Extensions and Improvements to the Solutions for Linear Tank Dynamics

2015 ◽  
Author(s):  
Arild Ludvigsen ◽  
Zhi Yuan Pan
Keyword(s):  
Stability Analysis ◽  
Velocity Potential ◽  
Linear Equations ◽  
Stable Solution ◽  
Equation System ◽  
Added Mass ◽  
Eigenvalue Analysis ◽  
Singular Equation ◽  
Global Response ◽  
Numerical Noise

Linear solvers for the flow exterior to the hull may be used to solve for the fluid dynamics also in the interior of a tank, as discussed in Newman (2005) and Ludvigsen et al. (2013). This introduces extra, erroneous terms in the radiation part of the pressure in the tank, but due to cancellation in restoring and radiation terms, the total representation of the pressure and the global response is correctly obtained. For some kinds of analysis, specific knowledge is needed of the radiation and restoring parts separately. The cancellation of the extra terms can then not be utilized. Examples of this are stability analysis and eigenvalue analysis. In stability analysis we need to know the actual real global restoring coefficients. In eigenvalue analysis, we should have the separately correct representations of the added mass and restoring coefficients, respectively, to be able to conveniently use them as input to standard eigenvalue solvers. Here, we develop the expressions for the corrected, actual terms of the total added mass and restoring coefficients for tanks. This is used in our computer program for performing eigenvalue analysis. Results for peak global response and natural periods of the structure with the influence of tank dynamics are presented. Comparisons are made with results obtained by a quasi-static method for an FPSO and a ship with more largely extensive tanks. For a completely filled tank, the boundary value problem (BVP) for the velocity potential is reduced to Laplace equation in the fluid domain, subject to a Neuman condition on the fixed boundary and it is not closed. The extra condition of having zero pressure at some point in the tank is then added. Direct re-use of the BVP solver for the external flow, gives an undetermined set of linear equations for the velocity potential in the tank fluid. A typical solver for sets of linear equations may still return a solution, but this will contain a random undetermined constant. After imposing zero pressure in the top of the tank, this solution is still unstable, contaminated by numerical noise. An improved method is introduced by imposing algebraically, in the equation system, the constraint of zero pressure in the top of the tank. This gives a non-singular equation system with a stable solution holding zero pressure in some selected point in the tank.

Effect of the Condition Number of the Inverse Fourier Transform Matrix on the Solution Behavior of the Time Spectral Equation System

2020 ◽  
Author(s):  
Sen Zhang ◽  
Dingxi Wang ◽  
Yi Li ◽  
Hangkong Wu ◽  
Xiuquan Huang
Keyword(s):  
Fourier Transform ◽  
Stability Analysis ◽  
Condition Number ◽  
Inverse Fourier Transform ◽  
Real Life ◽  
Equation System ◽  
Time Instant ◽  
Transform Matrix ◽  
Spectral Equation ◽  
The Impact

Abstract The time spectral method is a very popular reduced order frequency method for analyzing unsteady flow due to its advantage of being easily extended from an existing steady flow solver. Condition number of the inverse Fourier transform matrix used in the method can affect the solution convergence and stability of the time spectral equation system. This paper aims at evaluating the effect of the condition number of the inverse Fourier transform matrix on the solution stability and convergence of the time spectral method from two aspects. The first aspect is to assess the impact of condition number using a matrix stability analysis based upon the time spectral form of the scalar advection equation. The relationship between the maximum allowable Courant number and the condition number will be derived. Different time instant groups which lead to the same condition number are also considered. Three numerical discretization schemes are provided for the stability analysis. The second aspect is to assess the impact of condition number for real life applications. Two case studies will be provided: one is a flutter case, NASA rotor 67, and the other is a blade row interaction case, NASA stage 35. A series of numerical analyses will be performed for each case using different time instant groups corresponding to different condition numbers. The conclusion drawn from the two real life case studies will corroborate the relationship derived from the matrix stability analysis.

Stability of a String in Axial Flow

1985 ◽  
Vol 107 (4) ◽  
pp. 421-425 ◽  
Author(s):  
G. S. Triantafyllou ◽  
C. Chryssostomidis
Keyword(s):  
Stability Analysis ◽  
Frictional Coefficient ◽  
Axial Flow ◽  
Added Mass ◽  
Equation Of Motion ◽  
Bending Stiffness ◽  
Diameter Ratio ◽  
Constant Speed ◽  
Hamilton’S Principle ◽  
The Stability

The equation of motion of a long slender beam submerged in an infinite fluid moving with constant speed is derived using Hamilton’s principle. The upstream end of the beam is pinned and the downstream end is free to move. The resulting equation of motion is then used to perform the stability analysis of a string, i.e., a beam with negligible bending stiffness. It is found that the string is stable if (a) the external tension at the free end exceeds the value of a U2, where a is the “added mass” of the string and U the fluid speed; or (b) the length-over-diameter ratio exceeds the value 2Cf/π, where Cf is the frictional coefficient of the string.

scholarly journals Profil Berpikir Siswa dalam Menyelesaikan Soal Sistem Persamaan Linear

2019 ◽  
Vol 6 (1) ◽  
pp. 69-84
Author(s):  
K. Ayu Dwi Indrawati ◽  
Ahmad Muzaki ◽  
Baiq Rika Ayu Febrilia
Keyword(s):  
Qualitative Research ◽  
Linear Equation ◽  
Linear Equations ◽  
Equation System ◽  
Mathematical Ability ◽  
System Of Linear Equations ◽  
Mathematics Ability ◽  
Linear Equation System ◽  
Thinking Process ◽  
Descriptive Qualitative

This research aimed to describe the thinking process of students in solving the system of linear equations based on Polya stages. This study was a descriptive qualitative research involving six Year 10 students who are selected based on the teacher's advice and the initial mathematical ability categories, namely: (1) Students with low initial mathematics ability, (2) Students with moderate initial mathematics ability, and ( 3) students with high initial mathematics ability categories. The results indicated that students with low initial mathematical ability category were only able to solve the two-variable linear equation system problems. Students in the medium category of initial mathematics ability and students in the category of high initial mathematics ability were able to solve the problem in the form of a system of linear equations of two variables and a system of three-variable linear equations. However, students found it challenging to solve problems with complicated or unusual words or languages.

On the Radiation and Diffraction of Surface Waves by Submerged Spheroids

1989 ◽  
Vol 33 (02) ◽  
pp. 84-92
Author(s):  
G. X. Wu ◽  
R. Eatock Taylor
Keyword(s):  
Degrees Of Freedom ◽  
Velocity Potential ◽  
Three Dimensional ◽  
Added Mass ◽  
Wave Radiation ◽  
Legendre Functions ◽  
Incoming Wave ◽  
Six Degrees Of Freedom ◽  
Damping Coefficients ◽  
Exciting Forces

The problem of wave radiation and diffraction by submerged spheroids is analyzed using linearized three-dimensional potential-flow theory. The solution is obtained by expanding the velocity potential into a series of Legendre functions in a spheroidal coordinate system. Tabulated and graphical results are provided for added mass and damping coefficients of various spheroids undergoing motions in six degrees of freedom. Graphs are also provided for exciting forces and moments corresponding to a range of incoming wave angles.

Hydrodynamic Torque Converters in Mechanical Transmission Systems: Methods and Analysis

1994 ◽  
Author(s):  
Anders Hedman
Keyword(s):  
Linear Equations ◽  
Equation System ◽  
Torque Converter ◽  
Transmission System ◽  
Mechanical Transmission ◽  
Transmission Systems ◽  
Linear Relationships ◽  
Non Linear ◽  
Computer Calculations ◽  
System Power

Methods for analysis of general mechanical transmission systems with a hydrodynamic torque converter (HTC) are presented. The methods are adapted for computer calculations. The properties of the HTC must be known, either explicitly as speed and torque characteristics, or implicitly as internal geometry (blade angles, etc.). Linear relationships between the torques and between the speeds of the shafts of the transmission system (except the HTC) are easily formulated. The HTC has coupled, non-linear, relationships for torques and speeds. Different ways of including these non-linear equations are presented. This can be implemented in a computer program. Solving the equation system yields the torque and speed of each shaft of the transmission system. Power losses can be handled.

scholarly journals Surge motion on a floating cylinder in water of finite depth

2003 ◽  
Vol 2003 (57) ◽  
pp. 3643-3656 ◽  
Author(s):  
Dambaru D. Bhatta
Keyword(s):  
Velocity Potential ◽  
Separation Of Variables ◽  
Finite Depth ◽  
Added Mass ◽  
Computational Results ◽  
Complex Matrix ◽  
Interior Region ◽  
Exterior Region ◽  
Damping Coefficients ◽  
Calm Water

We derived added mass and damping coefficients of a vertical floating circular cylinder due to surge motion in calm water of finite depth. This is done by deriving the velocity potential for the cylinder by considering two regions, namely, interior region and exterior region. The velocity potentials for these two regions are obtained by the method of separation of variables. The continuity of the solutions has been maintained at the imaginary interface of these regions by matching the functions and gradients of each solution. The complex matrix equation is numerically solved to determine the unknown coefficients. Some computational results are presented for different depth-to-radius and draft-to-radius ratios.

scholarly journals IMPULSIVE AND REFLECTIVE STUDENTS’ UNDERSTANDING TO LINEAR EQUATIONS SYSTEM: AN ANALYSIS THROUGH APOS THEORY

MATHEdunesa ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 128-135
Author(s):  
Dinda Ayu Rachmawati ◽  
Tatag Yuli Eko Siswono
Keyword(s):  
Linear Equation ◽  
Cognitive Style ◽  
Junior High School ◽  
Linear Equations ◽  
Equation System ◽  
Mathematical Concept ◽  
Apos Theory ◽  
Process Stage ◽  
Linear Equation System ◽  
Action Process

Understanding is constructed or reconstructed by students actively. APOS theory (action, process, object, schema) is a theory that states that individuals construct or reconstruct a concept through four stages, namely: action, process, object, and scheme. APOS theory can be used to analyze understanding of a mathematical concept. This research is a qualitative research which aims to describe impulsive and reflective students’ understanding to linear equations system based on APOS theory. Data collection techniques were carried out by giving Matching Familiar Figure Test (MFFT) and concept understanding tests to 32 students of 8th grade in junior high school, then selected one subject with impulsive cognitive style and one subject with reflective cognitive style that can determine solutions set and solve story questions of linear equation system of two variables correctly, then the subjects were interviewed. The results show that there were differences between impulsive and reflective subjects at the stage of action in explaining the definition and giving non-examples of linear equation system of two variables, show the differences in initial scheme of two subjects. At the process stage, impulsive and reflective subjects determine solutions set of linear equation system of two variables. At the object stage, impulsive and reflective subjects determine characteristics of linear equation system of two variables. At the schema stage, impulsive and reflective subjects solve story questions of of linear equation system of two variables, show the final schematic similarity of two subjects.Keywords: understanding, APOS theory, linear equations system of two variables, impulsive cognitive style, reflective cognitive style.

scholarly journals PENGEMBANGAN MODUL MATEMATIKA BERBASIS MASALAH PADA MATERI SISTEM PERSAMAAN LINEAR DUA VARIABEL

2019 ◽  
Vol 6 (1) ◽  
pp. 57
Author(s):  
Siti Suprihatiningsih ◽  
Pradipta Annurwanda
Keyword(s):  
Student Learning ◽  
Learning Outcomes ◽  
Linear Equations ◽  
Equation System ◽  
Material System ◽  
Average Value ◽  
Linear Equation System ◽  
Eighth Grade Students ◽  
Class Viii ◽  
The Subject

This study aims to develop a problem-based mathematical module in the material system of two-variable linear equations of class VIII SMP students that are valid, practical and effective. This research is a development research with the method used is ADDIE. The subject of this study was the eighth grade students of SMP Negeri 1 Ngabang, Landak District. The instruments used were validation sheets, observation sheets and student learning outcomes tests. The result indicates that a problem-based mathematical module has been developed in the material system of two-variable linear equations of class VIII SMP students that are valid, practical and effective. Based on the results of the validity test, it can be stated that the problem-based mathematical module produced is valid with a validity value of 79.11. The implementation observation results show that the problem-based mathematics module is in the practical category with a practical value of 82.18. Then the results of student learning tests showed that the average value of the learning outcomes test was 83.00 and the effectiveness of the problem-based mathematics module was 87.50% and included in the effective category. When viewed from the results of the tests performed, the development of problem-based mathematical modules in the subject matter of the two-variable linear equation system has been successfully developed and is suitable for use in Ngabang 1 State Middle School.

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