Originally Posted by

**jfortiv** Hello,

I'm trying to understand the solution to the wave equation for an Euler-Bernoulli beam as written in Inman's "Engineering Vibration" text. The solution involves separation of variables, which leads to the following expression in X(x):

X''''(x) - b^4*X(x) = 0 , where b^4 = w^2/c^2 (1)

The book assumes a solution of the form X(x) = A*exp(sigma*x) and leaves the general solution derivation up to the reader, yielding:

X(x) = a1*sin(b*x) + a2*cos(b*x) + a3*sinh(b*x) + a4*cosh(b*x) (2)

Here is my calculation of the general solution:

Plugging A*exp(sigma*x) into (1) yields:

sigma^4 - b^4 = 0; sigma = b, -b, i*b, -i*b, where i=sqrt(-1)

I create solution from these sigma values:

X(x) = c1*exp(bx) + c2*exp(-bx) + c3*exp(ibx) + c4*exp(-ibx)

I use the following Euler equation forms:

cosh(x) + sinh(x) = exp(x)

cosh(x) - sinh(x) = exp(-x)

cos(x) + i*sin(x) = exp(ix)

cos(x) - i*sin(x) = exp(-ix)

When I work these out for X(x), I always end up with an i attached to the sign term:

X(x) = a1*i*sin(bx) + a2*cos(bx) + a3*sinh(bx) + a4*cosh(bx), where each an is a rearrangement of the cn coefficients.

How did they just drop the i out in the solution above (2)? Can someone make sense of this?

Thanks!