scholarly journals XVI. Researches in physical astronomy

1831 ◽  
Vol 121 ◽  
pp. 283-298
Keyword(s):  
Analytical Form ◽  
Numerical Results ◽  
Lunar Theory ◽  
Great Care ◽  
Planetary Theory ◽  
Disturbing Force

I Propose in this paper to extend the equations I have already given for determining the planetary inequalities, as far as the terms depending on the squares and products of the eccentricities, to the terms depending on the cubes of the eccentricities and quantities of that order, which is done very easily by a Table similar to Table II. in my Lunar Theory; and particularly to the determination of the great inequality of Jupiter, or at least such part of it as depends on the first power of the disturbing force. That part which depends on the square of the disturbing force may I think be most easily calculated by the methods given in my Lunar Theory; but not without great care and attention can accurate numerical results be expected. I have how­-ever given the analytical form of the coefficients of the arguments in the development of R, upon which that inequality principally depends. It is I think particularly convenient to designate the arguments of the planetary disturbances by indices. The system of indices adopted in this paper is given as appearing better adapted for the purpose than that used in my former paper on the Planetary Theory; but it is not advisable to make use of the same indices in this as in the Lunar Theory.

Researches in physical astronomy

1837 ◽  
Vol 3 ◽  
pp. 59-60
Keyword(s):  
Lunar Theory ◽  
Disturbing Force ◽  
The One

The author extends, in the present paper, the equations he has already given for determining the planetary inequalities, as far as the terms depending on the squares and products of the eccentricities, to the terms depending on the cubes of the eccentricities and quantities of that order, which he does by means of a table, similar to the one given in his lunar theory; and applies them particularly to the determination of the great inequality of Jupiter, or at least such part of it as depends on the first power of the disturbing force. That part which depends on the square of the disturbing force may, he thinks, be most easily calculated by the methods given in his lunar theory.

scholarly journals On Construction of a Long-Term Moon’s Theory

1999 ◽  
Vol 172 ◽  
pp. 415-416
Author(s):  
T.V. Ivanova
Keyword(s):  
Lunar Theory ◽  
Planetary Theory ◽  
The Moon ◽  
Planetary Orbits ◽  
Small Parameters ◽  
The One ◽  
General Planetary Theory ◽  
Laplace Type

An analytical long-term theory of the motion of the Moon is constructed within the framework of the general planetary theory (Brumberg, 1995). A method, different from the one of (Ivanova, 1997) designated below as (*), for the determination of the perturbations depending on the eccentricities and inclinations of lunar and planetary orbits is used which allows to obtain the solution of the problem in the purely trigonometric form up to any order with respect to the small parameters.The aim of this paper is to construct the long-term Lunar theory in the form consistent with the general planetary theory (Brumberg, 1995). For this purpose the Moon is considered as an additional planet in the field of eight major planets (Pluto being excluded). In the result the coordinates of the Moon may be represented by means of the power series in the evolutionary eccentric and oblique variables with trigonometric coefficients in mean longitudes of the Moon and the planets. The long-period perturbations are determined by solving a secular system in Laplace-type variables describing the secular motions of the lunar perigee and node and taking into account the secular planetary inequalities.

scholarly journals Determination of the number of radicals in the initial chain reactions by mathematical methods

2009 ◽  
Vol 63 (2) ◽  
pp. 121-127
Author(s):  
Branko Pejovic ◽  
Milovan Jotanovic ◽  
Vladan Micic ◽  
Milorad Tomic ◽  
Goran Tadic
Keyword(s):  
General Chemistry ◽  
Difference Equation ◽  
Analytical Form ◽  
Mathematical Methods ◽  
Linear Difference Equations ◽  
Efficient Procedure ◽  
Chain Reactions ◽  
Chemical Equation ◽  
The Difference

Starting from the fact that the real mechanism in a chemical equation takes places through a certain number of radicals which participate in simultaneous reactions and initiate chain reactions according to a particular pattern, the aim of this study is to determine their number in the first couple of steps of the reaction. Based on this, the numbers of radicals were determined in the general case, in the form of linear difference equations, which, by certain mathematical transformations, were reduced to one equation that satisfies a particular numeric series, entirely defined if its first members are known. The equation obtained was solved by a common method developed in the theory of numeric series, in which its solutions represent the number of radicals in an arbitrary step of the reaction observed, in the analytical form. In the final part of the study, the method was tested and verified using two characteristic examples from general chemistry. The study also gives a suggestion of a more efficient procedure by reducing the difference equation to a lower order.

Determination of the energy in body and surface waves

1959 ◽  
Vol 49 (1) ◽  
pp. 1-10
Author(s):  
John De Noyer
Keyword(s):  
Surface Waves ◽  
Seismic Waves ◽  
Numerical Results

Abstract Part II of this paper is concerned with the application of the methods for estimating the energy in seismic waves that were presented in Part I. Numerical results for the energy in the various phases of several earthquakes have been obtained.

scholarly journals Experimental and Numerical Study on Dynamics of Two Footbridges with Different Shapes of Girders

2020 ◽  
Vol 10 (13) ◽  
pp. 4505 ◽  
Author(s):  
Anna Banas ◽  
Robert Jankowski
Keyword(s):  
Natural Vibration ◽  
Numerical Study ◽  
Numerical Results ◽  
Vibration Exciter ◽  
Structural Vibrations ◽  
Impulse Load ◽  
Different Shapes ◽  
Comfort Criteria ◽  
Good Agreement

The paper presents the experimental and numerical results of the dynamic system identification and verification of the behavior of two footbridges in Poland. The experimental part of the study involved vibration testing under different scenarios of human-induced load, impulse load, and excitations induced by vibration exciter. Based on the results obtained, the identification of dynamic parameters of the footbridges was performed using the peak-picking method. With the impulse load applied to both structures, determination of their natural vibration frequencies was possible. Then, based on the design drawings, detailed finite element method (FEM) models were developed, and the numerical analyses were carried out. The comparison between experimental and numerical results obtained from the modal analysis showed a good agreement. The results also indicated that both structures under investigation have the first natural bending frequency of the deck in the range of human-induced excitation. Therefore, the risk of excessive structural vibrations caused by pedestrian loading was then analysed for both structures. The vibration comfort criteria for both footbridges were checked according to Sétra guidelines. In the case of the first footbridge, the results showed that the comfort criteria are fulfilled, regardless of the type of load. For the second footbridge, it was emphasized that the structure meets the assumptions of the guidelines for vibration severability in normal use; nevertheless, it is susceptible to excitations induced by synchronized users, even in the case of a small group of pedestrians.

Elastic Layer Pressed Against a Half Space

1974 ◽  
Vol 41 (3) ◽  
pp. 703-707 ◽  
Author(s):  
K. C. Tsai ◽  
J. Dundurs ◽  
L. M. Keer
Keyword(s):  
Eigenvalue Problem ◽  
Integral Equations ◽  
Half Space ◽  
Elastic Layer ◽  
Numerical Results ◽  
Fredholm Integral Equations ◽  
Auxiliary Functions ◽  
Receding Contact ◽  
Two Bodies

The paper considers the elastic layer which is pressed against a half space by loads that are not necessarily symmetric about the center of the loaded region. It is shown that the receding contact between the two bodies can be treated by means of superposition, leading to two homogeneous Fredholm integral equations for auxiliary functions that are directly related to the contact tractions. The determination of the extent of contact and the shift between the load and contact intervals can be viewed as an eigenvalue problem of the homogeneous integral equations. Specific numerical results are given for two types of triangular loads, and a comparison is made with certain symmetric loads.

Characteristics of a Self-Lubricated Stepped Thrust Pad of Infinite Width With Compressible Lubricant

1959 ◽  
Vol 81 (2) ◽  
pp. 135-145 ◽  
Author(s):  
Kikuo C. Kochi
Keyword(s):  
Optimum Design ◽  
Practical Interest ◽  
Numerical Data ◽  
Numerical Results ◽  
Design Parameters ◽  
Analytic Expressions ◽  
Thrust Pad ◽  
Compressible Lubricant

Harrison’s equation for the pressure in a gas-lubricated bearing of infinite width is solved for a thrust pad with stepped configuration. Analytic expressions for the pressure and load are developed. Numerical results are presented graphically. The analytic expressions together with the numerical data permit most of those characteristics of the stepped pad of practical interest to be completely determinable. Determination of optimum design parameters is given by a pair of graphs.

scholarly journals Determination of pressure in the near-ground space pile terminated and broadening of the surface

2018 ◽  
Vol 251 ◽  
pp. 04062
Author(s):  
Natalia Kupchikova
Keyword(s):  
Numerical Modeling ◽  
Circular Cross Section ◽  
Analytical Form ◽  
Soil Mass ◽  
Software Systems ◽  
Discrete Method ◽  
Building Foundation ◽  
Geological Conditions ◽  
Complex Engineering

The article deals with the problem of determining the stress state of a complex pile structure with end broadening in the form of a sphere in the soil mass in the analytical form by a discrete method. The calculation schemes for determining the stress tensor at the boundary of the pile of square and circular cross-section with expansions in the soil massif are shown. The elements of the polynomial are found by the discrete method in rectangular and spherical coordinates, which is a cumbersome complex mathematical apparatus for a modern design engineer. The stresses are determined. At present, as the analysis has shown, the solution of complex geotechnical problems of soil bases and foundations for different types of loads in numerical modeling is carried out using modern software. Numerical modeling and calculation with the help of specialized software systems allows to consider the system “building-foundation-ground foundation”, as dynamic, integrally developing. However, the interaction of the components of this system requires a theoretical justification of the resistance of foundations in the ground environment, especially in complex engineering-geological conditions.

An L1 adaptive closed-loop guidance law for an orbital injection problem

2009 ◽  
Vol 223 (6) ◽  
pp. 753-761 ◽  
Author(s):  
J Roshanian ◽  
M Zareh ◽  
H H Afshari ◽  
M Rezaei
Keyword(s):  
Closed Loop ◽  
Adaptive Strategy ◽  
Analytical Form ◽  
Open Loop ◽  
Control Input ◽  
Guidance Law ◽  
Non Linear ◽  
Non Linear System ◽  
Orbital Injection

The current paper presents the determination of a closed-loop guidance law for an orbital injection problem using two different approaches and, considering the existing time-optimal open-loop trajectory as the nominal solution, compares the advantages of the two proposed strategies. In the first method, named neighbouring optimal control (NOC), the perturbation feedback method is utilized to determine the closed-loop trajectory in an analytical form for the non-linear system. This law, which produces feedback gains, is in general a function of small perturbations appearing in the states and constraints separately. The second method uses an L1 adaptive strategy in determination of the non-linear closed-loop guidance law. The main advantages of this method include characteristics such as improvement of asymptotic tracking, guaranteed time-delay margin, and smooth control input. The accuracy of the two methods is compared by introducing a high-frequency sinusoidal noise. The simulation results indicate that the L1 adaptive strategy has a better performance than the NOC method to track the nominal trajectory when the noise amplitude is increased. On the other hand, the main advantage of the NOC method is its ability to solve a non-linear, two-point, boundary-value problem in the minimum time.

Sign in / Sign up

Export Citation Format

Share Document