Question 16.2: Find the fundamental natural frequency and period of vibrati...
Find the fundamental natural frequency and period of vibration of a cantilever beam of uniform cross section, Figure 16.6. Let L=length, in., EI=constant, lbs-in. ^{2}, w=uniform distributed weight, lbs/in. of length, m = mass=w/386.4, lbs- s ^{2}/in. ^{2}
After a number of trials, the deflection of the beam is assumed as shown in line (1) of Figure 16.6. The deflections at points 2–4 are assumed as 1.77, 5.60, and 10 in., respectively, multiplied by a constant K. The mass of the beam multiplied by this deflection and multiplied by the quantity \rho^{2} is assumed as an applied distributed inertia load as shown in line (2), where \rho is the natural frequency. The equivalent concentrated forces due to the distributed load at points 2–4 are calculated from Eqs. (1)–(3) and are shown in line (3). Lines (4) and (5) show the shear and moment diagrams. In line (5), the moment is divided by the quantity EI.
From structural analysis, the moment diagram divided by EI is applied on a conjugate beam as an equivalent load. The shear due to this load is the rotation of the actual beam, and the moment due to this load is the deflection of the actual beam. Also, the fixed and free ends of the actual beam become free and fixed, respectively, in a conjugate beam.
Line (6) shows the forces at points 1–3 of the equivalent M/EI load. Lines (7) and (8) show the rotation and deflection diagrams. Point 4 of the deflection diagram is adjusted to match the deflection assumed in line (1) as shown in line (9), and all other points in line (8) are adjusted accordingly as shown in line (9). The average variation of the deflection at points 2 and 3 is about 5%. Thus, the analysis is deemed complete. A slightly more accurate result can be obtained by performing a second cycle of analysis using the shape of line (9) as a starting point.
The natural frequency for the beam is obtained by equating the maximum deflections in lines (1) and (8).
10 K=0.837 K m \rho^{2} L^{4} / EIor,
\rho=3.46\left( EI / m L^{4}\right)^{0.5} rad / sf=0.55\left( EI / m L^{4}\right)^{0.5} cycles/s (cps)
and T=1.8\left(m L^{4} / EI \right)^{0.5} s
It is of interest to note that the frequency obtained from theoretical analysis Eq. (16.6a) is
T=0.091\left(w L^{4} / EI \right)^{0.5} for vertical cantilever vessel (16.6a)
f=0.56\left( EI / m L^{4}\right)^{0.5} cps, (16.8)
which is essentially the same as that obtained from the numerical analysis.