Modeling Vertical Planing Boat Motions using a Neural-Corrector Method

2021 ◽  
Author(s):  
Kyle E. Marlantes ◽  
Kevin J. Maki
Keyword(s):  
Differential Equations ◽  
High Speed ◽  
Nonlinear Differential Equations ◽  
Computational Cost ◽  
Higher Order ◽  
Operating Conditions ◽  
Irregular Waves ◽  
Linear Potential ◽  
Planing Craft ◽  
Wide Range

The dynamics of high-speed planing craft are complex and nonlinear. Standard analysis methods, such as linear potential theory, while convenient and computationally efficient, are often not suitable for use in predicting the dynamics of such craft because physical realities or design requirements invalidate the inherent assumptions. High-fidelity methods, such as state-of-the-art CFD simulations, can offer accurate solutions, but these methods are limited by computational cost and numerical sensitivity. In addition, these methods are not efficient enough to provide rapid evaluation of operability, i.e. simulations over a wide range of operating conditions and environments. This leaves few practical analysis options for small, high-speed craft designers who need to perform such predictions. In this paper, the authors present a neural-corrector method that shows promise in providing efficient predictions of vertical planing craft motions. The method retains higher-order terms typically truncated in the classical coupled 2-DOF system of ordinary differential equations using Long Short-Term Memory (LSTM) recurrent neural networks. In this manner, the robust solution provided by the linear model is retained, and the LSTM networks act as higher-order correctors. The correctors primarily regress on the solution, affording familiar numerical integration techniques for systems of nonlinear differential equations. Training and validation results from the method are compared to nonlinear simulations of 2-DOF motion of a Generic Prismatic Planing Hull (GPPH) at forward speed in head seas, with time histories given for both regular and irregular waves.

scholarly journals Oscillation Results for Nonlinear Higher-Order Differential Equations with Delay Term

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 446
Author(s):  
Alanoud Almutairi ◽  
Omar Bazighifan ◽  
Youssef N. Raffoul
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Higher Order ◽  
Middle Term ◽  
Delay Term ◽  
Integral Averaging Technique ◽  
Equations With Delay ◽  
Oscillation Of Solutions ◽  
Comparison Technique ◽  
Higher Order Differential Equations

The aim of this work is to investigate the oscillation of solutions of higher-order nonlinear differential equations with a middle term. By using the integral averaging technique, Riccati transformation technique and comparison technique, several oscillatory properties are presented that unify the results obtained in the literature. Some examples are presented to demonstrate the main results.

scholarly journals Asymptotic Dichotomy in a Class of Third-Order Nonlinear Differential Equations with Impulses

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Kun-Wen Wen ◽  
Gen-Qiang Wang ◽  
Sui Sun Cheng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Functional Differential Equations ◽  
Higher Order ◽  
Third Order ◽  
Order Nonlinear Differential Equation ◽  
Functional Differential

Solutions of quite a few higher-order delay functional differential equations oscillate or converge to zero. In this paper, we obtain several such dichotomous criteria for a class of third-order nonlinear differential equation with impulses.

Characteristics of air entrainment during dynamic wetting failure along a planar substrate

2014 ◽  
Vol 747 ◽  
pp. 119-140 ◽  
Author(s):  
E. Vandre ◽  
M. S. Carvalho ◽  
S. Kumar
Keyword(s):  
High Speed ◽  
Water Solution ◽  
Air Entrainment ◽  
Dynamic Contact ◽  
Operating Conditions ◽  
Dynamic Wetting ◽  
Glycerol Water ◽  
Planar Substrate ◽  
Wide Range ◽  
Stationary Plate

AbstractCharacteristic substrate speeds and meniscus shapes associated with the onset of air entrainment are studied during dynamic wetting failure along a planar substrate. Using high-speed video, the behaviour of the dynamic contact line (DCL) is recorded as a tape substrate is drawn through a bath of a glycerol/water solution. Air entrainment is identified by triangular air films that elongate from the DCL above some critical substrate speed. Meniscus confinement within a narrow gap between the substrate and a stationary plate is shown to delay air entrainment to higher speeds for a wide range of liquid viscosities, expanding upon the findings of Vandre, Carvalho & Kumar (J. Fluid Mech., vol. 707, 2012, pp. 496–520). A pressurized liquid reservoir controls the meniscus position within the confinement gap. It is found that liquid pressurization further postpones air entrainment when the meniscus is located near a sharp corner along the stationary plate. Meniscus shapes recorded near the DCL demonstrate that operating conditions influence the size of entrained air films, with smaller films appearing in the more viscous solutions. Regardless of size, air films become unstable to thickness perturbations and ultimately rupture, leading to the entrainment of air bubbles. Recorded critical speeds and air-film sizes compare well to predictions from a hydrodynamic model for dynamic wetting failure, suggesting that strong air stresses near the DCL trigger the onset of air entrainment.

Subsynchronous Vibration Patterns Under Reduced Oil Supply Flow Rates

2017 ◽  
Author(s):  
B. R. Nichols ◽  
R. L. Fittro ◽  
C. P. Goyne
Keyword(s):  
Flow Rate ◽  
High Speed ◽  
Operating Conditions ◽  
System Stability ◽  
Supporting Evidence ◽  
Test Rig ◽  
Flow Rates ◽  
Oil Supply ◽  
Bending Mode ◽  
Wide Range

Many high-speed, rotating machines across a wide range of industrial applications depend on fluid film bearings to provide both static support of the rotor and to introduce stabilizing damping forces into the system through a developed hydrodynamic film wedge. Reduced oil supply flow rate to the bearings can cause cavitation, or a lack of a fully developed film layer, at the leading edge of the bearing pads. Reducing oil flow has the well-documented effects of higher bearing operating temperatures and decreased power losses due to shear forces. While machine efficiency may be improved with reduced lubricant flow, little experimental data on its effects on system stability and performance can be found in the literature. This study looks at overall system performance of a test rig operating under reduced oil supply flow rates by observing steady-state bearing performance indicators and baseline vibrational response of the shaft. The test rig used in this study was designed to be dynamically similar to a high-speed industrial compressor. It consists of a 1.55 m long, flexible rotor supported by two tilting pad bearings with a nominal diameter of 70 mm and a span of 1.2 m. The first bending mode is located at approximately 5,000 rpm. The tiling-pad bearings consist of five pads in a vintage, flooded bearing housing with a length to diameter ratio of 0.75, preload of 0.3, and a load-between-pad configuration. Tests were conducted over a number of operating speeds, ranging from 8,000 to 12,000 rpm, and bearing loads, while systematically reducing the oil supply flow rates provided to the bearings under each condition. For nearly all operating conditions, a low amplitude, broadband subsynchronous vibration pattern was observed in the frequency domain from approximately 0–75 Hz. When the test rig was operated at running speeds above its first bending mode, a distinctive subsynchronous peak emerged from the broadband pattern at approximately half of the running speed and at the first bending mode of the shaft. This vibration signature is often considered a classic sign of rotordynamic instability attributed to oil whip and shaft whirl phenomena. For low and moderate load conditions, the amplitude of this 0.5x subsynchronous peak increased with decreasing oil supply flow rate at all operating speeds. Under the high load condition, the subsynchronous peak was largely attenuated. A discussion on the possible sources of this subsynchronous vibration including self-excited instability and pad flutter forced vibration is provided with supporting evidence from thermoelastohydrodynamic (TEHD) bearing modeling results. Implications of reduced oil supply flow rate on system stability and operational limits are also discussed.

scholarly journals On Global Dynamics in Duffing Equation with Quasiperiodic Perturbation

2020 ◽  
Vol 22 (2) ◽  
pp. 164-176
Author(s):  
Timofey N. Dragunov ◽  
Kirill E. Morozov ◽  
Albert D. Morozov
Keyword(s):  
Differential Equations ◽  
Dimensional Space ◽  
Nonlinear Differential Equations ◽  
Uniform Mesh ◽  
Global Dynamics ◽  
Nonlinear Operator Equations ◽  
Wide Range ◽  
Perturbations Of Initial Data ◽  
Analysis Of Stability ◽  
Stated Problem

An iterative method for solution of Cauchy problem for one-dimensional nonlinear hyperbolic differential equation is proposed in this paper. The method is based on continuous method for solution of nonlinear operator equations. The keystone idea of the method consists in transition from the original problem to a nonlinear integral equation and its successive solution via construction of an auxiliary system of nonlinear differential equations that can be solved with the help of different numerical methods. The result is presented as a mesh function that consists of approximate values of the solution of stated problem and is constructed on a uniform mesh in a bounded domain of two-dimensional space. The advantages of the method are its simplicity and also its universality in the sense that the method can be applied for solving problems with a wide range of nonlinearities. Finally it should be mentioned that one of the important advantages of the proposed method is its stability to perturbations of initial data that is substantiated by methods for analysis of stability of solutions of systems of ordinary differential equations. Solving several model problems shows effectiveness of the proposed method.

scholarly journals Homotopy Perturbation Method

2017 ◽  
pp. 24-61
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Perturbation Method ◽  
Exact Solutions ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Homotopy Perturbation ◽  
Wide Range ◽  
Transfer Problems

The homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.

scholarly journals Higher Order Haar Wavelet Method for Solving Differential Equations

2020 ◽  
Author(s):  
Jüri Majak ◽  
Mart Ratas ◽  
Kristo Karjust ◽  
Boris Shvartsman
Keyword(s):  
Differential Equations ◽  
Computational Cost ◽  
Haar Wavelet ◽  
Higher Order ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Convergence Results ◽  
Key Characteristics ◽  
Parameter Values ◽  
Excellent Accuracy

The study is focused on the development, adaption and evaluation of the higher order Haar wavelet method (HOHWM) for solving differential equations. Accuracy and computational complexity are two measurable key characteristics of any numerical method. The HOHWM introduced recently by authors as an improvement of the widely used Haar wavelet method (HWM) has shown excellent accuracy and convergence results in the case of all model problems studied. The practical value of the proposed HOHWM approach is that it allows reduction of the computational cost by several magnitudes as compared to HWM, depending on the mesh and the method parameter values used.

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