Modeling Helicopter Blade Sailing: Dynamic Formulation in the Planar Case

2007 ◽  
Vol 74 (6) ◽  
pp. 1104-1113 ◽  
Author(s):  
A. S. Wall ◽  
R. G. Langlois ◽  
F. F. Afagh
Keyword(s):  
Rigid Bodies ◽  
Operating Conditions ◽  
Ship Motion ◽  
Three Dimensions ◽  
Elastic Response ◽  
Helicopter Rotor ◽  
Planar Case ◽  
Geometrical Properties ◽  
Lagrange’S Equation ◽  
Blade Sailing

As part of a research project aimed at simulating rotor dynamic response during shipboard rotor startup and shutdown operations, a dynamic model of the ship–helicopter–rotor system that is appropriate for use in predicting rotor elastic response was developed. This planar model consists of a series of rigid bodies connected by rotational stiffness and damping elements that allow motion in the flapwise direction. The rotors were partitioned into an arbitrary number of rigid beam segments having the inertial and geometrical properties of a typical rotor. Helicopter suspension flexibility and damping were also modeled, although the helicopter was otherwise considered as a rigid body. Lagrange’s equation was used to derive the governing dynamic equations for the helicopter–rotor model. The effect of ship motion on blade deflection was also considered. The ship motion supplied as input to the model included representative frigate flight deck motion in three dimensions corresponding to an actual sea spectrum, ship particulars and ship operating conditions. This paper is intended to detail the dynamic approach adopted for this blade sailing study, and its conceptual validation in the planar case. The methodologies that have been developed lend themselves to easy expansion into three dimensions, and into torsion and lead/lag modeling. The amount of blade motion induced by ship motion on nonrotating helicopter blades is included. Although aerodynamic loads are a major contributor to blade sailing, this paper focuses on the dynamics aspect of the problem, and thus does not include aerodynamic effects.

Numerical Simulation of Multibody Systems With Time Dependant Structure

1993 ◽  
Author(s):  
Pierre Joli ◽  
Madeleine Pascal ◽  
René Gibert
Keyword(s):  
Numerical Simulation ◽  
Multibody Systems ◽  
Computational Algorithm ◽  
Rigid Bodies ◽  
Three Dimensions ◽  
Integration Step ◽  
General Systems ◽  
Articulated Systems ◽  
Jump Velocity ◽  
Assembly Tasks

Abstract Current dynamic simulation programs are able to calculate the continuous motions of articulated systems or more general systems of rigid bodies in the absence of contact between members of the system or between the system and its environment. Some are able to simulate the effects of isolated contacts and impacts but none are able to simulate the motion with unrestricted multiple concurrent contacts. However, in special robotic programs such as robots performing assembly tasks or walking, it would be very interesting to simulate appropriate commands before implementing them on the robots. This paper develops intrinsic problems of collision to produce an efficient computational algorithm. This algorithm handles the detection of collision in three dimensions, the reduction of the integration step in order to avoid interpenetration between the bodies before impact, the jump velocity caused by a new collision and indicator magnitudes which determine the addition or deletion of constraints.

Kinematic Analysis of Spherical Four-Bar Linkages via the Inflection Cubic and Thales Ellipse

2015 ◽  
Author(s):  
Giorgio Figliolini ◽  
Jorge Angeles
Keyword(s):  
Special Interest ◽  
Kinematic Analysis ◽  
Vector Fields ◽  
Three Dimensions ◽  
Unique Point ◽  
Planar Case ◽  
Acceleration Vector ◽  
The Subject ◽  
Four Bar Linkage ◽  
Spherical Motion

The subject of this paper is the formulation of a specific algorithm for the kinematic analysis of spherical four-bar linkages via the inflection spherical cubic and spherical Thales ellipse by devoting particular attention to the crossed four-bar linkage (anti-parallelogram). Moreover, both the inflection and the elliptic cones, which represent the equivalent of the Bresse cylinders of the planar case in three-dimensions, are obtained by showing the particular properties of the spherical motion in terms of the curvature of a coupler curve and both the velocity and acceleration vector fields. Of special interest are also the cases in which the three acceleration poles coincide at one unique point or in two plus one, which depends on the intersections of two spherical curves of third and second degree.

Geometrical Properties of Three Symmetric Matrices

1935 ◽  
Vol 31 (2) ◽  
pp. 174-182 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Explicit Form ◽  
Three Dimensions ◽  
Symmetric Matrices ◽  
Geometrical Properties ◽  
Cubic Surface ◽  
Quadric Surfaces ◽  
Quartic Curve ◽  
Polar Property ◽  
Linear Complexes ◽  
Image Position

In the early editions of the Geometry of Three Dimensions Salmon had stated that the equations of any three quadric surfaces could be simultaneously reduced to the sums of five squares. Such a reduction is not possible in general, but can be performed if and only if a certain combinant Λ, of the net of quadrics, vanishes. Algebraically the theory of such a net of quadrics is equivalent, as Hesse(2) showed, to that of a plane quartic curve: and the condition for the equation a quartic to be expressible to the sum of five fourth powers is equivalent to the condition Λ = 0(1). While Clebsch(3) was the first to establish this condition, Lüroth(4) gave it more explicit form by studying the quartic curvewhich satisfies the condition. Frahm(5) seems to have been the first to prove the impossibility of the above reduction of three general quadric surfaces, by remarking that the plane quartic curve obtained in Hesse's way from the locus of the vertices of cones of the net of quadrics would be a Lüroth quartic. Frahm further remarked that the three quadrics, so conditioned, could be regarded as the polar quadrics belonging to a cubic surface in ∞2 ways; but that for three general quadrics no such cubic surface exists. An explicit algebraical account of these properties was given by E. Toeplitz(6), who incidentally noticed that certain linear complexes associated with three general quadrics became special linear complexes when Λ = 0. This polar property of three quadrics in [3] was generalized to n dimensions by Anderson (7).

scholarly journals Complex Bogie Modeling Incorporating Advanced Friction Wedge Components

2009 ◽  
Author(s):  
Brian Sperry ◽  
Corina Sandu ◽  
Brent Ballew
Keyword(s):  
Dynamic Behavior ◽  
Degrees Of Freedom ◽  
Model Simulation ◽  
Surface Friction ◽  
Rigid Bodies ◽  
Operating Conditions ◽  
Normal Operation ◽  
Contact Points ◽  
Modeling Software ◽  
Simulation Results

This research focuses on the dynamic behavior of the three-piece bogie that supports the freight train car bodies. While the system is relatively simple, in that there are very few parts involved, the behavior of the bogie is somewhat more complex. Our research focuses primarily on the behavior of the friction wedges under different operating conditions that are seen under normal operation. The Railway Technologies Laboratory (RTL) at Virginia Tech has been developing a model to better capture the dynamic behavior of friction wedges using 3-D modeling software. In previous years, a quarter-truck model, and half-truck variably damped model have been developed using MathWorks MATLAB®. This year, research has focused on the development of a half-truck variably damped model with a new (curved surface) friction wedge, and a half-truck constantly damped model, both using the MATLAB® based software program. Currently a full-truck variably damped model has been created using LMS Virtual.Lab. This software allows for a model that is more easily created and modified, as well as allowing for a much shorter simulation time, which became a necessity as more contact points, and more complex inputs were needed to increase the accuracy of the simulation results. The new model consists of seven rigid bodies: the bolster, two sideframes, and four wedges. We have also implemented full spring nests on each sideframe, where in previous models equivalent spring forces were used. The model allows six degrees-of-freedom for the wedges and bolster: lateral, longitudinal, and vertical translations, as well as pitch, roll, and yaw. The sideframes are constrained to two degrees-of-freedom: vertical and longitudinal translations. The inputs to the model are vertical and longitudinal translations or forces on the sideframes, which can be set completely independent of each other. The model simulation results have been compared with results from NUCARS®, an industrially-used train modeling software developed by the Transportation Technology Center, Inc. (TTCI), a wholly owned subsidiary of the Association of American Railroads (AAR), for similar inputs, as well as experimental data from warping tests performed at TTCI.

scholarly journals Improved Bounds for Incidences Between Points and Circles

2014 ◽  
Vol 24 (3) ◽  
pp. 490-520 ◽  
Author(s):  
MICHA SHARIR ◽  
ADAM SHEFFER ◽  
JOSHUA ZAHL
Keyword(s):  
Three Dimensional ◽  
Combinatorial Geometry ◽  
Two Dimensions ◽  
Three Dimensions ◽  
List Type ◽  
Planar Case ◽  
Constant Degree ◽  
Common Plane ◽  
Formula Group ◽  
Image Position

We establish an improved upper bound for the number of incidences betweenmpoints andncircles in three dimensions. The previous best known bound, originally established for the planar case and later extended to any dimension ≥ 2, isO*(m2/3n2/3+m6/11n9/11+m+n), where theO*(⋅) notation hides polylogarithmic factors. Since all the points and circles may lie on a common plane (or sphere), it is impossible to improve the bound in ℝ3without first improving it in the plane.Nevertheless, we show that if the set of circles is required to be ‘truly three-dimensional’ in the sense that no sphere or plane contains more thanqof the circles, for someq≪n, then for any ϵ > 0 the bound can be improved to\[ O\bigl(m^{3/7+\eps}n^{6/7} + m^{2/3+\eps}n^{1/2}q^{1/6} + m^{6/11+\eps}n^{15/22}q^{3/22} + m + n\bigr). \]For various ranges of parameters (e.g., whenm= Θ(n) andq=o(n7/9)), this bound is smaller than the lower bound Ω*(m2/3n2/3+m+n), which holds in two dimensions.We present several extensions and applications of the new bound.(i)For the special case where all the circles have the same radius, we obtain the improved boundO(m5/11+ϵn9/11+m2/3+ϵn1/2q1/6+m+n).(ii)We present an improved analysis that removes the subpolynomial factors from the bound whenm=O(n3/2−ϵ) for any fixed ϵ < 0.(iii)We use our results to obtain the improved boundO(m15/7) for the number of mutually similar triangles determined by any set ofmpoints in ℝ3.Our result is obtained by applying the polynomial partitioning technique of Guth and Katz using a constant-degree partitioning polynomial (as was also recently used by Solymosi and Tao). We also rely on various additional tools from analytic, algebraic, and combinatorial geometry.

Collision Detection of Cylindrical Rigid Bodies Using Line Geometry

2005 ◽  
Author(s):  
John S. Ketchel ◽  
Pierre M. Larochelle
Keyword(s):  
Collision Detection ◽  
Nuclear Physics ◽  
Detection Algorithm ◽  
Rigid Bodies ◽  
Three Dimensions ◽  
Necessary Condition ◽  
Infinite Length ◽  
Line Geometry ◽  
Well Drilling ◽  
Necessary And Sufficient

This paper presents a novel methodology for detecting collisions of cylindrically shaped rigid bodies moving in three dimensions. This algorithm uses line geometry and dual number algebra to exploit the geometry of right circular cylindrical objects to facilitate the detection of collisions. First, the rigid bodies are modelled with infinite length cylinders and a necessary condition for collision is evaluated. If the necessary condition is not satisfied then the two bodies are not capable of collision. If the necessary condition is satisfied then a collision between the bodies may occur and we proceed to the next stage of the algorithm. In the second stage the bodies are modelled with finite length cylinders and a definitive necessary and sufficient collision detection algorithm is employed. The result is a straight-forward and efficient means of detecting collisions of cylindrically shaped bodies moving in three dimensions. This methodology has applications in spatial mechanism design, robot motion planning, workspace analysis of parallel kinematic machines such as Stewart-Gough platforms, nuclear physics, medical research, computer graphics and well drilling. A case study examining a spatial 4C robotic mechanism for self collisions is included.

Mid-gap invasion in two-layer slot coating

2009 ◽  
Vol 631 ◽  
pp. 397-417 ◽  
Author(s):  
JAEWOOK NAM ◽  
MARCIO S. CARVALHO
Keyword(s):  
Free Surface Flows ◽  
Operating Conditions ◽  
Three Dimensions ◽  
Separation Point ◽  
Complex Flow ◽  
Two Phase ◽  
Navier Stokes Equation ◽  
Slot Coating ◽  
Die Surface ◽  
Flow States

Multi-layer, continuous liquid coating is the most efficient way to manufacture films that require more than one layer for optimal performance. Dual-layer slot coating is one of different coating methods largely used to deposit two thin, uniform liquid layers on to a moving substrate. The two liquid phases are separated by an inter-layer that starts at the separation point (or line, in three dimensions) attached to the die surface. The stability of the two-phase flow and the location of the separation point are directly related to the quality of the final product. Ideally, the separation point should be attached to the downstream corner of the mid die piece of a dual slot-coating die. However, its location may change as operating conditions vary, leading to undesired flow states, with microvortices and periodic oscillation. The movement of the separation point from its desired location along the die surface is usually referred to as mid-gap invasion and can be associated with the onset of coating defects. It is crucial to determine the set of flow conditions at which it occurs. We study the evolution of the separation-point location and the inter-layer configuration as a function of operating conditions by flow visualization and by solving the two-dimensional Navier–Stokes equation for free-surface flows. The results reveal two different mechanisms for mid-gap invasion, depending on the viscosity ratio of the two liquid layers. They also show that the most critical parameter responsible for the onset of mid-gap invasion is the bottom-layer wet thickness (flow rate). Although the movement of the separation point involves an evolution of complex flow states, a simple but accurate criterion based on rectilinear flow approximation is proposed.

Analysis and Computation of Two Body Impact in Three Dimensions

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Yan-Bin Jia ◽  
Feifei Wang
Keyword(s):  
Closed Form ◽  
Contact Point ◽  
Positive Definiteness ◽  
Rigid Bodies ◽  
Three Dimensions ◽  
Sliding Velocity ◽  
Form Analysis ◽  
Coulomb’S Friction ◽  
The Matrix ◽  
Single Contact Point

A formal impulse-based analysis is presented for the collision of two rigid bodies at single contact point under Coulomb's friction in three dimensions (3D). The tangential impulse at the contact is known to be linear in the sliding velocity whose trajectory, parametrized with the normal impulse and referred to as the hodograph, is governed by a generally nonintegrable ordinary differential equation (ODE). Evolution of the hodograph is bounded by rays in several invariant directions of sliding in the contact plane. Exact lower and upper bounds are derived for the number of such invariant directions, utilizing the established positive definiteness of the matrix defining the governing ODE. If the hodograph reaches the origin, it either terminates (i.e., the contact sticks) or continues in a new direction (i.e., the contact resumes sliding) whose existence and uniqueness, only assumed in the literature, are proven. Closed-form integration of the ODE becomes possible as soon as the sliding velocity turns zero or takes on an invariant direction. Assuming Stronge's energy-based restitution, a complete algorithm is described to combine fast numerical integration (NI) with a case-by-case closed-form analysis. A number of solved collision instances are presented. It remains open whether the modeled impact process will always terminate under Coulomb's friction and Stronge's (or Poisson's) restitution hypothesis.

scholarly journals Geometrical Design of a Rotor Blade for a Small Scale Horizontal Axis Wind Turbine

2019 ◽  
Vol 8 (3) ◽  
pp. 3390-3400
Keyword(s):  
Wind Turbine ◽  
Rotor Blade ◽  
Operating Conditions ◽  
Horizontal Axis ◽  
Small Scale ◽  
Horizontal Axis Wind Turbine ◽  
Geometrical Properties ◽  
Momentum Theory ◽  
Bem Theory ◽  
Blade Element

In the present study, Blade Element Momentum theory (BEMT) has been implemented to heuristically design a rotor blade for a 2kW Fixed Pitch Fixed Speed (FPFS) Small Scale Horizontal Axis Wind Turbine (SSHAWT). Critical geometrical properties viz. Sectional Chord ci and Twist distribution θTi for the idealized, optimized and linearized blades are analytically determined for various operating conditions. Results obtained from BEM theory demonstrate that the average sectional chord ci and twist distribution θTi of the idealized blade are 20.42% and 14.08% more in comparison with optimized blade. Additionally, the employment of linearization technique further reduced the sectional chord ci and twist distribution θTi of the idealized blade by 17.9% and 14% respectively, thus achieving a viable blade bounded by the limits of economic and manufacturing constraints. Finally, the study also reveals that the iteratively reducing blade geometry has an influential effect on the solidity of the blade that in turn affects the performance of the wind turbine.

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