Page 296 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 9.5 Euler Equation for Beams 283

TABLE 9.5-1

(Hiiy
Beam Configuration Fundamental Second Mode Third Mode
Simply supported 9.87 39.5 88.9
Cantilever 3.52 22.0 61.7
Free-free 22.4 61.7 121.0
Clamped-clamped 22.4 61.7 121.0
Clamped-hinged 15.4 50.0 104.0
Hinged-free 0 15.4 50.0

To arrive at this result, we assume a solution of the form
y =
which will satisfy the differential equation when

a = ±/3, and a = ±ip
Because
c - = cosh px ± sinh

the solution in the form of Eq. (9.5-12) is readily established.
The natural frequencies of vibration are found from Eq. (9.5-10) to be
El El
<^n=Pn\l— (9.5-13)

where the number depends on the boundary conditions of the problem. Table
9.5-1 lists numerical values of for typical end conditions.
Example 9.5-1
Determine the natural frequencies of vibration of a uniform beam clamped at one
end and free at the other.
Solution: The boundary conditions are