Response to an External Force

Much of the analysis of helicopter flying qualities is involved with determining the response to a gust or to a control input. To illustrate the basic technique used for this type of analysis, let us suddenly impose a constant force on the system of Figure 9.3. Now the equation of equilibrium is:

kx + cx + mx = F

After using the substitution for x, this equation becomes

k + cs + ms2 = F(s)

The solution to this equation contains the roots of the homogeneous equation—that is, the equation with F(s) = 0, plus two new terms involving constants of integration:

x = x^jje’1′ + х2(г)е’2′ + Cx + C2t

where Jj and s2 are the same roots obtained earlier. The constants x^j), x2(j), C„ and C2 can be evaluated from conditions when time is equal to zero and when time is equal to infinity. At / = 0,

x = 0

x = 0

for a system with positive damping. At / = 00,

x = F/k

Using the last condition first, it is obvious that C2 must be zero and that since sx and s2 have negative real parts for a system with positive damping: at zero time:

Note that these equations are identical to those obtained for the free system except that xj2 has been replaced by —F/2k. The final solution is:

A typical time history based on this equation for a system with a damped oscillation is as follows:

It is often desirable to describe the response in terms of the time required to achieve 63% of its final value. For this purpose, it is only necessary to work with real portions of the solution:

for