Statistics of Combined Linear and Quadratic Springing Response of a TLP in Random Waves

1994 ◽  
Vol 116 (3) ◽  
pp. 127-136 ◽  
Author(s):  
A. Naess
Keyword(s):  
Frequency Response ◽  
Transfer Functions ◽  
Equations Of Motion ◽  
Calculated Data ◽  
Statistical Analyses ◽  
Random Waves ◽  
Sum Frequency ◽  
First Order ◽  
Time Invariant ◽  
The Waves

The paper presents the results of statistical analyses of combined first-order and second-order, sum-frequency response in heave, pitch, and roll of a TLP structure subjected to random, long-crested seas. The results are based on available numerically calculated data for the linear and quadratic transfer functions from the waves to the hydrodynamic loads on the TLP. It has also been assumed that the equations of motion in heave, pitch, and roll can be reasonably well approximated by a set of uncoupled, linear, and time-invariant equations.

Transfer Functions for Efficient Calculation of Multidimensional Transient Heat Transfer

1989 ◽  
Vol 111 (1) ◽  
pp. 5-12 ◽  
Author(s):  
J. E. Seem ◽  
S. A. Klein ◽  
W. A. Beckman ◽  
J. W. Mitchell
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Transfer Functions ◽  
Piecewise Linear ◽  
First Order ◽  
Efficient Calculation ◽  
Long Time ◽  
Transfer Problems ◽  
Time Invariant ◽  
First Order Differential Equations

Finite difference or finite element methods reduce transient multidimensional heat transfer problems into a set of first-order differential equations when thermal physical properties are time invariant and the heat transfer processes are linear. This paper presents a method for determining the exact solution to a set of first-order differential equations when the inputs are modeled by a continuous, piecewise linear curve. For long-time solutions, the method presented is more efficient than Euler, Crank–Nicolson, or other classical techniques.

Extreme Forces in Short-Crested Seas

1980 ◽  
Vol 20 (06) ◽  
pp. 567-578
Author(s):  
S.W. Huntington ◽  
G. Gilbert
Keyword(s):  
Transfer Functions ◽  
Extreme Values ◽  
Numerical Models ◽  
Wave Spectrum ◽  
Standard Theory ◽  
Total Force ◽  
Random Waves ◽  
Linear Superposition ◽  
Incident Waves ◽  
The Waves

Abstract The overall wave force on a large structure in a real multidirectional sea does not occur in a single direction but is a vector randomly varying in magnitude and direction. Although existing theories enable us to calculate the extremes of the orthogonal components of force, the designer needs the extreme resultant or total force. A theory is presented for estimating the extremes of the resultant and is confirmed by experimental measurement. Introduction Recent development of offshore resources in deep water and severe environmental conditions has led to the use of monolithic concrete structures. These structures have members large enough to modify the wave field, and are regarded as being in the diffraction regime of wave loading. Thus, the induced forces and moments are considered linear responses to the incident waves. Estimation of the forces and moments on such structures is based on the linear diffraction theory of Havelock. Several numerical models are used to compute the transfer functions between the incident waves and resuring forces and moments on large structures of arbitrary shape. In general, such models give the magnitude and phase of the transfer function between the waves and the loading at a range of discrete frequencies and angles of wave incidence. In parallel with the numerical approach, analytical methods have been developed to give directly the transfer functions for force and moment on a vertical cylinder in long-crested random waves. This analytical approach has also been extended to real seas that are multidirectional (short crested). In such seas, forces and moments are induced on structures both in line with and at right angles to the principal wave direction. The method gives the transfer functions between the components of force and moment and the total wave spectrum for any particular angular distribution of wave energy. The validity of this direct approach in short-crested seas has been confirmed by laboratory model tests in multidirectional random waves. These two approaches are complementary in that the numerical method allows estimation for regular waves on arbitrarily-shaped real structures; the analytical and laboratory studies allows the extension of results to real multidirectional random seas using the principle of superposition. By using this method, it is possible to compute the spectra of force and moment in two horizontal component directions on a real structure in a real short-crested sea. Since linear superposition is used both in the frequency and angular domains, the calculated component forces and moments also are linear with respect to the waves. However, the spectra of force and moment on a structure are of little direct value to designers concerned with primary failure. They are interested in the possible extremes and will want to set design limits on the forces and moments that are unlikely to be exceeded during the lift of the structure. Since the component forces and moments are linear responses to the waves, the statistical technique used to describe extreme wave elevations can be used to describe the extremes of the components of the loading. This method requires only the gross parameters of the spectra. Since the total (vector) force or moment combines the components and their probabilities in a nonlinear manner, the vital extreme values cannot be derived from the standard theory. This paper presents an analytical solution to this vector problem. SPEJ P. 567＾

Kinetic theory

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Distribution Function ◽  
Kinetic Theory ◽  
Liouville Equation ◽  
Vlasov Equation ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Hamiltonian Form ◽  
First Order ◽  
Number Of Particles

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

scholarly journals Development and Validation of Quasi-Eulerian Mean Three-Dimensional Equations of Motion Using the Generalized Lagrangian Mean Method

2021 ◽  
Vol 9 (1) ◽  
pp. 76
Author(s):  
Duoc Nguyen ◽  
Niels Jacobsen ◽  
Dano Roelvink
Keyword(s):  
Surface Waves ◽  
Deep Water ◽  
Surf Zone ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Breaking Waves ◽  
Successful Implementation ◽  
Random Waves ◽  
Mean Motion ◽  
Generalized Lagrangian

This study aims at developing a new set of equations of mean motion in the presence of surface waves, which is practically applicable from deep water to the coastal zone, estuaries, and outflow areas. The generalized Lagrangian mean (GLM) method is employed to derive a set of quasi-Eulerian mean three-dimensional equations of motion, where effects of the waves are included through source terms. The obtained equations are expressed to the second-order of wave amplitude. Whereas the classical Eulerian-mean equations of motion are only applicable below the wave trough, the new equations are valid until the mean water surface even in the presence of finite-amplitude surface waves. A two-dimensional numerical model (2DV model) is developed to validate the new set of equations of motion. The 2DV model passes the test of steady monochromatic waves propagating over a slope without dissipation (adiabatic condition). This is a primary test for equations of mean motion with a known analytical solution. In addition to this, experimental data for the interaction between random waves and a mean current in both non-breaking and breaking waves are employed to validate the 2DV model. As shown by this successful implementation and validation, the implementation of these equations in any 3D model code is straightforward and may be expected to provide consistent results from deep water to the surf zone, under both weak and strong ambient currents.

Complementary Sensitivity Design of PID Controllers for Arbitrary-Order Transfer Functions With Time Delay

2008 ◽  
Author(s):  
Tooran Emami ◽  
John M. Watkins
Keyword(s):  
Time Delay ◽  
Transfer Functions ◽  
Linear Time ◽  
Order System ◽  
Pid Controllers ◽  
Single Input Single Output ◽  
Linear Time Invariant ◽  
Single Output ◽  
Plant Transfer ◽  
Time Invariant

A graphical technique for finding all proportional integral derivative (PID) controllers that stabilize a given single-input-single-output (SISO) linear time-invariant (LTI) system of any order system with time delay has been solved. In this paper a method is introduced that finds all PID controllers that also satisfy an H∞ complementary sensitivity constraint. This problem can be solved by finding all PID controllers that simultaneously stabilize the closed-loop characteristic polynomial and satisfy constraints defined by a set of related complex polynomials. A key advantage of this procedure is the fact that it does not require the plant transfer function, only its frequency response.

Analysis of Linear Nonconservative Vibrations

1995 ◽  
Vol 62 (3) ◽  
pp. 685-691 ◽  
Author(s):  
F. Ma ◽  
T. K. Caughey
Keyword(s):  
Modal Analysis ◽  
Equations Of Motion ◽  
Substantial Reduction ◽  
Computational Effort ◽  
Equivalence Transformations ◽  
State Space Approach ◽  
Nonconservative System ◽  
First Order ◽  
Physical Insight ◽  
Space Approach

The coefficients of a linear nonconservative system are arbitrary matrices lacking the usual properties of symmetry and definiteness. Classical modal analysis is extended in this paper so as to apply to systems with nonsymmetric coefficients. The extension utilizes equivalence transformations and does not require conversion of the equations of motion to first-order forms. Compared with the state-space approach, the generalized modal analysis can offer substantial reduction in computational effort and ample physical insight.

scholarly journals Actions, topological terms and boundaries in first-order gravity: A review

2016 ◽  
Vol 25 (04) ◽  
pp. 1630011 ◽  
Author(s):  
Alejandro Corichi ◽  
Irais Rubalcava-García ◽  
Tatjana Vukašinac
Keyword(s):  
Boundary Conditions ◽  
Symplectic Structure ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Action Principle ◽  
Internal Boundary ◽  
Four Dimensions ◽  
First Order ◽  
Conserved Charges ◽  
Boundary Terms

In this review, we consider first-order gravity in four dimensions. In particular, we focus our attention in formulations where the fundamental variables are a tetrad [Formula: see text] and a [Formula: see text] connection [Formula: see text]. We study the most general action principle compatible with diffeomorphism invariance. This implies, in particular, considering besides the standard Einstein–Hilbert–Palatini term, other terms that either do not change the equations of motion, or are topological in nature. Having a well defined action principle sometimes involves the need for additional boundary terms, whose detailed form may depend on the particular boundary conditions at hand. In this work, we consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. We focus on the covariant Hamiltonian formalism where the phase space [Formula: see text] is given by solutions to the equations of motion. For each of the possible terms contributing to the action, we consider the well-posedness of the action, its finiteness, the contribution to the symplectic structure, and the Hamiltonian and Noether charges. For the chosen boundary conditions, standard boundary terms warrant a well posed theory. Furthermore, the boundary and topological terms do not contribute to the symplectic structure, nor the Hamiltonian conserved charges. The Noether conserved charges, on the other hand, do depend on such additional terms. The aim of this manuscript is to present a comprehensive and self-contained treatment of the subject, so the style is somewhat pedagogical. Furthermore, along the way, we point out and clarify some issues that have not been clearly understood in the literature.

Optimal Nonlinear Estimation of Linear Stochastic Systems: The Multivariable Extension

1996 ◽  
Vol 118 (2) ◽  
pp. 350-353 ◽  
Author(s):  
M. A. Hopkins ◽  
H. F. VanLandingham
Keyword(s):  
Stochastic Systems ◽  
Transfer Functions ◽  
Mimo Systems ◽  
Linear Time ◽  
Nonlinear Method ◽  
Single Input Single Output ◽  
Linear Time Invariant ◽  
Single Output ◽  
Time Invariant ◽  
Time Systems

This paper extends to multi-input multi-output (MIMO) systems a nonlinear method of simultaneous parameter and state estimation that appeared in the ASME JDSM&C (September, 1994), for single-input single-output (SISO) systems. The method is called pseudo-linear identification (PLID), and applies to stochastic linear time-invariant discrete-time systems. No assumptions are required about pole or zero locations; nor about relative degree, except that the system transfer functions must be strictly proper. In the earlier paper, proofs of optimality and convergence were given. Extensions of those proofs to the MIMO case are also given here.

Consistent section-averaged equations of quasi-one-dimensional laminar flow

2010 ◽  
Vol 656 ◽  
pp. 337-341 ◽  
Author(s):  
PAOLO LUCHINI ◽  
FRANÇOIS CHARRU
Keyword(s):  
Equations Of Motion ◽  
Stability Result ◽  
Momentum Equation ◽  
Dissipation Function ◽  
Dimensional Solution ◽  
First Order ◽  
Averaged Equations ◽  
Classical Stability ◽  
Momentum Integral ◽  
Free Falling

Section-averaged equations of motion, widely adopted for slowly varying flows in pipes, channels and thin films, are usually derived from the momentum integral on a heuristic basis, although this formulation is affected by known inconsistencies. We show that starting from the energy rather than the momentum equation makes it become consistent to first order in the slowness parameter, giving the same results that have been provided until today only by a much more laborious two-dimensional solution. The kinetic-energy equation correctly provides the pressure gradient because with a suitable normalization the first-order correction to the dissipation function is identically zero. The momentum equation then correctly provides the wall shear stress. As an example, the classical stability result for a free falling liquid film is recovered straightforwardly.

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