Mathematical Modelling of Vertical Looped Water Slide Using Matlab

2021 ◽  
Vol 9 (11) ◽  
pp. 1918-1931
Author(s):  
M. Rohan Kini
Keyword(s):  
Mathematical Model ◽  
Kinetic Energy ◽  
Potential Energy ◽  
Mathematical Modelling ◽  
Arc Length ◽  
Exposure Limits ◽  
Centripetal Acceleration ◽  
Conservation Of Energy ◽  
Euler’S Method ◽  
Force Variation

Abstract: A conceptual mathematical model of a water slide with vertical loops is developed. The principle used is the conservation of energy. The thrill experienced by a rider on a water slide is mainly due to the variation of G-force acting on the rider through the course of the ride. The geometry of the slide is developed by plotting G-force variation with the arc length of the loop. The G-force exposure limits should meet with the standards set by the F24 committee on amusement parks and rides. The coordinates of the slide geometry are determined by using Euler’s method of discretized equations. Keywords: G-Force, Centripetal acceleration, Clothoid curve, Weightlessness, Potential Energy, Kinetic Energy

scholarly journals A hydrostatic model of the Wirtz pump

2018 ◽  
Vol 474 (2211) ◽  
pp. 20170533 ◽  
Author(s):  
Jonathan H. B. Deane ◽  
Jonathan J. Bevan
Keyword(s):  
Mathematical Model ◽  
Dynamical Systems ◽  
Kinetic Energy ◽  
Mathematical Modelling ◽  
Systems Approach ◽  
Output Pressure ◽  
History Of ◽  
Hydrostatic Model ◽  
Constant Output ◽  
Discrete Mathematical Model

The Wirtz pump is not only an excellent example of alternative technology, using as it does the kinetic energy of a stream to raise a proportion of its water, but its mathematical modelling also poses several intriguing problems. We give some history of the Wirtz pump and describe its operation. Taking a novel dynamical systems approach, we then derive a discrete mathematical model in the form of a mapping that describes its hydrostatic behaviour. Our model enables us to explain several aspects of the behaviour of the pump as well as to design one that gives approximately maximal, and maximally constant, output pressure.

Aspects of Mathematical Modelling of Heat and Mass Exchange at Burning of Droplet of Water Fuel Oil Emulsion

2013 ◽  
Vol 371 ◽  
pp. 642-646 ◽  
Author(s):  
Ştefania Roxana Buzdugă ◽  
Tudor Sajin
Keyword(s):  
Mathematical Model ◽  
Computer Simulation ◽  
Mathematical Modelling ◽  
Mass Exchange ◽  
Fuel Oil ◽  
Gas Cell ◽  
Heat And Mass Exchange ◽  
Conservation Of Energy ◽  
Oil Emulsion ◽  
Thermo Physical Properties

This paper presents some aspects regarding modelling of heat and mass exchange at burning of droplet of water-oil emulsion (WOE). Based on nonstationary differential equations of conservation of energy and matter in the liquid and gaseous phases, it was formulated a physical and mathematical model of heat and mass exchange in system of WOE drop-elementary gas cell. Unlike other models, in this model is taken into account the dependence of thermo physical properties of phase by temperature. The heating, micro explosion, vaporization and combustion stages of droplet burning were considered. For numerical solution and computer simulation of mathematical model has been developed an algorithm in Mathcad14 software. The numerical results were validated experimentally.

Mathematical Modelling the Process of Cup Wheel Precision Grinding of Rotating Concave Paraboloid

2016 ◽  
Vol 693 ◽  
pp. 837-842
Author(s):  
Fu Yi Xia ◽  
Li Ming Xu ◽  
De Jin Hu
Keyword(s):  
Mathematical Model ◽  
Mathematical Modelling ◽  
Abrasive Particle ◽  
Influence Factors ◽  
Machining Process ◽  
Quadric Surface ◽  
Precision Grinding ◽  
Cup Wheel ◽  
The Mathematical Model ◽  
Trajectory Equation

A novel principle of cup wheel grinding of rotating concave quadric surface was proposed. The mathematical model of machining process was established to prove the feasibility of precision grinding of rotating concave paraboloid based on the introduced principle. The conditions of non-interference grinding of concave paraboloid were mathematically derived. The processing range and its influence factors were discussed. The trajectory equation of abrasive particle was concluded. Finally, the math expressions of numerical controlled parameters was put forward in the process of grinding of the concave paraboloid.

MULTI–INERT OSCILLATORY MECHANISM

2020 ◽  
Vol 2 ◽  
pp. 19-25
Author(s):  
I.P. POPOV
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Oscillation Frequency ◽  
Free Oscillation ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Harmonic Oscillations ◽  
Oscillatory Systems ◽  
Oscillatory Mechanism ◽  
Mutual Conversion

A mechanical oscillatory system with homogeneous elements, namely, with n massive loads (multi– inert oscillator), is considered. The possibility of the appearance of free harmonic oscillations of loads in such a system is shown. Unlike the classical spring pendulum, the oscillations of which are due to the mutual conversion of the kinetic energy of the load into the potential energy of the spring, in a multi–inert oscillator, the oscillations are due to the mutual conversion of only the kinetic energies of the goods. In this case, the acceleration of some loads occurs due to the braking of others. A feature of the multi–inert oscillator is that its free oscillation frequency is not fixed and is determined mainly by the initial conditions. This feature can be very useful for technical applications, for example, for self–neutralization of mechanical reactive (inertial) power in oscillatory systems.

Tracker-assisted modelling: simultaneous validation of the conservation law of energy and the work-energy theorem

2021 ◽  
Vol 57 (1) ◽  
pp. 015012
Author(s):  
Unofre B Pili ◽  
Renante R Violanda
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Gravitational Potential ◽  
Conservation Law ◽  
Gravitational Potential Energy ◽  
Time Data ◽  
Home Based ◽  
Basic Physics ◽  
Free Falling ◽  
Work Done

Abstract The video of a free-falling object was analysed in Tracker in order to extract the position and time data. On the basis of these data, the velocity, gravitational potential energy, kinetic energy, and the work done by gravity were obtained. These led to a rather simultaneous validation of the conservation law of energy and the work–energy theorem. The superimposed plots of the kinetic energy, gravitational potential energy, and the total energy as respective functions of time and position demonstrate energy conservation quite well. The same results were observed from the plots of the potential energy against the kinetic energy. On the other hand, the work–energy theorem has emerged from the plot of the total work-done against the change in kinetic energy. Because of the accessibility of the setup, the current work is seen as suitable for a home-based activity, during these times of the pandemic in particular in which online learning has remained to be the format in some countries. With the guidance of a teacher, online or face-to-face, students in their junior or senior high school—as well as for those who are enrolled in basic physics in college—will be able to benefit from this work.

Study of the Mechanisms of Vortex Variability in the Lofoten Basin Based on the Energy Analysis

2021 ◽  
Vol 37 (3) ◽  
Author(s):  
V. S. Travkin ◽  
◽  
T. V. Belonenko ◽  
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Baroclinic Instability ◽  
Positive Trend ◽  
Barotropic Instability ◽  
Available Potential Energy ◽  
Conversion Rates ◽  
Order Of Magnitude ◽  
The Mean ◽  
Mean Kinetic Energy

Purpose. The Lofoten Basin is one of the most energetic zones of the World Ocean characterized by high activity of mesoscale eddies. The study is aimed at analyzing different components of general energy in the basin, namely the mean kinetic and vortex kinetic energy calculated using the integral of the volume of available potential and kinetic energy of the Lofoten Vortex, as well as variability of these characteristics. Methods and Results. GLORYS12V1 reanalysis data for the period 2010–2018 were used. The mean kinetic energy and the eddy kinetic one were analyzed; and as for the Lofoten Vortex, its volume available potential and kinetic energy were studied. The mesoscale activity of eddies in winter is higher than in summer. Evolution of the available potential energy and kinetic energy of the Lofoten Vortex up to the 1000 m horizon was studied. It is shown that the vortex available potential energy exceeds the kinetic one by an order of magnitude, and there is a positive trend with the coefficient 0,23⋅1015 J/year. It was found that in the Lofoten Basin, the intermediate layer from 600 to 900 m made the largest contribution to the potential energy, whereas the 0–400 m layer – to kinetic energy. The conversion rates of the mean kinetic energy into the vortex kinetic one and the mean available potential energy into the vortex available potential one (barotropic and baroclinic instability) were analyzed. It is shown that the first type of transformation dominates in summer, while the second one is characterized by its increase in winter. Conclusions. The vertical profile shows that the kinetic energy of eddies in winter is higher than in summer. The available potential energy of a vortex is by an order of magnitude greater than the kinetic energy. An increase in the available potential energy is confirmed by a significant positive trend and by a decrease in the vortex Burger number. The graphs of the barotropic instability conversion rate demonstrate the multidirectional flows in the vortex zone with the dipole structure observed in a winter period, and the tripole one – in summer. The barotropic instability highest intensity is observed in summer. The baroclinic instability is characterized by intensification of the regime in winter that is associated with weakening of stratification in this period owing to winter convection.

Mathematical modelling.

2021 ◽  
Author(s):  
Ed Rutgers Durner
Keyword(s):  
Mathematical Model ◽  
Mathematical Modelling ◽  
Mathematical Models ◽  
Growth And Development ◽  
Fertilizer Recommendations ◽  
Mathematical Equations

Abstract Plants are studied to understand their growth and development so that their quality and productivity can be optimised. Models are developed that can be simple and descriptive, or quite complex with numerous mathematical equations; their level of complexity is linked to their purpose. This summary serves as an introduction to mathematical models in horticulture. It is not a manual for modelling itself, but rather an overview of how important mathematical models are in horticultural production. Mathematical models are used extensively in horticulture both extrinsically, i.e. when calculating chilling hour accumulations and intrinsically, i.e. when applying fertilizer to a crop. In chilling calculations, developed models are used directly. Fertilizer recommendations were probably developed using a mathematical model. The first part of this article discusses models in general and reviews general characteristics of mathematical models. The second part outlines the major uses of mathematical modelling in modern horticultural production. Presentations of specific models are limited in order to present a general discussion of models with examples that will interest most horticulturists.

scholarly journals Walking in simulated reduced gravity: mechanical energy fluctuations and exchange

1999 ◽  
Vol 86 (1) ◽  
pp. 383-390 ◽  
Author(s):  
Timothy M. Griffin ◽  
Neil A. Tolani ◽  
Rodger Kram
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Gravitational Potential ◽  
Inverted Pendulum ◽  
Mechanical Energy ◽  
Center Of Mass ◽  
Metabolic Cost ◽  
Metabolic Energy ◽  
Gravitational Potential Energy ◽  
Reduced Gravity

Walking humans conserve mechanical and, presumably, metabolic energy with an inverted pendulum-like exchange of gravitational potential energy and horizontal kinetic energy. Walking in simulated reduced gravity involves a relatively high metabolic cost, suggesting that the inverted-pendulum mechanism is disrupted because of a mismatch of potential and kinetic energy. We tested this hypothesis by measuring the fluctuations and exchange of mechanical energy of the center of mass at different combinations of velocity and simulated reduced gravity. Subjects walked with smaller fluctuations in horizontal velocity in lower gravity, such that the ratio of horizontal kinetic to gravitational potential energy fluctuations remained constant over a fourfold change in gravity. The amount of exchange, or percent recovery, at 1.00 m/s was not significantly different at 1.00, 0.75, and 0.50 G (average 64.4%), although it decreased to 48% at 0.25 G. As a result, the amount of work performed on the center of mass does not explain the relatively high metabolic cost of walking in simulated reduced gravity.

scholarly journals An Analysis of the Energetics of Tropical and Extra-Tropical Regions for Warm ENSO Composite Episodes

2017 ◽  
Vol 32 (1) ◽  
pp. 39-51
Author(s):  
Zayra Christine Sátyro ◽  
José Veiga
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
El Niño ◽  
Energy Production ◽  
El Nino ◽  
Southern Oscillation ◽  
Stable Condition ◽  
The Other ◽  
Tropical Regions ◽  
Using Data

Abstract This study focuses on the quantification and evaluation of the effects of ENSO (El Niño Southern Oscillation) warm phases, using a composite of five intense El Niño episodes between 1979 – 2011 on the Energetic Lorenz Cycle for four distinct regions around the globe: 80° S – 5° N (region 1), 50° S – 5° N (region 2), 30° S – 5° N (region 3), and 30° S – 30° N (region 4), using Data from NCEP reanalysis-II. Briefly, the results showed that zonal terms of potential energy and kinetic energy were intensified, except for region 1, where zonal kinetic energy weakened. Through the analysis of the period in which higher energy production is observed, a strong communication between the available zonal potential and the zonal kinetic energy reservoirs can be identified. This communication weakened the modes linked to eddies of potential energy and kinetic energy, as well as in the other two baroclinic conversions terms. Furthermore, the results indicate that for all the regions, the system itself works to regain its stable condition.

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