Equilibrium

The field of statics considers objects either at rest or moving with a constant velocity. These objects are said to be in equilibrium, meaning that the sum of the external forces and the sum of moments equal zero. In a three-dimensional problem the above equations both result in three scalar equations in the x, y, z directions.

Degrees of freedom
The number of independent variables required to specify the configuration of a mechanical system is referred to as the number of degrees of freedom of the system.
A three-dimensional rigid body has six degrees of freedom: the translations along the three axis x, y, z and the rotations about the three axis x, y, z. Each of the six scalar conditions of equilibrium inhibits one degree of freedom. In a two-dimensional problem the body has two translational degrees of freedom and one rotational degree of freedom only.

Equilibrium of multiple systems

Multiple systems are systems that consist of several rigid bodies that are mutually connected. The system as a whole must be in equilibrium under the effect of the external forces and of the reaction forces at the connections between the system and the environment. Each component must be in equilibrium under the effect of the external forces acting directly on the component and of the reaction forces at the connections between the component and the environment and between the component and the rest of the system.

Types of connections
In a multiple system of n objects the total number of degrees of freedom is given by the sum of the degrees of freedom of the n components minus the number of degrees of freedom inhibited by the connections present between the objects and between the object and the environment. If a connection prevents the translation of the body in a given direction a reaction force is developed on the body in order to establish force equilibrium in that direction. If rotation about an axis is prevented a moment is exerted in order to reach moment equilibrium. A few types of connections are shown below.

In an ideal free contact between two objects, or between an object and the environment, relative displacement within the plane tangential to both objects is allowed. Relative displacement normal to the common tangent plane with the two objects moving towards each other is forbidden. No resistance is made to the objects moving away from each other. The two objects exchange a force normal to the tangent plane as shown in Figure 1. If the free contact is not ideal when a tangential relative displacement takes place a friction force W is exchanged between the two objects opposing the relative motion as shown in Figure 2.

Figure 1: Ideal free contact

Figure 2: Free contact

An ideal hinge between an object and the environment allows rotation about the axis perpendicular to the plane of motion. Translation of the object is forbidden. Thus the reaction force has two components Fx and Fy (Figure 3).

Figure 3: Ideal hinge

A fixed support between an object and the environment is a connection that inhibits all degrees of freedom. The object experiences two components Fx and Fy of the reaction force F and a reaction moment M (Figure 4). The former inhibits the translation of the object and the latter the rotation of the object.

Figure 4: Fixed support

Single and multiple connections
In a single connection between two objects, object 1 and object 2, force F1 acting on object 1 and F2 acting on object 2 are equal in magnitude and opposite in direction (Figure 5).

Figure 5: Single connection

Multiple connections connect more than two objects. Considering three objects connected with a hinge (Figure 6) each object i experiences a reaction force Fi made of two components Fix and Fiy in the x and y directions. In order to evaluate the reaction force for each object the equilibrium at the joint must be imposed, considering that the forces on the joint are opposite to the forces on the objects. This results in the force equilibrium equations in the x and y directions -F1x+F2x-F3x=0 and -F1y+F2y-F3y=0.

Figure 6: Multiple connection

When external forces act on the joint these must be taken into account in the equilibrium of the joint (Figure 7). In this case the equilibrium equations for the joint are -F1x+F2x-F3x=0 and -F1y+F2y-F3y-F=0.

Figure 7: Multiple connection with external forces