous assumption that timing information in the brain has one or two time scales for a given interval, we found strong evidence for a continuous distribution of the exponential rate constants for both past and future cell populations. The real Laplace transformation of time predicts exponential firing with a continuous distribution of rate constants across the population. Therefore, the firing pattern of the past cells can be identified with the Laplace transform of time since the past event while the firing pattern of the future cells can be identified with the Laplace transform of time until the planned future event.

Stability analysis of random fractional-order nonlinear systems and its application

The research on stability analysis and control design for random nonlinear systems have been greatly popularized in recent ten years, but almost no literature focuses on the fractional-order case. This paper explores the stability problem for a class of random Caputo fractional-order nonlinear systems. As a prerequisite, under the globally and the locally Lipschitz conditions, it is shown that such systems have a global unique solution with the aid of the generalized Gronwall inequality and a Picard iterative technique. By resorting to Laplace transformation and Lyapunov stability theory, some feasible conditions are established such that the considered fractional-order nonlinear systems are respectively Mittag-Leffler noise-to-state stable, Mittag-Leffler globally asymptotically stable. Then, a tracking control strategy is established for a class of random Caputo fractional-order strict-feedback systems. The feasibility analysis is addressed according to the established stability criteria. Finally, a power system and a mass–spring-damper system modeled by the random fractional-order method are employed to demonstrate the efficiency of the established analysis approach. More critically, the deficiency in the existing literatures is covered up by the current work and a set of new theories and methods in studying random Caputo fractional-order nonlinear systems is built up.

SPGAN Optimized by Piranha Foraging Optimization for Thyroid Nodule Classification in Ultrasound Images.

In this research work, Semantic-Preserved Generative Adversarial Network optimized by Piranha Foraging Optimization for Thyroid Nodule Classification in Ultrasound Images (SPGAN-PFO-TNC-UI) is proposed. Initially, ultrasound images are gathered from the DDTI dataset. Then the input image is sent to the pre-processing step. During pre-processing stage, the Multi-Window Savitzky-Golay Filter (MWSGF) is employed to reduce the noise and improve the quality of the ultrasound (US) images. The pre-processed output is supplied to the Generalized Intuitionistic Fuzzy C-Means Clustering (GIFCMC). Here, the ultrasound image's Region of Interest (ROI) is segmented. The segmentation output is supplied to the Fully Numerical Laplace Transform (FNLT) to extract the features, such as geometric features like solidity, orientation, roundness, main axis length, minor axis length, bounding box, convex area, and morphological features, like area, perimeter, aspect ratio, and AP ratio. The Semantic-Preserved Generative Adversarial Network (SPGAN) separates the image as benign or malignant nodules. Generally, SPGAN does not express any optimization adaptation methodologies for determining the best parameters to ensure the accurate classification of thyroid nodules. Therefore, the Piranha Foraging Optimization (PFO) algorithm is proposed to improve the SPGAN classifier and accurately identify the thyroid nodules. The metrics, like F-score, accuracy, error rate, precision, sensitivity, specificity, ROC, computing time is examined. The proposed SPGAN-PFO-TNC-UI method attains 30.54%, 21.30%, 27.40%, and 18.92% higher precision and 26.97%, 20.41%, 15.09%, and 18.27% lower error rate compared with existing techniques, like Thyroid detection and classification using DNN with Hybrid Meta-Heuristic and LSTM (TD-DL-HMH-LSTM), Quantum-Inspired convolutional neural networks for optimized thyroid nodule categorization (QCNN-OTNC), Thyroid nodules classification under Follow the Regularized Leader Optimization based Deep Neural Networks (CTN-FRL-DNN), Automatic classification of ultrasound thyroids images using vision transformers and generative adversarial networks (ACUTI-VT-GAN) respectively.

Semi-analytical solutions for steady-state bending-bending coupled forced vibrations of a rotating wind turbine blade by means of Green's functions

Wind power has received significant attention in recent years as a renewable energy source. Wind turbine blades, characterized by their long lengths, are prone to damage as a result of aeroelastic instability. This paper aims to derive semi-analytical solutions for steady-state forced vibrations of rotating wind turbine blades, considering the coupling of bending, bending, and unsteady aerodynamic loads. The novelty of this work lies in the use of the Green's function method to solve differential dynamic equations with variable coefficients, which are induced by aerodynamic forces. To model a wind turbine blade, the Euler-Bernoulli beam model is employed, and Greenberg's expressions are taken into consideration. Governing equations for coupled vibrations of the blade are then determined. In order to solve the governing equations, displacement solutions are decomposed into quasi-static displacements and dynamic displacements. The Laplace transformation and Green's function methods are utilized to obtain fundamental solutions of the governing equations. Subsequently, employing the principle of superposition, Fredholm integral equations for steady-state forced vibration of a rotating wind turbine blade are derived. To spatially discretize Fredholm integral equations, compound trapezoid formulae and central finite difference approximations are employed. This discretization process leads to the formation of a system of algebraic equations. Semi-analytical solutions for coupled forced vibrations of the rotating wind turbine blade are obtained by solving the algebraic equations. In the numerical solution part, validation of proposed solutions are verified by comparing them with numerical and finite element solutions in the literature. Influences of some important physical parameters, such as the rotating velocity, the setting angle, the cone angle, the inflow ratio, and damping, on vibration responses of blades are discussed.

Exact Analysis of MHD Casson Fluid Flow Past an Exponentially Accelerated Vertical Plate in a Porous Medium with Radiation Absorption, Heat Generation, and Diffusion-Thermo Effects with Thermal and Solutal Ramped Conditions

The current investigation aims at to examine the effect of radiation absorption, heat generation and Dufour number on MHD Casson fluid flow past an exponentially accelerated vertical plate in a porous medium with chemical reaction. The governing equations for momentum, energy and concentration are solved by implementing the Laplace transformation method. Skin friction, rate of heat transfer and rate of mass transfer expressions are also extracted and depicted in tabular form. Investigation simulates that Casson parameter diminished the fluid velocity, whereas energy flux due to a mass concentration gradient improves the temperature field of the flow problem. In addition to this, temperature field is observed to be developed under the influence of radiation absorption and heat generation. Furthermore, the effects of different non-dimensional parameters on velocity field, temperature fluid and species concentration are exhibited graphically.

A transparent experimental modelling method for linear multiple-input multiple-output systems

This paper presents a transparent technique that simulates processes using input and output measurement data without the use of any complex optimization or iterative algorithms. The method is very simple and easy to understand as it comprises only of linear ordinary differential equations, Laplace transformations and least squares technique. The solution can be expressed only in two linear matrix equations. The main aim is to provide bachelor students of engineering with a tool to formulate transfer functions.The method is validated with artificially generated erroneous measurement data as well as using measurements obtained from a practical application at the university. Inclusion of dead times and estimation of initial conditions make it ideal to be used for various range of applications such as stable and unstable systems with and without damping, open and closed loop systems. Over-integration, normalizing and/or zeroing of different coefficients add more degrees of freedom to improve the model quality.

A fast and reliable semi-analytical method for assessing energy replenishment from fracturing-flooding in low-permeability and tight oil reservoirs

The development of low-permeability and tight oil reservoirs is challenged by insufficient natural energy and rapid production decline. Fracturing-flooding is a technique that relies on high-pressure and large-volume fluid injection to replenish reservoir energy, making it a significant method for rapidly boosting formation energy. To evaluate the energy replenishment effect of fracturing-flooding technology in low-permeability and tight reservoirs, this study proposes a semi-analytical method for quick calculation. This approach employs dimensionless simplification, Pedrosa's substitution, Laplace transformation, and Stehfest inversion methods to derive pressure solutions for both the stimulation region and the external matrix region, each with varying flow capacities. The average formation pressure (AFP) of the reservoir is determined using the area-weighted average method, and numerical verification is performed using a commercial simulator. A case study from the Binnan area, along with a sensitivity analysis, demonstrates that after 30 days of fracturing-flooding, the AFP of the reservoir increases to 46.97 MPa, the corresponding reservoir pressure coefficient rises from 1.2 to 1.68, and reservoir energy increases by 40%. The factors influencing energy replenishment are ranked as follows: reservoir thickness, injection rate, stress sensitivity coefficient, matrix permeability, stimulation region radius, and mobility ratio. This study provides theoretical guidance for optimizing fracturing-flooding development schemes in low-permeability and tight oil reservoirs and offers valuable reference for the industry.

On the Love Numbers of an Andrade Planet

AbstractThe Andrade rheological model is often employed to describe the response of solar system or extra‐solar planets to tidal perturbations, especially when their properties are still poorly constrained. While for uniform planets with steady‐state Maxwell rheology the analytical form of the Love numbers was established long ago, for the transient Andrade rheology no closed‐form solutions have been yet determined, and the planetary response is usually studied either semi‐analitically in the frequency domain or numerically in the time domain. Closed‐form expressions are potentially important since they could provide insight into the dependence of Love numbers upon the model parameters and the time‐scales of the isostatic readjustment of the planet. First, we focus on the Andrade rheological law in 1‐D and we obtain a previously unknown explicit form, in the time domain, for the relaxation modulus in terms of the higher Mittag‐Leffler transcendental function Eα,β(z) that generalizes the exponential function. Second, we consider the general response of an incompressible planetary model — often referred to as the “Kelvin sphere” — studying the Laplace domain, the frequency domain and the time domain Love numbers by analytical methods. Through a numerical approach, we assess the effect of compressibility on the Love numbers in the Laplace and frequency domains. Furthermore, exploiting the results obtained in the 1‐D case, we establish closed‐form — although not elementary — expressions of the time domain Love numbers and we discuss the frequency domain response of the Kelvin sphere with Andrade rheology analytically.

Exact analytical investigation of Duffing oscillator vibration spectra under time-periodic oscillatory external force

This work introduces an analytical technique for determining solutions to a highly nonlinear Duffing-harmonic oscillator model problem. The parametric solutions of Duffing oscillator vibrations for an undamped case are achieved analytically in terms of Adomian polynomials by implementing the straightforward approach of the Laplace Transformation, known as the Laplace Decomposition Procedure (LDP). Possible plots of both numerical and analytical results sketched for various parameters are also presented to support our discussion. These graphs demonstrate variations of position-time, speed-time, and speed-position for an undamped Duffing oscillator case. When analyzed in general, they can provide vibration pattern descriptions for a variety of physical and engineering system configurations. We also examine their physical structures and behavioral characteristics within the conceptual framework of chaotic formalism.

Analytical model development for vertical medium/deep ground heat exchangers with U-tube(s)

The U-shaped ground heat exchanger (UGHE) offers an effective solution for building heating and cooling purposes. In the practical applications of vertical UGHEs, the borehole wall temperature varies along the borehole depth due to the uneven initial ground temperature and transient heat transfer process. The assumption of a uniform temperature along the borehole wall in the existing quasi-3D model of UGHE causes a non-negligible error in the calculation of fluid temperature, especially for the cases of shallow-medium boreholes. Thus, two new analytical models are established to calculate the fluid temperatures for the single U-shaped GHE (1-UGHE) and double U-shaped GHE (2-UGHE), considering the depth-dependent borehole wall temperatures. The convolution theorem is employed to solve the mathematical problem of the nonlinear condition on the borehole wall, and the Laplace transformation is used for getting the final temperature expressions. The new analytical models are compared and verified by the on-site experiments. The results show that the two models exhibit promising prediction accuracies and similar variation trends in fluid temperatures. Then, parametrical and economical analyses for 1−/2- UGHEs have been carried out. Finally, the geographical thermal potentials of 1−/2- UGHEs across China have been comparatively investigated and the corresponding thermal map-based application potential recommendations have been given.

Magneto-photo-thermoelastic influences on a semiconductor hollow cylinder via a series-one-relaxation model

This article discusses the deformation of semiconductor cylinders in the context of photothermoelastic theory. The proposed model is used to describe thermal waves, plasma waves, and elastic waves and analyze the theoretical analysis of thermal deformation effects on semiconductor hollow cylinders. The interior of the hollow cylinder is clamped and unaffected by thermal loads and carrier concentrations, while the exterior is subject to sinusoidal heating and limited carrier density. In addition, the surface of the cylinder is surrounded by magnets in the direction of its axis. Initially, the governing equations are explained in Laplace domain and the Laplace inversion is used numerically. The results from thermal physics are presented graphically to investigate the impact of thermal relaxation and temperature on temperature of plasma thermoelastic waves. The effects of carrier diffusion coefficient and surface recombination rate on carrier concentration distribution are also discussed in detail.

Analytical Solution of Fractional Order Diffusion Equations Using Iterative Laplace Transform Method

In this present article, by using the Iterative Laplace Transform Method (ILTM), the diffusion equation of fractional order is solved. The ILTM, which works as a combination of two methods, the iterative method and the other is the Laplace transform method, is applied to several diffusion equations to obtain analytical solutions. The proposed method gives the closed-form of series solutions in terms of the Mittag-Leffler function, which is a queen of functions in fractional calculus. The main aim of this work is to present a simple but reliable algorithm for the solution of diffusion equations of the multi-dimensional type, which clearly describes the materials of density dynamics in the diffusion process. The results obtained by using the ILTM approach indicate that this approach is attractive computationally and implemented easily. Due to its straightforward approach and comfortable way of solving problems, the ILTM can be utilized to solve nonlinear fractional problems in various applied and engineering sciences.

Modelling transient unloading-triggered dynamic responses in rock mass under a non-hydrostatic geo-stress

Transient excavation unloading of in situ stress affects the stability of surrounding rock in the deep tunnel engineering. This paper focuses on the dynamic responses triggered by the transient unloading of a non-hydrostatic geo-stress field in deep-buried engineering. With resort to the integral transformation, the frequency-domain responses of stress, displacement and velocity components induced by the transient unloading are obtained under a non-hydrostatic in situ stress. Based on the numerical inversion of the Laplace transformation, the corresponding time-domain theoretical results are determined. Agreement of the current solutions with the existing results and the numerical simulations verifies the proposed scheme in this paper. The elastic analytical results indicate that the influence of unloading path on the stress redistribution is characterized by the unloading rate. The higher the unloading rate is, the larger the stress magnitude is. The smaller the unloading time is, the more remarkable vibration is. The elasto-plastic dynamic numerical responses are investigated on the basis of a self-developed code, elasto-plastic cellular automaton (EPCA), which is a module of CASRock. The numerical results demonstrate that the transient excavation unloading results in the stress redistribution and concentration of surrounding rock mass, inducing damage for the case of exceeding the capacity of rock mass. For the small lateral pressure coefficient (less than 0.25), the tensile-shear failure is the major damage mechanism of surrounding rock, while the shear failure is the major damage mechanism for the large lateral pressure coefficient. The analytical and numerical results can provide theoretical basis for the support and reinforcement of underground tunnel.

Bending of bidirectional functionally graded nonlocal stress-driven beam

This paper provides a comprehensive analysis, using nonlocal stress-driven integral theory, of the static behavior of a nanoscale beam of bidirectionally graded materials. After a brief explanation of the mathematical formulation of BDFGMs, the work done and strain energy expressions derived from the displacement field are discussed. Variational formulations and Hamilton's principle are used to develop the equilibrium equation. An analytical development of the nonlocal kernel for stress-driven integral theory and formulated governing equation which was nondimensionalized later. Explicit equations for displacement and moment are obtained by solving this equation using the Laplace transformation. Three different boundary conditions are examined, and differences in the maximum displacement with respect to the nonlocal parameter and the two material FGM parameters are displayed both visually and in table form. The results exhibit excellent agreement and provide a standard for further research when they are closely compared to the existing numerical data. This work contributes to the knowledge of BDFGMs under nonlocal effects generated by stress-driven integral theory and offers solutions that have been confirmed for further investigation.

Analytical solutions for out-of-plane response of curved beams resting on an elastic foundation under a random moving load

In this study, the vibration behaviour of a horizontally curved beam resting on an elastic foundation and subjected to a random moving force is studied. The governing coupled differential equations for the out-of-plane vibration of curved beams are derived based on the mode superposition method, Duhamel's integral, and the Laplace transformation. Note that the out-of-dynamic responses of curved beams induced by a random moving load are composed of the centred random and mean value. Analytical solutions are proposed for the beam’s vertical and torsional deflections and the bending moment, where the rotary inertia, warping constant, and foundation stiffness are incorporated in these formulations. Because analytical studies on the random vibration of a curved beam resting on an elastic foundation are very limited, the aim of this study is to investigate the effect of relative parameters, namely, high-mode vibrations, the radius of curvature, the moving velocity, and foundation vertical and torsional stiffnesses, on the centred random and mean values of the beam dynamic responses. The results show that high-mode vibrations have a greater impact on the beam responses at higher velocities. To obtain precise results, the first 30 modes are considered in the calculation for the dynamic responses of the beams. The critical velocity is an inherent property of bridges, and it increases as the radius of curvature and foundation stiffness increase. In addition, an increase in the velocity and foundation stiffness results in a reduction in the centred random values for responses.

Correlation function of thin-shell operators

In this study, we explore the correlation functions of thin-shell operators, represented semiclassically by a homogeneous, thin interface of dust particles. Employing the monodromy method, we successfully compute the contribution from the Virasoro vacuum block and present the monodromy equation in a closed form without assuming the probe limit. Although an analytical solution to the monodromy equation remains difficult, we demonstrate that it is perturbatively solvable within specific limits, including the probe limit, the heavy-shell limit, and the early-time limit. Moreover, we compare our results with gravitational calculations and find precise agreement. We strengthen our findings by proving that the thermal correlation functions in gravity, after an inverse Laplace transformation, satisfy the field theory’s monodromy equation. Additionally, we identify an infinite series of unphysical solutions to the monodromy equation and discuss their potential geometrical duals.