N

Figure 2.6 Example initial shape of wave.

We have used Eq. (2.75) to reduce the range ofmotion, which was initially —oo < x < +oo, down to the range 0 < x < I, our physical space (i.e., where the string is actually mounted).

Example 2: Traveling Wave

The initial string shape is given in Fig. 2.6. At subsequent times the string shape will appear as shown in Fig. 2.7. The absolute distance each of the half shapes has traveled at time t is denoted by x. Note that the faint lines are the displacements associated with the two constituent waves after transformation to bring them into the range 0 < x < t, while the bold line is the sum of these two displacements. The displacement during the time I'.-Jm/T < t < 21-Jm/T is a mirror image of the progression revealed in Fig. 2.7 with a return to the original shape at t — 2l*Jm/T. The motion is periodic thereafter with period 2(,^/m/T.

2.1.6 Generalized Equations of Motion

Once the free-vibration modes have been determined for a linear, conservative system it is a straightforward procedure to determine the system's response to any external loading. This is accomplished by treating each mode of vibration as a dimensional degree of freedom whose scalar coordinate is the mode's generalized coordinate. For each of these modal degrees of freedom a "generalized equation of motion" can be formulated from "Lagrange's Equations" (see Appendix). Lagrange's equations can be written as d ( dL\ dL

where L — K — P is called the "Lagrangean," the difference between the total kinetic energy, K, and the total potential energy, P, of the system. The generalized coordinates are ; and the term on the right-hand side, E, , is called the "generalized force." The latter represents the effects of all nonconservative forces, as well as any conservative forces that are not treated in the total potential energy.

Under many circumstances, the kinetic energy can be represented as a function of only the coordinate rates so that

Figure 2.7 Shape of traveling wave at various times.

Figure 2.7 Shape of traveling wave at various times.

The potential energy is a function of only the coordinates themselves; that is,

P = P(SuS 2,(2.81) Thus, Lagrange's equations can be written as d /dK \ dP _ _

0 0

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