Ci r

Thus, the string displacement can be written as a sum of the contributions from the odd-indexed modes. Recall that neither the excitation loading nor the initial conditions excite the even-indexed modes. Thus, v(x, t) — J2 HAt)<Pi{x) i = l,3,...

When the forcing frequency coincides with one of the natural frequencies, an interesting situation results. Take a typical term in the series solution ofEq. (2.129), and consider only its time-dependent part, for example, sin(ft)r) — sin(&), r)

When co —»• &), , the term becomes indeterminate. To see what its value in the limit is, we let a>i — co + e,-, which gives sin(&)f )

i[(co + eiT - co2] Invoking l'Hopital's rule to take the limit as e, sin(ft>r) — cot cos {cot)

0, one obtains

The second term tends to infinity as time increases, with a linearly increasing amplitude. This phenomenon is called resonance and, because of its destructive nature, should be avoided. That is, when one excites a structure using harmonic excitation, the forcing frequency must not be too near any of the structure's natural frequencies.

Example 4: Calculation of Forced Response with Nonzero Initial Conditions

A second example will be considered to illustrate the treatment of a concentrated force and finite initial conditions. In this case a concentrated step-function force of magnitude F0 will be applied to the center of the string as illustrated in Fig. 2.10. Recall that the unit step function, l(r), is defined by

0 0