1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

12.3 • VIBRATIONS IN MECHANICAL SYSTEMS 533

For damped mechanical systems driven by a periodic input F (t), the general
solution involves a transient part that vanishes as t -+ +oo, and a steady
state part that is periodic. We find the transient part of the solution u,. (t) by
solving the homogeneous differential equation

mu;: (t) + cUI. (t) + kU,.. (t) = o.

This homogeneous equation has the characteristic equation m).^2 +CA+ k = 0,
-c ± ./Cl -4mk
and its roots are ). =
2

m The coefficients m, c, and k are all

positive, and there are three cases to consider.

Case 1 If Cl - 4mk > 0, then the roots are real and distinct, and because the

inequality .,/ c^2 - 4mk < c holds, it follows that the roots ). 1 and ). 2 are negative

real numbers. Thus, for this case, we have

Case 2 If c^2 - 4mk = 0, then the roots are real and equal and ). 1 = A2 = A,

where A is a negative real number. Again, for this case we find that

Case 9 If c^2 - 4mk < 0, then the roots are complex and ). = -a± /3i, where a

and /3 are positive real numbers, and it follows that

In all three cases, the homogeneous solution u,. (t) decays to 0 as t-+ +oo.

We obtain the steady state solution Up (t) by representing Up (t) by its

Fourier series, substituting u; (t), u; (t), and Up (t) into t he nonhomogeneous

differential equation, and solving the resulting system for the Fourier coefficients

of Up (t). The general solution to Equation (12-18) then becomes

U (t) = U,. (t) +Up (t).

• EXAMPLE 12.4 Find the general solution to U" (t) + 2U' (t) + U (t) = F (t),
  where F (t) is given by the Fourier series F (t) = E

1

n=I (Zn - 1)^2 cos [(2n - 1) t].

Solution First, we solve Uf: (t) + ZU/. (t) +Un (t) = 0 for the transient solution.

The characteristic equation is X^2 + 2). + 1 = 0, which bas a double root).= - 1.

Hence