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Constraint (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 constraints * Scleronomic constraints (not depending on time) and rheonomic constraints (depending on time) *Ideal constraints: those for which the work done by the constraint forces under a virtual displacement In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) \delta \gamma shows how the mechanical system's trajectory can ''hypothetically'' (hence the ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friction Angle
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of the processes involved is called tribology, and has a history of more than 2000 years. Friction can have dramatic consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire. Another important consequence of many types of friction can be wear, which may lead to performance degradation or damage to components. It is known that frictional energy losses account for about 20% of the total energy expenditure of the world. As briefly discussed later, there are many different contributors to the retarding force in friction, ranging from asperity deformation to the generation of charges and changes in local structure. When two bodies in contact move relative to each other, due to these various ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tertiary Constraint
In Hamiltonian mechanics, a primary constraint is a relation between the coordinates and momenta that holds without using the equations of motion. A secondary constraint is one that is not primary—in other words it holds when the equations of motion are satisfied, but need not hold if they are not satisfied The secondary constraints arise from the condition that the primary constraints should be preserved in time. A few authors use more refined terminology, where the non-primary constraints are divided into secondary, tertiary, quaternary, etc. constraints. The secondary constraints arise directly from the condition that the primary constraints are preserved by time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ..., the tertiary constraints arise from the condition that the sec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virtual Displacement
In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) \delta \gamma shows how the mechanical system's trajectory can ''hypothetically'' (hence the term ''virtual'') deviate very slightly from the actual trajectory \gamma of the system without violating the system's constraints. For every time instant t, \delta \gamma(t) is a vector tangential to the configuration space at the point \gamma(t). The vectors \delta \gamma(t) show the directions in which \gamma(t) can "go" without breaking the constraints. For example, the virtual displacements of the system consisting of a single particle on a two-dimensional surface fill up the entire tangent plane, assuming there are no additional constraints. If, however, the constraints require that all the trajectories \gamma pass through the given point \mathbf at the given time \tau, i.e. \gamma(\tau) = \mathbf, then \delta\gamma (\tau) = 0. Notations Let M be the configu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rheonomic Constraint
A mechanical system is rheonomous if its equations of constraints contain the time as an explicit variable. Such constraints are called rheonomic constraints. The opposite of rheonomous is scleronomous. Example: simple 2D pendulum As shown at right, a simple pendulum is a system composed of a weight and a string. The string is attached at the top end to a pivot and at the bottom end to a weight. Being inextensible, the string has a constant length. Therefore, this system is scleronomous; it obeys the scleronomic constraint : \sqrt - L=0\,\!, where (x,\ y)\,\! is the position of the weight and L\,\! the length of the string. The situation changes if the pivot point is moving, e.g. undergoing a simple harmonic motion :x_t=x_0\cos\omega t\,\!, where x_0\,\! is the amplitude, \omega\,\! the angular frequency, and t\,\! time. Although the top end of the string is not fixed, the length of this inextensible string is still a constant. The distance between the top end and the weig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scleronomic Constraint
A mechanical system is scleronomous if the equations of constraints do not contain the time as an explicit variable and the equation of constraints can be described by generalized coordinates. Such constraints are called scleronomic constraints. The opposite of scleronomous is rheonomous. Application In 3-D space, a particle with mass m\,\!, velocity \mathbf has kinetic energy T T =\fracm v^2 . Velocity is the derivative of position r with respect to time t\,\!. Use chain rule for several variables: \mathbf = \frac = \sum_i\ \frac \dot_i + \frac . where q_i are generalized coordinates. Therefore, T = \frac m \left(\sum_i\ \frac\dot_i+\frac\right)^2 . Rearranging the terms carefully, \begin T &= T_0 + T_1 + T_2 : \\ exT_0 &= \frac m \left(\frac\right)^2 , \\ T_1 &= \sum_i\ m\frac\cdot \frac\dot_i\,\!, \\ T_2 &= \sum_\ \fracm\frac\cdot \frac\dot_i\dot_j, \end where T_0\,\!, T_1\,\!, T_2 are respectively homogeneous functions of degree 0, 1, and 2 in generalized velociti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Pfaffian constraint form. Derivation Given a holonomic system described by a set of holonomic constraint equations :f_r(u_1, u_2, u_3,\ldots, u_n, t) = 0;\; r = 1,\ldots, L where \ are the ''n'' generalized coordinates that describe the system, and where L is the number of equations in a system of constraints, we can differentiate by the chain rule for each equation: : \sum_^n\fracdu_s + \fracdt = 0;\; r = 1,\ldots, L By a simple substitution of nomenclature we arrive at: : \sum_^nA_du_s + A_rdt = 0;\; r = 1,\ldots, L Examples Pendulum Consider a pendulum. Because of how the motion of the weight is constrained by the arm, the velocity vector \overrightarrow of the weight must be perpendicular at all times to the position vector \overr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonholonomic System
A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints and non-linear constraints, such that when the system evolves along a path in its parameter space (the parameters varying continuously in values) but finally returns to the original set of parameter values at the start of the path, the system itself may not have returned to its original state. Nonholonomic mechanics is an autonomous division of Newtonian mechanics. Details More precisely, a nonholonomic system, also called an ''anholonomic'' system, is one in which there is a continuous closed circuit of the governing parameters, by which the system may be transformed from any given state to any other state. Because the final state of the system depends on the intermediate values of its trajectory through parameter space, the system cannot be represented by a conserv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternary Constraint
The Quaternary ( ) is the current and most recent of the three periods of the Cenozoic Era in the geologic time scale of the International Commission on Stratigraphy (ICS), as well as the current and most recent of the twelve periods of the Phanerozoic eon. It follows the Neogene Period and spans from 2.58 million years ago to the present. The Quaternary Period is divided into two epochs: the Pleistocene (2.58 million years ago to 11.7 thousand years ago) and the Holocene (11.7 thousand years ago to today); a proposed third epoch, the Anthropocene, was rejected in 2024 by IUGS, the governing body of the ICS. The Quaternary is typically defined by the Quaternary glaciation, the cyclic growth and decay of continental ice sheets related to the Milankovitch cycles and the associated climate and environmental changes that they caused. Research history In 1759 Giovanni Arduino proposed that the geological strata of northern Italy (geographical region), Italy could be di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
