Generalized Forces


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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 qi has a corresponding generalized force Qi. 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, …, Qs 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 Mf and a frictional moment Mf are applied to the shaft, then neglecting the weight of the cable we obtain δA1 = (MtMfPr)δø, where r is the radius of the shaft. Consequently, for this system, the generalized force corresponding to the coordinate ø will be Q1 = MtMfPr.

The dimensions of a generalized force depend on the dimensions of the generalized coordinate. For example, if the qi are lengths, then the Qi will have dimensions of an ordinary force; if the qi are angles, then the Qi 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, Mt = Mf + Pr.

S. M. TARG

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