Free and constrained bodies. A body treats as free if its displacements are not restricted by any other bodies, in opposite case a body is constrained. Examples of free body: 1. The airplane in the sky is free body.
2. The helicopter is free.
3. Bird treats as free body.
Examples of constrained body. 1. The motion of the balloon is restricted by four cords. Cord 3 Cord 4 Cord 1 Cord 2
2. The motion of the yellow rectangular plate ABCD is restricted by hinges at A and B and is supported by pined joint link DE.
• The bodies that restrict the motion of some body are called constraints imposed upon the body. • Mechanical effect of a constraint action is the same as action of a force. Therefore the constraints can be replaced by forces that are called reactions. • Axiom of constraint eliminating: any constrained body can be considered free if constraints applied to the body are mentally eliminated and replaced with the corresponding reaction forces.
Forces classification: applied forces and reactions In mechanics all forces acting on the body or mechanical system are subdivided into two classes in two ways. In the first way forces are subdivided into applied (given) and reactive forces. Active (applied) forces are independent from constraints and usually it is supposed that active forces are given in statement of a problem.
Some reactions at supports and connections for two-dimensional structure Smooth surfaces
Weightless flexible, nonstretchable cord or weightless rigid link Sliding joint or roller support
Pinned joint
Collar or sliding support Double stage sliding support Fixed support, сlamped joint
Some reactions at supports and connections for three-dimensional structure Ball-and-socket joint and footstep bearing
Hinge Bearing supports
Fixed support
• In the second way forces are subdivided into external and internal. • Forces are called external with respect to a given object (body or mechanical system) if they are produced by particles and other bodies not included into a given object (by external objects). Forces of interaction between particles or parts of the given object are called internal.
Concurrent force system If lines of action of all forces of a system intersect at the same point the force system is called concurrent. Concurrent force system resultant Concurrent force system is equivalent to single force (resultant). Resultant is equal to the vector sum of the forces of system. Resultant is applied at the point of intersection all forces lines of action.
The composition of forces can be accomplished: • geometrically, constructing a vector force polygon • analytically, summing projections of all forces on selected coordinate axes:
Equilibrium conditions for the concurrent force system To be balanced any concurrent force system must have zero resultant. Analytical conditions of concurrent force system equilibrium In geometrical form it can be formulated as follows: closure of the polygon of forces is a necessary and sufficient condition for the concurrent force system to be in balance.
Problems of statics There are two general problems of rigid body statics: 1. The composition of forces and the reduction of the force system acting on rigid bodies to the simplest possible form. 2. The determination of conditions for the equilibrium of the force system acting on rigid bodies.
Moment of a force about a point
Moment of a force about an axis • The projection of the moment about the point onto an axis that passes through the point is called the moment of force about the axis
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