Numerical Methods for Solving Dynamic Equilibrium Equations

2021 ◽  
pp. 194-214
Keyword(s):  
Numerical Methods ◽  
General Principle ◽  
Degrees Of Freedom ◽  
Dynamic Equilibrium ◽  
Continuous Media ◽  
Equilibrium Equations ◽  
Numerical Resolution ◽  
Large Size ◽  
One Step ◽  
Step Algorithm

Once the number of degrees of freedom exceeds a certain number, it would be impossible to solve the dynamic equilibrium equation manually, hence the need to switch to a numerical resolution, whose general principle is to convert a dynamic equation into a static one. We are interested, for the dynamic analysis of the structures and the continuous media, in “one-step” algorithms rather than “multi-step” one. It is mainly because the systems to be solved are of large size and that it is important to minimize the number of operations and value to be memorized to the detriment, if necessary, of precision. A “one-step” algorithm, like that of Newmark, makes it possible to calculate the solution at time tn+1, starting from the solution at time tn. In addition to the disadvantage of requiring the storage of several steps, the “multi-step” algorithms such as that of Houbolt requires a startup procedure. This chapter allows the reader to enumerate and understand different numerical method with different examples.

scholarly journals Dynamic Equilibrium Equations in Unified Mechanics Theory

2021 ◽  
Vol 2 (1) ◽  
pp. 63-80
Author(s):  
Noushad Bin Jamal Bin Jamal M ◽  
Hsiao Wei Lee ◽  
Chebolu Lakshmana Rao ◽  
Cemal Basaran
Keyword(s):  
Energy Loss ◽  
Entropy Generation ◽  
Equilibrium Equation ◽  
Dynamic Equilibrium ◽  
Free Vibration Analysis ◽  
Newtonian Mechanics ◽  
Equilibrium Equations ◽  
Time Coordinate ◽  
Proposed Model ◽  
Laws Of Thermodynamics

Traditionally dynamic analysis is done using Newton’s universal laws of the equation of motion. According to the laws of Newtonian mechanics, the x, y, z, space-time coordinate system does not include a term for energy loss, an empirical damping term “C” is used in the dynamic equilibrium equation. Energy loss in any system is governed by the laws of thermodynamics. Unified Mechanics Theory (UMT) unifies the universal laws of motion of Newton and the laws of thermodynamics at ab-initio level. As a result, the energy loss [entropy generation] is automatically included in the laws of the Unified Mechanics Theory (UMT). Using unified mechanics theory, the dynamic equilibrium equation is derived and presented. One-dimensional free vibration analysis with frictional dissipation is used to compare the results of the proposed model with that of a Newtonian mechanics equation. For the proposed entropy generation equation in the system, the trend of predictions is comparable with the reported experimental results and Newtonian mechanics-based predictions.

Facile one-step exfoliation of large-size 2D materials via simply shearing in triethanolamine

2017 ◽  
Vol 199 ◽  
pp. 124-127 ◽  
Author(s):  
Hao Chen ◽  
Bin Liu ◽  
Quanling Yang ◽  
Shan Wang ◽  
Wei Liu ◽  
...  
Keyword(s):  
2D Materials ◽  
Large Size ◽  
One Step

Turbulent Flows with Drops and Bubbles: What Numerical Simulations Can Tell Us – Freeman Scholar Lecture

2021 ◽  
Author(s):  
Giovanni Soligo ◽  
Alessio Roccon ◽  
Alfredo Soldati
Keyword(s):  
Numerical Methods ◽  
Best Practices ◽  
Numerical Simulations ◽  
Turbulent Flows ◽  
Multiphase Flows ◽  
Simulation Analysis ◽  
Droplet Size Distribution ◽  
Limited Range ◽  
Numerical Resolution ◽  
Multi Scale

Abstract Turbulent flows laden with large, deformable drops or bubbles are ubiquitous in nature and in a number of industrial processes. These flows are characterized by a physics acting at many different scales: from the macroscopic length scale of the problem down to the microscopic molecular scale of the interface. Naturally, the numerical resolution of all the scales of the problem, which span about eight to nine orders of magnitude, is not possible, with the consequence that numerical simulations of turbulent multiphase flows impose challenges and require methods able to capture the multi-scale nature of the flow. In this review, we start by describing the numerical methods commonly employed and discussing their advantages and limitations, and then we focus on the issues arising from the limited range of scales that can be possibly solved. Ultimately, the droplet size distribution, a key result of interest for turbulent multiphase flows, is used as a benchmark to compare the capabilities of the different methods and to discuss the main insights that can be drawn from these simulations. Based on this, we define a series of guidelines and best practices that we believe important in the simulation analysis and in the development of new numerical methods.

scholarly journals The determination of wormwheel toothing surface

2018 ◽  
Vol 157 ◽  
pp. 01013 ◽  
Author(s):  
Tadeusz Nieszporek ◽  
Rafał Gołębski ◽  
Piotr Boral
Keyword(s):  
Numerical Methods ◽  
Helical Surface ◽  
Heavy Industry ◽  
Manufacturing Costs ◽  
Rotary Tool ◽  
Large Size ◽  
Envelope Method ◽  
Worm Gears ◽  
Very High

In heavy industry (metallurgy, mining), large-size worm gears designed to carry large loads are often used. However, their technology is very difficult and their manufacturing costs are very high. In practice, cone-derivative worm gears are most often used, which are machined by the envelope method using a rotary tool. The literature has given much coverage to the determination of the worm helical surface. The surface of wormwheel teeth is much less commonly described. Therefore, this paper presents an analytical and a numerical methods for generating the wormwheel toothing by the tangential and radial methods with a special cutter and with a modular hob.

scholarly journals New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II

2018 ◽  
Vol 1 (3) ◽  
pp. 30 ◽  
Author(s):  
Hussein ALKasasbeh ◽  
Irina Perfilieva ◽  
Muhammad Ahmad ◽  
Zainor Yahya
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Trapezoidal Rule ◽  
Approximation Methods ◽  
System Of Differential Equations ◽  
Fuzzy Transform ◽  
Fuzzy Partition ◽  
Fuzzy Approximation ◽  
One Step ◽  
Basic Functions

In this research, three approximation methods are used in the new generalized uniform fuzzy partition to solve the system of differential equations (SODEs) based on fuzzy transform (FzT). New representations of basic functions are proposed based on the new types of a uniform fuzzy partition and a subnormal generating function. The main properties of a new uniform fuzzy partition are examined. Further, the simpler form of the fuzzy transform is given alongside some of its fundamental results. New theorems and lemmas are proved. In accordance with the three conventional numerical methods: Trapezoidal rule (one step) and Adams Moulton method (two and three step modifications), new iterative methods (NIM) based on the fuzzy transform are proposed. These new fuzzy approximation methods yield more accurate results in comparison with the above-mentioned conventional methods.

Dynamics Redesign of Marine Structures

1985 ◽  
Vol 29 (04) ◽  
pp. 285-295 ◽  
Author(s):  
Curtis J. Hoff ◽  
Michael M. Bernitsas
Keyword(s):  
Large Scale ◽  
Structural Changes ◽  
Dynamic Equilibrium ◽  
Degree Of Freedom ◽  
Attractive Alternative ◽  
Equilibrium Equations ◽  
Element Analysis ◽  
Solution Methods ◽  
Tower Model ◽  
Energy Equations

The dynamic response of a marine structure depends upon the exciting forces and the modal characteristics of the structure. Excessive vibratory response requires reduction of the exciting loads or redesign of the structure or both. In this paper the general redesign problem is formulated. It applies to large-scale structures and allows for large structural changes. Solution of the redesign problem is achieved through perturbation methods which are an attractive alternative to traditional trial-and-error methods. Perturbation solution methods are based on dynamic equilibrium equations or energy equations or both. A new method based on the energy equations which enforces the mode orthogonality conditions is developed and evaluated against all existing methods. Two test cases, a 191-degree-of-freedom two-dimensional ship model and a 810-degree-of-freedom offshore light tower model are used to compare the methods numerically. It is shown that the method developed in this paper can produce, with a single finite element analysis of the baseline system, a structure which satisfies within acceptable limits all nonconflicting design objectives.

Method for solving dynamic equilibrium equations based on displacement excitation

2020 ◽  
Vol 19 (2) ◽  
pp. 423-433
Author(s):  
Xu Guolin ◽  
Zhang Lingxin ◽  
Bai Yashuang ◽  
Sun Hao
Keyword(s):  
Dynamic Equilibrium ◽  
Equilibrium Equations

scholarly journals Single-Element Dual-Interferometer for Precision Inertial Sensing

Sensors ◽  
2020 ◽  
Vol 20 (17) ◽  
pp. 4986
Author(s):  
Yichao Yang ◽  
Kohei Yamamoto ◽  
Victor Huarcaya ◽  
Christoph Vorndamme ◽  
Daniel Penkert ◽  
...  
Keyword(s):  
High Precision ◽  
Degrees Of Freedom ◽  
Dynamic Range ◽  
Inertial Sensor ◽  
Single Element ◽  
Inertial Sensing ◽  
Optical Readout ◽  
One Step ◽  
Isolation Systems ◽  
Laser Interferometers

Tracking moving masses in several degrees of freedom with high precision and large dynamic range is a central aspect in many current and future gravitational physics experiments. Laser interferometers have been established as one of the tools of choice for such measurement schemes. Using sinusoidal phase modulation homodyne interferometry allows a drastic reduction of the complexity of the optical setup, a key limitation of multi-channel interferometry. By shifting the complexity of the setup to the signal processing stage, these methods enable devices with a size and weight not feasible using conventional techniques. In this paper we present the design of a novel sensor topology based on deep frequency modulation interferometry: the self-referenced single-element dual-interferometer (SEDI) inertial sensor, which takes simplification one step further by accommodating two interferometers in one optic. Using a combination of computer models and analytical methods we show that an inertial sensor with sub-picometer precision for frequencies above 10 mHz, in a package of a few cubic inches, seems feasible with our approach. Moreover we show that by combining two of these devices it is possible to reach sub-picometer precision down to 2 mHz. In combination with the given compactness, this makes the SEDI sensor a promising approach for applications in high precision inertial sensing for both next-generation space-based gravity missions employing drag-free control, and ground-based experiments employing inertial isolation systems with optical readout.

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