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Unilateral Contact
In contact mechanics, the term unilateral contact, also called unilateral constraint, denotes a mechanical constraint which prevents penetration between two rigid/flexible bodies. Constraints of this kind are omnipresent in non-smooth multibody dynamics applications, such as granular flows, legged robot, vehicle dynamics, particle damping, imperfect joints, or rocket landings. In these applications, the unilateral constraints result in impacts happening, therefore requiring suitable methods to deal with such constraints. Modelling of the unilateral constraints There are mainly two kinds of methods to model the unilateral constraints. The first kind is based on smooth contact dynamics, including methods using Hertz's models, penalty methods, and some regularization force models, while the second kind is based on the non-smooth contact dynamics, which models the system with unilateral contacts as variational inequalities. Smooth contact dynamics In this method, normal force ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contact Mechanics
Contact mechanics is the study of the deformation of solids that touch each other at one or more points.Johnson, K. L, 1985, Contact mechanics, Cambridge University Press.Popov, Valentin L., 2010, ''Contact Mechanics and Friction. Physical Principles and Applications'', Springer-Verlag, 362 p., . A central distinction in contact mechanics is between stresses acting perpendicular to the contacting bodies' surfaces (known as normal stress) and frictional stresses acting tangentially between the surfaces (shear stress). Normal contact mechanics or frictionless contact mechanics focuses on normal stresses caused by applied normal forces and by the adhesion present on surfaces in close contact, even if they are clean and dry. ''Frictional contact mechanics'' emphasizes the effect of friction forces. Contact mechanics is part of mechanical engineering. The physical and mathematical formulation of the subject is built upon the mechanics of materials and continuum mechanics and fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contact Dynamics Unilateral
Contact may refer to: Interaction Physical interaction * Contact (geology), a common geological feature * Contact lens or contact, a lens placed on the eye * Contact sport, a sport in which players make contact with other players or objects * Contact juggling * Contact mechanics, the study of solid objects that deform when touching each other * Contact process (mathematics), a model of an interacting particle system * Electrical contacts * ''Sparśa'', a concept in Buddhism that in Sanskrit/Indian language is translated as "contact", "touching", "sensation", "sense impression", etc. Social interaction * Contact (amateur radio) * Contact (law), a concept related to visitation rights * Contact (social), a person who can offer help in achieving goals * Contact Conference, an annual scientific conference * Extraterrestrial contact, see Search for extraterrestrial intelligence * First contact (anthropology), an initial meeting of two cultures * Language contact, the interaction of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Collision Response
In the context of classical mechanics simulations and physics engines employed within video games, collision response deals with models and algorithms for simulating the changes in the motion of two solid bodies following collision and other forms of contact. Rigid body contact Two rigid bodies in unconstrained motion, potentially under the action of forces, may be modelled by solving their equations of motion using numerical integration techniques. On collision, the kinetic properties of two such bodies seem to undergo an instantaneous change, typically resulting in the bodies rebounding away from each other, sliding, or settling into relative static contact, depending on the elasticity of the materials and the configuration of the collision. Contact forces The origin of the rebound phenomenon, or ''reaction'', may be traced to the behaviour of real bodies that, unlike their perfectly rigid idealised counterparts, do undergo minor compression on collision, followed by expa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root-finding Algorithms
In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number such that . As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form, root-finding algorithms provide approximations to zeros, expressed either as floating-point numbers or as small isolating intervals, or disks for complex roots (an interval or disk output being equivalent to an approximate output together with an error bound). Solving an equation is the same as finding the roots of the function . Thus root-finding algorithms allow solving any equation defined by continuous functions. However, most root-finding algorithms do not guarantee that they will find all the roots; in particular, if such an algorithm does not find any root, that does not mean that no root exists. Most num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex Algorithm
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are not actually used in the method, but one interpretation of it is that it operates on simplicial ''cones'', and these become proper simplices with an additional constraint. The simplicial cones in question are the corners (i.e., the neighborhoods of the vertices) of a geometric object called a polytope. The shape of this polytope is defined by the constraints applied to the objective function. History George Dantzig worked on planning methods for the US Army Air Force during World War II using a desk calculator. During 1946 his colleague challenged him to mechanize the planning process to distract him from taking another job. Dantzig formulated the problem as linear inequalities inspired by the work of Wassily Leontief, however, at that t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Complementarity Problem
In mathematical optimization theory, the linear complementarity problem (LCP) arises frequently in computational mechanics and encompasses the well-known quadratic programming as a special case. It was proposed by Cottle and Dantzig in 1968. Formulation Given a real matrix ''M'' and vector ''q'', the linear complementarity problem LCP(''q'', ''M'') seeks vectors ''z'' and ''w'' which satisfy the following constraints: * w, z \geqslant 0, (that is, each component of these two vectors is non-negative) * z^Tw = 0 or equivalently \sum\nolimits_i w_i z_i = 0. This is the complementarity condition, since it implies that, for all i, at most one of w_i and z_i can be positive. * w = Mz + q A sufficient condition for existence and uniqueness of a solution to this problem is that ''M'' be symmetric positive-definite. If ''M'' is such that has a solution for every ''q'', then ''M'' is a Q-matrix. If ''M'' is such that have a unique solution for every ''q'', then ''M'' is a P-matrix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Complementarity Problem
In applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ..., a nonlinear complementarity problem (NCP) with respect to a mapping ''ƒ'' : R''n'' → R''n'', denoted by NCP''ƒ'', is to find a vector ''x'' ∈ R''n'' such that : x \geq 0,\ f(x) \geq 0 \text x^f(x)=0 where ''ƒ''(''x'') is a smooth mapping. The case of a discontinuous mapping was discussed by Habetler and Kostreva (1978). References * * Applied mathematics {{applied-math-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of two solid surfaces in contact. Dry friction is subdivided into ''static friction'' ("stiction") between non-moving surfaces, and ''kinetic friction'' between moving surfaces. With the exception of atomic or molecular friction, dry friction generally arises from the interaction of surface features, known as asperities (see Figure 1). *Fluid friction describes the friction between layers of a viscous fluid that are moving relative to each other. *Lubricated friction is a case of fluid friction where a lubricant fluid separates two solid surfaces. *Skin friction is a component of drag, the force resisting the motion of a fluid across the surface of a body. *Internal friction is the force resisting motion between the elements making up a so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coefficient Of Restitution
The coefficient of restitution (COR, also denoted by ''e''), is the ratio of the final to initial relative speed between two objects after they collide. It normally ranges from 0 to 1 where 1 would be a perfectly elastic collision. A perfectly inelastic collision has a coefficient of 0, but a 0 value does not have to be perfectly inelastic. It is measured in the Leeb rebound hardness test, expressed as 1000 times the COR, but it is only a valid COR for the test, not as a universal COR for the material being tested. The value is almost always less than 1 due to initial translational kinetic energy being lost to rotational kinetic energy, plastic deformation, and heat. It can be more than 1 if there is an energy gain during the collision from a chemical reaction, a reduction in rotational energy, or another internal energy decrease that contributes to the post-collision velocity. \text (e) = \frac The mathematics were developed by Sir Isaac Newton in 1687. It is also known as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signorini Problem
The Signorini problem is an elastostatics problem in linear elasticity: it consists in finding the elastic equilibrium configuration of an anisotropic non-homogeneous elastic body, resting on a rigid frictionless surface and subject only to its mass forces. The name was coined by Gaetano Fichera to honour his teacher, Antonio Signorini: the original name coined by him is problem with ambiguous boundary conditions. History The problem was posed by Antonio Signorini during a course taught at the '' Istituto Nazionale di Alta Matematica'' in 1959, later published as the article , expanding a previous short exposition he gave in a note published in 1933. himself called it ''problem with ambiguous boundary conditions'', since there are two alternative sets of boundary conditions the solution ''must satisfy'' on any given contact point. The statement of the problem involves not only equalities ''but also inequalities'', and ''it is not a priori known what of the two se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constraint (classical Mechanics)
In classical mechanics, a constraint on a system is a parameter that the system must obey. For example, a box sliding down a slope must remain on the slope. There are two different types of constraints: holonomic and non-holonomic. Types of constraint *First class constraints and second class constraints *Primary constraints, secondary constraints, tertiary constraints, quaternary constraints. *Holonomic constraints, also called integrable constraints, (depending on time and the coordinates but not on the momenta) and Nonholonomic system *Pfaffian constraint In dynamics, a Pfaffian constraint is a way to describe a dynamical system in the form: : \sum_^nA_du_s + A_rdt = 0;\; r = 1,\ldots, L where L is the number of equations in a system of constraints. Holonomic systems can always be written in Pfa ...s * Scleronomic constraints (not depending on time) and rheonomic constraints (depending on time). *Ideal constraints: those for which the work done by the constraint forces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hertz Contact Animated
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that one hertz is the reciprocal of one second. It is named after Heinrich Rudolf Hertz (1857–1894), the first person to provide conclusive proof of the existence of electromagnetic waves. Hertz are commonly expressed in multiples: kilohertz (kHz), megahertz (MHz), gigahertz (GHz), terahertz (THz). Some of the unit's most common uses are in the description of periodic waveforms and musical tones, particularly those used in radio- and audio-related applications. It is also used to describe the clock speeds at which computers and other electronics are driven. The units are sometimes also used as a representation of the energy of a photon, via the Planck relation ''E'' = ''hν'', where ''E'' is the photon's energy, ''ν'' is its frequency ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
