# Equilibrium of a Mechanical System

## Equilibrium of a Mechanical System

If a mechanical system is acted on by forces, it is said to be in a state of equilibrium when all its points are at rest with respect to the frame of reference under consideration. If the frame of reference is inertial, we speak of absolute equilibrium; otherwise, we speak of relative equilibrium.

The study of the conditions for the equilibrium of a mechanical system is one of the basic tasks of statics. These conditions have the form of equations giving the relation between the forces acting on the system and the parameters determining the position of the system. The number of conditions is equal to the number of degrees of freedom of the system. The conditions for the relative equilibrium of a mechanical system are formed in the same way as the conditions for absolute equilibrium if the corresponding vehicle forces of intertia are added to the forces acting on the points.

Suppose a free rigid body in an *Oxyz* reference frame is acted on by external forces. The conditions of equilibrium of the body are that the sums of the projections of the forces on the three coordinate axes must equal zero and that the sums of the moments of the forces about each of the three axes must also equal zero:

(1) Σ*F _{kx}* = 0, Σ

*F*= 0, Σ

_{ky}*F*= 0

_{kz}Σ*m _{x}* (

*F*) = 0, Σ

_{k}*m*(

_{y}*F*) = 0, Σ

_{k}*m*(

_{z}*F*) = 0

_{k}When conditions (1) are satisfied, the body is at rest with respect to the given frame of reference if the velocities of all its points with respect to this frame at the moment the forces began to act were zero. Otherwise, when conditions (1) are satisfied, the body undergoes inertial motion—for example, it may move translationally, uniformly, and rectilinearly.

If the body is not free, its equilibrium conditions are given by those equations in (1) (or the consequences thereof) that do not contain reactions to imposed constraints. The remaining equations can be used to determine the unknown reactions. Suppose, for example, a body has a fixed axis of rotation *Oz.* The equilibrium condition is then Σ*m _{z}* (

*F*) = 0. The remaining equations from (1) are used to determine the reactions to the bearings securing the axle. If the body is rigidly fixed by the imposed constraints, all the equations (1) are used to determine the reactions to the imposed constraints. Problems of this type are frequently encountered in engineering.

_{k}It follows from the solidification principle that the equations in (1) that do not contain reactions to external constraints simultaneously give necessary, but not sufficient, conditions of equilibrium for any mechanical system and, in particular, a deformable body. The necessary and sufficient conditions of equilibrium for any mechanical system can be found by using the virtual work principle. For a system having *s* degrees of freedom, these conditions are that the corresponding generalized forces must be equal to zero:

(2) *Q*_{1} = 0, *Q*_{2} = 0,... *Q*_{s} = 0

Of the states of equilibrium defined by conditions (1) and (2), in practice only those that are stable are realized. The equilibrium of liquids and gases is considered in hydrostatics and aerostatics.

S. M. TARG