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e(x, t)

- e(x ) +

e(x, r),

T( x, t)

- T(x) +

Substituting the perturbation expressions of Eqs. (2.7) and (2.8) into Eqs. (2.1) while ignoring all squares and products of the ( (-quantities, one finds that the equations of motion can be reduced to two linear, partial differential equations:

dx2 dt2

To d2i>

l+eodx2 dt2

Thus, the two nonlinear equations of motion, Eqs. (2.1), for the free vibration of a string have been reduced to two uncoupled linear equations, one for longitudinal vibration and the other for transverse vibration. Since it is typically true that EA^> T0, longitudinal motions have much smaller amplitudes and much higher natural frequencies; thus, they are not usually of interest. Moreover, the fact that EA^>Tq also leads to the observations that <?o 1 and 8 « f'o [see Eqs. (2.6)]. Thus, the transverse motion is governed by

32v d2v

For convenience, we will drop the ( )s and the subscript, thus yielding the usual equation for string vibration found in texts on vibration:

d2V d2V

dx dt2

This is called the one-dimensional "wave equation" and governs the structural dynamic behavior of the string in conjunction with the boundary conditions. These conditions at the ends of the string correspond to zero displacement as described by u(0, /) = t) = 0, (2.12)

where it is noted that the distinction between i0 and i is no longer relevant. The general solution to the above wave equation with these homogeneous boundary conditions comprises a simple eigenvalue problem. The fact that the equation is of second order both temporally and spatially indicates that two boundary conditions and two initial conditions need to be specified.

2.1.2 Standing Wave (Modal) Solution

The preceding wave equation, which governs the dynamic behavior of the string, can be reduced from a partial differential equation with two independent variables to two ordinary differential equations by making a "separation of variables." The dependent variable of transverse displacement will be represented by v(x,t) = X(x)Y(t). (2.13)

This product form will now be substituted into the wave equation, Eq. (2.11). To simplify the notation let ( )' and ( ) denote ordinary derivatives with respect to x and t. Thus, the wave equation becomes

Rearranging terms as

one observes that the left-hand side of this equation is only a function of the single independent variable x and the right-hand side is a function of only t. This, of course, presumes that both m and T are constants. Constant m implies that the string is uniform, and constant T is consistent with the prior approximations used in deriving Eq. (2.11). Since each side of the equation is a function of different independent variables, then the only way the equality can be valid is for each side to be equal to a common constant. Let this constant be —a2, so that

This yields two ordinary differential equations, given by

Because the general solutions to these linear, second-order, differential equations are well known they will be written without any further justification as

These solutions are only valid when a ^ 0. The much simpler solutions for the special case of ci? = 0 will be discussed later. The boundary condition on the left end of the string, where a' = 0, can be written as u((U) = X(W(t) = o, (2.19)

which is satisfied when

and so

B = 0. (2.21) The boundary condition on the right end is v(e, t) = X(£)Y(t) = 0, (2.22) which is satisfied when

and so

If A — 0 the displacement will be identically zero for all x and t. Although this is an acceptable solution, it is of little interest and is thus called a trivial solution. Of more concern is when sin(af) = 0. (2.25)

This relation is called the "characteristic equation" and has a denumerably infinite set of solutions known as "eigenvalues." These solutions can be written as ai = j- (i = l,2,...). (2.26)

It should be noted that, although i = 0 (implying a0 = 0) appears to lead to a trivial solution, this is not true in general. Although a = 0 does lead to a trivial solution in this case, the only way to ascertain whether a nontrivial a = 0 solution exists in the general case is to return to Eqs. (2.17) and determine whether an a = 0 solution to those equations satisfies all the boundary conditions. Obviously, it does not here. Additional examples associated with the a = 0 solution will be addressed in more detail later when we consider problems other than strings.

Therefore, for each integer value of the index i there is an eigenvalue a, and an associated solution Xj, called the "eigenfunction." It contributes to the general solution based on the corresponding value of Y, . Thus, its total contribution can be written as v,(x, t) = Xi(x)Yi(t), (2.27)

where

Note that the constants A,-, C,, and D, may have different numerical values for each eigenvalue; thus they have been subscripted with the index. The most general solution for the string displacement would have contributions associated with all the eigenvalues. Thus, the general solution can be written as a sum of the complete set as v(x, t) — v,-(x, t)

Note that the original constants have been combined as

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