Constraint
A restriction on the natural degrees of freedom of a system. If n and m are the numbers of the natural and actual degrees of freedom, the difference n - m is the number of constraints. In principle n = 3N, where N is the number of particles, for example, atoms. In practice n is determined by the number of effectively rigid components.
A holonomic system is one in which the n original coordinates can be expressed in terms of m independent coordinates and possibly also the time. It is characterized by frictionless contacts and inextensible linkages. The new coordinates are called generalized coordinates. See Lagrange's equations
Nonholonomic systems cannot be reduced to independent coordinates because the constraints are not on the n coordinate values themselves but on their possible changes. For example, an ice skate may point in all directions but at each position it must point along its path. See Degree of freedom (mechanics)
| (programming, mathematics) | constraint - A Boolean relation, often an
equality or ineqality relation, between the values of one or
more mathematical variables (often two). E.g. x>3 is a
constraint on x. constraint satisfaction attempts to assign
values to variables so that all constraints are true. Usenet newsgroup: news:comp.constraints. FAQ. |
Constraint
(or reaction of a connection). For connections formed by bodies of any type, constraints are the forces of reaction of these bodies acting on points of the mechanical system. In contradistinction to active forces, constraints have values not known in advance. They depend not only on the type of connection but also on the active forces acting on the system; if the system is in motion, they depend additionally on how the system is moving. They are determined by solving the corresponding problems in mechanics. The directions of constraints or reactions in some cases are determined by the type of connection. Thus, if a point in the system is forced to remain always on a given smooth (frictionless) surface as a result of applied connections, then the reaction R is directed along the normal n
to this surface (Figure 1). Figure 2 shows a smooth cylindrical hinge (bearing), for which two components (Rx and Ry) of the reaction are unknown, and a smooth ball and socket joint for which all three components (Rx, Ry, and Rz) of the reaction are unknown. For a rough surface, the constraint has two components: a normal and a tangential component; the latter is called the frictional force.
In the general case, in solving problems in dynamics a restricted mechanical system is considered a free system if certain forces are applied to its points, such that the conditions imposed on the system by the connections are always satisfied when the system is in motion; these forces are called constraints.
S. M. TARG
Constraint
a restriction imposed on the position or motion of a mechanical system. Constraints are usually realized by bodies. Examples of constraints are a surface along which a body slides or rolls, a thread by which a weight is suspended, and joints connecting the links of mechanisms. If the positions of points of a mechanical system relative to a given reference system are determined by the points’ Cartesian coordinates xk, yk, zk (k = 1, 2,…,n, where n is the number of points in the system), then the restrictions imposed by the constraints can be expressed in the form of equalities or inequalities that give the relation between the time t, the coordinates xk, yk, zk, and the first derivatives of the coordinates with respect to time ẋk, ẏk, żk (that is, the velocities of the points of the system).
Constraints that impose restrictions only on the positions (coordinates) of the points of a system and are expressed by equations of the form
(1) f(…, xk, yk, zk,…,t) = 0
are called geometric constraints. If the constraints also impose restrictions on the velocities of the points of the system, they are called kinematic constraints, and their equations are of the form
(2) Φ(…, xk, yk, zk,…, ẋk, ẏk, żk,…,t) = 0
When equation (2) can be integrated with respect to time, the corresponding kinematic constraint is said to be integrable and is equivalent to a geometric constraint. Geometric and integrable kinematic constraints have the common name of holonomic constraints (seeHOLONOMIC SYSTEMS). Kinematic nonintegrable constraints are called nonholonomic (seeNONHOLONOMIC SYSTEMS).
Constraints that do not change with time are referred to as stationary constraints; their equations do not explicitly contain t. On the other hand, constraints that change with time are called moving constraints. Finally, two-way constraints are constraints such that to each virtual displacement of points of the system there corresponds a displacement in precisely the opposite direction. The equations of such constraints are of the form of equations (1) and (2). One-way constraints are constraints that do not satisfy the condition for two-way constraints. An example is a flexible thread, which permits displacement along the thread in only one direction. Such constraints are expressed by inequalities of the form f(…, xk, yk, zk,…)≥0.
The methods of solving problems in mechanics depend to a substantial degree on the nature of the constraints on the system. The effect of the action of constraints can be taken into account by introducing corresponding forces, called constraint forces. To determine these forces (or to eliminate the forces), constraint equations of the form (1) or (2) must be added to the equations of equilibrium or motion of the system. Ideal constraints are constraints for which the sum of the elementary works of all the forces in any virtual displacement of the system is equal to zero. Examples are a frictionless surface or flexible thread. For mechanical systems with ideal constraints, it is possible to obtain immediately equations of equilibrium or of motion that do not contain constraint forces by using the virtual work principle, the d’Alembert-Lagrange principle, or the Lagrange equations.
S. M. TARG
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