Generalized Forces

quantities that play the role of ordinary forces when the configuration of a mechanical system is defined by generalized coordinates in an investigation of the equilibrium or motion of the system. The number of generalized forces is equal to the number s of degrees of freedom of the system; here, each generalized coordinate q_i has a corresponding generalized force Q_i. The value of Q₁ corresponding to the coordinate q₁ can be found by computing the virtual work δA₁ performed by all the forces for a virtual displacement of the system in which only the coordinate q₁ is changed, receiving an increment δq₁. Then δA₁ = Q₁ δ q₁ that is, the coefficient for δq₁ in the expression for δA₁ will be the generalized force Q₁. The generalized forces Q₂, Q₃, …, Q_s are calculated in a similar manner.

For example, suppose we have a winch (see Figure 1) that is lifting a load P on a cable, which is a system having one degree of freedom. If the generalized coordinate q₁ is taken to be the angle of rotation ø of the shaft of the winch and if a torque M_f and a frictional moment M_f are applied to the shaft, then neglecting the weight of the cable we obtain δA₁ = (M_t – M_f – Pr)δø, where r is the radius of the shaft. Consequently, for this system, the generalized force corresponding to the coordinate ø will be Q₁ = M_t – M_f – Pr.

The dimensions of a generalized force depend on the dimensions of the generalized coordinate. For example, if the q_i are lengths, then the Q_i will have dimensions of an ordinary force; if the q_i are angles, then the Q_i will have dimensions of a moment of a force. In an investigation of the motion of a mechanical system, generalized forces appear instead of ordinary forces in the Lagrange equations of mechanics; when the system is in equilibrium, all the generalized forces are equal to zero. As an example, for the winch discussed above, when the load is being lifted uniformly, we must have Q₁ = O; that is, M_t = M_f + Pr.