Analytic Solutions to the Closed Bomb

1988 ◽  
Author(s):  
Frederick W. Robbins ◽  
Franz R. Lynn
Keyword(s):  
Analytic Solutions

scholarly journals Type Synthesis of 1T2R Parallel Mechanisms Using Structure Coupling-Reducing Method

2019 ◽  
Vol 32 (1) ◽  
Author(s):  
Haitao Liu ◽  
Ke Xu ◽  
Huiping Shen ◽  
Xianlei Shan ◽  
Tingli Yang
Keyword(s):  
Explicit Form ◽  
Parallel Mechanism ◽  
Analytic Solutions ◽  
Forward Kinematics ◽  
Parallel Mechanisms ◽  
Type Synthesis ◽  
Real Time Control ◽  
Time Control ◽  
Topological Characteristics ◽  
Zero Coupling

Abstract Direct kinematics with analytic solutions is critical to the real-time control of parallel mechanisms. Therefore, the type synthesis of a mechanism having explicit form of forward kinematics has become a topic of interest. Based on this purpose, this paper deals with the type synthesis of 1T2R parallel mechanisms by investigating the topological structure coupling-reducing of the 3UPS&UP parallel mechanism. With the aid of the theory of mechanism topology, the analysis of the topological characteristics of the 3UPS&UP parallel mechanism is presented, which shows that there are highly coupled motions and constraints amongst the limbs of the mechanism. Three methods for structure coupling-reducing of the 3UPS&UP parallel mechanism are proposed, resulting in eight new types of 1T2R parallel mechanisms with one or zero coupling degree. One obtained parallel mechanism is taken as an example to demonstrate that a mechanism with zero coupling degree has an explicit form for forward kinematics. The process of type synthesis is in the order of permutation and combination; therefore, there are no omissions. This method is also applicable to other configurations, and novel topological structures having simple forward kinematics can be obtained from an original mechanism via this method.

scholarly journals Singularity analysis and analytic solutions for the Benney–Gjevik equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andronikos Paliathanasis ◽  
Genly Leon ◽  
P. G. L. Leach
Keyword(s):  
Evolution Equations ◽  
Singularity Analysis ◽  
Analytic Solutions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Painlevé Test ◽  
Wave Solutions ◽  
Algebraic Solutions

Abstract We apply the Painlevé test for the Benney and the Benney–Gjevik equations, which describe waves in falling liquids. We prove that these two nonlinear 1 + 1 evolution equations pass the singularity test for the travelling-wave solutions. The algebraic solutions in terms of Laurent expansions are presented.

scholarly journals A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernels

2021 ◽  
Vol 47 (3) ◽  
Author(s):  
Timon S. Gutleb
Keyword(s):  
Open Source ◽  
Spectral Method ◽  
Jacobi Polynomial ◽  
Numerical Experiments ◽  
Exponential Convergence ◽  
Analytic Solutions ◽  
Convolution Type ◽  
Volterra Equations ◽  
Polynomial Bases ◽  
Sparsity Structure

AbstractWe present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator’s banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form K(x, y) = K(x − y) but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments based on an open-source implementation for problems with and without known analytic solutions and comparisons with other methods.

scholarly journals Mathematical Formulation and Analytic Solutions for Uncertainty Analysis in Probabilistic Safety Assessment of Nuclear Power Plants

Energies ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 929
Author(s):  
Gyun Seob Song ◽  
Man Cheol Kim
Keyword(s):  
Monte Carlo ◽  
Uncertainty Analysis ◽  
Probability Density ◽  
Nuclear Power ◽  
Safety Assessment ◽  
Power Plants ◽  
Mathematical Formulation ◽  
Analytic Solutions ◽  
Probabilistic Safety Assessment ◽  
Probabilistic Safety

Monte Carlo simulations are widely used for uncertainty analysis in the probabilistic safety assessment of nuclear power plants. Despite many advantages, such as its general applicability, a Monte Carlo simulation has inherent limitations as a simulation-based approach. This study provides a mathematical formulation and analytic solutions for the uncertainty analysis in a probabilistic safety assessment (PSA). Starting from the definitions of variables, mathematical equations are derived for synthesizing probability density functions for logical AND, logical OR, and logical OR with rare event approximation of two independent events. The equations can be applied consecutively when there exist more than two events. For fail-to-run failures, the probability density function for the unavailability has the same probability distribution as the probability density function (PDF) for the failure rate under specified conditions. The effectiveness of the analytic solutions is demonstrated by applying them to an example system. The resultant probability density functions are in good agreement with the Monte Carlo simulation results, which are in fact approximations for those from the analytic solutions, with errors less than 12.6%. Important theoretical aspects are examined with the analytic solutions such as the validity of the use of a right-unbounded distribution to describe the uncertainty in the unavailability/probability. The analytic solutions for uncertainty analysis can serve as a basis for all other methods, providing deeper insights into uncertainty analyses in probabilistic safety assessment.

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