Method of integral equations in the polytropic theory of stars with axial rotation. II. Polytropes with indices $n>1$

2021 ◽  
Vol 8 (3) ◽  
pp. 474-485
Author(s):  
M. V. Vavrukh ◽  
◽  
D. V. Dzikovskyi ◽  
Keyword(s):  
Integral Form ◽  
Axial Rotation ◽  
Geometrical Parameters ◽  
Surface Form ◽  
Equilibrium Equations ◽  
Method Of Integral Equations ◽  
Mass Moment ◽  
Self Consistent ◽  
Nonlinear Equilibrium ◽  
Mass Moment Of Inertia

A new method for finding solutions of the nonlinear equilibrium equations for rotational polytropes was proposed, which is based on a self-consistent description of internal region and periphery using the integral form of equations. Dependencies of geometrical parameters, surface form, mass, moment of inertia and integration constants on angular velocity were calculated for indices $n=2.5$ and $n=3$.

scholarly journals Method of integral equations in the polytropic theory of stars with axial rotation. I. Polytropes n=0 and n=1

2021 ◽  
Vol 8 (2) ◽  
pp. 338-358
Author(s):  
M. V. Vavrukh ◽  
◽  
D. V. Dzikovskyi ◽  
Keyword(s):  
Angular Velocity ◽  
Axial Rotation ◽  
Surface Shape ◽  
Equilibrium Equations ◽  
Polytropic Model ◽  
New Approach ◽  
Method Of Integral Equations ◽  
Integral Forms ◽  
Self Consistent ◽  
First Time

Calculations of characteristics of stars with axial rotation in the frame of polytropic model are based on the solution of mechanical equilibrium equation – differential equation of second order in partial derivatives. Different variants of approximate determinations of integration constants are based on traditional in the theory of stellar surface approximation, namely continuity of gravitational potential in the surface vicinity. We proposed a new approach, in which we used simultaneously differential and integral forms of equilibrium equations. This is a closed system and allows us to define in self-consistent way integration constants, the polytrope surface shape and distribution of matter over volume of a star. With the examples of polytropes n=0 and n=1, we established the existence of two rotation modes (with small and large eccentricities). It is proved that the polytrope surface is the surface of homogeneous rotational ellipsoid for the case n=0. The polytrope characteristics with n=1 in different approximations were calculated as the functions of angular velocity. For the first time it has been calculated the deviation of polytrope surface at fixed value of angular velocity from the surface of associated rotational ellipsoid.

The Effect of Porosity on Elastic Stability of Toroidal Shell Segments Made of Saturated Porous Functionally Graded Materials

2020 ◽  
Vol 143 (3) ◽  
Author(s):  
Hadi Babaei ◽  
Mohsen Jabbari ◽  
M. Reza Eslami
Keyword(s):  
Elastic Stability ◽  
Functionally Graded ◽  
Toroidal Shell ◽  
Geometrical Parameters ◽  
Equilibrium Equations ◽  
Closed Form Solutions ◽  
Porous Shell ◽  
Total Potential ◽  
Nonlinear Equilibrium ◽  
The Stability

Abstract This research deals with the stability analysis of shallow segments of the toroidal shell made of saturated porous functionally graded (FG) material. The nonhomogeneous material properties of porous shell are assumed to be functionally graded as a function of the thickness and porosity parameters. The porous toroidal shell segments with positive and negative Gaussian curvatures and nonuniform distributed porosity are considered. The nonlinear equilibrium equations of the porous shell are derived via the total potential energy of the system. The governing equations are obtained on the basis of classical thin shell theory and the assumptions of Biot's poroelasticity theory. The equations are a set of the coupled partial differential equations. The analytical method including the Airy stress function is used to solve the stability equations of porous shell under mechanical loads in three cases. Porous toroidal shell segments subjected to lateral pressure, axial compression, and hydrostatic pressure loads are analytically analyzed. Closed-form solutions are expressed for the elastic buckling behavior of the convex and concave porous toroidal shell segments. The effects of porosity distribution and geometrical parameters of the shell on the critical buckling loads of porous toroidal shell segments are studied.

Thermal Postbuckling of Imperfect Circular Functionally Graded Material Plates: Examination of Voigt, Mori–Tanaka, and Self-Consistent Schemes

2014 ◽  
Vol 137 (2) ◽  
Author(s):  
Y. Kiani ◽  
M. R. Eslami
Keyword(s):  
Functionally Graded Material ◽  
Initial Imperfection ◽  
Functionally Graded ◽  
Circular Plates ◽  
Equilibrium Equations ◽  
Simply Supported ◽  
Self Consistent ◽  
Nonlinear Equilibrium ◽  
Graded Material ◽  
Thermal Postbuckling

Thermal postbuckling of solid circular plates made of a through-the-thickness functionally graded material (FGM) is analyzed in this paper. Initial imperfection of the plate is also taken into account. Each thermomechanical property of the plate is assumed to be a function of the temperature and thickness coordinate. Equivalent properties of the FGM media are obtained based on three different homogenization schemes, namely, Voigt rule, Mori–Tanaka scheme, and self-consistent estimate. Temperature profile is assumed to be through-the-thickness direction only. The solution of the heat conduction equation is obtained using an iterative central finite difference scheme. Various types of thermal loadings, such as uniform temperature rise, temperature specified at surfaces, and heat flux, are considered. Nonlinear equilibrium equations of the plate are obtained by means of the conventional Ritz method. Solution of the resulting nonlinear equilibrium equations and temperature distribution are obtained simultaneously at each step of heating. It is shown that response of a perfect clamped FGM plate is of the bifurcation type of buckling with stable postbuckling equilibrium branch, whereas imperfect clamped and perfect/imperfect simply supported FGM plates do not reveal the bifurcation type of instability through the nonuniform heating process. Furthermore, amplitude of initial imperfection is an important factor on the equilibrium path of FGM circular plates, especially for simply supported ones.

Modeling and validation of crankshaft speed fluctuations of a single-cylinder four-stroke diesel engine

2021 ◽  
pp. 095440702110262
Author(s):  
Mustafa Babagiray ◽  
Hamit Solmaz ◽  
Duygu İpci ◽  
Fatih Aksoy
Keyword(s):  
Diesel Engine ◽  
Dynamic Model ◽  
Moment Of Inertia ◽  
Mass Motion ◽  
Cylinder Pressure ◽  
Motion Equations ◽  
Single Cylinder ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
Speed Fluctuations

In this study, a dynamic model of a single-cylinder four-stroke diesel engine has been created, and the crankshaft speed fluctuations have been simulated and validated. The dynamic model of the engine consists of the motion equations of the piston, conrod, and crankshaft. Conrod motion was modeled by two translational and one angular motion equations, by considering the kinetic energy resulted from the mass moment of inertia and conrod mass. Motion equations involve in-cylinder gas pressure forces, hydrodynamic and dry friction, mass inertia moments of moving parts, starter moment, and external load moment. The In-cylinder pressure profile used in the model was obtained experimentally to increase the accuracy of the model. Pressure profiles were expressed mathematically using the Fourier series. The motion equations were solved by using the Taylor series method. The solution of the mathematical model was performed by coding in the MATLAB interface. Cyclic speed fluctuations obtained from the model were compared with experimental results and found compitable. A validated model was used to analyze the effects of in-cylinder pressure, mass moment of inertia of crankshaft and connecting rod, friction, and piston mass. In experiments for 1500, 1800, 2400, and 2700 rpm engine speeds, crankshaft speed fluctuations were observed as 12.84%, 8.04%, 5.02%, and 4.44%, respectively. In simulations performed for the same speeds, crankshaft speed fluctuations were calculated as 10.45%, 7.56%, 4.49%, and 3.65%. Besides, it was observed that the speed fluctuations decreased as the average crankshaft speed value increased. In the simulation for 157.07, 188.49, 219.91, 251.32, and 282.74 rad/s crankshaft speeds, crankshaft speed fluctuations occurred at rates of 10.45%, 7.56%, 5.84%, 4.49%, and 3.65%, respectively. The effective engine power was achieved as 5.25 kW at an average crankshaft angular speed of 219.91 rad/s. The power of friction loss in the engine was determined as 0.68 kW.

Instability of Discrete Pseudo-Conservative Structural Systems

1991 ◽  
Vol 44 (11S) ◽  
pp. S194-S198 ◽  
Author(s):  
Anibal E. Mirasso ◽  
Luis A. Godoy
Keyword(s):  
Classical System ◽  
Potential Energy Function ◽  
Structural Systems ◽  
Equilibrium Path ◽  
Equilibrium Equations ◽  
Total Potential ◽  
Approximate Form ◽  
Nonlinear Equilibrium ◽  
Mechanical Models ◽  
New Formulation

Critical and postcritical states of pseudo-conservative discrete structural systems are studied by means of a new formulation leading to a classification of critical states and to an approximate form of the postcritical equilibrium path. The nonlinear equilibrium equations are derived from the total potential energy function of a classical system, but with the addition of at least one control parameter. The follower force effect is thus included by nonlinear constraints to the equilibrium equation. The nonlinear equations are solved by perturbation techniques. Finally the theory is applied to investigate the instability of some simple mechanical models.

Optimal Performance of the TLCD in Structural Pitching Vibration Control

2002 ◽  
Vol 8 (5) ◽  
pp. 619-642 ◽  
Author(s):  
S. D. Xue ◽  
J. M. Ko ◽  
Y. L. Xu
Keyword(s):  
Numerical Optimization ◽  
Sensitivity Study ◽  
Optimization Approach ◽  
Optimum Control ◽  
Head Loss Coefficient ◽  
Tuning Ratio ◽  
Mass Moment ◽  
Practical Design ◽  
Mass Moment Of Inertia ◽  
Structural Uncertainties

A detailed optimal parametric study is performed for a tuned liquid column damper (TLCD) in suppressing the pitching vibration of structures. Due to the difficulty of finding analytical solutions for the damped structure, a numerical optimization approach is proposed and applied to the system to find the optimum TLCD parameters. The variations of the optimum control parameter with system parameters are determined and discussed. Using various numerical searching data, a set of practical design formulas for the optimum tuning ratio and optimum head loss coefficient of the TLCD are then derived through regression analysis. The comparison between practical design formula and numerical optimization shows a very close agreement between the two results. The practical design formulas provide a convenient tool for designers. In order to account for the possible effects of structural uncertainties, a parametric sensitivity study on the de-tuning of optimum damper parameters is also carried out. It is found that the detuning effect is more severe for low damped structure with lower ratios of mass moment of inertia, especially for the detuning of tuning ratio.

scholarly journals Heavy Military Land Vehicle Mass Properties Estimation Using Hoisting and Pendulum Motion Method

2019 ◽  
Vol 69 (6) ◽  
pp. 550-556
Author(s):  
M. S. Risby ◽  
Khalis Suhaimi ◽  
Tan Kean Sheng ◽  
Arif Syafiq M. S. ◽  
Mohd Hafizi N
Keyword(s):  
Centre Of Gravity ◽  
Mass Moment ◽  
Mass Properties ◽  
Proper Mass ◽  
Vehicle Rollover ◽  
Mass Moment Of Inertia ◽  
Mobile Crane ◽  
Vector Mechanics ◽  
Rollover Crash ◽  
Modelling Process

Mass properties such as the centre of gravity location, moments of inertia, and total mass are of great importance for vehicle stability studies and deployment. Certain parameters are required when these vehicles need to be arranged inside an aircraft for the carrier to achieve proper mass balance and stability during a flight. These parameters are also important for the design and modelling process of vehicle rollover crash studies. In this study, the mass properties of a military armoured vehicle were estimated using hoisting and pendulum method. The gross total weight, longitudinal and vertical measurements were recorded by lifting the vehicle using a mobile crane and the data were used to estimate the centre of gravity. The frequency of vehicle oscillation was measured by applying swing motion with a small angle of the vehicle as it is suspended on air. The centre of gravity and mass moment of inertia were calculated using the vector mechanics approach. The outcomes and limitations of the approach as discussed in details.

scholarly journals Research Concerning the Dynamic Model of the Conventional Sucker Rod Pumping Units

2019 ◽  
Vol 70 (8) ◽  
pp. 2818-2821
Author(s):  
Georgeta Toma
Keyword(s):  
Computer Program ◽  
Dynamic Model ◽  
Angular Speed ◽  
Management Program ◽  
Programming Environment ◽  
Synthesis Parameters ◽  
Mass Moment ◽  
Sucker Rod ◽  
Mass Moment Of Inertia ◽  
Pumping Units

The study of the dynamic model of the conventional sucker rod pumping units requires first determining the variation on the cinematic cycle of the synthesis parameters (the reduced moment and the reduced mass moment of inertia) and then the variation of the angular speed of the cranks, in response to the dynamic and resistant actions on the component elements that appear during operation. The paper presents the way of determining the variation on the cinematic cycle of the synthesis parameters of the dynamic model corresponding to the conventional pumping unit mechanism and of the variation of the angular speed of its cranks. The experimental records have been processed with the Total Well Management program. The simulations have been performed with a computer program developed by the author using the Maple programming environment.

scholarly journals Pengaruh Variasi Radius-Dalam Rim Terhadap Pengurangan Massa dan Momen Inersia Massa Dengan Studi Kasus Benda Putar Berdiameter 10 cm

2018 ◽  
Vol 1 (1) ◽  
Author(s):  
Eka Taufiq Firmansjah
Keyword(s):  
Outer Radius ◽  
Moment Of Inertia ◽  
Mass Reduction ◽  
Inner Radius ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
Machine Elements ◽  
The Moment ◽  
The Masses

ABSTRAK Mesin terdiri dari sekumpulan elemen mesin yang diam dan bergerak. Elemen mesin yang bergerak dengan gerakan berputar disebut benda putar. Pada beberapa kasus seringkali diinginkan pengurangan massa dari benda putar tersebut untuk alasan ekonomis, biasanya untuk elemen mesin yag diproduksi massal. Namun pengurangan massa berakibat pada pengurangan momen inersia massa benda putar bersangkutan. Jika tuntutan perancangan tidak mempermasalahkan perubahan tersebut, maka pengurangan massa tidak menjadi masalah. Namun jika momen inersia massa tidak boleh terlalu rendah, maka harus dicari kompromi dimana pengurangan massa sebesar-besarnya namun penurunan momen inersia massa sekecil-kecilnya. Pada penelitian ini dilakukan studi kasus terhadap benda putar berjari- jari 10 cm jari-jari dalam hub 2 cm dan jari-jari luar hub 4 cm. Jumlah jari-jari ada 4 dengan lebar 1 cm dan tebal benda putar 0,5 cm. Variasi pengurangan massa dilakukan dengan memvariasikan jari-jari- dalam rim. Untuk tiap variasi, dilakukan perhitungan untuk mendapatkan jumlah massa yang dapat dikurangi dan momen inersia massa dari benda putar. Ternyata pada nilai jari-jari dalam tertentu, dapat diperoleh nilai kompromi dari permasalahan diatas. Kata kunci: benda putar, penghematan bahan, momen inersia massa.  ABSTRACT Machine consists of a set of machine elements that still and moving. Machine elements that move in a circular motion called rotary object. In some cases it is often desirable reduction in the mass of the rotating object for economic reasons, usually for a mass production of machine elements. But the mass reduction results in a reduction in moment of inertia of the mass. If the demands of the design allow this decrease of moment of inertia, mass reduction is not a problem. But if the moment of inertia of the masses should not be too low, it must find a compromise in which a mass reduction profusely but the decrease in the mass moment of inertia of the smallest. In this research conducted a case study of rotating element radius of 10 cm, radius of the hub 2 cm and outer radius hub 4 cm. The number of spoke are 4 with a width of 1 cm and uniform thickness 0.5 cm all over rotating element. Variations mass reduction is done by varying the inner radius of the rim. For each variation, calculation is performed to obtain the amount of mass that can be reduced and the mass moment of inertia of the rotating object. It turned out that in the certain value of inner radius of the rim in particular, can compromise the values obtained from the above problem. Keywords: rotating element, reducing material, mass moment of inertia.

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