Generalized Force

The generalized force, Ei (t)—which appears on the right-hand side of the general­ized equations of motion—represents the effective loading associated with all forces and moments not accounted for in P, which includes any nonconservative forces and moments. These forces and moments are most commonly identified as exter­nally applied loads, which may or may not be a function of modal response. They also include any dissipative loads such as those from dampers. To determine the contribution of distributed loads, denoted by f (x, t), the virtual work is computed from Eq. (2.38), repeated here for convenience as

(3.91)

The term Sv(x, t) represents a variation of the displacement field, typically re­ferred to as the “virtual displacement,” which can be written in terms of the

Figure 3.6. Concentrated force acting on string

generalized coordinates and mode shapes as

TO

Sv(x, t) = ^2 Фі (x)S^i (t) (3.92)

i=1

where S^i (t) is an arbitrary increment in the ith generalized coordinate. Thus, the virtual work becomes

______ /. г TO

S W = f (x, t)Фі (хЩі (t)dx

0 i=1

TO /. г

= J2 S^i (t) f (x, t)Фі (x)dx

0

Identifying the generalized force as

Si (t) = f (x, t)Фі (x)dx (3.94)

0

we find that the virtual work reduces to

TO

SW =J2 Si (t) S^i (t) (3.95)

І =1

The loading f (x, t) in this development is a distributed load with units of force per unit length. If instead this loading is concentrated at one or more points—say, as Fc(t) with units of force acting at x = xc as shown in Fig. 3.6—then its functional representation must include the Dirac delta function, S(x – xc), which is similar to the impulse function in the time domain. In this case, the distributed load can be written as

f (x, t) = Fc(t)S(x – Xc)

Recall that the Dirac delta function can be thought of as the limiting case of a rectangular shape with area held constant and equal to unity as its width goes to zero (Fig. 3.7). Thus, it may be defined by its integral property; for example, for a < x0 < b

b

S(x – x0)dx = 1

a

b

f (x)S(x – x0)dx = f (x0)

Figure 3.7. Approaching the Dirac delta function

As a consequence, this integral expression for the generalized force can be applied to the concentrated load so that

Si (t) = Fc(t)8(x – Xc)Фі (x)dx

0

= Fc (t) 8(x – Xc)фі (x)dx

0

= Fc (t)фі (xc)