Page 283 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

270 Vibration of Continuous Systems Chap. 9

By substitution into Eq. (9.1>2), we obtain
i i . (9.1-6)
^ dt^
Because the left side of this equation is independent of t, whereas the right side is
independent of x, it follows that each side must be a constant. Letting this
constant be -((o/cY, we obtain two ordinary differential equations:

(9.1-7)
dx^ U )
d^G 2^ « (9.1-8)
^ -f- o> G = 0
d r
with the general solutions
Y = A sin—jr + B cos—jc (9.1-9)
c c
G = C sin o)t D cos o)t (9.1-10)
The arbitrary constants A, B,C, and D depend on the boundary conditions
and the initial conditions. For example, if the string is stretched between two fixed
points with distance / between them, the boundary conditions are y(0, t) = y(/, t)

= 0. The condition that y(0, r) ==0 will require that 5 = 0, so the solution will
appear as
y = (C sin (ot D cos cot) sin ~ x (9.1-11)
The condition y(/, i) = 0 then leads to the equation
0)1
sm— = 0
c
or
_ _ 2itI
c A = /Î7T, n = 1,2,3,...
and \ = c / f is the wavelength and / is the frequency of oscillation. Each n
represents a normal mode vibration with natural frequency determined from the
equation
T
fn 21^ 21 P n 1,2,3,... (9.1-12)
The mode shape is sinusoidal with the distribution
X
Y = sin nir (9.1-13)
In the more general case of free vibration initiated in any manner, the
solution will contain many of the normal modes and the equation for the displace-