9.1 Fundamental Conceptions in Mechanics
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An ordered set of all particles of a mechanical system in the same instant in space
determines the location and shape of the system, which is called the configuration
of the system.
9.1.2 Constraints and Their Classification
If the configuration or velocity to the various particles of an unfree system applies
a certain restrictive condition of geometry or kinematics, then the restrictive condition is called a constraint. The acting force of the constraint on the constrained
unfree system is called the reactive force of constraint or constraint reaction
force, it is called the constraint force, force of constraint or constraining force
for short. The force except the reactive force of constraint is call the active force
or applied force. The restrictive condition of various particles in an unfree system
expressed by the mathematical equation is called the constraint equation. When the
constraint equation is expressed in the equality, the constraint is called the bilateral
constraint, it is also called the fixed constraint or irremoval constraint. When the
constraint equation is expressed in the inequality, the constraint is called the unilateral constraint, it is also called the unfixed constraint or removal constraint. When
the constraint equation does not explicitly contain time variable t, the constraint is
called the steady constraint, scleronomic constraint or scleronomous constraint,
otherwise it is called the unsteady constraint. When the constraint equation only
contains time variable t and radius vector r i , the constraint is called the geometrical constraint or finite constraint. The geometric constraint imposes a certain
restrictions on the position that the particle system may occupy, it is also called
the position constraint. When the constraint equation contains the velocity ˙
r i , the
constraint is called the differential constraint or kinematic constraint. The differential constraint not only restricts the location of the particle system, but also restricts
the velocity of the particle system. If the constraint equation contains only the term
of the first degree of the velocity, then it is called the linear kinematic constraint.
If the constraint equation can be integrated, then the constraint is called the integrable differential constraint. In fact, the integrable differential constraint is the
geometrical constraint.
The geometrical constraint and integrable differential constraint are collectively
called the holonomic constraint, sometimes they are called the semi-holonomic
constraint. Note that here defined holonomic constraint differs from defined holonomic constraint in Chap. 5 to some extent. The non-integrable differential constraint
is called the nonholonomic constraint. The particle system with the holonomic
constraint is called the holonomic system, otherwise it is called the nonholonomic
system.
513
An ordered set of all particles of a mechanical system in the same instant in space
determines the location and shape of the system, which is called the configuration
of the system.
9.1.2 Constraints and Their Classification
If the configuration or velocity to the various particles of an unfree system applies
a certain restrictive condition of geometry or kinematics, then the restrictive condition is called a constraint. The acting force of the constraint on the constrained
unfree system is called the reactive force of constraint or constraint reaction
force, it is called the constraint force, force of constraint or constraining force
for short. The force except the reactive force of constraint is call the active force
or applied force. The restrictive condition of various particles in an unfree system
expressed by the mathematical equation is called the constraint equation. When the
constraint equation is expressed in the equality, the constraint is called the bilateral
constraint, it is also called the fixed constraint or irremoval constraint. When the
constraint equation is expressed in the inequality, the constraint is called the unilateral constraint, it is also called the unfixed constraint or removal constraint. When
the constraint equation does not explicitly contain time variable t, the constraint is
called the steady constraint, scleronomic constraint or scleronomous constraint,
otherwise it is called the unsteady constraint. When the constraint equation only
contains time variable t and radius vector r i , the constraint is called the geometrical constraint or finite constraint. The geometric constraint imposes a certain
restrictions on the position that the particle system may occupy, it is also called
the position constraint. When the constraint equation contains the velocity ˙
r i , the
constraint is called the differential constraint or kinematic constraint. The differential constraint not only restricts the location of the particle system, but also restricts
the velocity of the particle system. If the constraint equation contains only the term
of the first degree of the velocity, then it is called the linear kinematic constraint.
If the constraint equation can be integrated, then the constraint is called the integrable differential constraint. In fact, the integrable differential constraint is the
geometrical constraint.
The geometrical constraint and integrable differential constraint are collectively
called the holonomic constraint, sometimes they are called the semi-holonomic
constraint. Note that here defined holonomic constraint differs from defined holonomic constraint in Chap. 5 to some extent. The non-integrable differential constraint
is called the nonholonomic constraint. The particle system with the holonomic
constraint is called the holonomic system, otherwise it is called the nonholonomic
system.