scholarly journals A Convenient Approximation of the Projectile Motion with Quadratic Air Resistance

2021 ◽  
Author(s):  
Marc Landon
Keyword(s):  
Analytic Solution ◽  
The Body ◽  
Integration Scheme ◽  
Reduction Principle ◽  
Initial State ◽  
Resistive Force ◽  
Projectile Motion ◽  
Integration Schemes ◽  
Air Resistance ◽  
Standard Integration

Abstract A convenient approximated analytic solution is proposed for the problem of the motion of a body under a resistive force, acting in the magnitude of the squared velocity of the body. This solution is an explicit function of time, that keeps a good behavior both near the initial state and far from the initial state. To obtain a general analytic solution, we firstly used a reduction principle to be able to manipulate scalar objects, and we analyzed limit behaviors, both near the initial state and far from the initial state. Secondly, we proposed an approximated analytic solution with heuristics based on the built knowledge. Finally, a robust and stable integration scheme is proposed, based on the obtained analytic solution. We compared the scheme with other standard integration schemes.

scholarly journals Investigation of Air Pocket Behavior in Pipelines Using Rigid Column Model and Contributions of Time Integration Schemes

Water ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 785
Author(s):  
Arman Rokhzadi ◽  
Musandji Fuamba
Keyword(s):  
Transient Flow ◽  
Time Integration ◽  
Driving Pressure ◽  
Implicit Time ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Time Step ◽  
Column Model ◽  
Integration Schemes ◽  
Backward Euler

This paper studies the air pressurization problem caused by a partially pressurized transient flow in a reservoir-pipe system. The purpose of this study is to analyze the performance of the rigid column model in predicting the attenuation of the air pressure distribution. In this regard, an analytic formula for the amplitude and frequency will be derived, in which the influential parameters, particularly, the driving pressure and the air and water lengths, on the damping can be seen. The direct effect of the driving pressure and inverse effect of the product of the air and water lengths on the damping will be numerically examined. In addition, these numerical observations will be examined by solving different test cases and by comparing to available experimental data to show that the rigid column model is able to predict the damping. However, due to simplified assumptions associated with the rigid column model, the energy dissipation, as well as the damping, is underestimated. In this regard, using the backward Euler implicit time integration scheme, instead of the classical fourth order explicit Runge–Kutta scheme, will be proposed so that the numerical dissipation of the backward Euler implicit scheme represents the physical dissipation. In addition, a formula will be derived to calculate the appropriate time step size, by which the dissipation of the heat transfer can be compensated.

scholarly journals A Semi-Implicit Version of the MPAS-Atmosphere Dynamical Core

2015 ◽  
Vol 143 (9) ◽  
pp. 3838-3855 ◽  
Author(s):  
Steven Sandbach ◽  
John Thuburn ◽  
Danail Vassilev ◽  
Michael G. Duda
Keyword(s):  
Time Integration ◽  
Standard Test ◽  
Newton Iteration ◽  
Atmospheric Modeling ◽  
Implicit Time ◽  
Integration Scheme ◽  
Baroclinic Wave ◽  
Time Step ◽  
Integration Schemes ◽  
Helmholtz Problem

Abstract An important question for atmospheric modeling is the viability of semi-implicit time integration schemes on massively parallel computing architectures. Semi-implicit schemes can provide increased stability and accuracy. However, they require the solution of an elliptic problem at each time step, creating concerns about their parallel efficiency and scalability. Here, a semi-implicit (SI) version of the Model for Prediction Across Scales (MPAS) is developed and compared with the original model version, which uses a split Runge–Kutta (SRK3) time integration scheme. The SI scheme is based on a quasi-Newton iteration toward a Crank–Nicolson scheme. Each Newton iteration requires the solution of a Helmholtz problem; here, the Helmholtz problem is derived, and its solution using a geometric multigrid method is described. On two standard test cases, a midlatitude baroclinic wave and a small-planet nonhydrostatic gravity wave, the SI and SRK3 versions produce almost identical results. On the baroclinic wave test, the SI version can use somewhat larger time steps (about 60%) than the SRK3 version before losing stability. The SI version costs 10%–20% more per step than the SRK3 version, and the weak and strong scalability characteristics of the two versions are very similar for the processor configurations the authors have been able to test (up to 1920 processors). Because of the spatial discretization of the pressure gradient in the lowest model layer, the SI version becomes unstable in the presence of realistic orography. Some further work will be needed to demonstrate the viability of the SI scheme in this case.

Interlaced Attitude Estimation for Spacecraft Using Vector Observations

2021 ◽  
Vol 01 (03) ◽  
Author(s):  
Lubin Chang
Keyword(s):  
Prior Information ◽  
State Of The Art ◽  
Estimation Method ◽  
Attitude Estimation ◽  
The Body ◽  
Body Frame ◽  
Initial Value ◽  
Initial State ◽  
Initial Errors ◽  
Initial Attitude

This paper proposes an interlaced attitude estimation method for spacecraft using vector observations, which can simultaneously estimate the constant attitude at the very start and the attitude of the body frame relative to its initial state. The arbitrary initial attitude, described by constant attitude at the very start, is determined using quaternion estimator which requires no prior information. The multiplicative extended Kalman filter (EKF) is competent for estimating the attitude of the body frame relative to its initial state since the initial value of this attitude is exactly known. The simulation results show that the proposed algorithms could achieve better performance compared with the state-of-the-art algorithms even with extreme large initial errors. Meanwhile, the computational burden is also much less than that of the advanced nonlinear attitude estimators.

Energy-Momentum Conserving Time Integration of Modally Reduced Flexible Multibody Systems

2013 ◽  
Author(s):  
Alexander Humer ◽  
Johannes Gerstmayr
Keyword(s):  
Angular Momentum ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Time Integration ◽  
Flexible Multibody Dynamics ◽  
Integration Scheme ◽  
Integration Schemes ◽  
Flexible Multibody ◽  
Floating Frame ◽  
Energy And Momentum

Many conventional time integration schemes frequently adopted in flexible multibody dynamics fail to retain the fundamental conservation laws of energy and momentum of the continuous time domain. Lack of conservation, however, in particular of angular momentum, may give rise to unexpected, unphysical results. To avoid such problems, a scheme for the consistent integration of modally reduced multibody systems subjected to holonomic constraints is developed in the present paper. As opposed to the conventional approach, in which the floating frame of reference formulation is combined with component mode synthesis for approximating the flexible deformation, an alternative, recently proposed formulation based on absolute coordinates is adopted in the analysis. Owing to the linear relationship between the generalized coordinates and the absolute displacement, the inertia terms in the equations of motion attain a very simple structure. The mass matrix remains independent of the current state of deformation and the velocity dependent term known from the floating frame approach vanishes due to the absence of relative coordinates. These advantageous properties facilitate the construction of an energy and momentum consistent integration scheme. By the mid-point rule, algorithmic conservation of both linear and angular momentum is achieved. In order to consistently integrate the total energy of the system, the discrete derivative needs to be adopted when evaluating the strain energy gradient and the derivative of the algebraic constraint equations.

scholarly journals Efficient Approximate Techniques for Integrating Stochastic Differential Equations

2006 ◽  
Vol 134 (10) ◽  
pp. 3006-3014 ◽  
Author(s):  
James A. Hansen ◽  
Cecile Penland
Keyword(s):  
Time Series ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Deterministic Model ◽  
Probability Distributions ◽  
Approximate Solutions ◽  
Integration Scheme ◽  
Time Step ◽  
Integration Schemes ◽  
Runge Kutta

Abstract The delicate (and computationally expensive) nature of stochastic numerical modeling naturally leads one to look for efficient and/or convenient methods for integrating stochastic differential equations. Concomitantly, one may wish to sensibly add stochastic terms to an existing deterministic model without having to rewrite that model. In this note, two possibilities in the context of the fourth-order Runge–Kutta (RK4) integration scheme are examined. The first approach entails a hybrid of deterministic and stochastic integration schemes. In these examples, the hybrid RK4 generates time series with the correct climatological probability distributions. However, it is doubtful that the resulting time series are approximate solutions to the stochastic equations at every time step. The second approach uses the standard RK4 integration method modified by appropriately scaling stochastic terms. This is shown to be a special case of the general stochastic Runge–Kutta schemes considered by Ruemelin and has global convergence of order one. Thus, it gives excellent results for cases in which real noise with small but finite correlation time is approximated as white. This restriction on the type of problems to which the stochastic RK4 can be applied is strongly compensated by its computational efficiency.

scholarly journals Sensorimotor feedback based on task-relevant error robustly predicts temporal recruitment and multidirectional tuning of muscle synergies

2013 ◽  
Vol 109 (1) ◽  
pp. 31-45 ◽  
Author(s):  
Seyed A. Safavynia ◽  
Lena H. Ting
Keyword(s):  
Sagittal Plane ◽  
Balance Control ◽  
Center Of Mass ◽  
The Body ◽  
Muscle Synergy ◽  
Muscle Synergies ◽  
Muscle Coordination ◽  
Initial State ◽  
Sensorimotor Feedback ◽  
Task Level

We hypothesized that motor outputs are hierarchically organized such that descending temporal commands based on desired task-level goals flexibly recruit muscle synergies that specify the spatial patterns of muscle coordination that allow the task to be achieved. According to this hypothesis, it should be possible to predict the patterns of muscle synergy recruitment based on task-level goals. We demonstrated that the temporal recruitment of muscle synergies during standing balance control was robustly predicted across multiple perturbation directions based on delayed sensorimotor feedback of center of mass (CoM) kinematics (displacement, velocity, and acceleration). The modulation of a muscle synergy's recruitment amplitude across perturbation directions was predicted by the projection of CoM kinematic variables along the preferred tuning direction(s), generating cosine tuning functions. Moreover, these findings were robust in biphasic perturbations that initially imposed a perturbation in the sagittal plane and then, before sagittal balance was recovered, perturbed the body in multiple directions. Therefore, biphasic perturbations caused the initial state of the CoM to differ from the desired state, and muscle synergy recruitment was predicted based on the error between the actual and desired upright state of the CoM. These results demonstrate that that temporal motor commands to muscle synergies reflect task-relevant error as opposed to sensory inflow. The proposed hierarchical framework may represent a common principle of motor control across motor tasks and levels of the nervous system, allowing motor intentions to be transformed into motor actions.

The Impact of Ecowas on Intraregional Trade Flows: An Empirical Investigation

1995 ◽  
Vol 23 (3) ◽  
pp. 113-129 ◽  
Author(s):  
Mamit Deme
Keyword(s):  
Economic Integration ◽  
Regional Integration ◽  
Empirical Investigation ◽  
West African ◽  
Trade Flows ◽  
Integration Scheme ◽  
General View ◽  
Economic Community ◽  
Integration Schemes ◽  
The Impact

There is a scarcity of published statistical analysis that examines the impact of economic integration schemes in Africa. This study fills the gap in the literature. The impact of Economic Community of West African States (ECOWAS) on trade flows of its member countries is examined applying an econometric analysis. The parameters of a multivariate trade-flows model are estimated using a panel data of the 1975 to 1991 period. The results suggest that the regional integration scheme has succeeded in increasing trade flows between member countries. The general view in the literature is that African integration schemes have failed to increase trade flows.

scholarly journals The Influence of Hybrid Surface Modification on the Selected Properties of CP Titanium Grade II Manufactured by Selective Laser Melting

Materials ◽  
2020 ◽  
Vol 13 (12) ◽  
pp. 2829
Author(s):  
Anna Woźniak ◽  
Marcin Adamiak ◽  
Grzegorz Chladek ◽  
Mirosław Bonek ◽  
Witold Walke ◽  
...  
Keyword(s):  
Surface Modification ◽  
Selective Laser Melting ◽  
Pure Titanium ◽  
Microscopic Investigation ◽  
The Body ◽  
Laser Melting ◽  
Initial State ◽  
Tin Coatings ◽  
Grade Ii ◽  
Titanium Grade

The human body is an extremely aggressive environment in terms of corrosion. Titanium and its alloys are one of the most popular biomaterials used for implant applications due to biocompatibility. However, every element introduced into the body is treated as a foreign body. The human body’s immune response may, therefore, lead to implant rejection and the need for reoperation. For this purpose, it seems important to carry out surface modifications by applying coatings and inter alia by texturing to implants. The objective of this paper is to investigate the effect of surface treatment on the chosen properties of the pure titanium (Grade II) samples obtained by selective laser melting (SLM) processing. The samples were divided into five groups: Initial state (after polishing), after surface modification by the physical vapour deposition (PVD) method—CrN and TiN coatings were deposited on the surface of the tested material, and after laser texturing. The paper presents the results of the microscopic investigation, chemical and phase compositions, and physicochemical and electrochemical properties of the tested samples. Based on the results obtained it can be concluded that the hybrid surface modification shows significant effects on the properties of the pure titanium. The samples with the textured PVD-deposited TiN coatings were characterized by favorable physicochemical properties and were the highest performing in terms of pitting corrosion resistance.

scholarly journals Parabolic Sandwiches for Functions on a Compact Interval and an Application to Projectile Motion

2019 ◽  
Vol 2019 ◽  
pp. 1-7 ◽  
Author(s):  
Robert Kantrowitz ◽  
Michael M. Neumann
Keyword(s):  
Present Article ◽  
Decisive Role ◽  
Valid Argument ◽  
Quadratic Polynomials ◽  
New Approach ◽  
Projectile Motion ◽  
Full Result ◽  
Air Resistance ◽  
Special Case ◽  
Natural Way

About a century ago, the French artillery commandant Charbonnier envisioned an intriguing result on the trajectory of a projectile that is moving under the forces of gravity and air resistance. In 2000, Groetsch discovered a significant gap in Charbonnier’s work and provided a valid argument for a certain special case. The goal of the present article is to establish a rigorous new approach to the full result. For this, we develop a theory of those functions which can be sandwiched, in a natural way, by a pair of quadratic polynomials. It turns out that the convexity or concavity of the derivative plays a decisive role in this context.

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